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 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 (and uniqueness) of solution to ($$P_{\lambda }$$) 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.