Positive Entire Solution To System Of Second Order Nonlinear Elliptic Equation

In this paper, the second order nonlinear elliptic systemwith α, γ < 1, β ≥ 1 is considered in R^N, N ≥ 3. Under a set of suitable hypotheses on function f_i, g_i h_i and P, it is shown that this system possesses an entire decaying positive solution (u,v) ∈ C_loc^2,θ(R^N) × C_loc^2,θ(R^N).The paper is mainly divided into four sections.In the fisrt section, we give an introduction of this paper. It includes the background of this problem, the main work in this paper and some open problems.Next, we privide some basal definitions and theories in the second section, such as the maximum principle, Holder continuity, Schauder theory and so on, which are foundation of the later work.The third section of this paper solves the existence and uniqueness of the minimal positive entire solutions to a certain second order nonlinear elliptic equation. To solve this problem, upper and lower method and comparison lemma are introduced.At last, we establish the theory of the existence of decaying positive solutions for system (*). And some sufficient conditions of the existence of positive solutions to similar predator-prey type and competition type are provided.