An obstacle problem with gradient term and asymptotically linear reaction

Abstract

We will consider the following obstacle problem

with the condition that $u\ge \psi$ a.e in $\Omega .$ Under suitable condition relating $g$,$h$ and $q$, we show the existence of a solution for all $f\in L^1(\Omega )$ .

The main feature is, assuming that $g(x,s)$ is asymptotically linear as $|s|\to \pm \infty$ and independently of the values of $\underset{s\to \pm \infty }{\lim }\frac{g(x,s)}{s},$ to obtain a solution for all $\lambda >0$ and $f\in L^1(\Omega )$. In this sense we could say that the first order term break down any resonant effect.