On The Study Of Standing Waves Of Schrodinger Equations

In this paper,we mainly study the existence,concentration phenomenon and asymptotic behaviour of solutions for fractional Schodinger equations and Schrodinger systems.The thesis consists of four 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 notations and definitions used in the whole thesis.In Chapter two,we study the following fractional Schrodinger equation?2s(-?)su + Vu?|u|p-2u,x?RN.We show that if the external potential V ? C(RN;[0,?))has a local minimum and p ?(2 + 2s(N-2s),2s*),where 2s*= 2N/(N-2s),N?2s,the problem has a family of solutions concentrating at the local minimum of V provided that lim inf|x|?? V(x)|x|2s>0.The proof is based on variational methods and penalized technique.In Chapters three and four,we study the existence,concentration phenomenon and asymptotic behaviour of solutions for the following weakly coupled Schrodinger system where ?>0,??R is a coupling constant,2p ?(2,2*)with 2*=2N/N-2 if N?3 and+? if N= 1,2,V1 and V2 belong to C(RN,[0,?)).This type of systems arise from models of nonlinear optics and Bose-Einstein condensates.In Chapter 3,we use the penalized idea to make semiclassical study for the system above when ?>0 is small.We construct a new penalized function to show the problem has a family of nonstandard solutions {??=(u?1,u?2):0<?<?0}concentrating synchronously for all nonnegative potentials V1 and V2 provided that p?2 and ??0 is suitably large and small.When V1 or V2 has compact support,we answer positively the conjecture proposed by Ambrosetti and Malchiodi in[9].Surprisingly,we also obtain nonstandard solutions for the problem for all ?>0 if 1<p<2,Moreover,we know the location of concentration points in the case?>0 small and p?2 by local Pohozaev identities,see their applications in[42,60]for example.In Chapter 4,we consider the existence of positive solutions and their asymp-totic behaviour of the above system when ?=1.By constructing two types of two-dimensional mountain-pass geometries,we obtain a positive synchronized solu-tion to the above problem for |?|>0 small and a positive segregated solution to the above problem with Vi(x)(i=1,2)satisfying some additional assumptions for ?<0 respectively.We show that when 1<p<2,the positive solutions are not unique if 0<?/?? is small.The asymptotic behavior of the solutions when ??0 and??? is also studied respectively.The proofs are based on variational methods and the uniqueness of positive solutions of the corresponding single equation(?=0).We want to emphasize that we do not add radial restriction when ?<0,which and the general N,ai(i=1,2)and p make all the methods before this paper not work well(see[12,13,66,69,74]and the reference therein).