Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 47, pp. 1-30.

Title: An eigenvalue problem for the infinity-Laplacian
        
Authors: Tilak Bhattacharya (Western Kentucky Univ., Bowling Green, KY, USA)
         Leonardo Marazzi (Univ. of Kentucky, Lexington, KY, USA)

Abstract:
 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 08, 2013.
Math Subject Classifications: 35J60, 35J70, 35P30.
Key Words: Infinity-Laplacian; first eigenvalue; comparison principle.