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

Borsuk, Michail ; Portnyagin, Dmitriy

Electronic journal of differential equations, Tome 1999 (1999)

Résumé

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.

Détail

Classification : 35B45, 35B65, 35D10, 35J25, 35J60, 35J65, 35J70
Keywords: quasilinear elliptic degenerate equations, barrier functions, conical points