## Differential and Integral Equations

- Differential Integral Equations
- Volume 31, Number 5/6 (2018), 403-434.

### Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth

João Marcos do Ó and José Carlos de Albuquerque

#### Abstract

In this paper, we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schrödinger equations with square root of the Laplacian $$ \begin{cases} (-\Delta)^{ \frac 12 } u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R},\\ (-\Delta)^{ \frac 12 } v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{cases} $$ where the nonlinearities $f_{1}(s)$ and $f_{2}(s)$ have exponential critical growth of the Trudinger-Moser type, the potentials $V_{1}(x)$ and $V_{2}(x)$ are nonnegative and periodic. Moreover, we assume that there exists $\delta\in (0,1)$ such that $\lambda(x)\leq\delta\sqrt{V_{1}(x)V_{2}(x)}$. We are also concerned with the existence of ground states when the potentials are asymptotically periodic. Our approach is variational and based on minimization technique over the Nehari manifold.

#### Article information

**Source**

Differential Integral Equations Volume 31, Number 5/6 (2018), 403-434.

**Dates**

First available in Project Euclid: 23 January 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1516676436

**Subjects**

Primary: 35J50: Variational methods for elliptic systems 35B33: Critical exponents 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

do Ó, João Marcos; de Albuquerque, José Carlos. Coupled elliptic systems involving the square root of the Laplacian and Trudinger-Moser critical growth. Differential Integral Equations 31 (2018), no. 5/6, 403--434. https://projecteuclid.org/euclid.die/1516676436