Existence Of Solutions To The Optical Vortex Schrödinger Equation

In this paper,we mainly study a class of doubly coupled optical vortex Schr?dinger equations.We mainly apply the method of constraint minimization to prove the existence of constraint solution for the equations.We obtain the existence of mountain-pass solution for the equations by the mountain pass theorem.In the first chapter,the research background and significance of the optical vortex Schr?dinger equation are briefly discussed,and the research status of the related equations and equation system,the main research results and some commonly used mathematical notations in this paper are described.In the second chapter,the preliminary knowledge of proving theorem 1.1.1 is introduced,and the propositions obtained are proved.Finally,Theorem 1.1.1 is proved completely.The third chapter consists of three sections.The first section introduces the basic definitions,theorems and basic theoretical knowledge which are needed to find mountain pass solution.The second section verifies the condition of the mountain pass theorem and proves Theorem 1.1.2.In the third section,the positive properties of the mountain path solution are proved by the Nehari manifold method.