A Study On The Existence Of Standing Wave Solutions For Generalized Jackiw-Pi Models

Chern-Simons gauge theory is a hot issue in applied mathematics and quantum mechanics.It has theoretical significance and application value in high-temperature superconductivity and fractional quantum Hall effect.Jackiw-Pi model is that Jackiw and others introduced Chern-Simons theory into physics to describe the long-term dynamics of multiple particles on the plane.These particles not only interact,but also interact through the electromagnetic field generated between particles,so their long-term dynamics are very complex.Because a large part of the actual problems are fixed state problems,so the existence of the equation standing wave solutions is also a very important research topic.This paper focuses on the existence of standing wave solutions for the generalized Jackiw-Pi model.We simplify the model into a semi-linear elliptic equation by assuming the standing wave solution of the model,and we derive the energy functional corresponding to the equation using the variational method.Furthermore,Pohozaev identity and Sobolev type inequality are constructed for the special structure of the model.We prove the existence and nonexistence of model standing wave solutions combined with critical point theory,functional analysis and minimization theory.Meanwhile,it is shown that the model has energy conservation and gauge invariance.This article contains five parts.The first part is the introduction,which introduces the physical background of the model and the current domestic and foreign research status related to this paper.The second part is the preparatory knowledge,which explains the symbols appearing in the article and defines the workspace.In addition,we briefly introduce the basic knowledge of the theorems,inequality and formulas involved in this paper,and also outline the theorem.In the third part,we prove the equation for energy conservation as well as gauge invariance.The fourth part is the proof of the paper lemma and theorem.In this chapter,a series of lemmas are first proposed,including the energy functional and properties of elliptic equation and the Pohozaev equality and Sobolev inequality with the help of divergence theorem and other tools.Secondly,by discussing the constant of the interaction potential in the equation,the proved lemma and critical point theory can obtain the existence and nonexistence of the standing wave solution of the equation.The fifth part is the study summary and the outlook for the future research.