Normalized solutions of NLS equations with mixed nonlocal nonlinearities

Zhenyu Zhang; Juntao Sun

doi:10.1515/anona-2024-0004

Article Open Access

Normalized solutions of NLS equations with mixed nonlocal nonlinearities

Zhenyu Zhang and Juntao Sun

Published/Copyright: May 31, 2024

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From the journal Advances in Nonlinear Analysis Volume 13 Issue 1

Abstract

We study the existence and nonexistence of normalized solutions for the nonlinear Schrödinger equation with mixed nonlocal nonlinearities:

− Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N , ∫ R N ∣ u ∣ 2 d x = c ,

where N ≥ 1 , N + α N ≤ p < q ≤ N + α + 2 N , the parameter μ ∈ R and λ is a Lagrange multiplier. Furthermore, we prove the relationship between minimizers and ground state solutions under the Nehari manifold, which seems to be the first result in the nonlocal context.

Keywords: normalized solutions; mixed nonlocal nonlinearities; minimizers; ground state solutions

MSC 2010: 35J20; 35J60

1 Introduction

Consider the Cauchy problem for the nonlinear Schrödinger equation (NLS) with mixed nonlocal nonlinearities:

(1) i ∂ t ψ + Δ ψ + μ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ + ( I α ∗ ∣ ψ ∣ q ) ∣ ψ ∣ q − 2 ψ = 0 , ψ ( 0 , x ) = ψ 0 ( x ) ,

where ψ ( t , x ) is the complex-valued function in the spacetime R × R N ( N ≥ 1 ), N + α N ≕ 2 α ≤ p < q ≤ 2 α # ≔ N + α + 2 N , the parameter μ ∈ R and I α is the Riesz potential of order α ∈ ( 0 , N ) defined by

I α = Γ N − α 2 π N 2 2 α Γ N − α 2 1 ∣ x ∣ N − α , ∀ x ∈ R N \ { 0 } .

An important feature related to the nonlinear evolution equations such as (1) is the study of standing waves. A standing wave of (1) is a solution of the form

ψ ( t , x ) = e − i λ t u ( x ) ,

where λ ∈ R and u satisfies the stationary equation

(2) − Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N .

In this article, we would like to look for solutions of (2) with the frequency λ unknown. In this case, λ ∈ R appears as a Lagrange multiplier and L 2 -norms of solutions are prescribed, which are usually called normalized solutions. This study seems particularly meaningful from the physical point of view, since standing waves of (1) conserve their mass along time. For c > 0 given, we consider the problem of finding solutions to

(3) − Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N , ∫ R N ∣ u ∣ 2 d x = c .

Solutions of (3) can be obtained as critical points of the energy functional E μ : H 1 ( R N ) → R given by

E μ ( u ) = 1 2 ∫ R N ∣ ∇ u ∣ 2 d x − μ 2 p ∫ R N ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p d x − 1 2 q ∫ R N ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q d x

on the constraint

S ( c ) = u ∈ H 1 ( R N ) ∣ ∫ R N ∣ u ∣ 2 d x = c .

It is easy to show that E μ is a well-defined and C 1 -functional on S ( c ) for 2 α ≤ p < q ≤ 2 α # .

In recent years, the study of normalized solutions to NLS equations of various forms has been extensively studied, such as [1,16] for fractional NLS equations, [26] for p -Laplacian equations, and [7,11,13,14,24,25,27,28] for NLS equations with combined nonlinearities. In particular, more and more researchers focus on the study of NLS equations with mixed nonlocal nonlinearities like (2). When the problem only involves the single Hartree-type nonlinearity, i.e., μ = 0 , (2) is usually called the nonlinear Choquard or the Choquard-Pekar equation. It has several physical origins. In the case where N = 3 and α = p = 2 , the problem

(4) − Δ u + λ u = ( I 2 ∗ u 2 ) u in R 3

appears as a model in quantum theory of a polaron at rest [22]. The time-dependent form of (4) also describes the self-gravitational collapse of a quantum mechanical wave function [23]. Lieb [17] proved the existence and uniqueness of normalized solutions for (4) on S ( c ) by using symmetrization techniques, and Lions [19] studied the existence and stability issues of normalized solutions for (4) on S ( c ) . Recently, for (2) with μ = 0 , Li and Ye [15] showed that if N ≥ 3 and 2 α # < p < 2 α ∗ ≔ 2 N N − α , then the energy functional E μ has an MP geometry on S ( c ) and there exists a mountain-pass normalized solution for each c > 0 .

When the problem involves double Hartree-type nonlinearities, the situation seems to become more complicated. As a similar version to (3), Cao et al. [5] studied the existence and multiplicity of solutions for the following NLS equation with van der Waals-type potentials:

− Δ u = λ u + μ ( ∣ x ∣ − β ∗ ∣ u ∣ 2 ) u + ( ∣ x ∣ − γ ∗ ∣ u ∣ 2 ) u in R N , ∫ R N ∣ u ∣ 2 d x = c ,

where N ≥ 3 , μ > 0 , 0 < β < γ = 2 , or 0 < β < 2 < γ < min { N , 4 } , or 2 < β < γ < min { N , 4 } . Note that L 2 -critical exponent is 2 for β and γ . Later, Jia and Luo [12] studied the case of μ > 0 and γ = 4 , and Bhimani et al. [2] studied the case of μ < 0 . Very recently, Ding and Wang [8] proved the existence of solutions for (3) with N ≥ 3 and 2 α < q < 2 α # < p < 2 α ∗ .

The study of normalized solutions for the NLS equation with mixed nonlocal nonlinearities has attracted increasing attention in recent years. However, so far only a very few results exist in our context, see [2,5,8,12]. In view of this, in this article, we focus on the solvability of the minimization

E c ≔ inf u ∈ S ( c ) E μ ( u ) . (P_c)

Furthermore, we explore the relationship between the minimizers of ( P c ) and the ground state solutions of the associated energy functional under the Nehari manifold, which seems to be the first result in the nonlocal context.

We now summarize on our main results.

Theorem 1.1

Let N ≥ 1 , μ ∈ R and 2 α < p < q < 2 α # . Then, the following statements are true.

E c > − ∞ for all c > 0 .
The map c ↦ E c is non-increasing and continuous.
There exists a number c ∗ ∈ [ 0 , ∞ ) such that
E c = 0 if 0 < c ≤ c ∗ and E c < 0 if c > c ∗ .
When 0 < c < c ∗ , E c = 0 is not achieved.
When c > c ∗ , the global infimum E c is achieved, i.e., there exists a ground state v ∈ S ( c ) with E μ ( v ) = E c < 0 . Moreover, v ∈ C 2 ( R N ) has constant sign and is radially symmetry if μ > 0 .
If μ ≥ 0 , then E c < 0 is achieved for all c > 0 .
If μ > 0 , then the associated Lagrange multiplier λ < 0 ; and if μ < 0 , then the associated Lagrange multiplier λ > 0 .

