Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

TL;DR

This paper introduces new inequalities and embeddings for the logarithmic p-Laplacian, proves existence of solutions for related boundary value problems, and analyzes asymptotic behavior of fractional p-Laplacian solutions as the fractional parameter approaches zero.

Contribution

It develops a new p-Logarithmic Sobolev inequality, establishes optimal embeddings, and proves existence results for critical growth problems involving the logarithmic p-Laplacian, extending previous nonlinear analyses.

Findings

Established a new p-Logarithmic Sobolev inequality

Proved existence of nontrivial weak solutions for boundary value problems

Analyzed asymptotic behavior of fractional p-Laplacian solutions as s approaches 0

Abstract

In this paper, we derive a new $p$-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic $p$-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic $p$-Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional $p$-Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic $p$-Laplacian as the fractional parameter $s \to 0^+$. The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.