Homogeneous Groups Of Hardy Inequalities, Oscillation Theory, The Pohozaev Identity And Its Applications

This thesis is finished under the guiding of my tutor Professor Niu PengCheng. Through discussing some natures of homogeneous group, we obtained the results of this thesis. At first, using the Garofalo and Lanconelli's method that is to use function formula to prove Hardy type inequality, We have got a class of Hardy type inequality and uncertainly priciple on the homogeneous group. Then we discuss the question of eigenvalues' existence of a sub-Laplacian partial differential equation with indefinite potential on the homogeneous group. In Chapter Two, along the thought of Allegretto and Huang, we give a Picone type identity on the homogeneous group, and then as an application we prove some Hardy type inequalities inside and outside of ball domain on the Rn and Heisenberg group. In Chapter Three, using the method of establishing classic Sturmian comparison theorem, we set up a more extensive Picone type identity and a Sturmian type comparison theorem for a semilinear partial differential equation system on the homogeneous group. As an application, we give an oscillation theorem on the homogeneous group and a more extensive Hardy type inequality on the Heisenberg group. In Chapter Four, through discussing the papers in which Garofalo and lanconelli had established some nonexistence results on the Heisenberg group, we give a Pohozaev type identity and a nonexistence result by discussing the question of characteristic set on the homogeneous group. In above, we state our main results in this paper.