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.