SCHRODINGER OPERATORS WITH NEGATIVE POTENTIALS AND LANE-EMDEN DENSITIES

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SCHRODINGER OPERATORS WITH NEGATIVE POTENTIALS AND LANE-EMDEN DENSITIES

Lorenzo Brasco, Giovanni Franzina, Berardo Ruffini

To cite this version:

Lorenzo Brasco, Giovanni Franzina, Berardo Ruffini. SCHRODINGER OPERATORS WITH NEG-

ATIVE POTENTIALS AND LANE-EMDEN DENSITIES. Journal of Functional Analysis, Elsevier,

2018. �hal-01819321�

WITH NEGATIVE POTENTIALS AND LANE-EMDEN DENSITIES

LORENZO BRASCO, GIOVANNI FRANZINA, AND BERARDO RUFFINI

Abstract. We consider the Schr¨odinger operator−∆ +V for negative potentialsV, on open sets with positive first eigenvalue of the Dirichlet-Laplacian. We show that the spectrum of−∆ +V is positive, provided thatV is greater than a negative multiple of the logarithmic gradient of the solution to the Lane-Emden equation−∆u=u^q−1 (for some 1≤q <2). In this case, the ground state energy of −∆ +V is greater than the first eigenvalue of the Dirichlet-Laplacian, up to an explicit multiplicative factor. This is achieved by means of suitable Hardy-type inequalities, that we prove in this paper.

Contents

1. Introduction 1

1.1. Foreword 1

1.2. Aim of the paper 3

1.3. Main results 4

1.4. Plan of the paper 5

2. Preliminaries 5

2.1. Notation 5

2.2. Lane-Emden densities: bounded sets 6 2.3. Lane-Emden densities: general sets 8

3. Hardy-Lane-Emden inequalities 9

4. Sobolev embeddings and densities 11 5. Hardy-Lane-Emden inequalities for sets with positive

spectrum 16

6. Lower bounds for the ground state energy 17

7. Applications 20

7.1. N−dimensional ball 20

7.2. An infinite slab 21

7.3. A rectilinear wave-guide 22

Appendix A. A local L^∞ estimate for Lane-Emden

densities 25

References 29

1. Introduction

1.1. Foreword. Let V ∈ L

( R

) be a real-valued potential such that V ≤ 0 and let us consider the Schr¨ odinger operator H

:= − ∆ + V , acting on the domain

D( H

) := H

(R

) ∩ { u ∈ L

(R

) : V u ∈ L

(R

) } .

2010Mathematics Subject Classification. 35P15, 47A75, 49S05.

Key words and phrases. Schr¨odinger operators, ground state energy, Hardy inequalities, Lane-Emden equation.

1

Observe that the hypothesis V ∈ L

(R

) entails the inclusion C

( R

) ⊂ D( H

),

thus D( H

) is dense in L

( R

). The operator H

: D( H

) → L

( R

) is symmetric and self- adjoint as well, thanks to the fact that V is real-valued (see [16, Example p. 68]). The spectrum of H

is the set

σ( H

) = R \ ρ( H

),

where ρ( H

) is the resolvent set of H

, defined as the collection of real numbers λ such that H

− λ is bijective and its inverse is a bounded linear operator.

A distinguished subset of σ( H

) is given by the collection of those λ such that the kernel of H

− λ is nontrivial. In this case, the stationary Schr¨ odinger equation

(1.1) H

u = λ u,

admits a nontrivial solution u ∈ D( H

). Whenever this happens, λ is called an eigenvalue of the Schr¨ odinger operator. Correspondingly, the solution is said to be an eigenfunction corresponding to λ.

The operator H

comes with the associated quadratic form ϕ 7→ Q

(ϕ) =

ˆ

R^N

|∇ ϕ |

dx + ˆ

R^N

V ϕ

dx, ϕ ∈ D( H

).

From classical Spectral Theory, we have (see [16, Theorem 2.20])

(1.2) inf σ( H

) = inf

ϕ∈D(H_V)

Q

(ϕ) : ˆ

R^N

ϕ

dx = 1

. We call such a value ground state energy of H

.

This quantity is important in classical Quantum Mechanics, since it is the lowest energy that a particle in R

interacting with the force field generated by the potential V can attain (and which will eventually attain by emitting energy). From a mathematical point of view, we observe that the stationary Schr¨ odinger equation (1.1) is precisely the Euler-Lagrange equation of problem (1.2).

An issue of main interest is providing a lower bound on the ground state energy (and thus on the spectrum) of H

.

It is well-known that when V ≡ 0, then inf σ( H

) = 0. On the other hand, if we take V ≤ 0, the kinetic energy ´

R^N

|∇ ϕ |

dx and the potential energy ´

R^N

V ϕ

dx are in competition in the quadratic form Q

and one could expect that

inf σ( H

) < 0.

