## 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