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

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
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