Actually, this depends on the potential V . For example, by recalling the Hardy inequality on R

(for N ≥ 3)

N − 2 2

ˆ

R^N

ϕ

| x |

dx ≤ ˆ

R^N

|∇ ϕ |

dx, ϕ ∈ C

( R

\ { 0 } ), we get that if the potential V is such that

0 ≥ V ≥ −

N − 2 2

1

| x |

,

then the spectrum of H

is still non-negative. This is an example of how Hardy-type inequalities

can be exploited in order to identify classes of negative potentials with non-negative spectrum.

1.2. Aim of the paper. In this paper we deal with a confined version of this problem. More precisely, we turn our attention to prescribed open sets Ω ⊂ R

. We fix a potential V ∈ L

(Ω) such that V ≤ 0 and consider the localized Schr¨ odinger operator with homogeneous boundary conditions H

= − ∆ + V , this time acting on the domain

(1.3) D( H

) := H

(Ω) ∩ H

(Ω) ∩ { u ∈ L

(Ω) : V u ∈ L

(Ω) } .

Here H

(Ω) is the closure of C

(Ω) in the Sobolev space H

(Ω). This is still a symmetric and self-adjoint operator H

: D( H

) → L

(Ω), with real spectrum σ( H

). Observe that the hypothesis V ∈ L

(Ω) entails as before the inclusion

C

(Ω) ⊂ D( H

),

thus the operator is densely defined. We define the associated quadratic form Q

(ϕ) =

ˆ

Ω

|∇ ϕ |

dx + ˆ

Ω

V ϕ

dx, ϕ ∈ D( H

).

The stationary equation (1.1) now reads

(1.4)

H

u = λ u in Ω, u = 0, in R

\ Ω.

Equation (1.4) can be formally considered as a peculiar form of (1.1), where the potential V has the trapping property V = + ∞ in R

\ Ω. This models the physical situation where the particle is

“trapped” in the confining region Ω.

The issue we tackle is the following

“find explicit pointwise bounds on the potential V assuring that the ground state energy of H

stays positive ”

In the vein of the example discussed above using Hardy’s inequality in the entire space, we will approach this problem by proving localized Hardy-type inequalities with suitable weights. A typical instance of these inequalities occurs when we limit ourselves to consider functions supported in a proper open subset Ω ⊂ R

and we use the distance d

(x) := dist(x, ∂Ω) as a weight. In other words, one has

1 C

ˆ

Ω

ϕ

d

dx ≤

ˆ

Ω

|∇ ϕ |

dx, ϕ ∈ C

(Ω).

However, the existence of such a constant C > 0 typically requires some conditions on the geometry

of the set Ω or on the regularity of its boundary. In this paper on the contrary, we will prove

alternative Hardy-type inequalities, with weights depending on solutions of peculiar elliptic partial

differential equations.

Roughly speaking, we will consider the solution w

to the Lane-Emden equation

with 1 ≤ q < 2 (1.5)







− ∆u = u

in Ω, u = 0, in R

\ Ω, u > 0, in Ω,

prove a Hardy inequality with weight depending on w

and show that the condition 0 ≥ V & −

∇ w

w

2

, a. e. in Ω, leads to positivity of the spectrum of the Schr¨ odinger operator H

.

The function w

will be called the Lane-Emden q − density of Ω, we refer to Definitions 2.5 and 2.8 below.

1.3. Main results. Let us now try to be more precise about our results. We first need to fix some definitions. For γ ≥ 1, we denote

(1.6) λ

(Ω) = inf

ϕ∈C₀^∞(Ω)

ˆ

Ω

|∇ ϕ |

dx : k ϕ k

= 1

. Henceforth we shall often work with the following class of sets.

Definition 1.1. We say that Ω ⊂ R

is an open set with positive spectrum if it is open and (1.7) λ

(Ω) := λ

(Ω) = inf

ϕ∈C₀^∞(Ω)

ˆ

Ω

|∇ ϕ |

dx : k ϕ k

= 1

> 0.

The main result of the paper is the following lower bound on the ground state energy of H

. We refer to Theorem 6.2 and Corollary 6.3 for its proof.

Theorem 1.2. Let Ω ⊂ R

be an open set with positive spectrum, and let V ∈ L

(Ω). For an exponent 1 ≤ q < 2, we assume that

0 ≥ V ≥ − 1 4

∇ w

w

2

, a. e. in Ω.

Then the spectrum σ( H

) of H

is positive and we have that inf σ( H

) = inf

ϕ∈C₀^∞(Ω)

Q

(ϕ) : ˆ

Ω

ϕ

dx = 1

≥ 1

C λ

(Ω), where C = C(N, q) > 0 is an explicit dimensional constant.

