Non-radial Solutions For A Class Of Nonlinear Schr(o|¨ )dinger Equations In Electromagnetic Fields

In the study of partial differential equations, the study of the singularly per-turbed elliptic problems has been very mature. For the nonlinear Schrodinger equation without magnetic field, there are many results about the existence and multiplicity of solutions. And once with the magnetic field, the solution is no longer real valued, therefore it becomes more complex, and the corresponding research is not so rich. In this paper, we consider a class of non radially symmetric solutions of nonlinear Schrodinger equations with magnetic field.We consider the equation of the following from ε2(iâ–½+A(|x|))2+u=|u|p-1u,x∈RN,u:RNâ†'C. where A (x)=(A1(x),…, AN (x)), Aj (x) j= 1.2…N is real function, A (|x|) is radially symmetric. We mainly study the existence and multiplicity of solutions of the above equations. Precisely, the solution of the original equation is constructed by the local energy method, which is a new kind of symmetric complex valued solution. Moreover, there are infinite number of non radial symmetric complex solutions under the conditions (T1) and (T2).