scholarly journals Multipolar Hardy inequalities on Riemannian manifolds

2018 ◽  
Vol 24 (2) ◽  
pp. 551-567 ◽  
Author(s):  
Francesca Faraci ◽  
Csaba Farkas ◽  
Alexandru Kristály
Keyword(s):  
Variational Methods ◽  
Riemannian Manifolds ◽  
Quantitative Information ◽  
Beltrami Operator ◽  
Hadamard Manifolds ◽  
Hardy Inequalities ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Complete Riemannian Manifolds

We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that our inequalities deeply depend on the curvature, providing (quantitative) information about the deflection from the flat case. By using these inequalities together with variational methods and group-theoretical arguments, we also establish non-existence, existence and multiplicity results for certain Schrödinger-type problems involving the Laplace-Beltrami operator and bipolar potentials on Cartan-Hadamard manifolds and on the open upper hemisphere, respectively.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Gabriele Bonanno ◽  
Pasquale Candito ◽  
Giuseppina D’Aguí
Keyword(s):  
Variational Methods ◽  
Nonlinear Term ◽  
Asymptotic Condition ◽  
Multiplicity Results ◽  
Order Difference ◽  
Existence And Multiplicity ◽  
Finite Dimensional ◽  
Superlinear Growth ◽  
Discrete Problems ◽  
Minimum Theorem

AbstractIn this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.

scholarly journals Quantitative Rellich inequalities on Finsler—Hadamard manifolds

2016 ◽  
Vol 18 (06) ◽  
pp. 1650020 ◽  
Author(s):  
Alexandru Kristály ◽  
Dušan Repovš
Keyword(s):  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Finsler Geometry ◽  
Flag Curvature ◽  
Hadamard Manifolds ◽  
Hardy Inequalities

In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].

scholarly journals EXISTENCE OF MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC SYSTEMS INVOLVING CRITICAL HARDY–SOBOLEV EXPONENTS AND SIGN-CHANGING WEIGHT FUNCTION

2012 ◽  
Vol 17 (3) ◽  
pp. 330-350 ◽  
Author(s):  
Nemat Nyamoradi
Keyword(s):  
Weight Function ◽  
Variational Methods ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Sobolev Exponent ◽  
Quasilinear Elliptic

In this paper, we consider a class of quasilinear elliptic systems with weights and the nonlinearity involving the critical Hardy–Sobolev exponent and one sign-changing function. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiple positive solutions for a critical elliptic system with concave—convex nonlinearities

2009 ◽  
Vol 139 (6) ◽  
pp. 1163-1177 ◽  
Author(s):  
Tsing-San Hsu ◽  
Huei-Li Lin
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiple Positive Solutions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System

We consider a semilinear elliptic system with both concave—convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

Multiplicity of Positive Solutions for Critical Singular Elliptic Systems With Concave-Convex Nonlinearities

2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Tsing-San Hsu
Keyword(s):  
Variational Methods ◽  
Elliptic System ◽  
Positive Solutions ◽  
Elliptic Systems ◽  
Critical Growth ◽  
Bounded Domains ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractIn this paper, we consider a singular elliptic system with both concave-convex nonlinearities and critical growth terms in bounded domains. The existence and multiplicity results of positive solutions are obtained by variational methods.

scholarly journals Multiple Symmetric Results for Quasilinear Elliptic Systems Involving Singular Potentials and Critical Sobolev Exponents inRN

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Zhiying Deng ◽  
Yisheng Huang
Keyword(s):  
Variational Methods ◽  
Elliptic Systems ◽  
Singular Potentials ◽  
Multiplicity Results ◽  
Critical Sobolev Exponents ◽  
Existence And Multiplicity ◽  
Quasilinear Elliptic Systems ◽  
Quasilinear Elliptic

This paper deals with a class of quasilinear elliptic systems involving singular potentials and critical Sobolev exponents inRN. By using the symmetric criticality principle of Palais and variational methods, we prove several existence and multiplicity results ofG-symmetric solutions under certain appropriate hypotheses on the potentials and parameters.

scholarly journals MULTIPLE SOLUTIONS FOR A FRACTIONAL LAPLACIAN SYSTEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND HOMOGENEOUS TERM

2020 ◽  
Vol 25 (1) ◽  
pp. 1-20
Author(s):  
Jinguo Zhang ◽  
Tsing-San Hsu
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Multiple Solutions ◽  
Fractional Laplacian ◽  
Nehari Manifold ◽  
Weight Functions ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
The Nehari Manifold

In this paper, we deal with a class of fractional Laplacian system with critical Sobolev-Hardy exponents and sign-changing weight functions in a bounded domain. By exploiting the Nehari manifold and variational methods, some new existence and multiplicity results are obtain.

scholarly journals Sharp Caffarelli–Kohn–Nirenberg inequalities on Riemannian manifolds: the influence of curvature

2021 ◽  
pp. 1-26
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Remainder Term ◽  
Sharp Constant ◽  
Hadamard Manifolds ◽  
Best Constant ◽  
Quantitative Version ◽  
Rigidity Results ◽  
Ckn Inequalities ◽  
Complete Riemannian Manifolds

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

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