Heriot-Watt Mathematics Report Series
HWM06-17, 21 Apr 2006
The spectrum of the periodic $p$-Laplacian
P A Binding and B P Rynne
Abstract
We consider one dimensional $p$-Laplacian eigenvalue problems
of the form
\begin{gather*}
-\De_p u = (\la - q) |u|^{p-1} \sgn u,
\quad \text{ on $(0,\pi_p)$},
\end{gather*}
together with periodic or separated boundary conditions,
where $p > 1$, $\De_p$ is the $p$-Laplacian,
$q\in C^1[0,b]$, and $\pi_p > 0,\, \lambda \in R$.
It will be shown that when $p \ne 2$ and $q \ne 0$,
the structure of the spectrum in the periodic case
(that is, with periodic boundary conditions),
can be completely different
from those of the following known cases:
(i) the general periodic case with $p = 2$,
(ii) the periodic case with $p \ne 2$ and $q=0$, and
(iii) the general separated case for all $p > 1$.
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