The Existence Of Positive Solutions Of A Strongly Coupled Elliptic System

In this paper, a strongly coupled elliptic system with Holling-Tanner responce is considered:whereΩis a bounded domain in R~N(N≥1) with smooth boundary (?)Ω.Here the parametersα_i,β_i,γ_i,a,b,a_i,b_i,(i= 1,2,) are all positive constants. Making use of the monotone property and Schauder fixed point theory, we give some sufficient conditions for this elliptic problem to have a coexistence state.The paper is organized as follows:First, the introduction of the whole paper presents the background of this paper and plans for the research of the problem.Next section of the paper is composed of some basic definitions and theorems, including the Holder continuity, upper and lower solutions theory and some known results which are useful in later sections.The third section provide the priori estimate of the solution and existence of the semi-coexistence state.The fourth section is the main part of this paper, where we discuss the existence of positive solutions by the upper and lower solutions property and Schauder fixed point theory. Moreover the first part of this section also discuss how to construct the upper and lower solutions of the equation.At last,we summarize the work of the whole paper and give some open problems about the equation.