Existence of infinitely many high energy solutions for a class of fractional Schrödinger systems

In this paper, we investigate a class of nonlinear fractional Schrödinger systems {(−△)su+V(x)u=Fu(x,u,v),x∈RN,(−△)sv+V(x)v=Fv(x,u,v),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left \{ \textstyle\begin{array}{l@{\quad}l}(-\triangle)^{s} u +V(x)u=F_{u}(x,u,v),& x\in \mathbb{R}^{N}, \\(-\triangle)^{s} v +V(x)v=F_{v}(x,u,v),& x\in\mathbb{R}^{N}, \end{array}\displaystyle \right . $$\end{document} where s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\in(0, 1)$\end{document}, N>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$N>2$\end{document}. Under relaxed assumptions on V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$V(x)$\end{document} and F(x,u,v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(x, u, v)$\end{document}, we show the existence of infinitely many high energy solutions to the above fractional Schrödinger systems by a variant fountain theorem.


Introduction
In the work, we are concerned with the existence of infinitely many high energy solutions for the following fractional Schrödinger systems: where s ∈ (0, 1), N > 2 and F u (x, u, v), F v (x, u, v) ∈ C(R N × R × R, R). We assume that there exists F(x, u, v) ∈ C(R N × R × R, R) such that ∇F = (F u , F v ), where ∇F denotes the gradient of F in (u, v) ∈ R 2 . The operator (-) s is the fractional Laplacian of order s, which can be defined by the Fourier transform (-) s u = F -1 (|ξ | 2s F u). On the calculation and application of classical fractional differential equations and other aspects in mathematics, we refer the reader to [1][2][3][4][5] and the references therein. Over the past years, the fractional Laplacian (-) s (0 < s < 1), as one of the fundamental nonlocal operators, has increasingly had impact on a number of important fields in science, technology and other fields. As a result, much attention has been focused on the problem of fractional Laplacians. For instance, Teng [6] studied the following fractional Schrödinger equation: (1.2) and proved the existence of infinitely many nontrivial high or small energy solutions by variant fountain theorems. Du and Mao [7] obtained a sufficient condition for the existence of infinitely many nontrivial high energy solutions by variant fountain theorems for (1.2). Some interesting results can be found in [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and the references therein.
Recently, Di Nezza et al. [16] have proved that (-) s can be reduced to the standard Laplacian -as s → 1. If s = 1, Eq. (1.2) reduces to the classical Schrödinger equation With the aid of variational method and critical theorems, for the potential V (x) and nonlinearity f (x, u) under various conditions, the results of existence and multiplicity for Eq. (1.3) have been extensively investigated in the literature; see [24][25][26][27] and the references therein.
In recent decades, extensive attention of researchers has been devoted to the existence of solutions to the elliptic systems. Zhang and Zhang [28] considered some nonlinear elliptic systems and obtained the existence of weak solutions by using variational methods. Cao and Tang [29] considered the superlinear elliptic system. They presented the existence of infinitely many solutions which were characterized by the number of nodes of each component under some conditions on the nonlinear term. Pomponio [30] discussed the asymptotically linear cooperative elliptic system at resonance. They proved the existence of a non-zero solution and the existence of N -1 pairs of nontrivial solutions due to the difference between the Morse index at zero and the Morse index at infinity by a penalization technique. In recent years, many interesting results have been presented on the class of systems; see [31][32][33][34][35][36][37][38] and the references therein. However, the above literature is concerned with the problem of integer order Laplacian and there is little literature which discusses the Schrödinger systems with fractional order Laplacian. Based on the situation, we consider fractional Schrödinger systems (1.1). In this work, we will show the existence of infinitely many nontrivial high energy solutions by variant fountain theorems.
For convenience, we firstly present the following hypotheses: (f 5 ) There exist μ > 2 and c > 0 such that The paper is arranged as follows. In Sect. 2, we introduce preliminaries for proof of main results and variational setting. In Sect. 3, we present our main results and their proofs.

Preliminaries
Let us address a Hilbert space The space is endowed with the natural norm and with the inner product By means of the Fourier transform, the space H s (R N ) can be defined by For Eq. (1.2), the Hilbert space H is defined by with the following inner product and norm: and Then H × H is a Hilbert space with the following the inner product ·, · and norm for any Under the hypothesis (V 1 ), we have the following lemma.

Lemma 2.1 The Hilbert space H × H is compactly embedded in
holds for all (ϕ, ψ) ∈ H × H. A weak solution of the systems (1.1) corresponds to a critical point of the energy functional that is well defined. Furthermore, I is C 1 (H × H, R) functional with derivative given by Let H × H be Banach space with the norm (·, ·) and let {H j } be a sequence of subspace Now, we state two variant fountain theorems which come from the idea of Zou in [39].
In order to present our main work by the above variant fountain theorems, we define the functional A, B and Φ λ (u, v) on the space H × H by

Proofs of the main results
In this section, we will present the main results and their proofs.
Proof We argue by contradiction. Assume (u n , v n ) ∈ E such that meas x ∈ R N : (u n , v n ) ≥ 1 n (u n , v n ) < 1 n , for any n ∈ N.
This implies B(u, v) → +∞ as (u, v) → +∞ on any finite dimensional subspace of H × H. The proof is completed.