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

We consider the problem $(P)$, $-Δu =c(x)u+μ|\nabla u|^2 +f(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω),$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $N \geq 3$, $μ>0, \, c \in \mathcal{C}(\overlineΩ),$ and $f \in L^q(Ω)$ 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 $c^+ +μf$ 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.

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