As stated above, the main tool we use to prove this result is an Hardy-type inequality, in which a weight involving the solution w

of the Lane-Emden equation (1.5) enters. This is the content of the next result. For questions related to optimal choices of weights in Hardy-type inequalities, see [10] and the references therein.

1The terminology comes from astrophysics, where the Lane-Emden equation is 1

%² d d%

%²du d%

+u^γ= 0,

for a radially symmetric functionu:R³→R. The positive numberγis usually calledpolytropic index. Observe that for a radial functionudefined inR³, this is equivalent to

−∆u=u^γ.

Though our paper is not concerned with astrophysics, we found useful to give a name to the equation and its solution.

Theorem 1.3 (Hardy-Lane-Emden inequality). Let 1 ≤ q < 2 and let Ω ⊂ R

be an open set with positive spectrum. Then for every u ∈ C

(Ω) and δ > 0 we have that

1 δ

1 − 1

δ ˆ

Ω

∇ w

w

2

ϕ

dx + 1 δ

ˆ

Ω

ϕ

w

dx ≤

ˆ

Ω

|∇ ϕ |

dx.

We refer to Remark 3.2 for some comments about the proof of this result.

1.4. Plan of the paper. The paper is organized as follows: in Section 2, we define the Lane- Emden q − density of a set Ω ⊂ R

, first under the assumption that Ω is bounded and then for a general open set. Then in Section 3 we prove the Hardy-Lane-Emden inequality of Theorem 1.3 for bounded open sets.

In Section 4 we show how the summability properties of the Lane-Emden densities are equivalent to the embedding of D

(Ω) into suitable Lebesgue spaces. This part generalizes some results contained in the recent paper [3], by replacing the torsion function with any Lane-Emden q − density.

Though this section may appear unrelated to ground state energy estimates for H

, some of its outcomes are used to extend (in Section 5) the Hardy-Lane-Emden inequality to open sets with positive spectrum.

The proof of Theorem 1.2 is then contained in Section 6, while Section 7 contains some appli- cations of our main result to some particular geometries (a ball, an infinite slab and a rectilinear wave-guide with circular cross-section).

We conclude the paper with an Appendix, containing a local L

estimate for subsolutions of the Lane-Emden equation, which is necessary in order to get the explicit lower bound on the ground state energy of H

.

Acknowledgments. The first author would like to thank Douglas Lundholm for a discussion on Hardy inequalities and the so-called Ground State Representation in February 2017, during a visit to the Department of Mathematics of KTH (Stockholm). He also wishes to thank Erik Lindgren for the kind invitation. Remark 4.4 comes from an informal discussion with Guido De Philippis in December 2015, we wish to thank him.

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

2. Preliminaries

2.1. Notation. Let Ω ⊂ R

be an open set and define the norm on C

(Ω) k ϕ k

0 (Ω)

= ˆ

Ω

|∇ ϕ |

dx

2

, ϕ ∈ C

(Ω).

We consider the homogeneous Sobolev space D

(Ω), obtained as the completion of C

(Ω) with respect to the norm k · k

. For N ≥ 3 this is always a functional space, thanks to Sobolev inequality but in dimension N = 1 or N = 2, this may fail to be even a space of distributions if Ω is “too big”, see for example [9, Remark 4.1].

Remark 2.1. For an open set with positive spectrum Ω, we have automatically continuity of the embedding D

(Ω) , → L

(Ω). Thus in this case D

(Ω) is a functional space. Moreover, we have that

D

(Ω) = H

(Ω),

thanks to the fact that in this case ˆ

Ω

|∇ ϕ |

dx

and

ˆ

Ω

|∇ ϕ |

dx

+ ˆ

Ω

ϕ

dx

, are equivalent norms on C

(Ω).

2.2. Lane-Emden densities: bounded sets. We start with the following auxiliary result.

Lemma 2.2. Let Ω ⊂ R

be an open bounded set. For 1 ≤ q < 2, the variational problem

(2.1) min

ϕ∈D₀^1,2(Ω)

1 2

ˆ

Ω

|∇ ϕ |

dx − 1 q

ˆ

Ω

ϕ

dx : ϕ ≥ 0 a. e. in Ω

,

admits a unique solution.

Proof. Since the absolute value of every minimizer of the functional ϕ 7→ 1

2 ˆ

Ω

|∇ ϕ |

dx − 1 q

ˆ

Ω

| ϕ |

dx, is also a minimizer of (2.1), problem (2.1) is equivalent to

min

ϕ∈D₀^1,2(Ω)

1 2

ˆ

Ω

|∇ ϕ |

dx − 1 q

ˆ

Ω

| ϕ |

dx

.

The existence of a solution follows then by the Direct Methods in the Calculus of Variations, since the embedding D

(Ω) , → L

(Ω) is compact and D

(Ω) is weakly closed.

