Heriot-Watt Mathematics Report Series
HWM06-44, 29 Nov 2006
Variational and non-variational eigenvalues of the $p$-Laplacian
B P Rynne and P A Binding
Abstract
It is well known that all the eigenvalues of the linear
eigenvalue problem
\[
\Delta u = (q - \lambda r) u ,
\quad \text{ in } \Omega\subset R^N,
\]
can (under appropriate conditions on $q$, $r$ and $\Omega$) be characterized by
minimax principles, but it has been a long-standing question whether that
remains true for analogous equations involving the $p$-Laplacian $\Delta_p$.
It will be shown that there are corresponding nonlinear eigenvalue problems
$$
\Delta_p u = (q - \lambda r) |u|^{p-1}\sgn u,
\quad \text{ in } \Omega\subset R^N,
$$
with $1 < p \ne 2$ and $q,\, r \in C^1(\overline\Omega)$,
for which not all eigenvalues are of variational type.
As far as we know, this is the first observation of such a phenomenon, and
examples will be given for one and higher dimensional equations. The question
of exactly which eigenvalues are variational is also discussed when $N=1$.
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