Existence of nontrivial solutions for fractional Schrödinger equations with electromagnetic fields and critical or supercritical nonlinearity

Abstract

In this paper, we study the following fractional Schrödinger equation with electromagnetic fields and critical or supercritical nonlinearity: ${(-\mathrm{\Delta })}_A^{s}u+V(x)u=f(x,|u{|}^2)u+\lambda |u{|}^{p-2}u,x\in {\mathbb{R}}^N,$ where ${(-\mathrm{\Delta })}_A^s$ is the fractional magnetic operator with $0<s<1$, $N>2s$, $\lambda >0$, $2_s^{\ast }=\frac{2N}{N-2s}$, $p\ge 2_s^{\ast }$, f is a subcritical nonlinearity, and $V\in C({\mathbb{R}}^N,\mathbb{R})$ and $A\in C({\mathbb{R}}^N,{\mathbb{R}}^N)$ are the electric and magnetic potentials, respectively. Under some suitable conditions, by variational methods we prove that the equation has a nontrivial solution for small $\lambda >0$. Our main contribution is related to the fact that we are able to deal with the case $p>2_s^{\ast }$.