Estimates for $p$-Laplace type equation in a limit case

TL;DR

This paper investigates the existence, uniqueness, and continuity of solutions to a $p$-Laplacian type equation within Orlicz-Zygmund spaces, focusing on how parameter choices affect these properties.

Contribution

It provides new conditions on the parameter $eta$ in Orlicz-Zygmund spaces that ensure well-posedness of the $p$-Laplacian type problem.

Findings

Established existence and uniqueness criteria based on $eta$

Identified parameter ranges for solution continuity

Extended analysis to Orlicz-Zygmund space setting

Abstract

We study some Dirichlet problem for a $p$--Laplacian type operator in the setting of Orlicz--Zygmund space $L^q\log^{-\alpha}L(\Omega,\mathbb R^N)$, $q >1$ and $\alpha>0$. More precisely, our aim is to establish which assuptions on the parameter $\alpha>0$ lead to existence, uniqueness of the solution and continuity of the associated nonlinear operator.