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

Let Omega subset of RN, N >= 2, be a smooth bounded domain. We consider the boundary value problem where c lambda and h belong to Lq(Omega) for some q>N/2, mu belongs to R\textbackslash{0} and we write c lambda under the form c lambda:=lambda c+-c- with c+?0, c->= 0, c+c-equivalent to 0 and lambda is an element of 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 <= 0. Then, assuming that (P0) 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.