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 $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D = (\partial / \partial x_1, \ldots, \partial / \partial x_n)$ 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 $$ C_1 |\xi|^p \le \xi A (x, \xi), \quad |A (x, \xi)| \le C_2 |\xi|^{p-1}, \quad C_1, C_2 = const > 0, $$ 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.