Multiple solutions for fractional p-Laplace equation with concave-convex nonlinearities

In this paper, we investigate the existence of solutions for the fractional p-Laplace equation (−Δ)psu+V(x)|u|p−2u=h1(x)|u|q−2u+h2(x)|u|r−2uin RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (-\Delta)_{p}^{s}u+V(x) \vert u \vert ^{p-2}u=h_{1}(x) \vert u \vert ^{q-2}u+h_{2}(x) \vert u \vert ^{r-2}u \quad \mbox{in } \mathbb{R}^{N}, $$\end{document} where N>sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N>sp$\end{document}, 0<s<1<p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< s<1<p$\end{document}, 1<q<p<r<ps∗:=NpN−sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< q< p< r< p_{s}^{*}:=\frac{Np}{N-sp}$\end{document}, and the potential function V(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)>0$\end{document} and h1(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{1}(x)$\end{document}, h2(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{2}(x)$\end{document} are allowed to change sign in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{N}$\end{document}. By using variant fountain theorem, we prove that the above equation admits infinitely many small and high energy solutions.


Introduction and main result
In this paper, we consider the existence and multiplicity of solutions for the following elliptic problem: (-) s p u + V (x)|u| p-2 u = h 1 (x)|u| q-2 u + h 2 (x)|u| r-2 u in R N , (1.1) where (-) s p is the fractional p-Laplacian operator with 0 < s < 1 < p and sp < N , 1 < q < p < r < p * s := Np N-sp and potential function V (x) > 0, h 1 and h 2 are sign-changing weight functions. The exact assumptions will be given below.
The fractional p-Laplacian operator (-) s p is defined along a function u ∈ C ∞ 0 (R N ) as follows: |u(x)u(y)| p-2 (u(x)u(y)) |x -y| N+ps dy (1.2) for x ∈ R N , where B ε (x) = {y ∈ R N : |x -y| < ε}, see [1][2][3] and the references therein. In the last years, since the nonlinear equations involving fractional powers of the Laplacian played an increasingly important role in physics, probability, and finance, a great at-tention has been focused on the study of problems involving the fractional Laplacian. So there has been a lot of interest in the study of the fractional Schrödinger equation (1.3) where the nonlinearity f satisfies some general conditions, see for instance [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein. More recently, Xiang et al. [3] investigated the fractional p-Laplacian equation where λ > 0, p < r < min{q, p s * }, and a(x) and b(x) are related by the condition a(a/b) (r-p)(q-r) ∈ L N/ps (R N ). By using a direct variational method and the mountain pass theorem, the authors proved the existence of two nontrivial weak solutions of (1.4) for λ > λ * (λ * > 0 is a given constant).
There are also a lot of works about problem (1.3) with concave-convex nonlinearities. Goyal and Sreenadh in [20] considered the following p-fractional Laplace equation: where Ω is a bounded domain in R N with Lipschitz boundary, p ≥ 2, n > pα, 1 < q < p < r < p * s , λ > 0, h and b are sign-changing smooth functions. They proved that there exists λ 0 > 0 such that problem (1.5) has at least two nonnegative solutions for all λ ∈ (0, λ 0 ).
In [21], the authors considered the problem as follows: where M(t) = a + bt θ-1 , θ > 1, a, b ≥ 0, a + b > 0, λ > 0, and 1 < q < p < θ p < r < p * s . The functions h 1 (x), h 2 (x), and h(x) may change sign on R N . Note that problem (1.6) is reduced to the fractional p-Laplacian equation with a = 1 and b = 0. Under some suitable conditions, they obtained the existence of two nontrivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle.
In this paper, we are interested in the multiplicity of solutions to equation (1.1) and find sufficient conditions to guarantee the existence of infinitely many solutions.
The present article is motivated by the papers [6,7], as well as by the fact that we do not find in the literature any paper dealing with the existence of infinitely many solutions to equation (1.1). The main tools employed in our works are the variant fountain theorems established in [22]. They are effective tools for studying the existence of infinitely many large or small energy solutions. Moreover, the results about the existence of solutions in the above papers are all related to the number λ. So we are also interested in whether the restriction on λ can be taken out.
Throughout this paper, we make the following assumptions: Our main results in this paper are stated as follows. The functional I that appeared in Theorems 1.1-1.2 is the energy functional for problem (1.1), which will be given below.
The rest of the paper is organized as follows. In the forthcoming section, we set up the variational framework for (1.1) and state the variant fountain theorems that will be used later. In Sect. 3, we study problem (1.1) and give the proof of Theorem 1.1. Section 4 is devoted to the proof of Theorem 1.2.

Preliminaries
First of all, we give some basic results of fractional Sobolev space that will be used in the next sections. Let 0 < s < 1 < p be real numbers. The fractional Sobolev space W s,p (R N ) is defined as follows: ) is a uniformly convex Banach space and the embedding Moreover, the embedding is locally compact whenever 1 ≤ t < p * s , see [1] for details. For our problem (1.1), consider the subspace X ⊂ W s,p (R N ) given by Then X is a separable Banach space with the norm As usual, for t ≥ 1, we let By the embedding X → L t (R N ), we know that there exists a constant S t > 0 such that Let I(u) : X → R be the energy functional associated with (1.1) defined by Using (2.6), it follows from conditions (H 1 ), (H 2 ), and (H 3 ) (or (H 3 )) that the functional I is well defined and I ∈ C 1 (X, R) with for any v ∈ X. It is standard to verify that the weak solutions of (1.1) correspond to the critical points of I. Finally, we give the variant fountain theorems. Let X be a Banach space with the norm · and Consider the following C 1 -functional I λ : X → R defined by The following two variant fountain theorems were established in [22].

