Nontrivial solutions for a fourth-order elliptic equation of Kirchhoff type via Galerkin method

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Abstract

In this paper we study the existence of nontrivial solutions of the nonlocal elliptic problem
$$\begin{aligned} \left\{ \begin{array}{lcr} \Delta ^2 u-(a+b\int _\Omega |\nabla u|^2\mathrm{d}x)\Delta u+u=(\int _\Omega g(x,u)\mathrm{d}x)^\gamma f(x,u),\;in\;\Omega ,\\ u=\Delta u=0,\;\;on\;\;\partial \Omega \end{array}\right. \end{aligned}$$
via Galerkin method, where \(\Omega \subset \mathbb {R}^N (N\ge 5)\) is a smooth bounded domain.

Keywords

Nonlocal elliptic problems biharmonic operator Galerkin method 

Mathematics Subject Classification

Primary 35J58 35J65 

Notes

Acknowledgements

The authors would like to thank the referees for the helpful suggestions.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Depatment of Mathematics, College of ScienceHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of ScienceChina pharmaceutical UniversityNanjingPeople’s Republic of China

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