MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 173-184 (1999)

On weighted estimates of solutions of
nonlinear elliptic problems

Igor V. Skrypnik, Dmitry V. Larin

Igor V. Skrypnik, Dmitry V. Larin, Institute Appl. Math. & Mech. NAS Ukraine, R. Luxemburg Str. 74, 340114 Donetsk, Ukraine, e-mail: skrypnik @iamm.

Abstract: The paper is devoted to the estimate $$ \vert u(x,k)\vert\leq K\vert k\vert\left\{\mathop cap\nolimits_{p,w}(F)\frac{\rho^p}{w(B(x,\rho))}\right\} ^{\frac{1}{p-1}}, $$ $2&lt;p<n$ for a solution of a degenerate nonlinear elliptic equation in a domain ${B(x_0,1)\setminus F}$, $F\subset B(x_0,d)=\{x\in\Bbb R^n |x_0-x|<d\}$, $d<\frac{1}{2}$, 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.

Keywords: degeneracy, Muckenhoupt class, pointwise estimate, nonlinear elliptic equation, capacity, a-priori estimate

Classification (MSC2000): 35J70, 35B45

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