Existence and asymptotic behaviour of ground state solution for quasilinear Schrödinger-Poisson systems in \({\mathbb R}^3\). (English) Zbl 1367.35153

Ding, Ling; Li, Lin; Meng, Yi-Jie; Zhuang, Chang-Ling

Topol. Methods Nonlinear Anal. 47, No. 1, 241-264 (2016).

Summary: In this paper, we are concerned with existence and asymptotic behavior of ground state in the whole space \(\mathbb{R}^3\) for quasilinear Schrödinger-Poisson systems \[ \begin{aligned} -\Delta u+ u+ K(x)\phi(x)u= a(x) f(u),\quad & x\in\mathbb{R}^3,\\ -\text{div}[(1+ \varepsilon^4|\nabla \phi|^2)\nabla\phi]= K(x)u^2,\quad & x\in\mathbb{R}^3,\end{aligned} \] when the nonlinearity coefficient \(\varepsilon> 0\) goes to zero, where \(f(t)\) is asymptotically linear with respect to \(t\) at infinity. Under appropriate assumptions on \(K\), \(a\) and \(f\), we establish existence of a ground state solution \((u_\varepsilon,\phi_{\varepsilon,K}(u_\varepsilon))\) of the above system.
Furthermore, for all a sufficiently small, we show that \((u_\varepsilon, \phi_{\varepsilon,K}(u_\varepsilon))\) converges to \((u_\varepsilon,\phi_{\varepsilon,K}(u_\varepsilon))\) which is the solution of the corresponding system for \(\varepsilon= 0\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
82D99	Applications of statistical mechanics to specific types of physical systems
35B25	Singular perturbations in context of PDEs
35B40	Asymptotic behavior of solutions to PDEs
35J50	Variational methods for elliptic systems
47J30	Variational methods involving nonlinear operators
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35D35	Strong solutions to PDEs

Keywords:

Schrödinger-Poisson system; variational methods; ground state solution; asymptotically linear; asymptotic behavior