Multiple Solutions for a Class of Fractional Equations

In this paper we study a class of fractional Laplace equations with asymptoti-cally linear right-hand side. The existence results of three nontrivial solutions under the resonance and non-resonance conditions are established by using the minimax method and Morse theory.


Introduction
In this article, we are interested in the following non-local fractional equations: where s ∈ (0, 1) is a fixed parameter, Ω is a bounded domain in R N with smooth boundary ∂Ω, N > 2s and (−∆) s is the fractional Laplace operator.
In recent years, many papers are devoted to the study of non-local fractional Laplacian with superlinear and subcritical or critical growth (see [2,4,13,14,16,17] and references therein).Particularly, in [8], Fiscella et al. studied equations (1.1) with asymptotically linear right-hand side and obtained some existence results by using saddle point theorem; in [9], Iannizzotto et al. also studied fractional p-Laplacian equations with asymptotically p-linear and obtained two nontrivial solutions by the use of the mountain pass theorem.
There are many interesting problems in the standard framework of the Laplacian (or higher order Laplacian), widely studied in the literature.A natural question is whether or not the existence results of multiple solutions obtained in the classical context can be extended to the non-local framework of the fractional Laplacian operator.Chang et al. [7] showed the existence of three nontrivial solutions for asymptotically linear Dirichlet problem via the mountain pass Corresponding author.Email: prc211@163.comtheorem and Morse theory.In [11] Qian et al. did similar work for fourth-order asymptotically linear elliptic problem.
Motivated by their work, we study the following non-local problem with homogeneous Dirichlet boundary conditions investigated by Servadei et al. [15] and the related works [12,14]: where L k is the integro-differential operator defined as follows: with the kernel K : R N \0 → (0, +∞) such that For narrative convenience, in this paper, we only consider the particular case of problem (1.2), i.e., we let K be given by the singular kernel K(x) = |x| −(N+2s) which leads to the fractional Laplace operator −(−∆) s , which, up to normalization factors, may be defined as Obviously, the corresponding fractional equation in the above model (1.2) changes problem (1.1).In fact, our methods and results in this paper also adapt for the general problem (1.2).
Let f (x, 0) = 0 and F(x, t) = t 0 f (x, s)ds.Moreover, suppose that the non-linearity f satisfy the following conditions: = l uniformly for x ∈ Ω, where f 0 and l are constants; Now, we give our main results.
• are the eigenvalues of (−∆) s with homogeneous Dirichlet boundary data and φ 1 (x) > 0 be the eigenfunction corresponding to λ 1 .
In view of the condition ( f 3 ), problem (1.1) is called asymptotically linear at both zero and infinity, which means that usual Ambrosetti-Rabinowitz condition (see [1]) is not satisfied.This will bring some difficulty if the mountain pass theorem is used to seek nontrivial solutions of problem (1.1).For the standard Laplacian Dirichlet problem, Zhou [18] have overcome it by using some monotonicity condition.Novelties of our this paper are as following.
We consider multiple solutions of problem (1.1) in the cases of resonance and nonresonance by using the mountain pass theorem and Morse theory.First, we use the truncated technique and the mountain pass theorem to obtain a positive solution and a negative solution of problem (1.1) under our more general conditions ( f 1 ), ( f 2 ) and ( f 3 ) with respect to the conditions (H 1 ) and (H 3 ) in [18].In the course of proving the existence of a positive solution and a negative solution, the monotonicity condition (H 2 ) of [18] on the nonlinear term f is not necessary, this point is very important because we can directly prove existence of positive solution and negative solution by using Rabinowitz's mountain pass theorem.That is, the proof of our compact condition is more simple than that in [18].Furthermore, we can obtain a nontrivial solution when the nonlinear term f is resonant or non-resonant at the infinity by using Morse theory.
The paper is organized as follows.In Section 2, we present some necessary preliminary knowledge about the working space.In Section 3, we prove some lemmas in order to prove our main results.In Section 4, we give the proofs for our main results.

