A GENERALIZED DISTRIBUTIONAL INEQUALITY AND APPLICATIONS

The distributional inequality recently introduced by Tran and Nguyen has been used to investigate gradient estimates for solutions to partial differential equations. In particular, the authors established several sufficient conditions under which two measurable functions can be compared via their norms in general Lebesgue spaces. The results are then applied to some classes of p-Laplace type problems. This paper extends this inequality to make it applicable to a broader range of equations. Specifically, we propose a generalized distributional inequality that can be applied to the p(x)-Laplace equation, the typical version of quasi-linear elliptic equations with variable exponents.