Existence And Stability Of Standing Waves For A Planar Gauged Nonlinear Schr?dinger Equation

In this thesis,we mainly study the existence,stability,quantitative property and asymptotic behavior of standing waves for the planar gauged nonlinear Schr?dinger equation arising from the Chern-Simons theory.The thesis consists of five chapters:In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.We also give some preliminary results and notations used in the whole thesis.In Chapter Two,we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear C1er1-Si1ons-Schr?dinger equations in R2 where p ?[4,+?)andTo get such solutions we look for critical points of the energy functional on the constraintsWhen p = 4,we prove a sufficient condition for the nonexistence of constrain critical points of I on Sr(c)for certain c and get infinitely many minimizers of I on Sr(87r).For the value p ?(4,+?)considered,the functional I is unbounded from below on Sr(c).By using the constrained minimization method on a suitable submanifold of Se(C),we prove that for certain c>0,I has a critical point on S,(c).After that,we get an H1-bifurcation result of our problem.Moreover,by using a minimax procedure,we prove that there axe infinitely many critical points of I restricted on Sr(c)for any c ?(0,(?))The main results in this chapter extend the main results in Byeon,Huh and Seok(J.Funct.Analysis.2012),Huh(J.Math.Phys.2012)and have been published in(Ann.Acad.Sci.Fenn.Math.2017).In Chapter Three,we study the existence and asymptotic behavior of the least energy sign-changing solutions for a gauged nonlinear Schr?dinger equation where ?,?>0,p>6 andCombining constraint minimization method and quantitative deformation lemma,we prove that the problem possesses at least one least energy sign-changing solution u?,which changes sign exactly once.Moreover,we show that the energy of u? is strictly larger than two times of the ground state energy related to(E2).Finally,the asymptotic behavior of u? as ??0 is also analyzed.The main results in this chapter have been published in(J.Math.Anal.Appl.2017).In Chapter Four,we prove the existence,stability and quantitative property of standing waves for the following planar gauged nonlinear Schr?dinger equationwhere i denotes the imaginary unit,(?)0 =(?)/(?)t.(?)1=(?)/(?)x1,(?)2=(?)/(?)x2 for(t,x1,x2)? R1+2,?:R1+2 ? C is the complex scalar field,A?:R1+2 ? R is the gauge field and D?=(?)?+iA? is the covariant derivative for ? = 0,1,2 and ?'>0 is a constarnt representing the strength of interaction potential.This is a covariant nonlinear Schr?dinger type problem,which is mass-critical and arises from the Chern-Simons theory.In the defocusing case(?'<1),we prove the existence of stable standing waves with arbitrary mass.However,in the focusing case(?'? 1),stable standing waves with small mass exist and we prove a mass collapse behavior of the standing waves.The main results in this chapter can be viewed as a complement of the main results in Berge,Bouard and Saut(Nonlinearity 1995),Huh(Nonlinearity 2009)about the dynamic of system(E3).In Chapter Five,we study the existence,multiplicity,quantitative property and asymptotic behavior of normalized solutions for a gauged nonlinear Schr?dinger equationwhere ??R,?>0,p>4 and Combining constraint minimization method and minimax principle,we prove that the problem possesses at least two normalized solutions,one is a ground state and the other is an excited state.Furthermore,the asymptotic behavior and quantitative property of the ground state are analyzed.The main results in this chapter extend the results in Bellazzini,Boussaid,Jeanjean and Visciglia(Comm.Math.Phys.2017),which dealt with semi-linear Schr?dinger equations with a cigar-shaped po-tential,to the harmonic trapped Chern-Simons-Schr?dinger equation(E4),and have been accept for publication in(Z.Angew.Math.Phys.2018).