Baptiste Devyver
email: baptiste.devyver@univ-nantes.fr

Yehuda Pinchover

OPTIMAL L p HARDY-TYPE INEQUALITIES

 for the linear case (p = 2), and in particular, to obtain "as large as possible" nonnegative (optimal) Hardy-type weight W satisfying

Our main results deal with the case where V = 0, and Ω is a general punctured domain (for V = 0 we obtain only some partial results). In the case 1 < p ≤ n, an optimal Hardy-weight is given by

where G is the associated positive minimal Green function with a pole at 0. On the other hand, for p > n, several cases should be considered, depending on the behavior of G at infinity in Ω. The results are extended to annular and exterior domains.

Introduction

In a recent paper [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], the authors studied a general second-order linear elliptic operator P ≥ 0 in a general domain Ω ⊂ R n (or a noncompact smooth manifold of dimension n), where n ≥ 2, and obtained an optimal improvement of the inequality P ≥ 0. The improved inequality is of the form P ≥ W , where W is "as large as possible" weight function, and (in the self-adjoint case) the inequality P ≥ W is meant in the quadratic form sense. The weight W is given explicitly using a simple construction called the supersolution construction; any two linearly independent positive (super)solutions u 0 , u 1 of the equation P u = 0 give rise to a one-parameter family of Hardy-type weights {W α } {0≤α≤1} satisfying the inequality P ≥ W α (for more details on this construction see Section 4). The optimal weight is obtained by a careful choice of u 0 , u 1 and α.

In the case of a Schrödinger type operator P , the main result of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] reads as follows.

Theorem 1.1. Consider a symmetric second-order linear elliptic operator P of the form P u := -div A(x)∇u + V (x)u which is subcritical in Ω. Let q be the associated quadratic form. Then there exists a nonzero, nonnegative weight W satisfying the following properties: (a) The following Hardy-type inequality holds true

q(ϕ) ≥ λ Ω W (x)|ϕ(x)| 2 dx ∀ϕ ∈ C ∞ 0 (Ω), (1.1)
with λ > 0. Denote by λ 0 := λ 0 (P, W, Ω) the best constant satisfying (1.1). (b) The operator P -λ 0 W is critical in Ω; that is, the inequality

q(ϕ) ≥ Ω W 1 (x)ϕ 2 (x) dx ∀ϕ ∈ C ∞ 0 (Ω)
is not valid for any W 1 λ 0 W . (c) The constant λ 0 is also the best constant for (1.1) with test functions supported in the exterior of any fixed compact set in Ω. (d) The operator P -λ 0 W is null-critical in Ω; that is, the corresponding Rayleigh-Ritz variational problem inf ϕ∈D 1,2 P (Ω) q(ϕ)

Ω W (x)|ϕ(x)| 2 dx (1.2)
admits no minimizer. Here D 1,2 P (Ω) is the completion of C ∞ 0 (Ω) with respect to the norm u → q(u). (e) If furthermore, W > 0, then the spectrum and the essential spectrum of the Friedrichs extension of the operator W -1 P on L 2 (Ω, W dx) are both equal to [λ 0 , ∞).

In the present paper we consider the quasilinear case. Let 1 < p < ∞, and denote by ∆ p (u) := div(|∇u| p-2 ∇u) the p-Laplace operator. Throughout the paper, Ω is either a domain in R n , or a noncompact smooth Riemannian manifold of dimension n, n ≥ 2, such that 0 ∈ Ω. Let V ∈ L ∞ loc (Ω) be a real valued potential, and let Q V be the quasilinear operator

Q V (u) = Q(u) := -div(|∇u| p-2 ∇u) + V (x)|u| p-2 u (1.3)
defined on Ω. Denote by

Q V (ϕ) = Q(ϕ) := Ω |∇ϕ| p + V |ϕ| p dν
the associated energy defined on C ∞ 0 (Ω). We say that Q ≥ 0 in Ω if Q(ϕ) ≥ 0 for all ϕ ∈ C ∞ 0 (Ω). Let W ≥ 0 in Ω. We denote

λ 0 (Q V , W, Ω) := sup{λ ∈ R | Q V -λW ≥ 0 in Ω}, λ ∞ (Q V , W, Ω) := sup{λ ∈ R | ∃K ⊂⊂ Ω s.t. Q V -λW ≥ 0 in Ω \ K},
respectively, the best constant and best constant at infinity in the Hardy-type inequality

Q V (ϕ) ≥ λ Ω ⋆ W |ϕ| p dν, ∀ϕ ∈ C ∞ 0 (Ω).
The aim of the present article is to generalize Theorem 1.1 (obtained in the linear case), to the quasilinear case and to obtain "as large as possible" nonnegative (optimal) weight W satisfying

Q(ϕ) ≥ λ Ω W (x)|ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).
In particular, we answer affirmatively a problem posed by the authors in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] (see Problem 13.12 therein).

