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

• Main Authors: Michail Borsuk, Dmitriy Portnyagin
• Format: Article
• Language: English
• Published: Texas State University 1999-06-01
• Series: Electronic Journal of Differential Equations
• Subjects: quasilinear elliptic degenerate equations; barrier functions; conical points.
• Online Access: http://ejde.math.txstate.edu/Volumes/1999/23/abstr.html

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 $$ -frac{d}{dx_i} (|x|^au |u|^q |abla u|^{m-2} u_{x_i})+ frac{a_0|x|^au }{(x_{n-1}^2+x_n^2)^{m/2}} u|u|^{q+m-2} -mu |x|^au u |u| ^{q-2} |abla u|^m = f_0(x)-frac{partial f_i}{partial x_i}, $$ with $a_0ge 0$, $qge 0$, $le mu <1$, $1<mle n$, and $au >m-n$ in a domain with a boundary conical point. We obtain the exact H"older exponent of the solution near the conical point.