Local estimates for gradients of solutions to elliptic equations with variable exponents

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 $$ left| abla uight|in L^{infty}_{loc} $$ for all weak solutions of $$ operatorname{div} (g(|abla u|^2,x) abla u )=0quad ext{in } Omega. $$

Keywords

• Regularity
• divergence
• nonlinear
• elliptic equation
• gradient
• variable exponent
• p(x)-Laplacian