Electronic Journal of Differential Equations,
Vol. 1999(1999), No. 23, pp. 1-25.
Title: On the Dirichlet problem for quasilinear elliptic second order
equations with triple degeneracy and singularity in a domain with
a boundary conical point
Authors: Michail Borsuk (Olsztyn Univ., Poland)
Dmitriy Portnyagin (Lvov State Univ., Ukraine)
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
$$
-\frac{d}{dx_i} (|x|^\tau |u|^q |\nabla u|^{m-2} u_{x_i})+
\frac{a_0|x|^\tau }{(x_{n-1}^2+x_n^2)^{m/2}} u|u|^{q+m-2}
-\mu |x|^\tau u |u| ^{q-2} |\nabla u|^m =
f_0(x)-\frac{\partial f_i}{\partial x_i},
$$
with $a_0\ge 0$, $q\ge 0$, $\le \mu <1$, $1m-n$
in a domain with a boundary conical point. We obtain the exact
H\"older 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.