The extension of Theorem 1.1 to the quasilinear case is not a straightforward task. First, due to the nonlinearity of the operator Q V , the supersolution construction has to be modified, and in particular in the case p > n, the supersolution construction leading to optimal potentials is essentially different. In fact, we could not extend Theorem 1.1 to operators Q V with V = 0. Secondly, the proof of Theorem 1.1 given in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] is mostly of linear nature, and therefore a new approach is needed for the quasilinear case. Moreover, the proof of Theorem 1.5 actually provides us with an alternative proof for parts (b) and (c) of Theorem 1.1. On the other hand, it seems that there is no analog to part (e) of Theorem 1.1 concerning the essential spectrum of the corresponding operator. We note that in the linear case, the proof of part (e) relies on a construction of a family of generalized eigenfunctions, and this construction does not apply to the quasilinear case.

Let us introduce first our definition of optimal Hardy-weights for Q V in a punctured domain.

Definition 1.2. Suppose that Q V ≥ 0 in Ω, and denote Ω ⋆ := Ω \ {0}. Assume that a nonzero nonnegative function W satisfies the following Hardytype inequality

Q V (ϕ) ≥ λ Ω ⋆ W |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω ⋆ ), (1.4)
where λ is a positive constant. Set

λ 0 := λ 0 (Q V , W, Ω ⋆ ).
We say that W is an optimal Hardy-weight for the operator Q V in Ω if the following conditions hold true.

(

) The functional Q V -λ 0 W is critical in Ω ⋆ , i.e. for any W 1 λ 0 W , the Hardy-type inequality Q V (ϕ) ≥ Ω ⋆ W 1 |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω ⋆ ) 1
does not hold. In particular, the equation Q V -λ 0 W (u) = 0 in Ω ⋆ admits, up to a multiplicative positive constant, a unique positive (super)solution v; such a v is called the Agmon ground state. (2) λ 0 is also the best constant for inequality (1.4) restricted to functions ϕ that are compactly supported either in a fixed punctured neighborhood of the origin, or in a fixed neighborhood of infinity in Ω. In particular, λ

∞ (Q V , W, Ω ⋆ ) = λ 0 . (3) Suppose further that V ≥ 0. For an open set Ω ⊂ Ω, let D 1,p Q V ( Ω) be the completion of C ∞ 0 ( Ω) with respect to the norm Q V (•) 1/p . Then the functional Q V -λ 0 W
is null-critical at 0 and at infinity in the following sense: for any pre-compact open set O containing 0, the (Agmon) ground state

v of Q V -λ 0 W in Ω ⋆ belongs neither to D 1, p Q V (Ω \ Ō) nor to D 1, p Q V (O \ {0}).
In particular, the variational problem inf

v∈D 1, p (Ω ⋆ ) Q V (ϕ) Ω ⋆ |ϕ| p W dν (1.5)
does not admit a minimizer.

Remark 1.3. It is natural to ask whether all the above properties of an optimal Hardy-weight are independent. It is indeed the case; in fact, in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] we gave the following example which shows that, in general, (3) is not a consequence of ( 1) and [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF].

Let 0 ≤ V ∈ C ∞ 0 (R n ) be a potential such that the operator -∆ -V (x) is critical in R n . Consider the operator Q := -∆ + 1 -V (x), and the potential W (x) := 1. Then λ 0 (Q, W, R n ) = λ ∞ (Q, W, R n ) = 1. On the other hand, the operator Q -W is null-critical in R n for n ≤ 4, and positive-critical if n > 4. Remark 1.4. If p = 2, the definition of D 1,p Q V (Ω)
cannot be applied to the case where V ≥ 0, since the positivity of the functional Q V on C ∞ 0 (Ω) does not necessarily imply its convexity, and thus it does not give rise to a norm (see the discussion in [START_REF] Pinchover | On the Hardy-Sobolev-Maz'ya inequality and its generalizations[END_REF]).

