| Abstract: |
| Consider the nonlinear eigenvalue problem for the $p$-Laplacian
\begin{equation*}
-\Delta_p u = \lambda |u|^{p-2} u
~~\mathrm{in}~~ \Omega,
\quad
u = 0
~~\mathrm{on}~~ \partial\Omega,
\end{equation*}
and denote by $(\lambda(p), \varphi_p) \in \mathbb{R} \times W_0^{1,p}(\Omega)$ its eigenpairs.
In the present talk, we give a survey of various properties of $\lambda(p)$ and $\varphi_p$ obtained recently in co-authorship with B.~Audoux, P.~Dr\`abek, S.~Kolonitskii, E.~Parini, M.~Tanaka.
In particular, we show that $\varphi_p$ and $\varphi_q$ are linearly independent provided $p\neq q$; the nodal set of any second eigenfunction $\varphi_p$ intersects with the boundary $\partial \Omega$ provided $\Omega$ is Steiner symmetric; if $\Omega$ is a disk, then there can occur eigenfunctions $\varphi_p$ whose nodal structure is impossible in the linear case $p=2$.
Also, we show that $\lambda(p)$ can be non-monotone with respect to $p$, and if $\Omega$ is a disk, then radial and nonradial eigenfunctions can be associated with the same eigenvalue, which is again impossible for $p=2$. |
|