MULTIPLE SOLUTIONS FOR AN INDEFINITE ELLIPTIC PROBLEM WITH CRITICAL GROWTH IN THE GRADIENT

We consider the problem (P) -Delta u = c(x)u+mu vertical bar Delta u vertical bar(2) + f(x), u is an element of H-0(1)(Omega) boolean AND L-infinity(Omega), where Omega is a bounded domain of R-N, N >= 3, mu > 0 and c, f is an element of L-q(Omega) for some q > N/2 with f not greater than or equal to 0. Here c is allowed to change sign and we assume that c(+) not equivalent to 0. We show that when c(+) and mu f 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.