On the behavior of solutions of quasilinear elliptic inequalities near a boundary point

Abstract

Assume that $p>1$ and $p-1\le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb{R}}^n$, $n\ge 2$. We consider the inequality $$\mathrm{div}A(x,Du)+b(x)|Du{|}^{\alpha }\ge 0,$$ where $D=(\partial /\partial x_1,\dots ,\partial /\partial x_n)$ is the gradient operator and $A:\Omega \times {\mathbb{R}}^n\to {\mathbb{R}}^n$ and $b:\Omega \to [0,\infty )$ are some functions with $$C_1|\xi {|}^p\le \xi A(x,\xi ),|A(x,\xi )|\le C_2|\xi {|}^{p-1},C_1,C_2=const>0,$$ for almost all $x\in \Omega$ and for all $\xi \in {\mathbb{R}}^n$. For solutions of this inequality we obtain estimates depending on the geometry of $\Omega$. In particular, these estimates imply regularity conditions of a boundary point.