On some unexpected properties of radial and symmetric eigenvalues and eigenfunctions of the $p$-Laplacian on a disk

Abstract

We discuss several properties of eigenvalues and eigenfunctions of the $p$-Laplacian on a ball subject to zero Dirichlet boundary conditions. Among main results, in two dimensions, we show the existence of nonradial eigenfunctions which correspond to the radial eigenvalues. Also we prove the existence of eigenfunctions whose shape of the nodal set cannot occur in the linear case $p=2$. Moreover, the limit behavior of some eigenvalues as $p\to 1+$ and $p\to +\infty$ is studied.

arXiv comment15 pages, 3 figures; minor typos fixed, two references added. To appear in Journal of Differential Equations

References in corpus (1)

• Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains

Cited by in corpus (2)

• Asymptotic relation for zeros of cross-product of Bessel functions and applications
• On the Structure of Periodic Eigenvalues of the Vectorial $p$-Laplacian