Using a modified supersolution construction, we obtain the main result of our paper: Theorem 1.5. Let ∞ denote the ideal point in the one-point compactification of Ω. Suppose that -∆ p admits a positive p-harmonic function G in Ω ⋆ := Ω \ {0} satisfying one of the following conditions (1.6) and (1.7):

1 < p ≤ n, lim x→0 G(x) = ∞,
and

lim x→ ∞ G(x) = 0, (1.6) p > n, lim x→0 G(x) = γ ≥ 0, and lim x→ ∞ G(x) = ∞ if γ = 0, 0 if γ > 0. (1.7)
Define a positive function v and a nonnegative weight W on Ω ⋆ as follows:

(1) If either (1.6) is satisfied, or (1.7) is satisfied with γ = 0, then v := G (p-1)/p , and

W := p-1 p p ∇G G p .
(2) If (1.7) is satisfied with γ > 0, then v := [G(γ -G)] (p-1)/p , and

W := p -1 p p ∇G G(γ -G) p |γ -2G| p-2 2(p -2)G(γ -G) + γ 2 .
Then the following Hardy-type inequality holds in Ω ⋆ :

Ω ⋆ |∇ϕ| p dν ≥ Ω ⋆ W |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω ⋆ ), (1.8)
and W is an optimal Hardy-weight for -∆ p in Ω.

Moreover, up to a multiplicative constant, v is the unique positive supersolution of the equation

Q -W (w) = 0 in Ω ⋆ .
Remark 1.6. Let us discuss hypotheses (1.6) and (1.7). Suppose first that Ω is a C 1,α -bounded domain with 0 < α ≤ 1. Let G Ω (x, 0) be the positive minimal p-Green function of the operator -∆ p in Ω with a pole at 0. Then G := G Ω (•, 0) satisfies either (1.6), or (1.7) with γ > 0. This assertion follows, for example, from the results in [START_REF] Fraas | Isolated singularities of positive solutions of p-Laplacian type equations in R d[END_REF][START_REF] García-Melián | Maximum and comparison principles for operators involving the p-Laplacian[END_REF] and is valid more generally for any subcritical operator

Q V with V ∈ L ∞ (Ω).
Suppose further that Ω is a C 1,α -subdomain of a noncompact Riemannian manifold M (where α ∈ (0, 1]), with a positive p-Green function G M that satisfies lim x→ ∞ G M (x, 0) = 0. Using a standard exhaustion argument, the monotonicity of the Green functions as a function of the domain, and the above remark, it follows that G := G Ω (•, 0) satisfies either (1.6), or (1.7) with γ > 0.

If Ω = R n , Q = -∆ p , and 1 < p < n (resp., p > n) , then G(x) := |x| p-n p-1 satisfies assumption (1.6) (resp., assumption (1.7) with γ = 0). In this case, Ω ⋆ = R n \ {0} is the punctured space, and W (x) = p-1 p p |x| -p is the classical Hardy potential. We note that the criticality of the operator

Q -W (u) = -div(|∇u| p-2 ∇u) - p -1 p p |u| p-2 u |x| p in Ω ⋆
follows also from the proof of [START_REF] Poliakovsky | Uniqueness of positive solutions for singular problems involving the p-Laplacian[END_REF]Theorem 1.3] given by Poliakovsky and Shafrir.

Remark 1.7. In our study, the domain Ω ⋆ should be viewed as a manifold with two ends: the origin and ∞, the ideal point obtained by the one-point compactification of Ω. In particular, the notion of optimal Hardy-weight can be extended analogously to the case of any manifold with two ends (see Section 6, for an extension of Theorem 1.5 to annular or exterior domains).

The outline of the present paper is as follows. In Section 2 we review the theory of positive solutions for p-Laplacian type equations. Section 3 is devoted to a coarea formula which is a key result in our study (see Proposition 3.1). Section 4 explains the supersolution construction of Hardy-weights in various situations. Section 5 is devoted to the proof of Theorem 1.5. Section 6 we present extensions of Theorem 1.5 to the case of annular and exterior domains. In Section 7 we present some L p -Rellich-type inequalities and discuss the optimality of the obtained constants. Finally, in Section 8 we study the supersolution construction for general operators Q V of the form (1.3), where the obtained weight is in general not optimal.

