Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 51, pp. 1-8.

Title: Local estimates for gradients of solutions to elliptic equations 
       with variable exponents 
        
Author: Fengping Yao (Shanghai Univ., Shanghai, China)

Abstract:
 In this article we  present local $L^\infty$ estimates for the  gradient  
 of  solutions to  elliptic equations with variable exponents. 
 Under proper conditions on the coefficients, we prove that
 $$
  \left| \nabla u\right|\in L^{\infty}_{loc}
  $$
 for all  weak solutions of
 $$
 \operatorname{div}  (g(|\nabla u|^2,x) \nabla u )=0\quad \text{in } \Omega.
 $$

Submitted September 3, 2012. Published February 18, 2013.
Math Subject Classifications: 35J60, 35J70.
Key Words: Regularity; divergence; nonlinear; elliptic equation;
           gradient; variable exponent; p(x)-Laplacian.