Startseite Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
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Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation

  • Jiali Lan , Xiaoming He EMAIL logo und Yuxi Meng
Veröffentlicht/Copyright: 18. November 2023

Abstract

In this article, we study the fractional critical Choquard equation with a nonlocal perturbation:

( Δ ) s u = λ u + α ( I μ * u q ) u q 2 u + ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u , in R N ,

having prescribed mass

R N u 2 d x = c 2 ,

where s ( 0 , 1 ) , N > 2 s , 0 < μ < N , α > 0 , c > 0 , and I μ ( x ) is the Riesz potential given by

I μ ( x ) = A μ x μ with A μ = Γ μ 2 2 N μ π N 2 Γ N μ 2 ,

and 2 N μ N < q < 2 μ , s * = 2 N μ N 2 s is the fractional Hardy-Littlewood-Sobolev critical exponent. Under the L 2 -subcritical perturbation α ( I μ * u q ) u q 2 u with exponent 2 N μ N < q < 2 N μ + 2 s N , we obtain the existence of normalized ground states and mountain-pass-type solutions. Meanwhile, for the L 2 -critical and L 2 -supercritical cases 2 N μ + 2 s N q < 2 N μ N 2 s , we also prove that the equation has ground states of mountain-pass-type.

MSC 2010: 35J62; 35J50; 35B65

1 Introduction and main results

In this article, we consider the fractional critical Choquard equation with a nonlocal perturbation:

(1.1) ( Δ ) s u = λ u + α ( I μ * u q ) u q 2 u + ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u , x R N ,

having prescribed L 2 -norm

(1.2) R N u 2 d x = c 2 ,

where s ( 0 , 1 ) , N > 2 s , 0 < μ < N , 2 N μ N < q < 2 μ , s * = 2 N μ N 2 s , α is a positive parameter, and I μ : R N \ { 0 } R is the Riesz potential, which is defined by:

I μ ( x ) = A α x μ ,

with A α = Γ μ 2 2 N μ π N 2 Γ N μ 2 , and the fractional Laplacian ( Δ ) s is defined for u S ( R N ) by

( Δ ) s u ( x ) C N , s P.V. R N u ( x ) u ( y ) x y N + 2 s d y , x R N ,

where S ( R N ) denotes the Schwartz space of rapidly decreasing smooth functions, P.V. stands for the principle value of the integral, and C N , s is the positive normalization constant. The nature function space associated with ( Δ ) s in N dimension is H s ( R N ) , which is a Hilbert space equipped with the inner product and norm, respectively, given by:

u , v R N ( ( Δ ) s 2 u ( Δ ) s 2 v + u v ) d x , u H s ( R N ) 2 = u , u .

The working space is the homogeneous fractional Sobolev space D s , 2 ( R N ) defined by:

D s , 2 ( R N ) = u L 2 s * ( R N ) : R 2 N u ( x ) u ( y ) 2 x y N + 2 s d x d y < + ,

a completion of C 0 ( R N ) under the norm:

u 2 u D s , 2 ( R N ) 2 = R 2 N u ( x ) u ( y ) 2 x y N + 2 s d x d y ,

where 2 s * = 2 N ( N 2 s ) is the fractional Sobolev exponent. From Propositions 3.4 and 3.6 in [46], we have that

(1.3) u 2 = ( Δ ) s 2 u 2 2 = R 2 N u ( x ) u ( y ) 2 x y N + 2 s d x d y .

Problems (1.1) and (1.2) arise in seeking standing waves for the following nonlinear fractional Schrödinger equation:

(1.4) i t ψ + ( Δ ) s ψ = α ( I μ * ψ p ) ψ p 2 ψ + ( I μ * ψ 2 α , s * ) ψ 2 α , s * 2 ψ in R × R N .

A standing wave of (1.4) is a solution having the form ψ ( t , x ) = e i λ t u ( x ) for some t R and u satisfying (1.1). So (1.1) is the stationary equation of the time-dependent equation (1.4).

We say that a function u H s ( R N ) is a weak solution to (1.1) provided φ H s ( R N ) ,

(1.5) R N [ ( Δ ) s 2 u ( Δ ) s 2 φ λ u φ ] d x = R N ( ( α I μ * ψ q ) ψ q 2 ψ φ + ( I μ * ψ 2 α , s * ) ψ 2 α , s * 2 ψ φ ) d x = 0 .

For fixed a > 0 , we aim at finding a real number λ R and a function u H s ( R N ) weakly solving (1.1) with u 2 = c . To study the normalized solutions to (1.1), we need to study the following energy functional and the constraint:

(1.6) J α ( u ) = 1 2 u 2 α 2 q R N ( I μ * u q ) u q d x 1 2 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x

and

S c = { u H s ( R N ) : R N u 2 d x = c 2 } ,

with λ being the Lagrange multipliers. We shall prove the existence of normalized solutions to equation (1.1) in different cases by comparing q and the L 2 -critical exponent.

Equation (1.1) is a special standing wave problem from the following fractional Schrödinger equation:

(1.7) i t ψ + ( Δ ) s ψ = f ( ψ ) ψ in R × R N .

It is a fundamental equation of the space-fractional quantum mechanics, where the fractional power of the Laplacian s ( 0 , 1 ) was introduced by Laskin in [28], which comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths [2]. The operator has a wide range of applications and appears in various areas of pure and applied mathematics such as charge transport in biopolymers, water waves, crystal dislocations, neural systems, and Bose-Einstein. For more details about the application background on fractional Laplacian, we refer to [20,31,46] and references therein.

Condition (1.2) is called as the normalization condition, which imposes a normalization on the L 2 -masses of u . In order to obtain the solution to (1.1)–(1.2), one needs to consider the critical point on S ( c ) ; in this case, λ appears as Lagrange multipliers with respect to the mass constraint, which cannot be determined a priori, but are part of the unknown. The prescribed mass approach that we shall follow here, has seen an increasing interest in these last few years and has been applied to various related problems (see, for example, [3,5,25,55] and references within). This approach is particularly relevant from a physical point of view. Indeed, the L 2 -norm is a preserved quantity of the evolution, and the variational characterization of such solutions is often a strong help to analyze their orbital stability/instability (see, for example, [7,8,24,53,54], for future references).

Recently, the normalized solution of fractional Schrödinger equations with the Choquard term as follows:

(1.8) ( Δ ) s u = λ u + u q 2 u + μ ( I α * u p ) u p 2 u , x R N ,

with the constraint R N u 2 d x = a 2 , I α ( x ) = x α N , has gradually attracted many researchers. Bhattarai in [6] considered (1.5) for μ 0 ; q ( 2 , 2 + 4 s N ) and p [ 2 , 1 + ( 2 s + α ) N ) , and Feng et al. in [14,17] studied the existence of stable standing waves of (1.8). In [58], Yang studied the existence and asymptotic properties of normalized solutions for (1.5) with s ( 0 , 1 ) , q ( 2 + 4 s N , 2 N N 2 s ] and p [ 2 , 1 + s + α N , N + α N 2 s ) , for μ , a small, by using a refined version of the minmax principle, the author obtained a mountain-pass-type solution. Furthermore, he gave some asymptotic properties of the solutions. In [34], Li et al. obtained the existence and asymptotic properties of normalized solutions for (1.8) with μ > 0 , q ( 2 , 2 + 4 s N ) and p [ 2 , 1 + ( 2 s + α ) N ) . The authors mainly extended the results of [6] and [14] concerning the aforementioned problem from L 2 -subcritical and L 2 -critical setting to L 2 -supercritical setting with respect to q , involving Sobolev critical case especially.

In the critical case: p = 2 α , s * , He et al. in [23] studied the existence results and asymptotically behavior of normalized ground states of (1.8) with the subcritical perturbation μ u q 2 u , and the parameter μ > 0 is suitable small. Meanwhile, when μ > 0 is large enough, Feng et al. [16] obtained the multiplicity of normalized solutions using the concentration-compactness principle and truncation technique. For more results for the fractional Schrödinger equations or Schrödinger-Choquard equations (1.8), we refer to [1,9,10,13,15,35,38,44,45,57] and references therein.

It is well known that when s = 1 , equation (1.8) is related to the nonlinear Choquard equation:

(1.9) Δ u + V ( x ) u = ( I α * u p ) u p 2 u , x R N ,

where N 3 , N + α N p N + α N 2 . When p = 2 , the problem goes back to the description of quantum theory of a polaron at rest by Pekar [47] in 1954. And it is a certain approximation to Hartree-Fock theory of one-component plasma (see [18,19,29]) in the work [29] of Choquard on the modeling of an electron trapped in its own hole in 1976. In 1996, Penrose also derived the same equation in discussing the self-gravitational collapse of a quantum mechanical wave function in [4850]. The equation was also called Schrödinger-Newton system [32]. For more results without prescribed mass obtained by variational methods, we refer to [3943] and the references therein. As far as we know, there are a few studies on normalized solutions of Choquard equations in the literature. Li and Ye [33] studied the following equation:

(1.10) Δ u = λ u + ( I α * F ( u ) ) F ( u ) , x R N ,

where λ R , N 3 and I α : R N R is the Riesz potential. Using the minimax method and the concentration compactness principle, it is shown that equation (1.10) has at least a pair of weak solution ( u a , λ a ) that satisfies R N u a 2 d x = a . Bartsch et al. [4] first proved the existence of the minimum energy solution in all dimensions for (1.10) and the authors used the fountain theorem to prove the existence of infinitely many solutions for (1.10) when f ( u ) is odd. Recently, Li [36] considered the normalized solutions to the Choquard equation with a local perturbation:

(1.11) Δ u = λ u + μ u q 2 u + ( I α * u p ) u p 2 u , x R N , u H 1 ( R N ) , R N u 2 d x = a 2 .

The author proved the existence, multiplicity, qualitative properties, and orbital stability of the ground states of (1.11) with the upper critical exponent p = 2 α * = N + α N 2 . Ye et al. in [59], and Shang and Ma [52] studied the effect of lower-order nonlocal perturbations in the existence of positive ground state solutions to (1.11). Bellazzini et al. [7] verified the existence of standing wave solutions with L 2 -norm for the following Schrödinger-Poisson equations:

(1.12) Δ u ( x 1 u 2 ) u = λ u + μ u q 2 u , x R 3 ,

where q ( 10 3 , 6 ) . They first showed that the energy functional has a mountain path geometry under constraint conditions and then proved the boundedness of the Palais-Smale sequence in a special case. Furthermore, they proved that critical points exist when a > 0 is sufficient small; on the contrary, a non-exist result is expected. For more results on the normalized solution for the Schrödinger equations or Choquard equations, we refer to [26,27] and references therein.

Motivated by the aforementioned works, especially by [5254,59], in this article, we aim to prove several existence and non-existence results for problems (1.1)–(1.2) by distinguishing the three cases: (i) L 2 -subcritical case: 2 N μ N < q < p ¯ 2 N μ + 2 s N ; (ii) L 2 -critical case: q = p ¯ and (iii) L 2 -supercritical case: p ¯ < q < 2 μ , s * , respectively.

In the sequel, we give some preliminary materials that will be useful in our approach. To begin with, we recall that the key point in applying variational method for Problem (1.1)–(1.2) is the following standard estimates for the Riesz potential (Theorem 4.3 [30]).

Proposition 1.1

(Hardy-Littlewood-Sobolev inequality [30]) Let t , r > 1 and 0 < μ < N be such that 1 r + 1 t + μ N = 2 , f L r ( R N ) and h L t ( R N ) . Then, there exists a constant C ( N , μ , r , t ) > 0 such that

(1.13) R N ( I μ * f ) h d x C ( N , μ , r , t ) f r h t .

The equation holds if and only if t = r = 2 N 2 N μ , then

C ( N , μ , r , t ) = C ( N , μ ) = π μ 2 Γ N μ 2 Γ N μ 2 Γ N 2 Γ ( N ) 1 + μ N ,

with f ( x ) = C h ( x ) and

h ( x ) = A ( γ 2 + x a 2 ) 2 N μ 2

for some A C , 0 γ R and a R N .

Remark 1.1

By a direct calculation from the Hardy-Littlewood-Sobolev inequality, we have

(1.14) R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x C N , μ , s u 2 s * 2 2 μ , s * , u D s , 2 ( R N ) ,

where C N , μ , s > 0 is a constant defined depending only on N , μ , and s .

Consequently, we can define the best constant S h , l as:

(1.15) S h , l inf u D s , 2 ( R N ) \ { 0 } u 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x 1 2 μ , s * ,

and the authors in [22,45] have proved that the best constant can be attained by the function:

(1.16) U ˜ ε , y ( x ) = S ( N μ ) ( 2 s N ) 4 s ( N μ + 2 s ) C N , μ , s 2 s N 2 ( N μ + 2 s ) U ε , y ( x ) C ˜ N , μ , s U ε , y ( x ) , x , y R N ,

and U ˜ ε , y ( x ) satisfies the following equation:

( Δ ) s u = ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u , x R N ,

for any fixed y R N and ε > 0 with

(1.17) R N ( Δ ) s 2 U ˜ ε , y 2 d x = R N ( I μ * U ˜ ε , y 2 μ , s * ) U ˜ ε , y 2 μ , s * d x = S h , l 2 N μ N μ + 2 s .

In addition, the function

(1.18) U ε , y ( x ) = κ ( ε 2 + x y 2 ) N 2 s 2

satisfies the equation ( Δ ) s u = u 2 s * 2 u , x R N , and achieves the best Sobolev constant S defined as:

(1.19) S inf u D s , 2 ( R N ) \ { 0 } u 2 u 2 s * 2 ,

and from [12,22], we know that the constant κ is given by:

κ = S N ( 2 s ) Γ ( N ) π N 2 Γ ( N 2 ) N 2 s 2 N .

Moreover, the constants S h , l , S , and C N , μ , s have the relationship:

(1.20) S h , l = S C ˜ N , μ , s 1 2 μ , s * .

We recall the fractional Gagliardo-Nirenberg inequality.

Lemma 1.1

[33] Let 0 < s < 1 , N > 2 s , and q ( 2 , 2 s * ) . Then, there exists a constant D ( N , q , s ) = S δ q , s 2 > 0 such that

(1.21) u q q D ( N , q , s ) ( Δ ) s 2 u 2 q δ q , s u 2 q ( 1 δ q , s ) , u H s ( R N ) ,

where δ q , s = N ( q 2 ) 2 q s .

