  We consider the stationary semilinear Schr\"odinger equation $-\Delta u +
a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions
converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as
$|x|\to\infty$. In the indefinite setting where the Schr\"odinger operator
$-\Delta +a$ has negative eigenvalues, we combine a reduction method with a
topological argument to prove the existence of a solution of our problem under
weak one-sided asymptotic estimates. The minimal energy level need not be
attained in this case. In a second part of the paper, we prove the existence of
ground-state solutions under more restrictive assumptions on $a$ and $f$. We
stress that for some of our results we also allow zero to lie in the spectrum
of $-\Delta + a$.
