On weighted estimates of solutions of nonlinear elliptic problems

Skrypnik, Igor V. ; Larin, Dmitry V.

Mathematica Bohemica, Tome 124 (1999) no. 2-3, pp. 173-184.

Résumé

The paper is devoted to the estimate \vert u(x,k)\vert\leq K\vert k\vert\left\{\mathop cap\nolimits_{p,w}(F)\frac{\rho^p}{w(B(x,\rho))}\right\} ^{\frac1{p-1}}, $2\$ for a solution of a degenerate nonlinear elliptic equation in a domain ${B(x_0,1)\setminus F}$, $F\subset B(x_0,d)=\{x\in\Bbb R^n |x_0-x|$, $d\frac12$, under the boundary-value conditions $u(x,k)=k$ for $x\in\partial F$, $ u(x,k)=0$ for $x\in\partial B(x_0,1)$ and where $0\rho\leq\mathop dist(x,F)$, $w(x)$ is a weighted function from some Muckenhoupt class, and $\mathop cap_{p,w}(F)$, $w(B(x,\rho))$ are weighted capacity and measure of the corresponding sets.

Détail

DOI : 10.21136/MB.1999.126242

Classification : 35B45, 35J70
Keywords: degeneracy; Muckenhoupt class; pointwise estimate; nonlinear elliptic equation; capacity; a-priori estimate