For any u H s ( R N ) , take t = r = 2 N ( 2 N μ ) , f = h = u q in Proposition 1.1, using (1.21), we have

(1.22) R N ( I μ * u q ) u q d x C ( N , μ ) u 2 N q 2 N μ 2 q C N , q , s ( Δ ) s 2 u 2 2 q γ q , s u 2 2 q ( 1 γ q , s ) ,

where

C N , q , s = C ( N , μ ) S q γ q , s 1 2 q = π μ 2 S q γ q , s Γ N μ 2 Γ ( 2 N μ 2 ) Γ ( N 2 ) Γ ( N ) 1 + μ N 1 2 q > 0

and γ q , s = N ( q 2 ) + μ 2 q s . Let S q = C ( N , q , s ) 1 , then Inequality (1.22) is equivalent to

S q = inf u D s , 2 ( R N ) \ { 0 } ( Δ ) s 2 u 2 γ q , s u 2 1 γ q , s R N ( I μ * u q ) u q d x 1 2 q ,

and Feng [17] has proved that the S q is achieved.

Lemma 1.2

[30] (Weak Young’s inequality). Let μ ( 0 , N ) , q , s > 1 , and 1 p = 1 s + N μ N . If u L p ( R N ) , then I μ * u L s ( R N ) and

R N I μ * u s d x 1 s C ( N , μ , p , s ) R N u p d x 1 p .

In particular, we can set p = N N μ and s = + .

In addition, it is easy to enumerate that

q γ q , s < 1 , 2 N μ N < q < q ¯ ; = 1 , q = q ¯ ; > 1 , q ¯ < q < 2 N μ N 2 s ,

where q ¯ 2 + 2 s μ N is the L 2 -critical exponent. We also introduce two constants defined as:

C 1 ( 1 q γ q , s ) 2 μ , s * S h , l 2 μ , s * 2 μ , s * q γ q , s 1 q γ q , s 2 μ , s * 1 q ( 2 μ , s * 1 ) C N , q , s ( 2 μ , s * q γ q , s )

and

C 2 N μ + 2 s 4 N 2 μ 2 2 μ , s * C N , q , s γ q , s ( 2 μ , s * q γ q , s ) q γ q , s 1 q γ q , s S h , l 2 N μ N μ + 2 s 1 q γ q , s .

Now, the main results can be formulated as the following.

Theorem 1.1

Let N > 2 s , 0 < μ < N , 2 N μ N < q < q ¯ = 2 + 2 s μ N , and c , α > 0 , if α c 2 q ( 1 γ q , s ) < C * , where C * = min { C 1 , C 2 } , then

  1. J α restricted to S c admits a ground state u α + with J α ( u α + ) < 0 ; and u α + is a local minimizer of J α on the set M k { u S c u < k , k > 0 } for some suitable k small enough. Moreover, any ground state for J α S c is a local minimizer of J α on the set M k .

  2. The solution u α + is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for some λ α + < 0 .

Note that J α S c is unbounded from blew for 2 N μ N < q < q ¯ = 2 + 2 s μ N , then it is natural for us to try to obtain a second critical point of mountain-pass type on S c . To be more precisely, we have the following assertion:

Theorem 1.2

Let N > 2 s , 2 < q < p ¯ = 2 + 2 s μ N , 0 < μ < 2 s , and c , α > 0 , if N > 2 s ( q 1 ) μ q 2 , α c 2 q ( 1 γ q , s ) < C * , where C * = min { C 1 , C 2 } , then

  1. J α restricted to S c has another critical point u α S c satisfying J α ( u α ) < 0 .

  2. The solution u α is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for some λ α < 0 .

In the case of L 2 -critical perturbation, J α S c may change its structure, which will influence the number of critical points of J α . In fact, we have the following existence result.

Theorem 1.3

Let q = q ¯ = 2 + 2 s μ N , and c , α > 0 , if α c 2 q ( 1 γ q , s ) < C * = q ¯ C N , q ¯ , s , then

  1. If N 4 s and 0 < μ < N , then J α S c has a ground state u with 0 < J α ( u ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

  2. If 2 2 s < N < 4 s and μ 2 ( 4 s 2 + ( 2 s N ) N ) 4 s N , N , it is still true that J α S c has a ground state u with 0 < J α ( u ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

  3. u is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for suitable λ < 0 .

In the case of L 2 -supercritical perturbation, the existence of ground states can be given by the following theorem, which is similar to Theorem 1.3.

Theorem 1.4

Let 2 + 2 s μ N = q ¯ < q < 2 N μ N 2 s , 0 < μ < N and c , α > 0 , then

  1. If N 4 s and α c 2 q ( 1 γ q , s ) < C , where C > 0 is a certain constant, then J α S c has a ground state u with 0 < J α ( u ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

  2. If 2 s < N < 4 s and q max { q ¯ , 2 N μ N 2 s 4 s 2 ( N 2 s ) ( 4 s N ) } , 2 N μ N 2 s , it is also true that J α S c has a ground state u with 0 < J α ( u ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

  3. u is real-valued, positive, radially symmetric, and radially decreasing and solves (1.1) for suitable λ < 0 .

This article consists of the following sections: in Section 2, some preliminary results that will be used frequently in the sequel. Especially, we study the convergence of the Palais-Smale sequences. In Section 3, we consider the existence of ground state and mountain-pass-type solutions, respectively, for the case of L 2 -subcritical, and prove Theorems 1.1 and 1.2. In Section 4, we study L 2 -critical and L 2 -supercritical perturbations, and then complete the proof of Theorems 1.3 and 1.4.

Notation. In the sequel of this article, we denote by C , C i > 0 , i = 1 , 2 , , different positive constants whose values may vary from line to line and are not essential to the problem. We denote by L p = L p ( R N ) ( 1 < p ) the Lebesgue space with the standard norm u p = R N u p d x 1 p .

2 Pohozaev manifold and compactness lemma

2.1 Pohozaev manifold

In this section, we introduce the Pohozaev mainfold and the decomposition of it. To begin with, we introduce the following Pohozaev identity, which can be derived from [11,37].

Proposition 2.1

Let u H s ( R N ) L ( R N ) be a weak solution of (1.1), then u satisfies the equality:

(2.1) N 2 s 2 u 2 = N λ 2 R N u 2 d x + ( 2 N μ ) α 2 q R N ( I μ * u q ) u q d x + 2 N μ 2 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Lemma 2.1

Let u H s ( R N ) be a weak solution of (1.1), then we can construct the following Pohozaev manifold:

P c , α = { u S c : P α ( u ) = 0 } ,

where

(2.2) P α ( u ) = s u 2 s α γ q , s R N ( I μ * u q ) u q d x s R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Proof

From Proposition 2.1, we know that u satisfies the Pohozaev identity as follows:

(2.3) N 2 s 2 u 2 = N λ 2 R N u 2 d x + ( 2 N μ ) α 2 q R N ( I μ * u q ) u q d x + 2 N μ 2 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Moreover, since u is the weak solution of Problem (1.1), we have

(2.4) u 2 = λ R N u 2 d x + α R N ( I μ * u q ) u q d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Combining with (2.3) and (2.4), we obtain

s u 2 = s α γ q , s R N ( I μ * u q ) u q d x + s R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ,

which finishes the proof.□

From Lemma 2.1, it is clear that any critical point of J α ( u ) S c belongs to P c , α . Consequently, the properties of the manifold P c , α have relation to the mini-max structure of J α ( u ) S c . For u S c and t R , we introduce the transformation:

(2.5) ( t u ) ( x ) = e N t 2 u ( e t x ) , x R N , t R .

It is easy to check that the dilations preserve the L 2 -norm such that t u S c , hence, we introduce the fiber map naturally as follows:

(2.6) Ψ u α ( t ) J α ( t u ) = e 2 s t 2 u 2 α 2 q e 2 q γ q , s s t R N ( I μ * u q ) u q d x 1 2 2 μ , s * e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

A direct calculus shows that ( Ψ u α ) ( t ) = P α ( t u ) ; hence, we obtain the proposition immediately.

Proposition 2.2

For every u S c , t is the critical point of Ψ u α if and only if t u P c , α . And we can split Ψ u α into three disjoint part such that

P c , α = P c , α + P c , α 0 P c , α ,

where

(2.7) P c , α + { u P c , α u 2 > α q γ q , s 2 R N ( I μ * u q ) u q d x + 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x } , = { u S c ( Ψ u α ) ( 0 ) = 0 , ( Ψ u α ) ( 0 ) > 0 } P c , α 0 { u P c , α u 2 = α q γ q , s 2 R N ( I μ * u q ) u q d x + 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x } = { u S c ( Ψ u α ) ( 0 ) = 0 , ( Ψ u α ) ( 0 ) = 0 } , P c , α { u P c , α u 2 < α q γ q , s 2 R N ( I μ * u q ) u q d x + 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x } = { u S c ( Ψ u α ) ( 0 ) = 0 , ( Ψ u α ) ( 0 ) < 0 } .

2.2 Palais-Smale sequence

In this subsection, we shall study the convergence of the Palais-Smale sequence of the energy function J α ( u ) .

Lemma 2.2

Let N > 2 s , 2 N μ N < q < 2 N μ N 2 s , and c , α > 0 . Assume that { u n } is a Palais-Smale sequence for J α S c at level m 0 . If P α ( u n ) 0 as n , then { u n } is bounded in H s ( R N ) .

Proof

We prove the lemma in three different cases.

Case 1: 2 N μ N < q < q ¯ = 2 + 2 s μ N . In this case, we have q γ q , s < 1 , then from the condition P α ( u n ) 0 and the Inequality (1.12), we can derive that

J α ( u n ) = N μ + 2 s 4 N 2 μ u n 2 α 2 q 1 q γ q , s 2 μ , s * R N ( I μ * u n q ) u n q d x + o n ( 1 ) N μ + 2 s 4 N 2 μ u n 2 α 2 q 1 q γ q , s 2 μ , s * C ( N , q , s ) ( Δ ) s 2 u n 2 2 q γ q , s u n 2 2 q ( 1 γ q , s ) + o n ( 1 ) = N μ + 2 s 4 N 2 μ u n 2 α 2 q 1 q γ q , s 2 μ , s * C ( N , q , s ) c 2 q ( 1 γ q , s ) u n 2 q γ q , s + o n ( 1 ) .

Since J α ( u n ) m as n , we know that J α ( u n ) m + C for n , C > 0 large enough, such that m + C > 0 . Then,

N μ + 2 s 4 N 2 μ u n 2 α 2 q 1 q γ q , s 2 μ , s * C N , q , s c 2 q ( 1 γ q , s ) u n 2 q γ q , s + m + C ,

which means that { u n } is bounded in H s ( R N ) .

Case 2: q = q ¯ = 2 + 2 s μ N . In this case, we have q ¯ γ q ¯ , s = 1 . From P α ( u n ) 0 , we have

J α ( u n ) = 1 2 1 2 2 μ , s * R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x + o n ( 1 ) ,

which implies that

R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x C

for a constant C > 0 . By Inequality (1.22), we have

R N ( I μ * u n q ¯ ) u n q ¯ d x C N , q ¯ , s c 2 N μ + 2 s N u n 2 .

Combining this with P α ( u n ) 0 , we infer that

u n 2 = α γ q ¯ , s R N ( I μ * u n q ¯ ) u n q ¯ d x + R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x α γ q ¯ , s C N , q ¯ , s c 2 N μ + 2 s N u n 2 + C + o n ( 1 ) ,

which means that { u n } is bounded in H s ( R N ) .

Case 3: q > q ¯ = 2 + 2 s μ N . In this case, we obtain q γ q , s > 1 ; using P α ( u n ) 0 again, we have

J α ( u n ) = 1 2 1 2 2 μ , s * R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x + α 2 q ( q γ q , s 1 ) R N ( I μ * u n q ) u n q d x + o n ( 1 ) ,

which implies that both R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x and R N ( I μ * u n q ) u n q d x are bounded. Consequently,

u n 2 = α γ q , s R N ( I μ * u n q ) u n q d x + R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x + o n ( 1 ) C .

Then, the conclusion follows immediately.□

Lemma 2.3

Assume that the conditions of Lemma 2.2are satisfied, and m < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s , then the sequence { u n } S c , r = S c H r s ( R N ) has a nontrivial weak limit, where H r s ( R N ) , consisting of radially symmetric functions, is the subspace of H s ( R N ) .

Proof

From Lemma 2.2, we know that { u n } is bounded in H s ( R N ) , then there is a subsequence, still denoted by itself, such that u n u weakly in H s ( R N ) as n . We show that the weak limit u 0 . Suppose by contradiction that u 0 . Since the Sobolev embedding H r s ( R N ) L p ( R N ) for q ( 2 , 2 s * ) is compacted where 2 s * = 2 N N 2 s , then u n u strongly in L 2 N q 2 N μ ( R N ) as n by q ( 2 N μ N , 2 N μ N 2 s ) . Hence, from the the Hardy-Littlewood-Sobolev inequality, we can derive that

(2.8) R N ( I μ * u n q ) u n q d x 0 ,

as n .

Since { u n } is bounded in H s ( R N ) , then we can find a subsequence such that u n 2 0 . Therefore, from P α ( u n ) 0 and (2.8), we obtain

R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x = u n 2 α γ q , s R N ( I μ * u n q ) u n q d x ,

as n . Moreover, by Inequality (1.15), we derive

S h , l 1 2 μ , s * .

Hence, = 0 or S h , l 2 N μ N μ + 2 s .

If = 0 , then u n 2 0 , and so, R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x 0 , which implies that J α ( u n ) 0 , which is a contradiction.

If S h , l 2 N μ N μ + 2 s , by J α ( u n ) m and P α ( u n ) 0 , we arrive that

m + o n ( 1 ) = J α ( u n ) = N + 2 s μ 4 N 2 μ u n 2 α 2 q 1 q γ q , s 2 μ , s * R N ( I μ * u n q ) u n q d x + o n ( 1 ) = N + 2 s μ 4 N 2 μ u n 2 + o n ( 1 ) = N + 2 s μ 4 N 2 μ + o n ( 1 ) N + 2 s μ 4 N 2 μ S h , l 2 N μ N μ + 2 s + o n ( 1 ) ,

as n , which is a contradiction to the assumption m < N + 2 s μ 4 N 2 μ S h , l 2 N μ N μ + 2 s .□

Proposition 2.3

Assume that N > 2 s , 2 N μ N < q < 2 N μ N 2 s , and c , α > 0 . Let { u n } S c , r = S c H r s be a Palais-Smale sequence for J α S c at level m 0 with

m < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s ,

which satisfies P α ( u n ) 0 as n . Then, one of the following alternatives occurs:

  1. u n u weakly in H s ( R N ) but not strongly, up to a subsequence. Moreover, u solves equation (1.1) for some λ < 0 with J α ( u ) m N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

  2. Up to a subsequence u n u strongly in H s ( R N ) with J α ( u ) = m , and u is a solution of (1.1), satisfying R N u 2 d x = c 2 for some λ < 0 .

Proof

From Lemma 2.2, we know that { u n } H s ( R N ) is bounded, and we may assume that u n u weakly in H s ( R N ) as n . Set v n = u n u , then v n 0 weakly in H s ( R N ) . We first prove the following claim:

(2.9) lim n R N ( I μ * u n q ) u n q d x = R N ( I μ * u q ) u q d x .

