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On Schrödinger-Poisson systems. (English) Zbl 1181.35257

Summary: We discuss some recent results dealing with the existence of bound states of the nonlinear Schrödinger-Poisson system
\[ \begin{cases} -\Delta u + V(x)u + \lambda K(x)\varphi (x)u = |u|^{{p - 1}} u, \\ -\Delta \varphi = K(x)u^{2}, \end{cases} \]
as well as of the corresponding semiclassical limits. The proofs are based upon critical point theory and perturbation methods.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J10 Schrödinger operator, Schrödinger equation
35J20 Variational methods for second-order elliptic equations
35J47 Second-order elliptic systems
35B20 Perturbations in context of PDEs
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