Licois, Jean René; Véron, Laurent 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 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) Cited in 8 Documents 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 PDF BibTeX XML Cite \textit{J. R. Licois} and \textit{L. Véron}, C. R. Acad. Sci., Paris, Sér. I 320, No. 11, 1337--1342 (1995; Zbl 0839.53031)