As for uniqueness, we first suppose that Ω is connected. We observe that for q = 1 problem (2.1) is strictly convex, thus the solution is unique. For 1 < q < 2, we can use a trick by Brezis and Oswald based on the so-called Picone’s inequality, see [6, Theorem 1]. We reproduce their argument here for completeness. We first observe that a minimizer is a positive solution of the Lane-Emden equation

(2.2) − ∆u = u

, in Ω,

with homogeneous Dirichlet boundary conditions. More precisely, for every ϕ ∈ D

(Ω) it holds (2.3)

ˆ

Ω

h∇ u, ∇ ϕ i dx = ˆ

Ω

u

ϕ dx.

We now suppose that (2.1) admits two minimizers u

, u

∈ D

(Ω). By the minimum principle for superharmonic functions, u

> 0 and u

> 0 on Ω. Moreover, by standard Elliptic Regularity, u

, u

∈ L

(Ω). We fix ε > 0, then we test equation (2.3) for u

with

ϕ = u

u

+ ε − u

, and equation (2.3) for u

with

ϕ = u

u

+ ε − u

.

Summing up, we get that ˆ

Ω

∇ u

, ∇ u

u

+ ε

dx − ˆ

Ω

|∇ u

|

dx + ˆ

Ω

∇ u

, ∇ u

u

+ ε

dx − ˆ

Ω

|∇ u

|

dx

= ˆ

Ω

u

u

+ ε u

dx − ˆ

Ω

u

dx + ˆ

Ω

u

u

+ ε u

dx − ˆ

Ω

u

dx.

We now recall that (2.4)

∇ u, ∇ v

u

≤ |∇ v |

,

for v and u > 0 differentiable. This is precisely Picone’s inequality, see for example [2]. By observing that ∇ u

= ∇ (u

+ ε) and using (2.4) in the identity above, we conclude that

ˆ

Ω

u

u

+ ε u

dx − ˆ

Ω

u

dx + ˆ

Ω

u

u

+ ε u

dx − ˆ

Ω

u

dx ≤ 0.

We now take the limit as ε goes to 0. By Fatou’s Lemma, we obtain that ˆ

Ω

u

u

dx − ˆ

Ω

u

dx + ˆ

Ω

u

u

dx − ˆ

Ω

u

dx ≤ 0.

The previous terms can be recast into inequality ˆ

Ω

(u

− u

) (u

− u

) dx ≥ 0.

By using the fact the function t 7→ t

is monotone decreasing, we get that u

= u

as desired.

Finally, if Ω is not connected, it is sufficient to observe that a solution of (2.1) must minimize the same functional on every connected component, due to the locality of the functional; since the solution is unique on every connected component, we get the conclusion in this case as well.

Remark 2.3 (About uniqueness). Uniqueness of the solution to (2.1) can also be inferred directly at the level of the minimization problem. It is sufficient to observe that the functional to be minimized is convex along curves of the form

γ

=

(1 − t) ϕ

+ t ϕ

q

, t ∈ [0, 1], ϕ

, ϕ

∈ D

(Ω) positive,

see [2, Proposition 2.6]. Then one can reproduce the uniqueness proof of [1]. For a different proof of the uniqueness for (2.2), we also refer to [11, Corollary 4.2].

Remark 2.4. It is useful to keep in mind that if u ∈ D

(Ω) solves equation

− ∆u = t u

, in Ω, for some t > 0, then the new function

v

= t

u, solves (2.2).

Definition 2.5. Let Ω ⊂ R

be an open bounded set. For 1 ≤ q < 2, we define the Lane-Emden

q − density of Ω as the unique solution of (2.1). We denote such a solution by w

. In the case

q = 1, we simply write w

and call it torsion function of Ω.

The variational problem defining w

is related to the optimal Poincar´e constant λ

(Ω) defined in (1.6). This is the content of the next result, that we record for completeness. We omit the proof since it is based on a straightforward scaling argument.

Lemma 2.6. Let 1 ≤ q < 2 and let Ω ⊂ R be an open bounded set. Then we have

(2.5) min

ϕ∈D₀^1,2(Ω)

1 2

ˆ

Ω

|∇ ϕ |

dx − 1 q

ˆ

Ω

ϕ

dx : ϕ ≥ 0 in Ω

= q − 2 2 q

1 λ

(Ω)

and

(2.6)

ˆ

Ω

| w

|

dx

= 1

λ

(Ω) .

2.3. Lane-Emden densities: general sets. We now want to define the Lane-Emden densities for a general open set, where the variational problem

inf

ϕ∈D^1,2₀ (Ω)

1 2

ˆ

Ω

|∇ ϕ |

dx − 1 q

ˆ

Ω

ϕ

dx : ϕ ≥ 0 in Ω

, may fail to admit a solution.