Preliminaries

Let Ω be a domain in R n (or in a noncompact Riemannian manifold of dimension n), where n ≥ 2. We equip Ω with the one-point compactification, and denote by ∞ the added ideal point which we call the infinity in Ω. So, x n → ∞ if and only if the sequence {x n } n∈N ⊂ Ω eventually exits any compact subset of Ω. For example, if Ω ⊂ R n is bounded, then the infinity in Ω is just ∂Ω, and x n → ∞ if and only if dist(x n , ∂Ω) → 0, where dist(•, ∂Ω) is the distance function to ∂Ω.

Throughout the paper we assume that Ω is equipped with an absolutely continuous measure ν with respect to the Lebesgue measure in R n (or with respect to the Riemannian measure in the case of a Riemannian manifold), and that the corresponding density is positive and smooth.

We write

Ω 1 ⋐ Ω 2 if Ω 2 is open, Ω 1 is compact and Ω 1 ⊂ Ω 2 . Let f, g ∈ C(D) be nonnegative functions, we denote f ≍ g on D if there exists a positive constant C such that C -1 g(x) ≤ f (x) ≤ Cg(x)
for all x ∈ D.

For 1 < p < ∞, we consider a quasilinear operator

Q V (u) = Q(u) := -∆ p (u) + V |u| p-2 u, (2.1)
where V ∈ L ∞ loc (Ω). Here, the p-Laplacian ∆ p is defined by

∆ p (u) := div(|∇u| p-2 ∇u),
where div is the divergence with respect to the measure ν, so, the integration by parts formula Ω -div(X)ϕ dν = Ω X • ∇ϕ dν holds for any smooth vector field X and function ϕ that are compactly supported in Ω. Associated to Q V there is the energy functional

Q V (ϕ) = Q(ϕ) := Ω |∇ϕ| p + V |ϕ| p dν ϕ ∈ C ∞ 0 (Ω). (2.2)
We say that u ∈ W 1, p loc (Ω) is a (weak) solution of the equation

Q(u) = f in Ω if for every ϕ ∈ C ∞ 0 (Ω), Ω |∇u| p-2 ∇u • ∇ϕ + V |u| p-2 uϕ dν = Ω f ϕ dν . (2.3)
We define in a similar way the notions of subsolution and supersolution of Q(u) = f . Weak solutions of the equation Q(u) = 0 admit Hölder continuous first derivatives, and nonnegative solutions of the equation Q(u) = 0 satisfy the Harnack inequality (see for example [START_REF] Ladyženskaja | ceva, A variational problem for quasilinear elliptic equations with many independent variables[END_REF][START_REF] Pucci | The Maximum Principle[END_REF][START_REF] Serrin | Local behavior of solutions of quasi-linear equations[END_REF][START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF][START_REF] Tolksdorf | Regularity for a more general class of quasilinear elliptic equations[END_REF]). Therefore, in the definition (2.3) with f = 0, one can equivalently take test functions in C 1 0 (Ω) instead of C ∞ 0 (Ω). The notions of criticality and subcriticality of Q V have been studied in this context, and we refer to [START_REF] Pinchover | Ground state alternative for p-Laplacian with potential term[END_REF] for an account on this. For completeness, we recall the essential notions and results that we need throughout the present paper.

The operator Q is said to be nonnegative in Ω (and we denote it by Q ≥ 0) if the equation Q(u) = 0 in Ω admits a positive (super)solution. As in the (selfadjoint) linear case, the following Allegretto-Piepenbrink type theorem holds:

Theorem 2.1 ([13, Theorem 2.3]). Q V ≥ 0 in Ω if and only if Q V (ϕ) ≥ 0 for every ϕ ∈ C ∞ 0 (Ω).
Throughout the paper, we assume that Q is nonnegative in Ω. As in the linear case, there is a dichotomy for nonnegative operators: Q of the form (2.1) is either critical or subcritical in Ω. We note that in the case of Q = -∆ p on a Riemannian manifold M equipped with its Riemannian measure, criticality (resp., subcriticality) is often called p-parabolicity (resp., p-hyperbolicity). Criticality/subcriticality has several equivalent definitions, which we recall below, but first we need to introduce some notions. Definition 2.3. Let K 0 be a compact set in Ω. A positive solution u of the equation Q(w) = 0 in Ω \ K 0 is said to be a positive solution of minimal growth in a neighborhood of infinity in Ω (or u ∈ M Ω,K 0 for brevity) if for any compact set K in Ω, with a smooth boundary, such that K 0 ⋐ int(K), and any positive supersolution v ∈ C((Ω\K)∪∂K) of the equation

Q(w) = 0 in Ω \ K, the inequality u ≤ v on ∂K implies that u ≤ v in Ω \ K.
Similarly, for x 0 ∈ Ω, we define the notion of a positive solution of the equation Q(w) = 0 in a punctured neighborhood of x 0 of minimal growth at x 0 .

