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

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  • 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 $S^1(T)\times S^{n-1}$.
    Mathematics Subject Classification: Primary: 35J62, 35J47, 58J05; Secondary: 35J10.


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