A finiteness theorem for the space of Lp harmonic sections

In this paper we give a unified and improved treatment to finite dimensionality results for subspaces of Lp harmonic sections of Riemannian or Hermitian vector bundles over complete manifolds. The geometric conditions on the manifold are subsumed by the assumption that the Morse index of a related Schro \u308dinger operator is finite. Applications of the finiteness theorem to concrete geometric situations are also presented

Some non-linear function theoretic properties of Riemannian manifolds

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the $p$-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators

A Liouville theorem for a class of superlinear elliptic equations on cones

We give sufficient conditions for non-existence of positive solutions of the equation \u394u + a(x)u + b(x)up = 0 on a cone of \u211dn We further analyze the existence of positive solutions in the radial, subcritical case, and show that under suitable conditions on the coefficients, every radial solution whose value in 0 is sufficiently large must vanish. 2000 Mathematics Subject Classification: Primary: 35J60 Secondary: 35B05, 35R45

On the Lsp2Lsp 2 form spectrum of the Laplacian on nonnegatively curved manifolds

Let $(M,g o)$ be a complete, noncompact Riemannian manifold with a pole, and let $g=fg o$ be a conformally related metric. We obtain conditions on the curvature of $g o$ and on $f$ under which the Laplacian on $p$-forms on $(M,g)$ has no eigenvalues

Vanishing and Finiteness Results in Geometric Analysis \ub7 A Generalization of the Bochner Technique

Accettato per la pubblicazione in Progress in Mathematics, Birkhauser Verla

Existence and non-existence results for a logistic-type equation on manifolds

We study the steady state solutions of a generalized logistic-type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls