Nonlinear Schrödinger Equations with Sign-Changing Potential

Abstract

Abstract In this paper, we show that the following system −Δu + λV(x)u = g(x, v), −Δv + λV(x)v = f (x, u), x ∈ ℝN (0.1) possesses at least one non-trivial solution pair (u, v) for λ > 0 large enough, where f (x, t), g(x, t) are continuous functions on ℝN × ℝ and super-linear at t = 0 as well as at t = +∞, V(x) is allowed to be sign-changing. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition.