数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (2): 551-560.doi: 10.1007/s10473-022-0209-6

• 论文 • 上一篇    下一篇

THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN

王文清1, 毛安民2   

  1. 1. Department of Mathematics, Wuhan University of Technology, Wuhan 430071, China;
    2. School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
  • 收稿日期:2019-05-16 修回日期:2021-04-19 出版日期:2022-04-25 发布日期:2022-04-22
  • 通讯作者: Wenqing WANG,E-mail:wangwenqing-1234@163.com E-mail:wangwenqing-1234@163.com
  • 作者简介:Anmin MAO,E-mail:maoam@163.com
  • 基金资助:
    This work was supported by NSFC (11931012; 11871387; 11471187).

THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN

Wenqing WANG1, Anmin MAO2   

  1. 1. Department of Mathematics, Wuhan University of Technology, Wuhan 430071, China;
    2. School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China
  • Received:2019-05-16 Revised:2021-04-19 Online:2022-04-25 Published:2022-04-22
  • Supported by:
    This work was supported by NSFC (11931012; 11871387; 11471187).

摘要: We investigate the bi-harmonic problem $$\left\{\begin{array}{ll} \Delta^{2}u - \alpha\nabla \cdot (f(\nabla u)) - \beta\Delta_{p}u = g(x,u) &\hbox{in}\ \ \Omega,\\[2mm] \frac{\partial u}{\partial n}=0, \frac{\partial(\Delta u)}{\partial n}=0 &\hbox{on}\ \ \partial\Omega,\end{array} \right. $$ where $\Delta^{2}u = \Delta(\Delta u), \Delta_{p}u =\div\left(|\nabla u|^{p-2}\nabla u\right)$ with $p > 2.$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N \geq 1.$ By using a special function space with the constraint $\int_{\Omega}u {\rm d}x = 0$, under suitable assumptions on $f$ and $g(x,u)$, we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.

关键词: Bi-harmonic, sign-changing solution, Fountain theorem

Abstract: We investigate the bi-harmonic problem $$\left\{\begin{array}{ll} \Delta^{2}u - \alpha\nabla \cdot (f(\nabla u)) - \beta\Delta_{p}u = g(x,u) &\hbox{in}\ \ \Omega,\\[2mm] \frac{\partial u}{\partial n}=0, \frac{\partial(\Delta u)}{\partial n}=0 &\hbox{on}\ \ \partial\Omega,\end{array} \right. $$ where $\Delta^{2}u = \Delta(\Delta u), \Delta_{p}u =\div\left(|\nabla u|^{p-2}\nabla u\right)$ with $p > 2.$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N \geq 1.$ By using a special function space with the constraint $\int_{\Omega}u {\rm d}x = 0$, under suitable assumptions on $f$ and $g(x,u)$, we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.

Key words: Bi-harmonic, sign-changing solution, Fountain theorem

中图分类号: 

  • 35J05