Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

In this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.


Introduction
In this paper, we are interested in the existence of nontrivial solutions for the following fractional quasi-linear problem: where P.V. refers to the principle value, see [1] for details. In recent years, there has been growing interest in the study of fractional elliptic equations. Concerning the existence result for this kind of equations, some well-known results for classical Laplace operators have been extended to the nonlocal fractional setting, and there are a lot of works on the quasi-linear problem where the nonlinearity f satisfies some general growth conditions, see [2][3][4][5][6][7][8][9][10][11][12]. For instance, in [2,4] the authors studied the fractional p-eigenvalue problems. In [5] the authors studied the local behavior of fractional p-minimizers. In [6] the authors studied problem (1. 3) under different growth assumptions on the reaction term and obtained various existence results by Morse theory, while in [7] the authors studied problem (1.3) in an unbounded domain with weight, and symmetry results were given by authors in [8]. Moreover, by the variant fountain theorem the authors in [9] studied problem (1. 3) in R N and obtained infinitely many solutions, while similar results were obtained by authors in [3], Typically, when f (x, u) = |u| p * s -2 u + λ|u| p-2 u, problem (1.3) turns into the Brezis-Nirenberg problem (1. 4) which has been studied by the authors in [10][11][12]. In [10], the authors proved multiplicity results of problem (1.4) by cohomological index and abstract critical theorem, while in [11], the authors obtained nontrivial solutions of problem (1.4) by an abstract linking theorem. In [12], replacing |u| p-2 u with a subcritical nonlinearity g(x, u), the authors proved the existence of one weak solution of problem (1.4) provided λ is sufficiently small, and multiplicity result was established if the perturbation term g vanishes at the origin. Moreover, Kirchhoff type equations involving fractional p-Laplacian and critical nonlinearities were studied by authors in [13][14][15] by using variational methods. Inspired by the above papers, we tend to investigate the existence and multiplicity result of problem (1.1). From our analysis, it is clear that under different assumptions of the growth condition on nonlinearities near infinity and origin, the existence and multiplicity results are quite different. We consider the Banach space X = W s,p 0 ( ), where the fractional Sobolev space W s,p 0 ( ) = {u ∈ W s,p ( )|u = 0 ∈ R N \ } is defined as follows: equipped with the norm Our approach to study problem (1.1) is variational, including the mountain pass theorem and the critical point theorems of G. Bonanno and R. Kajikiya. Generally, we check the geometric structure of the functional and prove the compactness results of the functional to meet the conditions of the critical point theorems. Due to the presence of critical nonlinearity, the energy functional no longer satisfies global compactness conditions but on certain ranges, thus we apply different variational theorems for existence results. We assume that the nonlinearities f , g, h ∈ C(R, R) and satisfy the following assumptions: where r is defined as r := sup W s,p ( ) Our main results read as follows.
The present paper is organized as follows: in Sect. 2 we prove the existence of mountain pass solution, in Sect. 3 we prove the existence of three solutions, and in Sect. 4 we give infinitely many solutions for the critical case.

Existence of mountain pass solution
It is well known that the solution of problem (1.1) is a critical point of the functional I : X → R is defined by and satisfies I (u), ϕ = 0, i.e., We first check the mountain pass geometry of I.
Proof Suppose that {u n } n∈N is a Palais-Smale sequence of I, i.e., there exists C > 0 such that then we have Thus {u n } n∈N is bounded in X. Up to a subsequence, still denoted by {u n } n∈N , there exists u 0 ∈ X satisfying u n u 0 in W s,p ( ), u n → u 0 in L p ( ), u n (x) → u 0 (x) a.e. on . (2.7) From (f 1 ), (f 2 ), (g 1 ),we have by the Lebesgue convergence theorem f (x, u n )(u nu 0 ) dx → 0, as n → ∞, g(x, u n )(u nu 0 ) dx → 0, as n → ∞.
Note that I (u n ), u nu 0 = |u n (x)u n (y)| p-2 (u n (x)u n (y)) |x -y| N+sp u n (x)u n (y)u 0 (x) + u 0 (y) and I (u n ), u nu 0 → 0 as n → ∞, thus we have Combined with weak convergence of u n u 0 in W s,p ( ), we have u n → u 0 inW 1,p ( ) as n → ∞, thus I satisfies the Palais-Smale condition.
Proof of Theorem 1.1 In view of Lemma 2.1 and Lemma 2.2, Theorem 1.1 follows from the mountain pass theorem [16].

