Electron. J. Diff. Equ., Vol. 2010(2010), No. 77, pp. 1-12.

Infinity Laplace equation with non-trivial right-hand side

Guozhen Lu, Peiyong Wang

Abstract:
We analyze the set of continuous viscosity solutions of the infinity Laplace equation $-\Delta^N_{\infty}w(x) = f(x)$, with generally sign-changing right-hand side in a bounded domain. The existence of a least and a greatest continuous viscosity solutions, up to the boundary, is proved through a Perron's construction by means of a strict comparison principle. These extremal solutions are proved to be absolutely extremal solutions.

Submitted July 2, 2009. Published June 8, 2010.
Math Subject Classifications: 35J70, 35B35.
Key Words: Infinity Laplace equation; inhomogeneous equation; viscosity solutions; least solution; greatest solution; strict comparison principle; existence; uniqueness; local Lipschitz continuity.

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Guozhen Lu
Department of Mathematics, Wayne State University
656 W. Kirby, 1150 FAB
Detroit, MI 48202, USA
email: gzlu@math.wayne.edu
Peiyong Wang
Department of Mathematics, Wayne State University
656 W. Kirby, 1150 FAB
Detroit, MI 48202, USA
email: pywang@math.wayne.edu

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