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

Assume that $p>1$ and $p-1\le \alpha \le p$ are real numbers and $\mathrm{\Omega }$ is a non-empty open subset of ${\mathbb{R}}^n$, $n\ge 2$. We consider the inequality $$\mathrm{d}\mathrm{i}\mathrm{v}A(x,Du)+b(x)\mathrm{\mid }Du{\mathrm{\mid }}^{\alpha }\ge 0,$$ where $D=(\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_1,\dots ,\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}_n)$ is the gradient operator and $A : \Omega \times {\mathbb R}^n \to {\mathbb R}^n$and$b : \Omega \to [0, \infty)$aresomefunctionswith$$ C_1 |\xi|^p \le \xi A (x, \xi), \quad |A (x, \xi)| \le C_2 |\xi|^{p-1}, \quad C_1, C_2 = const > 0, $$foralmostall$x \in \Omega$andforall$\xi \in {\mathbb R}^n$. For solutions of this inequality we obtain estimates depending on the geometry of $\mathrm{\Omega }$. In particular, these estimates imply regularity conditions of a boundary point.