Zero mass case for a fractional Berestycki-Lions type problem

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 $r=|x|$.