It is obviously that

R N ( I μ * u n q ) u n q d x R N ( I μ * u q ) u q d x = R N ( I μ * ( u n q + u q ) ) ( u n q u q ) d x .

For every p , q R and r > 2 , there is a simple inequality that

p r q r p r q r = q p r x r 1 d x = r 0 1 ( t p + ( 1 t ) q ) r 1 ( p q ) d t r p q 0 1 t p + ( 1 t ) q r 1 d t r 2 r 2 p q ( p r 1 + q r 1 ) .

Then, by the Hardy-Littlewood-Sobolev Inequality (1.13), the Hölder inequality, and the last inequality, we infer to

R N ( I μ * u n q ) u n q d x R N ( I μ * u q ) u q d x C R N ( u n q + u q ) 2 N 2 N μ d x 2 N μ 2 N R N u n q u q 2 N 2 N μ d x 2 N μ 2 N C R N ( u n q 1 + u q 1 ) 2 N 2 N μ u n u 2 N 2 N μ d x 2 N μ 2 N × R N ( u n q + u q ) 2 N 2 N μ d x 2 N μ 2 N C 1 R N ( u n 2 N q 2 N μ + u 2 N q 2 N μ ) d x ( 2 N μ ) ( 2 q 1 ) 2 N q R N u n u 2 N q 2 N μ d x 2 N μ 2 N q 0

as n ; in view of 2 < 2 N q 2 N μ < 2 s * , the compactness of Sobolev embedding H r s ( R N ) L p ( R N ) , p ( 2 , 2 s * ) , and the claim is proved.

Since { u n } is a bounded Palais-Smale sequence restricted on S c for J α , then there exists { λ n } R that satisfies that

(2.10) R N ( Δ ) s 2 u n ( Δ ) s 2 φ d x λ n R N u n φ d x α R N ( I μ * u n q ) u n q 2 u n φ d x R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * 2 u n φ d x = o n ( 1 ) φ , φ H s ( R N ) ,

by the Lagrange multipliers rule. Taking the text function φ = u n , then we derive { λ n } is bounded, and we obtain that λ n λ for some λ R . Using P α ( u n ) 0 , we have

λ c 2 = lim n λ n u n 2 2 = lim n u n 2 R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x α R N ( I μ * u n q ) u n q d x = lim n α ( γ q , s 1 ) R N ( I μ * u n q ) u n q d x = α ( γ q , s 1 ) R N ( I μ * u q ) u q d x .

Then, we can derive that λ < 0 by virtue of α > 0 , u 0 , and γ q , s < 1 .

Since the Sobolev embedding H r s ( R N ) L p ( R N ) for p [ 2 , 2 s * ] is continuous, we know that u n u weakly in L 2 s * ( R N ) and u n u a.e. in R N . Then,

u n 2 μ , s * u 2 μ , s * weakly in  L 2 N 2 N μ ( R N )

as n . Based on the aforementioned fact and using Lemma 1.2, we have

I μ * u n 2 μ , s * I μ * u 2 μ , s * weakly in  L 2 N μ ( R N )

as n . Combining this with the following fact that

u n 2 μ , s * 2 u n u 2 μ , s * 2 u weakly in  L 2 N N μ + 2 s ( R N )

as n , we derive that

( I μ * u n 2 μ , s * ) u n 2 μ , s * 2 u n ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u weakly in  L 2 N N + 2 s ( R N )

as n . Then, by the definition of weak convergence, we obtain that for any φ H s ( R N ) ,

R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * 2 u n φ d x R N ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u φ d x .

Therefore, passing to the limit on both sides of (2.10), we obtain that

(2.11) ( Δ ) s u = λ u + α ( I μ * u q ) u q 2 u + ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u , in R N ,

and then P α ( u ) = 0 by the Pohozaev identity.

Since v n 0 weakly in H s ( R N ) , then by the Brezis-Lieb lemma and [22], we have

(2.12) u n 2 = v n 2 + u 2 + o n ( 1 )

and

(2.13) R N ( I μ * u n 2 μ , s * ) u n 2 μ , s * d x = R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x + o n ( 1 ) .

Consequently, from the fact that P α ( u n ) 0 , (2.9), (2.12), and (2.13), we infer that

u 2 + v n 2 = α γ q , s R N ( I μ * u q ) u q d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x + R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x + o n ( 1 ) .

Combining this with P α ( u ) = 0 , we deduce that

lim n v n 2 = lim n R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x .

Assume that lim n v n 2 = ξ and by Inequality (1.15), we infer to

ξ S h , l ξ 1 2 μ , s * ,

which implies that ξ = 0 or ξ S h , l 2 N μ N μ + 2 s .

If ξ S h , l 2 N μ N μ + 2 s , then by (2.12) and (2.13), we obtain

m = lim n J α ( u n ) = lim n J α ( u ) + 1 2 v n 2 1 2 2 μ , s * R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x = J α ( u ) + N + 2 s μ 4 N 2 μ ξ J α ( u ) + N + 2 s μ 4 N 2 μ S h , l 2 N μ N μ + 2 s ,

which yields Conclusion (i) of this proposition.

If ξ = 0 , then u n u in H s ( R N ) . Let φ = u n u in (2.10) and multiply u n u on both sides of (2.11), and integrating by part, then we have

(2.14) u n u 2 R N ( λ n u n λ u ) ( u n u ) d x = α R N [ ( I μ * u n q ) u n q 2 u n ( I μ * u q ) u q 2 u ] ( u n u ) d x + R N [ ( I μ * u n 2 μ , s * ) u n 2 μ , s * 2 u n ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u ] ( u n u ) d x + o n ( 1 ) .

Combining with (2.9) and (2.13), and Lemma 1.2, we see that the right-hand side of (2.14) tends to zero, which implies that

0 = lim n R N ( λ n u n λ u ) ( u n u ) d x = λ lim n R N u n u 2 d x ,

and then, u n u strongly in L 2 ( R N ) by λ < 0 , and Conclusion (ii) is completed.□

3 L 2 -subcritical case

In this section, we consider that N > 2 s , c , α > 0 , and α c 2 q ( 1 γ q , s ) < C * , where C * = min { C 1 , C 2 } . We shall study Problems (1.1)–(1.2) with L 2 -subcritical perturbation: 2 N μ N < q < q ¯ = 2 + 2 s μ N . First, we analyze the structure and properties of the Pohozaev manifold P c , α .

Lemma 3.1

P c , α 0 = and P c , α is a smooth manifold of codimension 2 in H s ( R N ) .

Proof

We first show that P c , α 0 = . Otherwise, there exists a u ˜ P c , α 0 , which means that

u ˜ 2 = α q γ q , s 2 R N ( I μ * u ˜ q ) u ˜ q d x + 2 μ , s * R N ( I μ * u ˜ 2 μ , s * ) u ˜ 2 μ , s * d x .

Thus by P μ ( u ˜ ) = 0 , we obtain

α γ q , s ( 1 q γ q , s ) R N ( I μ * u ˜ q ) u ˜ q d x = ( 2 μ , s * 1 ) R N ( I μ * u ˜ 2 μ , s * ) u ˜ 2 μ , s * d x .

Moreover, using P μ ( u ˜ ) = 0 again, we have

(3.1) u ˜ 2 = 2 μ , s * q γ q , s 1 q γ q , s R N ( I μ * u ˜ 2 μ , s * ) u ˜ 2 μ , s * d x 2 μ , s * q γ q , s S h , l 2 μ , s * ( 1 q γ q , s ) u ˜ 2 2 μ , s *

and

(3.2) u ˜ 2 = α γ q , s ( 2 μ , s * q γ q , s ) 2 μ , s * 1 R N ( I μ * u ˜ q ) u ˜ q d x C N , q , s α γ q , s ( 2 μ , s * q γ q , s ) 2 μ , s * 1 c 2 q ( 1 γ q , s ) u ˜ 2 q γ q , s .

Combining with (3.1) and (3.2), we have

C N , q , s α γ q , s ( 2 μ , s * q γ q , s ) 2 μ , s * 1 c 2 q ( 1 γ q , s ) 1 1 q γ q , s ( 1 q γ q , s ) S h , l 2 μ , s * 2 μ , s * q γ q , s 1 2 μ , s * 1 .

Hence

(3.3) α c 2 q ( 1 γ q , s ) ( 1 q γ q , s ) S h , l 2 μ , s * 2 μ , s * q γ q , s 1 q γ q , s 2 μ , s * 1 2 μ , s * 1 C N , q , s γ q , s ( 2 μ , s * q γ q , s ) .

Next, we show that the right-hand of (3.3) is greater than or equal to C 1 . To show

(3.4) ( 1 q γ q , s ) S h , l 2 μ , s * 2 μ , s * q γ q , s 1 q γ q , s 2 μ , s * 1 2 μ , s * 1 C N , q , s γ q , s ( 2 μ , s * q γ q , s ) ( 1 q γ q , s ) 2 μ , s * S h , l 2 μ , s * 2 μ , s * q γ q , s 1 q γ q , s 2 μ , s * 1 q ( 2 μ , s * 1 ) C N , q , s ( 2 μ , s * q γ q , s ) C 1 ,

we only need to prove that

(3.5) ( 2 μ , s * ) 1 q γ q , s ( q γ q , s ) 2 μ , s * 1 1 , for every 2 < q < q ¯ < 2 s * .

Define the function

f ( x ) ( 2 μ , s * ) 1 x x 2 μ , s * 1 , x > 0 .

It is easy to check that f ( x ) is monotone increasing for x ( 0 , ( 2 μ , s * 1 ) ln 2 μ , s * ) and monotone decreasing for x ( ( 2 μ , s * 1 ) ln 2 μ , s * , + ) . Moreover,

0 < q γ q , s < 1 < 2 μ , s * 1 ln 2 μ , s * .

This implies that f ( q γ q , s ) f ( 1 ) = 1 and Inequality (3.5) holds. Combining (3.3)–(3.4), we obtain a contradiction to α c 2 q ( 1 γ q , s ) < C 1 ; therefore, P c , α 0 = .

Next, we intend to show that P c , α is a smooth manifold of codimension 2 in H s ( R N ) . We can rewrite the manifold P c , α as follows:

P c , α = { u H s ( R N ) : P α ( u ) = 0 , Q ( u ) = 0 } ,

where Q ( u ) = R N u 2 d x c 2 , and P α , Q C 1 in H s ( R N ) . Thus, we have to show that the map ( d Q ( u ) , d P α ( u ) ) : H s ( R N ) R 2 is surjective. For every η T u S c , which denotes the tangent space to S c in u , we have d Q ( u ) [ η ] = 0 . In addition, we claim that for every u P c , α , there exists η such that d P α ( u ) [ η ] 0 . Otherwise, d P α ( u ) [ η ] = 0 for every η T u S c , which implies that u is a constrained critical point for P α ( u ) on the set S c ; then, there exists τ R that satisfies

( Δ ) s u = τ u + α q γ q , s ( I μ * u q ) u q 2 u + 2 μ , s * ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u , x R N ,

by the Lagrange multiplies rule. By the Pohozaev identity of the aforementioned equation, we infer that u satisfies the following equality:

u 2 = α q γ q , s 2 R N ( I μ * u q ) u q d x + 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Hence, u P c , α 0 , which contradicts with the claim that P c , α 0 = . Therefore, for any ( x , y ) R 2 , the following system

d Q ( u ) [ a η + b u ] = x d P α ( u ) [ a η + b u ] = y b c 2 = x a d P α ( u ) [ η ] + b d P α ( u ) [ u ] = y

implies that a and b can be solved easily. Consequently, the surjectivity is proved, and P c , α is a smooth manifold of codimension 2 in H s ( R N ) .□

Proposition 3.1

Suppose the conditions of Theorem 1.1are satisfied, then P c , α is a smooth manifold of codimension 1 in S c . Moreover, if u P c , α is a critical point for J α constrained on P c , α , then u is a critical point for J α S c .

Proof

From Lemma 3.1, we know that P c , α 0 = and P c , α is a smooth manifold of codimension 2 in H s ( R N ) . If u P c , α is a critical point for J α constrained on P c , α , then for every φ H s ( R N ) , there exists λ 1 , λ 2 R that satisfies that

d J α ( u ) [ φ ] λ 1 R N u φ d x λ 2 d P α ( u ) [ φ ] = 0

by the Lagrange multipliers rule. Hence, we can derive that

( 1 2 s λ 2 ) ( Δ ) s u = λ 1 u + ( α 2 s α q γ q , s λ 2 ) ( I μ * u q ) u q 2 u + ( 1 2 2 μ , s * λ 2 ) ( I μ * u 2 μ , s * ) u 2 μ , s * 2 u .

By the Pohozaev identity, we derive that

( 1 2 s λ 2 ) u 2 = α γ q , s ( 1 2 s q γ q , s λ 2 ) R N ( I μ * u q ) u q d x + ( 1 2 s 2 μ , s * λ 2 ) R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Combining this with the fact that P α ( u ) = 0 , we infer to

λ 2 u 2 α q γ q , s 2 R N ( I μ * u q ) u q d x 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x = 0 .

Since u P c , α 0 , we deduce that λ 2 = 0 , and the conclusion is proved.□

To complete the proof of Theorem 1.1 in the L 2 -subcritical perturbation, we need to investigate the behaviors of the constrained functional J α S c . By the definition of J α and Inequality (1.15) and (1.22), we have

(3.6) J α S c 1 2 u 2 α 2 q C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s 1 2 2 μ , s * S h , l 2 μ , s * u 2 2 μ , s * .

In order to learn more better properties of the right-hand side of Inequality (3.6), we introduce the following function F : R + R :

(3.7) F ( t ) 1 2 t 2 α 2 q C N , q , s c 2 q ( 1 γ q , s ) t 2 q γ q , s 1 2 2 μ , s * S h , l 2 μ , s * t 2 2 μ , s * .

Lemma 3.2

Suppose that the inequality α c 2 q ( 1 γ q , s ) < C 1 holds, then the function F ( t ) has only two critical points with a local strict minimum at the negative level and a global maximum at the positive level. Besides, there are two zero points that satisfy F ( t 1 ) = F ( t 2 ) = 0 and F ( t ) > 0 if and only if t ( t 1 , t 2 ) .

Proof

It is obvious that for t > 0 , F ( t ) > 0 if and only if

φ ( t ) q C N , q , s t 2 ( 1 q γ q , s ) q C N , q , s 2 μ , s * S h , l 2 μ , s * t 2 ( 2 μ , s * q γ q , s ) > α c 2 q ( 1 γ q , s ) .

