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

Emmanuel Hebey

doi:10.3934/cpaa.2010.9.955

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2010, Volume 9, Issue 4: 955-962. Doi: 10.3934/cpaa.2010.9.955

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

Emmanuel Hebey^1,

1.
Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received: October 2009

Revised: January 2010

Published: July 2010

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Abstract

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 $S^1(T)\times S^{n-1}$.

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Mathematics Subject Classification: Primary: 35J62, 35J47, 58J05; Secondary: 35J10.

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