Ground state solutions for asymptotically periodic linearly coupled Schrödinger equations with critical exponent

Abstract

We consider the following system of coupled nonlinear Schrödinger equations $$\left\{ \begin{array}\\-\Delta u + a(x)u = \vert u \vert^{p-2}u + \lambda(x)v, \quad x \in \mathbf{R}^{N},\\ -\Delta v + b(x)v = \vert v \vert^{2^{*}-2}v + \lambda(x)u, \quad x \in \mathbf{R}^{N},\\ u, v \in H^{1} (\mathbf{R}^{N}), \end{array} \right.$$ where $N \geq 3, 2 \lt p \lt 2^{*}, 2^{*} = 2N / (N - 2)$ is the Sobolev critical exponent, $a, b, \lambda \in C(\mathbf{R}^{N}, \mathbf{R}) \cap L^{\infty} (\mathbf{R}^{N}, \mathbf{R})$ and $a(x)$, $b(x)$ and $\lambda(x)$ are asymptotically periodic, and can be sign-changing. By using a new technique, we prove the existence of a ground state of Nehari type solution for the above system under some mild assumptions on $a, b$ and $\lambda$. In particular, the common condition that $\vert\lambda(x)\vert \lt \sqrt{a(x)b(x)}$ for all $x \in \mathbf{R}^{N}$ is not required.