It is easy to check that φ ( t ) has a unique critical point at

t 0 = 1 q γ q , s 2 μ , s * q γ q , s 2 μ , s * S h , l 2 μ , s * 1 2 ( 2 μ , s * 1 ) ,

where t 0 is the strict maximum. Moreover,

φ ( t 0 ) = ( 1 q γ q , s ) 2 μ , s * S h , l 2 μ , s * 2 μ , s * q γ q , s 1 q γ q , s 2 μ , s * 1 q ( 2 μ , s * 1 ) C N , q , s ( 2 μ , s * q γ q , s ) = C 1 .

Therefore, F ( t ) is positive on an open interval ( t 1 , t 2 ) and F ( t 1 ) = F ( t 2 ) = 0 from α c 2 q ( 1 γ q , s ) < C 1 , and the fact that F ( t ) 0 as t 0 + and F ( t ) as t . Therefore, F ( t ) attains its global maximum at the positive level in ( t 1 , t 2 ) and a local minimum at the negative level at ( 0 , t 1 ) . Moreover,

F ( t ) = t 2 γ q , s 1 [ t 2 ( 1 γ q , s ) α γ q , s C N , q , s c 2 q ( 1 γ q s ) S h , l 2 μ , s * t 2 ( 2 μ , s * γ q , s ) ] = 0

if and only if

ψ ( t ) t 2 ( 1 γ q , s ) S h , l 2 μ , s * t 2 ( 2 μ , s * γ q , s ) = α γ q , s C N , q , s c 2 q ( 1 γ q s ) .

Obviously, ψ ( t ) has only one strict maximum. If max t 0 ψ ( t ) α γ q , s C N , q , s c 2 q ( 1 γ q s ) , then F ( t ) is monotonously decreasing for t 0 which is impossible, since F is positive on an open interval ( t 1 , t 2 ) . Hence, max t 0 ψ ( t ) > α γ q , s C N , q , s c 2 q ( 1 γ q s ) , which means that F ( t ) has only a global strict maximum at the positive level and a local strict minimum at the negative level, and no other critical points.□

From the aforementioned analysis, we can derive the following lemma, which is important to prove Theorem 1.2.

Lemma 3.3

For each u S c , the function Ψ u α ( t ) has exactly two critical points t u , 1 , t u , 2 R and two zero points c u , d u R with t u , 1 < c u < t u , 2 < d u . Moreover,

  1. t u , 1 u P c , α + , t u , 2 u P c , α and if t u P c , α , then either t = t u , 1 or t = t u , 2 .

  2. t u t 1 for each t c u , and

    J α ( t u , 1 u ) = min { J α ( t u ) : t R and t u t 1 } < 0 .

  3. J α ( t u , 2 u ) = max { J α ( t u ) t R } > 0 and Ψ u α ( t ) is strictly decreasing and concave on ( t u , 2 , + ) . Moreover, if t u , 2 < 0 , then P α ( u ) < 0 .

  4. The maps u t u , 1 and u t u , 2 for u S c are of class C 1 .

Proof

For u S c , then t u P c , α if and only if ( Ψ u α ) ( t ) = 0 . For Inequality (3.6), we have

Ψ u α ( t ) = J α ( t u ) F ( t u ) = F ( e s t u ) .

Hence, the C 2 function Ψ u α ( t ) is positive on ( s 1 ln ( t 1 u 1 ) , s 1 ln ( t 2 u 1 ) ) with Ψ u α ( ) = 0 , Ψ u α ( + ) = ; hence, Ψ u α ( t ) has a local minimum point t u , 1 at the negative level in ( 0 , s 1 ln ( t 1 u 1 ) ) and a global maximum t u , 2 at the positive level on ( s 1 ln ( t 1 u 1 ) , s 1 ln ( t 2 u 1 ) ) . Moreover, Ψ u α ( t ) has no other critical points. In fact, ( Ψ u α ) ( t ) = 0 shows that

g ( t ) = α γ q , s R N ( I μ * u q ) u q d x ,

where

g ( t ) = e 2 ( 1 q γ q , s ) s t u 2 e 2 ( 2 μ , s * q γ q , s ) s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

It is easy to check that g ( t ) has only one critical point, which is a global maximum point; therefore, ( Ψ u α ) ( t ) = 0 has at most two solutions. Note that, for each u S c , t R , ( Ψ u α ) ( t ) = 0 if and only if t u P c , α . Since t u , 1 u , t u , 2 u P c , α and so, t u P c , α if and only if t = t u , 1 or t = t u , 2 . Since t u , 1 is a local minimum point of Ψ u α ( t ) , we have ( Ψ t u , 1 u α ) ( 0 ) = ( Ψ u α ) ( t u , 1 ) 0 . From Lemma 3.1, we know P c , α 0 = ; therefore, ( Ψ t u , 1 u α ) ( 0 ) = ( Ψ u α ) ( t u , 1 ) > 0 and t u , 1 u P c , α + . Similarly, we obtain that t u , 2 u P c , α . By the monotonicity and the behavior of Ψ u α ( t ) at + , we obtain that Ψ u α ( t ) has only two zero points c u and d u with t u , 1 < c u < t u , 2 < d u . Moreover, Ψ u α ( t ) has exactly two inflection points, especially Ψ u α ( t ) is concave on [ t u , 2 , + ) ; hence, we can deduce that if t u , 2 < 0 , then P α ( u ) = ( Ψ u α ) ( 0 ) < 0 . So far, we have checked that items (i)–(iii) are true.

To prove Conclusion (iv), we study the C 1 function J ( t , u ) ( Ψ u α ) ( t ) . Since J ( t u , 1 , u ) = 0 , t J ( t u , 1 , u ) = ( Ψ u α ) ( t u , 1 ) > 0 , then by the implicit function theorem, we obtained that u t u , 1 , u S c is of class C 1 . By the same way, we can obtain that u t u , 2 , u S c is of class C 1 .□

For k > 0 , we set

M k { u S c : u < k } , and m c , α inf u M t 1 J α ( u ) .

Then, by Lemma 3.3, we can obtain the following conclusion.

Corollary 3.1

The set P c , α + M t 1 { u S c : u < t 1 } , and

sup u P c , α + J α 0 inf u P c , α J α .

Lemma 3.4

The level m c , α ( , 0 ) , and satisfies

m c , α = inf u P c , α J α = inf u P c , α + J α

and

m c , α < inf M ¯ t 1 \ M t 1 ρ J α

for ρ small enough.

Proof

By (3.6) and (3.7) for any u M t 1 , we have

J α ( u ) F ( u ) min t [ 0 , t 1 ] F ( t ) > .

Thus, m c , α > . Moreover, for any u S c , we have t u < t 1 and J α ( t u ) < 0 for t 1 small enough; hence, m c , α < 0 . From Corollary 3.1, we know that P c , α + M t 1 , and so m c , α inf u P c , α + J α . On the other hand, if u M t 1 , then by Lemma 3.3, we know that t u , 1 u P c , α + M t 1 and

J α ( t u , 1 u ) = min { J α ( t u ) : t R and u < t 1 } J α ( u ) .

Therefore, inf u P c , α + J α m c , α . Using Corollary 3.1 again, we have J α > 0 on P c , α so that inf u P c , α J α = inf u P c , α + J α .

Finally, by the continuity of F ( t ) , there exists ρ > 0 that satisfies F ( t ) m c , α 2 for t [ t 1 ρ , t 1 ] . Then, for every u S c with t 1 ρ u t 1 , we have

J α ( u ) F ( u ) m c , α 2 > m c , α

by (3.6) and m c , α < 0 . Therefore, we can infer that m c , α < inf M ¯ t 1 \ M t 1 ρ J α .

Proof of Theorem 1.1

Take a minimizing sequence { w n } H s ( R N ) S c for J α M t 1 . We can assume that { w n } H r s ( R N ) are radially decreasing for every n . Otherwise, we can take the Schwarz rearrangement of w n if necessary. By Lemma 3.3, we know that there exist sequence { t w n , 1 w n } P c , α + and t w n , 1 w n t 1 such that

J α ( t w n , 1 w n ) = min { J α ( t w n ) : t R and t w n t 1 } J α ( w n ) .

We denote by v n t w n , 1 w n , then the radially decreasing sequence { v n } S c , r P c , α + is a new minimizing sequence for J α M t 1 with v n < t 1 ρ for every n , by Lemma 3.4. By Ekeland’s variational principle, we can obtain a new minimizing sequence { u n } M t 1 . Obviously, v n are all real-valued, non-negative, radially symmetric, and decreasing in r = x as well as a Palais-Smale sequence for J α S c at level m c , α such that u n v n H s ( R N ) 0 . According to the Brézis-Lieb Lemma and Sobolev embedding theorem, we have that

u n 2 = u n v n 2 + v n 2 2 R 2 N ( [ u n ( x ) v n ( x ) ] [ u n ( y ) v n ( y ) ] ) ( v n ( x ) v n ( y ) ) x y N + 2 s d x d y + o n ( 1 ) = v n 2 + o n ( 1 ) .

Furthermore, for every 2 N μ N p 2 μ , s * = 2 N μ N 2 s , there exists ξ n ξ n ( x ) ( 0 , 1 ) , x R N , such that

R N ( I μ * u n p ) u n p d x = R N ( I μ * v n p ) v n p d x + p R N ( I μ * v n + ξ n ( u n v n ) p 1 ) v n + ξ n ( u n v n ) p ( u n v n ) d x + p R N ( I μ * v n + ξ n ( u n v n ) p ) v n + ξ n ( u n v n ) p 1 ( u n v n ) d x + o n ( 1 ) = R N ( I μ * v n p ) v n p d x + o n ( 1 ) .

Hence, P α ( u n ) = P α ( v n ) + o n ( 1 ) 0 as n .

Next, we show that up to a subsequence, u n u strongly in H s ( R N ) and J α ( u ) = m c , α . Since m c , α < 0 , then one of the conclusions in Proposition 2.3 holds. If the first case holds, we have u n u weakly in H s ( R N ) ,

J α ( u ) m c , α N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s ,

and u is a solution for (1.1) with some λ < 0 . By the Pohozaev identity, we obtain P α ( u ) = 0 , and combined with Inequality (1.22) we have

m c , α N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + J α ( u ) = N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + N μ + 2 s 4 N 2 μ u 2 α 2 q 1 q γ q , s 2 μ , s * R N ( I μ * u q ) u q d x N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + N μ + 2 s 4 N 2 μ u 2 α 2 q 1 q γ q , s 2 μ , s * C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s ,

where we have used the lower semi-continuity of the L 2 -norm that u 2 c by weak convergence.

Since α c 2 q ( 1 γ q , s ) < C * = min { C 1 , C 2 } < C 2 , we have

N μ + 2 s 4 N 2 μ u 2 α 2 q 1 q γ q , s 2 μ , s * C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s N μ + 2 s 4 N 2 μ u 2 N μ + 2 s 4 N 2 μ 1 q γ q , s q γ q , s 1 q γ q , s S H , L 2 N μ N μ + 2 s 1 q γ q , s u 2 q γ q , s .

We set

h ( t ) = N μ + 2 s 4 N 2 μ t N μ + 2 s 4 N 2 μ 1 q γ q , s q γ q , s 1 q γ q , s S H , L 2 N μ N μ + 2 s 1 q γ q , s t q γ q , s .

Since q γ q , s < 1 , it is easy to check that h ( t ) has a unique global minimum at

t min = q γ q , s 1 q γ q , s S h , l 2 N μ N μ + 2 s

and

h ( t min ) = N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Therefore, we have that

m c , α N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + h ( u 2 ) N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + h ( t min ) > 0 ,

which is a contradiction with m c , α < 0 . Hence, the second alternative holds that, up to a sequence, u n u α + strongly in H s ( R N ) . In addition, for some λ α + < 0 , u α + is a normalized solution to (1.1) with J α ( u α + ) = m c , α = inf u P c , α J α , which implies that u α + is a ground state solution for J α S c by Lemma 3.4.

Finally, we show that any ground state is a local minimizer of J α S c . Assume that u is a critical point for J α S c at level m c , α , then we have u P c , α and J α ( u ) < 0 . By Lemma 3.3, we know u P c , α + M t 1 and

J α ( u ) = m c , α = inf u M t 1 J α , u t 1 .

Then, the proof is completed.□

Lemma 3.5

There holds m c , α = inf u P c , α J α ( u ) > 0 .

Proof

From Lemma 3.2, we know that F ( t ) has a unique global maximum at the positive level we denote by t max . Hence, for every u P c , α , we can find τ u such that τ u u = t max and

J α ( u ) = Ψ u α ( 0 ) Ψ u α ( τ u ) = J α ( τ u u ) F ( τ u u ) = F ( t max ) > 0 .

Since u P c , α could be chosen arbitrarily, we infer to inf u P c , α J α ( u ) max t R F ( t ) > 0 as desired.□

Lemma 3.6

Let N > 2 s , 2 < q < p ¯ = 2 + 2 s μ N , 0 < μ < 2 s , and c , α > 0 , if N > 2 s ( q 1 ) μ q 2 , α c 2 q ( 1 γ q , s ) < C * , where C * = min { C 1 , C 2 } , then

m c , α < m c , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Proof

It is well known that the best Sobolev constant S is achieved at

U ε ( x ) = C ( N , s ) ε N 2 s 2 ( ε 2 + x 2 ) N 2 s 2 ,

with C ( N , s ) chosen such that

U ε 2 = R N U ε 2 s * d x = S N 2 s .

We introduce a cut-off function η ( x ) C 0 ( R N , [ 0 , 1 ] ) such that 0 η 1 , η = 1 in B 1 ( 0 ) , η = 0 in B 2 c ( 0 ) , and let

u ε η ( x ) U ε ( x )

for ε > 0 . Then, by [51], we have

(3.8) u ε 2 = R 2 N u ε ( x ) u ε ( y ) 2 x y N + 2 s d x d y S N 2 s + O ( ε N 2 s ) .

(3.9) R N u ε 2 d x = C ε 2 s + O ( ε N 2 s ) , if N > 4 s ; C ε 2 s log ε + O ( ε 2 s ) , if N = 4 s ; C ε N 2 s + O ( ε 2 s ) , if N < 4 s .

(3.10) R N u ε 2 s * d x = S N 2 s + O ( ε N ) .

By Lemma 4.5 [23], we have

(3.11) R N u ε q d x = O ( ε N N 2 s 2 q ) , if q > N N 2 s ; O ( ε N 2 log ε ) , if q = N N 2 s ; O ( ε N 2 s 2 q ) , if q < N N 2 s .

By the analogous arguments as Lemma 4.5 of [23], we have

(3.12) R 2 N u ε ( x ) 2 μ , s * u ε ( y ) 2 μ , s * x y μ d x d y C N , μ , s S 2 N μ 2 s + O ( ε N ) ,

(3.13) R 2 N u ε ( x ) 2 μ , s * u ε ( y ) 2 μ , s * x y μ d x d y 1 C ˜ N , μ , s 2 2 μ , s * S h , l 2 N μ N μ + 2 s O ( ε 2 N μ 2 ) .

