Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients

Abstract

Let $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_{\lambda }$} -Δu = c_λ(x) u + μ|\nabla u|^2 + h(x)\,, \quad u \in H_0^1(Ω) \cap L^{\infty}(Ω)\,, \end{equation} where $c_{\lambda }$ and $h$ belong to $L^q(\Omega )$ for some $q>N/2$, $\mu$ belongs to $\mathbb{R}\setminus \{0\}$ and we write $c_{\lambda }$ under the form $c_{\lambda }:=\lambda c_{+}-c_{-}$ with $c_{+}\gneqq 0$, $c_{-}\ge 0$, $c_{+}{c}_{-}\equiv 0$ and $\lambda \in \mathbb{R}$. Here $c_{\lambda }$ and $h$ are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence of a unique solution to \eqref{Plambda-Abstract-ch3} when $\lambda \le 0$. Then, assuming that $(P_0)$ has a solution, we prove existence and multiplicity results for $\lambda >0$. Our proofs rely on a suitable change of variable of type $v=F(u)$ and the combination of variational methods with lower and upper solution techniques.