Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 16, pp. 1-11.
Title: The eigenvalue problem for a singular quasilinear elliptic equation
Author: Benjin Xuan (Univ. Science and Technology of China)
Abstract:
We show that many results about the eigenvalues and
eigenfunctions of a quasilinear elliptic equation
in the non-singular case can be extended to the singular case.
Among these results, we have the first eigenvalue is associated
to a $C^{1,\alpha}(\Omega)$ eigenfunction which is positive and
unique (up to a multiplicative constant), that is, the first
eigenvalue is simple. Moreover the first eigenvalue is isolated
and is the unique positive eigenvalue associated to a non-negative
eigenfunction. We also prove some variational properties of the
second eigenvalue.
Submitted August 15, 2003. Published February 6, 2004.
Math Subject Classifications: 35J60.
Key Words: Singular quasilinear elliptic equation; eigenvalue problem;
Caffarelli-Kohn-Nirenberg inequality.