And by the proof of Lemma 5.3 in [58], we have

(3.14) R 2 N u ε ( x ) q u ε ( y ) q x y μ d x d y O ( ε 2 N μ ( N 2 s ) q ) .

Furthermore, we set u ˆ θ , ε = u α + + θ u ε and u ˜ θ , ε = τ N 2 s 2 u ˆ θ , ε ( τ x ) , where u α + is given in Lemma 3.4, then

(3.15) u ˜ θ , ε 2 = u ˆ θ , ε 2 ,

(3.16) R N ( I μ * u ˜ θ , ε 2 μ , s * ) u ˜ θ , ε 2 μ , s * d x = R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x ,

(3.17) R N ( I μ * u ˜ θ , ε q ) u ˜ θ , ε q d x = τ 2 q s ( γ q , s 1 ) R N ( I μ * u ˆ θ , ε q ) u ˆ θ , ε q d x ,

with

(3.18) u ˜ θ , ε 2 2 = τ 2 s u ˆ θ , ε 2 2 .

Now, we choose a suitable τ that satisfies τ s = u ˆ θ , ε 2 c , then u ˜ θ , ε S c . By Lemma 3.3, there exists t θ , ε > 0 such that t θ , ε u ˜ θ , ε P c , α . Therefore,

(3.19) u ˜ θ , ε 2 e 2 s t θ , ε ( 1 q γ q , s ) = α γ q , s R N ( I μ * u ˜ θ , ε q ) u ˜ θ , ε q d x + e 2 s t θ , ε ( 2 μ , s * q γ q , s ) R N ( I μ * u ˜ θ , ε 2 μ , s * ) u ˜ θ , ε 2 μ , s * .

Since u α + P c , α + , then we have t 0 , ε > 0 by Proposition 2.2 and Lemma 3.3. In addition, from (3.16) and (3.8)–(3.10), we have t θ , ε as θ + uniformly for ε > 0 small enough. Moreover, we can obtain a θ ε > 0 such that t θ ε , ε = 0 by applying Lemma 3.3 repeatedly. Therefore,

(3.20) m c , α = inf u P c , α J α ( u ) sup θ 0 J α ( u ˜ θ , ε ) .

Next, we show that

(3.21) sup θ 0 J α ( u ˜ θ , ε ) < m c , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

From u α + S c , u ε > 0 , and note that

J α ( u ˜ θ , ε ) m c , α as θ 0 ; J α ( u ˜ θ , ε ) as θ ,

we see that there exists θ 0 > 0 large enough, such that for θ ( θ 0 , ) ( 0 , 1 θ 0 ) , one has

J α ( u ˜ θ , ε ) < m c , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Therefore, we only need to estimate J α ( u ˜ θ , ε ) for 1 θ 0 θ θ 0 . Using the inequalities

( a + b ) r a r + b r + r a r 1 b , a > 0 , b > 0 , r 2 ,

and

( a + b ) r a r + b r + r a r 1 b + r a b r 1 , a > 0 , b > 0 , r 3 ,

we have the following assertions:

(3.22) u ˆ θ , ε 2 = u α + 2 + θ u ε 2 + 2 θ R N ( Δ ) s 2 u α + ( Δ ) s 2 u ε d x ;

(3.23) R N ( I μ * u ˆ θ , ε q ) u ˆ θ , ε q d x R N ( I μ * u α + q ) u α + q d x + R N ( I μ * θ u ε q ) θ u ε q d x + 2 q R N ( I μ * u α + q ) u α + q 1 θ u ε ,

(3.24) R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x R N ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * d x + θ 2 2 μ , s * R N ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * 1 u ε d x + 2 2 μ , s * R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * 1 u α + d x + R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * d x ,

for 2 s < N < 4 s , and

(3.25) R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x R N ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * d x + θ 2 2 μ , s * R N ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * 1 u ε d x + R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * d x ,

for N 4 s . For notational simplicity, denote by

H ( u α + , ε ) R N ( I μ * u α + q ) u α + q d x + R N ( I μ * θ u ε q ) θ u ε q d x + 2 q R N ( I μ * u α + q ) u α + q 1 θ u ε .

Note that u α + satisfies the equation

(3.26) ( Δ ) s u α + = λ u α + + α ( I μ * u α + q ) u α + q 2 u α + + ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * 2 u α + , x R N ,

with λ < 0 . Combining P μ ( u α + ) = 0 , we have that

(3.27) λ a 2 = λ u α + 2 2 = α ( γ q , s 1 ) R N ( I μ * u α + q ) u α + q d x .

By a direct calculus, we obtain

(3.28) J α ( u ˜ θ , ε ) = 1 2 u ˆ θ , ε 2 α 2 q τ 2 q s ( γ q , s 1 ) R N ( I μ * u ˆ θ , ε q ) u ˆ θ , ε q d x 1 2 2 μ , s * R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x 1 2 u ˆ θ , ε 2 α 2 q τ 2 q s ( γ q , s 1 ) H ( u α + , ε ) 1 2 2 μ , s * R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x T 1 + T 2 ,

where

T 1 = 1 2 u ˆ θ , ε 2 α 2 q H ( u α + , ε ) 1 2 2 μ , s * R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x

and

T 2 = α 2 q ( 1 τ 2 q s ( γ q , s 1 ) ) H ( u α + , ε ) .

From (3.21)–(3.23) and (3.25) we deduce that

T 1 = 1 2 u ˆ θ , ε 2 α 2 q H ( u α + , ε ) 1 2 2 μ , s * R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x J α ( u α + ) + J α ( θ u ε ) + θ R N ( Δ ) s 2 u α + ( Δ ) s 2 u ε α R N ( I μ * u α + q ) u α + q 1 θ u ε d x R N ( I μ * u α + 2 μ , s * ) u α + 2 μ , s * 1 θ u ε d x R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * 1 u α + d x = J α ( u α + ) + J α ( θ u ε ) + θ λ R N u α + u ε d x R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * 1 u α + d x .

We use the fact that

sup t 0 t 2 2 a t 2 2 μ , s * 2 2 μ , s * b = N + 2 s μ 2 ( 2 N μ ) a b 1 2 μ , s * 2 N μ N + 2 s μ , for any a , b > 0 ,

and we can deduce that

(3.29) max θ 1 θ 0 , θ 0 J α ( θ u ε ) max θ 1 θ 0 , θ 0 θ 2 u ε 2 θ 2 2 μ , s * 2 2 μ , s * R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x max θ 0 θ 2 u ε 2 θ 2 2 μ , s * 2 2 μ , s * R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x = N + 2 s μ 2 ( 2 N μ ) u ε 2 R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x 1 2 μ , s * 2 N μ N + 2 s μ N + 2 s μ 2 ( 2 N μ ) S N 2 s + O ( ε N 2 s ) 1 C ˜ N , μ , s 2 2 μ , s * S h , l 2 N μ N μ + 2 s O ( ε 2 N μ 2 ) 1 2 μ , s * 2 N μ N + 2 s μ = N + 2 s α 2 ( 2 N α ) S N 2 s 2 N α N + 2 s α C ˜ N , α , s 2 ( 2 N α ) N + 2 s α ( 1 + O ( ε N 2 s ) ) S h , l ( 2 N α ) ( N 2 s ) ( N α + 2 s ) 2 1 O ( ε 2 N α 2 ) = N + 2 s μ 2 ( 2 N μ ) S h , l ( 2 N μ ) 2 ( N μ + 2 s ) 2 ( 1 + O ( ε N 2 s ) ) S h , l ( 2 N μ ) ( N 2 s ) ( N μ + 2 s ) 2 1 O ( ε 2 N μ 2 ) = N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ + O ( ε N 2 s ) .

By direct calculation, we have

(3.30) R N ( I μ * θ u ε 2 μ , s * ) θ u ε 2 μ , s * 1 u α + d x 1 θ 0 2 2 μ , s * 1 min B 1 u α + B 1 ( I μ * U ε 2 μ , s * ) U ε 2 μ , s * 1 u α + d x C B 1 B 1 U ε 2 μ , s * ( x ) 2 μ , s * U ε ( y ) 2 μ , s * 1 x y μ d x d y = C ε N 2 s 2 B 1 ε B 1 ε d x d y ( 1 + x 2 ) N 2 s 2 2 μ , s * x y μ ( 1 + y 2 ) N 2 s 2 ( 2 μ , s * 1 ) C 1 ε N 2 s 2 .

Consequently, by (3.28) and (3.29), we infer that

(3.31) T 1 = 1 2 u ˆ θ , ε 2 α 2 q H ( u α + , ε ) 1 2 2 μ , s * R N ( I μ * u ˆ θ , ε 2 μ , s * ) u ˆ θ , ε 2 μ , s * d x J α ( u α + ) + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ + O ( ε N 2 s ) + θ λ R N u α + u ε d x C 1 ε N 2 s 2 .

From (3.9), u ˆ θ , ε 2 a , and τ s = u ˆ θ , ε 2 a , we infer

(3.32) τ 2 q s ( γ q , s 1 ) = u ˆ θ , ε 2 2 a 2 q ( γ q , s 1 ) = 1 + 2 θ a 2 R N u α + u ε d x + O ( ε N 2 s ) q ( γ q , s 1 ) = 1 + q ( γ q , s 1 ) 2 θ a 2 R N u α + u ε d x + O ( ε N 2 s ) , if 2 s < N < 4 s ; O ( ε 2 s ln ε ) , if N = 4 s ; O ( ε 2 s ) , if N > 4 s .

From (1.22) and

(3.33) R N ( I μ * θ u ε q ) θ u ε q d x C ( N , μ ) θ 0 2 q u 2 N q 2 N μ 2 q = O ( ε 2 N μ ( N 2 s ) q ) , if 2 N q 2 N μ > N N 2 s ; O ( ε 2 N μ 2 log ε 2 N μ N ) , if 2 N q 2 N μ = N N 2 s ; O ( ε ( N 2 s ) q ) , if 2 N q 2 N μ < N N 2 s ;

and

(3.34) R N ( I μ * u α + q ) u α + q 1 θ d x C u ε 2 N q 2 N μ = O ( ε N 2 s 2 ) .

From (3.33) and (3.34), we obtain

H ( u α + , ε ) = R N ( I μ * u α + q ) u α + q d x + R N ( I μ * θ u ε q ) θ u ε q d x + 2 q R N ( I μ * u α + q ) u α + q 1 θ u ε = R N ( I μ * u α + q ) u α + q d x + O ( ε 2 N μ ( N 2 s ) q ) , if 2 N q 2 N μ > N N 2 s ; O ( ε 2 N μ 2 log ε 2 N μ N ) , if 2 N q 2 N μ = N N 2 s ; O ( ε ( N 2 s ) q ) , if 2 N q 2 N μ < N N 2 s .

Thus, using (3.41), and θ [ 1 θ 0 , θ 0 ] , we obtain

(3.35) T 2 = α 2 q ( 1 τ 2 q s ( γ q , s 1 ) ) H ( u α + , ε ) θ α ( 1 γ q , s ) a 2 R N u α + u ε d x R N ( I μ * u α + q ) u α + q d x + O ( ε N 2 s ) , if 2 s < N < 4 s ; O ( ε 2 s ln ε ) , if N = 4 s ; O ( ε 2 s ) , if N > 4 s ; = λ θ R N u α + u ε d x + O ( ε N 2 s ) , if 2 s < N < 4 s ; O ( ε 2 s ln ε ) , if N = 4 s ; O ( ε 2 s ) , if N > 4 s . .

If 2 s < N < 4 s , then by (3.28), (3.28), (3.31), and (3.35), we derive that

(3.36) J α ( u ˜ θ , ε ) m a , α + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ + O ( ε N 2 s ) C 1 ε N 2 s 2 < m a , α + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ .

If N 4 s , we can calculate that

(3.37) R N ( I μ * u ˆ θ , ε q ) u ˆ θ , ε q d x R N ( I μ * u α + q ) u α + q d x + R N ( I μ * θ u ε q ) θ u ε q d x + 2 q R N ( I μ * u α + q ) u α + q 1 θ u ε d x + 2 R N ( I μ * u α + q ) θ u ε + q d x .

Using (3.37), arguing as we have done for (3.36), we can derive that

(3.38) J α ( u ˜ θ , ε ) m a , α + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ + O ( ε N 2 s ) C 1 ε N 2 s 2 2 R N ( I μ * u α + q ) θ u ε + q + O ( ε 2 s ln ε ) .

By Lemma 3.3 in [13], I μ * u α + q C 0 ( R N ) . Therefore, for θ [ 1 θ 0 , θ 0 ] , we have

R N ( I μ * u α + q ) θ u ε + q 1 θ 0 q min B 1 u α + B 1 ( I μ * u α + q ) R N U ε q d x C θ 0 q R N U ε q d x C θ 0 q ε N ( 2 q ) + 2 s q 2 ,

where we have used the fact that

I μ * u α + q = R N u α + ( y ) q x y μ d y B 2 ( x ) u α + ( y ) q x y μ d y = B 2 ( 0 ) u α + ( x + z ) q z μ d z C B 2 ( 0 ) u α + ( x + z ) q d z C 1 B 1 ( 0 ) u α + ( z ) q d z = C 2 , for x B 1 ( 0 ) .

From (3.37)–(3.39), we infer that

(3.39) J α ( u ˜ θ , ε ) m a , α + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ + O ( ε N 2 s ) C 1 ε N 2 s 2 2 C θ 0 q ε N ( 2 q ) + 2 s q 2 + O ( ε 2 s ln ε ) < m a , α + N + 2 s μ 2 ( 2 N μ ) S h , l 2 N μ N + 2 s μ

by virtue of N ( N 2 s ) q 2 < min { N 2 s , 2 s } for q > 2 . In conclusion,

m c , α < m c , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N + 2 s μ .

Hence, we complete the proof.□

Remark 3.1

In view of m c , α < 0 , we see that m c , α < N μ + 2 s 4 N 2 μ S h , l 2 N μ N + 2 s μ

Now, we define the function

v b = b c u S b

for all b > 0 , where u P c , α ± , c > 0 and α c 2 q ( 1 γ q , s ) < C * . By Lemma 3.3, there exists t ± ( b ) R that satisfies t ± ( b ) v b P b , α ± for b > 0 such that α b 2 q ( 1 γ q , s ) < C * . Clearly, t ± ( c ) = 0 .

Lemma 3.7

Let N > 2 s , 2 N μ N < q < q ¯ = 2 + 2 s μ N , and c , α > 0 . If α c 2 q ( 1 γ q , s ) < C * , then t ± ( c ) exists and

(3.40) t ± ( c ) = α q γ q , s R N ( I μ * u q ) u q d x + 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x u 2 c s ( u 2 α q γ q , s 2 R N ( I μ * u q ) u q d x 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ) .

