Multiple Peak Solutions For Linearly Coupled Schr(?)dinger Systems

In this thesis,we study the following linearly coupled Schr(?)dinger systems:(?)where ?>0,??R,? is a smooth and bounded domain in R3,and n is the outer normal vector defined on(?)?.the boundary of ?.By using Lyapunov-Schmidt reduction method,local analysis,and variational tech-niques,we prove the following results.1.There exist 0<?0<1 sufficiently small and ?1<0 such that for any 0<?<?0,0<?<1 and A??1/?1-2,the problem(A?)has O(1/?3|ln ?|3)many synchronized vector solutions.2.For the above ? and ? in the first result,the problem(A?)has O(1/?3)many syn-chronized vector solutions,and the number of synchronized vector solutions to(A?)is optimal.3.For the above ? and ? in the first result,we prove the existence of solutions with multi-peaks to the system(A?)with all peaks being on the boundary and all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain under the condition that the mean curvature H(P)of the boundary(?)? has several local minimums or local maximums.4.For the above ? and ? in the first result,the problem(A?)has a synchronized vector solution with multiple spikes both on the boundary and in the interior of the domain ?,where the interior spikes concentrate at sphere packing points in ? and the boundary spikes concentrate at multiple critical points of the mean curvature function H(P),P ?(?).