Existence of infinitely many solutions for a class of fractional Schr\"odinger equations in $\mathbb{R}^N$ with combined nonlinearities

TL;DR

This paper proves the existence of infinitely many solutions for a class of fractional Schrödinger equations with combined nonlinearities using variational methods, extending and improving recent results in the field.

Contribution

It establishes the existence of infinitely many high energy solutions for fractional Schrödinger equations with combined superlinear and sublinear nonlinearities, employing the Fountain theorem.

Findings

Existence of infinitely many solutions proven

Solutions have high energy levels

Results extend and improve previous work

Abstract

This paper is devoted to the following class of nonlinear fractional Schr\"odinger equations: \begin{equation*} (-\Delta)^{s} u + V(x)u = f(x,u) + \lambda g(x,u), \quad \text{in}\: \mathbb{R}^N, \end{equation*} where $s\in (0,1)$, $N>2s$, $(-\Delta)^{s}$ stands for the fractional Laplacian, $\lambda\in \mathbb{R}$ is a parameter, $V\in C(\mathbb{R}^N,R)$, $f(x,u)$ is superlinear and $g(x,u)$ is sublinear with respect to $u$, respectively. We prove the existence of infinitely many high energy solutions of the aforementioned equation by means of the Fountain theorem. Some recent results are extended and sharply improved.