Existence and multiplicity of solutions for the fractional $p$-Laplacian Choquard logarithmic equation involving a nonlinearity with exponential critical and subcritical growth

TL;DR

This paper proves the existence and multiplicity of solutions for a fractional p-Laplacian Choquard logarithmic equation with nonlinearities exhibiting exponential critical and subcritical growth, using variational methods.

Contribution

It introduces new existence and multiplicity results for a fractional p-Laplacian Choquard equation with exponential growth nonlinearities, including ground states and infinitely many solutions.

Findings

Existence of a nontrivial solution at the mountain pass level.

Existence of a nontrivial ground state solution.

Infinitely many solutions when the nonlinearity has subcritical growth.

Abstract

In the present work we obtain the existence and multiplicity of nontrivial solutions for the Choquard logarithmic equation $(-\Delta)_{p}^{s}u + |u|^{p-2}u + (\ln|\cdot|\ast |u|^{p})|u|^{p-2}u = f(u) \textrm{ \ in \ } \mathbb{R}^N $ , where $ N=sp $, $ s\in (0, 1) $, $ p>2 $, $ a>0 $, $ \lambda >0 $ and $f: \mathbb{R}\rightarrow \mathbb{R} $ a continuous nonlinearity with exponential critical and subcritical growth. We guarantee the existence of a nontrivial solution at the mountain pass level and a nontrivial ground state solution under critical and subcritical growth. Morever, when $ f $ has subcritical growth we prove the existence of infinitely many solutions, via genus theory.