Entire Solutions to Nonlinear Scalar Field Equations with Indefinite Linear Part

Abstract

Abstract We consider the stationary semilinear Schrödinger equation −Δu + a(x)u = f (x, u), u ∈ H1(ℝN), where a and f are continuous functions converging to some limits a∞ > 0 and f∞ = f∞(u) as |x| → ∞. In the indefinite setting where the Schrödinger operator −Δ + 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 −Δ+a.