The Existence And Properties Of Solutions For Two Classes Of Nonlinear Equations In Field Theory

This paper is attempted to discuss the boundary value problem of a class of ordinary differential equations derived by 't Hooft-Polyakov Magnetic monopole model and the boundary value problem of a class of Gross-Pitaevskii equation.For the existence of the solution of the first class of ordinary differential equation,we need to determine an initial slope to transform the boundary value problem of the original problem into the initial value problem,and then explore the existence of the solution in the cases of different parameters.For second class equations,we consider radial symmetric solutions.The problem is turned into the boundary value problem of ordinary differential equations.Two methods are used here to prove the existence of the solution.Taking the first method,we consider the symmetric mountain lemma.The relevant P-S conditions and the mountain structure need to be verified to prove the existence of the critical point.In addition,the existence of solutions can be proved by the shooting method in reference[8].