Zero mass case for a fractional Berestycki-Lions type problem

TL;DR

This paper proves the existence of a positive, spherically symmetric, and decreasing solution to a fractional scalar field equation with zero mass term, using variational methods.

Contribution

It establishes the existence of solutions for a fractional scalar field problem with zero mass, a case not extensively studied before.

Findings

Existence of positive solutions with zero mass term.

Solutions are spherically symmetric and decreasing.

Application of variational methods to fractional problems.

Abstract

In this work we study the following fractional scalar field equation \begin{equation*}\label{P} \left\{ \begin{array}{ll} (-\Delta)^{s} u = g'(u) \mbox{ in } \mathbb{R}^{N} \\ u> 0 \end{array} \right. \end{equation*} where $N\geq 2$, $s\in (0,1)$, $(-\Delta)^{s}$ is the fractional Laplacian and the nonlinearity $g\in C^{2}(\mathbb{R})$ is such that $g''(0)=0$. By using variational methods, we prove the existence of a positive solution which is spherically symmetric and decreasing in $r=|x|$.