We start with a sort of comparison principle for Lane-Emden densities.

Lemma 2.7. Let 1 ≤ q < 2 and let Ω

⊂ Ω

⊂ R

be two open bounded sets. Then we have w

≤ w

.

Proof. We test the minimality of w

against ϕ = min { w

, w

} . After some simple manipu- lations, this gives

1 2

ˆ

{w_q,Ω2<wq,Ω1}

|∇ w

|

dx − 1 q

ˆ

{w_q,Ω2<wq,Ω1}

w

2

dx

≥ 1 2

ˆ

{w_q,Ω2<wq,Ω1}

|∇ w

|

dx − 1 q

ˆ

{w_q,Ω2<wq,Ω1}

w

1

dx.

We now add on both sides the term 1

2 ˆ

{w_q,Ω2>wq,Ω1}

|∇ w

|

dx − 1 q

ˆ

{w_q,Ω2>wq,Ω1}

w

2

dx, thus if set U = max { w

, w

} , we get that

1 2

ˆ

Ω2

|∇ w

|

dx − 1 q

ˆ

Ω2

w

dx ≥ 1 2

ˆ

Ω2

|∇ U |

dx − 1 q

ˆ

Ω2

U

dx.

By uniqueness of the minimizer w

, this gives U = w

. By recalling the definition of U, this

in turn yields the desired conclusion.

Thanks to the previous property, we can define the Lane-Emden density for every open set. In what follows, we set

Ω

= Ω ∩ B

(0), R > 0,

where B

(0) is the N − dimensional open ball, with radius R and centered at the origin.

Definition 2.8. Let Ω ⊂ R

be an open set. For 1 ≤ q < 2 we define w

= lim

R→+∞

w

.

We observe that this definition is well-posed, since each w

∈ D

(Ω

) exists thanks to the boundedness of Ω

and the function

R 7→ w

(x), is monotone, thanks to Lemma 2.7.

Remark 2.9 (Consistency). When Ω ⊂ R

is an open bounded set or, more generally, is such that the embedding D

(Ω) , → L

(Ω) is compact, then the definition of w

above coincides with the variational one. For q = 1 this is proved in [3, Lemma 2.4], the other cases can be treated in exactly the same way. We skip the details.

3. Hardy-Lane-Emden inequalities

The following theorem, which is a generalization of [3, Theorem 4.3], is the main result of the present section. For simplicity, we state and prove the result just for open bounded sets, but it is easily seen that the same proof works for every open set Ω ⊂ R

such that the embedding D

(Ω) , → L

(Ω) is compact.

Theorem 3.1. Let 1 ≤ q < 2 and let Ω ⊂ R

be an open bounded set. Then for every ϕ ∈ C

(Ω) and δ > 0 we have

(3.1) 1

δ ˆ

Ω

"

1 − 1

δ

∇ w

w

2

+ 1

w

#

ϕ

dx ≤ ˆ

Ω

|∇ ϕ |

dx.

Proof. We recall that (3.2)

ˆ

Ω

h∇ w

, ∇ ψ i dx = ˆ

Ω

w

ψ dx,

for any ψ ∈ D

(Ω). Let ϕ ∈ C

(Ω) and let ε > 0, by taking in (3.2) the test function ψ = ϕ

w

+ ε , we get

(3.3)

ˆ

Ω

"

|∇ w

|

+ w

(w

+ ε) (w

+ ε)

#

ϕ

dx = 2 ˆ

Ω

ϕ

∇ w

(w

+ ε) , ∇ ϕ

dx.

By Young’s inequality, it holds ϕ

∇ w

(w

+ ε) , ∇ ϕ

≤ δ

2 |∇ ϕ |

+ 1 2 δ

|∇ w

|

(w

+ ε)

ϕ

for δ > 0. Thus we get

ˆ

Ω

"

|∇ w

|

+ w

(w

+ ε) (w

+ ε)

#

ϕ

dx ≤ δ ˆ

Ω

|∇ ϕ |

dx + 1 δ

ˆ

Ω

|∇ w

|

(w

+ ε)

ϕ

dx.

The previous inequality gives 1

δ ˆ

Ω

"

1 − 1

δ

|∇ w

|

(w

+ ε)

+ w

(w

+ ε)

#

ϕ

dx ≤ ˆ

Ω

|∇ ϕ |

dx.

By recalling that ϕ is compactly supported in Ω and observing that

∇ w

w

2

∈ L

(Ω),

we conclude the proof by taking the limit as ε goes to 0 and appealing to the Monotone Convergence

Theorem.

Remark 3.2 (A comment on the proof). The idea of the previous proof comes from that of Moser’s logarithmic estimate for elliptic partial differential equations, see [14, page 586]. In regularity theory, this is an essential tool in order to establish the validity of Harnack’s inequality for solutions.

