Concentration phenomenon of semiclassical states to reaction–diffusion systems

Abstract

In this paper, we consider concentration phenomenon of semiclassical states to the following 2M-component reaction–diffusion system in \(\mathbb {R}\times \mathbb {R}^N\), $$\begin{aligned} \left\{ \begin{aligned} \partial _t u&=\varepsilon ^2 \Delta _x u-u-V(x)v + \partial _v H(u, v),\\ \partial _t v&=-\varepsilon ^2 \Delta _x v+v + V(x)u - \partial _u H(u, v), \end{aligned} \right. \end{aligned}$$where \(M \ge 1\), \(N \ge 1\), \(\varepsilon >0\) is a small parameter, \(V \in C^1(\mathbb {R}^N, \, \mathbb {R})\), \(H \in C^1(\mathbb {R}^M \times \mathbb {R}^M, \, \mathbb {R})\) and \((u, v): \mathbb {R}\times \mathbb {R}^N \rightarrow \mathbb {R}^M \times \mathbb {R}^M\). It is proved that there exist semiclassical states concentrating around the local minimum points of V under mild assumptions. The approach is variational, which is mainly based upon a new linking-type argument, iterative techniques and interior estimates for nonlinear parabolic equations.