Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction

We consider the following elliptic system:

$$\left\{\begin{array}{l} \varepsilon^2 \Delta u-\lambda_1 u+\mu_1... ... \mbox{in} \Omega, u=v=0 \mbox{on} \partial\Omega, \end{array}\right.$$

where $\Omega\subset \mathbb{R}^N (N\leq 3)$ is a smooth and bounded domain, $\varepsilon>0$ is a small parameter, $\lambda_1, \lambda_2, \mu_1, \mu_2 >0$ are positive constants and $\beta \ne 0$ is a coupling constant. We show that there exists an interval $I=[a_0, b_0]$ and a sequence of numbers $0<\beta_1 <\beta_2 <...<\beta_n <...$ such that for any $\beta \in (0, +\infty) \backslash (I \cup \{ \beta_1,..., \beta_n, ...\})$, the above problem has a solution such that both $u$ and $v$ develop a spike layer at the innermost part of the domain. Central to our analysis is the nondegeneracy of radial solutions in $\mathbb{R}^N$.