Abstract In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term −ϵ2Δψ+ϵb→⋅∇ψ+ψ+V(x)φ=f(|η|)φ in RN,−ϵ2Δφ−ϵb→⋅∇φ+φ+V(x)ψ=f(|η|)ψ in RN, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ where η = (ψ, φ) : ℝN → ℝ2, ϵ is a small positive parameter and b⃗ is a constant vector. We require that the potential V only satisfies certain local condition. Combining this with other suitable assumptions on f, we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of V, convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.

中文翻译：

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哈密​​顿椭圆系统半经典解的集中行为

摘要 在本文中，我们研究了以下非线性哈密顿椭圆系统，其梯度项为−ϵ2Δψ+ϵb→⋅∇ψ+ψ+V(x)φ=f(|η|)φ，-ϵ2Δφ−ϵb→⋅∇ φ+φ+V(x)ψ=f(|η|)ψ 在 RN, $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2} {\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~ \mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\ psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ 其中 η = (ψ, φ) : ℝN → ℝ2, ϵ 是一个小的正参数，而 b⃗ 是一个常数向量。我们要求势 V 只满足一定的局部条件。将其与 f 的其他合适假设相结合，我们构建了一系列半经典解。此外，V 的局部最小值附近的集中现象，还探讨了半经典解的收敛和指数衰减。在证明中，由于势的局部性质和能量泛函的强不定性，我们应用了惩罚方法、链接论证和一些分析技术。