Theorem 1.2

Let N ≥ 1 , μ > 0 and 2 α = p < q < 2 α # . Then, E c is achieved by v ∈ H 1 ( R N ) such that

E μ ( v ) = E c < − μ N 2 ( N + α ) S α − ( N + α ) ∕ N c ( N + α ) ∕ N .

Theorem 1.3

Let N ≥ 1 and μ ∈ R . Then, the following statements are true.

If p = 2 α , q = 2 α # and N N + α + 2 C G ( 2 α # , N , α ) c ( 2 + α ) ∕ N ≤ 1 , then (2) has no normalized solution.
If q = 2 α , p = 2 α # and ∣ μ ∣ N N + α + 2 C G ( 2 α # , N , α ) c ( 2 + α ) ∕ N ≤ 1 , then (2) has no normalized solution.

Theorem 1.4

Let N ≥ 1 , μ ∈ R and 2 ≤ p < q ≤ 2 α # . Then, for all ψ 0 ( x ) ∈ H 1 ( R N ) , there exists a unique global solution ψ ( t , x ) ∈ C ( R , H 1 ( R N ) ) ∩ C 1 ( R , H − 1 ( R N ) ) for the Cauchy problem (1) with the conversation of energy and mass

E μ ( ψ ( t , x ) ) = E μ ( ψ 0 ) and ‖ ψ ( t , x ) ‖ 2 2 = ‖ ψ 0 ‖ 2 2 for all t > 0 .

Moreover, the set of ground states

(5) Z ( c ) ≔ { ψ ( t , x ) = e − i λ t u ( x ) ∣ u ∈ S ( c ) and E μ ( u ) = E c }

is orbitally stable, that is, ∀ ε > 0 , there exists δ > 0 such that for all ψ 0 ∈ H 1 ( R N ) with inf v ∈ Z ( c ) ‖ ψ 0 − v ‖ H 1 < δ , it satisfies

inf v ∈ Z ( c ) ‖ ψ ( t , x ) − v ‖ H 1 < ε , ∀ t > 0 .

Definition 1.5

For given λ < 0 , a nontrivial solution w ∈ H 1 ( R N ) of the free problem

− Δ u = λ u + μ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + ( I α ∗ ∣ u ∣ q ) ∣ u ∣ q − 2 u in R N , u ∈ H 1 ( R N ) , ( Q λ )

is said to be a ground state solution if it achieves the infimum of the C 1 -energy functional

J λ ( u ) ≔ E μ ( u ) − λ 2 ∫ R N ∣ u ∣ 2 d x

among all the nontrivial solutions, namely,

J λ ( w ) = H λ ≔ inf u ∈ N λ J λ ( u ) ,

where

(6) N λ ≔ { u ∈ H 1 ( R N ) \ { 0 } ∣ ⟨ J λ ′ ( u ) , u ⟩ = 0 }

is usually called the Nehari manifold.

Theorem 1.6

Let N ≥ 1 and μ > 0 . Let λ ( v ) be the Lagrange multiplier corresponding to an arbitrary minimizer v ∈ S m of ( P c ) . Then, the following statements hold:

Any minimizer v of ( P c ) is a ground state solution of ( Q λ ) with λ = λ ( v ) < 0 .
For given λ ∈ { λ ( v ) ∣ v ∈ S ( c ) is a minimizer of ( P c ) } , any ground state solution w ∈ H 1 ( R N ) of ( Q λ ) is a minimizer of ( P c ) , namely
‖ w ‖ 2 2 = c and E μ ( w ) = E c .
The minimizer of ( P c ) on S ( c ) is unique if and only if the ground state solution of ( Q λ ) with λ < 0 is unique.

2 Preliminary results

For the reader’s convenience, we set

A p ( u ) ≔ ∫ R N ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p d x , ∀ 2 α ≤ p ≤ 2 α #

and

L T s L x r ≔ L s ( ( − T , T ) , L r ( R N ) ) ,

and A ≲ B means that there exists a constant C > 0 such that A < C B .

Lemma 2.1

(Hardy-Littlewood-Sobolev inequality [18]) Let N ≥ 1 , α ∈ ( 0 , N ) , k , l > 1 with 1 ∕ k + ( N − α ) ∕ N + 1 ∕ l = 2 . Then, for any u ∈ L k ( R N ) and v ∈ L l ( R N ) , there exists a constant C ( N , α , k ) > 0 such that

∫ R N ∫ R N u ( x ) v ( y ) ∣ x − y ∣ N − α d x d y ≤ C ( N , α , k ) ‖ u ‖ k ‖ v ‖ l .

Lemma 2.2

(Gagliardo-Nirenberg inequality of Hartree type [21]) Let N ≥ 1 , α ∈ ( 0 , N ) , 2 α < p < 2 α # . Then, there exists a constant C G ( N , α , p ) > 0 such that

A p ( u ) ≤ C G ( N , α , p ) ‖ ∇ u ‖ 2 2 p η p ‖ u ‖ 2 2 p ( 1 − η p ) ,

where η p ≔ N p − N − α 2 p , and the constant C G is defined by

C G ( N , α , p ) = p ‖ W p ‖ 2 2 p − 2 ,

where W p is a radially ground state solution of equation

− p η p Δ W + p ( 1 − η p ) W = ( I α ∗ ∣ W ∣ p ) ∣ W ∣ p − 2 W in R N .

Lemma 2.3

[18, Theorem 4.3] Let N ≥ 1 and α ∈ ( 0 , N ) . Then, the minimization problem

S α ≔ inf u ∈ H 1 ( R N ) ∫ R N ∣ u ∣ 2 d x ∫ R N ( I α ∗ ∣ u ∣ N + α N ) ∣ u ∣ N + α N d x N N + α > 0

is achieved by

v = a δ δ 2 + ∣ x − y ∣ 2 N ∕ 2 ,

where a , δ > 0 , y ∈ R N .

Lemma 2.4

[21] Let u ∈ H 1 ( R N ) be a weak solution of (2). Then, u satisfies the Pohozaev identity

N − 2 N ‖ ∇ u ‖ 2 2 = λ N 2 ‖ u ‖ 2 2 + μ ( N + α ) 2 p A p ( u ) + N + α 2 q A q ( u ) .

