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

Abstract: We consider the problem $(P)$, $$ -\Delta u =c(x)u+\mu|\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega),$$ where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $\mu>0, \, c \in \mathcal{C}(\overline{\Omega}),$ 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 $c \leq 0$, 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.