Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

Caifeng Zhang

doi:10.1515/ans-2018-2026

Article Open Access

Trudinger–Moser Inequalities in Fractional Sobolev–Slobodeckij Spaces and Multiplicity of Weak Solutions to the Fractional-Laplacian Equation

Caifeng Zhang

Published/Copyright: September 10, 2018

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From the journal Advanced Nonlinear Studies Volume 19 Issue 1

Abstract

In line with the Trudinger–Moser inequality in the fractional Sobolev–Slobodeckij space due to [S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 2017, 4, 871–884] and [E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 2018, 2, 315–319], we establish a new version of the Trudinger–Moser inequality in Ws,p⁢(ℝN). Define

∥ u ∥ 1 , τ = ( [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ) 1 p for any ⁢ τ > 0 .

There holds

sup u ∈ W s , p ⁢ ( ℝ N ) , ∥ u ∥ 1 , τ ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) < + ∞ ,

where s∈(0,1), s⁢p=N, α∈[0,α*) and

Φ N , s ⁢ ( t ) = e t - ∑ i = 0 j p - 2 t j j ! .

Applying this result, we establish sufficient conditions for the existence of weak solutions to the following quasilinear nonhomogeneous fractional-Laplacian equation:

( - Δ ) p s ⁢ u ⁢ ( x ) + V ⁢ ( x ) ⁢ | u ⁢ ( x ) | p - 2 ⁢ u ⁢ ( x ) = f ⁢ ( x , u ) + ε ⁢ h ⁢ ( x ) in ⁢ ℝ N ,

where V⁢(x) has a positive lower bound, f⁢(x,t) behaves like eα⁢|t|N/(N-s), h∈(Ws,p⁢(ℝN))* and ε>0. Moreover, we also derive a weak solution with negative energy.

Keywords: Trudinger–Moser inequality; exponential growth; fractional Sobolev–Slobodeckij spaces, fractional Laplacians

MSC 2010: 35A23; 35J35; 35J60

1 Introduction and Main Results

Let Ω be a bounded, open domain of ℝN (N≥2). The standard Sobolev space W0k,p⁢(Ω) is defined by the completion of 𝒞c∞⁢(Ω) equipped with the norm

∥ u ∥ W k , p = ( ∥ u ∥ p p + ∑ j = 1 k ∥ ∇ j ⁡ u ∥ p p ) 1 p .

The well-known Sobolev embedding theorem states that W0k,p⁢(Ω) embeds continuously into LN⁢p/(N-k⁢p)⁢(Ω) for a positive integer k<N and 1≤p<Nk. When it comes to the limiting case p=Nk, the embedding W0k,N/k⁢(Ω)⊆L∞⁢(Ω) fails. To overcome this difficulty, Trudinger [61] (see also [59, 26]) proved that the functions in W01,N⁢(Ω) enjoy summability of exponential type, that is,

W 0 1 , N ⁢ ( Ω ) ⊂ { u ∈ L 1 ⁢ ( Ω ) : E β ⁢ ( u ) := ∫ Ω e β ⁢ | u | N N - 1 ⁢ 𝑑 x < + ∞ } for any ⁢ β < ∞ .

Furthermore, the functional Eβ is continuous on W01,N⁢(Ω). In the 1970s, Moser found the optimal β and proved the following sharp form of the Trudinger inequality in [53].

Theorem A.

Let Ω⊆RN be a smooth bounded domain. Then there exist a positive constant CN and a sharp constant αN=N⁢ωN-11/(N-1) such that

(1.1) 1 | Ω | ⁢ ∫ Ω exp ⁡ ( α ⁢ | u | N N - 1 ) ⁢ 𝑑 x ≤ C N

for any α≤αN and u∈Cc∞⁢(Ω) with ∫Ω|∇⁡u|N⁢𝑑x≤1, where ωN-1 is the area of the surface of the unit ball.

When Ω is unbounded, inequality (1.1) fails. Adachi and Tanach [1] and do Ó [8] gave a subcritical Trudinger–Moser-type inequality by replacing the integral function exp⁡(α⁢|u|N/(N-1)) by Φ⁢(α⁢|u⁢(x)|N/(N-1)). More precisely, they proved the following subcritical Trudinger–Moser inequality.

Theorem B.

For 0<α<αN, there exists a positive constant CN such that

sup u ∈ W 1 , N ⁢ ( ℝ N ) , ∫ ℝ N | ∇ ⁡ u | N ⁢ 𝑑 x ≤ 1 ⁡ ∫ ℝ N Φ ⁢ ( α ⁢ | u ⁢ ( x ) | N N - 1 ) ⁢ 𝑑 x ≤ C N ⁢ ∫ ℝ N | u ⁢ ( x ) | N ⁢ 𝑑 x ,

where

Φ ⁢ ( t ) := e t - ∑ i = 0 N - 2 t i i ! .

Moreover, the constant αN is sharp in the sense that if α≥αN, the supremum will become infinite.

If α=αN, the above Trudinger–Moser inequality makes no sense. By using the standard Sobolev norm in W1,N⁢(ℝN) instead of the Dirichlet norm, Ruf [60] (N=2) and Li and Ruf [41] (N≥2) proved the criticalness of such inequality. Proofs of both subcritical and critical Trudinger–Moser inequalities use the symmetrization argument and the Polyá–Szegö inequality in Euclidean spaces. When it comes to the higher-order Adams inequality on ℝn or Trudinger–Moser inequalities in settings where a rearrangement argument fails such as in the Heisenberg group ℍn, the method becomes invalid. In [31, 29, 34, 35, 38], Lam, Lu and their collaborators developed a rearrangement-free approach to establish the higher-order Adams inequality in Euclidean spaces, which was first established by [2] and the Trudinger–Moser inequality on Heisenberg groups. Furthermore, Lam, Lu and Zhang showed the asymptotic behaviors of the supremums of the subcritical and critical Trudinger–Moser inequalities and established their equivalence of such inequalities in [36]. For more results about Trudinger–Moser and Adams inequalities, we refer to, e.g., [7, 10, 15, 13, 14, 19, 23, 28, 30, 37, 39, 42, 40, 44, 45, 46, 47, 48, 49, 51, 54, 55, 65].

Based on these Trudinger–Moser inequalities, many authors consider the existence of weak solutions for the following singular quasilinear N-Laplacian equation as an application:

- div ⁡ ( | ∇ ⁡ u | N - 2 ⁢ ∇ ⁡ u ) + V ⁢ ( x ) ⁢ | u | N - 2 ⁢ u = f ⁢ ( x , u ) | x | β + ε ⁢ h ⁢ ( x ) in ⁢ ℝ N .

To get existence results, they assume f is a continuous function which maps ℝN×ℝ to ℝ and behaves like exp⁡(α⁢|t|N/(N-1)) as t→+∞, and h⁢(x)∈(W1,N⁢(ℝN))*. By assuming condition (V1) that the function 1V⁢(x) is in L1⁢(ℝN), or condition (V2) that V⁢(x)→∞ as |x|→∞ for some continuous V⁢(x) satisfying V⁢(x)≥V0>0, one can get that the imbedding

E := { u : ∫ ℝ N | ∇ ⁡ u | N + V ⁢ ( x ) ⁢ | u | N ⁢ d ⁢ x < + ∞ } ↪ L p ⁢ ( ℝ N )

is compact. With the help of the standard mountain-pass procedure, the authors of [3, 5, 8, 16, 32, 38] established the existence of nontrivial weak solutions. Notice that the assumptions (V1) and (V2) do not include the basic case when V⁢(x) is a constant. For this case, the authors of [18, 52, 64] gave the existence of ground state solutions. In their proofs, the Ambrosetti–Rabinowitz (AR) condition plays an important role. In [32, 33], Lam and Lu used a weaker condition to replace the Ambrosetti–Rabinowitz condition and got existence results for N-Laplacian and polyharmonic operators without the (AR) condition.

In a recent work, Martinazzi [50] studied the Adams inequality in a fractional Sobolev space H~s,p⁢(Ω) and further extended this result to the case of Lorentz spaces. His proofs were simple, by using Green’s functions for fractional-Laplacian operators and suitable cut-off procedures to reduce fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai [62]. In order to state his result, we recall the fractional Sobolev space H~s,p⁢(Ω) defined by

H ~ s , p ⁢ ( Ω ) := { u ∈ L p ⁢ ( Ω ) : u = 0 ⁢ on ⁢ ℝ N ∖ Ω , ( - Δ ) s 2 ⁢ u ∈ L p ⁢ ( ℝ N ) } ,

where

( - Δ ) s ⁢ φ = ℱ - 1 ⁢ ( | ξ | 2 ⁢ s ⁢ φ ^ ) , φ ∈ 𝒮 ⁢ ( ℝ N ) .

The following theorem was proved in [50].

Theorem C.

For any open set Ω⊂RN with finite measure and p∈(1,∞), we have

sup u ∈ H ~ N p , p ⁢ ( Ω ) , ∥ ( - Δ ) N 2 ⁢ p ⁢ u ∥ L p ⁢ ( Ω ) ≤ 1 ⁡ ∫ Ω e α N , p ⁢ | u | p ′ ⁢ 𝑑 x ≤ C N , p ⁢ | Ω | ,

where the constant αN,p is given by

α N , p = N | S N - 1 | ⁢ ( Γ ⁢ ( N 2 ⁢ p ) ⁢ 2 N p ⁢ π N 2 Γ ⁢ ( N ⁢ p - N 2 ⁢ p ) ) p ′ .

Moreover, the constant αN,p is sharp in the sense that we cannot replace it by any larger one without making the above supremum infinite.

When Ω is unbounded, there are many works devoted to it. In 2013, Lam and Lu [31] proved that for Ω=ℝN and ∥(τ⁢I-Δ)N/2⁢p⁢u∥Lp⁢(ℝN)≤1 (here τ>0 is fixed),

sup u ∈ H ~ N p , p ⁢ ( ℝ N ) , ∥ ( τ ⁢ I - Δ ) N 2 ⁢ p ⁢ u ∥ L p ⁢ ( ℝ N ) ≤ 1 ⁡ ∫ ℝ N Φ ⁢ ( α N , p ⁢ | u | p ′ ) ⁢ 𝑑 x < + ∞ ,

where αN,p is the same as that in Theorem C and

Φ ⁢ ( t ) := e t - ∑ j = 0 j p - 2 t j j ! , j p := min ⁡ { j ∈ ℕ : j ≥ p } .

For more results related to the Adams inequality in a fractional setting, we refer to [21, 25, 31].

Now, we focus our attention on the Sobolev–Slobodeckij space W0s,p⁢(Ω) or W~0s,p⁢(Ω). First, we recall the definition of the Sobolev–Slobodeckij space. For s∈(0,1), the space W~0s,p⁢(Ω) is defined by the completion of 𝒞c∞⁢(Ω) with respect to the norm

u ↦ ( ∥ u ∥ L p ⁢ ( Ω ) p + [ u ] W s , p ⁢ ( ℝ N ) p ) 1 p ,

where

[ u ] W s , p ⁢ ( ℝ N ) := ( ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + s ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y ) 1 p .