Furthermore, it holds

P ( u ) ≔ ‖ ∇ u ‖ 2 2 − μ η p A p ( u ) − η q A q ( u ) = 0 .

To show the compactness of minimizing sequence, we need the following P.-L. Lions lemma [19].

Lemma 2.5

Let 2 ≤ q < 2 ∗ . If { u n } is a bounded sequence in H 1 ( R N ) and satisfies

sup y ∈ R N ∫ B ( y , 1 ) ∣ u n ∣ p d x → 0 as n → ∞ ,

then u n → 0 in L p ( R N ) for 2 < p < 2 ∗ .

3 Existence and nonexistence

Lemma 3.1

Assume that N ≥ 1 and 2 α < p < q < 2 α # . Then, for any bounded sequence { u n } ∈ H 1 ( R N ) satisfying lim n → ∞ ‖ u n ‖ 2 N p N + α = 0 , we have

limsup n → ∞ A p ( u n ) = 0 .

Proof

For any bounded sequence { u n } ∈ H 1 ( R N ) satisfying lim n → ∞ ‖ u n ‖ 2 N p N + α = 0 , it follows from Lemma 2.1 that

A p ( u n ) ≤ C ‖ u n ‖ 2 N p N + α 2 ∕ p → 0 as n → ∞ .

The proof is complete.□

Lemma 3.2

Let N ≥ 1 , μ ∈ R and 2 α < p < q < 2 α # . Then, the following statements are true.

− ∞ < E c ≤ 0 for all c > 0 . Furthermore, if μ ≥ 0 , then E c < 0 for all c > 0 .
E c c ( N + α ) ∕ N ≤ E c ¯ c ¯ ( N + α ) ∕ N for any c > c ¯ > 0 .
The map c ↦ E c is non-increasing and continuous.
If E c ¯ < 0 , then we have
E c c ≤ E c ¯ c ¯ for any c > c ¯ > 0 .
In particular, if E c ¯ is achieved, then the inequality is strict. Furthermore, if either E c < 0 or E c ¯ < 0 is achieved, then
(7) E c + c ¯ < E c + E c ¯ .
If μ < 0 , then there exists c ∗ > 0 such that E c < 0 for c > c ∗ , and E c = 0 for c ≤ c ∗ .
If c < c ∗ , then E c = 0 is not achieved.

Proof

( i ) By Lemma 2.2, we have

(8) E μ ( u ) ≥ 1 2 ‖ ∇ u ‖ 2 2 − ∣ μ ∣ 2 p A p ( u ) − 1 2 q A q ( u ) ≥ 1 2 ‖ ∇ u ‖ 2 2 − ∣ μ ∣ 2 p C G ( N , α , p ) ‖ ∇ u ‖ 2 2 p η p ‖ u ‖ 2 2 p ( 1 − η p ) − 1 2 q C G ( N , α , q ) ‖ ∇ u ‖ 2 2 q η q ‖ u ‖ 2 2 q ( 1 − η q ) = 1 2 ‖ ∇ u ‖ 2 2 − ∣ μ ∣ 2 p C G ( N , α , p ) c p ( 1 − η p ) ‖ ∇ u ‖ 2 2 p η p − 1 2 q C G ( N , α , q ) c q ( 1 − η q ) ‖ ∇ u ‖ 2 2 q η q ,

which implies that E c > − ∞ , since 2 p η p < 2 for 2 α < p < 2 α # . Fix a u ∈ S ( c ) ∩ L ∞ ( R N ) . Then, by a direct calculation, we have

‖ u t ‖ 2 2 = c , ‖ ∇ u t ‖ 2 2 = t 2 ‖ ∇ u ‖ 2 2 and A p ( u t ) = t N p − N − α A p ( u ) ,

where u t ( x ) ≔ t N ∕ 2 u ( t x ) . This indicates that

E c ≤ lim t → 0 E μ ( u t ) = 1 2 lim t → 0 t 2 ‖ ∇ u ‖ 2 2 − μ p t N p − N − α A p ( u ) − 1 q t N q − N − α A q ( u ) = 0 for all c > 0 .

Let μ ≥ 0 and fix a u ∈ S ( c ) ∩ L ∞ ( R N ) . Define the constant

D ≔ μ 2 p + 1 2 q − 1 ‖ ∇ u ‖ 2 2 A 2 α # ( u ) > 0 .

Then, there exists δ 1 > 0 such that for any v ∈ H 1 ( R N ) with ‖ v ‖ ∞ < δ 1 ,

(9) ∣ v ∣ p ∣ v ∣ 2 α # > 1 ‖ v ‖ ∞ 2 α # − p > D .

Similarly, there exists δ 2 > 0 such that for any v ∈ H 1 ( R N ) with ‖ v ‖ ∞ < δ 2 ,

(10) I α ∗ ∣ v ∣ p I α ∗ ∣ v ∣ 2 α # > D .

Now, let us choose a t > 0 such that ‖ u t ‖ ∞ < min { δ 1 , δ 2 } . Thus, by (9)–(10), we have

E c ≤ E μ ( u t ) = 1 2 ‖ ∇ u t ‖ 2 2 − μ 2 p A p ( u t ) − 1 2 q A q ( u t ) < t 2 2 ‖ ∇ u ‖ 2 2 − μ D 2 p t 2 A 2 α # ( u ) − D 2 q t 2 A 2 α # ( u ) = t 2 2 ‖ ∇ u ‖ 2 2 − D t 2 μ 2 p + 1 2 q A 2 α # ( u ) = t 2 2 ‖ ∇ u ‖ 2 2 − t 2 ‖ ∇ u ‖ 2 2 = − t 2 2 ‖ ∇ u ‖ 2 2 < 0 .

( i i ) Let c > c ¯ > 0 . For any ε > 0 , there exists u ∈ S ( c ¯ ) such that E μ ( u ) ≤ E c ¯ + ε , and we define

v ( x ) ≔ u ( t − 1 ∕ N x ) ,

where t ≔ c c ¯ > 1 . Then, we have

(11) E c ≤ E μ ( v ) = t N + α N E μ ( u ) + 1 2 t N − 2 N ( 1 − t α + 2 N ) ‖ ∇ u ‖ 2 2 < t N + α N E μ ( u ) ≤ c c ¯ ( N + α ) ∕ N ( E c ¯ + ε ) ,

which shows that

E c c ( N + α ) ∕ N ≤ E c ¯ c ¯ ( N + α ) ∕ N for c > c ¯ > 0 ,

since ε > 0 is arbitrary.

