Multiple solutions for an indefinite elliptic problem with critical growth in the gradient

Abstract

We consider the problem $(P)$, $$-\Delta u=c(x)u+\mu |\nabla u{|}^2+f(x),u\in H_0^1(\Omega )\cap L^{\infty }(\Omega ),$$ where $\Omega$ is a bounded domain of ${\mathbb{R}}^N$, $N\ge 3$, $\mu >0,c\in \mathcal{C}(\overline{\Omega }),$ and $f\in L^q(\Omega )$ for some $q>\frac{N}{2}$ with $f\gneqq 0.$ Here $c$ is allowed to change sign. We show that when $c^{+}\not\equiv 0$ and $c^{+}+\mu f$ is suitably small, this problem has at least two positive solutions. This result contrasts with the case $c\le 0$, where uniqueness holds. To show this multiplicity result we first transform $(P)$ into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so called slow growth at infinity. The key point is to show that the Palais-Smale condition holds.