In this paper, we use a sufficient and necessary condition of weak solutions of $A$-harmonic equation to obtain the Caccioppoli type inequality and reverse Hölder inequality. As some applications of the above results, we derive the $L^p$-integrability, Hölder continuity and almost everywhere differentiability for weak solutions of $A$-harmonic equation. The same results were given for the component functions of quasiregular mappings.

Keywords: $A$-harmonic equation, reverse Hö,lder inequality, quasiregular mapping