We have Theorem 2.4 ( [START_REF] Pinchover | Ground state alternative for p-Laplacian with potential term[END_REF][START_REF] Fraas | Isolated singularities of positive solutions of p-Laplacian type equations in R d[END_REF]). Suppose that Q is nonnegative in Ω, and fix x 0 ∈ Ω. Then the equation Q(w) = 0 has (up to a multiplicative constant) a unique positive solution u ∈ M Ω,{x 0 } of minimal growth in a neighborhood of infinity in Ω.

Moreover, u is either a global positive solution of Q(w) = 0 in Ω (such a solution is called Agmon's ground state), or u has singularity at x 0 with the following asymptotic:

u(x) ∼ x→x 0        |x -x 0 | p-n p-1 if 1 < p < n, -log |x -x 0 | if p = n, 1 if p > n.
In the latter case, the appropriately normalized solution is called the positive minimal Green function of Q in Ω with a pole at x 0 , and is denoted by

G Ω Q (x, x 0 ) = G(x).
Furthermore, any positive solution v of Q(w) = 0 in a punctured neighborhood of x 0 of minimal growth at x 0 has the following asymptotic near

x 0 : v(x) ∼ x→x 0 1 if 1 < p ≤ n, |x -x 0 | p-n p-1 if p > n. Definition 2.5. Suppose that Q ≥ 0 in Ω. Then Q is said to be critical in Ω if the equation Q(u) = 0
in Ω admits a (Agmon) ground state, and subcritical in Ω otherwise.

Lemma 2.6 ([13]

). Suppose that Q ≥ 0 in Ω. Then the following assertions are equivalent:

(1) Q is critical in Ω. ( 2
)
The equation Q(w) = 0 in Ω admits a unique positive supersolution (up to a multiplicative constant).

(3) The only nonnegative function W such that the inequality

Q(ϕ) ≥ Ω W (x)|ϕ| p dν holds for every ϕ ∈ C ∞ 0 (Ω) is W = 0. (4) Q admits a null sequence in Ω.
A nonnegative functional Q might contain an indefinite term (if the potential has indefinite sign). Although, by Picone identity [START_REF] Allegretto | A Picone's identity for the p-Laplacian and applications[END_REF], such functional Q can be represented as the integral of a nonnegative Lagrangian L, this L still contains an indefinite term. It was proved in [START_REF] Pinchover | A Liouville-type theorem for the p-Laplacian with potential term[END_REF] that Q is equivalent to a simplified energy containing only nonnegative terms, as we explain now. Definition 2.7. Let v be a positive solution of the equation

Q(u) = 0 in Ω. The simplified energy is defined for nonnegative functions w ∈ C ∞ 0 (Ω) by Q sim (w) :=          Ω v 2 |∇w| 2 (v|∇w| + w|∇v|) p-2 dν if 1 < p ≤ 2, Ω v p |∇w| p + v 2 |∇v| p-2 w p-2 |∇w| 2 dν if p > 2.
(2.4)

Since Picone identity holds also on manifolds (cf. [15, Section 2]), it follows that Lemma 2.2 in [START_REF] Pinchover | A Liouville-type theorem for the p-Laplacian with potential term[END_REF] is valid also on manifolds. Therefore, we obtain the following equivalence between the functional Q and the simplified energy

Q sim : Lemma 2.8 ([15, Lemma 2.2]). Assume that Q = Q V ≥ 0 in Ω. Let v ∈ C 1,α loc (Ω) be a fixed positive solution of the equation Q(u) = 0 in Ω. Then for all w ∈ C ∞ 0 (Ω) we have Q(w) ≍ Q sim w v .
Lemma 2.8 is a generalization of the ground state transform (see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]) to the nonlinear case. In the nonlinear case, one obtains the equivalence (and not equality, as in the linear case) between Q and a functional containing only positive terms. As a corollary of Lemma 2.8, we state the following obvious upper estimate for the simplified energy, which will be of use later.

