- Research Article
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Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials
Journal of Inequalities and Applications volume 2011, Article number: 917201 (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.
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.
Let be an integer,
, and let
be the ball in
of radius
centered at 0. In the first part of the paper, we study nonnegative solutions to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ1_HTML.gif)
where is a varying parameter. By a standard definition, a solution to (1.1) is a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ2_HTML.gif)
for any nonnegative . Notice that the weights in (1.1) derive from the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ3_HTML.gif)
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.
Let and let
be a distributional solution to (1.1). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ4_HTML.gif)
Then almost everywhere in
.
We remark that Theorem 1.1 is sharp, in view of the explicit counterexample in Remark 2.6.
Let us point out some consequences of Theorem 1.1. We use the Hardy-Leray inequality (1.3) to introduce the space as the closure of
with respect to the scalar product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ5_HTML.gif)
(see, e.g., [3]). It turns out that strictly contains the standard Sobolev space
, unless
.
Take in Theorem 1.1. Then problem (1.1) has no nontrivial and nonnegative solutions
if
. Therefore, if in the dual space
, a function
, solves
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ6_HTML.gif)
then in
.
Next take and
. From Theorem 1.1 it follows that problem (1.1) has no nontrivial and nonnegative solutions
. In particular, if
and if
is a weak solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ7_HTML.gif)
then in
. Thus Theorem 1.1 improves some of the nonexistence results in [2] and in [4].
The case of boundary singularities has been little studied. In Section 2, we prove sharp nonexistence results for inequalities in cone-like domains in ,
, having a vertex at 0. A special case concerns linear problems in half-balls. For
, we let
, where
is any half-space. Notice that
or
if
. A necessary and sufficient condition for the existence of nonnegative and nontrivial distributional solutions to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ8_HTML.gif)
is that (see Theorem B.3 and Remark 3.3), and the following result holds.
Theorem 1.2.
Let ,
, and let
be a distributional solution to (1.8). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ9_HTML.gif)
Then almost everywhere in
.
The key step in our proofs consists in studying the ordinary differential inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ10_HTML.gif)
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)
We start by fixing some terminologies. Let be the Hilbert space obtained via the Hardy inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ11_HTML.gif)
as the completion of with respect to the scalar product
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ12_HTML.gif)
Notice that with a continuous embedding and moreover
by Sobolev embedding theorem. By Hölder inequality, the space
is continuously embedded into the dual space
.
Finally, for any we put
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ13_HTML.gif)
We need two technical lemmata.
Lemma 2.1.
Let and
be a function satisfying
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ14_HTML.gif)
Put . Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ15_HTML.gif)
Proof.
We first show that and that (2.5) holds. Let
be a cutoff function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ16_HTML.gif)
and put . Then
and
. Multiply (2.4) by
and integrate by parts to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ17_HTML.gif)
Notice that for some constant depending only on
it results that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ18_HTML.gif)
as , since
. Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ19_HTML.gif)
by Lebesgue theorem, as by Hölder inequality. In conclusion, from (2.7) we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ20_HTML.gif)
since on
. By Fatou's Lemma, we get that
and (2.5) readily follows from (2.10). To prove that
, it is enough to notice that
in
. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ21_HTML.gif)
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.
Let and
. Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ22_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ23_HTML.gif)
Proof.
We start by noticing that almost everywhere. Then we use Hölder inequality to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ24_HTML.gif)
for any . Since
, then
in
. Thus
in
by the (generalized) Lebesgue Theorem, and (2.12) follows.
To prove (2.13), we first argue as before to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ25_HTML.gif)
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.
Let and let
be a distributional solution to (1.10). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ26_HTML.gif)
Then almost everywhere in
.
Proof.
We start by noticing that with a continuous embedding for any
. In addition, we point out that we can assume
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ27_HTML.gif)
Let be a standard sequence of mollifiers, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ28_HTML.gif)
Then in
and almost everywhere, and
in
by Lemma 2.2. Moreover,
is a nonnegative solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ29_HTML.gif)
We assume by contradiction that . We let
such that
. Up to a scaling and after replacing
with
, we may assume that
. We will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ30_HTML.gif)
leads to a contradiction. We fix a parameter
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ31_HTML.gif)
and for large we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ32_HTML.gif)
Clearly, and one easily verifies that
is a bounded sequence in
by (2.20) and (2.21). Finally, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ33_HTML.gif)
so that and
. In addition,
solves
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ34_HTML.gif)
where . Notice that
and that all the terms in the right-hand side of (2.24) belong to
, by (2.21). Thus Lemma 2.1 gives
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ35_HTML.gif)
since is bounded in
and
in
. By (2.17) and Hardy's inequality (2.1), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ36_HTML.gif)
Thus, for any fixed we get that
almost everywhere in
as
, since
is bounded away from 0 by (2.20). Finally, we notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ37_HTML.gif)
Since and
almost everywhere in
, and since
, we infer that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ38_HTML.gif)
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.