In 2016, Parini and Ruf [56] established the existence of the optimal exponent α*>0 such that for α∈[0,α*), the corresponding version of inequality (1.1) holds. Their proof is based on the facts from [57]. Furthermore, they applied a slightly modified version of the Trudinger–Moser sequence and gave an upper bound for α*. More precisely, they showed the following theorem.

Theorem D ([56]).

Let Ω be a bounded, open domain of RN (N≥2) with Lipschitz boundary, and let s∈(0,1), s⁢p=N. Then there exists an exponent α of the fractional Trudinger–Moser inequality such that

sup u ∈ W ~ 0 s , p ⁢ ( Ω ) , [ u ] W s , p ⁢ ( ℝ N ) ≤ 1 ⁡ ∫ Ω exp ⁡ ( α ⁢ | u | N N - s ) < + ∞ .

Set

α * = α * ⁢ ( s , Ω ) := sup ⁡ { α : sup u ∈ W ~ 0 s , p ⁢ ( Ω ) , [ u ] W s , p ⁢ ( ℝ N ) ≤ 1 ⁡ ∫ Ω exp ⁡ ( α ⁢ | u | N N - s ) < + ∞ } .

Moreover, α*≤αs,N*, where

α s , N * := N ⁢ ( 2 ⁢ ( N ⁢ ω N ) 2 ⁢ Γ ⁢ ( p + 1 ) N ! ⁢ ∑ k = 0 + ∞ ( N + k - 1 ) ! k ! ⁢ 1 ( N + 2 ⁢ k ) p ) s N - s .

By replacing the norm [u]Ws,p⁢(ℝN) by ∥u∥Ws,p⁢(ℝN), Iula [24] extended the results to the whole space ℝ1.

Our first result is to extend Theorem D to the case of the whole space ℝN.

Theorem 1.1.

Let s∈(0,1) and s⁢p=N. Then for every 0≤α<α* the following inequality holds:

(1.2) sup u ∈ W s , p ⁢ ( ℝ N ) , ∥ u ∥ W s , p ⁢ ( ℝ N ) ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x < + ∞ ,

where ΦN,s⁢(t)=et-∑i=0jp-2tjj! and jp:=min⁡{j∈N:j≥p}. Moreover, for α>αs,N*,

sup u ∈ W s , p ⁢ ( ℝ N ) , ∥ u ∥ W s , p ⁢ ( ℝ N ) ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x = + ∞ .

Remark 1.2.

In the theorem, αs,N* is just an upper bound of α*, we don’t give the precise value of α*.

By utilizing the method of the change of variables, we generalize inequality (1.2) to a larger class of functions. More precisely, we have

Theorem 1.3.

For any τ>0, let s∈(0,1) and s⁢p=N. Then for every 0≤α<α* the following inequality holds:

sup u ∈ W s , p ⁢ ( ℝ N ) , [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x < + ∞ .

Moreover, for α>αs,N*,

sup u ∈ W s , p ⁢ ( ℝ N ) , [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x = ∞ .

Motivated by [3, 17, 27, 35, 63], we derive an application of Theorem 1.3. Indeed, we consider the existence of weak solutions for the following fractional-Laplacian equation:

(1.3) ( - Δ ) p s ⁢ u ⁢ ( x ) + V ⁢ ( x ) ⁢ | u ⁢ ( x ) | p - 2 ⁢ u ⁢ ( x ) = f ⁢ ( x , u ) + ε ⁢ h ⁢ ( x ) in ⁢ ℝ N ,

where V⁢(x)≥c0, h∈(Ws,p⁢(ℝN))*, s∈(0,1), s⁢p=N and ε>0. The fractional operator is denoted as follows:

( - Δ ) p s ⁢ u ⁢ ( x ) := 2 ⁢ P . V . ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p - 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) | x - y | N + s ⁢ p ⁢ 𝑑 y , x ∈ ℝ N .

Moreover, we assume that f⁢(x,t) satisfies the following conditions:

The nonlinearity f⁢(x,t):ℝN×ℝ→ℝ is continuous, f⁢(x,t)=0 for all (x,t)∈ℝN×(-∞,0), and it has exponential growth as t→+∞, which means there exists a constant α0>0 such that

lim t → + ∞ ⁡ f ⁢ ( x , t ) ⁢ e - α ⁢ | t | N N - s = { 0 for all ⁢ α > α 0 , + ∞ for all ⁢ α < α 0 ,

uniformly in x∈ℝN.
There exist constants α0, b1, b2>0 such that for any (x,t)∈ℝN×(0,+∞),

0 < f ⁢ ( x , t ) ≤ b 1 ⁢ t p - 1 + b 2 ⁢ Φ N , s ⁢ ( α 0 ⁢ | t | N N - s ) ,

where

Φ N , s ⁢ ( y ) = e y - ∑ i = 0 j p - 2 y j j ! .
There exists a constant μ>p such that for all x∈ℝN and t>0,

0 < μ ⁢ F ⁢ ( x , t ) := μ ⁢ ∫ 0 t f ⁢ ( x , r ) ⁢ 𝑑 r ≤ f ⁢ ( x , t ) ⁢ t .
There exist constants t0 and M0>0 such that

0 < F ⁢ ( x , t ) ≤ M 0 ⁢ f ⁢ ( x , t ) for all ⁢ ( x , t ) ∈ ℝ N × [ t 0 , + ∞ ) .
One has

lim sup t → 0 + ⁡ p ⁢ F ⁢ ( x , t ) | t | p < λ

uniformly in x∈ℝN, where

λ = inf u ∈ E ⁡ [ u ] W s , p ⁢ ( ℝ N ) p + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ∫ ℝ N | u | p ⁢ 𝑑 x ,

E is defined by

E := { u ∈ W s , p ⁢ ( ℝ N ) : ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x < + ∞ }

and the norm in E is denoted by

∥ u ∥ := ( [ u ] W s , p ⁢ ( ℝ N ) p + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p .

We claim that Ws,p⁢(ℝN) equipped with the norm [u]Ws,p⁢(ℝN) is uniformly convex, and hence reflexive. Indeed, we define a linear isometry mapping

u ↦ u ~ : u ⁢ ( x ) - u ⁢ ( y ) | x - y | N p + s

which maps Ws,p⁢(ℝN) to Lp⁢(ℝ2⁢N); then the uniform convexity of Lp⁢(ℝ2⁢N) infers the claim. As for E, we regard it as a closed subspace of Ws,p⁢(ℝN), therefore it is a reflexive space. As in [8, 27], we put more assumptions on V⁢(x) which is continuous and satisfies V⁢(x)≥V0 for all x∈ℝn. Precisely, we assume the following conditions on the potential V:

V ⁢ ( x ) → ∞ as |x|→∞; or more generally, for every M>0,

μ ⁢ ( { x ∈ ℝ N : V ⁢ ( x ) ≤ M } ) < ∞ .
The function [V⁢(x)]-1 belongs to L1⁢(ℝN).

Thanks to these assumptions, one can get that E↪↪Lq(ℝN) for q≥p. Then it follows from (H2) and (H3) that for all (x,t)∈ℝN×[0,+∞) there exists a constant a>0 such that

0 < F ⁢ ( x , t ) ≤ a ⁢ f ⁢ ( x , t ) .

Combining this with (H1) and Corollary 2.4, we obtain that for any u,v∈E,

F ⁢ ( x , u ) ∈ L 1 ⁢ ( ℝ N ) , f ⁢ ( x , u ) ⁢ v ∈ L 1 ⁢ ( ℝ N ) ,

which implies that the functional related to the fractional-Laplacian equation (1.3), given by

I s , ε ⁢ ( u ) = 1 p ⁢ [ u ] W s , p ⁢ ( ℝ N ) p + 1 p ⁢ ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x - ∫ ℝ N F ⁢ ( x , u ) ⁢ 𝑑 x - ε ⁢ ∫ ℝ N h ⁢ ( x ) ⁢ u ⁢ ( x ) ⁢ 𝑑 x ,

is well defined. By direct calculations, one can derive that for any u,v∈E one has Is,ε∈𝒞1⁢(E,ℝ). More precisely,

I s , ε ′ ⁢ ( u ) ⁢ v = ∫ ℝ 2 ⁢ N J p ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ ( v ⁢ ( x ) - v ⁢ ( y ) ) | x - y | N + s ⁢ p ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ J p ⁢ ( u ) ⁢ v ⁢ 𝑑 x - ∫ ℝ N f ⁢ ( x , u ) ⁢ v ⁢ 𝑑 x - ε ⁢ ∫ ℝ N h ⁢ v ⁢ 𝑑 x ,

where we use Jp⁢(t)=|t|p-2⁢t to simplify it.

Recently, there have been considerable results about the existence of weak solutions for fractional-Laplacian operators. In 2014, Iannizzotto and Squassina [22] studied the case N=1 and s=12 (see also [20]). In fact, they proved the existence of weak solutions to the problem

- C s 2 ⁢ ∫ ℝ u ⁢ ( x + y ) + u ⁢ ( x - y ) - 2 ⁢ u ⁢ ( x ) | y | 1 + 2 ⁢ s ⁢ 𝑑 y = f ⁢ ( u ) , u ∈ W 0 1 2 , 2 ⁢ ( - 1 , 1 ) .

Furthermore, do Ó, Miyagaki and Squassina [17, 18] overcame the possible failure of the lack of compactness and investigated the existence of ground state solutions to the following problem:

- C s 2 ⁢ ∫ ℝ u ⁢ ( x + y ) + u ⁢ ( x - y ) - 2 ⁢ u ⁢ ( x ) | y | 1 + 2 ⁢ s ⁢ 𝑑 y + u = f ⁢ ( u ) , u ∈ W 0 1 2 , 2 ⁢ ( ℝ ) .

In 2017, de Souza and Araújo [12] investigated the existence and multiplicity of weak solutions for the following class of equations:

- 1 2 ⁢ π ⁢ ∫ ℝ u ⁢ ( x + y ) + u ⁢ ( x - y ) - 2 ⁢ u ⁢ ( x ) | y | 1 + 2 ⁢ s ⁢ 𝑑 y + V ⁢ ( x ) ⁢ u = f ⁢ ( x , u ) + h in ⁢ ℝ ,

where V:ℝ→ℝ is a continuous potential and may change sign, the nonlinearity f⁢(x,t) behaves like exp⁡(α0⁢t2) when t→∞ for some α0>0, and h belongs to the dual of an appropriate functional space. For general N≥1 and s∈(0,1), Perera and Squassina [58] showed that under some conditions the following equation has a nontrivial solution:

{ ( - Δ ) p s ⁢ u = λ ⁢ | u | N - 2 ⁢ s s ⁢ u ⁢ e | u | N N - s in ⁢ Ω , u = 0 in ⁢ ℝ N ∖ Ω .

