Global bifurcation result for the p-biharmonic operator

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.

Keywords

• p-biharmonic operator
• principal eigenvalue
• global bifurcation.