Abstract
In this paper, we study the following Hamiltonian elliptic system with gradient term and critical growth: \begin{equation*} \begin{cases} -\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi=K(x)f(|\eta|)\varphi+W(x)|\eta|^{2^*-2}\varphi &\text{in } \mathbb{R}^{N},\\ -\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi=K(x)f(|\eta|)\psi+W(x)|\eta|^{2^*-2}\psi &\text{in } \mathbb{R}^{N}, \end{cases} \end{equation*} where $\eta=(\psi,\varphi)\colon \mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $K, W\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. We require that the nonlinear potentials $K$ and $W$ have at least one global maximum. Combining this with other suitable assumptions on $f$, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon> 0$
Citation
Jian Zhang. Xianhua Tang. Wen Zhang. "On semiclassical ground states for Hamiltonian elliptic system with critical growth." Topol. Methods Nonlinear Anal. 49 (1) 245 - 272, 2017. https://doi.org/10.12775/TMNA.2016.069
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