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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N\ge 2$, be a smooth bounded domain. We consider the boundary value problem \begin{equation} \label{Plambda-Abstract-ch3} \tag{$P_{\lambda }$} -\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 $L^q(\mathrm{\Omega })$ for some $q

N/2$,$\mu$belongsto$$\mathbb{R}$ \setminus {0}$andwewrite$c_{\lambda}$undertheform$c_{\lambda}:= \lambda c_{+} - c_{-}$with$c_{+} \gneqq 0$,$c_{-} \geq 0$,$c_{+} c_{-} \equiv 0$and$\lambda \in $\mathbb{R}$$. 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 $\lambda \leq 0$\mathrm{.}Then,assumingthat$(P_0)$ 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.