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Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in \( \mathbb{R} ^{3} \). (English) Zbl 1412.35135

Summary: We consider the existence of positive solutions for the following fractional Schrödinger-Poisson system \[ \begin{cases} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u+{\phi}(x)u = K(x)f(u)+|u|^{2_{s}^{*}-2}u, \quad & x\in \mathbb{R} ^3, \\ \varepsilon^{2s}(-\Delta)^{s}{\varphi} = u^{2}, & x \in \mathbb{R} ^3, \end{cases} \] where \( s \in (\frac{3}{4}, 1) \), \( \varepsilon \) is a small and positive parameter, \( V \) and \( K \) are nonnegative potential functions. \( 2_{s}^{*} \) is the critical exponent with respect to fractional Sobolev embedding theorem. Under some suitable conditions on the nonlinearity \( f \) and potential functions \( V \) and \( K \), we prove that for \( \varepsilon \) small, the system has a positive ground state solution concentrating around a concrete set related to \( V \) and \( K \). This result generalizes the result for fractional Schrödinger-Poisson system with subcritical exponent by Y. Yu et al. [Calc. Var. Partial Differ. Equ. 56, No. 4, Paper No. 116, 25 p. (2017; Zbl 1437.35699)] to critical exponent. Moreover, when \( V \) attains its minimum and \( K \) attains its maximum, we also obtain multiple solutions by Ljusternik-Schnirelmann theory.

MSC:

35J62 Quasilinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations

Citations:

Zbl 1437.35699
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