# On the Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with General Potentials and Super-quadratic Nonlinearity

On the Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with... In this article, we are concerned with the following fractional Schrödinger–Poisson system: \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u+V(x)u+\phi u=f(u)&{} \quad \hbox {in}~\mathbb {R}^{3},\\ (-\Delta )^{t}\phi =u^2&{} \quad \hbox {in}~\mathbb {R}^{3},\\ \end{array} \right. \end{aligned} ( - Δ ) s u + V ( x ) u + ϕ u = f ( u ) in R 3 , ( - Δ ) t ϕ = u 2 in R 3 , where $$0<s\le t<1$$ 0 < s ≤ t < 1 , $$2s+2t>3$$ 2 s + 2 t > 3 , and $$f\in C(\mathbb {R},\mathbb {R})$$ f ∈ C ( R , R ) . Under more relaxed assumptions on potential V(x) and f(x), we obtain the existence of ground state solutions for the above problem by adopting some new tricks. Our results here extend the existing study. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

# On the Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with General Potentials and Super-quadratic Nonlinearity

, Volume 15 (3) – May 30, 2018
15 pages

/lp/springer_journal/on-the-existence-of-ground-state-solutions-for-fractional-schr-dinger-ifw0mN5PaR
Publisher
Springer Journals
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-018-1179-8
Publisher site
See Article on Publisher Site

### Abstract

In this article, we are concerned with the following fractional Schrödinger–Poisson system: \begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u+V(x)u+\phi u=f(u)&{} \quad \hbox {in}~\mathbb {R}^{3},\\ (-\Delta )^{t}\phi =u^2&{} \quad \hbox {in}~\mathbb {R}^{3},\\ \end{array} \right. \end{aligned} ( - Δ ) s u + V ( x ) u + ϕ u = f ( u ) in R 3 , ( - Δ ) t ϕ = u 2 in R 3 , where $$0<s\le t<1$$ 0 < s ≤ t < 1 , $$2s+2t>3$$ 2 s + 2 t > 3 , and $$f\in C(\mathbb {R},\mathbb {R})$$ f ∈ C ( R , R ) . Under more relaxed assumptions on potential V(x) and f(x), we obtain the existence of ground state solutions for the above problem by adopting some new tricks. Our results here extend the existing study.

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: May 30, 2018

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