Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 48, pp. 1-19.
Title: Global bifurcation result for the p-biharmonic operator
Authors: Pavel Drabek (Univ. of West Bohemia, Czech Republic)
Mitsuharu Otani (Waseda Univ., Tokyo, Japan)
Abstract:
We prove that the nonlinear eigenvalue problem for the p-biharmonic
operator with $p > 1$, 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
$\partial \Omega$, $\Delta u_1 < 0$ 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.