齐次群上二阶半线性偏微分方程的一类Picone型恒等式和Sturmian比较定理

A PICONE TYPE IDENTITY AND STURMIAN COMPARISON THEOREM OF SECOND ORDERSEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS ON THE HOMOGENEOUS GROUP

本文给出了齐次群上的一类广义Picone型恒等式,由此证明了以下半线性方程组(其中 表示齐次群上的广义梯度)的Sturmian比较定理及一类振荡定理,并用于Heisenberg群上一类半线性方程.

In this paper, we prove a generalized Picone type identity on the homogeneous group. Then, we give a Sturmian comparison theorem and a oscillation theorem of the following semilinear partial differential equation system where is the generalizing gradient of the homogeneous group. Moreover, we apply the oscillation theorem to a semilinear partial differential equation on the Heisenberg group. Finally, by using the method of Allegretto and Huang, we prove a generalized Hardy type inequality.