Degree Theory And The Existence Of Solutions For A Class Of Elliptic Equations

In this paper,we mainly study the existence of solutions fbr the fbllowing ODE systems under different conditions (critical,supercritical and subcritical). where u(r), v(r)>0,u’(0) =v’(0)=0,α is any positive real number,and u(r),u(r)∈C1[0,∞]∩C2(0,∞).Under critical and supercritical situations,we mainly refer to the degree shoot-ing method introduced in paper [25] by Liu,Guo,Zhang and in [23] by Li and we proved the global existence of positive solutions. In subcritical case, we refer to [18] and [26] by Mitidieri. We also deduced the Rellich-Pohozaev Identity and thus proved the nonexistence of solutions.This paper consists of five parts:Part Ⅰ: We sort out the ODE systems corresponding to a class of elliptic equations which evolved from HLS inequality.We introduce the previous research on this topic and propose our research problem.Part Ⅱ: We briefly introduce some basic definitions and theorems which are useful fbr readers to understand this paper.Part Ⅲ: We prove the global existence of positive solutions fbr the ODE systems (0.2).Part Ⅳ: This chapter fbcus on the global nonexistence of the systems under subcritical condition.Part Ⅴ: With the previous study, we further discover the weighted cases.