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

We consider the problem $(P)$, $$-\mathrm{\Delta }u=c(x)u+\mu \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+f(x),u\in H_0^1(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega }),$$ where $\mathrm{\Omega }$ 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)$forsome$ q>\frac{N}{2}$with$ f\gneqq 0. $Here$c$ is allowed to change sign. We show that when $c^{+}\equiv ̸0$ 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.