Ground state solutions for fractional Schrödinger equations with critical exponents

Abstract

This paper investigates the following critical fractional Schrödinger equation

$$\begin{aligned}&(-\Delta )^s u+ V(x) u +\alpha [(-\Delta )^s|u|^{2\alpha }] |u|^{2\alpha -2} u \\&\quad =\lambda |u|^{p-2} u +|u|^{2^{*}_{s}-2} u , \ x \in {\mathbb {R}}^N, \end{aligned}$$

where $N >2s$ with $s \in (0,1),$ $2<p<2^{*}_{s} ,$ $(-\Delta )^s$ denotes the fractional Laplacian of order s,  $\lambda$ is a positive parameter, and $2^{*}_{s}=\frac{2N}{N-2s}$ is the fractional critical exponent, V(x) is a positive continuous potential function satisfying some conditions. We obtain the existence of ground state solutions for the fractional Schrödinger equation at critical growth.