Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian

TL;DR

This paper establishes the existence of an infinite sequence of eigenvalues for the logarithmic p-Laplacian and applies these results to prove new solutions for nonlocal Dirichlet problems with nonlinearities.

Contribution

It introduces a novel approach to eigenvalue problems for the logarithmic p-Laplacian using cohomological index theory and applies this to nonlocal Dirichlet problems.

Findings

Existence of an unbounded sequence of minimax eigenvalues.

New existence results for nonlocal Dirichlet problems.

Application of p-logarithmic Sobolev inequality to nonlinear problems.

Abstract

In this work, we show the existence of an unbounded sequence of minimax eigenvalues for the logarithmic $p$-Laplacian via the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz. As an application of these minimax eigenvalues and $p$-logarithmic Sobolev inequality proved in [4], we prove new existence results for nonlocal Dirichlet problems involving logarithmic $p$-Laplacian and nonlinearities with $p$-superlinear and subcritical growth at infinity.