Positive Ground State Solutions To Two Classes Of Quadratically Coupled Schr(?)dinger Systems

In nonlinear optic theory,the propagation of soliton in quadratic nonlinear fiber cou-plers can be described by a set of coupled nonlinear Schr(?)dinger equation(CNLSE).The existence of solitary waves to CNLSE has been widely investigated in recent years.In this paper,the quadratically coupled of Schr(?)dinger equations are discussed in R6.Therefore,all the quadratical nonlinearities and coupling terms in the systems are of growth since 2*= 2N/(N-2)= 3.Mathematically,it becomes more interesting and challenging because of the lack of the compactness in the critical situation.Therefore,the systems in this paper provide a series of operational theoretical basis in the application of physics and other fields.Hence,further research and exploration of the quadratical nonlinear schrodinger systems have a highly investigative value.To be specific,we consider existence of positive ground state solutions for the following quadratically coupled Schr(?)dinger systems in present paperwhere Ω(?)Z C R6 is a smooth domain,-λ(Ω)<λ1,λ2<0,μ1,μ2,α,γ>0,andλ(Ω)is the first eigenvalue of-Δ with the Dirichlet boundary condition.Depending on the value of λ1andλ2,The systems are divided into two types:λ1 =λ2 or λ1 ≠λ2 respectively.Our aim is to determine the relationship between the existence of the positive ground state solutions and parameters.Therefore,we have the following conditions:(C)μ1-2γ ≤ 3/γ1/3α2/3,μ2-2α ≤3α1/3γ2/3When λ1 =λ2,the above Schr(?)dinger equations are called the quadratically coupled homomorphic Schr(?)dinger equations;When λ1 ≠λ2,the above Schr(?)dinger equations are called the quadratically coupled heteromorphic Schr(?)dinger equations.This paper is divided into two chapters.In the first chapter,under the(C)conditions,we investigate the existence of positive ground state solutions to the quadratically coupled homomorphic Schr6dinger equations.In the second chapter,under the(C)conditions,the existence of positive ground state solutions is discussed to the quadratically coupled heteromorphic Schr(?)dinger equa-tions.