Existence and Nonexistence of Positive Solutions of Indefinite Elliptic Problems in $\rz^N$

Abstract

Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the semilinear elliptic problem $-\laplace u - λV(x) u = h(x) u^{p-1}$ for $2<p$. The functions $V$ and $h$ may have an indefinite sign and the linearized operator need not to have a first (principal) eigenvalue, e.g. we allow $V\equiv 1$. We give precise existence and nonexistence criteria, which depend on $λ$ 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.