Electron. J. Diff. Eqns., Vol. 2004(2004), No. 16, pp. 1-11.

The eigenvalue problem for a singular quasilinear elliptic equation

Benjin Xuan

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.

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Benjin Xuan
Department of Mathematics
University of Science and Technology of China

Dept. de Matematicas, Universidad Nacional
Bogota, Colombia
e-mail: wenyuanxbj@yahoo.com

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