Nonlocal Dirichlet problems involving the Logarithmic $p$-Laplacian
Rakesh Arora, Hichem Hajaiej, Kanishka Perera
TL;DR
This work develops a variational framework for nonlocal Dirichlet problems involving the logarithmic $p$-Laplacian $L_{\Delta_p}$, introducing an unbounded sequence of minimax eigenvalues $(\lambda_k)$ via the $\mathbb{Z}_2$-cohomological index. It leverages a newly established $p$-logarithmic Sobolev inequality and Orlicz-type embeddings to obtain compactness and critical point results for nonlinearities with $p$-superlinear and subcritical growth. Existence results are proved for parameter ranges below the first eigenvalue and between consecutive eigenvalues, using linking arguments and Cerami-type compactness. The work advances the theory of the logarithmic $p$-Laplacian in the nonlinear, nonlocal setting and suggests directions for further spectral and variational analysis.
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.