To have an overview over the fractional p-Laplacian operator, we refer to [52].

In this paper, we focus on the existence of weak solutions of equation (1.3). Our method relies heavily on the mountain-pass procedure and Ekeland’s variational principle. Since we put some assumptions on the potential V⁢(x), the imbedding becomes compact. Thus, we are prepared to derive our main results.

Theorem 1.4.

Assume f⁢(x,t) satisfies (H0)–(H4), V⁢(x) is continuous in RN, V⁢(x)≥V0>0 in RN, and (V1) or (V2) holds. Moreover, h⁢(x) belongs to the dual space of Ws,p⁢(RN). Then there exists a positive constant ε1 such that for 0≤ε<ε1 the fractional-Laplacian equation has a weak solution of mountain-pass type.

Theorem 1.5.

Suppose the assumptions of Theorem 1.4, and suppose h≢0. There exists a positive constant ε2 such that if 0<ε<ε2, the fractional equation has a weak solution with negative energy.

We organize this paper as follows: Section 2 is devoted to establishing the Trudinger–Moser inequality on the Sobolev–Slobodeckij space Ws,p⁢(ℝN). As an application of Theorem 1.3, in Section 3, we give sufficient conditions to guarantee the existence of weak solutions for the fractional-Laplacian equation. Then, with the help of Ekeland’s variational principle, we derive a weak solution with negative energy in Section 4.

2 The Proof of Theorem 1.1

Before our work, we recall some useful lemmas which play an important role in deducing our proof to the radially symmetric case.

Lemma 2.1 ([43]).

Given a measurable function F:R→R and a non-negative function f:RN→R, it holds

∫ ℝ N F ⁢ ( f ) ⁢ 𝑑 x = ∫ ℝ N F ⁢ ( f * ) ⁢ 𝑑 x .

Lemma 2.2 ([4]).

Let 0<s<1 and 1≤p<+∞. If f∈Ws,p⁢(RN), then we have

∥ | f | * ∥ W s , p ⁢ ( ℝ N ) ≤ ∥ f ∥ W s , p ⁢ ( ℝ N ) .

Proof of Theorem 1.1.

With the help of Lemma 2.1 and Lemma 2.2, one can easily obtain that for any u∈Ws,p⁢(ℝN),

∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x = ∫ ℝ N Φ N , s ⁢ ( α ⁢ ( | u | * ) N N - s ) ⁢ 𝑑 x

and

∥ | u | * ∥ W s , p ⁢ ( ℝ N ) ≤ ∥ u ∥ W s , p ⁢ ( ℝ N ) ≤ 1 .

Without loss of generality, we may assume that u∈Ws,p⁢(ℝN) with the norm ∥u∥Ws,p⁢(ℝN)≤1 is non-negative and radially decreasing in ℝN. By splitting the integral into two parts, we obtain that

∫ ℝ N Φ N , s ( α | u | N N - s ) d x = ∫ B r 0 c ⁢ ( 0 ) Φ N , s ( α | u | N N - s ) d x + ∫ B r 0 ⁢ ( 0 ) Φ N , s ( α | u | N N - s ) d x = : I + II ,

where r0 is a positive constant which needs to be chosen later and Br0⁢(0) is a ball in ℝN centered at 0 with radius r0. Since u is non-negative and radially decreasing in ℝN, one can estimate u as follows:

(2.1) | u ⁢ ( x ) | p ≤ 1 | B | x | ⁢ ( 0 ) | ⁢ ∫ B | x | ⁢ ( 0 ) | u ⁢ ( y ) | p ⁢ 𝑑 y ≤ ∥ u ∥ p p | B | x | ⁢ ( 0 ) | for ⁢ x ≠ 0 .

Notice that ∥u∥pp≤1. One can pick r0>(NωN-1)1/N to get that

| u ⁢ ( x ) | ≤ 1 for ⁢ x ∈ B r 0 c ⁢ ( 0 ) .

Taking these results into account, one can get that

(2.2) I = ∑ k = j p - 1 ∞ ∫ B r 0 c ⁢ ( 0 ) α k k ! ⁢ | u | k ⁢ p p - 1 ⁢ 𝑑 x ≤ ∑ k = j p - 1 ∞ ∫ B r 0 c ⁢ ( 0 ) α k k ! ⁢ | u | p ⁢ 𝑑 x ≤ e α .

As for II, we introduce a function v denoted by

v ⁢ ( x ) = { u ⁢ ( x ) - u ⁢ ( r 0 ) , | x | ≤ r 0 , 0 , | x | > r 0 .

Then we claim that [v]Ws,p⁢(ℝN)≤[u]Ws,p⁢(ℝN). For x∈Br0⁢(0), one can apply the monotonicity of u to derive that

∫ ℝ N | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y = ∫ B r 0 ⁢ ( 0 ) | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y + ∫ B r 0 c ⁢ ( 0 ) | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y

= ∫ B r 0 ⁢ ( 0 ) | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y + ∫ B r 0 c ⁢ ( 0 ) | u ⁢ ( x ) - u ⁢ ( r 0 ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y

(2.3) ≤ ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y .

Similarly, we can get that for x∈Br0c⁢(0),

(2.4) ∫ ℝ N | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y ≤ ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y .

Thus it follows from (2.3) and (2.4) that

[ v ] W s , p ⁢ ( ℝ N ) ≤ [ u ] W s , p ⁢ ( ℝ N ) .

Recall an elementary inequality which states that (a+b)t≤λ1-t⁢at+(1-λ)1-t⁢bt for a,b,t≥0 and λ∈(0,1). Pick a=v⁢(x), b=u⁢(r0), t=NN-s and λ=1/(1+∥u∥pp). One can obtain that

| u ⁢ ( x ) | N N - s ≤ ( 1 + ∥ u ∥ p p ) s N - s ⁢ v ⁢ ( x ) N N - s + ( 1 + ∥ u ∥ p p ) s N - s ∥ u ∥ p N N - s ⁢ u ⁢ ( r 0 ) N N - s .

This together with (2.1) yields that

| u ⁢ ( x ) | N N - s ≤ ( 1 + ∥ u ∥ p p ) s N - s ⁢ v ⁢ ( x ) N N - s + ( 1 + ∥ u ∥ p p ) s N - s ⁢ C ⁢ ( r 0 ) ≤ ( 1 + ∥ u ∥ p p ) s N - s ⁢ v ⁢ ( x ) N N - s + C ⁢ ( r 0 ) .

As a consequence,

II = ∫ B r 0 ⁢ ( 0 ) Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x

≤ ∫ B r 0 ⁢ ( 0 ) exp ⁡ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x

≲ ∫ B r 0 ⁢ ( 0 ) exp ⁡ ( α ⁢ ( 1 + ∥ u ∥ p p ) s N - s ⁢ v ⁢ ( x ) N N - s ) ⁢ 𝑑 x

(2.5) ≲ 1 ,

where we apply the fact that

[ ( 1 + ∥ u ∥ p p ) s N ⁢ v ⁢ ( x ) ] W s , p ⁢ ( ℝ N ) p ≤ ( 1 + ∥ u ∥ p p ) ⁢ ( 1 - ∥ u ∥ p p ) < 1 .

Combining (2.5) with (2.2), we derive (1.2). Then it suffices to show that αs,N* is an upper bound for α*. Denote the same sequence introduced in [56] by

u ε ⁢ ( x ) = { | ln ⁡ ε | N - s N if ⁢ | x | ≤ ε , | ln ⁡ | x | | | ln ⁡ ε | s N if ⁢ ε ≤ | x | < 1 , 0 if ⁢ | x | ≥ 1 .

Direct calculations yield that

(2.6) ∥ u ε ∥ p p = ∫ ℝ N | u ε | p ⁢ 𝑑 x = O ⁢ ( | log ⁡ ε | - 1 ) .

Moreover, calculations in [56] give that

[ u ε ] W s , p ⁢ ( ℝ N ) → ( α s , N * N ) N - s N .

This together with (2.6) implies that

∥ u ε ∥ W s , p ⁢ ( ℝ N ) → ( α s , N * N ) N - s N .

Setting α>αs,N* and ε sufficiently close to 0, we have

α ⁢ ∥ u ε ∥ W s , p ⁢ ( ℝ N ) - N N - s ≥ β > N and Φ N , s ⁢ ( α ⁢ | u ε ⁢ ( x ) | N N - s ) ≥ 1 2 ⁢ e α ⁢ | u ε ⁢ ( x ) | N N - s

for x∈Bε⁢(0). Then we obtain

∫ ℝ N Φ N , s ⁢ ( α ⁢ ( | u ε ⁢ ( x ) | ∥ u ε ∥ W s , p ⁢ ( ℝ N ) ) N N - s ) ⁢ 𝑑 x > ∫ B ε ⁢ ( 0 ) Φ N , s ⁢ ( α ⁢ ( | u ε ⁢ ( x ) | ∥ u ε ∥ W s , p ⁢ ( ℝ N ) ) N N - s ) ⁢ 𝑑 x

≥ 1 2 ⁢ ∫ B ε ⁢ ( 0 ) exp ⁡ ( α ⁢ ( | u ε ⁢ ( x ) | ∥ u ε ∥ W s , p ⁢ ( ℝ N ) ) N N - s ) ⁢ 𝑑 x

≥ 1 2 ⁢ ∫ B ε ⁢ ( 0 ) exp ⁡ ( - β ⁢ ln ⁡ ε ) ⁢ 𝑑 x

→ ∞ ,

which finishes the proof. ∎

Remark 2.3.

In Theorem 1.1, we derive the Trudinger–Moser inequality in the Sobolev–Slobodeckij space Ws,p⁢(ℝN). In this proof, we apply the monotonicity of the function instead of applying [56, Proposition 4.3], which seems to be very convenient in dimension one but complex in the higher-dimensions case, and we get that [v]Ws,p⁢(ℝN)≤[u]Ws,p⁢(ℝN).

Now, we extend Theorem 1.1 to a larger Sobolev–Slobodeckij space and give the proof of Theorem 1.3.

Proof of Theorem 1.3.

For any τ>0, 0≤α<α* and u∈Ws,p⁢(ℝN) satisfying

∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y ⁢ 𝑑 x + τ ⁢ ∫ ℝ N | u ⁢ ( x ) | p ⁢ 𝑑 x ≤ 1 ,

denote a new function v⁢(x) given by v⁢(τ1/N⁢x)=u⁢(x). With direct computations, one can get that

∫ ℝ N ∫ ℝ N | v ⁢ ( x ) - v ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y ⁢ 𝑑 x + ∫ ℝ N | v ⁢ ( x ) | p ⁢ 𝑑 x = ∫ ℝ N ∫ ℝ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y ⁢ 𝑑 x + τ ⁢ ∫ ℝ N | u ⁢ ( x ) | p ⁢ 𝑑 x

and

∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x = 1 τ ⁢ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | v | N N - s ) ⁢ 𝑑 x .

