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Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger–Poisson system in \({\mathbb {R}}^{3}\)

Article

Abstract

In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schrödinger–Poisson system:
$$\begin{aligned} \left\{ \begin{aligned}&(-\Delta )^{s} u+V(x)u+\lambda \phi (x)u=f(x, u),\quad&\text {in}\, \ {\mathbb {R}}^{3},\\&(-\Delta )^{t}\phi =u^{2},&\text {in}\,\ {\mathbb {R}}^{3}, \end{aligned} \right. \end{aligned}$$
where \(\lambda \in {\mathbb {R}}^{+}\) is a parameter, \(s, t\in (0, 1)\) and \(4s+2t>3\), \((-\Delta )^{s}\) stands for the fractional Laplacian. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-changing solution. Moreover, for any \(\lambda >0\), we show that the energy of the least energy sign-changing solutions is strictly larger than two times the ground state energy. Finally, we consider \(\lambda \) as a parameter and study the convergence property of the least energy sign-changing solutions as \(\lambda \searrow 0\).

Keywords

Fractional Schrödinger–Poisson system Sign-changing solutions Constraint variational method Quantitative deformation lemma 

Mathematics Subject Classification

35J61 58E30 

Notes

Acknowledgements

Chao Ji is supported by Shanghai Natural Science Foundation (18ZR1409100), NSFC (No. 11301181) and China Postdoctoral Science Foundation.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsEast China University of Science and TechnologyShanghaiChina

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