A Liouville Type Theorem For The Dirichlet Problems In A Half Space Of Weighted Elliptic Systems

In this dissertation, we mainly consider the Liouville type theorem of a weighted elliptic system in a half space, there are two principle parts in this paper:the equivalence between the weighted elliptic system and the integral system, the Liouville type theorem of the weighted elliptic systems proved by the method of moving plane. We intend to separate the dissertation into four chapters.In chapter one, we introduce the backgrounds of the weighted elliptic system and the fundamental methods of the dissertation is also illustrated.In chapter two, firstly, we recall the definitions of the fundamental solutions of elliptic differential equations and the Green’s function with the Dirichlet boundary conditions on the unit ball. Secondly, we extend the Green’s function to an arbitrary ball and the half space. Thirdly, we introduce the concepts of Sobolev spaces. Furthermore, we use the Possion—Green representation to rewrite the weighted elliptic system into an integral system on an arbitrary ball. In the end, a deeper estimate can help us to transform the weighted elliptic system into the integral system on a half space.In chapter three, we first recall the conceptions about the distribution and δ function. Then we introduce the test function and the definition of generalized derivative. Finally, we use the Fubini theorem to transform the integral system into the elliptic system.In chapter four, first, we illustrate the Kelvin transform, then we introduce an equiv-alence of the weighted Hardy—Littlewood—Sobolev inequality and Holder inequality; Next, we use the moving plane method in integral forms to establish a Liouville type theorem for the system.