Positive Least Energy Solutions and Phase Separation for Coupled Schrödinger Equations with Critical Exponent

Abstract

In this paper we study the following coupled Schrödinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem:

$$\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.$$

Here, ${\Omega\subset \mathbb{R}^4}$ is a smooth bounded domain, ${-\lambda_1(\Omega) < \lambda_1,\lambda_2 < 0 , \mu_1,\mu_2 > 0 }$ and ${\beta\neq 0}$ , where ${\lambda_1(\Omega)}$ is the first eigenvalue of −Δ with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (that is, ${\frac{2N}{N-2}=4}$ when N = 4). We show that this critical system has a positive least energy solution for negative β, positive small β and positive large β. For the case in which ${\lambda_1=\lambda_2}$ , we obtain the uniqueness of positive least energy solutions. We also study the limit behavior of the least energy solutions in the repulsive case ${\beta\to -\infty}$ , and phase separation is expected. These seem to be the first results for this Schrödinger system in the critical case.