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., Lp(1<p<4) integrability of Tu) and then the second order weak differentiability of weak solution in the horizontal directions (i.e., L2 integrability of ∇H2u) and weak differentiability of gradient of weak solution in the vertical direction (i.e., L2 integrability of T∇Hu)).