Entire solutions to nonlinear scalar field equations with indefinite linear part

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 $a_{\mathrm{\infty }}>0$ and $f_{\mathrm{\infty }}=f_{\mathrm{\infty }}(u)$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. In the indefinite setting where the Schr"odinger operator $-\mathrm{\Delta }+a$ 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 $-\mathrm{\Delta }+a$.