# The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system

The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

# The concentration behavior of ground state solutions for a fractional Schrödinger–Poisson system

, Volume 56 (4) – Jul 13, 2017
25 pages

/lp/springer_journal/the-concentration-behavior-of-ground-state-solutions-for-a-fractional-rXRqcl30hr
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1199-4
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Calculus of Variations and Partial Differential EquationsSpringer Journals

Published: Jul 13, 2017

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