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

### The eigenvalue problem for a singular quasilinear elliptic equation Benjin Xuan

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 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