Electron. J. Diff. Equ.,
Vol. 2013 (2013), No. 47, pp. 130.
An eigenvalue problem for the infinityLaplacian
Tilak Bhattacharya, Leonardo Marazzi
Abstract:
In this work, we study an eigenvalue problem for the infinityLaplacian
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 starshaped
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: InfinityLaplacian; 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 405060027, USA
email: leonardo.marazzi@uky.edu

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