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Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

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arxiv 2203.03345 v2 pith:YJ2BS3X2 submitted 2022-03-07 math.AP math.DG

Giovanni Catino , Dario Daniele Monticelli This is my paper · ORCID
classification math.APmath.DG
keywords nonnegativesolutionscasecriticalcurvatureellipticequationsmanifolds
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In this paper we prove classification results for solutions to subcritical and critical semilinear elliptic equations with a nonnegative potential on noncompact manifolds with nonnegative Ricci curvature. We show in the subcritical case that all nonnegative solutions vanish identically. Moreover, under some natural assumptions, in the critical case we prove a strong rigidity result, namely we classify all nontrivial solutions showing that they exist only if the potential is constant and the manifold is isometric to the Euclidean space.

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  1. Liouville theorem of the subcritical biharmonic equation on complete manifolds

    math.AP 2025-08 unverdicted novelty 5.0 of 10

    For n>=5 and 1<alpha<(n+4)/(n-4), the equation Delta^2 u = u^alpha admits no positive C^4 solution on complete noncompact manifolds with nonnegative Ricci curvature.

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