Existence And Orbital Stability Of Normalized Solutions To Nonlinear Schr(?)dinger Equations And Systems

In this thesis,we study the existence and orbital stability of normalized solutions to two classes of nonlinear Schr(?)dinger systems in RN and a class of fourth-order nonlinear Schr(?)dinger equations in RN.Here normalized solutions correspond to critical points of the underlying energy functional restricted to the L2-norm constraint.The main ingredients of research are variational.Basically,the thesis is divided into 6 chapters.The first chapter consists in an introduction of the problems we treat and a presentation of the results we obtain.Then,we consider a type of nonlinear Schr(?)dinger systems in RN on the constraint?RN|u1|2dx=a1>0,?RN|u2|2dx=a2>0,(2)where N ?1,?1,?2,?>0,2<p1,p2<2*,r1,r2>1,r1+ r2<2*,and 2*:=2N/(N-2)+.In particular,the parameters ?1,?2 are unknown and appear as Lagrange multipliers.Firstly,we study the existence and orbital stability of solutions of the problem under the assumption N>1,?1,?2,?>0,2<p1,p2<2 +4/Nr1,r2>1,r1+ r2<2 +4/N.In this case,the associated energy functional subject to the constraint(2)is bounded from below.Thus we are able to introduce a global minimization problem,whose minimizers are solutions of the problem.By making use of the coupled rearrangement arguments,we establish the compactness of any minimizing sequence and the orbital stability of the set of global minimizers.Secondly,we concern the existence and orbital stability of solutions of the problem under the assumptions N ? 1,?1,?2,,?>0,2<p1,p2<2+4/N,r1,r2>1,r1+r2>2 +4/N,and N>1,?1,?2,?>0,2<p1,p2>2 +4/N,r1,r2>1,r1 + r2<2 + 4/N.In both cases,the energy functional restricted to the constraint becomes unbounded from below.In such situation,we prove that there are two distinct solutions,one is a local minimizer and the other is a saddle type solution.In addition,we obtain the orbital stability of the set of local minimizers.Subsequently,we consider a class of nonlinear Schr(?)dinger systems in RN with a partial confinement on the constraint?R3|u1|2dx=a1>0,?R3|u2|2dx=a2>0,where ?1,?2,?>0,2<p1,p2<10/3,r1,r2>1,r1+r2<10/3,and ?1,?2 are Lagrange multipliers.Note that the associated energy functional on the constraint is bounded from below,by applying the coupled rearrangement arguments,we prove that any minimizing sequence of the associated minimization problem is compact and the set of global minimizers is orbitally stable.Later,we deal with a class of fourth-order nonlinear Schr(?)dinger equations in RN??2u-?u+?u=|u|2?u,on the constraint?RN|u|2dx=c>0,where N>1,?>0,aN>4,and ? is a Lagrange multiplier.Based upon a Po-hozaev type manifold,the existence of ground state solution and the multiplicity of bound state solutions are established.In addition,we prove that any radially symmetric ground state solution is unstable.Finally,we give some related remarks and put forward some interesting prob-lems to research.