Sobolev inequalities for the Hardy-Schrödinger operator: extremals and critical dimensions

DOI10.1007/s13373-015-0075-9zbMath1336.35145arXiv1506.05787OpenAlexW2136368562WikidataQ59465553 ScholiaQ59465553MaRDI QIDQ270064

Nassif Ghoussoub, Frédéric Robert

Publication date: 7 April 2016

Published in: Bulletin of Mathematical Sciences (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1506.05787

zbMATH Keywords

positive mass theoremSobolev inequalitiesboundary singularityCaffarelli-Kohn-Nirenberg inequalitiesHardy-Schrödinger operator

Mathematics Subject Classification ID

Nonlinear elliptic equations (35J60) Elliptic equations on manifolds, general theory (58J05) Variational methods for higher-order elliptic equations (35J35) Blow-up in context of PDEs (35B44)

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Cites Work

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• Minimizing problems for the Hardy-Sobolev type inequality with the singularity on the boundary
• Existence and bifurcation of solutions for an elliptic degenerate problem
• Solutions for semilinear elliptic equations with critical exponents and Hardy potential
• Positive solutions for some singular critical growth nonlinear elliptic equations
• Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
• Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent
• Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential
• The effect of curvature on the best constant in the Hardy-Sobolev inequalities
• The concentration-compactness principle in the calculus of variations. The limit case. I
• The concentration-compactness principle in the calculus of variations. The limit case. II
• Optimal Hardy inequalities for general elliptic operators with improvements
• Revisiting an idea of Brézis and Nirenberg
• Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary
• A compactness theorem for the Yamabe problem
• Conformal deformation of a Riemannian metric to constant scalar curvature
• Elliptic equations with nearly critical growth
• Best constant in Sobolev inequality
• Problèmes isoperimetriques et espaces de Sobolev
• The role played by space dimension in elliptic critical problems
• A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms.
• Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents
• Elliptic partial differential equations of second order
• Elliptic equations with critical Sobolev exponents in dimension 3
• Optimizing improved Hardy inequalities
• The Hardy-Schrödinger operator with interior singularity: the remaining cases
• Positive solutions for singular critical elliptic problems.
• Hardy-Sobolev critical elliptic equations with boundary singularities
• Elliptic problems with critical exponents and Hardy potentials.
• The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds
• Prescribed scalar curvature on the \(n\)-sphere
• Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities
• On the best possible remaining term in the Hardy inequality
• Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
• Elliptic equations with critical growth and a large set of boundary singularities
• Concentration estimates for Emden-Fowler equations with boundary singularities and critical growth
• On the symmetry of extremals for the Caffarelli-Kohn-Nirenberg inequalities
• The Yamabe problem
• Positive Solutions of Semilinear Equations in Cones
• Symmetry of extremal functions for the Caffarelli-Kohn-Nirenberg inequalities
• Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents
• YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS
• Optimal Sobolev inequalities and extremal functions. The three dimensional case
• Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
• The 𝐴𝐵 program in geometric analysis: sharp Sobolev inequalities and related problems
• Functional Inequalities: New Perspectives and New Applications
• Variational Methods