Lemma 2.9. Denote

X(w) := Ω ⋆ v p |∇w| p dν, Y (w)
:= Ω ⋆ |w| p |∇v| p dν.
Then there exists C > 0 such that for all w ∈ C ∞ 0 (Ω) we have

Q sim (w) ≤        CX(w) if 1 < p ≤ 2, C X(w) + X(w) Y (w) 2/p Y (w) if p > 2.
(

We conclude this section with the following useful lemma Lemma 2.10. Let u ∈ C 1,α loc (Ω) for some α ∈ (0, 1), and f ∈ C 2 . Then the following formula holds in the weak sense:

-∆ p (f (u)) = -|f ′ (u)| p-2 (p -1)f ′′ (u)|∇u| p + f ′ (u)∆ p (u) .
(2.6)

Proof. Denote g := -∆ p (u), and let ϕ ∈ C ∞ 0 (Ω). Then, by Leibniz's product rule and the chain rule we have

Ω |∇f (u)| p-2 ∇f (u) • ∇ϕ dν = Ω |∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u)ϕ dν - Ω |∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u) ϕdν
Note that for p ≥ 2, the function ψ(s) := |s| p-2 s is continuously differentiable, and ψ ′ (s) := (p -1)|s| p-2 . Moreover, for 1 < p < 2 the function ψ is not differentiable at zero but its derivative near zero is integrable. Recall that by our assumptions u ∈ C 1,α (Ω). Therefore if p ≥ 2, then the func-

tion |f ′ (u)| p-2 f ′ (u)ϕ belongs to C 1 0 (Ω). On the other hand, for 1 < p < 2, ∇ |f ′ (u)| p-2 f ′ (u)ϕ is integrable. Hence in both cases, |f ′ (u)| p-2 f ′ (u)ϕ is a legitimate test function. Consequently, Ω |∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u)ϕ dν = Ω g|f ′ (u)| p-2 f ′ (u)ϕ dν. Therefore, Ω |∇f (u)| p-2 ∇f (u) • ∇ϕ dν = Ω g|f ′ (u)| p-2 f ′ (u)ϕ dν - Ω |∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u) ϕ dν.
Consequently, in the weak sense we have

-∆ p (f (u)) = -|∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u) -∆ p (u)|f ′ (u)| p-2 f ′ (u).
But since ψ ′ (s) := (p -1)|s| p-2 for s = 0, and ψ ′ is integrable at 0, we have that in the weak sense

|∇u| p-2 ∇u • ∇ |f ′ (u)| p-2 f ′ (u) = (p -1)|f ′ (u)| p-2 |∇u| p f ′′ (u). (2.7)
This completes the proof of Lemma 2.10.

The coarea formula

The present section is devoted to the proof of a coarea formula associated with the p-Laplacian. It seems that this key result in our study cannot be extended to the case of an operator Q V of the form (1.3) with V = 0 and p = 2 (cf. [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]Lemma 9.2], where an analogue coarea formula is obtained for any linear symmetric operator). Proposition 3.1. Let G be a positive p-harmonic function in Ω ⋆ := Ω \ {0}. Define v := G (p-1)/p . Then there exists positive constants c and c such that for every real functions f and g, defined on (0, ∞) such that f (v) and g(v) have compact support in Ω, the following formulae hold:

Ω ⋆ f (v)|∇v| p dν = c sup v inf v f (τ ) τ dτ, (3.1)
and

Ω ⋆ g(G)|∇G| p dν = c sup G inf G g(t) dt. (3.2)
Proof. The idea is the same as in [6, Lemma 9.2]. Setting g(t) := f (t (p-1)/p ) and performing the change of variable τ := t (p-1)/p , it follows that (3.1) is equivalent to (3.2). By the coarea formula, we have

Ω ⋆ g(G)|∇G| p dν = sup G inf G {G=t} g(G) |∇G| p |∇G| dσ dt = sup G inf G g(t) {G=t} |∇G| p-1 dσ dt, (3.3)
where dσ denotes the Hausdorff measure of dimension n -1. Indeed, G ∈ C 1,α loc , in particular |∇G| p-1 ∈ L 1 loc and the use of the coarea formula is licit. We claim that {G=t} |∇G| p-1 dσ does not depend on t. This essentially follows from Green's formula, but since G is not smooth, we have to be careful. Let us fix t 1 , t 2 such that inf G < t 1 < t 2 < sup G, and define A to be the "annulus"