Preliminaries
In this section, we give some preliminary results which will be used in the sequel.We briefly recall the related definitions and notes for functional space X 0 introduced in [15].
The functional space X denotes the linear space of Lebesgue measurable functions from R N to R such that the restriction to Ω of any function g in X belongs to L 2 (Ω) and the map ( Also, we denote by X 0 the following linear subspace of X X 0 := {g ∈ X : g = 0 a.e. in R N \Ω}. Note that X and X 0 are non-empty, since C 2 0 (Ω) ⊆ X 0 by [15].Moreover, the space X is endowed with the norm defined as where We equip X 0 with the following norm which is equivalent to the usual one defined in (2.1) (see [14]).It is easy to see that (X 0 , • X 0 ) is a Hilbert space with scalar product Denote by H s (Ω) the usual fractional Sobolev space with respect to the Gagliardo norm . (2.4) Now, we give a basic fact which will be used later.
Lemma 2.1 ([14]).The embedding j : Next, we state some propositions for the operator (−∆) s .Let the sequence of the eigenvalues of (−∆) s (see [8]) and φ k be the k-th eigenfunction corresponding to the eigenvalues λ k .Moreover, we will set ).The following inequality holds true for all u ∈ H k and k ∈ N.
Next, we recall some definitions for compactness condition and a version of the mountain pass theorem.Definition 2.4.Let (X 0 , • X 0 ) be a real Banach space with its dual space (X * 0 , • X * 0 ) and J ∈ C 1 (X 0 , R).For c ∈ R, we say that J satisfies the (PS) c condition if for any sequence {x n } ⊂ X 0 with J (x n ) → c, DJ (x n ) → 0 in X * 0 , there is a subsequence {x n k } such that {x n k } converges strongly in X 0 .Also, we say that J satisfy the (C) c condition stated in [5] if for any sequence {x n } ⊂ X 0 with

Some lemmas
First, we observe that problem (1.1) has a variational structure, indeed it is the Euler-Lagrange equation of the functional J : X 0 → R defined as follows: It is well know that the functional J is Fréchet differentiable in X 0 and for any ϕ ∈ X 0 Thus, critical points of J are solutions of problem (1.1).
Consider the following problem where Define a functional J + : X 0 → R by where follows that v n X 0 is bounded.Without loss of generality, we assume that v n v in X 0 , then v n → v in L 2 (Ω).Hence, v n → v a.e. in Ω. Dividing both sides of (3.1) by |u n | L 2 (Ω) , we get , ∀ϕ ∈ X 0 . (3.2) Then for a.e.x ∈ Ω, we deduce that f + (x,u n ) → lv + as n → ∞, where v + = max{v, 0}.In fact, when v(x) > 0, by ( f 3 ) we have When v(x) < 0, we have 2) and the Lebesgue dominated convergence theorem, we arrive at From the strong maximum principle (see [9]) we deduce that v > 0. Choosing ϕ = φ 1 in (3.3), we obtain l This is a contradiction.
Proof.For u ∈ P k+1 , by ( f 3 ), for any ε > 0, there exists So, from Proposition 2.3 we have and It follows from (3.7) that for every M > 0, there exists a constant T > 0 such that For τ > 0, we have In the case of λ k < l < λ k+1 , we needn't the assumption ( f 4 ) and it is easy to see that the conclusion also holds .(3.13), it follows that v n X 0 is bounded.Without loss of generality, we assume v n v in X 0 , then v n → v in L 2 (Ω).Hence, v n → v a.e. in Ω. Dividing both sides of (3.13) by |u n | L 2 (Ω) , we get , ∀ϕ ∈ X 0 . (3.14) Then for a.e.x ∈ Ω, we have f (x,u n ) → lv as n → ∞.In fact, if v(x) = 0, by ( f 3 ), we have ≤ c|v n |, by (3.14) and the Lebesgue dominated convergence theorem, we arrive at Obviously v = 0, hence, l is an eigenvalue of (−∆) s .This contradicts our assumption.
Proof.Suppose u n ∈ X 0 satisfies In view of ( f 3 ), it suffices to prove that u n is bounded in X 0 .Similar to the proof of Lemma 3.4, we have which contradicts (3.17).Hence u n is bounded.
It is well known that critical groups and Morse theory are the main tools in solving elliptic partial differential equation.Let us recall some results which will be used later.We refer the readers to the book [6] for more information on Morse theory.
Let X be a Hilbert space and J ∈ C 1 (X, R) be a functional satisfying the (PS) condition or (C) condition, and H q (X, Y) be the q-th singular relative homology group with integer coefficients.Let u 0 be an isolated critical point of J with J (u 0 ) = c, c ∈ R, and U be a neighborhood of u 0 .The group q ∈ Z is said to be the q-th critical group of J at u 0 , where J c = {u ∈ X : J (u) ≤ c}.
Let K := {u ∈ X : J (u) = 0} be the set of critical points of J and a < inf J (K), the critical groups of J at infinity are formally defined by (see [3]) The following result comes from [3,6] and will be used to prove the results in this paper.

Proposition 3.6 ([3]
). Assume that X = V ⊕ W, J is bounded from below on W and J (u) → −∞ as u → ∞ with u ∈ V. Then