The Concentration of Multiplicity Solutions for a Class Hamiltonian Elliptic Systems

In this paper, we are concerned with the concentration of multiplicity solutions for the following Hamiltonian elliptic systems \(\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^2\Delta u+u=K(x)f(v),~~x\in \mathbb {R}^N,\\ -\varepsilon ^2\Delta v+v=K(x)g(u),~~x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-{\epsilon }^2\mathrm{\Delta }u+u=K(x)f(v),x\in {\mathbb{R}}^N,\hfill \\ -{\epsilon }^2\mathrm{\Delta }v+v=K(x)g(u),x\in {\mathbb{R}}^N,\hfill \end{array}\right)\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(\varepsilon >0\) $\epsilon >0$ is a small parameter, \(K:\mathbb {R}^N\rightarrow \mathbb {R}\) $K:{\mathbb{R}}^N\to \mathbb{R}$ is bounded positive continuous function, f and g are continuous but are not necessarily of class \(C^1\) $C^1$ . By establishing a strongly indefinite variational setting, we prove the number of solutions is at least the number of global maximum points of K, and the maximum points of K is the concentration position of these solutions as \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ .