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

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.