## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 40, Number 3 (2017), 562-576.

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

Sitong Chen, XianHua Tang, and Jianxiong Li

#### 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.

#### Note

This work is partially supported by the National Natural Science Foundation of China (No: 11571370).

#### Article information

**Source**

Kodai Math. J., Volume 40, Number 3 (2017), 562-576.

**Dates**

Received: 24 October 2016

Revised: 11 January 2017

First available in Project Euclid: 31 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1509415233

**Digital Object Identifier**

doi:10.2996/kmj/1509415233

**Mathematical Reviews number (MathSciNet)**

MR3718498

**Zentralblatt MATH identifier**

1383.35077

**Subjects**

Primary: 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations 58E50: Applications

**Keywords**

linearly coupled Schrödinger system Nehari-type ground state solutions Sobolev critical exponent

#### Citation

Chen, Sitong; Tang, XianHua; Li, Jianxiong. Ground state solutions for asymptotically periodic linearly coupled Schrödinger equations with critical exponent. Kodai Math. J. 40 (2017), no. 3, 562--576. doi:10.2996/kmj/1509415233. https://projecteuclid.org/euclid.kmj/1509415233