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Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent. (English) Zbl 1350.35086

Summary: Existence of one positive solution of the generalized Schrödinger-Poisson system \[ \begin{cases} - \Delta u + V(x) u - K(x) \phi | u |^3 u = f(x, u) \quad & \text{in } \mathbb{R}^3, \\ - \Delta \phi = K(x) | u |^5 \quad & \text{in } \mathbb{R}^3, \end{cases} \] where \(V, K, f\) are asymptotically periodic functions of \(x\), is proved by the mountain pass theorem and the concentration-compactness principle. The system with subcritical nonlocal term has been studied extensively in the last twenty years, while the system with critical nonlocal term has seldom been studied. It turns out that new techniques are needed in dealing with the case of critical nonlocal term.

MSC:

35J47 Second-order elliptic systems
35B09 Positive solutions to PDEs
35B33 Critical exponents in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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