Segregated and synchronized vector solutions to linearly coupled systems of Schrödinger equations

摘要

In this paper, we study the following linearly coupled system −ε2Δui+Pi(x)ui=ui3+∑j≠iNλijuj,ui∈H1(R3),i=1,…,N, where ε > 0 is a small parameter, Pi(x) are positive potentials, and λij = λji > 0 (i ≠ j) are coupling constants for i, j = 1, …, N. We investigate the effect of potentials to the structure of the solutions. More precisely, we construct multi-spikes solutions concentrating near the local maximum point x0i of Pi(x). When x0i=x0j, Pi(x0i)=Pj(x0j)=a,i≠j, i,j=1,…,N, the components have spikes clustering at the same point as ε → 0⁺. When x0i≠x0j, i≠j, the components have spikes clustering at the different points as ε → 0⁺.