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The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth condition. (English. Russian original) Zbl 0949.35048

Differ. Equations 33, No. 12, 1653-1663 (1997); translation from Differ. Uravn. 33, No. 12, 1651-1660 (1997).
The paper is devoted to studying interior properties of solutions to elliptic equations of the form \(\sum^n_{i=1}{\partial\over\partial x}_i \left(|\nabla u|^{p(x)- 2}{\partial u\over\partial x_i}\right)= 0\), \(x\in D\), \(n\geq 2\), with a piecewise continuous exponent \(p(x)\) such that \(1< p_1\leq p(x)\leq p_2<\infty\), where \(D\) is a bounded domain in \(\mathbb{R}^n\), and the unknown function \(u\in W^1_{1,\text{loc}}(D)\) satisfies some integral identity related to this equation. Modifying the Moser method the author proves an interior a priori estimate for the Hölder norm of solutions and derives the weak Harnarck inequality for positive solutions. The obtained results are extended to solutions of equations \(\sum^n_{i,j= 1} \left(a_{ij}(x)|\nabla u|^{p(x)-2} {\partial u\over\partial x_i}\right)= 0\) with a uniformly positive definite matrix \(\|a_{ij}\|\). References, 16 in number, reflect the topic very well.

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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