A note on p(x)-harmonic maps

This article is concerned with L^{p(x)} estimates of the gradient of p(x)-harmonic maps. It is known that p(x)-harmonic maps are the weak solutions of a system with natural growth conditions, but it is difficult to use the classical elliptic techniques to find gradient estimates. In this article, we use the monotone inequality to show that the minimum p(x)-energy can be expressed by the L^{p(x)} norm of a gradient of a function Phi, which is a weak solution of a single equation.