Consider the characteristic equation of the ordinary differential equation (1.10):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ39_HTML.gif)
For , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ40_HTML.gif)
be the largest roof of the above equation. Then it is not difficult to see that the proof of Theorem 2.3 highlights that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ41_HTML.gif)
for some constant . Moreover, one can easily verify that the function
belongs to
if and only if
.
2.2. Conclusion of the Proof
We will show that any nonnegative distributional solution to problem (1.1) gives rise to a function
solving (1.10), and such that
if and only if
. To this aim, we introduce the Emden-Fowler transform
by letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ42_HTML.gif)
By change of variable formula, for any , it results than
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ43_HTML.gif)
so that for any
. Now, for an arbitrary
we define the radially symmetric function
by setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ44_HTML.gif)
so that . By direct computations, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ45_HTML.gif)
Thus we are led to introduce the function defined in
by setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ46_HTML.gif)
We notice that for any
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ47_HTML.gif)
by Hölder inequality. Moreover, from (2.35) it immediately follows that is a distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ48_HTML.gif)
By Theorem 2.3, we infer that in
, and hence
in
. The proof of Theorem 1.1 is complete.
Remark 2.6.
The assumptions on the integrability of in Theorem 1.1 are sharp. If
, use the results in Appendix B. For
, let
be defined in (2.30) and notice that the function
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ49_HTML.gif)
solves
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ50_HTML.gif)
Moreover, if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ51_HTML.gif)
Finally we notice that, by Remark 2.5, for every solution , there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ52_HTML.gif)
3. Cone-Like Domains
Let . To any Lipschitz domain
, we associate the cone
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ53_HTML.gif)
For any given , we introduce also the cone-like domain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ54_HTML.gif)
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
.
It has been noticed in [11, 12], that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ55_HTML.gif)
The infimum is the best constant in the Hardy inequality for maps having compact support in
. In particular, for any half-space
, it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ56_HTML.gif)
The aim of this section is to study the elliptic inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ57_HTML.gif)
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.
Let be a Lipschitz domain properly contained in
,
, and let
be a distributional solution to (3.5). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ58_HTML.gif)
Then almost everywhere in
.
Proof.
Let be the positive eigenfunction of the Laplace-Beltrami operator
in
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ59_HTML.gif)
Let be as in the statement, and put
. We let
be the Emden-Fowler transform, as in (2.32). We further let
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ60_HTML.gif)
Next, for being an arbitrary nonnegative test function, we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ61_HTML.gif)
In essence, our aim is to test (3.5) with to prove that
satisfies (1.10) in
. To be more rigorous, we use a density argument to approximate
in
by a sequence of smooth maps
. Then we define
accordingly with (3.9), in such a way that
. By direct computation, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ62_HTML.gif)
Since is an admissible test function for (3.5), using also (3.3) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ63_HTML.gif)
Since and
in
, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ64_HTML.gif)
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.
Let and let
be a distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ65_HTML.gif)
Assume that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ66_HTML.gif)
Then almost everywhere in
.
Proof.
Write for a function
and then notice that
is a distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ67_HTML.gif)
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.
In case , the assumptions on
and on the integrability of
in Theorems 3.1 and 3.2 are sharp. Fix
, let
be defined in (2.30) and define the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ68_HTML.gif)
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].
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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.
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Appendices
Appendices
A. Hardy-Leray Inequalities on Cone-Like Domains
In this appendix, we give a simple proof of an improved Hardy inequality for mappings having support in a cone-like domain. We recall that for we have set
and that
.
Proposition A.1.
Let be a domain in
, with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ69_HTML.gif)
for any .
Proof.
We start by fixing an arbitrary function . We apply the Hardy inequality to the function
, for any fixed
, and then we integrate over
to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ70_HTML.gif)
In addition, notice that for any
. Thus, the Poincaré inequality for maps in
plainly implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ71_HTML.gif)
Adding these two inequalities, we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ72_HTML.gif)
for any . We use once more the Emden-Fowler transform
in (2.32) by letting
for
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ73_HTML.gif)
then (2.33) readily leads to the conclusion.
Remark A.2.
The arguments we have used to prove Proposition A.1 and the fact that the best constant in the Hardy inequality for maps in is not achieved show that the constants in inequality (A.1) are sharp, and not achieved.
Remark A.3.
Notice that for , we have
and
. Thus (A.1) gives (1.3) for
.
In the next proposition, we extend the inequality (A.1) to cover the case .
Proposition A.4.
It holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ74_HTML.gif)
for any . The constants are sharp, and not achieved.
Proof.
Write for a function
and then apply the Hardy inequality to
.
Next, let be a given parameter and let
be a Lipschitz domain in
, with
. For an arbitrary
, we put
. Then the Hardy-Leray inequality (A.1) and integration by parts plainly imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ75_HTML.gif)
for any , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ76_HTML.gif)
It is well known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ77_HTML.gif)
is the Hardy constant relative to the operator . For the case
, one can obtain in a similar way the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ78_HTML.gif)
which holds for any and for any
.
B. A General Necessary Condition
In this appendix, we show in particular that a necessary condition for the existence of nontrivial and nonnegative solutions to (1.1) and (3.5) is that . We need the following general lemma, which naturally fits into the classical Allegretto-Piepenbrink theory (see for instance, [13, 14]).
