Existence of solutions of scalar field equations with fractional operator

TL;DR

This paper proves the existence of multiple solutions, including least energy and positive solutions, for scalar field equations involving a fractional operator, using variational methods and mountain pass characterization.

Contribution

It establishes new existence results for solutions of fractional scalar field equations with Berestycki-Lions type nonlinearities, including least energy and infinitely many solutions.

Findings

Existence of least energy solutions proved.

Infinitely many solutions established.

Positive solutions under certain conditions shown.

Abstract

In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation $(1-\Delta)^\alpha u = f(u)$ in $\mathbf{R}^N$ where $0<\alpha<1$, $N \geq 2$ and $f(s)$ is a Berestycki-Lions type nonlinearity. The characterization of the least energy by the mountain pass value is also considered and the existence of optimal path is shown. Finally, exploiting these results, the existence of positive solution for the equation $(1-\Delta)^\alpha u = f(x,u)$ in $\mathbf{R}^N$ is established under suitable conditions on $f(x,s)$.