Positive Solutions To Some Schr(?)dinger Systems With Critical Growth

In this thesis,we study three coupled Schr(?)dinger systems with critical growth.Our results are summarized as follows.1.We study the system of nonlinear Schr(?)dinger equations with critical expo-nent:where the coefficients μ(x),ν(x),λ(x)are nonnegative functions.The existence of a positive solution to the system is proved using Nehari manifold method and concentration-compactness type arguments.2.We study the multiplicity of positive solutions to the following linearly cou-pled Schr(?)dinger system with critical exponent:where f,g ∈ L2N/N(+2)(Ω)are nonnegative functions,-λ1(Ω)<μ,ν<0 for N≥4 and-λ1(Ω)<μ,ν<-1/4λ1(Ω)for N=3.We prove thath the system has at least two positive solutions:one positive ground state solution and one positive bound state solution.3.We study the fol owing coupled Schr(?)dinger system with critical nonlineari-ties where β>0,f,g are differentiable functions with critical growth,and H,K are primitive functions of h and k respectively.Under certain assumptions on f,g,h and k,we obtain the existence of a positive ground state solution of the system for N≥2.