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

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} -Δu+λ_1u=μ_1u(e^{u^2}-1)+βv\left(e^{uv}-1\right) \text{ in } Ω, &\\ -Δv+λ_2v=μ_2v(e^{v^2}-1)+βu\left(e^{uv}-1\right)\text{ in } Ω, &\\ u,v\in H^1_0(Ω) \end{cases} \end{equation} where $\Omega$ is a bounded smooth domain, ${\lambda }_1,{\lambda }_2>-{\Lambda }_1$ (the first eigenvalue of $(-\Delta ,H_0^1(\Omega ))$, ${\mu }_1,{\mu }_2>0$ 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.