Existence Of Positive Solutions To Elliptic Systems

In this paper, some semilinear systems are investigated using sup-sub solutionsmethod, maximum principle, moving plane method, the regular theory and inner es-timate theory of elliptic equations of second order. The main results are existence,non-existence, uniqueness, radial symmetry and monotonicity.In Chapter 2, we discuss the existence and uniqueness of positive solution to semi-linear elliptic systems in global space. We consider the simple form of systems, andobtain maximal and minimal solutions by monotonous method. Moreover, we provethat maximal and minimal solutions are equal by maximum principle and compari-son principle. So systems have unique positive solution. we also obtain the other twosystems have unique positive solutions by same method.In Chapter 3, we consider semilinear elliptic systems with the weight functions. Weconvert the partial differential equations into ordinary differential equations, and getglobal solution with the related ODE knowledge. In three cases, we obtain the necessaryand suffcient conditions of existence for maximal solution by Young inequality, Gronwallinequality and other inequalities. At the same time, we get the non-existence conditionof maximal solution.In Chapter 4, this charter is devoted to the study of the positive solutions of asystem of integral equations with the weight functions. Firstly, we consider systemsincluding three equations. Under some proper assumptions, it has been proved that allthe solutions are radial symmetric and monotone decreasing about some point by themethod of moving planes integral forms, HLS and ho¨lder inequalities. Then, by thesame method, we can get the same results of systems including m(m > 3) equations.