An alternative proof is based on Picone’s inequality (2.4). This goes as follows: one observes that the function W = w

locally solves

− ∆W = − 1 δ w

1 δ−1

q,Ω

∆w

− 1 δ

1 δ − 1

w

1 δ−2

q,Ω

|∇ w

|

= W

"

1

δ w

+ 1 δ

1 − 1

δ

∇ w

w

2

# . Thus we have

ˆ

Ω

"

1

δ w

+ 1 δ

1 − 1

δ

∇ w

w

2

#

W ψ dx = ˆ

Ω

h∇ W, ∇ ψ i dx,

for every ψ ∈ C

(Ω). If we now take the test function ψ = ϕ

/W and use inequality (2.4), we get the desired inequality.

This technique to obtain Hardy-type inequalities is sometimes referred to as Ground State Rep- resentation, see for example [12, Proposition 1].

As a consequence of the Hardy-Lane-Emden inequality, we record the following integrability properties of functions in D

(Ω).

Corollary 3.3. Let 1 ≤ q < 2 and let Ω ⊂ R

be an open bounded set. Then for every ϕ ∈ D

(Ω) (3.4)

ˆ

Ω

∇ w

w

2

ϕ

dx < + ∞ and

ˆ

Ω

ϕ

w

dx < + ∞ . Moreover, if { ϕ

}

⊂ D

(Ω) converges strongly to ϕ ∈ D

(Ω), then

n→∞

lim ˆ

Ω

∇ w

w

2

| ϕ

− ϕ |

dx = 0 and lim

n→∞

ˆ

Ω

| ϕ

− ϕ |

w

dx = 0.

2It is sufficient to remark that∇wq,Ω∈L²(Ω) and that by the strong minimum principle, we have wq,Ω≥cK>0 for everyKbΩ.

Proof. Let ϕ ∈ D

(Ω), then there exists { ϕ

}

⊂ C

(Ω) converging to ϕ in D

(Ω). By choosing δ = 2 in (3.1), we have that

1 4

ˆ

Ω

"

∇ w

w

2

+ 2

w

#

ϕ

dx ≤ ˆ

Ω

|∇ ϕ

|

dx.

By using the norm convergence in the right-hand side and Fatou’s Lemma in the left-hand side, we deduce the validity of (3.4) for ϕ.

In order to prove the second part of the statement, we observe that the first part of the proof also implies the validity of inequality (3.1) in D

(Ω), for δ = 2. Plugging in ϕ

− ϕ gives that

lim sup

n→∞

1 4

ˆ

Ω

"

∇ w

w

2

+ 2

w

#

| ϕ

− ϕ |

dx ≤ lim

n→∞

ˆ

Ω

|∇ ϕ

− ∇ ϕ |

dx = 0,

as desired.

As a consequence of Corollary 3.3 and thanks to the definition of D

(Ω), we get the following Corollary 3.4. The Hardy-Lane-Emden inequality (3.1) is valid for every δ > 0 and u ∈ D

(Ω).

4. Sobolev embeddings and densities

In this section, we consider general open sets and study the connections between the integrability of w

and the embeddings of D

(Ω) into Lebesgue spaces. For the case of the torsion function, i.e. when q = 1, related studies can be found in [3, 5, 7] and [8].

We start with a simple consequence of Theorem 3.1. This is valid for a general open set.

Lemma 4.1. Let Ω ⊂ R

be an open set and 1 ≤ q < 2. Then for any ϕ ∈ C

(Ω) it holds that ˆ

{x∈Ω :wq,Ω(x)<+∞}

ϕ

w

dx ≤

ˆ

Ω

|∇ ϕ |

dx.

Proof. Let B

(0) be the ball of radius R centered in 0, we set Ω

= Ω ∩ B

(0) and w

= w

. Let ϕ ∈ C

(Ω), then for every R large enough the support of ϕ is contained in Ω

. By using (3.1) on Ω

with δ = 1, we get

ˆ

Ω

ϕ

w

dx ≤

ˆ

Ω

|∇ ϕ |

dx.

We conclude by letting R → + ∞ and by Fatou’s Lemma.

The following result is a generalization of [3, Theorem 1.2]. We point out that the equivalence between 1. and 2. below is a known fact in Sobolev spaces theory, see [13, Theorems 15.6.2].

Theorem 4.2. Let 1 ≤ q < 2 and let Ω ⊂ R

be an open set. Then for every q ≤ γ < 2 the following three facts are equivalent

1. the embedding D

(Ω) , → L

(Ω) is continuous;

2. the embedding D

(Ω) , → L

(Ω) is compact;

3. w

∈ L

2−q 2−γγ

(Ω).

