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

Abstract

Let $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_{\lambda}$} -\Delta u = c_{\lambda}(x) u + \mu |\nabla u|^2 + h(x)\,, \quad u \in H_0^1(\Omega) \cap L^{\infty}(\Omega)\,, \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 $c_{+} \gneqq 0$, $c_{-} \geq 0$, $c_{+} c_{-} \equiv 0$ 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 $\lambda \leq 0$. 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.