Mountain pass solutions for the fractional Berestycki-Lions problem

TL;DR

This paper establishes the existence of multiple solutions, including least energy and positive radially symmetric solutions, for a nonlinear fractional Laplacian equation using mountain pass techniques under Berestycki-Lions conditions.

Contribution

It extends the mountain pass method to fractional Laplacian problems with Berestycki-Lions type nonlinearities, including the zero mass case.

Findings

Existence of least energy solutions.

Existence of infinitely many solutions.

Existence of positive radially symmetric solutions.

Abstract

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb{R}^{N}, where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian and $g: \mathbb{R} \rightarrow \mathbb{R}$ is an odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in \cite{HIT}. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above problem when $g$ satisfies suitable growth conditions which make our problem fall in the so called "zero mass" case.