The Existence Of Positive Stationary Solutions For Several Nonlinear Reaction-Diffusion Equations

Several typical kinds of PDE models called reaction-diffusion equations have beenconsidered in this thesis and the existence results of the respective elliptic equations are obtained.This thesis is mainly composed of three parts of contents:In the first part, a class of reaction-diffusion system discrbes the morphogenesis inmathematical biology is studied, by means of calculating the indices of fixed points of compactmaps in cones, in combination with the maximum principles and the decoupling method, westudy the coexistence and uniqueness of positive stationary solutions for this system. The processindicates that the decoupling method for the class of differential equations is similar to theelimination method in algebra.The model considered in the second part is arising from the nuclear reactors, which is asemilinear parabolic system. By an abstract fixed point theorem on positive cone together withthe technique of a priori estimate, we study the initial-boundary value problem for a class ofreaction-diffusion system, and the coexistence of positive stationary solutions and uniqueness ofordered positive stationary solutions are obtained. Moreover, by the existence results ofstationary equations we just obtained, we observe the global e,,dstence and blowup of positivesolutions and obtain a threshold result. At last we deal with the blowup rate estimates of positiveblowup solutions.The third part studies the positive solutions for a class of multiparameter p-Laplacianproblem which is a quasilinear equation. Via the upper and lower solution method and the criticalpoint theory, the existence and multiplicity of positive solutions are obtained when bothparameters in their given intervals.Differential equations are transformed to the operator equation or the functional equation inall the above parts. By the existence of the fixed point for the operaor equation or the minimumof the variational functional, we obtain the existence of positive solutions for the differentialequations, in which the various kinds of fixed point theorem and the critical point theory play animportant role.