Ground state solutions for a fractional Schr\"odinger equation with critical growth

TL;DR

This paper establishes the existence of ground state solutions for a fractional Schrödinger equation with critical growth, using variational methods and extending results to non-autonomous potentials.

Contribution

It introduces new existence results for fractional Schrödinger equations with critical growth, including non-autonomous potentials, using a Jeanjean-Tanaka approach.

Findings

Existence of ground state solutions for constant potential.

Extension to non-autonomous potential cases.

Novel application of Jeanjean-Tanaka argument.

Abstract

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation \begin{align*} (-\Delta)^{s} u+V(x)u= f(u) \mbox{ in } \mathbb{R}^{N}, \end{align*} where $s\in (0,1)$, $N> 2s$, $(-\Delta)^{s}$ is the fractional Laplacian, $V: \mathbb{R}^{N}\rightarrow \mathbb{R}$ is a bounded potential satisfying suitable assumptions, and $f\in C^{1, \beta}(\mathbb{R}, \mathbb{R})$ has critical growth. We first analyze the case $V$ constant, and then we develop a Jeanjean-Tanaka argument \cite{JT} to deal with the non autonomous case. As far as we know, all results presented here are new.