Moreover, we have the double-sided estimates

(4.1) 1 ≤ λ

(Ω)

ˆ

Ω

w

2−q 2−γγ q,Ω

dx

γ

≤ 2 − γ

γ − 2 (q − 1)

2 − q 2 − γ

, where λ

(Ω) is the optimal Poincar´ e constant defined in (1.6).

Proof. As announced above, the equivalence 1. ⇐⇒ 2. is already known, see also [3, Theorem 1.2]

for a different proof. It is sufficient to prove the equivalence 1. ⇐⇒ 3.

Let us suppose that the embedding D

(Ω) , → L

(Ω) is continuous. As always, we set B

(0) the ball of radius R centered in 0, Ω

= Ω ∩ B

(0) and w

= w

. Then by testing (2.3) with w

for some β ≥ 1, we get

ˆ

ΩR

w

dx = β 2

β + 1

ˆ

ΩR

∇ w

β+1 2

R

2

dx

≥ β 2

β + 1

λ

(Ω

) ˆ

ΩR

w

β+1 2 γ

R

dx

≥ β 2

β + 1

λ

(Ω) ˆ

ΩR

w

β+1 2 γ

R

dx

. By choosing

β = γ − 2 (q − 1) 2 − γ , from the previous estimate we get

ˆ

ΩR

w

2−q 2−γγ

R

dx

≤ 2 − γ γ − 2 (q − 1)

2 − q 2 − γ

1 λ

(Ω)

By Fatou’s Lemma, we can take the limit as R goes to + ∞ and get the desired integrability of w

, together with the upper estimate in (4.1).

Suppose now that w

∈ L

(Ω), this implies that w

< + ∞ almost everywhere in Ω. We take u ∈ C

(Ω), then by H¨ older’s inequality and Lemma 4.1 we have

ˆ

Ω

| ϕ |

dx = ˆ

Ω

| ϕ |

w

γ 2

q,Ω

w

γ 2

q,Ω

dx ≤ ˆ

Ω

ϕ

w

dx

!

ˆ

Ω

w

2−q 2−γγ q,Ω

dx

≤ ˆ

Ω

|∇ ϕ |

dx

ˆ

Ω

w

2−q 2−γγ q,Ω

dx

.

We conclude by density of C

(Ω) in D

(Ω) that the embedding D

(Ω) , → L

(Ω) is continuous.

Moreover, we also obtain the lower bound in (4.1).

The following result generalizes [3, Theorem 1.3] and [4, Theorem 9], by allowing any Lane- Emden densities in place of the torsion function.

3Observe thatβ≥1 thanks to the fact thatq≤γ <2.

Proposition 4.3. Let 1 ≤ q < 2 and let Ω ⊂ R

be an open set. Then we have that λ

(Ω) > 0 ⇐⇒ w

∈ L

(Ω).

Moreover, we have that

(4.2) λ

(Ω)

≤ k w

k

≤

2

C

2 C

+ 4

λ

(Ω)

, where C is the same constant appearing in (A.1).

Proof. We suppose that w

∈ L

(Ω). This in particular implies that w

< + ∞ almost every- where in Ω. Then for any ϕ ∈ C

(Ω) we have that

ˆ

Ω

ϕ

dx = ˆ

Ω

ϕ

w

w

dx

≤ ˆ

Ω

ϕ

w

dx

!

k w

k

≤ k w

k

ˆ

Ω

|∇ ϕ |

dx, the last inequality being due to Lemma 4.1. This shows that

w

∈ L

(Ω) for 1 ≤ q < 2 = ⇒ λ

(Ω) > 0, together with the lower bound in (4.2).

The converse implication is more involved and we adapt the proof of [4, Theorem 9], which deals with the case q = 1. Without loss of generality we can suppose Ω to be bounded and smooth;

indeed, the general case can be then covered by considering a family of smooth bounded sets approaching Ω from inside.

For ease of notation we set w := w

and we suppose that w(0) = k w k

. This can be done up to translating Ω. Moreover we can extend w to 0 outside Ω. Since ∂Ω is regular, we get by means of Hopf’s Lemma that the extended function, which we still denote by w, satisfies

(4.3) − ∆w ≤ w

,

in the weak sense. Let R > 0 to be fixed, and let ζ be a cut-off Lipschitz function such that 0 ≤ ζ ≤ 1, ζ = 1 in B

(0), ζ = 0 in R

\ B

(0), |∇ ζ | ≤ 1

R . From the variational characterization of λ

(Ω), we have

(4.4) λ

(Ω) ≤ ˆ

Ω

|∇ (w ζ) |

dx ˆ

Ω

(w ζ)

dx

= ˆ

Ω

|∇ w |

ζ

+ 2 w ζ h∇ w, ∇ ζ i + |∇ ζ |

w

dx ˆ

Ω

w

ζ

dx

.

