Ground state solutions of Nehari-Pohozaev type for a fractional Schrödinger-Poisson system with critical growth

Abstract

We study the following nonlinear fractional Schrödinger-Poisson system with critical growth:

$$\left\{ {\begin{array}{*{20}{c}} {{{( - \Delta )}^s}u + u + \phi u = f(u) + {{\left| u \right|}^{2_s^*}}u,}&{x \in {\mathbb{R}^3},} \\ {{{( - \Delta )}^t}\phi = {u^2},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}&{x \in {\mathbb{R}^3},} \end{array}} \right.$$
((0.1))

where 0 < s, t < 1, 2s + 2t>3 and \(2_s^* = {6 \over {3 - 2s}}\) is the critical Sobolev exponent in ℝ3. Under some more general assumptions on f, we prove that (0.1) admits a nontrivial ground state solution by using a constrained minimization on a Nehari-Pohozaev manifold.

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Acknowledgements

The authors would like to thank Professor Yinbin Deng for useful suggestions and comments.

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Correspondence to Li Wang.

Additional information

The first author was supported by the Science and Technology Project of Education Department in Jiangxi Province (GJJ180357) and the second author was supported by NSFC (11701178).

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Cite this article

Huang, W., Wang, L. Ground state solutions of Nehari-Pohozaev type for a fractional Schrödinger-Poisson system with critical growth. Acta Math Sci 40, 1064–1080 (2020). https://doi.org/10.1007/s10473-020-0413-1

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Key words

  • fractional Schrödinger-Poisson system
  • Nehari-Pohozaev manifold
  • ground state solutions
  • critical growth

2010 MR Subject Classification

  • 35R11
  • 35A15
  • 35B33