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Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

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  • In this paper, we study the following Hamiltonian elliptic system with gradient term \begin{eqnarray} &-\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi+V(x)\varphi=K(x)f(|\eta|)\varphi \ \ \hbox{in}~\mathbb{R}^{N},\\ &-\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi+V(x)\psi=K(x)f(|\eta|)\psi \ \ \hbox{in}~\mathbb{R}^{N}, \end{eqnarray} where $\eta=(\psi,\varphi):\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $V, K\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. Suppose that $V(x)$ is sign-changing and has at least one global minimum, and $K(x)$ has at least one global maximum, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon>0$.
    Mathematics Subject Classification: 35J50, 58E05.


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