Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

TL;DR

This paper investigates positive least energy solutions and phase separation phenomena for coupled nonlinear Schrödinger equations with critical exponent in higher dimensions, extending previous results from the four-dimensional case.

Contribution

It proves the existence of positive least energy solutions for all nonzero coupling parameters in dimensions five and higher, and analyzes their limit behavior and phase separation as the coupling tends to negative infinity.

Findings

Existence of positive least energy solutions for any nonzero coupling in dimensions N≥5.

Asymptotic behavior of solutions as coupling parameter approaches -∞.

Phase separation and convergence to sign-changing solutions in higher dimensions.

Abstract

We study the following nonlinear Schr\"{o}dinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-\Delta u +\la_1 u = \mu_1 u^{2^\ast-1}+\beta u^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in \Omega, -\Delta v +\la_2 v =\mu_2 v^{2^\ast-1}+\beta v^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in $\om$},\quad u=v=0 \,\,\hbox{on $\partial\om$}.{cases}{displaymath} Here $\om\subset \R^N$ is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\la_1(\om)<\la_1,\la_2<0$, $\mu_1,\mu_2>0$ and $\beta\neq 0$, where $\lambda_1(\om)$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When $\bb=0$, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution $(u_\bb, v_\bb)$ {\it for any $\beta\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of $(u_\bb, v_\bb)$ as $\beta\to -\infty$ and phase separation is expected. In particular, $u_\bb-v_\bb$ will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case $\la_1=\la_2$, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.