Then it follows from Theorem 1.1 that

sup [ u ] W s , p ⁢ ( ℝ N ) p + τ ⁢ ∥ u ∥ p p ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x ≤ 1 τ ⁢ sup [ v ] W s , p ⁢ ( ℝ N ) p + ∥ v ∥ p p ≤ 1 ⁡ ∫ ℝ N Φ N , s ⁢ ( α ⁢ | v | N N - s ) ⁢ 𝑑 x < ∞ .

To get the upper bound αs,N*, one can set vε⁢(x)=uε⁢(τ1/N⁢x) to check it. For simplicity, we omit the details. Thus, we complete the proof. ∎

As a direct consequence of Theorem 1.1, using the density of 𝒞c∞⁢(ℝN) in Ws,p⁢(ℝN) and [56, Proposition 3.2], we derive the following conclusion.

Corollary 2.4.

If u∈Ws,p⁢(RN), then for any α>0 it holds

∫ ℝ N Φ N , s ⁢ ( α ⁢ | u | N N - s ) ⁢ 𝑑 x < + ∞ .

3 The Proof of Theorem 1.4

In this section, we focus our attention on the existence of weak solutions. During our proof, the mountain-pass method plays a key role. First, we show that the functional Is,ε satisfies the geometric conditions of the mountain-pass theorem. Then, by utilizing the mountain-pass theorem, one can obtain a Palais–Smale sequence {un} and some convergence results. Finally, we show that the weak limit of un is just a weak solution. At the beginning of this section, we introduce the mountain-pass theorem and the Brezis–Lieb lemma.

Lemma 3.1 ([6]).

Let X be a Banach space, let φ∈C2⁢(X,R), let e∈X and r>0 such that ∥e∥>r, and let b:=inf∥u∥=r⁡φ⁢(u)>φ⁢(0)≥φ⁢(e). Define

c = inf g ∈ Γ ⁡ max s ∈ [ 0 , 1 ] ⁡ φ ⁢ ( g ⁢ ( s ) ) ,

where

Γ := { g ∈ C ⁢ ( [ 0 , 1 ] , X ) : g ⁢ ( 0 ) = 0 , g ⁢ ( 1 ) = e } .

Then there exists a sequence {uk}k∈X such that φ⁢(uk)→c and φ′⁢(uk)→0 as k→+∞.

Lemma 3.2 ([9]).

Let 1<q<+∞ and let (fn)n⊂Lq⁢(Rk) be a bounded sequence such that fn→f almost everywhere. Then

lim n → ∞ ⁡ ( ∥ f n ∥ L q ⁢ ( ℝ k ) q - ∥ f n - f ∥ L q ⁢ ( ℝ k ) q ) = ∥ f ∥ L q ⁢ ( ℝ k ) .

Furthermore,

lim n → ∞ ⁡ ∫ ℝ k | J q ⁢ ( f n ) - J q ⁢ ( f n - f ) - J q ⁢ ( f ) | = 0 .

Now, we check that Is,ε satisfies geometric conditions of the mountain-pass theorem without the Palais–Smale condition.

Lemma 3.3.

Assume (H2) holds. Then for any non-negative, compactly supported function u∈Ws,p⁢(RN)∖{0}, there holds Is,ε⁢(l⁢u)→-∞ as l→∞.

Proof.

By (H2), there exists a positive constant R0 such that

μ ⁢ F ⁢ ( x , t ) ≤ t ⁢ f ⁢ ( x , t ) ⁢ for ⁢ ( x , t ) ∈ ℝ N × [ R 0 , ∞ ) ,

which yields that

F ⁢ ( x , t ) ≥ F ⁢ ( x , R 0 ) ⁢ R 0 - μ ⁢ t μ for ⁢ ( x , t ) ∈ ℝ N × [ R 0 , ∞ ) .

Since u is compactly supported in a bounded domain Ω, then for any (x,t)∈Ω×[0,+∞) there exist positive constants c1, c2 such that

F ⁢ ( x , t ) ≥ c 1 ⁢ t μ - c 2 .

Thus, it follows that

I s , ε ⁢ ( l ⁢ u ) = l p p ⁢ [ u ] W s , p ⁢ ( ℝ N ) p + l p p ⁢ ∫ ℝ N V ⁢ | u | p ⁢ 𝑑 x - ∫ ℝ N F ⁢ ( x , l ⁢ u ) ⁢ 𝑑 x - ε ⁢ l ⁢ ∫ Ω h ⁢ u ⁢ 𝑑 x

≤ l p p ⁢ ∥ u ∥ p - c 1 ⁢ ∫ ℝ N ( l ⁢ u ) μ ⁢ 𝑑 x + c + O ⁢ ( l ) .

Since μ>p, we conclude that Is,ε⁢(l⁢u)→-∞ as l→∞. ∎

Lemma 3.4.

Assume (H1) and (H4) are satisfied. Then there exists a positive constant ε1 such that for 0<ε<ε1 there exist positive constants rε and νε such that Is,ε⁢(u)≥νε for all u with ∥u∥=rε.

Proof.

From (H4) one can derive that there exist τ>0 and δ>0 such that for any |t|≤δ,

(3.1) F ⁢ ( x , t ) ≤ 1 p ⁢ ( λ - τ ) ⁢ | t | p for ⁢ x ∈ ℝ N .

Then assumption (H1) yields that for any |t|≥δ and x∈ℝN,

F ⁢ ( x , t ) ≤ b 1 p ⁢ | t | p + c 2 ⁢ | t | ⁢ Φ N , s ⁢ ( α 0 ⁢ | t | N N - s ) ≤ C δ ⁢ | t | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | t | N N - s ) ,

where

C δ = b 1 δ ⁢ p ⁢ Φ N , s ⁢ ( α 0 ⁢ | δ | N N - s ) + b 2 δ p .

This together with (3.1) yields that

(3.2) F ⁢ ( x , t ) ≤ 1 p ⁢ ( λ - τ ) ⁢ | t | p + C ⁢ | t | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | t | N N - s ) for all ⁢ ( x , t ) ∈ ℝ N × ℝ .

Now we claim that for sufficiently small ∥u∥ there holds

∫ ℝ N | u | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | r | N N - s ) ⁢ 𝑑 x ≤ C ⁢ ∥ u ∥ p + 1 .

With the help of the Hölder inequality and the continuous embedding E↪Lq⁢(ℝN) (q≥p), we arrive at

∫ ℝ N | u | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | u | N N - s ) ⁢ 𝑑 x ≲ ( ∫ ℝ N Φ N , s ⁢ ( α 0 ⁢ t ⁢ | u | N N - s ) ⁢ 𝑑 x ) 1 t ⁢ ( ∫ ℝ N | u | ( p + 1 ) ⁢ t ′ ⁢ 𝑑 x ) 1 t ′

≲ ( ∫ ℝ N Φ N , s ⁢ ( α 0 ⁢ t ⁢ | u | N N - s ) ⁢ 𝑑 x ) 1 t ⁢ ∥ u ∥ p + 1 ,

where t>1 and 1t+1t′=1. Since ∥u∥ is sufficiently small, one can choose t>1 sufficiently close to 1 such that α0⁢t⁢∥u∥N/(N-s)<α*. Then it follows from Theorem 1.1 that

(3.3) ∫ ℝ N | u | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | u | N N - s ) ⁢ 𝑑 x ≤ C ⁢ ∥ u ∥ p + 1 .

Combining (3.2) with (3.3), we have

I s , ε ⁢ ( u ) = 1 p ⁢ ∥ u ∥ p - ∫ ℝ N F ⁢ ( x , u ) ⁢ 𝑑 x - ε ⁢ ∫ ℝ N h ⁢ u ⁢ 𝑑 x

≥ 1 p ⁢ ∥ u ∥ p - λ - τ p ⁢ ∫ ℝ N | u | p ⁢ 𝑑 x - C ⁢ ∫ ℝ N | u | p + 1 ⁢ Φ N , s ⁢ ( α 0 ⁢ | u | N N - s ) ⁢ 𝑑 x - ε ⁢ ∥ h ∥ p p - 1 ⁢ ∥ u ∥ p

≥ 1 p ⁢ ∥ u ∥ p - 1 p ⁢ λ - τ λ ⁢ ∥ u ∥ p - C ⁢ ∥ u ∥ p + 1 - ε V 0 1 / p ⁢ ∥ h ∥ p p - 1 ⁢ ∥ u ∥

= ∥ u ∥ ⁢ ( τ p ⁢ λ ⁢ ∥ u ∥ p - 1 - C ⁢ ∥ u ∥ p - ε V 0 1 / p ⁢ ∥ h ∥ p p - 1 ) .

Since τ>0, for sufficiently small ∥u∥=r>0 such that

τ p ⁢ λ ⁢ r p - 1 - C ⁢ r p ≥ τ 2 ⁢ p ⁢ λ ⁢ r p - 1 ,

let ε be sufficiently small such that we can get that

τ p ⁢ λ ⁢ r ε p - 1 - C ⁢ r ε p ≥ τ 2 ⁢ p ⁢ λ ⁢ r ε p - 1 = 2 ⁢ ν ε = 2 ⁢ ε V 0 1 / p ⁢ ∥ h ∥ p p - 1 ,

which finishes the proof. ∎

Remark 3.5.

In Lemma 3.4, rε can be sufficiently close to 0 as ε→0.

Lemmas 3.3 and 3.4 and the mountain-pass theorem give the existence of a Palais–Smale sequence {un}, i.e.

I s , ε ⁢ ( u n ) → c , I s , ε ′ ⁢ ( u n ) → 0 in ⁢ E ′ ⁢ as ⁢ n → ∞ ,

where

c = min g ∈ Γ ⁡ max x ∈ [ 0 , 1 ] ⁡ I s , ε ⁢ ( g ⁢ ( x ) )

and

Γ := { g ∈ C ⁢ ( [ 0 , 1 ] , X ) : g ⁢ ( 0 ) = 0 , g ⁢ ( 1 ) = e } .

Now, we are in the position to give some convergence results which play a crucial role in getting the existence of weak solutions and the energy estimation.

Lemma 3.6.

Suppose V⁢(x)≥V0>0 in RN satisfies (V1) or (V2) and the nonlinear term f satisfies (H1)–(H4). Then for any Palais–Smale sequence {un} there exist a subsequence of {un} (for simplicity, we still denote it by {un}) and u∈E such that

{ u n → u a.e. in ⁢ ℝ N , f ⁢ ( x , u n ) → f ⁢ ( x , u ) strongly in ⁢ L loc 1 ⁢ ( ℝ N ) , F ⁢ ( x , u n ) → F ⁢ ( x , u ) strongly in ⁢ L 1 ⁢ ( ℝ N ) .

Proof.

