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Papers by Bruno Bianchini

Research paper thumbnail of Non periodic solutions of fourth order nonlinear equations

Research paper thumbnail of Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

Journal of Differential Equations, 2016

Research paper thumbnail of Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds

Journal of Functional Analysis, 2015

1 In this work, we study the existence problem for positive solutions of the Yamabe type equation

Research paper thumbnail of Yamabe type equations with sign-changing nonlinearities on the Heisenberg group, and the role of Green functions

Contemporary Mathematics, 2013

Research paper thumbnail of Some generalizations of Calabi compactness theorem

Arxiv preprint arXiv:1112.3703, 2011

1 In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian ma... more 1 In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian manifolds that allow the presence of negative amounts of Ricci curvature. These, in turn, can be rephrased as new conditions for the positivity, for the existence of a first zero and for the nonoscillatory-oscillatory behaviour of a solution g(t) of g ′′ + Kg = 0, subjected to the initial condition g(0) = 0, g ′ (0) = 1. A unified approach for this ODE, based on the notion of critical curve, is presented. With the aid of suitable examples, we show that our new criteria are sharp and, even for K ≥ 0, in borderline cases they improve on previous works of Calabi, Hille-Nehari and Moore.

Research paper thumbnail of Nonexistence and Uniqueness of Positive Solutions of Yamabe Type Equations on Nonpositively Curved Manifolds

We prove nonexistence and uniqueness of positive C2{solutions of the elliptic equation u + a(x)u ... more We prove nonexistence and uniqueness of positive C2{solutions of the elliptic equation u + a(x)u K(x)u =0 , >1, on a nonpositively curved, complete manifold (M;g ).

Research paper thumbnail of Spectral radius, index estimates for Schrödinger operators and geometric applications

Journal of Functional Analysis, 2009

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of... more In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation (vz ′ ) ′ +Avz = 0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.

Research paper thumbnail of On some aspects of oscillation theory and geometry

Memoirs of the American Mathematical Society, 2012

ABSTRACT The aim of this paper is to analyze some of the relationships between oscillation theory... more ABSTRACT The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation we prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE’s that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep our investigation basically self-contained we also collect some, more or less known, material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.

Research paper thumbnail of Non periodic solutions of fourth order nonlinear equations

Research paper thumbnail of Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

Journal of Differential Equations, 2016

Research paper thumbnail of Yamabe type equations with sign-changing nonlinearities on non-compact Riemannian manifolds

Journal of Functional Analysis, 2015

1 In this work, we study the existence problem for positive solutions of the Yamabe type equation

Research paper thumbnail of Yamabe type equations with sign-changing nonlinearities on the Heisenberg group, and the role of Green functions

Contemporary Mathematics, 2013

Research paper thumbnail of Some generalizations of Calabi compactness theorem

Arxiv preprint arXiv:1112.3703, 2011

1 In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian ma... more 1 In this paper we obtain generalized Calabi-type compactness criteria for complete Riemannian manifolds that allow the presence of negative amounts of Ricci curvature. These, in turn, can be rephrased as new conditions for the positivity, for the existence of a first zero and for the nonoscillatory-oscillatory behaviour of a solution g(t) of g ′′ + Kg = 0, subjected to the initial condition g(0) = 0, g ′ (0) = 1. A unified approach for this ODE, based on the notion of critical curve, is presented. With the aid of suitable examples, we show that our new criteria are sharp and, even for K ≥ 0, in borderline cases they improve on previous works of Calabi, Hille-Nehari and Moore.

Research paper thumbnail of Nonexistence and Uniqueness of Positive Solutions of Yamabe Type Equations on Nonpositively Curved Manifolds

We prove nonexistence and uniqueness of positive C2{solutions of the elliptic equation u + a(x)u ... more We prove nonexistence and uniqueness of positive C2{solutions of the elliptic equation u + a(x)u K(x)u =0 , >1, on a nonpositively curved, complete manifold (M;g ).

Research paper thumbnail of Spectral radius, index estimates for Schrödinger operators and geometric applications

Journal of Functional Analysis, 2009

In this paper we study the existence of a first zero and the oscillatory behavior of solutions of... more In this paper we study the existence of a first zero and the oscillatory behavior of solutions of the ordinary differential equation (vz ′ ) ′ +Avz = 0, where A, v are functions arising from geometry. In particular, we introduce a new technique to estimate the distance between two consecutive zeros. These results are applied in the setting of complete Riemannian manifolds: in particular, we prove index bounds for certain Schrödinger operators, and an estimate of the growth of the spectral radius of the Laplacian outside compact sets when the volume growth is faster than exponential. Applications to the geometry of complete minimal hypersurfaces of Euclidean space, to minimal surfaces and to the Yamabe problem are discussed.

Research paper thumbnail of On some aspects of oscillation theory and geometry

Memoirs of the American Mathematical Society, 2012

ABSTRACT The aim of this paper is to analyze some of the relationships between oscillation theory... more ABSTRACT The aim of this paper is to analyze some of the relationships between oscillation theory for linear ordinary differential equations on the real line (shortly, ODE) and the geometry of complete Riemannian manifolds. With this motivation we prove some new results in both directions, ranging from oscillation and nonoscillation conditions for ODE’s that improve on classical criteria, to estimates in the spectral theory of some geometric differential operator on Riemannian manifolds with related topological and geometric applications. To keep our investigation basically self-contained we also collect some, more or less known, material which often appears in the literature in various forms and for which we give, in some instances, new proofs according to our specific point of view.