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

Abstract

We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-{\Delta }_p{)}^{s}u=g(u)\text{ in }{\mathbb{R}}^N,u\in W^{s,p}({\mathbb{R}}^N),$$ where $s\in (0,1)$, $2\le p<\infty$, $sp\le N$, $2\le N\in \mathbb{N}$ 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.

16 pages