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Ground state solutions for the periodic fractional Schrödinger-Poisson systems with critical Sobolev exponent. (English) Zbl 1440.35121

Summary: We study the fractional Schrödinger-Poisson system with critical Sobolev exponent \[\begin{cases} (-\Delta )^s u + V (x) u + \phi u = f (x,u) + K (x) | u |^{2_{s}^{\ast}-2} u & \text{ in } \mathbb{R}^3, \\ (- \Delta )^t \phi = u^2 & \text{ in } \mathbb{R}^3, \end{cases}\] where \((-\Delta)^\alpha\) denotes the fractional Laplacian of order \(\alpha = s\), \(t \in (0, 1)\); \(V (x)\), \(f(x,u )\) and \(K(x)\) are \(1\)-periodic in the \(x\)-variables; \( 2_s^\ast = 6 / (3-2 s)\) is the fractional critical Sobolev exponent in dimension \(3\). Under some weaker conditions on \(f\), we prove the existence of ground state solutions for such a system via the mountain pass theorem in combination with the concentration-compactness principle. Our results are new even for \(s = t = 1\).

MSC:

35J60 Nonlinear elliptic equations
35R11 Fractional partial differential equations
35B33 Critical exponents in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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