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

TL;DR

This paper proves the existence of ground state solutions for coupled Schrödinger systems involving the square root of the Laplacian with exponential critical growth, using variational methods and Nehari manifold minimization.

Contribution

It establishes the existence of ground states for coupled fractional Schrödinger systems with Trudinger-Moser critical nonlinearities and periodic or asymptotically periodic potentials.

Findings

Existence of nonnegative ground state solutions under periodic potentials.

Extension to asymptotically periodic potentials.

Application of variational methods and Nehari manifold techniques.

Abstract

In this paper we prove the existence of a nonnegative ground state solution to the following class of coupled systems involving Schr\"{o}dinger equations with square root of the Laplacian $$ \left\{ \begin{array}{lr} (-\Delta)^{1/2}u+V_{1}(x)u=f_{1}(u)+\lambda(x)v, & x\in\mathbb{R}, (-\Delta)^{1/2}v+V_{2}(x)v=f_{2}(v)+\lambda(x)u, & x\in\mathbb{R}, \end{array} \right. $$ 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.