Ground state solutions for a non-autonomous nonlinear Schrödinger-KdV system

Abstract

We study the Schrödinger-KdV system

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + {\lambda _1}(x)u = {u^3} + \beta uv,}&{u \in {H^1}({\mathbb{R}^N}),} \\ { - \Delta v + {\lambda _2}(x)v = \frac{1}{2}{v^2} + \frac{\beta }{2}{u^2},}&{v \in {H^1}({\mathbb{R}^N}),} \end{array}} \right.$$

where N = 1, 2, 3, λ_i(x) ∈ C(ℝ^N, ℝ), lim_{∣x∣→∞}λ_i(x) = λ_i(∞), and λ_i(x) ⩽ λ_i(∞), i = 1, 2, a.e. x ∈ ℝ^N. We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.