Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials
© M. M. Fall and R. Musina. 2011
Received: 4 January 2011
Accepted: 4 March 2011
Published: 15 March 2011
We deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
In recent years, a great deal work has been made to find necessary and sufficient conditions for the existence of distributional solutions to linear elliptic equations with singular weights. Most of the papers deal with weak solutions belonging to suitable Sobolev spaces. We quote for instance, [1–4] and references therein.
In the present paper, we focus our attention on a class of model elliptic inequalities involving singular weights and we adopt the weakest possible concept of solution, that is, that one of distributional solution.
which holds for any . It is well known that the constants and are sharp and not achieved (see, e.g., [5–8] and Appendix A). Inequality (1.3) was firstly proved by Leray  in the lower-dimensional case .
Due to the sharpness of the constants in (1.3), a necessary and sufficient condition for the existence of nontrivial and nonnegative solutions to (1.1) is that (compare with Theorem B.2 in Appendix B and with Remark 2.6).
We remark that Theorem 1.1 is sharp, in view of the explicit counterexample in Remark 2.6.
(see, e.g., ). It turns out that strictly contains the standard Sobolev space , unless .
where . In our crucial Theorem 2.3, we prove a nonexistence result for (1.10), under suitable weighted integrability assumptions on . Secondly, thanks to an "averaged Emden-Fowler transform", we show that distributional solutions to problems of the form (1.1) and (1.8) give rise to solutions of (1.10); see Sections 2.2 and 3, respectively. Our main existence results readily follow from Theorem 2.3. A similar idea, but with a different functional change, was already used in  to obtain nonexistence results for a large class of superlinear problems.
In Appendix A, we give a simple proof of the Hardy-Leray inequality for maps with support in cone-like domains that includes (1.3) and that motivates our interest in problem (1.8).
In the last appendix, we point out some related results and some consequences of our main theorems.
We denote by the half real line . For , we put .We denote by the Lebesgue measure of the domain . Let and let be a nonnegative measurable function on . The weighted Lebesgue space is the space of measurable maps in with finite norm . For we simply write . We embed into via null extension.
2. Proof of Theorem 1.1
The proof consists of two steps. In the first one, we prove a nonexistence result for a class of linear ordinary differential inequalities that might have some interest in itself.
2.1. Nonexistence Results for Problem (1.10)
We need two technical lemmata.
The following result for solutions to (1.10) is a crucial step in the proofs of our main theorems.
2.2. Conclusion of the Proof
3. Cone-Like Domains
The conclusion readily follows from Theorem 2.3.
Nonexistence results for linear inequalities involving the differential operator were already obtained in .
The authors thank the Referee for his carefully reading of the paper and for his valuable comments. M. M. Fall is a research fellow from the Alexander-von-Humboldt Foundation.
- Brezis H, Marcus M: Hardy's inequalities revisited. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. IV 1997,25(1–2):217–237.MathSciNetMATHGoogle Scholar
- Adimurthi , Sandeep K: Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator. Proceedings of the Royal Society of Edinburgh A 2002,132(5):1021–1043.MathSciNetView ArticleMATHGoogle Scholar
- Dávila J, Dupaigne L: Comparison results for PDEs with a singular potential. Proceedings of the Royal Society of Edinburgh A 2003,133(1):61–83. 10.1017/S0308210500002286View ArticleMATHGoogle Scholar
- Gkikas KT: Existence and nonexistence of energy solutions for linear elliptic equations involving Hardy-type potentials. Indiana University Mathematics Journal 2009,58(5):2317–2345. 10.1512/iumj.2009.58.3626MathSciNetView ArticleMATHGoogle Scholar
- Adimurthi , Chaudhuri N, Ramaswamy M: An improved Hardy-Sobolev inequality and its application. Proceedings of the American Mathematical Society 2002,130(2):489–505. 10.1090/S0002-9939-01-06132-9MathSciNetView ArticleMATHGoogle Scholar
- Barbatis G, Filippas S, Tertikas A: Series expansion for Hardy inequalities. Indiana University Mathematics Journal 2003,52(1):171–190. 10.1512/iumj.2003.52.2207MathSciNetView ArticleMATHGoogle Scholar
- Chaudhuri N: Bounds for the best constant in an improved Hardy-Sobolev inequality. Zeitschrift für Analysis und ihre Anwendungen 2003,22(4):757–765.View ArticleMATHGoogle Scholar
- Ghoussoub N, Moradifam A: On the best possible remaining term in the Hardy inequality. Proceedings of the National Academy of Sciences of the United States of America 2008,105(37):13746–13751. 10.1073/pnas.0803703105MathSciNetView ArticleMATHGoogle Scholar
- Leray J: Étude de diverses équations integrales nonlinéaires et de quelques problèmes que pose l'hydrodinamique. Journal de Mathématiques Pures et Appliquées 1933,9(12):1–82.MathSciNetGoogle Scholar
- Birindelli I, Mitidieri E: Liouville theorems for elliptic inequalities and applications. Proceedings of the Royal Society of Edinburgh A 1998,128(6):1217–1247. 10.1017/S0308210500027293MathSciNetView ArticleMATHGoogle Scholar
- Pinchover Y, Tintarev K: Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality. Indiana University Mathematics Journal 2005,54(4):1061–1074. 10.1512/iumj.2005.54.2705MathSciNetView ArticleMATHGoogle Scholar
- Fall MM, Musina R: Hardy-Poincaré inequalities with boundary singularities. Prépublication Département de Mathématique Université Catholique de Louvain-La-Neuve 364, 2010, http://www.uclouvain.be/38324.html Prépublication Département de Mathématique Université Catholique de Louvain-La-Neuve 364, 2010,Google Scholar
- Allegretto W: On the equivalence of two types of oscillation for elliptic operators. Pacific Journal of Mathematics 1974, 55: 319–328.MathSciNetView ArticleMATHGoogle Scholar
- Piepenbrink J: Nonoscillatory elliptic equations. Journal of Differential Equations 1974, 15: 541–550. 10.1016/0022-0396(74)90072-2MathSciNetView ArticleMATHGoogle Scholar
- Baras P, Goldstein JA: The heat equation with a singular potential. Transactions of the American Mathematical Society 1984,284(1):121–139. 10.1090/S0002-9947-1984-0742415-3MathSciNetView ArticleMATHGoogle Scholar
- Dupaigne L: A nonlinear elliptic PDE with the inverse square potential. Journal d'Analyse Mathématique 2002, 86: 359–398.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.