July  2010, 9(4): 955-962. doi: 10.3934/cpaa.2010.9.955

The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case

1. 

Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

Received  October 2009 Revised  January 2010 Published  April 2010

We prove that critical vector-valued Schrödinger equations on compact Riemannian manifolds possess only constant solutions when the potential is sufficiently small. We prove the result in dimension $n = 3$ for arbitrary manifolds and in dimension $n \ge 4$ for manifolds with positive curvature. We also establish a gap estimate on the smallness of the potentials for the specific case of $S^1(T)\times S^{n-1}$.
Citation: Emmanuel Hebey. The Lin-Ni's conjecture for vector-valued Schrödinger equations in the closed case. Communications on Pure and Applied Analysis, 2010, 9 (4) : 955-962. doi: 10.3934/cpaa.2010.9.955
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