The limiting profile of solutions for semilinear elliptic systems with a shrinking self-focusing core

Abstract

In this paper, we consider the semilinear elliptic equation systems

$$\left\{ {\matrix{{ - \Delta u + u = \alpha {Q_n}(x)|u{|^{\alpha - 2}}|v{|^\beta }u\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr { - \Delta v + v = \beta Q(x)|u{|^\alpha }|v{|^{\beta - 2}}v\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfill \cr } } \right.$$

where $N\geqslant 3,\,\,\alpha ,\,\,\beta > 1,\,\alpha + \beta < {2^ * },\,{2^ * } = {{2N} \over {N - 2}}$ and Q_n are bounded given functions whose self-focusing cores {x ∈ ℍ^NQ_n(x) > 0} shrink to a set with finitely many points as n → ∞. Motivated by the work of Fang and Wang [13], we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points, and we build the localized concentrated bound state solutions for the above equation systems.