Electron. J. Diff. Eqns., Vol. 2001(2001), No. 48, pp. 1-19.

Global bifurcation result for the p-biharmonic operator

Pavel Drabek & Mitsuharu Otani

Abstract:
We prove that the nonlinear eigenvalue problem for the p-biharmonic operator with $p greater than 1$, and $\Omega$ a bounded domain in $\mathbb{R}^N$ with smooth boundary, has principal positive eigenvalue $\lambda_1$ which is simple and isolated. The corresponding eigenfunction is positive in $\Omega$ and satisfies $\frac{\partial u}{\partial n} < 0$ on $\partial \Omega$, $\Delta u_1 less than 0$ in $\Omega$. We also prove that $(\lambda_1,0)$ is the point of global bifurcation for associated nonhomogeneous problem. In the case $N=1$ we give a description of all eigenvalues and associated eigenfunctions. Every such an eigenvalue is then the point of global bifurcation.

Submitted February 9, 2001. Published July 3, 2001.
Math Subject Classifications: 35P30, 34C23.
Key Words: p-biharmonic operator, principal eigenvalue, global bifurcation.

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Pavel Drabek
Centre of Applied Mathematics
University of West Bohemia
Univerzitni 22, 306 14 Plzen
Czech Republic
e-mail: pdrabek@kma.zcu.cz
Mitsuharu Otani
Department of Applied Physics
School of Science and Engineering
Waseda University
3-4-1, Okubo Tokyo, Japan, 169-8555
e-mail: otani@mn.waseda.ac.jp

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