Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case

Abstract

We study the following nonlinear Schrödinger system which is related to Bose–Einstein condensate:

$$\begin{aligned} \left\{ {\begin{array}{lll} -\Delta u +\lambda _1 u = \mu _1 u^{2^*-1}+\beta u^{\frac{2^*}{2}-1}v^{\frac{2^*}{2}}, \quad x\in \Omega ,\\ -\Delta v +\lambda _2 v =\mu _2 v^{2^*-1}+\beta v^{\frac{2^*}{2}-1} u^{\frac{2^*}{2}}, \quad x\in \Omega ,\\ u\ge 0, v\ge 0 \,\,\hbox {in } \Omega ,\quad u=v=0 \,\,\hbox {on } \partial \Omega . \end{array}}\right. \end{aligned}$$

Here $\Omega \subset \mathbb R^N$ is a smooth bounded domain, $2^*:=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\lambda _1(\Omega )<\lambda _1,\lambda _2<0$, $\mu _1,\mu _2>0$ and $\beta \ne 0$, where $\lambda _1(\Omega )$ is the first eigenvalue of $-\Delta$ with the Dirichlet boundary condition. When $\beta =0$, 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 the higher dimensional case $N\ge 5$. It is interesting that we can prove the existence of a positive least energy solution $(u_\beta , v_\beta )$ for any $\beta \ne 0$ (which can not hold in the special case $N=4$). We also study the limit behavior of $(u_\beta , v_\beta )$ as $\beta \rightarrow -\infty$ and phase separation is expected. In particular, $u_\beta -v_\beta$ will converge to sign-changing solutions of the Brezis–Nirenberg problem, provided $N\ge 6$. In case $\lambda _1=\lambda _2$, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case $N=4$.