Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds. (English) Zbl 1329.58020
Let \(\Delta\) be the scalar Laplacian on a complete non-compact Riemannian manifold \((M,g)\) of dimension \(m\). For \(\sigma>1\) and for \(V(x)\) a potential function, the authors study the problem \(\Delta u+V(x)u^\sigma\leq0\). The authors investigate the relationship between the geometry of \((M,g)\), the behavior of the potential \(V\) at infinity, and the existence or non-existence of non-negative solutions. The introduction presents the problem in historical context, introduces the basic notational conventions, and summarizes the results of the paper. Section 2 gives some preliminary results. The remaining 2 sections prove the results of the paper.
Reviewer: Peter B. Gilkey (Eugene)
MSC:
| 58J05 | Elliptic equations on manifolds, general theory |
| 35B09 | Positive solutions to PDEs |
| 35B53 | Liouville theorems and Phragmén-Lindelöf theorems in context of PDEs |
| 35J62 | Quasilinear elliptic equations |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
Keywords:
Laplacian with potential; complete Riemannian manifold; weighted volume growth; volume growth geodesic ball; spherically symmetric manifold; hyperbolic spaceReferences:
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