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

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

(1.1)

where is a varying parameter. By a standard definition, a solution to (1.1) is a function such that

(1.2)

for any nonnegative . Notice that the weights in (1.1) derive from the inequality

(1.3)

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

(1.4)

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

(1.5)

(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

(1.6)

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

(1.7)

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

(1.8)

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

(1.9)

Then almost everywhere in .

The key step in our proofs consists in studying the ordinary differential inequality

(1.10)

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

(2.1)

as the completion of with respect to the scalar product

(2.2)

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

(2.3)

We need two technical lemmata.

Lemma 2.1.

Let and be a function satisfying and

(2.4)

Put . Then and

(2.5)

Proof.

We first show that and that (2.5) holds. Let be a cutoff function satisfying

(2.6)

and put . Then and . Multiply (2.4) by and integrate by parts to get

(2.7)

Notice that for some constant depending only on it results that

(2.8)

as , since . Moreover,

(2.9)

by Lebesgue theorem, as by Hölder inequality. In conclusion, from (2.7) we infer that

(2.10)

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,

(2.11)

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

(2.12)

(2.13)

Proof.

We start by noticing that almost everywhere. Then we use Hölder inequality to get

(2.14)

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

(2.15)

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

(2.16)

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

(2.17)

Let be a standard sequence of mollifiers, and let

(2.18)

Then in and almost everywhere, and in by Lemma 2.2. Moreover, is a nonnegative solution to

(2.19)

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

(2.20)

leads to a contradiction. We fix a parameter

(2.21)

and for large we put

(2.22)

Clearly, and one easily verifies that is a bounded sequence in by (2.20) and (2.21). Finally, we define

(2.23)

so that and . In addition, solves

(2.24)

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

(2.25)

since is bounded in and in . By (2.17) and Hardy's inequality (2.1), we conclude that

(2.26)

Thus, for any fixed we get that almost everywhere in as , since is bounded away from 0 by (2.20). Finally, we notice that

(2.27)

Since and almost everywhere in , and since , we infer that

(2.28)

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):

(2.29)

For , let

(2.30)

be the largest roof of the above equation. Then it is not difficult to see that the proof of Theorem 2.3 highlights that

(2.31)

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

(2.32)

By change of variable formula, for any , it results than

(2.33)

so that for any . Now, for an arbitrary we define the radially symmetric function by setting

(2.34)

so that . By direct computations, we get

(2.35)

Thus we are led to introduce the function defined in by setting

(2.36)

We notice that for any , since

(2.37)

by Hölder inequality. Moreover, from (2.35) it immediately follows that is a distributional solution to

(2.38)

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

(2.39)

solves

(2.40)

Moreover, if then

(2.41)

Finally we notice that, by Remark 2.5, for every solution , there exists a constant such that

(2.42)

3. Cone-Like Domains

Let . To any Lipschitz domain , we associate the cone

(3.1)

For any given , we introduce also the cone-like domain

(3.2)

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

(3.3)

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

(3.4)

The aim of this section is to study the elliptic inequality

(3.5)

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

(3.6)

Then almost everywhere in .

Proof.

Let be the positive eigenfunction of the Laplace-Beltrami operator in defined by

(3.7)

Let be as in the statement, and put . We let be the Emden-Fowler transform, as in (2.32). We further let defined as

(3.8)

Next, for being an arbitrary nonnegative test function, we put

(3.9)

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

(3.10)

Since is an admissible test function for (3.5), using also (3.3) we get

(3.11)

Since and in , we conclude that

(3.12)

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

(3.13)

Assume that there exists such that

(3.14)

Then almost everywhere in .

Proof.

Write for a function and then notice that is a distributional solution to

(3.15)

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

(3.16)

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].

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

(A.1)

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

(A.2)

In addition, notice that for any . Thus, the Poincaré inequality for maps in plainly implies

(A.3)

Adding these two inequalities, we conclude that

(A.4)

for any . We use once more the Emden-Fowler transform in (2.32) by letting for . Since

(A.5)

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

(A.6)

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

(A.7)

for any , where

(A.8)

It is well known that

(A.9)

is the Hardy constant relative to the operator . For the case , one can obtain in a similar way the inequality

(A.10)

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

(B.1)

Then

(B.2)

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

(B.3)

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

(B.4)

Since , then we can use as test function for (B.3) to get

(B.5)

by (B.4). Since , we readily infer

(B.6)

and Fatou's Lemma implies that

(B.7)

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

(B.8)

(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

(B.9)

(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

(C.1)

for all . It can be proved by induction that is well defined on and . We are interested in distributional solutions to

(C.2)

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

(C.3)

Then almost everywhere in .

Proof.

We start by introducing the th Emden-Fowler transform ,

(C.4)

Notice that for any it results that

(C.5)

so that for any . This can be easily checked by noticing that . Next we set

(C.6)

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

(C.7)

where if , and

(C.8)

if . Since , then

(C.9)

provided that is nonnegative. In addition, it results that

(C.10)

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

(C.11)

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

(C.12)

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

(C.13)

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

(C.14)

Assume that there exists such that

(C.15)

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

(C.16)

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

(C.17)

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.