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

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We study the following nonlinear Schrodinger 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 $$N=4$$ .