First of all, we claim that

(3.4) ∫ ℝ N F ⁢ ( x , u n ) ⁢ 𝑑 x ≤ C and ∫ ℝ N f ⁢ ( x , u n ) ⁢ u n ⁢ 𝑑 x ≤ C .

Let {un} be a Palais–Smale sequence of the function Is,ε, i.e.

(3.5) 1 p ⁢ ∥ u n ∥ p - ∫ ℝ N F ⁢ ( x , u n ) ⁢ 𝑑 x - ε ⁢ ∫ ℝ N h ⁢ u n ⁢ 𝑑 x → c as ⁢ n → ∞

and

(3.6) | I s , ε ′ ⁢ ( u n ) ⁢ v | ≤ τ n ⁢ ∥ v ∥ for all ⁢ v ∈ E ,

where τn→0 as n→∞. By picking v=un in (3.6), one can easily obtain that

(3.7) ∫ ℝ N f ⁢ ( x , u n ) ⁢ u n ⁢ 𝑑 x - ∥ u n ∥ p ≤ τ n ⁢ ∥ u n ∥ + ε ⁢ ∥ h ∥ p p - 1 ⁢ ∥ u n ∥ p .

Combining this with (3.5) and (H2), we have

( μ p - 1 ) ⁢ ∥ u n ∥ p ≲ 1 + ∥ u n ∥ ,

which implies that ∥un∥ is bounded. Then (3.4) comes from (3.5) and (3.7). With the help of (V1) or (V2), the continuous embedding E↪Lq⁢(ℝN) is compact for q≥p. Since ∥un∥ is bounded and E is reflexive, we can assume that up to a subsequence,

u n ⇀ u ⁢ weakly in ⁢ E ,

u n → u ⁢ strongly in ⁢ L q ⁢ ( ℝ N ) ⁢ for all ⁢ q ≥ p ,

u n ⁢ ( x ) → u ⁢ ( x ) ⁢ for almost every ⁢ x ∈ ℝ N .

By hypothesis (H1), using the same arguments as [11, Lemma 2.1], we obtain that

f ⁢ ( x , u n ) → f ⁢ ( x , u ) strongly in ⁢ L loc 1 ⁢ ( ℝ N ) .

For simplicity, we omit the proof. Now, we show that F⁢(x,un)→F⁢(x,u) in L1⁢(ℝN). By splitting the integral into two parts, we have

For the first part, we recall conditions (H2) and (H3) and obtain that there exists a constant R0>0 such that

(3.8) F ⁢ ( x , u n ) ≤ R 0 ⁢ f ⁢ ( x , u n ) for all ⁢ x ∈ ℝ N .

Thus, it follows from the generalized Lebesgue dominated convergence theorem that

lim R → + ∞ ⁡ lim n → + ∞ ⁡ ∫ B R | F ⁢ ( x , u n ) - F ⁢ ( x , u ) | ⁢ 𝑑 x = 0 .

As a consequence, it suffices to show that

lim R → + ∞ ⁡ lim n → + ∞ ⁡ ∫ ℝ N ∖ B R | F ⁢ ( x , u n ) - F ⁢ ( x , u ) | ⁢ 𝑑 x = 0 .

To acquire this aim, we can write

∫ ℝ N ∖ B R | F ⁢ ( x , u n ) - F ⁢ ( x , u ) | ⁢ 𝑑 x

= ∫ { | x | ≥ R } ∩ { | u n | > A } | F ⁢ ( x , u n ) - F ⁢ ( x , u ) | ⁢ 𝑑 x + ∫ { | x | ≥ R } ∩ { | u n | ≤ A } | F ⁢ ( x , u n ) - F ⁢ ( x , u ) | ⁢ 𝑑 x

= : I A + II A .

For IA, we use inequality (3.8) to derive that

∫ { | x | ≥ R } ∩ { | u n | > A } | F ⁢ ( x , u n ) | ⁢ 𝑑 x ≤ R 0 A ⁢ ∫ { | x | ≥ R } ∩ { | u n | > A } | f ⁢ ( x , u n ) ⁢ u n | ⁢ 𝑑 x ≲ R 0 A ,

which implies limA→+∞⁡limR→+∞⁡limn→+∞⁡IA=0.

As for IIA, we discuss the following two cases:

Case I: The function 1V⁢(x) belongs to L1⁢(ℝN). From (H1) we apply the compact embedding E↪↪Lq(ℝN) for q≥p, the Hölder inequality and the absolute continuity of ∫ℝN1V⁢(x)⁢𝑑x to get that

lim A → + ∞ ⁡ lim R → + ∞ ⁡ lim n → + ∞ ⁡ II A ≤ lim A → + ∞ ⁡ lim R → + ∞ ⁡ lim n → + ∞ ⁡ C ⁢ ( α 0 , A ) ⁢ ∫ { | x | ≥ R } ∩ { | u n | ≤ A } | u n | p ⁢ 𝑑 x

≤ lim A → + ∞ ⁡ lim R → + ∞ ⁡ lim n → + ∞ ⁡ C ⁢ ( α 0 , A ) ⁢ ∫ { | x | ≥ R } ∩ { | u n | ≤ A } | u n | p 2 ⁢ 𝑑 x

≤ lim A → + ∞ ⁡ lim R → + ∞ ⁡ lim n → + ∞ ⁡ C ⁢ ( α 0 , A ) ⁢ ( ∫ { | x | ≥ R } 1 | V ⁢ ( x ) | ⁢ 𝑑 x ) 1 2 ⁢ sup n ⁡ ∥ u n ∥ p 2

= 0 .

Case II: V ⁢ ( x ) → ∞ as |x|→∞. Thanks to hypothesis (H1) and the compact embedding E↪↪Lq(ℝN) for q≥p, one can utilize the property of V⁢(x) to derive that

≤ lim A → + ∞ ⁡ lim R → + ∞ ⁡ lim n → + ∞ ⁡ C ⁢ ( α 0 , A ) ⁢ sup { | x | ≥ R } ⁡ 1 | V ⁢ ( x ) | ⁢ sup n ⁡ ∥ u n ∥ p

= 0 .

Thus, we show that F⁢(x,un)→F⁢(x,u) strongly in L1⁢(ℝN). ∎

Based on these convergence results, we obtain the following corollary.

Corollary 3.7.

The u from Lemma 3.6 is a weak solution of (1.3).

Proof.

Obviously, it is sufficient to prove that for any φ∈𝒞c∞⁢(ℝN),

(3.9) I s , ε ′ ⁢ ( u n ) ⁢ φ → I s , ε ′ ⁢ ( u ) ⁢ φ as ⁢ n → + ∞ .

Since φ∈𝒞c∞⁢(ℝN), there exists a ball BR⁢(0) such that supp⁡φ⊂BR⁢(0). Then it follows from f⁢(x,un)→f⁢(x,u) in Lloc1⁢(ℝN) that

lim n → + ∞ ⁡ ∫ ℝ N f ⁢ ( x , u n ) ⁢ φ ⁢ 𝑑 x → ∫ ℝ N f ⁢ ( x , u ) ⁢ φ ⁢ 𝑑 x .

Notice that un→u a.e. in ℝN. By the Egoroff theorem, we obtain that for any δ>0 there exists

E δ ⊂ B R ⁢ ( 0 ) satisfying m ⁢ ( E δ ) < δ

such that

u n ⇉ u ⁢ in ⁢ B R ⁢ ( 0 ) ∖ E δ .

Hence, we have

lim n → + ∞ ⁡ ∫ ℝ N V ⁢ ( x ) ⁢ ( | u n | p - 2 ⁢ u n - | u | p - 2 ⁢ u ) ⁢ φ ⁢ 𝑑 x

= lim δ → 0 ⁡ lim n → + ∞ ⁡ ∫ ℝ N V ⁢ ( x ) ⁢ ( | u n | p - 2 ⁢ u n - | u | p - 2 ⁢ u ) ⁢ φ ⁢ 𝑑 x

= lim δ → 0 ⁡ lim n → + ∞ ⁡ ( ∫ B R ⁢ ( 0 ) ∖ E δ + ∫ E δ ) ⁢ V ⁢ ( x ) ⁢ ( | u n | p - 2 ⁢ u n - | u | p - 2 ⁢ u ) ⁢ φ ⁢ d ⁢ x

≤ lim δ → 0 ⁡ sup n ⁡ ( ( ∫ E δ V ⁢ ( x ) ⁢ | u n | p ⁢ 𝑑 x ) 1 p ′ + ( ∫ E δ V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p ′ ) ⁢ ( ∫ E δ V ⁢ ( x ) ⁢ | φ | p ⁢ 𝑑 x ) 1 p

= 0 .

As for the convergence of

∫ ℝ 2 ⁢ N J p ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N ⁢ 𝑑 x ⁢ 𝑑 y ,

it is more complicated. Indeed, we split it into four parts to obtain that

∫ ℝ 2 ⁢ N G p ⁢ ( u n - u ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N ⁢ 𝑑 x ⁢ 𝑑 y = ( ∫ Ω 1 + ∫ Ω 2 + ∫ Ω 3 + ∫ Ω 4 ) ⁢ G p ⁢ ( u n - u ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N ⁢ d ⁢ x ⁢ d ⁢ y

(3.10) = : I 1 + I 2 + I 3 + I 4 ,

where

Ω 1 = B R c ⁢ ( 0 ) × B R c ⁢ ( 0 ) , ⁢ Ω 2 = B R c ⁢ ( 0 ) × B R ⁢ ( 0 ) ,

Ω 3 = B R ⁢ ( 0 ) × B R c ⁢ ( 0 ) , ⁢ Ω 4 = B R ⁢ ( 0 ) × B R ⁢ ( 0 )

and

G p ⁢ ( u n - u ) := ( J p ⁢ ( u n ⁢ ( x ) - u n ⁢ ( y ) ) - J p ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ) .

For I1, it follows from φ⁢(x)=φ⁢(y)=0 on BRc⁢(0) that

(3.11) lim n → ∞ ⁡ I 1 = 0 .

To estimate I4, we notice that Gp⁢(un-u)→0 almost everywhere on Ω4 . By the Egoroff theorem, one can derive that for any δ>0 there exists a subset Eδ⊂Ω4 such that |Eδ|<δ and Gp⁢(un-u)⇉0 in Ω4∖Eδ. Therefore, it follows from the Hölder inequality that

lim n → ∞ ⁡ I 4 = lim δ → 0 ⁡ lim n → ∞ ⁡ I 4

= lim δ → 0 ⁡ lim n → ∞ ⁡ ( ∫ Ω 4 ∖ E δ + ∫ E δ ) ⁢ G p ⁢ ( u n - u ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N ⁢ d ⁢ x ⁢ d ⁢ y

≤ lim δ → 0 ⁡ ( sup n ⁡ ∥ u n ∥ 1 p ′ + ∥ u ∥ 1 p ′ ) ⁢ ( ∫ E δ | φ ⁢ ( x ) - φ ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 x ⁢ 𝑑 y ) 1 p

(3.12) = 0 .

