## 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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