Electron. J. Diff. Eqns., Vol. 1999(1999), No. 23, pp. 1-25.

On the Dirichlet problem for quasilinear elliptic second order equations with triple degeneracy and singularity in a domain with a boundary conical point

Michail Borsuk & Dmitriy Portnyagin

Abstract:
In this article we prove boundedness and Holder continuity of weak solutions to the Dirichlet problem for a second order quasilinear elliptic equation with triple degeneracy and singularity. In particular, we study equations of the form
$-{d \over dx_i} (|x|^\tau |u|^q |\nabla u|^{m-2} u_{x_i})+ {a_0|x|^\tau \over (x_{n-1}^2+x_n^2)^{m/2}} u|u|^{q+m-2} -\mu |x|^\tau u |u| ^{q-2} |\nabla u|^m =f_0(x)-{\partial f_i \over \partial x_i}, $
with $a_0 \ge 0$, $q\ge 0$, $0\le \mu$ < 1, 1 < $m\le n$, and $\tau$ > m-n in a domain with a boundary conical point. We obtain the exact Holder exponent of the solution near the conical point.

Submitted April 23, 1999. Published June 24, 1999.
Math Subject Classification: 35B45, 35B65, 35D10, 35J25, 35J60, 35J65, 35J70.
Key Words: quasilinear elliptic degenerate equations, barrier functions, conical points.

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Note This article is related to another article published by EJDE. Michail Borsuk & Dmitriy Portnyagin, Barriers on cones for degenerate quasilinear elliptic operators, Vol. 1998(1998), No. 11, pp. 1-8.

photo Michail Borsuk
Department of Applied Mathematics
Olsztyn University of Agriculture and Technology
10-957 Olsztyn-Kortowo, Poland
e-mail: borsuk@art.olsztyn.pl
photo Dmitriy Portnyagin
Department of Physics, Lvov State University
290602 Lvov, Ukraine
e-mail: mitport@hotmail.com

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