A := {x ∈ Ω ⋆ | t 1 < G < t 2 }. The boundary of A is the disjoint union of ∂ -:= {G = t 1 } and of ∂ + := {G = t 2 }.
We claim that A has finite perimeter, i.e., χ A , the characteristic function of A, has bounded variation. Indeed,

χ A = χ (t 1 ,t 2 ) • G, therefore, ∇χ A = χ ′ (t 1 ,t 2 ) (G) ∇G = (δ G=t 1 -δ G=t 2 )∇G.
Since ∇G is continuous, we obtain that χ A ∈ BV , hence A has finite perimeter. Since |∇G| p-2 ∇G is continuous, and has divergence which vanishes in A in the weak sense, Theorems 5.2 and 7.2 in [START_REF] Chen | Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws[END_REF] imply that the Gauss-Green formula is valid on A:

0 = - A div(|∇G| p-2 ∇G) dν = ∂ ⋆ + |∇G| p-2 ∇G • n dσ + ∂ ⋆ - |∇G| p-2 ∇G • n dσ, (3.4)
where ∂ ⋆ + and ∂ ⋆ -are the reduced boundaries (see [START_REF] Chen | Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws[END_REF]), n is the measure theoretic exterior unit normal, and σ is the (n -1)-dimensional Hausdorff measure. If x ∈ ∂ + (resp., x ∈ ∂ -) is such that |∇G(x)| = 0, then the boundary of A is C 1 in a neighborhood of x, and the vector field ∇G/|∇G| is well-defined near x; it is equal to n (resp., -n) around x. Furthermore, we can write around x

|∇G| p-1 = |∇G| p-2 |∇G| = ±|∇G| p-2 ∇G • n if x ∈ ∂ ± . (3.5) Since G is C 1,α ,
we may use a generalization of Sard's theorem due to Bojarski, Haj lasz and Strzelecki [START_REF] Bojarski | Sard's theorem for mappings in Hölder and Sobolev spaces[END_REF] to infer that for almost every t ∈ (0, ∞)

σ ({G = t} ∩ Crit(G)) = 0,
where Crit(G) is the set of critical points of G. This implies that for almost all t , (3.5) holds σ-almost everywhere on {G = t}, and that for almost all t 1 and t 2 , the reduced boundaries

∂ ⋆ + and ∂ ⋆ -coincide with ∂ + = {G = t 2 }
and ∂ -= {G = t 1 }, respectively, up to a set of zero measure for σ. Since |∇G| p-1 and |∇G| p-2 ∇G are continuous, we obtain that for almost all t 1 and t 2 we have

∂ ⋆ + |∇G| p-2 ∇G • n dσ + ∂ ⋆ - |∇G| p-2 ∇G • n dσ = ∂ + |∇G| p-1 dσ - ∂ - |∇G| p-1 dσ,
and therefore by (3.4),

{G=t 2 } |∇G| p-1 dσ = {G=t 1 } |∇G| p-1 dσ.
Thus, {G=t} |∇G| p-1 dσ is equal (almost everywhere) to a constant independent of t.

The supersolution construction for the p-Laplacian

In this section, we show how to extend the supersolution construction, which was a primary tool in the study of the linear case in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF], to the p -Laplace operator. As in the linear case, in some cases this construction will give us optimal Hardy weights. We postpone the study of the supersolution construction for Q V with V = 0 to Section 8, and here we present two particular supersolution constructions which apply to the p-Laplace operator. These constructions will lead us to the optimal weights of Theorems 1.5.

For completeness, we recall the supersolution construction for linear (not necessarily symmetric) elliptic operators: Lemma 4.1 ([12, Theorem 3.1] and [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]Remark 5.4]). Let P be a secondorder linear elliptic operator with real coefficients defined in Ω. For j = 0, 1, let V j be real valued potentials, and suppose that v j are positive (super)solutions of the equations

(P + V j )u = 0 in Ω. Then for 0 ≤ α ≤ 1 the function v α := (v 1 )) α (v 0 ) 1-α
is a positive (super)solution of the linear equation

