Existence Of Entire Large Solutions To Nonlinear Elliptic Equations With Gradient Term

In this paper, we consider the entire large solution of the following two kinds of nonlinear elliptic partial differential equations:whereα≤0,p(x)is a nonnegative,continuous function in R~N,f(x)∈C~1[0,+∞),f'≥0,f(0)=0.Under a set of suitable conditions, with the aid of explosive super-subsolution method and the inner estimation theory of elliptic equations, some results on the existence of entire positive large solutions are obtained.The paper is divided into four sections.The first section is an introduction of the whole paper, where we talk about the background of this paper, and the plans for the research of the problem.The next section consists of some basic definitions and theorems, including the maximum principle, Holder continuity, Arzela-Ascoli Theorem, and so on. They are the foundations and tools of the later work.The third section is the main part of this paper, where we discuss the existence of entire large solutions by using the super-subsolution method, the inner estimation theory and the perturbed method. In this section, we extend the previous results partially. Then, a necessary condition is given.In the last section, we summarize the work of the whole paper and point out some open problems.