Now, we begin to estimate I2. Similarly, we divide the integral into two parts.

(3.13) I 2 = ( ∫ ( B R 1 ⁢ ( 0 ) ∖ B R ⁢ ( 0 ) ) × B R ⁢ ( 0 ) + ∫ B R 1 c ⁢ ( 0 ) × B R ⁢ ( 0 ) ) G p ⁢ ( u n - u ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N d x d y = : II 1 + II 2 .

Then it follows from the Egoroff lemma and the Hölder inequality that

lim R 1 → ∞ ⁡ lim δ → 0 ⁡ lim n → ∞ ⁡ II 1 = 0 .

As for II2, we arrive at

lim R 1 → ∞ ⁡ lim δ → 0 ⁡ lim n → ∞ ⁡ II 2 ≤ lim R 1 → ∞ ⁡ lim δ → 0 ⁡ lim n → ∞ ⁡ ( [ u n ] W s , p ⁢ ( ℝ ℕ ) 1 / p ′ + [ u ] W s , p ⁢ ( ℝ ℕ ) 1 / p ′ ) ⁢ ( 1 R 1 - R ) p - N p ⁢ ( ∫ B R 1 c ⁢ ( 0 ) × B R ⁢ ( 0 ) | φ ⁢ ( y ) | p | x - y | N + p ⁢ 𝑑 x ⁢ 𝑑 y ) 1 p

≲ lim R 1 → ∞ ⁡ sup n ⁡ ( [ u n ] W s , p ⁢ ( ℝ ℕ ) 1 / p ′ + [ u ] W s , p ⁢ ( ℝ ℕ ) 1 / p ′ ) ⁢ ( 1 R 1 - R ) p - N p

(3.14) = 0 .

Combining (3.10)–(3.14) and I2=I3, we derive that

lim n → + ∞ ⁡ ∫ ℝ 2 ⁢ N G p ⁢ ( u n - u ) ⁢ ( φ ⁢ ( x ) - φ ⁢ ( y ) ) | x - y | 2 ⁢ N ⁢ 𝑑 x → 0 .

Then the weak convergence of un⇀u in E gives that

lim n → + ∞ ⁡ ∫ ℝ N h ⁢ u n ⁢ 𝑑 x → ∫ ℝ N h ⁢ u ⁢ 𝑑 x .

Summing up the results above, one can get (3.9). Thanks to the density of 𝒞c∞⁢(ℝN) in Ws,p⁢(ℝN), we prove that u is a weak solution. ∎

Keeping these results in mind, we are going to prove Theorem 1.4.

Proof of Theorem 1.4.

Thanks to Lemma 3.3 and Lemma 3.4, one can easily get that there exists a constant ε1>0 such that Is,ε, for all 0≤ε<ε1, satisfies geometric conditions of the mountain-pass theorem without the Palais–Smale condition. Thus, we get a sequence {un} of E such that

I s , ε ⁢ ( u n ) → c > 0 , I s , ε ′ ⁢ ( u n ) → 0 in ⁢ E ′ ,

where

c = min g ∈ Γ ⁡ max x ∈ [ 0 , 1 ] ⁡ I s , ε ⁢ ( g ⁢ ( x ) ) , Γ = { g ∈ 𝒞 ⁢ ( [ 0 , 1 ] , E ) : g ⁢ ( 0 ) = 0 , g ⁢ ( 1 ) = e } .

Then it follows from Lemma 3.6 and Corollary 3.7 that there exists a subsequence of {un} that converges weakly to a solution u of equation (1.3) in E. ∎

4 The Proof of Theorem 1.5

In this section, we find another weak solution with negative energy. In order to achieve this goal, we need the following results.

Definition 4.1 ([6]).

A functional I:X→ℝ on a Banach space X is called coercive if for every sequence {uk}k∈N⊂X,

∥ u k ∥ → + ∞ implies I ⁢ ( u k ) → + ∞ .

Lemma 4.2 ([6]).

Let X be a reflexive Banach space and let I:X→R be a continuous, convex and coercive functional. Then I has a global minimum point.

Based on these two lemmas, we are going to study the geometry of the functional Is,ε in more detail. In fact, we prove the following lemma.

Lemma 4.3.

If h≢0 and (H1) is satisfied, then there exist a positive constant τ and v∈E with ∥v∥=1 such that for all l∈(0,τ) one has Is,ε⁢(l⁢v)<0. In fact,

inf ∥ u ∥ ≤ τ ⁡ I s , ε ⁢ ( u ) < 0 .

Proof.

Let us begin with the existence of the weak solutions of the following equation:

( - Δ ) p s ⁢ w ⁢ ( x ) + V ⁢ ( x ) ⁢ | w ⁢ ( x ) | p - 2 ⁢ w ⁢ ( x ) = h ⁢ ( x ) in ⁢ ℝ N .

Obviously, we can easily find the functional related to it, given by

I ⁢ ( w ) = 1 p ⁢ ∫ ℝ N ∫ ℝ N | w ⁢ ( x ) - w ⁢ ( y ) | p | x - y | 2 ⁢ N ⁢ 𝑑 y ⁢ 𝑑 x + 1 p ⁢ ∫ ℝ N V ⁢ ( x ) ⁢ | w ⁢ ( x ) | p ⁢ 𝑑 x - ∫ ℝ N h ⁢ w ⁢ 𝑑 x .

Direct calculations show that I∈𝒞1⁢(E,ℝ) and I is coercive. Note that I is convex. Recalling an elementary inequality which states that |t⁢a+(1-t)⁢b|p≤|t|⁢a⁢|+(1-t)⁢|b||p≤t⁢|a|p+(1-t)⁢|b|p for any t∈[0,1], a,b∈ℝ, we have

I ⁢ ( t ⁢ u + ( 1 - t ) ⁢ v ) = [ t ⁢ u + ( 1 - t ) ⁢ v ] W s , p ⁢ ( ℝ N ) p p + ∫ ℝ N V ⁢ | t ⁢ u + ( 1 - t ) ⁢ v | p ⁢ 𝑑 x p - ∫ ℝ N h ⁢ ( t ⁢ u + ( 1 - t ) ⁢ v ) ⁢ 𝑑 x

≤ t ⁢ [ u ] W s , p ⁢ ( ℝ N ) p + ( 1 - t ) ⁢ [ v ] W s , p ⁢ ( ℝ N ) p p + ∫ ℝ N V ⁢ ( x ) ⁢ ( t ⁢ u p + ( 1 - t ) ⁢ v p ) ⁢ 𝑑 x - ∫ ℝ N t ⁢ h ⁢ u + ( 1 - t ) ⁢ h ⁢ v ⁢ d ⁢ x

= t ⁢ I ⁢ ( u ) + ( 1 - t ) ⁢ I ⁢ ( v ) .

This fact together with Lemma 4.2 implies that I has a global minimum which gives the existence of w. Notice that h≢0. We have

∫ ℝ N h ⁢ w ⁢ 𝑑 x = ∥ w ∥ p .

For t>0, we write

d d ⁢ t ⁢ I s , ε ⁢ ( t ⁢ w ) = t p - 1 ⁢ ∥ w ∥ p - ∫ ℝ N f ⁢ ( x , t ⁢ w ) ⁢ w ⁢ 𝑑 x - ε ⁢ ∫ ℝ N h ⁢ w ⁢ 𝑑 x ,

which yields that

d d ⁢ t ⁢ I s , ε ⁢ ( t ⁢ w ) | t = 0 < 0 .

Since dd⁢t⁢Is,ε⁢(t⁢w) is continuous, one can easily obtain that there exists a positive constant τ such that

d d ⁢ t ⁢ I s , ε ⁢ ( t ⁢ w ) < 0 for all ⁢ t ∈ ( 0 , τ ) .

Combining this with Is,ε⁢(0)=0, we accomplish the desired conclusion. ∎

Next, we analyze the Palais–Smale compactness condition which is a key step in the energy estimation.

Lemma 4.4.

Suppose V⁢(x)≥V0>0 in RN, suppose (V1) or (V2) hold and suppose (H1) and (H2) hold. For any Palais–Smale sequence {un} for Is,ε satisfying

lim ⁡ inf n → + ∞ ⁡ ∥ u n ∥ N N - s < α * α 0 ,

{ u n } has a subsequence (still denoted by {un}) which converges strongly to a weak solution of equation (1.3).

Proof.

Select a subsequence which satisfies

lim n → + ∞ ⁡ ∥ u n ∥ N N - s < α * α 0 .

With the help of Lemma 3.6 and Corollary 3.7, we may get that up to a subsequence, un converges weakly to u0 which is a weak solution of (1.3). Let un=u0+wn. We can easily get that wn⇀0 in E. Then it follows from the compact embedding E↪↪Lq(ℝN) that wn→0 in Lq⁢(ℝN) (q≥p), which implies un→u a.e. By the Brezis–Lieb Lemma, one can get that

(4.1) ∥ u n ∥ p = ∥ u 0 ∥ p + ∥ w n ∥ p + o n ⁢ ( 1 ) .

Now, we claim that

(4.2) lim n → + ∞ ⁡ ∫ ℝ N f ⁢ ( x , u n ) ⁢ u 0 ⁢ 𝑑 x = ∫ ℝ N f ⁢ ( x , u 0 ) ⁢ u 0 ⁢ 𝑑 x .

For the continuity of the statement, we postpone the proof of (4.2). If (4.2) holds, we can apply (4.1) to obtain that

〈 I s , ε ′ ⁢ ( u n ) , u n 〉 = 〈 I s , ε ′ ⁢ ( u 0 ) , u 0 〉 + ∥ w n ∥ p - ∫ ℝ N f ⁢ ( x , u n ) ⁢ w n ⁢ 𝑑 x + o n ⁢ ( 1 ) .

Hence,

(4.3) ∥ w n ∥ p = ∫ ℝ N f ⁢ ( x , u n ) ⁢ w n ⁢ 𝑑 x + o n ⁢ ( 1 ) .

Now, we estimate ∫ℝNf⁢(x,un)⁢wn⁢𝑑x. By hypothesis (H1), we arrive at

| ∫ ℝ N f ⁢ ( x , u n ) ⁢ w n ⁢ 𝑑 x | ≤ b 1 ⁢ ∫ ℝ N | u n | p - 1 ⁢ | w n | ⁢ 𝑑 x + b 2 ⁢ ∫ ℝ N Φ N , s ⁢ ( α 0 ⁢ | u n | N N - s ) ⁢ | w n | ⁢ 𝑑 x

≤ b 1 ⁢ ∥ u n ∥ p p - 1 ⁢ ∥ w n ∥ p + b 2 ⁢ ∥ Φ N , s ⁢ ( α 0 ⁢ | u n | N N - s ) ∥ q ⁢ ∥ w n ∥ q ′

(4.4) ≤ b 1 ⁢ ∥ u n ∥ p p - 1 ⁢ ∥ w n ∥ p + b 2 ⁢ ∥ Φ N , s ⁢ ( q ⁢ α 0 ⁢ | u n | N N - s ) ∥ 1 1 q ⁢ ∥ w n ∥ q ′ .

