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Semiclassical limit and well-posedness of nonlinear Schrödinger-Poisson systems. (English) Zbl 1055.35111
The authors deal with the Cauchy problem for the nonlinear Schrödinger-Poisson system $\begin{gathered} i\varepsilon\psi^\varepsilon_t+ {\varepsilon^2\over 2} \Delta\psi^\varepsilon- (V^\varepsilon(x,t)+ f'(|\psi^\varepsilon|^2) \psi^\varepsilon- (\arg\psi^\varepsilon) \psi^\varepsilon= 0,\\ -\Delta V^\varepsilon= |\psi^\varepsilon|^2- C(x),\quad V\to 0\text{ as }| x|\to\infty,\end{gathered}\tag{1}$ subject to the rapidly oscillating (WKB) initial condition $\psi^\varepsilon(x,0)= \psi^\varepsilon_0(x)= A^\varepsilon_0(x) \exp\biggl({i\over\varepsilon} S_0(x)\biggr),\tag{2}$ where $$f$$, $$S_0$$, $$A^\varepsilon_0$$ are given functions. Under some suitable assumptions on the data (1)–(2) the authors show the local well-posedness. Moreover, they obtain global existence and study the time-asymptotic behaviour of classical solutions of (1)–(2).

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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