Vector Solutions with Prescribed Component-Wise Nodes for a Schrodinger System

For the Schrodinger system{-△uj+λjuj+k∑i=1βijui^2uj in R^N,uj(x)→0 as|x|→∞,j=1,…,k where k≥2 and N=2,3,we prove that for anyλj>0 andβjj>0 and any positive integers pj,j=1,2,…,k,there exists b>0 such that ifβij=βji≤b for all i≠j then there exists a radial solution(u1,u2,…uk)with uj having exactly Pj-1 zeroes.Moreover,there exists a positive constant Co such that ifβij=βji≤b(i≠j)then any solution obtained satisfies k∑i,j=1|βij|∫R^Nui^2uj^2≤C0.Therefore,the solutions exhibit a trend of phase separations asβij→-∞for i≠j.

For the Schr?dinger system ■where k ≥ 2 and N = 2,3, we prove that for any λj> 0 and βjj> 0 and any positive integers pj, j = 1,2,···,k, there exists b > 0 such that if βij= βji≤ b for all i ≠ j then there exists a radial solution（u1,u2,···,uk） with ujhaving exactly pj-1 zeroes. Moreover,there exists a positive constant C0 such that if βij= βji≤ b（i ≠ j） then any solution obtained satisfies ■Therefore, the solutions exhibit a trend of phase separations as βij→-∞ for i ≠ j.