( i i i ) By ( i ) and ( i i ) , it is clear that E c is non-increasing on c > 0 . To show the continuity, we define the real function

Φ u ( c ) ≔ 1 c E μ ( u ( c − 1 ∕ N x ) ) = 1 2 c − 2 ∕ N ‖ ∇ u ‖ 2 2 − μ 2 p c α ∕ N A p ( u ) − 1 2 q c α ∕ N A q ( u )

for given u ∈ S ( 1 ) and any c > 0 . Clearly,

E c c = inf u ∈ S ( 1 ) Φ u ( c ) .

Then, it follows that E c ∕ c is continuous on c > 0 and so is E c .

( i v ) If E c ¯ < 0 , then we choose ε > 0 small enough such that E c ¯ + ε < 0 . By (11), we have

E c ≤ c c ¯ ( N + α ) ∕ N ( E c ¯ + ε ) ≤ c c ¯ ( E c ¯ + ε ) for any c > c ¯ > 0 .

If E c ¯ is achieved, we can let ε = 0 , and then

(12) E c c < E c ¯ c ¯ for any c > c ¯ > 0 .

Furthermore, if either E c < 0 or E c ¯ < 0 is achieved, then by (12) one has

E c + c ¯ = c c + c ¯ E c + c ¯ + c ¯ c + c ¯ E c + c ¯ < E c + E c ¯ .

( v ) Choosing a u ∈ H 1 ( R N ) such that μ 2 p A p ( u ) + 1 2 q A q ( u ) > 0 . Let

u c ( x ) ≔ u ‖ u ‖ 2 2 ∕ N c 1 ∕ N x .

Clearly, u c ∈ S ( c ) and

E μ ( u c ) = 1 2 ‖ ∇ u c ‖ 2 2 − μ 2 p A p ( u c ) − 1 2 q A q ( u c ) = c ( N − 2 ) ∕ N 2 ‖ u ‖ 2 2 ( N − 2 ) ∕ N ‖ ∇ u ‖ 2 2 − c ( N + α ) ∕ N ‖ u ‖ 2 2 ( N + α ) ∕ N μ 2 p A p ( u ) + 1 2 q A q ( u ) ,

which implies that E c ≤ E μ ( u c ) < 0 for c > 0 large enough. Define

c ∗ ≔ inf { c > 0 ∣ E c < 0 } .

Since μ < 0 , it follows from ( i ) and ( i i i ) that c ∗ > 0 and

E c = 0 if 0 < c ≤ c ∗ and E c < 0 if c > c ∗ .

( v i ) Let 0 < c < c ∗ , assuming by contradiction that E c = 0 is achieved, i.e., there exists a v ∈ S ( c ) such that E μ ( v ) = E c . Then, there holds

E c ∗ < c c ∗ ( N + α ) ∕ N E c = 0 .

This is a contradiction with the fact of E c ∗ = 0 . The proof is complete.□

Lemma 3.3

If v ∈ S ( c ) is a ground state solution of ( P c ) , then v ∈ C 2 ( R N ) is constant sign. Moreover, if μ > 0 , then v is radially symmetry.

Proof

By the regularity theory and the strong maximum principle, we obtain that v ∈ C 2 ( R N ) is constant sign. Assume that μ > 0 and ∣ v ∣ ∗ is Schwarz rearrangement of ∣ v ∣ . Then, a direct calculation shows that

∫ R N ∣ v ∣ 2 d x = ∫ R N ( ∣ v ∣ ∗ ) 2 d x , ∫ R N ∣ ∇ v ∣ 2 d x ≥ ∫ R N ∣ ∇ ∣ v ∣ ∗ ∣ 2 d x and A p ( ∣ v ∣ ) ≤ A p ( ∣ v ∣ ∗ ) ,

which implies that ∣ v ∣ ∗ ∈ S ( c ) and E μ ( ∣ v ∣ ) ≥ E μ ( ∣ v ∣ ∗ ) ≥ E c . Next, we replace ∣ v ∣ by ∣ v ∣ ∗ , and the proof is complete.□

Lemma 3.4

Let N ≥ 1 and 2 α < p < q < 2 α # . If μ > 0 , then the associated Lagrange multiplier λ of the solution u of ( P c ) is negative, and if μ < 0 , then the associated Lagrange multiplier λ of the solution u of ( P c ) is positive.

Proof

Let u ∈ S ( c ) be a solution of ( P c ) and λ be the associated Lagrange multiplier. Since 2 α < p < q < 2 α # , we have 0 < η p < η q < 1 . Then, it follows from Lemma 2.4 that

λ c = μ ( η p − 1 ) A p ( u ) + ( η q − 1 ) A q ( u ) < 0 if μ > 0 ,

and

λ η q c = ( 1 − η q ) ‖ ∇ u ‖ 2 2 + μ ( N + α ) ( p − q ) 2 p q A p ( u ) > 0 if μ < 0 .

The proof is complete.□

We are ready to prove Theorem 1.1: ( i ) – ( i v ) and ( v i ) – ( v i i ) in Theorem 1.1 can be followed from Lemmas 3.2 to 3.4. The rest is to prove that the global infimum E c < 0 is achieved when c > c ∗ . Fix c > c ∗ and let { u n } ⊂ S ( c ) be a minimizing sequence of E c < 0 . Clearly, { u n } is bounded in H 1 ( R N ) and then one may assume that

lim n → ∞ ∫ R N ∣ ∇ u n ∣ 2 d x and lim n → ∞ μ 2 p A p ( u n ) + 1 2 q A q ( u n )

both exist. First of all, we prove that { u n } is non-vanishing, namely

lim n → ∞ sup y ∈ R N ∫ B ( y , 1 ) ∣ u n ∣ 2 d x > 0 .

Assume on the contrary. Then u n → 0 in L 2 p N N + α ( R N ) and L 2 q N N + α ( R N ) by Lions [20, Lemma I.1] and thus

lim n → ∞ μ 2 p A p ( u n ) + 1 2 q A q ( u n ) ≤ 0

via Lemma 2.1. This indicates that

0 > E c = lim n → ∞ E μ ( u n ) ≥ 0 ,

which is a contradiction. So { u n } is non-vanishing. There exist a sequence { y n } ⊂ R N and v ∈ H 1 ( R N ) such that up to a subsequence u n ( x + y n ) ⇀ v in H 1 ( R N ) and u n ( x + y n ) → v a.e. on R N . Set

c ′ ≔ ‖ v ‖ 2 2 ∈ ( 0 , c ] and w n ( x ) ≔ u n ( x + y n ) − v .

