Structure Of Positive Solutions Of A Class Of Elliptic Systems

Firstly, we study the structure and properties of positive solutions of the semilinear elliptic system: which models the changes of the densities of two competing species in the domainΩ, whereΩ(?)R~N(N≥2) is a bounded and connected domain with smooth boundary (?)Ω, u(x) and v(x) are the densities of the two species, a and c are the increasing and decreasing rates of the providence of food, b and d are the competing rates between the species. We already know the existence of semi-trivial solutions (u, 0), (0, v) of (2.1.1). In this paper we mainly are interested in the structure and properties of non-trivial positive solution (u, v) of (2.1.1). By using the theory of bifurcation from infinity and the monotone techniques, we successfully obtain the existence and multiplicity of non-trivial positive solutions (2.1.1). Moreover, the continuity of solutions relying on parameters is also studied.Secondly, we study the structure and properties of positive solutions of the quasilinear elliptic system We also obtain the existence and multiplicity of non-trivial positive solutions of (3.1.1). Since the strong comparison principle is lost for our operators-△_p,-△_q with p≠2, q≠2, we cannot provide the accurate structure of the non-trivial positive solutions of (3.1.1).