Existence Of Entire Positive Solutions Of P-q Laplace Systems

p-q Laplace system is closely related to fluid mechanics, arises in the study of non-Newtonian fluids or in the theory of quasiregular and quasiconformal mappings. With extremely extensive application background and deep research value, the existence and nonexistence of solutions of p-q Laplace systems have received more and more attention recently by domestic and foreign scholars.This paper mainly from the view of Partial Differential Equations is specifically divided into three parts to study the existence of entire positive solutions of p-q Laplace systems.In the preface we introduce the actual background, research progress and the main work of this paper. In Chapter1, we first discuss the existence of entire large positive radial solution for systems in the form of div((|▽u|p-2▽u)=m(|x|).f(v),<flv((|▽u|q-2▽u)=m(|x|).f(v), x∈RN, N≥3,where nonlinear absorption terms f and g are superlinear. In Chapter2, we use the upper and lower solutions and successive approximation methods to get the existence conditions of entire bounded positive solution for systems in the form of div((|▽u|p-2▽u)=m(|x|).f(v), div(|▽u|p-2▽u)=m(|x|).f(v), x∈RN, N≥3,where m and n are not radial. In Chapter3, we mainly study the more general type systems in the form of div((|▽u|p-2▽u)=m(|x|).f(v), div((|▽u|p-2▽u)=m(|x|).f(v), x∈RN, N≥3,where f and g satisfy superlinear or sublinear case and respectively give the existence results of entire positive radial solution, bounded and large. On this basis, we generalize the existence results to the following systems with nonlinear gradient term in the form of div((|▽u|p-2▽u)=m(|x|).f(v),div((|▽u|p-2▽u)=m(|x|).f(v), where x∈RN,B≥3.