Electron. J. Diff. Eqns., Vol. 2005(2005), No. 87, pp. 19.
Steklov problem with an indefinite weight for the pLaplacian
Olaf Torne
Abstract:
Let
, with
,
be a Lipschitz
domain and let
.
We consider the eigenvalue problem
in
and
on
,
where
is the eigenvalue and
is an associated eigenfunction.
The weight
is assumed to lie in an appropriate Lebesgue space
and may change sign. We sketch how a sequence of eigenvalues may
be obtained using infinite dimensional LjusternikSchnirelman
theory and we investigate some of the nodal properties of
eigenfunctions associated to the first and second eigenvalues.
Amongst other results we find that if
and
then the first positive
eigenvalue is the only eigenvalue associated to an eigenfunction
of definite sign and any eigenfunction associated to the second
positive eigenvalue has exactly two nodal domains.
Submitted August 10, 2004. Published August 14, 2005.
Math Subject Classifications: 35J70, 35P30.
Key Words: Nonlinear eigenvalue problem; Steklov problem; pLaplacian;
nonlinear boundary condition; indefinite weight.
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Olaf Torné
Université Libre de Bruxelles
Campus de la Plaine, ULB CP214
Boulevard du Triomphe, 1050 Bruxelles, Belgium
email: otorne@ulb.ac.be

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