Fractional magnetic Schr\"{o}dinger equations with potential vanishing at infinity and supercritical exponents

TL;DR

This paper investigates fractional magnetic Schr"odinger equations with potentials that vanish at infinity and supercritical nonlinearities, employing variational and penalization methods to establish existence results.

Contribution

It introduces a novel approach combining variational methods, penalization, and $L^{}$-estimates to handle supercritical growth and vanishing potentials in fractional magnetic Schr"odinger equations.

Findings

Established existence of solutions under supercritical conditions.

Handled potentials vanishing at infinity.

Developed new variational and penalization techniques.

Abstract

This paper focuses on the following class of fractional magnetic Schr\"{o}dinger equations \begin{equation*} (-\Delta)_{A}^{s}u+V(x)u=g(\vert u\vert^{2})u+\lambda\vert u\vert^{q-2}u, \quad \mbox{in } \mathbb{R}^{N}, \end{equation*} where $(-\Delta)_{A}^{s}$ is the fractional magnetic Laplacian, $A :\mathbb{R}^N \rightarrow \mathbb{R}^N$ is the magnetic potential, $s\in (0,1)$, $N>2s$, $\lambda \geq0$ is a parameter, $V:\mathbb{R}^N \rightarrow \mathbb{R}$ is a potential function that may decay to zero at infinity and $g: \mathbb{R}_{+} \rightarrow \mathbb{R}$ is a continuous function with subcritical growth. We deal with supercritical case $q\geq 2^*_s:=2N/(N-2s)$. Our approach is based on variational methods combined with penalization technique and $L^{\infty}$-estimates.