Lemma B.1.
Let be a domain in
,
. Let
and
in
. Assume that
is a nonnegative, nontrivial solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ79_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ80_HTML.gif)
Proof.
Let be a measurable set such that
and
in
. Fix any function
and choose a domain
such that
and
. For any integer
large enough, put
. Let
be the unique solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ81_HTML.gif)
Notice that for any
. Since the function
is nonnegative and nontrivial, then
. Actually it turns out that
by the Harnack inequality. Finally, a convolution argument and the maximum principle plainly give
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ82_HTML.gif)
Since , then we can use
as test function for (B.3) to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ83_HTML.gif)
by (B.4). Since , we readily infer
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ84_HTML.gif)
and Fatou's Lemma implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ85_HTML.gif)
The conclusion readily follows.
The sharpness of the constants in (1.3) (compare with Remark A.2) and Lemma B.1 plainly imply the following result.
Theorem B.2.
Let ,
, and
. Let
be a nonnegative distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ86_HTML.gif)
(i)If , then
.
(ii)If and
, then
.
We notice that proposition (i) in Theorem B.2 was already proved in [15] (see also [16]).
Finally, from Remark A.2 and Lemma B.1, we obtain the next nonexistence result.
Theorem B.3.
Let be a domain properly contained in
,
, and
. Let
be a nonnegative distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ87_HTML.gif)
(i)If , then
.
(ii)If and
, then
.
Remark B.4.
It would be of interest to know if the sign assumption on the coefficient in Lemma B.1 can be weakened.
C. Extensions
In this appendix, we state some nonexistence theorems that can be proved by using a suitable functional change and Theorem 2.3. We shall also point out some corollaries of our main results.
C.1. The
-Improved Weights
We define a sequence of radii by setting
and
. Then we use induction again to define two sequences of radially symmetric weights
and
in
by setting
for
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ88_HTML.gif)
for all . It can be proved by induction that
is well defined on
and
. We are interested in distributional solutions to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ89_HTML.gif)
for . The next result includes Theorem 1.1 by taking
.
Theorem C.1.
Let ,
and let
be a distributional solution to (C.2). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ90_HTML.gif)
Then almost everywhere in
.
Proof.
We start by introducing the th Emden-Fowler transform
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ91_HTML.gif)
Notice that for any it results that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ92_HTML.gif)
so that for any
. This can be easily checked by noticing that
. Next we set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ93_HTML.gif)
By (C.5), we have that for any
. Thanks to Theorem 2.3, to conclude the proof, it suffices to show that
is a distributional solution to
in the interval
, where
. To this end, fix any test function
and define the radially symmetric mapping
such that
. By direct computation, one can prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ94_HTML.gif)
where if
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ95_HTML.gif)
if . Since
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ96_HTML.gif)
provided that is nonnegative. In addition, it results that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ97_HTML.gif)
Since was arbitrarily chosen, the conclusion readily follows.
By similar arguments as above and in Section 2, we can prove a nonexistence result of positive solutions to the problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ98_HTML.gif)
where is a Lipschitz proper cone in
,
, and
. We shall skip the proof of the following result.
Theorem C.2.
Let ,
, and let
be a distributional solution to (C.11). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ99_HTML.gif)
Then almost everywhere in
.
Some related improved Hardy inequalities involving the weight and which motivate the interest of problems (C.2) and (C.11) can be found in [5, 7, 8] and also [6].
C.2. Exterior Cone-Like Domains
The Kelvin transform
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ100_HTML.gif)
can be used to get nonexistence results for exterior domains in .
Let be a domain in
,
, and let
be the cone defined in Section 2. We recall that
. Since the inequality in (1.1) is invariant with respect to the Kelvin transform, then Theorems 1.1 and 3.1 readily lead to the following nonexistence result.
Theorem C.3.
Let be a Lipschitz domain in
, with
. Let
,
, and let
be a distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ101_HTML.gif)
Assume that there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ102_HTML.gif)
Then almost everywhere in
.
A similar statement holds in case for ordinary differential inequalities in unbounded intervals
with
, and for problems involving the weight
.
C.3. Degenerate Elliptic Operators
Let be a given real parameter. We notice that
is a distributional solution to (3.5) if and only if
is a distributional solution to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ103_HTML.gif)
where is defined in Remark A.2. Therefore, Theorems 1.1 and 3.1 imply the following nonexistence result for linear inequalities involving the weighted Laplace operator
.
Theorem C.4.
Let be a Lipschitz domain in
. Let
,
,
, and let
be a distributional solution to (C.16). Assume that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F917201/MediaObjects/13660_2011_Article_2367_Equ104_HTML.gif)
Then almost everywhere in
.
A nonexistence result for the operator similar to Theorem C.3 or to Theorem C.1 can be obtained from Theorem C.4, via suitable functional changes.
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Fall, M., Musina, R. Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials. J Inequal Appl 2011, 917201 (2011). https://doi.org/10.1155/2011/917201
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DOI: https://doi.org/10.1155/2011/917201