Ground state solutions for a fractional Schrödinger equation with critical growth

Abstract

In this paper we investigate the existence of nontrivial ground state solutions for the following fractional scalar field equation \begin{align*} (-Δ)^{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, $V:{\mathbb{R}}^N\to \mathbb{R}$ 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.

References in corpus (3)

• Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
• Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials
• Ground state solutions for fractional scalar field equations under a general critical nonlinearity

Cited by in corpus (3)

• Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian
• Sign-changing solutions for a class of zero mass nonlocal Schrödinger equations
• Bound state solutions for the supercritical fractional Schrödinger equation