Moreover, if b > c and α b 2 q ( 1 γ q , s ) < C * , then J α ( t ± ( b ) v b ) < J α ( u ) .

Proof

The proof mainly refers to [56]. We define

W ( b , t ) b c e s t 2 u 2 b c 2 q ( e s t ) 2 q γ q , s α γ q , s R N ( I μ * u q ) u q d x b c e s t 2 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Since t ± ( b ) v b P b , α ± , then W ( b , t ± ( b ) ) = 0 . Furthermore, it is easy to check that

t W ( c , 0 ) = 2 s u 2 2 s α q γ q , s 2 R N ( I μ * u q ) u q d x 2 s 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x 0

by u P c , α ± . Hence, t ± ( c ) exists and (3.40) holds by applying the implicit function theorem.

For t ± ( b ) v b P b , α ± and u P c , α ± , using the Taylor expansion, we can deduce that

J α ( t ± ( b ) v b ) = 1 2 1 2 q γ q , s t ± ( b ) v b 2 + 1 2 q γ q , s 1 2 2 μ , s * R N ( I μ * t ± ( b ) v b 2 μ , s * ) t ± ( b ) v b 2 μ , s * d x = b c e t ± ( b ) s 2 q γ q , s 1 2 q γ q , s u 2 + b c e t ± ( b ) s 2 2 μ , s * 2 μ , s * q γ q , s 2 2 μ , s * q γ q , s R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x = q γ q , s 1 2 q γ q , s u 2 + 2 μ , s * q γ q , s 2 2 μ , s * q γ q , s R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x + 1 + c t ± ( c ) s c 2 q γ q , s 1 2 q γ q , s u 2 + 2 2 μ , s * 2 μ , s * q γ q , s 2 2 μ , s * q γ q , s R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ( b c ) + o ( b c ) ,

where

1 + c t ± ( c ) s c = α q γ q , s ( 1 γ q , s ) R N ( I μ * u q ) u q d x c ( u 2 α q γ q , s 2 R N ( I μ * u q ) u q d x 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ) .

Therefore,

J α ( t ± ( b ) v b ) = J α ( u ) α ( 1 γ q , s ) R N ( I μ * u q ) u q d x c ( b c ) + o ( b c ) .

Moreover, for c > 0 and t ± ( b ) v b P b , α ± , there holds

d J α ( t ± ( b ) v b ) d b b = c = α ( 1 γ q , s ) R N ( I μ * u q ) u q d x c < 0 ,

and so J α ( t ± ( b ) v b ) < J α ( u ) for b > c .□

Lemma 3.8

Let N > 2 s and c > 0 , then P c , 0 = P c , 0 and

inf u P c , 0 E 0 ( u ) = inf u S c max t R E 0 ( t u ) = inf u S c N μ + 2 s 4 N 2 μ u 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x 1 2 μ , s * 2 μ , s * 2 μ , s * 1 .

Proof

A direct calculation shows that Ψ u 0 has a unique maximum point at t u , 0 such that

(3.41) e 2 s t u , 0 ( 2 μ , s * 1 ) = u 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ,

for every u S c , which implies that P c , 0 + = . If there exists some u P c , 0 0 , then we have

u 2 = 2 μ , s * R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x = 2 μ , s * u 2 .

Therefore, we can obtain that u 2 = 0 , which is a contradiction with u S c , and then, P c , 0 = P c , 0 .

If u P c , 0 , then t u , 0 = 0 , and we have

E 0 ( u ) = max t R E 0 ( t u ) inf u S c max t R E 0 ( t u ) .

On the other hand, if u S c , then t u , 0 u P c , 0 and

max t R E 0 ( t u ) = E 0 ( t u , 0 u ) inf u P c , 0 E 0 ( u ) .

Combining with the aforementioned two inequalities and (3.41), we have

inf u P c , 0 E 0 ( u ) = inf u S c max t R E 0 ( t u ) = inf u S c 1 2 u 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x 1 2 μ , s * 1 u 2 1 2 2 μ , s * u 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x 2 μ , s * 2 μ , s * 1 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x = inf u S c N μ + 2 s 4 N 2 μ u 2 ( R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ) 1 2 μ , s * 2 μ , s * 2 μ , s * 1 .

Proof of Theorem 1.2

Similar to the proof of Theorem 1.1, there exists a minimizing sequence { u n } P c , α , and by Schwarz rearrangement if necessary to ensure that { u n } S c , r being real valued, non-negative, radially symmetric and decreasing in r = x . Moreover, by Lemmas 2.2, 2.3, and Remark 3.1, we know that { u n } P c , α is bounded in H s ( R N ) and up to a subsequence u n u weakly in H s ( R N ) , where u 0 0 . Note that q γ q , s < 1 , then the method of using Proposition 2.3 and the Pohozaev identity to prove the compactness of the Palais-Smale sequence in Theorem 1.1 and the following Theorems 1.3 and 1.4 is no longer applicable. To overcome this difficulty, we shall prove the compactness of the Palais-Smale sequence mainly by Lemma 3.7, and the method mainly due to [56].

Set v n = u n u 0 , arguing as in the proof of Proposition 2.3, we have

lim n v n 2 = lim n R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x = β ,

then either β = 0 or β S h , l 2 N μ N μ + 2 s . If β = 0 , the second conclusion of Proposition 2.3 holds. Thus, it remains to consider the case of β S h , l 2 N μ N μ + 2 s . By the Fatou lemma, we see that u 0 2 2 c ¯ , with 0 < c ¯ c . Now, choose t n R such that

e 2 s ( 2 μ , s * 1 ) t n = v n 2 R N ( I μ * v n 2 μ , s * ) v n 2 μ , s * d x .

Then, by (1.6), we have

t n v n 2 = R N ( I μ * t n v n 2 μ , s * ) t n v n 2 μ , s * d x S h , l 2 N μ N μ + 2 s ,

and { t n } is bounded. Since u 0 2 2 c ¯ 2 , 0 < c ¯ c , then there exists ρ 0 such that ρ 0 u 0 P c ¯ , α . We want to prove t n ρ 0 , up to a subsequence. Otherwise, there is a subsequence still written as { t n } , which satisfies t n < ρ 0 for all n . For u n P c , α , by Lemmas 3.3, 3.7, the nonlocal Brezis-Lieb lemma, and the fact lim n R N ( I μ * u n q ) u n q d x = R N ( I μ * u q ) u q d x , we infer that

m c , α + o n ( 1 ) = J α ( u n ) J α ( t n u n ) = J α ( t n u 0 ) + J 0 ( t n v n ) + o n ( 1 ) m c ¯ , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + o n ( 1 ) m c , α + N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + o n ( 1 ) ,

which is a contradiction with Lemma 3.6. Without loss of generality, we can assume t n ρ 0 for all n . Furthermore, we have

m c , α + o n ( 1 ) = J α ( u n ) J α ( ρ 0 u n ) = J α ( ρ 0 u 0 ) + J 0 ( ρ 0 v n ) + o n ( 1 ) J α ( ρ 0 u 0 ) + o n ( 1 ) m c ¯ , α + o n ( 1 ) m c , α + o n ( 1 ) ,

where we used Lemma 3.3, the fact that t n ι 0 and J 0 ( ρ 0 v n ) 0 by Lemma 3.8. Hence, we deduce that c ¯ = c and m c , α = J α ( ρ 0 u 0 ) . Furthermore, u α ρ 0 u 0 solves (1.1), which a positive, real-valued, radially symmetric, and decreasing in r = x for some λ α < 0 .□

4 L 2 -critical and supercritical case

In this section, we will prove Theorems 1.3 and 1.4. The method used mainly stems from [53]. Note that the change in q reduces the difficulty of the proof. We still have the same thing that P c , α is a smooth manifold of codimension 1 in S c . Compared with the previous section, it is found that the geometry structure of J α S c changes due to q q ¯ = 2 + 2 s μ N , and the most significant is the number of critical points of Ψ u α ( s ) .

We first prove the following lemma.

Lemma 4.1

P c , α 0 = and P c , α is a smooth manifold of codimension 1 in H s ( R N ) .

Proof

Assume, by contradiction, that there exists a u ¯ P c , α 0 , which implies that

u ¯ 2 = α q γ q , s 2 R N ( I μ * u ¯ q ) u ¯ q d x + 2 μ , s * R N ( I μ * u ¯ 2 μ , s * ) u ¯ 2 μ , s * d x ,

and by P α ( u ¯ ) = 0 , we obtain

(4.1) α γ q , s ( 1 q γ q , s ) R N ( I μ * u ¯ q ) u ¯ q d x = ( 2 μ , s * 1 ) R N ( I μ * u ¯ 2 μ , s * ) u ¯ 2 μ , s * d x .

If q = q ¯ = 2 + 2 s μ N , then q γ q , s = 1 . Hence, from (4.1), we have R N ( I μ * u ¯ 2 μ , s * ) u ¯ 2 μ , s * d x = 0 , and so u ¯ = 0 , which is a contradiction with u S c .

If q > q ¯ = 2 + 2 s μ N , then q γ q , s > 1 which implies that the left-hand side of (4.1) is non-positive. On the other hand, since 2 μ , s * 1 > 0 , then the right-hand side of (4.1) is non-negative and we can deduce that u ¯ = 0 , which is a contradiction to u ¯ S c .

The rest of the proof is similar to Proposition 3.1, and we omit it here.□

Lemma 4.2

For each u S c , the function Ψ u α ( t ) has a unique critical point t u , 2 R , which is a strict maximum point at the positive level, and t u , 2 u P c , α . Moreover,

  1. P c , α = P c , α ;

  2. Ψ u α ( t ) is strictly decreasing and concave on ( t u , 2 , + ) . Moreover, if t u , 2 < 0 , then P α ( u ) < 0 ;

  3. The map u t u , 2 is of class C 1 ;

  4. If P α ( u ) < 0 , then t u , 2 < 0 .

Proof

If q = q ¯ = 2 + 2 s μ N , then q ¯ γ q ¯ , s = 1 and

(4.2) Ψ u α ( t ) = e 2 s t 2 u 2 α q ¯ R N ( I μ * u q ¯ ) u p ¯ d x 1 2 2 μ , s * e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

From Inequality (1.22), if α c 2 N μ + 2 s N < C * = q ¯ C N , q ¯ , s , then

u 2 α q ¯ R N ( I μ * u q ¯ ) u q ¯ d x 1 α q ¯ C N , q ¯ , s c 2 N μ + 2 s N u 2 > 0 .

By Proposition 2.2, monotonicity, and convexity Ψ u α ( t ) , we can obtain the existence and uniqueness of t u , 2 which is a strict maximum point at the positive level.

If q ¯ = 2 + 2 s μ N < q < 2 N μ N 2 s , then q γ q , s > 1 and

(4.3) Ψ u α ( t ) = e 2 s t 2 u 2 α 2 q e 2 q γ q , s s t R N ( I μ * u q ) u q d x 1 2 2 μ , s * e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Since q γ q , s > 1 and α > 0 , for every u S c , r , it is easy to check that

Ψ u α ( t ) 0 + , as t , and Ψ u α ( t ) , as t + .

Therefore, Ψ u α has a global maximum point t u , 2 at the positive level. In addition, since

( Ψ u α ) ( t ) = s e 2 s t u 2 s α γ q , s e 2 q γ q , s s t R N ( I μ * u q ) u q d x s e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x ,

( Ψ u α ) ( t ) = 0 if and only if

e 2 s t u 2 = α γ q , s e 2 q γ q , s s t R N ( I μ * u q ) u q d x + e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x f ( t ) .

It is easy to see that f ( t ) is positive, continuous, and increasing; furthermore, f ( t ) 0 + as t and f ( t ) + as t + . Therefore, t u , 2 is the unique critical point of Ψ u α . As in Lemma 3.3, we can obtain the map u t u , 2 is of class C 1 by the implicit function theorem.

Using the same way as Lemma 4.1, we can check that Ψ u α has only one inflection point and is concave on ( t u , 2 , + ) . Note that ( Ψ u α ) ( t ) < 0 holds if and only if t > t u , 2 ; hence, P μ ( u ) = ( Ψ u α ) ( 0 ) < 0 if and only if t u , 2 < 0 . Again by Lemma 4.1, we have t u , 2 u P c , α , ( Ψ u α ) ( t u , 2 ) 0 and P c , α 0 = ; thus, t u , 2 u P c , α . Consequently, by Proposition 2.2, we conclude that P c , α = P c , α .□

Lemma 4.3

m c , α inf u P c , α J α ( u ) > 0 .

Proof

If q = q ¯ = 2 + 2 s μ N , then for every u P c , α , by Inequalities (1.15) and (1.22), we have

u 2 = α γ q ¯ , s R N ( I μ * u q ¯ ) u q ¯ d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x α γ q ¯ , s C N , q ¯ , s c 2 N μ + 2 s N u 2 + S h , l 2 μ , s * u 2 2 μ , s * .

Since α c 2 N μ + 2 s N < C * = q ¯ C N , q ¯ , s , then

u 2 2 μ , s * 2 1 α γ q ¯ , s C N , q ¯ , s c 2 N μ + 2 s N S h , l 2 μ , s * > 0 ,

which implies that inf u P c , α u 2 > 0 . Therefore, for P α ( u ) = 0 ,

J α ( u ) = 1 2 1 1 2 μ , s * u 2 α 2 q ¯ 1 1 2 μ , s * R N ( I μ * u q ¯ ) u q ¯ d x 1 2 1 1 2 μ , s * u 2 α 2 q ¯ 1 1 2 μ , s * C N , q ¯ , s c 2 N μ + 2 s N u 2 = 1 2 1 1 2 μ , s * 1 α q ¯ C N , q ¯ , s c 2 N μ + 2 s N u 2 > 0 .

If q ¯ = 2 + 2 s μ N < q < 2 N μ N 2 s , then for every u P c , α , using Inequalities (1.15) and (1.22), we infer to

u 2 = α γ q , s R N ( I μ * u q ) u q d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x α γ q , s C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s + S h , l 2 μ , s * u 2 2 μ , s * ,

which implies that

α γ q , s C N , q , s c 2 q ( 1 γ q , s ) u 2 ( q γ q , s 1 ) + S h , l 2 μ , s * u 2 ( 2 μ , s * 1 ) 1 .

Since q γ q , s > 1 , then we obtain inf u P c , α u 2 > 0 . Moreover, by the fact that P α ( u ) = 0 , we can derive

inf u P c , α R N ( I μ * u q ) u q d x + R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x > 0 .

Therefore, by P α ( u ) = 0 and the aforementioned inequality, we obtain

inf u P c , α J α ( u ) = inf u P c , α α 2 q ( q γ q , s 1 ) R N ( I μ * u q ) u q d x + N μ + 2 s 4 N 2 μ R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x > 0 .

