- Research Article
- Open Access
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.
- Lipschitz Domain
- Nonnegative Solution
- Distributional Solution
- Hardy Inequality
- Nonexistence Result
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.
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
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.
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