By the Brezis-Lieb lemma, we have

(13) lim n → ∞ ‖ w n ‖ 2 2 = c − c ′

and

(14) E c = lim n → ∞ E μ ( u n ) = lim n → ∞ E μ ( w n + v ) = lim n → ∞ E μ ( w n ) + E μ ( v ) .

Let t n ≔ ‖ w n ‖ 2 2 for each n ∈ N + . If lim n → ∞ t n > 0 , then by (13) one has c ′ ∈ ( 0 , c ) . According to the definition of E t n and Lemma 3.2 ( i i i ), we obtain

(15) lim n → ∞ E μ ( w n ) ≥ lim n → ∞ E t n = E c − c ′ .

From (14) and (15), it follows that

(16) E c ≥ E μ ( v ) + E c − c ′ ≥ E c ′ + E c − c ′ .

If either E c ′ = 0 or E c − c ′ = 0 , then by Lemma 3.2 ( i i i ) and (16), we have

(17) E c ≥ E μ ( v ) + E c − c ′ ≥ E c ′ + E c − c ′ ≥ E c .

If both E c ′ < 0 and E c − c ′ < 0 , then it follows from Lemma 3.2 ( i v ) and (16) that

(18) E c ≥ E μ ( v ) + E c − c ′ ≥ E c ′ + E c − c ′ ≥ c ′ c E c + c − c ′ c E c = E c .

By (17) and (18), we deduce that E μ ( v ) = E c ′ , i.e., E c ′ is achieved at v ∈ S ( c ) . If E c ′ < 0 , then by (7) and (16), one has

E c ≥ E c ′ + E c − c ′ > E c ,

which is a contradiction. If E c ′ = 0 and E c − c ′ = 0 , then by (16) we obtain a contradiction:

0 > E c ≥ E c ′ + E c − c ′ = 0 .

If E c ′ = 0 and E c − c ′ < 0 , then by Lemma 3.2 ( i v ) and (16) one has

E c ≥ E c ′ + E c − c ′ = E c − c ′ ≥ c − c ′ c E c ,

which is a contradiction, since E c < 0 . Hence, lim n → ∞ t n = 0 , i.e., c = c ′ , which implies that the minimum E c < 0 is achieved at v ∈ S ( c ) .

Next, we prove Theorem 1.2. The following two lemmas will be used to exclude the vanishing and dichotomy of the minimizing sequence.

Lemma 3.5

Let N ≥ 1 , μ > 0 and 2 α = p < q < 2 α # . Then, we have

− ∞ < E c < − μ N 2 ( N + α ) S α − N + α N c N + α N for c > 0 .

Proof

By Lemma 2.2, we have

E μ ( u ) ≥ 1 2 ‖ ∇ u ‖ 2 2 − μ N 2 ( N + α ) S α − ( N + α ) ∕ N c ( N + α ) ∕ N − 1 2 q ‖ ∇ u ‖ 2 2 q η q c q ( 1 − η q ) ,

which shows that E μ ( u ) is bounded below on S ( c ) . By Lemma 2.3, we choose a function v such that

S α ∫ R N ( I α ∗ ∣ v ∣ N + α N ) ∣ v ∣ N + α N d x N N + α = ∫ R N ∣ v ∣ 2 d x .

Define u ≔ c v ‖ v ‖ 2 . By a direct calculation, we deduce that

E μ ( u t ) = t 2 2 ‖ ∇ u ‖ 2 2 − μ N 2 ( N + α ) S α − N + α N c N + α N − t N q − N − α 2 q A q ( u ) < − μ N 2 ( N + α ) S α − N + α N c N + α N

for t > 0 small enough. The proof is complete.□

Lemma 3.6

Let N ≥ 1 , μ > 0 and 2 α = p < q < 2 α # . Then, for every c > c ¯ > 0 , there holds

E c + c ¯ < E c + E c ¯ .

Proof

For c > c ¯ > 0 , let { u n } ⊂ S ( c ¯ ) be a bounded minimizing sequence of E c ¯ and let ν ≔ c c ¯ > 1 . Then ν u n ∈ S ( c ) and

E μ ( ν u n ) − ν E μ ( u n ) = μ ( ν − ν 2 α ) 2 2 α A 2 α ( u n ) + ν − ν q η q 2 q A q ( u n ) ,

which implies that E μ ( ν u n ) < ν E μ ( u n ) . So, E c ≤ ν E c ¯ , where the equality holds if and only if

lim n → ∞ μ 2 2 α A 2 α ( u n ) + 1 2 q A q ( u n ) = 0 .

But this is not possible, since otherwise we find that

0 > E c ¯ = lim n → ∞ E μ ( u n ) = lim n → ∞ 1 2 ‖ ∇ u n ‖ 2 2 + o ( 1 ) ≥ 0 ,

where we have used Lemma 3.5. Hence, E c < ν E c ¯ , i.e.

E c c < E c ¯ c ¯ ,

which implies that

E c + c ¯ = c c + c ¯ E c + c ¯ + c ¯ c + c ¯ E c + c ¯ < E c + E c ¯ .

The proof is complete.□

We are ready to prove Theorem 1.2: Let { u n } ⊂ S ( c ) be a minimizing sequence of E c . Then, by Lemma 2.2, we have

E μ ( u n ) ≥ 1 2 ‖ ∇ u ‖ 2 2 − μ N 2 ( N + α ) S α − N + α N c N + α N − C G ( q , N , α ) 2 q ‖ ∇ u ‖ 2 2 q η q c q ( 1 − η q ) ,

which shows that { u n } is bounded in H 1 ( R N ) . First of all, we verify the vanishing cannot occur, namely

lim n → ∞ sup y ∈ R N ∫ B ( y , 1 ) ∣ u n ∣ 2 d x > 0 .

Assume on the contrary. Then it follows from Lemma 2.5 that u n → 0 in L p ( R N ) for 2 < p < 2 ∗ , and together with Lemma 2.1, leading to

E c = lim n → ∞ E μ ( u n ) ≥ lim n → ∞ 1 2 ‖ ∇ u n ‖ 2 2 − μ N 2 ( N + α ) S α − N + α N c N + α N ≥ − μ N 2 ( N + α ) S α − ( N + α ) ∕ N c ( N + α ) ∕ N ,

which contradicts with Lemma 3.5.

