Global bifurcation result for the \(p\)-biharmonic operator
Electronic journal of differential equations, Tome 2001 (2001)
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.
Classification :
35P30, 34C23
Keywords: p-biharmonic operator, principal eigenvalue, global bifurcation
Keywords: p-biharmonic operator, principal eigenvalue, global bifurcation
@article{EJDE_2001__2001__a138,
author = {Dr\'abek, Pavel and \^Otani, Mitsuharu},
title = {Global bifurcation result for the \(p\)-biharmonic operator},
journal = {Electronic journal of differential equations},
year = {2001},
volume = {2001},
zbl = {0983.35099},
language = {en},
url = {http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a138/}
}
Drábek, Pavel; Ôtani, Mitsuharu. Global bifurcation result for the \(p\)-biharmonic operator. Electronic journal of differential equations, Tome 2001 (2001). http://geodesic.mathdoc.fr/item/EJDE_2001__2001__a138/