Multiple Solutions for an Indefinite Elliptic Problem with Critical Growth in the Gradient

We consider the problem (Formula Presented) where Ω is a bounded domain of (Formula Presented) for some (Formula Presented) Here c is allowed to change sign and we assume that c+ ≡ 0. We show that when c+ and μƒ are suitably small this problem has at least two positive solutions. This result contrasts with the case c ≤ 0, where uniqueness holds. To show this multiplicity result we first transform (P) into a semilinear problem having a variational structure. Then we are led to the search of two critical points for a functional whose superquadratic part is indefinite in sign and has a so-called slow growth at infinity. The key point is to show that the Palais-Smale condition holds. © 2015 American Mathematical Society.