Next, we verify the dichotomy cannot occur. Assume on the contrary. Then there exist 0 < ζ < c and { u n ( 1 ) } , { u n ( 2 ) } bounded in H 1 ( R N ) such that

‖ u n − u n ( 1 ) − u n ( 2 ) ‖ s → 0 ∀ 2 ≤ s < 2 ∗ and dist ( suppu n ( 1 ) , suppu n ( 2 ) ) → + ∞ ;

‖ u n ( 1 ) ‖ 2 2 → ζ and ‖ u n ( 2 ) ‖ 2 2 → c − ζ ;

liminf n → ∞ ∫ R N ( ∣ ▿ u n ∣ 2 − ∣ ▿ u n ( 1 ) ∣ 2 − ∣ ▿ u n ( 2 ) ∣ 2 ) d x ≥ 0 .

But

E c = lim n → ∞ E μ ( u n ) ≥ lim n → ∞ ( E μ ( u n ( 1 ) ) + E μ ( u n ( 2 ) ) ) ≥ E c − ζ + E ζ ,

which contradicts with Lemma 3.6. Hence, the compactness holds, and then there exist a sequence { y n } ⊂ R and v ∈ S ( c ) such that u n ( ⋅ + y n ) → v in L s ( R N ) for all 2 ≤ s < 2 ∗ , which implies that

E c ≤ E μ ( v ) ≤ lim n → ∞ E μ ( u n ( x + y n ) ) = E c .

This shows E c is achieved by v ∈ S ( c ) . The proof is complete.

We are ready to prove Theorem 1.3: ( i ) Assume that (2) has a normalized solution v ∈ S ( c ) . Then, by Lemmas 2.2 and 2.4, we have

0 = ∫ R N ∣ ∇ v ∣ 2 d x − η q A q ( v ) ≥ ‖ ∇ v ‖ 2 2 − N N + α + 2 C G ( 2 α # , N , α ) c 2 + α N ‖ ∇ v ‖ 2 2 = 1 − N N + α + 2 C G ( 2 α # , N , α ) c 2 + α N ‖ ∇ v ‖ 2 2 > 0

if N N + α + 2 C G ( 2 α # , N , α ) c ( 2 + α ) ∕ N < 1 . Clearly, this is a contradiction. If N N + α + 2 C G ( 2 α # , N , α ) c ( 2 + α ) ∕ N = 1 , then by the scaling transformation, v must be the unique positive radial solution to the equation

− Δ u + λ u = ( I α ∗ ∣ u ∣ ( N + α + 2 ) ∕ N ) ∣ u ∣ ( N + α + 2 ) ∕ N − 2 u .

But v is not the solution to (2). That shows that (2) has no normalized solution.

( i i ) The proof is similar to that of ( i ) , and we omit it here.

4 Global well-posedness and orbital stability

In order to prove the orbital stability of the set of ground state, we first show the well-posedness of the Cauchy problem (1).

Definition 4.1

If a pair of real number s , r ∈ [ 2 , ∞ ) satisfy

1 s = N 1 2 − 1 r , ∀ ( s , N , r ) ≠ ( 2 , ∞ , 2 ) ,

then ( s , r ) is called Schödinger admissible.

Lemma 4.2

Assume that ψ is the solution of the Cauchy problem (1). Then, ψ satisfies the following Duhamel formula:

ψ = e − i t Δ φ − i ∫ 0 t e − i ( t − τ ) Δ [ μ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ + ( I α ∗ ∣ ψ ∣ q ) ∣ ψ ∣ q − 2 ψ ] d τ ,

where e − i t Δ is the heat kernel which makes ϕ = e − i t Δ φ is unique solution of the following Cauchy problem:

i ∂ t ψ − Δ ψ = 0 , ∀ ( t , x ) ∈ R × R N , ϕ ( 0 , x ) = ψ 0 ( x ) .

Lemma 4.3

(Strichartz estimate [6]) Assume that ( s , r ) and ( s ¯ , r ¯ ) are Schödinger admissible. Then, there exists a constant T > 0 such that the following statements hold:

for all ψ 0 ∈ L T s L x r , we have
‖ e − i t Δ ψ 0 ‖ L T s L x r ≲ ‖ ψ 0 ‖ L T s L x r ;
for all f ∈ L T s ¯ ′ L x r ¯ ′ , we have
∫ 0 t e − i ( t − τ ) Δ f ( τ ) d τ L T s L x r ≲ ‖ f ‖ L T s ¯ ′ L x r ¯ ′ .

We are ready to prove Theorem 1.4: Define the space

X T , M ≔ { ψ ∈ C ( ( − T , T ) ) , H 1 ( R N ) ∩ L T s 1 W x 1 , r 1 ∩ L T s 2 W x 1 , r 2 ∣ ‖ ψ ‖ X T , M ≤ M } ,

where

‖ ψ ‖ X T , M ≔ ‖ ψ ‖ L T ∞ H x 1 + ‖ ψ ‖ L T s 1 W x 1 , r 1 + ‖ ψ ‖ L T s 2 W x 1 , r 2 ,

here ( s 1 , r 1 ) = 4 p N p − ( N + α ) , 2 N p N + α , ( s 2 , r 2 ) = 4 q N q − ( N + α ) , 2 N q N + α . Moreover, we define the operator

Φ ( ψ ) : e − i t Δ φ − i ∫ 0 t e − i ( t − τ ) Δ [ μ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ + ( I α ∗ ∣ ψ ∣ q ) ∣ ψ ∣ q − 2 ψ ] d τ .

First of all, we prove the operator Φ : X T , M → X T , M for some T , M > 0 . It is easy to show that

∇ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ = ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ∇ ψ + ∇ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ + ( I α ∗ ∣ ψ ∣ p ) ∇ ∣ ψ ∣ p − 2 ψ .

Moreover, for all ψ , ϕ ∈ X T , M and f ∈ S ( R N ) , we have

⟨ ( I α ∗ ∣ ψ ∣ p ψ ) ∣ ψ ∣ p − 2 ψ − ( I α ∗ ∣ ϕ ∣ p ϕ ) ∣ ϕ ∣ p − 2 ϕ , f ⟩ = ∫ R N [ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ f − ( I α ∗ ∣ ϕ ∣ p ) ∣ ϕ ∣ p − 2 ϕ f ] d x = ∫ R N ( ( I α ∗ ∣ ψ ∣ p ) − ∣ ϕ ∣ p ) ∣ ψ ∣ p − 2 ψ f d x + ∫ R N ( I α ∗ ∣ ϕ ∣ p ) ( ∣ ψ ∣ p − 2 ψ − ∣ ϕ ∣ p − 2 ϕ ) f d x ≲ ‖ ψ − ϕ ‖ 2 N p N + α ‖ ψ ‖ 2 N p N + α ∨ ‖ ψ ‖ 2 N p N + α 2 ( p − 1 ) ‖ f ‖ 2 N p N + α .

