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

Abstract:

We consider the problem \begin{equation} -\Delta u =c(x)u+\mu |\nabla u|^2 +f(x), \quad u \in H^1_0(\Omega ) \cap L^{\infty }(\Omega ), \tag {(P)} \end{equation} where $\Omega$ is a bounded domain of $\mathbb {R}^N$, $N \geq 3$, $\mu >0$ and $c, f \in L^q(\Omega ) \text { for some}$ $q>\frac {N}{2}$ with $f\gneqq 0.$ Here $c$ is allowed to change sign and we assume that $c^+ \not \equiv 0$. We show that when $c^+$ and $\mu f$ are 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.