Fractional magnetic Schrödinger equations with potential vanishing at infinity and supercritical exponents

Abstract

This paper focuses on the following class of fractional magnetic Schrödinger equations \begin{equation*} (-Δ)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+λ\vert u\vert^{q-2}u, \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $(-\Delta {)}_A^s$ is the fractional magnetic Laplacian, $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ is the magnetic potential, $s\in (0,1)$, $N>2s$, $\lambda \ge 0$ is a parameter, $V:{\mathbb{R}}^N\to \mathbb{R}$ is a potential function that may decay to zero at infinity and $g:{\mathbb{R}}_{+}\to \mathbb{R}$ is a continuous function with subcritical growth. We deal with supercritical case $q\ge 2_s^{\ast }:=2N/(N-2s)$. Our approach is based on variational methods combined with penalization technique and $L^{\infty }$-estimates.