Electron. J. Diff. Equ., Vol. 2013 (2013), No. 47, pp. 1-30.

An eigenvalue problem for the infinity-Laplacian

Tilak Bhattacharya, Leonardo Marazzi

In this work, we study an eigenvalue problem for the infinity-Laplacian on bounded domains. We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. This work also contains existence results, related to this problem, when a parameter is less than the first eigenvalue. A comparison principle applicable to these problems is also proven. Some additional results are shown, in particular, that on star-shaped domains and on C^2 domains higher eigenfunctions change sign. When the domain is a ball, we prove that the first eigenfunction has one sign, radial principal eigenfunction exist and are unique up to scalar multiplication, and that there are infinitely many eigenvalues.

Submitted July 20, 2012. Published February 8, 2013.
Math Subject Classifications: 35J60, 35J70, 35P30.
Key Words: Infinity-Laplacian; first eigenvalue; comparison principle.

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Tilak Bhattacharya
Department of Mathematics, Western Kentucky University
Bowling Green, KY 42101, USA
email: tilak.bhattacharya@wku.edu
Leonardo Marazzi
Department of Mathematics, University of Kentucky
Lexington, KY 40506-0027, USA
email: leonardo.marazzi@uky.edu

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