In fact, the last inequality follows from [23, Lemma A.2]. Since limn→+∞⁡∥un∥N/(N-s)<α*α0, we get that for n sufficiently large, ∥un∥N/(N-s)<α*α0. Choose q>1 sufficiently close to 1. We arrive at

q ⁢ α 0 ⁢ ( [ u n ] W s , p ⁢ ( ℝ 2 ⁢ N ) p + V 0 ⁢ ∥ u n ∥ p p ) N p ⁢ ( N - s ) ≤ q ⁢ α 0 ⁢ ∥ u n ∥ N N - s < α * .

This estimate together with wn→0 in Lq⁢(ℝN) for q≥p and Theorem 1.3 implies that

lim n → + ∞ ⁡ ∫ ℝ N f ⁢ ( x , u n ) ⁢ w n ⁢ 𝑑 x = 0 .

Thus, it follows from (4.3) that ∥wn∥→0 as n→+∞, that is, ∥un-u0∥→0 as n→+∞.

As a result, we only need to show (4.2). We write

lim n → + ∞ ⁡ ∫ ℝ N | f ⁢ ( x , u n ) - f ⁢ ( x , u 0 ) | ⁢ u 0 ⁢ 𝑑 x = lim R → + ∞ ⁡ lim M → + ∞ ⁡ lim n → + ∞ ⁡ ∫ ℝ N | f ⁢ ( x , u n ) - f ⁢ ( x , u 0 ) | ⁢ u 0 ⁢ 𝑑 x

= lim R → + ∞ ⁡ lim M → + ∞ ⁡ lim n → + ∞ ⁡ ( ∫ B R c ⁢ ( 0 ) + ∫ E u n M + ∫ B R ⁢ ( 0 ) ∖ E u n M ) ⁢ | f ⁢ ( x , u n ) - f ⁢ ( x , u 0 ) | ⁢ | u 0 | ⁢ d ⁢ x

= : lim R → + ∞ lim M → + ∞ lim n → + ∞ I 1 + I 2 + I 3 ,

where M>0 and EuM={|x|≤R:|u|>M} for a measurable function u. For I1, we show that

lim R → + ∞ ⁡ lim n → + ∞ ⁡ ∫ B R c ⁢ ( 0 ) | f ⁢ ( x , u n ) | ⁢ | u 0 | ⁢ 𝑑 x = 0 .

In fact, by condition (H1), using similar arguments as for inequality (4.4), we can obtain that

lim R → + ∞ ⁡ lim n → + ∞ ⁡ ∫ B R c ⁢ ( 0 ) f ⁢ ( x , u n ) ⁢ u 0 ⁢ 𝑑 x ≤ lim R → + ∞ ⁡ lim n → + ∞ ⁡ b 1 ⁢ ∫ B R c ⁢ ( 0 ) | u n | p - 1 ⁢ | u 0 | ⁢ 𝑑 x + b 2 ⁢ ∫ B R c ⁢ ( 0 ) Φ N , s ⁢ ( α 0 ⁢ | u n | N N - s ) ⁢ | u 0 | ⁢ 𝑑 x

≲ lim R → + ∞ ⁡ ( ∫ B R c ⁢ ( 0 ) | u 0 | p ⁢ 𝑑 x ) 1 p + lim R → + ∞ ⁡ ( ∫ B R c ⁢ ( 0 ) | u 0 | q ⁢ 𝑑 x ) 1 q

(4.5) = 0 .

Similarly,

(4.6) lim R → + ∞ ⁡ lim n → + ∞ ⁡ ∫ B R c ⁢ ( 0 ) | f ⁢ ( x , u 0 ) | ⁢ | u 0 | ⁢ 𝑑 x = 0 .

Combining (4.5) and (4.6), we derive that

(4.7) I 1 = 0 .

By Lebesgue’s dominate convergence theorem, one can easily check that |EunM|→|Eu0M|. Now, we estimate I2 as follows:

lim M → + ∞ ⁡ lim n → + ∞ ⁡ ∫ E u n M | f ⁢ ( x , u n ) | ⁢ | u 0 | + | f ⁢ ( x , u 0 ) | ⁢ | u 0 | ⁢ d ⁢ x

≤ lim M → + ∞ ⁡ lim n → + ∞ ⁡ ∥ f ⁢ ( x , u n ) ∥ p ⁢ ( ∫ E u n M | u 0 | p ′ ⁢ 𝑑 x ) 1 p ′ + ∥ f ⁢ ( x , u 0 ) ∥ p ⁢ ( ∫ E u n M | u 0 | p ′ ⁢ 𝑑 x ) 1 p ′

≲ lim M → + ∞ ⁡ lim n → + ∞ ⁡ ( ∫ E u n M | u 0 | p ′ ⁢ 𝑑 x ) 1 p ′

(4.8) = 0 ,

where we use Theorem 1.1 and condition (H1) in the above estimation. As for I3, it follows from Lebesgue’s dominate convergence theorem that

lim n → + ∞ ⁡ ∫ B R ⁢ ( 0 ) ∖ E u n M | f ⁢ ( x , u n ) - f ⁢ ( x , u 0 ) | ⁢ | u 0 | ⁢ 𝑑 x = 0 .

This together with (4.7) and (4.8) yields (4.2). Thus, we accomplish the proof. ∎

Now, let us begin to give the proof of Theorem 1.5.

Proof of Theorem 1.5.

With the help of Lemma 4.3, we get that Is,ε⁢(u)>0 for all u such that ∥u∥=rε. Pick ε2 with 0<ε2<ε1 such that when 0<ε<ε2, we have

(4.9) r ε < ( α * α 0 ) N - s N .

With precise calculations and condition (H2), one can obtain that

I s , ε ⁢ ( u ) ≥ C for all ⁢ u ⁢ such that ⁢ ∥ u ∥ ≤ r ε .

Then it follows from Lemma 4.3 that

inf ∥ u ∥ ≤ r ε ⁡ I s , ε ⁢ ( u ) < 0 .

Since Brε¯ is a complete metric space with the metric given by the norm of E, it is convex and the functional Is,ε is of class 𝒞1 and has a lower bound on Brε¯. By applying Ekeland’s variational principle, we derive that there exists some sequence (un)⊂Brε¯ such that

I s , ε ⁢ ( u n ) → c 0 = inf ∥ u ∥ ≤ r ε ⁡ I s , ε ⁢ ( u )

and

I s , ε ′ ⁢ ( u n ) → 0 in ⁢ E ′ ⁢ as ⁢ n → ∞ .

Based on inequality (4.9) and Lemma 3.6, we conclude that up to a subsequence, un→u0⁢ in ⁢E, where u0 is a weak solution of (1.3). Moreover, Is,ε⁢(u0)=limn→∞⁡Is,ε⁢(un)<0. ∎

Remark 4.5.

The convergence F⁢(x,un)→F⁢(x,u) in L1⁢(ℝN) is very crucial for getting the negative energy. Indeed, Is,ε⁢(u0)=limn→∞⁡Is,ε⁢(un) follows from F⁢(x,un)→F⁢(x,u) in L1⁢(ℝN) and un→u0 in E.

Remark 4.6.

The weak solution u from Theorem 1.4 and u0 from Theorem 1.5 are actually different solutions. Indeed, we can get Is,ε⁢(u0)=c0<0 from Theorem 1.5 and Is,ε⁢(u)=c>0 with the help of Lemma 4.4 and Lemma 3.6.

Communicated by Guozhen Lu

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11371056

Funding statement: The author was partly supported by a grant from the NNSF of China (no. 11371056).

References

[1] S. Adachi and K. Tanaka, Trudinger type inequalities in 𝐑N and their best exponents, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2051–2057. 10.1090/S0002-9939-99-05180-1Search in Google Scholar

[2] D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. (2) 128 (1988), no. 2, 385–398. 10.2307/1971445Search in Google Scholar

[3] Adimurthi and Y. Yang, An interpolation of Hardy inequality and Trundinger–Moser inequality in ℝN and its applications, Int. Math. Res. Not. IMRN 2010 (2010), no. 13, 2394–2426. Search in Google Scholar

[4] F. J. Almgren, Jr. and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc. 2 (1989), no. 4, 683–773. 10.1090/S0894-0347-1989-1002633-4Search in Google Scholar

[5] C. O. Alves and G. M. Figueiredo, On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in ℝN, J. Differential Equations 246 (2009), no. 3, 1288–1311. 10.1016/j.jde.2008.08.004Search in Google Scholar

[6] M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, Universitext, Springer, London, 2011. 10.1007/978-0-85729-227-8Search in Google Scholar

[7] J. Bao, N. Lam and G. Lu, Polyharmonic equations with critical exponential growth in the whole space ℝn, Discrete Contin. Dyn. Syst. 36 (2016), no. 2, 577–600. 10.3934/dcds.2016.36.577Search in Google Scholar

[8] J. A. M. B. do Ó, N-Laplacian equations in 𝐑N with critical growth, Abstr. Appl. Anal. 2 (1997), no. 3–4, 301–315. 10.1155/S1085337597000419Search in Google Scholar

[9] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490. 10.1007/978-3-642-55925-9_42Search in Google Scholar

[10] L. Chen, J. Li, G. Lu and C. Zhang, Sharpened Adams Inequality and Ground State Solutions to the Bi-Laplacian Equation in ℝ4, Adv. Nonlinear Stud. 18 (2018), no. 3, 429–452. 10.1515/ans-2018-2020Search in Google Scholar

[11] D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in 𝐑2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139–153. 10.1007/BF01205003Search in Google Scholar

[12] M. de Souza and Y. L. Araújo, Semilinear elliptic equations for the fractional Laplacian involving critical exponential growth, Math. Methods Appl. Sci. 40 (2017), no. 5, 1757–1772. 10.1002/mma.4095Search in Google Scholar

[13] M. Dong, N. Lam and G. Lu, Sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities and their extremal functions, Nonlinear Anal. 173 (2018), 75–98. 10.1016/j.na.2018.03.006Search in Google Scholar

[14] M. Dong and G. Lu, Best constants and existence of maximizers for weighted Trudinger–Moser inequalities, Calc. Var. Partial Differential Equations 55 (2016), no. 4, Article ID 88. 10.1007/s00526-016-1014-7Search in Google Scholar

[15] Y. Q. Dong and Q. H. Yang, An interpolation of Hardy inequality and Moser–Trudinger inequality on Riemannian manifolds with negative curvature, Acta Math. Sin. (Engl. Ser.) 32 (2016), no. 7, 856–866. 10.1007/s10114-016-5129-8Search in Google Scholar

[16] J. A. M. do Ó, M. de Souza, E. de Medeiros and U. Severo, An improvement for the Trudinger–Moser inequality and applications, J. Differential Equations 256 (2014), no. 4, 1317–1349. 10.1016/j.jde.2013.10.016Search in Google Scholar