Then by density, one has

(19) ‖ ( I α ∗ ∣ ψ ∣ p ψ ) ∣ ψ ∣ p − 2 ψ − ( I α ∗ ∣ ϕ ∣ p ϕ ) ∣ ϕ ∣ p − 2 ϕ ‖ r 1 ′ ≲ ‖ ψ − ϕ ‖ 2 N p N + α ‖ ψ ‖ 2 N p N + α ∨ ‖ ψ ‖ 2 N p N + α 2 ( p − 1 ) .

It follows from the Hardy-Littlewood-Sobolev inequality, (19) and Lemma 4.3 that

(20) ∫ 0 t e − i ( t − τ ) Δ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ∇ ψ d τ L T s L x r ≲ ‖ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ∇ ψ ‖ L T s 1 ′ L x r 1 ′ ≲ ∫ 0 T ‖ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ∇ ψ ‖ r 1 ′ s 1 ′ d t 1 ∕ s 1 ′ ≲ ∫ 0 T ‖ ∇ ψ ‖ r 1 s 1 ′ ‖ ψ ‖ r 1 s 1 ′ ( 2 p − 2 ) d t 1 ∕ s 1 ′ ≲ T 1 − η p ‖ ∇ ψ ‖ L T s 1 L x r 1 ‖ ψ ‖ L T ∞ L x r 1 2 p − 2 ≲ T 1 − η p ‖ ∇ ψ ‖ L T s 1 L x r 1 ‖ ψ ‖ L T ∞ H x 1 2 p − 2 ≲ T 1 − η p M 2 p − 1 .

Similarly, we can obtain

(21) ∫ 0 t e − i ( t − τ ) Δ ∇ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 2 ψ d τ L T s L x r ≲ p ∫ 0 t e − i ( t − τ ) Δ ( I α ∗ ∣ ψ ∣ p − 1 ∇ ψ ) ∣ ψ ∣ p − 2 ψ d τ L T s L x r ≲ p T 1 − η p M 2 p − 1 ,

and

(22) ∫ 0 T e − i ( t − s ) Δ ( I α ∗ ∣ ψ ∣ p ) ∇ ∣ ψ ∣ p − 2 ψ d s L T s L x r ≲ ∫ 0 t e − i ( t − s ) Δ ( I α ∗ ∣ ψ ∣ p ) ∣ ψ ∣ p − 3 ∇ ψ ψ d s L T s L x r ≲ T 1 − η p M 2 p − 1 .

Thus, by Lemma 4.3 and (20)–(22), we have

‖ Φ ( ψ ) ‖ X T , M ≲ ‖ ψ 0 ‖ X T , M + ( T 1 − η p M 2 p − 1 + T 1 − η q M 2 q − 1 ) ‖ ψ ‖ X T , M ,

which shows Φ : X T , M → X T , M by choosing T , M small enough.

Next, we show the operator Φ is a contraction. Following the above argument, we replace ∇ ψ by ( ∇ ψ − ∇ ϕ ) , and together with (19), leading to

‖ Φ ( ψ ) − Φ ( ϕ ) ‖ X T , M ≲ T 1 − η p M 2 p − 2 + T 1 − η q M 2 q − 2 ‖ ψ − ϕ ‖ X T , M ,

which indicates that the operator Φ is a contraction. Finally, by using the Banach fixed-point principle, there exists a fixed-point ψ ∈ X T , M of Φ , which is the unique solution of (1). By the regularity theory, we have ψ ∈ C 1 ( I , H − 1 ( R N ) ) . And in view of (4.3), we obtain

E μ ( ψ ( t ) ) ≥ 1 4 ‖ ∇ ψ ( t ) ‖ 2 2 − C , ∀ t ∈ R N ,

where C is the minimum of the following function in R +

k ( y ) = 1 4 y 2 − ∣ μ ∣ 2 p C G ( N , α , p ) c p ( 1 − η p ) y 2 p η p − 1 2 q C G ( N , α , q ) c q ( 1 − η q ) y 2 q η q .

By the conversation of energy, we obtain that ‖ ∇ ψ ( t ) ‖ 2 2 is bounded for all t ∈ R N . The proof of global well-posedness is complete.

The rest is to prove that the set of ground states (5) is orbitally stable. Assume that there exist ε 0 , ζ > 0 , a decreasing sequence { δ n } ⊂ R + converging to 0, and { ψ n } ⊂ H 1 ( R N ) satisfying inf u ∈ Z c ‖ ψ n , 0 − u ‖ H 1 < δ n such that

inf u ∈ Z c ‖ ψ n ( ζ ) − u ‖ H 1 ≥ ε 0 ,

where ψ n ( t ) is the unique solution of (1) with the initial value ψ n , 0 . We observe that ‖ ψ n , 0 ‖ 2 2 → c as n → ∞ and that E μ ( ψ n ( 0 , x ) ) → E c as n → ∞ by the continuity of E μ . By the conservation laws of the energy and mass, we have

(23) ‖ ψ n ( t ) ‖ 2 2 = ‖ ψ n , 0 ‖ 2 2 → c as n → ∞

and

(24) E μ ( ψ n ( t ) ) = E μ ( ψ n , 0 ) → E c as n → ∞ .

Now, let ϕ n ( ζ ) = c ψ n ( ζ ) ‖ ψ n ( ζ ) ‖ 2 . Then by (23) one has ‖ ϕ n ‖ 2 2 = c . Moreover, it follows from (24) that

E μ ( ϕ n ) = 1 2 ∫ R N ∣ ∇ ϕ n ( t ) ∣ 2 d x − μ 2 p A p ( ϕ n ) − 1 2 q A q ( ϕ n ) = c 2 ‖ ψ n ( t , x ) ‖ 2 2 ∫ R N ∣ ∇ ψ n ( ζ ) ∣ 2 d x − μ 2 p c ‖ ψ n ( t ) ‖ 2 2 p A p ( ψ n ( t ) ) − 1 2 q c ‖ ψ n ( t ) ‖ 2 2 q A q ( ψ n ( t ) ) → E c as n → ∞ ,

which shows that { ϕ n } ⊂ S ( c ) is a minimizing sequence to E c . Thus, there exists u ˜ ∈ S ( c ) such that

(25) ‖ ϕ n − u ˜ ‖ H 1 → 0 as n → ∞ .

Note that

‖ ϕ n − u ˜ ‖ H 1 = c ψ n ( ζ ) ‖ ψ n ( ζ ) ‖ 2 − u ˜ H 1 ≥ inf u ∈ Z c ‖ ψ n ( ζ ) − u ˜ ‖ ℋ ≥ ε 0 .

