Multiple high energy solutions for fractional Schrödinger equation with critical growth

Abstract

In this work, we study the following fractional Schrödinger equation with critical growth

$$\begin{aligned} (-\Delta )^{s}u+V(x)u=|u|^{2_{s}^{*}-2}u, \ \ x\in {\mathbb {R}}^N, \end{aligned}$$

where $s\in (0,1)$, $N>4s$, $(-\Delta )^{s}$ is the fractional Laplacian operator of order s, potential function $V(x):{\mathbb {R}}^N\rightarrow {\mathbb {R}}$, $2_{s}^{*}=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent. In virtue of a barycenter function, quantitative deformation lemma and Brouwer degree theory, we prove the existence and multiplicity of positive high energy solutions. Our results extend and improve the recent work on the existence of high energy solutions for fractional Schrödinger equation by Correia and Figueiredo (Calc Var Part Differ Equ 58:63, 2019).