Theorem 2.2
Assume that the functional I λ defined above satisfies (A 1 ) I λ maps bounded sets into bounded sets uniformly for λ ∈ [1,2], and and Then there exist λ n → 1, u n (λ n ) ∈ Y n such that In particular, if {u(λ n )} has a convergent subsequence for every k, then I 1 has infinitely many nontrivial critical points {u k } ∈ X \ {0} satisfying I 1 (u k ) → 0as k → ∞.

Proof of Theorem 1.1
In this section, we use Theorem 2.2 to complete the proof. For the notation in Theorem 2.2, we define the functional A, B, and I λ on our working space X by and Since X is a separable and reflexive Banach space, then there exist and

(3.4)
Let X i = Re i , and Y k and Z k be defined as (2.10).
In the proof of our results, we need the following limits.

Lemma 3.1 Assume (H 1 ), (H 2 ), and (H 3 ) (or (H 3 )), and let
Proof It is clear that 0 < α k+1 ≤ α k , so α k → α 0 ≥ 0 as k → ∞. For every k ∈ N + , taking u k ∈ Z k such that u k = 1 and As X is reflexive, {u k } has a weakly convergent subsequence, without loss of generality, suppose u k u weakly in X. Then, for every i ∈ N + , we have which implies that u = 0 and u k 0 weakly in X. Let (H 2 ) hold, then for any given small ε > 0, we may find R > 0 big enough such that where B R = {x ∈ R N : |x| < r}, B c R = R N \ B R , and S qτ is the embedding constant in (2.6).
If p * s p * s -q < τ ≤ p p-q , since the embedding X → L qτ (B R ) is compact, then u k → 0 in L qτ (B R ) and hence there exists K 1 > 0 such that for k > K 1 . Using (3.8) and (3.9), for all k > K 1 , we get Then {|h 1 ||u k | q } is uniformly integrable, and the Vitali convergence theorem implies So, for k big enough, (3.10) still holds. Then, from (3.6) and (3.10), we conclude that α k → 0 as k → ∞. Assume (H 3 ). Since μ ∈ [ p * s p * s -r , ∞) implies p < rμ ≤ p * s , arguing as in the above proof, one has β k → 0 as k → ∞.
Similarly, if assumption (H 3 ) holds, it follows Since h 2 (x) ∈ L ∞ (R N ) and h 2 (x) → 0 as |x| → ∞, β k → 0 can be obtained in a similar way, and we complete the proof.
In order to apply Theorem 2.2, we give the following lemma first.

Lemma 3.2 Let (H 1 ), (H 2 ), and (H 3 ) (or (H 3 )) hold. Then there exist two sequences
Proof From Lemma 3.1 we see that, for every u ∈ Z k , it holds (3.14) Thus Fix K 2 > 0 big enough such that 1 r β r k < 1 2p for k > K 2 , then for u ∈ Z k and u < 1, we have If we choose ρ k = (8pα q k /q) 1 p-q , then ρ k → 0 + as k → ∞ and for any u ∈ Z k with u = ρ k , we get that This inequality implies that In addition, for all λ ∈ [1, 2], k > K 2 and u ∈ Z k with u ≤ ρ k , we have Obviously, 1,2], by the equivalence of any norm in a finite dimensional space, we can derive where C 1 ≥ 0, C 2 > 0. Notice q < p < r, so we can choose r k > 0 small enough satisfying The proof is completed.  (2.6) we have Since p > 1, p > q > 0, the above inequality implies that {u(λ n )} is bounded in X. Using the Hölder inequality, we have (3.28) Denote P n := I λ n (u n )(u nu) and Q n := Then the fact I λ n (u(λ n )) → 0 shows that P n → 0 as n → ∞. Moreover, {u n } is a bounded sequence and u n u in X, which imply Q n → 0. Equations (3.29) and (3.30) show that, for large n, By using the standard inequalities (see [23]) given by and where C p is a positive constant and ·, · denotes the inner product in R N , we can easily deduce that u nu → 0 as n → ∞. Now, from Theorem 2.2, we see that I = I 1 possesses infinitely many nontrivial critical points u k for k ∈ Z + satisfying I(u k ) → 0as k → ∞. Therefore, problem (1.1) possesses infinitely many nontrivial solutions, the proof of Theorem 1.1 is completed.

Proof of Theorem 1.2
For the notation in Theorem 2.2, we define the functional A, B and I λ on our working space X by and I λ (u) = A(u) -λB(u) Proof Similar to the beginning of the proof of Lemma 3.2, by (3.14) we have Fix K 3 > 0 big enough such that 2 q α q k < 1 2p for k > K 3 , then for u ∈ Z k and u > 1, we have If we choose r k = ( r 8pβ r k ) 1 r-p , then r k → ∞ as k → ∞, and for any u ∈ Z k with u = r k , we get that For all u ∈ Y k , λ ∈ [1,2], by the equivalence of any norm in a finite dimensional space, we can derive where C 1 ≥ 0, C 2 > 0. Notice q < p < r, then I λ (u) → -∞ as u → ∞. So we can choose ρ k > r k big enough such that a k (λ) = max u∈Y k , u =ρ k I λ (u) < 0.