Existence Of Multiple Positive Solutions For A Class Of Semilinear Elliptic Systems

In this thesis,we study the following semilinear elliptic system:(?)where ?>1,?>1,?+?<2*:=2N/N-2(N?3).We use the variational method to prove if h1(x),h2(x)satisfies:(?)where C?,?=(?+?-2)(?+?-1)-?+?-1/?+?-2,and,(?)Then there are at least two positive solutions for the system(1).When we prove the first positive solution of the system(1),we first introduce the system:(?)We use the Ekeland variational principle on the Nehari manifold to prove that when h1(x),h2(x)satisfies(H1),(H2),system(2)has at least one positive solution,whereC?,?(?+?-2-)(?+?-1)-?+?-1?+?-2(1-?)?+?-1/?+?-2,??[0,1).It is easy to know that ?=0 in the system(2)is the system(1).At the same time,we consider the following system:(?)We use the method of subsolutions and supersolutions to prove that when h1(x),h2(x)satisfies(Hl),(H2),and F ?C1(R2)satisfies:(?)(H3)0?Fu(u,v)??/?+?|u|?-2u|v|?+?u,0?Fu(u,v)??/?+?|u|?|u|?-1v+?u System(3)has at least one positive solution.This thesis is divided into four chapters:The first chapter mainly explains the research progress of the above semilinear elliptic system and the main results of this thesis.The second chapter mainly introduces some basic concepts,basic principles,and several important inequalities that will be used in this thesis.In the third chapter,we respectively prove the existence of the positive solution of the system(2)and the system(3).The fourth chapter mainly proves that the system(1)have at least two positive solu-tions:take ?=0 in system(2)to get the first positive solution of system(1),then proved the existence of the second positive solution.