Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction

Abstract:

We consider the following elliptic system: \[ \left \{\begin {array}{l} \varepsilon ^2 \Delta u-\lambda _1 u+\mu _1 u^3 + \beta u v^2 =0 \ \mbox {in} \ \Omega , \varepsilon ^2 \Delta v-\lambda _2 v+\mu _2 v^3 + \beta u^2 v =0 \ \mbox {in} \ \Omega , u,v >0 \ \mbox {in} \ \Omega , \ u=v=0 \ \mbox {on} \ \partial \Omega , \end {array} \right . \] 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 $\mathbb {R}^N$.