  Our purpose is to find positive solutions $u \in D^{1,2}(\rz^N)$ of the
semilinear elliptic problem $-\laplace u - \lambda V(x) u = h(x) u^{p-1}$ 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 $\lambda$ 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.
