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

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.