Lemma 4.4

There exists k > 0 small enough, if u M ¯ k such that J α ( u ) , P α ( u ) > 0 and sup M ¯ k J α < m c , α , where M k { u S c : u < k , k > 0 } .

Proof

If q = q ¯ = 2 + 2 s μ N , by Inequalities (1.15) and (1.22), we have that both

J α ( u ) 1 2 1 α q ¯ C N , q ¯ , s c 2 q ¯ ( 1 γ q ¯ , s ) u 2 1 2 2 μ , s * S h , l 2 μ , s * u 2 2 μ , s * > 0

and

P α ( u ) s ( 1 α γ q ¯ , s C N , q ¯ , s c 2 q ¯ ( 1 γ q ¯ , s ) ) u 2 s S h , l 2 μ , s * u 2 2 μ , s * > 0

are true for α c 2 N μ + 2 s N < q ¯ C N , q ¯ , s and u M ¯ k with k small enough. Moreover, since m c , α > 0 by Lemma 4.3, then we have

J α ( u ) 1 2 u 2 < m c , α

for k small enough.

If q ¯ = 2 + 2 s μ N < q < 2 N μ N 2 s , then for all u P c , α and u M ¯ k with k small enough, we have

J α ( u ) 1 2 u 2 α 2 q C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s 1 2 2 μ , s * S h , l 2 μ , s * u 2 2 μ , s * > 0 , P α ( u ) u 2 α γ q , s C N , q , s c 2 q ( 1 γ q , s ) u 2 q γ q , s S h , l 2 μ , s * u 2 2 μ , s * > 0 ,

by (1.15) and (1.22). Similarly, we can choose k small enough such that

J α ( u ) 1 2 u 2 < m c , α .

In order to recover the compactness of the Palais-Smale sequence by applying Proposition 2.3, we need to consider

m c , α , r inf u P c , α S c , r J α ,

where S c , r = S c H r s is the subset of the radial functions in S c .

Lemma 4.5

m c , α , r = m c , α .

Proof

Obviously, m c , α m c , α , r . If m c , α m c , α , r , then there exists u P c , α \ S c , r such that

J α ( u ) < inf u P c , α S c , r J α ( u ) .

We let v u * , the symmetric decreasing rearrangement of modulus u , which lies in S c , r . Then, we have

v 2 u 2 , J α ( v ) J α ( u ) , P α ( v ) P α ( u ) = 0 .

If P α ( v ) = 0 , then J α ( u ) < inf u P c , α S c , r J α ( u ) J α ( v ) , which is a contradiction. Hence, we have P α ( v ) < 0 , and from Lemma 4.2, we have t v , 2 < 0 . If q = q ¯ = 2 + 2 s μ N , we have

J α ( u ) < J α ( t v , 2 v ) = N μ + 2 s 4 N 2 μ e 2 2 μ , s * s t v , 2 R N ( I μ * v 2 μ , s * ) v 2 μ , s * d x = N μ + 2 s 4 N 2 μ e 2 2 μ , s * s t v , 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x = e 2 2 μ , s * s t v , 2 J α ( u ) < J α ( u ) .

If q ¯ = 2 + 2 s μ N < q < 2 N μ N 2 s , we have

J α ( u ) < J α ( t v , 2 v ) = α 2 q ( q γ q , s 1 ) e 2 q γ q , s s t v , 2 R N ( I μ * v q ) v q d x + N μ + 2 s 4 N 2 μ e 2 2 μ , s * s t v , 2 R N ( I μ * v 2 μ , s * ) v 2 μ , s * d x = α 2 q ( q γ q , s 1 ) e 2 q γ q , s s t v , 2 R N ( I μ * u q ) u q d x + N μ + 2 s 4 N 2 μ e 2 2 μ , s * s t v , 2 R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x < J α ( u ) .

Combining the aforementioned two cases, we infer that m c , α , r = m c , α .□

To prove Theorems 1.3 and 1.4, we need to introduce some preliminary results from [21].

Definition 4.1

Let B be a closed subset of X . We say that a class of compact subsets of X is a homotopy-stable family with extended boundary B if

  1. each set in contains B ;

  2. for any set A in and any η C ( [ 0 , 1 ] × X ; X ) , satisfying η ( t , x ) = x for all ( t , x ) ( { 0 } × X ) ( [ 0 , 1 ] × B ) , we have η ( 1 × A ) .

Lemma 4.6

[21]. Let φ be a C 1 -functional on a complete connected C 1 -Finsler manifold X and consider a homotopy-stable family with an extended closed boundary B. Set c = c ( φ , ) = inf A max x A φ ( x ) and let F be a closed subset of X, satisfying

  1. ( A F ) \ B for every A , and

  2. sup φ ( B ) c inf φ ( F ) .

Then, for any sequence of sets ( A n ) n in such that lim n sup A n φ = c , there exists a sequence { x n } n in X such that

lim n + φ ( x n ) = c , lim n + d φ ( x n ) = 0 , lim n + dist ( x n , F ) = 0 , lim n + dist ( x n , A n ) = 0 .

Next, we introduce J ˜ α ( t , u ) : R × H s R

J ˜ α ( t , u ) J α ( t u ) = e 2 s t 2 u 2 α 2 q e 2 q γ q , s s t R N ( I μ * u q ) u q d x 1 2 2 μ , s * e 2 2 μ , s * s t R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x .

Obviously, J ˜ α ( u ) is of class C 1 and is an invariant under rotations applied to u , and a Palais-Smale sequence for J ˜ α R × S c , r is a Palais-Smale sequence for J ˜ α R × S c . Next, we introduce the minimax class

(4.4) Γ { γ = ( ϕ ( κ ) , ψ ( κ ) ) C ( [ 0 , 1 ] , R × S c , r ) : γ ( 0 ) ( 0 , M ¯ k ) , γ ( 1 ) ( 0 , J α 0 ) } ,

where we denote J α b { u S c : J α ( u ) b } , and k is given in Lemma 4.4. Since for every u S c , r , we have t u 0 + as t and J α ( t u ) as t + , so the family Γ is not empty. Therefore, there exist t 0 1 and t 1 1 such that

(4.5) γ u : θ [ 0 , 1 ] ( 0 , ( ( 1 θ ) t 0 + θ t 1 ) u ) R × S c , r

is a path in Γ , then the minimax value

σ ( c , α ) inf γ Γ max ( t , u ) γ ( [ 0 , 1 ] ) J ˜ α ( t , u )

is a real number.

Lemma 4.7

σ ( c , α ) = m c , α , r inf u P c , α S c , r J α ( u ) .

Proof

Let u P c , α S c , r ; then, γ u defined by (4.5) is a path in Γ such that

J α ( u ) = max ( t , u ) γ u ( [ 0 , 1 ] ) J ˜ α σ ( c , α ) ,

which implies that σ ( c , α ) m c , α , r . On the other hand, for any γ = ( ϕ , ψ ) Γ , we see that the function

P α , γ : θ [ 0 , 1 ] P α ( ϕ ( θ ) ψ ( θ ) ) R .

From Lemma 4.4, we have P α , γ ( 0 ) = P α ( ψ ( 0 ) ) > 0 . In addition, since Ψ ψ ( 1 ) α ( t ) > 0 for every t ( , t ψ ( 1 ) , 2 ] and Ψ ψ ( 1 ) α ( 0 ) = J α ( ψ ( 1 ) ) 0 , we obtain t ψ ( 1 ) , 2 < 0 , then by Lemma 4.2, we have P α , γ ( 1 ) = P α ( φ ( 1 ) ) < 0 . Moreover, the map θ ϕ ( θ ) ψ ( θ ) is continuous from [ 0 , 1 ] to H s ( R N ) ; then, for every γ Γ , there exists θ γ ( 0 , 1 ) such that

P α , γ ( θ γ ) = 0 and ϕ ( θ γ ) ψ ( θ γ ) P c , α .

It follows that

(4.6) max ( t , u ) γ ( [ 0 , 1 ] ) J ˜ α ( t , u ) J ˜ α ( γ ( θ γ ) ) = J ˜ α ( ϕ ( θ γ ) ψ ( θ γ ) ) inf u P c , α S c , r J α ( u ) = m c , α , r .

Therefore, σ ( c , α ) m c , α , r , and we complete the proof.□

Lemma 4.8

For the functional J α S c , i.e., J α restricted to S c , there exists a Palais-Smale sequence { u n } S c , r at level σ ( c , α ) with P α ( u n ) 0 .

Proof

We use Lemma 4.6 to complete the proof. To this aim, take

X = R × S c , r , = { γ ( [ 0 , 1 ] ) : γ Γ } , B = ( 0 , M ¯ k ) ( 0 , J α 0 ) ,

F = { ( t , u ) R × S c , r : J ˜ α ( t , u ) σ ( c , α ) } , A = γ ( [ 0 , 1 ] ) , A n = γ n ( [ 0 , 1 ] ) .

Clearly, for any A , there exists a γ 0 Γ such that A = γ 0 ( [ 0 , 1 ] ) and

σ ( c , α ) inf γ Γ max θ [ 0 , 1 ] J ˜ α ( γ ( θ ) ) max θ [ 0 , 1 ] J ˜ α ( γ 0 ( θ ) ) .

Hence, there exists a θ 0 [ 0 , 1 ] such that σ ( c , α ) J ˜ α ( γ 0 ( θ 0 ) ) . We derive that γ 0 ( θ 0 ) F . By Lemmas 4.4 and 4.7, we have

(4.7) σ ( c , α ) = m c , α , r > sup u ( M ¯ k J α 0 ) S c , r J α ( u ) = sup ( s , u ) ( ( 0 , M ¯ k ) ( 0 , J α 0 ) ) ( R × S c , r ) J ˜ α ( s , u ) .

Moreover, by the definition of F , we have

σ ( c , α ) inf ( s , u ) F J ˜ α ( s , u ) .

From (4.7) and (4.6), we infer that F B = , and γ ( θ γ ) = ( ϕ ( θ γ ) , ψ ( θ γ ) ) A F . Furthermore, ( A F ) \ B , for A F . We next check that is a homotopy stable family of compact subsets of X with extended closed boundary B and that F is a dual set for . Indeed, for every γ Γ , we have γ ( 0 ) ( 0 , M ¯ k ) and γ ( 1 ) ( 0 , J α 0 ) and then γ ( 0 ) , γ ( 1 ) B . Therefore, for any set A and any η C ( [ 0 , 1 ] × X ; X ) satisfying η ( t , x ) = x for all ( t , x ) ( { 0 } × X ) ( [ 0 , 1 ] × B ) , there is η ( 1 , γ ( 0 ) ) = γ ( 0 ) , η ( 1 , γ ( 1 ) ) = γ ( 1 ) . Hence, we have η ( { 1 } × A ) . Therefore Assumptions (i) and (ii) of Lemma 4.6 are satisfied. Then, we can we take a minimizing sequence γ n ( θ ) = ( ϕ ( θ ) , ψ ( θ ) ) for σ ( c , α ) with the property that ϕ n ( θ ) = 0 and ψ n ( θ ) 0 a.e. in R N for every θ [ 0 , 1 ] , and there exists a Palais-Smale sequence { ( t n , w n ) } for J ˜ α R × S c , r at level σ ( c , α ) and satisfies

(4.8) t J ˜ α ( t n , w n ) 0 , u J ˜ α ( t n , w n ) ( T w n S c , r ) * 0

as n with the property that

(4.9) t n + dist H s ( w n , ψ n ( [ 0 , 1 ] ) ) 0 .

By the first condition in (4.8) and the definition of J ˜ α , we have that P α ( t n w n ) 0 , and the second condition in (4.8) shows that

e 2 s t n R N ( Δ ) s 2 w n ( Δ ) s 2 ξ d x α e 2 q γ q , s s t n R N ( I μ * w n q ) w n q 2 w n ξ d x e 2 2 μ , s * s t n R N ( I μ * w n 2 μ , s * ) w n 2 μ , s * 2 w n ξ d x = o ( 1 ) ξ H s

for every ξ T w n S c , r . Since { t n } is bounded from above and below by (4.9), we have

(4.10) d J α ( t n w n ) [ t n ξ ] = o ( 1 ) ξ H s = o ( 1 ) t n ξ H s

as n , which implies that { u n t n w n } S c , r is a Palais-Smale sequence for J α restricted on S c , r at level σ ( c , α ) = m c , α , r with P α ( u n ) 0 by Lemma 5.4 of [53] and (4.10). Note that a Palais-Smale sequence for J ˜ α R × S c , r is a Palais-Smale sequence for J α R × S c since the J ˜ α is invariant under rotations applied to u .□

Lemma 4.9

Assume that q = q ¯ = 2 + 2 s μ N , and α c 2 N μ + 2 s N < C * = q ¯ C N , q ¯ , s , then m c , α , r < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Proof

As in the proof of Lemma 3.6, choose χ C 0 ( R N ) a radial cut-off function such that χ ( x ) = 1 for x 1 , χ ( x ) = 0 for x 2 , and 0 χ ( x ) 1 for any x R N . For ε > 0 and any x R N , we define

u ε ( x ) = χ ( x ) U ε ( x ) , v ε ( x ) = c u ε u ε 2 ,

where U ε ( x ) = C ( N , s ) ε N 2 s 2 ( ε 2 + x 2 ) N 2 s 2 , with C ( N , s ) chosen such that U ε 2 = R N U ε 2 s * d x = S N 2 s , achieving the best Sobolev constant S by [51]. Clearly, v ε S c , r , by Lemma 4.2, for every ε > 0 , there exists a unique t v ε , 2 , α such that

m c , α , r = inf u P c , α S c , r J α J α ( t v ε , 2 , α v ε ) = max t R J α ( t v ε ) = max t R Ψ v ε α ( t ) .

Next, we aim to show that max t R Ψ v ε α ( t ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s . Since t v ε , 2 , α is the unique maximum point of Ψ v ε α , then P α ( t v ε , 2 , α v ε ) = 0 and

e 2 ( 2 μ , s * 1 ) s t v ε , 2 , α = v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x α q ¯ R N ( I μ * v ε q ¯ ) v ε q ¯ d x R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x α q ¯ C N , q ¯ , s c 2 N μ + 2 s N v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x = 1 α q ¯ C N , q ¯ , s c 2 N μ + 2 s N v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x ,

from which, we infer that

(4.11) max t R Ψ v ε α ( t ) = Ψ v ε α ( t v ε , 2 , α ) = Ψ v ε 0 ( t v ε , 2 , α ) α 2 q ¯ e 2 s t v ε , 2 , α R N ( I μ * v ε q ¯ ) v ε q ¯ d x max t R Ψ v ε 0 ( t ) α 2 q ¯ 1 α q ¯ C N , q ¯ , s c 2 N μ + 2 s N 1 2 μ , s * 1 × v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x 1 2 μ , s * 1 R N ( I μ * v ε q ¯ ) v ε q ¯ d x = max t R Ψ v ε 0 ( t ) 1 2 q ¯ 1 α q ¯ C N , q ¯ , s c 2 N μ + 2 s N 1 2 μ , s * 1 × α c 2 N μ + 2 s N u ε 2 2 μ , s * 1 R N ( I μ * u ε q ¯ ) u ε q ¯ d x u ε 2 2 N μ + 2 s N ( R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x ) 1 2 μ , s * 1 .

