Existence and nonexistence of positive solutions of indefinite elliptic problems in RN

Our purpose is to find positive solutions u is an element of D-1,D-2 (RN) of the semilinear elliptic problem - Deltau - lambdaV(x)u = h(x)u(p-1) for 2 < p. The functions V and h may have an indefinite sign and the linearized operator need not have a first (principal) eigenvalue, e.g. we allow V equivalent to 1. We give precise existence and nonexistence criteria, which depend on lambda and on the growth of h(-) and h(+)/V+. Existence theorems are obtained by constrained minimization. The mountain pass theorem leads to a second solution.