Some existence results for a quasilinear problem with source term in Zygmund-space

Abstract

In this paper we study the existence of solution to the problem

where $\Omega$ is an open bounded set of ${\mathbb{R}}^2$, $A(x)$ a coercive matrix with coefficients in $L^{\infty }(\Omega )$, $H(x,s,\xi )$ a Carathéodory function satisfying, for some $\gamma >0$,

Here $f$ belongs to $L^1(\log L^1)(\Omega )$ and $a_0\ge 0$ to $L^q(\Omega )$, $q>1$. For $f$ and $a_0$ sufficiently small, we prove the existence of at least one solution $u$ of this problem which is such that $e^{{\delta }_0|u|}-1$ belongs to $H_0^1(\Omega )$ for some ${\delta }_0\ge \gamma$ and satisfies an a priori estimate.