Multiplicity of solutions for a scalar field equation involving a fractional $p$-Laplacian with general nonlinearity

TL;DR

This paper proves the existence of infinitely many radially symmetric solutions and a ground state for a fractional p-Laplacian equation with general nonlinearities, covering critical exponential and polynomial growth cases.

Contribution

It establishes the multiplicity of solutions for a fractional p-Laplacian problem with general nonlinearities, including critical growth scenarios, which is a novel extension.

Findings

Infinitely many radially symmetric solutions exist.

A ground state solution is also proven to exist.

Results cover both critical exponential and polynomial growth cases.

Abstract

We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-\Delta_p)^s u=g(u) \ \ \textrm{ in } \ \ \mathbb{R}^N, \ \ u\in W^{s,p}(\mathbb{R}^N),$$ where $s\in (0,1)$, $2 \leq p < \infty$, $sp \leq N $, $2 \leq N \in \mathbb{N}$ and $(-\Delta_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.