From Lemma 3.8, we know that Ψ v ε 0 has a unique maximum point t v ε , 2 , 0 that satisfies

e 2 ( 2 μ , s * 1 ) s t v ε , 2 , 0 = v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x ,

and so

max t R Ψ v ε 0 = Ψ v ε 0 ( t v ε , 2 , 0 ) = 1 2 v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x 1 2 μ , s * 1 v ε 2 1 2 2 μ , s * v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x 2 μ , s * 2 μ , s * 1 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x = N μ + 2 s 4 N 2 μ v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x 1 2 μ , s * 2 μ , s * 2 μ , s * 1 = N μ + 2 s 4 N 2 μ u ε 2 R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x 1 2 μ , s * 2 μ , s * 2 μ , s * 1 .

By applying (3.29), we have

(4.12) max t R Ψ v ε 0 = Ψ v ε 0 ( t v ε , 2 , 0 ) N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + O ( ε N 2 s ) .

When N > 2 s , we have

R N ( I μ * u ε q ¯ ) u ε q ¯ d x O ( ε 2 N μ ( N 2 s ) q ¯ ) .

Hence, we can derive

(4.13) R N ( I μ * u ε q ¯ ) u ε q ¯ d x u ε 2 2 N μ + 2 s N O ( ε ( 2 N μ ) + ( 2 s N q ¯ ) ) , if N > 4 s ; O ( log ε ( q ¯ 1 ) ) , if N = 4 s ; O ( ε ( 2 N μ ) ( N 2 s ) ( 2 q ¯ 1 ) ) , if 2 s < N < 4 s .

If N > 4 s , it is easy to check that ( 2 N μ ) + ( 2 s N q ¯ ) = 0 < N 2 s ; if N = 4 s , we have lim ε 0 ε 2 s log ε q ¯ 1 = 0 ; if 2 2 s < N < 4 s , let ( 2 N μ ) ( N 2 s ) ( 2 q ¯ 1 ) < N 2 s , which is equivalent to

q ¯ > 2 N μ 2 N 4 s ,

then we have μ > 2 ( 4 s 2 + ( 2 s N ) N ) 4 s N . In conclusion, if N 4 s and 0 < μ < N or 2 2 s < N < 4 s and μ ( 2 ( 4 s 2 + ( 2 s N ) N ) 4 s N , N ) , then by (4.11), (4.12), and (4.13), we have

m c , α , r max t R Ψ v ε α ( t ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Proof of Theorem 1.3

From Lemma 4.8, we know there exists a Palais-Smale sequence { u n t n w n } S c , r at level σ ( c , α ) with the property that P α ( u n ) 0 . By Lemmas 4.7 and 4.9, we have

m c , α , r = σ ( c , α ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Hence, one of the two alternatives in Proposition 2.3 holds. If Case (i) occurs, then up to a subsequence u n u weakly in H s ( R N ) such that for some λ < 0

J α ( u ) σ ( c , α ) N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s < 0 ,

where u 0 is a solution to (1.1). By the Pohozaev identity, we obtain P α ( u ) = 0 , which leads to

J α ( u ) = N μ + 2 s 4 N 2 μ R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x > 0 ,

which is a contradiction. Therefore, the second case of Proposition 2.3 holds, namely, u n u strongly in H s ( R N ) , such that

J α ( u ) = σ ( c , α ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s ,

and u 0 is a real-valued radial normalized solution to (1.1) for some λ < 0 . Moreover, by the proof of Lemma 4.8 that ψ n ( θ ) 0 a.e. in R N for every θ [ 0 , 1 ] and the convergence we obtain u is non-negative. By the regularity results from [11], we have u C 0 , γ ( R N ) , γ ( 0 , min { 1 , 2 s } ) . Assume that u ( x 0 ) = 0 for some x 0 R N , then ( Δ ) s u ( x 0 ) = 0 , by the definition of ( Δ ) s in [46], we have

( Δ ) s u ( x 0 ) = C s 2 R N u ( x 0 + y ) + u ( x 0 y ) 2 u ( x 0 ) y N + 2 s d y .

Hence, R N u ( x 0 + y ) + u ( x 0 y ) y N + 2 s d y = 0 , which implies that u 0 , a contradiction with u 2 2 = c 2 > 0 . Therefore, u ( x ) > 0 in R N . Finally, we show that u is a ground state. According to the previous analysis, we know any normalized solutions stay on P c , α and J α ( u ) = m c , α , r = inf u P c , α S c , r J α . By Lemma 4.5, we obtain

inf u P c , α S c , r J α = inf u P c , α J α = m c , α .

Hence, u is a ground state.□

Lemma 4.10

Assume that q ¯ = 2 + 2 s μ N < q < 2 N μ N 2 s , and α c 2 N μ + 2 s N < C * , where C * is a certain constant, then m c , α , r < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Proof

Note that q γ q , s > 1 , as in Lemma 4.9, we have

m c , α , r = inf u P c , α S c , r J α J α ( t v ε , 2 , α v ε ) = max t R J α ( t v ε ) = max t R Ψ v ε α ( t ) ,

and we aim to prove that max t R Ψ v ε α ( t ) < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s . By a direct calculation, we know that Ψ v ε α has a unique maximum point at t v ε , 2 , α such that

e 2 ( 2 μ , s * 1 ) s t v ε , 2 , α = v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x α γ q , s R N ( I μ * v ε q ) v ε q d x R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x e 2 ( q γ q , s 1 ) s t v ε , 2 , α .

Since t v ε , 2 , α v ε P c , α , we have

e 2 2 μ , s * s t v ε , 2 , α R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x = e 2 s t v ε , 2 , α v ε 2 α γ q , s e 2 q γ q , s s t v ε , 2 , α R N ( I μ * v ε q ) v ε q d x e 2 s t v ε , 2 , α v ε 2 ,

and then e 2 s t v ε , 2 , α v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x 1 2 μ , s * 1 . By the fact that q γ q , s > 1 , we can deduce

(4.14) e 2 ( 2 μ , s * 1 ) s t v ε , 2 , α v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x α γ q , s R N ( I μ * v ε q ) v ε q d x R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x v ε 2 R N ( I μ * v ε 2 μ , s * ) v ε 2 μ , s * d x q γ q , s 1 2 μ , s * 1 = u ε 2 2 ( 2 μ , s * 1 ) c 2 ( 2 μ , s * 1 ) u ε 2 R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x α γ q , s u ε 2 2 ( 2 μ , s * q ) c 2 ( 2 μ , s * q ) R N ( I μ * u ε q ) u ε q d x R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x × u ε 2 2 ( 2 μ , s * 1 ) c 2 ( 2 μ , s * 1 ) u ε 2 R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x q γ q , s 1 2 μ , s * 1 = u ε 2 2 ( 2 μ , s * 1 ) c 2 ( 2 μ , s * 1 ) u ε 2 q γ q , s 1 2 μ , s * 1 R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x u ε 2 2 μ , s * q γ q , s 2 μ , s * 1 α γ q , s c 2 q ( 1 γ q , s ) R N ( I μ * u ε q ) u ε q d x R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x q γ q , s 1 2 μ , s * 1 u ε 2 2 q ( 1 γ q , s ) .

By (3.8), (3.12), and (3.13), we can infer that there exist C 1 and C 2 such that

(4.15) u ε 2 q γ q , s 1 2 μ , s * 1 C 1 , C 2 ( R N ( I μ * u ε 2 μ , s * ) u ε 2 μ , s * d x ) q γ q , s 1 2 μ , s * 1 1 C 2 .

Using Proposition 1.1, the Hardy-Littlewood-Sobolev Inequality (1.22) and Estimations (3.8)–(3.9), we can deduce that there exists C 3 > 0 such that

(4.16) R N ( I μ * u ε q ) u ε q d x u ε 2 2 q ( 1 γ q , s ) C N , q , s u ε 2 q γ q , s C 3 .

Using (3.9) and (3.14), we have

(4.17) R N ( I μ * u ε q ) u ε q d x u ε 2 2 q ( 1 γ q , s ) O ( ε ( 2 N μ + q ( 2 s γ q , s N ) ) ) , if N > 4 s ; O ( log ε q ( 1 γ q , s ) ) , if N = 4 s ; O ( ε ( 2 N μ ) ( N 2 s ) q ( 2 γ q , s ) ) , if 2 s < N < 4 s .

Moreover, by (4.14), (4.15), and (4.16), we can infer that

(4.18) e 2 ( 2 μ , s * 1 ) s t v ε , 2 , α C u ε 2 2 ( 2 μ , s * 1 ) c 2 ( 2 μ , s * 1 ) C 1 α γ q , s c 2 q ( 1 γ q , s ) C 3 C 2 C u ε 2 2 ( 2 μ , s * 1 ) c 2 ( 2 μ , s * 1 )

holds if α c 2 q ( 1 γ q , s ) < C 2 ( C 1 1 ) C 3 γ q , s .

Finally, for ε > 0 small enough, by (4.12) and (4.18), we can conclude that

(4.19) max t R Ψ v ε α = Ψ v ε α ( t v ε , 2 , α ) = Ψ v ε 0 ( t v ε , 2 , α ) α 2 q e 2 q γ q , s s t v ε , 2 , α R N ( I μ * v ε q ) v ε q d x N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + O ( ε N 2 s ) C 2 q α c 2 q ( 1 γ q , s ) R N ( I μ * u ε q ) u ε q d x u ε 2 2 q ( 1 γ q , s ) N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s + O ( ε N 2 s ) C R N ( I μ * u ε q ) u ε q d x u ε 2 2 q ( 1 γ q , s ) .

If N > 4 s , it is easy to check that 2 N μ + q ( 2 s γ q , s N ) < N 2 s ; if N = 4 s , we have lim ε 0 ε N 2 s log ε q ( 1 γ q , s ) = 0 ; if 2 s < N < 4 s , let ( 2 N μ ) ( N 2 s ) q ( 2 γ q , s ) < N 2 s , then

q > 2 N μ N 2 s 2 s 4 s N .

Hence, if N 4 s , 0 < μ < N or 2 s < N < 4 s , 0 < μ < N and q max { q ¯ , 2 N μ N 2 s 2 s 4 s N } , 2 N μ N 2 s , then we have

m c , α , r max t R Ψ v ε α < N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s .

Proof of Theorem 1.4

The whole proof is very similar to Theorem 1.3, and the only difference is: since q γ q , s > 1 , by P α ( u ) = 0 , we can deduce that

J α ( u ) = α 2 q ( q γ q , s 1 ) R N ( I μ * u q ) u q d x + N μ + 2 s 4 N 2 μ R N ( I μ * u 2 μ , s * ) u 2 μ , s * d x > 0 ,

which is a contradiction with J α ( u ) σ ( c , α ) N μ + 2 s 4 N 2 μ S h , l 2 N μ N μ + 2 s < 0 .

Acknowledgements

The authors are very grateful for the anonymous reviewers for their careful reading of the manuscript and valuable comments.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (121714971, 11771468, and 11971027).

  2. Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.

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Received: 2023-08-28
Revised: 2023-09-22
Accepted: 2023-09-30
Published Online: 2023-11-18

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

  1. Regular Articles
  2. On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
  3. On the critical Choquard-Kirchhoff problem on the Heisenberg group
  4. On the local behavior of local weak solutions to some singular anisotropic elliptic equations
  5. Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
  6. Double-phase parabolic equations with variable growth and nonlinear sources
  7. Logistic damping effect in chemotaxis models with density-suppressed motility
  8. Bifurcation diagrams of one-dimensional Kirchhoff-type equations
  9. Standing wave solution for the generalized Jackiw-Pi model
  10. Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
  11. Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
  12. Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
  13. Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
  14. Bautin bifurcation with additive noise
  15. Small solitons and multisolitons in the generalized Davey-Stewartson system
  16. Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
  17. A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
  18. On a strongly damped semilinear wave equation with time-varying source and singular dissipation
  19. Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
  20. Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
  21. Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
  22. Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
  23. Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
  24. Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
  25. Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
  26. Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
  27. On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
  28. Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
  29. Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
  30. Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
  31. Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
  32. Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
  33. Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
  34. Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
  35. Approximations of center manifolds for delay stochastic differential equations with additive noise
  36. Periodic solutions to a class of distributed delay differential equations via variational methods
  37. Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
  38. Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
  39. Global Sobolev regular solution for Boussinesq system
  40. Normalized solutions for the p-Laplacian equation with a trapping potential
  41. Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
  42. Blow-up for compressible Euler system with space-dependent damping in 1-D
  43. High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
  44. On the dynamics of grounded shallow ice sheets: Modeling and analysis
  45. A survey on some vanishing viscosity limit results
  46. Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
  47. Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
  48. Front propagation in a double degenerate equation with delay
  49. Positive solutions for a class of singular (pq)-equations
  50. Higher integrability for anisotropic parabolic systems of p-Laplace type
  51. The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
  52. On a system of multi-component Ginzburg-Landau vortices
  53. Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
  54. Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
  55. On double phase Kirchhoff problems with singular nonlinearity
  56. Estimates for eigenvalues of the Neumann and Steklov problems
  57. Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
  58. Dirichlet problems involving the Hardy-Leray operators with multiple polars
  59. Incompressible limit for compressible viscoelastic flows with large velocity
  60. Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
  61. Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
  62. Noncoercive parabolic obstacle problems
  63. Touchdown solutions in general MEMS models
  64. Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
  65. Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
  66. Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
  67. Symmetries of Ricci flows
  68. Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
  69. On the topological gradient method for an inverse problem resolution
  70. Supersolutions to nonautonomous Choquard equations in general domains
  71. Uniform complex time heat Kernel estimates without Gaussian bounds
  72. Global existence for time-dependent damped wave equations with nonlinear memory
  73. Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
  74. Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
  75. Lamé system with weak damping and nonlinear time-varying delay
  76. Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
  77. Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
  78. Boundary regularity results for minimisers of convex functionals with (p, q)-growth
  79. Parametric singular double phase Dirichlet problems
  80. Special Issue on Nonlinear analysis: Perspectives and synergies
  81. Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
  82. Identification of discontinuous parameters in double phase obstacle problems
  83. Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
  84. On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
  85. On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
  86. The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
  87. Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
  88. On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
  89. Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
Heruntergeladen am 16.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/anona-2023-0112/html?lang=de
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