Entire solutions to nonlinear scalar field equations with indefinite linear part

Abstract

We consider the stationary semilinear Schrödinger equation $-\Delta u+a(x)u=f(x,u)$, $u\in H^1({\mathbb{R}}^N)$, 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 Schrödinger 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 $-\Delta +a$.

31 pages