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

Abstract

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