Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term

Abstract

In this paper we consider the problem

where $\Omega$ is an open bounded set of ${\mathbb{R}}^N$, $N\ge 3$, $A(x)$ is a coercive matrix with coefficients in $L^{\infty }(\Omega )$, $H(x,s,\xi )$ is a Carathéodory function which satisfies for some $\gamma >0$

$f$ belongs to $L^{N/2}(\Omega )$ and $a_0\ge 0$ to $L^q(\Omega )$, $q>N/2$ . 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 ${\delta }_0\ge \gamma$. This solution satisfies some a priori estimate.