Weak Differentiability to Nonuniform Nonlinear Degenerate Elliptic Systems under $p,q$-growth Condition on the Heisenberg Group

Abstract

The paper concerns the weak differentiability of weak solutions to two kinds of nonuniform nonlinear degenerate elliptic systems under the $p,q$-growth condition on the Heisenberg Group. We use the iteration to fractional difference quotients on the Heisenberg Group to get the weak differentiability of weak solution $u$ in the vertical direction (i.e., ${L^p}$($1<p<4$) integrability of $Tu$) and then the second order weak differentiability of weak solution in the horizontal directions (i.e., ${L^2}$ integrability of $\nabla_H^2u$) and weak differentiability of gradient of weak solution in the vertical direction (i.e., ${L^2}$ integrability of $T{\nabla_H}u$)).