Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients
Résumé
Let Ω ⊂ RN, N≥ 2 , be a smooth bounded domain. We consider the boundary value problem [Figure not available: see fulltext.]where cλ and h belong to Lq(Ω) for some q> N/ 2 , μ belongs to R\ { 0 } and we write cλ under the form cλ: = λc+- c- with c+≩ 0 , c-≥ 0 , c+c-≡ 0 and λ∈ R. Here cλ 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 (Pλ) when λ≤ 0. Then, assuming that (P) has a solution, we prove existence and multiplicity results for λ> 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.