By using the positive test function w ζ

into the weak formulation of (4.3), we get that ˆ

Ω

|∇ w |

ζ

dx + 2 ˆ

Ω

w ζ h∇ w, ∇ ζ i dx = ˆ

Ω

h∇ w, ∇ (w ζ

) i dx ≤ ˆ

Ω

w

ζ

dx.

Thus, by recalling that w attains its maximum in 0 and using the properties of ζ, from (4.4) we obtain that

(4.5) λ

(Ω) ≤ ˆ

Ω

|∇ ζ |

w

+ w

ζ

dx ˆ

Ω

w

ζ

dx

≤ 2

ω

w(0)

ˆ R

+ w(0)

R

B_R(0)

w

dx .

We use now the local L

− L

estimate of Lemma A.1 to handle the denominator. Indeed, by (A.1) with α = 2 we have that

ˆ

BR(0)

w

dx ≥ ω

R

1

C w(0) − R

2

!

.

By spending this information in (4.5), we end up with

λ

(Ω) ≤ 2

R

w(0)

+ w(0)

1

C w(0) − R

2

!

.

By choosing

R = 2

w(0) 2 C

, we obtain the inequality

λ

(Ω) ≤ 2

4 C

w(0)

1 4

2 C

+ 1

, and thus

w(0) ≤

2

C

2 C

+ 4

λ

(Ω)

.

This concludes the proof.

Remark 4.4 (Super-homogeneous embeddings). A closer inspection of the proof reveals that with exactly the same argument we can prove the following stronger statement: for every 1 ≤ q < 2 and 2 ≤ γ < 2

, we have that

(4.6) λ

(Ω) > 0 ⇐⇒ w

∈ L

(Ω),

where

2

= 2 N

N − 2 , for N ≥ 3 and 2

= + ∞ , for N ∈ { 1, 2 } . Observe that (4.6) implies in particular that

(4.7) D

(Ω) , → L

(Ω) ⇐⇒ D

(Ω) , → L

(Ω), for 2 < γ < 2

.

For the implication = ⇒ , it is sufficient to reproduce the proof above, using the variational charac- terization of λ

(Ω) and the L

estimate (A.1), this time with α = γ.

For the converse implication, it is sufficient to use the Gagliardo-Nirenberg interpolation inequal- ity (see for example [3, Proposition 2.6])

ˆ

R^N

| u |

dx

≤ C ˆ

R^N

| u |

dx

ˆ

R^N

|∇ u |

dx

,

where C = C(N, γ) > 0 and

ϑ =

1 − 2 γ

N

2 , 2 < γ < 2

.

This shows that if D

(Ω) , → L

(Ω) is continuous, then D

(Ω) , → L

(Ω) is continuous as well.

We leave the details to the interested reader.

We point out that the equivalence (4.7) can also be found in [13, Theorem 15.4.1]. The proof there is different.

We conclude this section with the following simple result which we record for completeness.

Proposition 4.5. Let 1 ≤ q < 2 and let Ω ⊂ R

be an open set such that the embedding D

(Ω) , → L

(Ω) is continuous. Then the embedding D

(Ω) , → L

(Ω) is compact.

Proof. We already know by Theorem 4.2 that the continuity of the embedding D

(Ω) , → L

(Ω) is equivalent to its compactness. Then it is sufficient to use the Gagliardo-Nirenberg inequality

ˆ

R^N

| u |

dx

2

≤ C ˆ

R^N

| u |

dx

q

ˆ

R^N

|∇ u |

dx

2

, where C = C(N, q) > 0 and

ϑ = 1 − q

2

2 N (2 − q) N + 2 q .

This guarantess that every bounded sequence { u

}

⊂ D

(Ω) strongly converging in L

(Ω), strongly converges in L

(Ω) as well. This gives the desired conclusion.

Remark 4.6. The converse implication of the previous proposition does not hold. Indeed, let { r

}

⊂ R be a sequence of strictly positive numbers, such that

i→∞

lim r

= 0 and

∞

X

i=0

r

2 2−γ+N

i

= + ∞ , for every 1 ≤ γ < 2.

For example, one could take r

= 1/ log(2 + i). We then define the sequence of points { x

}

⊂ R

by

x

= (0, . . . , 0),

x

= (r

+ r

, 0, . . . , 0) + x

, and the disjoint union of balls

Ω =

∞

[

i=0

B

(x

).

Thanks to the choice of the radii r

we have

w

∈ L

(Ω) and for every ε > 0, there exists R > 0 such that k w

k

< ε, thus the embedding D

(Ω) , → L

(Ω) is compact, see [3, Theorem 1.3].

On the other hand, w

6∈ L

(Ω) for every γ ∈ [1, + ∞ ) (see [3, Example 5.2]). Thus, by

Theorem 4.2, D

(Ω) is not continuously embedded in any L

(Ω), with 1 ≤ γ < 2.