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Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

Authors
  • Zhang, Jian1, 2, 3
  • Chen, Jianhua4
  • Li, Quanqing5
  • Zhang, Wen1, 2
  • 1 Hunan University of Technology and Business, 410205 Hunan , (China)
  • 2 Hunan University of Technology and Business, 410083 Hunan , (China)
  • 3 Central South University, 410083 Hunan , (China)
  • 4 Nanchang University, 330031 Jiangxi , (China)
  • 5 Honghe University, 661100 Yunnan , (China)
Type
Published Article
Journal
Advances in Nonlinear Analysis
Publisher
De Gruyter
Publication Date
Jul 17, 2020
Volume
10
Issue
1
Pages
233–260
Identifiers
DOI: 10.1515/anona-2020-0126
Source
De Gruyter
Keywords
License
Green

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.

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