Bose fluids and positive solutions to weakly coupled systems with critical growth in dimension two

TL;DR

This paper establishes the existence of positive ground state solutions for a coupled Bose-Einstein system with critical exponential nonlinearities in two dimensions using variational methods.

Contribution

It proves the existence of positive solutions for a weakly coupled system with critical exponential growth in 2D, extending variational techniques to this setting.

Findings

Existence of positive ground state solutions in 2D for Bose-Einstein systems.

Solutions are obtained under various signs and magnitudes of the coupling parameter.

The system's nonlinearities are of critical exponential type, similar to higher-dimensional power nonlinearities.

Abstract

We prove, using variational methods, the existence in dimension two of positive vector ground states solutions for the Bose-Einstein type systems \begin{equation} \begin{cases} -\Delta u+\lambda_1u=\mu_1u(e^{u^2}-1)+\beta v\left(e^{uv}-1\right) \text{ in } \Omega, &\\ -\Delta v+\lambda_2v=\mu_2v(e^{v^2}-1)+\beta u\left(e^{uv}-1\right)\text{ in } \Omega, &\\ u,v\in H^1_0(\Omega) \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain, $\lambda_1,\lambda_2>-\Lambda_1$ (the first eigenvalue of $(-\Delta,H^1_0(\Omega))$, $\mu_1,\mu_2>0$ and $\beta$ is either positive (small or large) or negative (small). The nonlinear interaction between two Bose fluids is assumed to be of critical exponential type in the sense of J. Moser. For `small' solutions the system is asymptotically equivalent to the corresponding one in higher dimensions with power-like nonlinearities.