In this paper, we study the following fractional Schrödinger–Poisson system $$\begin{aligned} \left\{ \begin{array}{ll} \varepsilon ^{2s}(-\Delta )^s u +V(x)u+\phi u=K(x)|u|^{p-2}u,\,\,\text {in}~\mathbb {R}^3,\\ \\ \varepsilon ^{2s}(-\Delta )^s \phi =u^2,\,\,\text {in}~\mathbb {R}^3, \end{array} \right. \end{aligned}$$ ε 2 s ( - Δ ) s u + V ( x ) u + ϕ u = K ( x ) | u | p - 2 u , in R 3 , ε 2 s ( - Δ ) s ϕ = u 2 , in R 3 , where $$\varepsilon >0$$ ε > 0 is a small parameter, $$\frac{3}{4}<s<1$$ 3 4 < s < 1 , $$4<p<2_s^*:=\frac{6}{3-2s}$$ 4 < p < 2 s ∗ : = 6 3 - 2 s , $$V(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)$$ V ( x ) ∈ C ( R 3 ) ∩ L ∞ ( R 3 ) has positive global minimum, and $$K(x)\in C(\mathbb {R}^3)\cap L^\infty (\mathbb {R}^3)$$ K ( x ) ∈ C ( R 3 ) ∩ L ∞ ( R 3 ) is positive and has global maximum. We prove the existence of a positive ground state solution by using variational methods for each $$\varepsilon >0$$ ε > 0 sufficiently small, and we determine a concrete set related to the potentials V and K as the concentration position of these ground state solutions as $$\varepsilon \rightarrow 0$$ ε → 0 . Moreover, we considered some properties of these ground state solutions, such as convergence and decay estimate.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Jul 13, 2017
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