A vanishing theorem for nonlinear elliptic equations on compact Riemannian manifolds. (Un théorème d’annulation pour des équations elliptiques non linéaires sur des variétés riemanniennes compactes.) (French. Abridged English version) Zbl 0839.53031

Licois, Jean René; Véron, Laurent

C. R. Acad. Sci., Paris, Sér. I 320, No. 11, 1337-1342 (1995).

Assuming the inequality \(\text{Ricc}_g\geq Rg\) \((R\geq 0)\) for the Ricci tensor of the metric \(g\), and inequalities \(1< q\leq (n+2)/ (n-2)\) (which may be omitted if \(n\leq 2)\) \[ (q-1) \lambda\leq \lambda_1+ {{q(n-1)n} \over {q+n (n+2)}} \biggl( R- {{n-1} \over n} \lambda_1 \biggr) \] (which should be strict if \(q= (n+2)/ (n-2)\) and the underlying space is conformally diffeomorphic to the sphere), where \(\lambda_1\) is the first nonzero eigenvalue of the Laplacian, the authors prove that any positive solution of the equation \(\Delta u+ u^q= \lambda u\) is a constant.

Reviewer: J.Chrastina (Brno)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J05	Elliptic equations on manifolds, general theory
58J50	Spectral problems; spectral geometry; scattering theory on manifolds

Keywords:

Ricci curvature; first nonzero eigenvalue; Laplacian