An Application Of The Shooting Principle To A High-order Elliptic System

In this paper, we consider the following Hardy-Sobolev type equations where k€N, αj≥0,βj,γj>0,j=1,2. With shooting principle, an improved version of the classical shooting method, we obtain the existence of global radial positive solutions in the critical and supercritical cases n+α1/γ1+1+n+α2/β2+1+2k-n≤0.Combining the traditional shooting method and the beautiful topological degree the-ory, the shooting principle proves to be a powerful tool in dealing with global situation problems. The basic idea is to connect PDEs defined over Rn with a same system with Dirichlet boundary condition on balls.More detailed process is as follows;First, we construct the target map. As the core of the new shooting principle, the tar-get map connects the initial value with the solution for a fixed point or a second boundary condition for an ordinary differential equation. With the basic property of the topological degree, we will prove that the map is both continuous and onto.Second, under some mild assumptions, we will show that the particular partial differ-ential equations we are interested in are non-degenerate. With the aid of the traditional shooting method, we will obtain equivalence between the existence of global solutions and the non-existence of solutions for local boundary problem. Finally, by the virtue of the Pohozaev identity, we derive the non-existence of positive solutions to the Dirichlet boundary value problem in the critical and supercritical cases. By far, we will obtain the desired result for global solutions.Meanwhile, we consider the scalar form of the Hardy-Sobolev type equations, that is where α≥0,β>0. Obviously, this special form of the Hardy-Sobolev system admits a global positive solution. In addition, we study the energy-finite solutions, or solutions of certain global integrability. By virtue of the classic method of moving planes in integral forms and Kelvin type transform, when α>√2kn, we derive the exact behavior of positive solutions at infinity. Further, we obtain the desired integrability at infinity of solutions to the integral equations equivalent to the differential system that we are interested in, and then arrive at the conclusions of energy finite.