- Research Article
- Open Access

# Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials

- MouhamedMoustapha Fall
^{1}and - Roberta Musina
^{2}Email author

**2011**:917201

https://doi.org/10.1155/2011/917201

© M. M. Fall and R. Musina. 2011

**Received:**4 January 2011**Accepted:**4 March 2011**Published:**15 March 2011

## Abstract

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.

## Keywords

- Lipschitz Domain
- Nonnegative Solution
- Distributional Solution
- Hardy Inequality
- Nonexistence Result

## 1. Introduction

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 [9] 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).

In case , we provide necessary conditions on the parameter to have the existence of nontrivial solutions satisfying suitable integrability properties.

Theorem 1.1.

Then almost everywhere in .

We remark that Theorem 1.1 is sharp, in view of the explicit counterexample in Remark 2.6.

(see, e.g., [3]). It turns out that strictly contains the standard Sobolev space , unless .

then in .

then in . Thus Theorem 1.1 improves some of the nonexistence results in [2] and in [4].

is that (see Theorem B.3 and Remark 3.3), and the following result holds.

Theorem 1.2.

Then almost everywhere in .

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 [10] 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).

Appendix B deals in particular with the case . The nonexistence Theorems B.2 and B.3 follow from an Allegretto-Piepenbrink type result (Lemma B.1).

In the last appendix, we point out some related results and some consequences of our main theorems.

Notation 1.

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)

Notice that with a continuous embedding and moreover by Sobolev embedding theorem. By Hölder inequality, the space is continuously embedded into the dual space .

We need two technical lemmata.

Lemma 2.1.

Proof.

as and .

Through the paper, we let be a standard mollifier sequence in , such that the support of is contained in the interval .

Lemma 2.2.

Proof.

for any . Since , then in . Thus in by the (generalized) Lebesgue Theorem, and (2.12) follows.

for any . Thus converges to in by Lebesgue's Theorem. In addition, in by (2.12). Thus in and the Lemma is completely proved.

The following result for solutions to (1.10) is a crucial step in the proofs of our main theorems.

Theorem 2.3.

Then almost everywhere in .

Proof.

This conclusion contradicts the assumption , as was arbitrarily chosen. Thus (2.20) cannot hold and the proof is complete.

Remark 2.4.

If , then every nonnegative solution to problem (1.10) vanishes. This is an immediate consequence of Lemma B.1 in Appendix B and the sharpness of the constant in the Hardy inequality (2.1).

Remark 2.5.

for some constant . Moreover, one can easily verify that the function belongs to if and only if .

### 2.2. Conclusion of the Proof

By Theorem 2.3, we infer that in , and hence in . The proof of Theorem 1.1 is complete.

Remark 2.6.

## 3. Cone-Like Domains

Notice that and . If is an half-sphere , the is an half-space and is a half-ball , as in Theorem 1.2.

Assume that is properly contained in . Then we let be the principal eigenvalue of the Laplace operator on . If , we put .

Notice that (3.5) reduces to (1.1) if . Problem (3.5) is related to an improved Hardy inequality for maps supported in cone-like domains which will be discussed in Appendix A.

Theorem 3.1.

Then almost everywhere in .

Proof.

By the arbitrariness of , we can conclude that is a distributional solution to (1.10). Theorem 2.3 applies to give , that is, in .

The next result extends Theorem 3.1 to cover the case . Notice that is a cone and is a cone-like domain in .

Theorem 3.2.

Then almost everywhere in .

Proof.

The conclusion readily follows from Theorem 2.3.

Remark 3.3.

If , then every nonnegative solution to problem (3.5) vanishes by Theorem B.3.

Here solves (3.7) if . If , we agree that and . By direct computations, one has that solves (3.5). Moreover, if and then if and only if .

Remark 3.4.

Nonexistence results for linear inequalities involving the differential operator were already obtained in [12].

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

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