Existence of three solutions to a double eigenvalue problem for the $p$-biharmonic equation

Volume 104 / 2012

Lin Li, Shapour Heidarkhani Annales Polonici Mathematici 104 (2012), 71-80 MSC: Primary 35J35; Secondary 58E05. DOI: 10.4064/ap104-1-5

Abstract

Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem $$ \begin{cases} \varDelta(|\varDelta u|^{p-2}\varDelta u) - \text{div} (|\nabla u|^{p-2} \nabla u)=\lambda f(x,u) + \mu g(x,u) &\hbox{in }\varOmega,\\ u=\varDelta u=0 &\hbox{on }\partial \varOmega,\end{cases} $$ where $\varOmega \subset \mathbb{R}^N$ $(N \geq 1)$ is a non-empty bounded open set with a sufficiently smooth boundary $\partial \varOmega$, $\lambda>0$, $\mu>0$ and $f,g : \varOmega \times \mathbb{R} \to \mathbb{R}$ are two $L^1$-Carathéodory functions.

Authors

  • Lin LiDepartment of Science
    Sichuan University of Science and Engineering
    643000 Zigong, P.R. China
    e-mail
  • Shapour HeidarkhaniDepartment of Mathematics
    Faculty of Sciences
    Razi University
    67149 Kermanshah, Iran
    e-mail

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