Boundary Estimate for the Gradient of a Solution to the Dirichlet Problem for (p,q)-Nonlinear Equations

(p, q)-Nonlinear elliptic equations are considered, where p, q, p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ < $$ \end{document} q, characterize the growth with respect to the gradient of eigenvalues of the principle matrix. Under the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$2 \leqslant p < q, q - p < \frac{2}{{n^2 + n}}p$$ \end{document} an a priori estimate for the maximum of the modulus of the gradient of a solution to the Dirichlet problem is obtained. Bibliography: 8 titles.