Clearly, this is a contradiction with (25). The proof is complete.

5 Relationship between minimizers and ground state solutions

The Nehari manifold N λ defined as (6) is closely related to the behavior of the function f u : t → J λ ( t u ) for t > 0 . Such maps are known as fibering maps and were introduced by Drábek and Pohozaev [9] and were also discussed by Brown and Zhang [4] and by Brown and Wu [3]. For u ∈ H 1 ( R N ) \ { 0 } , we have

f u ( t ) = t 2 2 ∫ R N ( ∣ ∇ u ∣ 2 − λ ∣ u ∣ 2 ) d x − μ 2 p t 2 p A p ( u ) − 1 2 q t 2 q A q ( u ) , f u ′ ( t ) = t ∫ R N ( ∣ ∇ u ∣ 2 − λ ∣ u ∣ 2 ) d x − μ t 2 p − 1 A p ( u ) − t 2 q − 1 A q ( u ) .

It is easy to verify that

t f u ′ ( t ) = t 2 ∫ R N ( ∣ ∇ u ∣ 2 − λ ∣ u ∣ 2 ) d x − μ t 2 p A p ( u ) − t 2 q A q ( u ) ,

and so, for any u ∈ H 1 ( R N ) \ { 0 } and t > 0 , f u ′ ( t ) = 0 if and only if t u ∈ N λ , i.e. positive critical points of f u correspond to points on the Nehari manifold N λ . In particular, f u ′ ( 1 ) = 0 if and only if u ∈ N λ . Moreover, we have the following result.

Lemma 5.1

Let N ≥ 1 , λ < 0 and μ > 0 . For each u ∈ H 1 ( R N ) \ { 0 } , there exists a unique t ( u ) > 0 such that t ( u ) u ∈ N λ and

(26) max t > 0 J λ ( t u ) = J λ ( t ( u ) u ) .

Proof

Fix u ∈ H 1 ( R N ) \ { 0 } and let

g u ( t ) = μ t 2 p − 2 A p ( u ) + t 2 q − 2 A q ( u ) for t > 0 .

Since 2 α < p < q < 2 α # and μ > 0 , it is clear that g u ( t ) is increasing on ( 0 , ∞ ) . We note that t u ∈ N λ if and only if

(27) g u ( t ) − ∫ R N ( ∣ ∇ u ∣ 2 − λ ∣ u ∣ 2 ) d x = 0 .

Thus, there exists a unique t ( u ) > 0 such that the equality (27) holds, since λ < 0 . This indicates that

f u ′ ( t ( u ) ) = t ( u ) g u ( t ( u ) ) − ∫ R N ( ∣ ∇ u ∣ 2 − λ ∣ u ∣ 2 ) d x = 0 ,

namely, t ( u ) u ∈ N λ . Moreover, by the profile of g u ( t ) , one has f u ( t ) is strictly increasing on ( 0 , t ( u ) ) and strictly decreasing on ( t ( u ) , ∞ ) . Therefore, (26) holds. The proof is complete.□

Theorem 5.2

Let N ≥ 1 and μ > 0 . For any c > 0 and u ∈ N λ , we have

(28) J λ ( u ) ≥ E c − 1 2 λ c .

In particular, the equality holds if and only if u is a minimizer of ( P c ) , and u is a ground state solution of ( Q λ ) .

Proof

Following the idea of [10]. Since u ∈ N λ , it follows from Lemma 5.1 that

J λ ( u ) = max t > 0 J λ ( t u ) ≥ J λ ( t u ) ,

and J λ ( u ) = J λ ( t u ) if and only if t = 1 . Then, according to the definition of E c , one has

(29) J λ ( u ) ≥ J λ c ‖ u ‖ 2 u = E μ c ‖ u ‖ 2 u − 1 2 λ c ≥ E c − 1 2 λ c .

Thus (28) holds. On the one hand, if the equality holds, then by (29) one has E μ c ‖ u ‖ 2 u = E c and J λ ( u ) = J λ c ‖ u ‖ 2 u . By Lemma 5.1, the latter implies that ‖ u ‖ 2 2 = c , leading to E μ ( u ) = E c . Hence, u is a minimizer of ( P c ) . Furthermore, by (28), for any v ∈ N λ , we have

J λ ( v ) ≥ E c − 1 2 λ c = E μ ( u ) − 1 2 λ ‖ u ‖ 2 2 = J λ ( u ) ,

which implies that u is a ground state solution of ( Q λ ) . On the other hand, if u is a minimizer of ( P c ) , then we have

J λ ( u ) = E μ ( u ) − 1 2 λ ‖ u ‖ 2 2 = E c − 1 2 λ c ,

which indicates that the equality holds. The proof is complete.□

We are ready to prove Theorem 1.6: ( i ) It follows from Theorem 5.2.

( i i ) For given λ ∈ { λ ( v ) ∣ v ∈ S ( c ) is a minimizer of ( P c ) } , let w ∈ H 1 ( R N ) be any ground state solution w ∈ H 1 ( R N ) of ( Q λ ) . Then, we have

J λ ( w ) ≤ E c − 1 2 λ c ,

and together with (28), leading to J λ ( w ) = E c − 1 2 λ c . Using Theorem 5.2, we obtain that u is a minimizer of ( P c ) .

( i i i ) Let the minimizer u ∈ S ( c ) of ( P c ) be unique. Then, u is a ground state solution of ( Q λ ) with λ = λ ( u ) < 0 by ( i ) . Assume that v ∈ N λ is another ground state solution of ( Q λ ) . Then, by Theorem 5.2, we have

J λ ( u ) = J λ ( v ) = E c − 1 2 λ c ,

which shows v is a minimizer of ( P c ) on S ( c ) . This is a contradiction. Similarly, we easily prove that the minimizer of ( P c ) on S ( c ) is unique if the ground state solution of ( Q λ ) with λ < 0 is unique. The proof is complete.

Funding information: J. Sun was supported by the National Natural Science Foundation of China (Grant No. 11671236) and Shandong Provincial Natural Science Foundation (Grant No. ZR2020JQ01).
Author contributions: The first author is responsible for the paper writing. The corresponding author is responsible for the global structure of the paper and for ensuring that the descriptions are accurate.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2023-07-27

Revised: 2023-11-20

Accepted: 2024-03-01

Published Online: 2024-05-31

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

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https://doi.org/10.1515/anona-2024-0004

Keywords for this article

normalized solutions; mixed nonlocal nonlinearities; minimizers; ground state solutions

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