Entire solutions to nonlinear scalar field equations with indefinite linear part

TL;DR

This paper proves the existence of solutions to a class of nonlinear Schrödinger equations with indefinite linear parts, using topological and reduction methods, even when the minimal energy level is not attained.

Contribution

It introduces a novel combination of reduction and topological methods to establish solutions in indefinite settings with weak asymptotic conditions.

Findings

Existence of solutions under weak asymptotic estimates.

Ground-state solutions under more restrictive assumptions.

Allowing zero in the spectrum of the linear operator.

Abstract

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 $-\Delta + a$.