Existence of N on-trivial S olutions to a C lass of F ractional p - Laplacian E quations of Schrödinger-T ype with a C ombined N onlinearity

. In this paper, we consider the following nonlinear fractional p -Laplacian equation of Schr¨odinger-type


Introduction
The purpose of this paper is to examine the existence of non-trivial solutions to the following nonlinear Schrödinger equation where (−∆) s p is the fractional p-Laplacian operator, 0 < s < 1 < η < p, N ≥ 2, V (x) is a real continuous function on R N .Furthermore, V , k and f satisfying the appropriate assumptions that will be formulated later.
In recent years, great attention has been concentrated on the study of the nonlinear problems involving the fractional Laplacian because it has played an increasingly important role in physics, probability, and finance.So, authors are interested in studying fractional Schrödinger equations like below where the nonlinearity f satisfies some general conditions, for instance see ( [4]- [15]) and the references therein.
In [16], authors studied the following Schrödinger-Poisson system where 1 < q < 2, λ > 0 is a positive parameter and f (x, u) is linearly bounded in u at infinity.
In this work, the authors establish the existence and multiplicity of solutions under suitable assumptions on V ,k and f .
In [17], Kexue Li studied the following nonlinear fractional Schrödinger-Poisson system and, by using the perturbation method and mountain pass theorem, got the existence of nontrivial solutions.
Goyal and Sreenadh [18], investigated the following fractional p-Laplace equation: where Ω is a bounded domain in R N with Lipschitz boundary, p ≥ 2, n > p, 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 ).Chen et al. [19] considered the following fractional p-Laplacian equation: where N > sp, 0 < s < 1 < q < p < r < p * s and the potential function V (x) and h 1 (x), h 2 (x) are allowed to change the sign in R N .By using variant fountain theorem, they prove that the problem (1.6) admits infinitely many small and high energy solutions.
The study of the existence and multiplicity of solutions was motivated by the above works for the problem (1.1) .The main tools employed in our work are Ekeland's variational principle and Mountain Pass theorem.
Before expressing our main results, we give the following assumptions: All over this paper, C > 0 is arbitrarily used to denote a suitable positive constant whose value may change from line to line.Also, we will use o(1) for a quantity that goes to zero, and As follow, the rest of the paper is organized.In section 2, we give some preliminary results concerning the variational structure of the system, and The proof of Theorems 1.1-1.2 and Corollary 1.3 is given in section 3, .

Variational setting and preliminaries
First, for any p ∈ (1, ∞) and s ∈ (0, 1), we define fractional p-Laplacian operator and fractional Sobolev space that will be used in the next section.
The fractional p-Laplacian operator (−∆) s p is defined as follows: ) and references therein.The fractional sobolev space W s,p (R N ) is defined as follows: (2.1) ) is a uniformly convex Banach space and the embedding W s,p (R N ) ֒→ In addition, the embedding is locally compact whenever t ∈ [p, p * s ).For details see [1] .For our problem, consider the subspace E ⊂ W s,p (R N ) given by Then E is a separable Banach space with the norm By the embedding E ֒→ L t (R N ), we know that there exists a constant C t > 0 such that Definition 2.1.u ∈ E is said to be a (weak) solution of (1.1), if for any v ∈ E, we have Let I : E → R be the energy functional associated (1.1) defined by It is easy to see that I is well defined in E and I ∈ C 1 (E, R), and for any v ∈ E. It is normal to verify that the weak solutions of (1.1) equal to the critical point of I.
Definition 2.2.We say a C 1 functional I satisfies Palais-Smale condition (PS) condition for abbreviate if any sequence {u n } ⊂ E such that contains a convergent subsequence, and such a sequence is called a Palais-Smale sequence (PS) sequence .
Proof.Let {u n } ⊂ E be a (PS) sequence of I, i.e., I(u n ) is bounded and I ′ (u n ) → 0. We will indicate that u n has a convergent subsequence in E. Then Since 1 < η < p, the above inequality implies that u n is bounded in E.
Up to a subsequence, we may assume that u n ⇀ u for some u ∈ E. Since E is compactly 3), we obtain Obviously, we have By (k1), Hölder inequality and Minkowski inequality, as n → ∞.
Furthermore, there exists K > 0 such that for |k| p p−η < K, we have max t∈(0,+∞) By ρ = t 0 , the proof will be finished.

□
Proof of Theorem 1.1.By Lemma 3.1 , we define Then we have ), so I is lower semicontinuous and bounded from below on Y ρ .Let Via (k2), we can select v ∈ C ∞ 0 (Ω).Since k(x) > 0 on Ω and 1 < η < p, it can be easily obtained that I(tv) < 0, for t small.Thus c 1 < 0. Now by (3.3), Lemma 2.3 and Ekeland's variational principle, c 1 can be obtained at some inner point u 1 ∈ Y ρ and u 1 is a critical point of I. ✷ Lemma 3.2.Assume that (f 3) hold.there exists e ∈ E with ∥e∥ > ρ such that I(e) < 0, where ρ is the same as in Lemma 3.1.
Proof.By (f 3), there exists a constant C > 0 such that For β > 0 and v ∈ C ∞ 0 (R N ), by 3.4 , we have Conflict of interest.The authors state no conflict of inerest.
Funding.No fund.
Data availability.The datasets generated and/or analysed during the current study is available from the correspond author on reasonable request.