[17] J. A. M. do Ó, O. H. Miyagaki and M. Squassina, Nonautonomous fractional problems with exponential growth, NoDEA Nonlinear Differential Equations Appl. 22 (2015), no. 5, 1395–1410. 10.1007/s00030-015-0327-0Search in Google Scholar

[18] J. A. M. do Ó, O. H. Miyagaki and M. Squassina, Ground states of nonlocal scalar field equations with Trudinger–Moser critical nonlinearity, Topol. Methods Nonlinear Anal. 48 (2016), no. 2, 477–492. 10.12775/TMNA.2016.045Search in Google Scholar

[19] L. Fontana, Sharp borderline Sobolev inequalities on compact Riemannian manifolds, Comment. Math. Helv. 68 (1993), no. 3, 415–454. 10.1007/BF02565828Search in Google Scholar

[20] J. Giacomoni, P. K. Mishra and K. Sreenadh, Fractional elliptic equations with critical exponential nonlinearity, Adv. Nonlinear Anal. 5 (2016), no. 1, 57–74. 10.1515/anona-2015-0081Search in Google Scholar

[21] A. Hyder, Moser functions and fractional Moser–Trudinger type inequalities, Nonlinear Anal. 146 (2016), 185–210. 10.1016/j.na.2016.08.024Search in Google Scholar

[22] A. Iannizzotto and M. Squassina, 1/2-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl. 414 (2014), no. 1, 372–385. 10.1016/j.jmaa.2013.12.059Search in Google Scholar

[23] M. Ishiwata, M. Nakamura and H. Wadade, On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, 297–314. 10.1016/j.anihpc.2013.03.004Search in Google Scholar

[24] S. Iula, A note on the Moser–Trudinger inequality in Sobolev–Slobodeckij spaces in dimension one, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 4, 871–884. 10.4171/RLM/789Search in Google Scholar

[25] S. Iula, A. Maalaoui and L. Martinazzi, A fractional Moser–Trudinger type inequality in one dimension and its critical points, Differential Integral Equations 29 (2016), no. 5–6, 455–492. 10.57262/die/1457536887Search in Google Scholar

[26] V. I. Judovič, Some estimates connected with integral operators and with solutions of elliptic equations (in Russian), Dokl. Akad. Nauk SSSR 138 (1961), 805–808. Search in Google Scholar

[27] N. Lam and G. Lu, Existence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in ℝN, J. Funct. Anal. 262 (2012), no. 3, 1132–1165. 10.1016/j.jfa.2011.10.012Search in Google Scholar

[28] N. Lam and G. Lu, Sharp Adams type inequalities in Sobolev spaces Wm,nm⁢(ℝn) for arbitrary integer m, J. Differential Equations 253 (2012), no. 4, 1143–1171. 10.1016/j.jde.2012.04.025Search in Google Scholar

[29] N. Lam and G. Lu, Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications, Adv. Math. 231 (2012), no. 6, 3259–3287. 10.1016/j.aim.2012.09.004Search in Google Scholar

[30] N. Lam and G. Lu, The Moser–Trudinger and Adams inequalities and elliptic and subelliptic equations with nonlinearity of exponential growth, Recent Developments in Geometry and Analysis, Adv. Lect. Math. (ALM) 23, International Press, Somerville (2012), 179–251. Search in Google Scholar

[31] N. Lam and G. Lu, A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument, J. Differential Equations 255 (2013), no. 3, 298–325. 10.1016/j.jde.2013.04.005Search in Google Scholar

[32] N. Lam and G. Lu, N-Laplacian equations in ℝN with subcritical and critical growth without the Ambrosetti–Rabinowitz condition, Adv. Nonlinear Stud. 13 (2013), no. 2, 289–308. 10.1515/ans-2013-0203Search in Google Scholar

[33] N. Lam and G. Lu, Elliptic equations and systems with subcritical and critical exponential growth without the Ambrosetti–Rabinowitz condition, J. Geom. Anal. 24 (2014), no. 1, 118–143. 10.1007/s12220-012-9330-4Search in Google Scholar

[34] N. Lam, G. Lu and H. Tang, Sharp subcritical Moser–Trudinger inequalities on Heisenberg groups and subelliptic PDEs, Nonlinear Anal. 95 (2014), 77–92. 10.1016/j.na.2013.08.031Search in Google Scholar

[35] N. Lam, G. Lu and H. Tang, Sharp affine and improved Moser–Trudinger–Adams type inequalities on unbounded domains in the spirit of Lions, J. Geom. Anal. 27 (2017), no. 1, 300–334. 10.1007/s12220-016-9682-2Search in Google Scholar

[36] N. Lam, G. Lu and L. Zhang, Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities, Rev. Mat. Iberoam. 33 (2017), no. 4, 1219–1246. 10.4171/RMI/969Search in Google Scholar

[37] J. Li, G. Lu and Q. Yang, Fourier analysis and optimal Hardy–Adams inequalities on hyperbolic spaces of any even dimension, Adv. Math. 333 (2018), 350–385. 10.1016/j.aim.2018.05.035Search in Google Scholar

[38] J. Li, G. Lu and M. Zhu, Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions, Calc. Var. Partial Differential Equations 57 (2018), no. 3, Article ID 84. 10.1007/s00526-018-1352-8Search in Google Scholar

[39] Y. Li, Moser–Trudinger inequality on compact Riemannian manifolds of dimension two, J. Partial Differential Equations 14 (2001), no. 2, 163–192. Search in Google Scholar

[40] Y. Li and C. B. Ndiaye, Extremal functions for Moser–Trudinger type inequality on compact closed 4-manifolds, J. Geom. Anal. 17 (2007), no. 4, 669–699. 10.1007/BF02937433Search in Google Scholar

[41] Y. Li and B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in ℝn, Indiana Univ. Math. J. 57 (2008), no. 1, 451–480. 10.1512/iumj.2008.57.3137Search in Google Scholar

[42] Y. X. Li, Remarks on the extremal functions for the Moser–Trudinger inequality, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 545–550. 10.1007/s10114-005-0568-7Search in Google Scholar

[43] E. H. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Search in Google Scholar

[44] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoam. 1 (1985), no. 1, 145–201. 10.4171/RMI/6Search in Google Scholar

[45] G. Lu and H. Tang, Best constants for Moser–Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), no. 4, 1035–1052. 10.1515/ans-2013-0415Search in Google Scholar

[46] G. Lu and H. Tang, Sharp singular Trudinger–Moser inequalities in Lorentz–Sobolev spaces, Adv. Nonlinear Stud. 16 (2016), no. 3, 581–601. 10.1515/ans-2015-5046Search in Google Scholar

[47] G. Lu, H. Tang and M. Zhu, Best constants for Adams’ inequalities with the exact growth condition in ℝn, Adv. Nonlinear Stud. 15 (2015), no. 4, 763–788. 10.1515/ans-2015-0402Search in Google Scholar

[48] G. Lu and Q. Yang, Sharp Hardy–Adams inequalities for bi-Laplacian on hyperbolic space of dimension four, Adv. Math. 319 (2017), 567–598. 10.1016/j.aim.2017.08.014Search in Google Scholar

[49] G. Lu and Y. Yang, Adams’ inequalities for bi-Laplacian and extremal functions in dimension four, Adv. Math. 220 (2009), no. 4, 1135–1170. 10.1016/j.aim.2008.10.011Search in Google Scholar

[50] L. Martinazzi, Fractional Adams–Moser–Trudinger type inequalities, Nonlinear Anal. 127 (2015), 263–278. 10.1016/j.na.2015.06.034Search in Google Scholar

[51] N. Masmoudi and F. Sani, Trudinger–Moser inequalities with the exact growth condition in ℝN and applications, Comm. Partial Differential Equations 40 (2015), no. 8, 1408–1440. 10.1080/03605302.2015.1026775Search in Google Scholar

[52] S. Mosconi and M. Squassina, Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Mathematical Analysis Seminar 2016, Bruno Pini Math. Anal. Semin. 2016, University Bologna, Bologna (2016), 147–164. Search in Google Scholar

[53] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1970/71), 1077–1092. 10.1512/iumj.1971.20.20101Search in Google Scholar

[54] T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl. 155 (1991), no. 2, 531–540. 10.1016/0022-247X(91)90017-TSearch in Google Scholar

[55] T. Ozawa, On critical cases of Sobolev’s inequalities, J. Funct. Anal. 127 (1995), no. 2, 259–269. 10.1006/jfan.1995.1012Search in Google Scholar

[56] E. Parini and B. Ruf, On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 2, 315–319. 10.4171/RLM/808Search in Google Scholar

[57] J. Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), no. 1, 279–317. 10.5802/aif.232Search in Google Scholar

[58] K. Perera and M. Squassina, Bifurcation results for problems with fractional Trudinger–Moser nonlinearity, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 561–576. 10.3934/dcdss.2018031Search in Google Scholar

[59] S. I. Pohozaev, The Sobolev embedding in the special case p⁢l=n, Proceeding of the Technical Scientific Conference on Advances of Scientific Research 1964-1965. Mathematics Sections, Moscov. Eberget. Inst., Moscow (1965), 158–170. Search in Google Scholar

[60] B. Ruf, A sharp Trudinger–Moser type inequality for unbounded domains in ℝ2, J. Funct. Anal. 219 (2005), no. 2, 340–367. 10.1016/j.jfa.2004.06.013Search in Google Scholar

[61] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473–483. 10.1512/iumj.1968.17.17028Search in Google Scholar

[62] J. Xiao and Z. Zhai, Fractional Sobolev, Moser–Trudinger Morrey–Sobolev inequalities under Lorentz norms, J. Math. Sci. (N.Y.) 166 (2010), no. 3, 357–376. 10.1007/s10958-010-9872-6Search in Google Scholar

[63] Y. Yang, Adams type inequalities and related elliptic partial differential equations in dimension four, J. Differential Equations 252 (2012), no. 3, 2266–2295. 10.1016/j.jde.2011.08.027Search in Google Scholar

[64] C. Zhang and L. Chen, Concentration-Compactness Principle of Singular Trudinger–Moser Inequalities in ℝn and n-Laplace Equations, Adv. Nonlinear Stud. 18 (2018), no. 3, 567–585. 10.1515/ans-2017-6041Search in Google Scholar

[65] C. Zhang, J. Li and L. Chen, Ground state solutions of poly-harmonic equation with potentials of positive lower bound, preprint. Search in Google Scholar

Received: 2018-04-08

Revised: 2018-07-09

Accepted: 2018-07-11

Published Online: 2018-09-10

Published in Print: 2019-02-01

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

Articles in the same Issue

https://doi.org/10.1515/ans-2018-2026

Keywords for this article

Trudinger–Moser inequality; exponential growth; fractional Sobolev–Slobodeckij spaces, fractional Laplacians

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