Existence of three solutions
In this section we consider multiplicity results of problem (1.1) when h = 0, We first recall the following theorem by G. Bonnano.

Lemma 3.1 ([17]
) Let X be a separable and reflexive real Banach space, and let φ, ψ : X → R be two continuously Gâteaux differentiable functionals. Assume that φ is sequentially weakly lower semicontinuous and even, that ψ is sequentially weakly continuous and odd, and that, for some b > 0 and for each λ ∈ [-b, b], the functional ψ + λφ satisfies the Palais-Smale condition and Finally, assume that there exists k > 0 such that admits at least three solutions in X whose norms are less than σ .
Consider the functional Proof of Theorem 1.2 It suffices to check that I satisfies all the assumptions in Lemma 3.1. By ( f 2 ), given > 0, we have thus the functional ψ(u) is continuously Gâteaux differentiable and weakly sequentially continuous. From (g 1 ) we know that φ(u) is weakly sequentially continuous. By (3.5) and (g 1 ), we derive Since p > r, taking sufficiently small, we have Following a similar argument in Lemma 2.2, I satisfies the Palais-Smale condition. By (f 1 ), we have Hence there exists k > 0 such that inf |φ(u)|<k ψ(u) = 0.
Due to (f 3 ), for any u ∈ W s,p ( ), let t → ∞, there holds Then we have Thus completes the proof.

Existence of infinitely many solutions
In this section, we consider the critical case for problem (3.1), i.e., Under assumption (g 3 ) of Theorem 1.3, it is easy to see that the Euler-Lagrange functional of (4.1) is even, thus we tend to use the symmetric mountain pass theorem of Kajikiya for existence of infinitely many solutions. Due to the presence of critical term, we first prove the local compactness result. If a = 0, the proof is complete. Otherwise, a ≥ S N/ps λ (ps-N)/ps . Combined with (4.2), (4.3), and (4.4), as n → ∞ we derive provided is sufficiently small. Then, given any M > 0, there exists λ * such that for all λ ∈ (0, λ * ), thus completes the proof. Now we introduce Krasnoselski's genus. Let E be a real Banach space. A closed subset A of E is called symmetric if x ∈ A implies -x ∈ A. Denote by the family of all symmetric closed sets of E. The genus of A is defined to be the smallest integer n if there is an odd map ϕ ∈ C(A, R n \{0}). If n does not exist, then γ (A) = ∞. Typically, γ (φ) = 0. x -A ≤ δ} of A such that γ (N δ (A)) = γ (A). We then give the symmetric mountain pass lemma due to Kajikiya [18].

Lemma 4.3
Let E be an infinite dimensional Banach space and I ∈ C 1 (E, R) be a functional satisfying the conditions below: (C 1 ) I(u) is even, bounded from below, I(0) = 0, and I(u) satisfies the local Palais-Smale condition, i.e., for some d * > 0, in the case when every sequence {u n } n∈R N in E satisfying I(u n ) → d < d * and I (u n ) → 0 in E * has a convergent subsequence; (C 2 ) For each n ∈ N, there exists A n ∈ n such that sup u∈A n I(u) < 0. Then either (i) or (ii) below holds. (i) There exists a sequence {u n } n∈R N such that I (u n ) = 0, I(u n ) = 0, and {u n } n∈R N converges to 0.