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

Hamilton Bueno; Olimpio Miyagaki; Ailton Vieira

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arXiv · 2102.13436

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

Hamilton Bueno·Olimpio Miyagaki·Ailton Vieira

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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.

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Hamilton Bueno, Olimpio Miyagaki, Ailton Vieira. 2021-02-26. Multiplicity of solutions for a scalar field equation involving a fractional $p$-Laplacian with general nonlinearity. https://arxiv.org/abs/2102.13436

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