Vector Solutions with Prescribed Component-Wise Nodes for a Schrödinger System
Year: 2019
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 3 : pp. 288–311
Abstract
For the Schrödinger system
{−Δuj+λjuj=k∑i=1βiju2iujin RN,uj(x)→0 as |x|→∞,j=1,⋯,k,
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 uj having exactly pj−1 zeros. Moreover, there exists a positive constant C0 such that if βij=βji≤b (i≠j) then any solution obtained satisfies
k∑i,j=1|βij|∫RNu2iu2j≤C0.
Therefore, the solutions exhibit a trend of phase separations as βij→−∞ for i≠j.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.OA-0009
Analysis in Theory and Applications, Vol. 35 (2019), Iss. 3 : pp. 288–311
Published online: 2019-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Vector solution prescribed component-wise nodes Schrödinger system variational methods.
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