[P + (1 -α)V 0 + αV 1 -α(1 -α)W ] u = 0 in Ω, (4.1)
where

W := ∇ log v 0 v 1 2 A , (4.2)
A = A(x) is the nonnegative definite matrix associated with the principal part of the operator P , and for ξ ∈ R n , |ξ| 2 A := ξ • Aξ. We notice that since the proof of Lemma 4.1 is purely local and algebraic, we obtain in fact the following pointwise result. Corollary 4.2. Let P be a second-order linear elliptic operator with real coefficients defined in Ω. For j = 0, 1, let V j be real valued potentials, and suppose that v j are positive functions satisfying the differential (in)equality

(P + V j )v j = (≥) 0 at x 0 ∈ Ω.
Then for 0 ≤ α ≤ 1 the function v α = (v 1 ) α (v 0 ) 1-α satisfies the differential (in)equality

[P + (1 -α)V 0 + αV 1 -α(1 -α)W ] u = (≥) 0 at x 0 ∈ Ω, (4.3)
where W is the function defined by (4.2).

A related -but weaker -convexity result is known in the case of p-Laplacian type equations:

Lemma 4.3 ([13, Proposition 4.3]). Let V 0 , V 1 ∈ L ∞ loc (Ω), V 0 = V 1 . For α ∈ [0, 1] we denote Q α (u) := Q (1-α)V 0 +αV 1 (u) = (1 -α)Q V 0 (u) + αQ V 1 (u), (4.4)
and suppose that The supersolution construction has been extended to the p-Laplacian itself by several authors, with v α := (v 1 ) α (v 0 ) 1-α , in the particular case where v 0 is a positive p-harmonic function, and v 1 = 1 (see for example [START_REF] Adimurthi | Role of the fundamental solution in Hardy-Sobolev-type inequalities[END_REF][START_REF] Ambrosio | Hardy inequalities on Riemannian manifolds and applications[END_REF][START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] and references therein). In particular, the following Caccioppoli-type inequality has been obtained in [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]: Proposition 13.11]). Assume that G is a positive supersolution (resp., solution) of the equation -∆ p (w) = 0 in Ω. Then for α ∈ (0, 1), G α is a positive supersolution (resp., solution) of the equation Q -Wα (w) = 0 in Ω, where

Q V i ≥ 0 in Ω for i = 0, 1. Then Q α ≥ 0 in Ω for all α ∈ [0, 1]. Moreover, Q α is subcritical in Ω for all α ∈ (0, 1).
Proposition 4.5 ([6,
W α := α p-1 (1 -α)(p -1) ∇G G p .
In particular, by taking the optimal value α = p-1 p we obtain the following logarithmic Caccioppoli inequality:

Ω |∇ϕ| p dν ≥ p -1 p p Ω ∇v v p |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω), (4.5)
where v is any positive p-superharmonic function in Ω.

Proof. The first assertion of the proposition follows from Lemma 2.10 and in particular from (2.6) with f (s) := s α . Hence using the Allegretto-Piepenbrink theorem 2.1, we obtain (4.5).

Remark 4.6. Inequality (4.5) has been independently proved in [START_REF] Ambrosio | Hardy inequalities on Riemannian manifolds and applications[END_REF] by L. D'Ambrosio and S. Dipierro, using a different approach.

R n \{0} |∇ϕ| p dx ≥ p -n p p R n \{0} |ϕ(x)| p |x| p dx ∀ϕ ∈ C ∞ 0 (Ω). (4.6)
We will see later that Proposition 4.5 yields an optimal Hardy weight if G further satisfies either assumption (1.6) or (1.7) with γ = 0 (see Theorem 1.5). However, as we shall see in Section 8, this supersolution construction does not provide us with an optimal Hardy weight if Ω is a bounded, C 1,αdomain if G satisfies (1.7) with γ > 0. In this case and also in other cases (see Section 6), an optimal Hardy weight will be obtained using a different supersolution construction given by the following proposition.

Proposition 4.8. Suppose that G is a C 1,β -positive supersolution (resp., solution) of -∆ p w = 0 in Ω satisfying 0 ≤ m < G < M < ∞ in Ω, where 0 < β ≤ 1. Set v α := [(G -m)(M -G)] α ,

and define

W α := (p-1)α p-1 ∇G v 1 p |m+M-2G| p-2 2(2α-1)v 1 +(1-α)(M -m) 2 ≥ 0. (4.7)
Then for α satisfying

α ∈ [1/2, 1] if m > 0, [0, 1] if m = 0,
the function v α is a positive supersolution (resp., solution) of the equation

Q -Wα (w) = 0 in Ω.
In particular, let α = (p -1)/p, and assume that either α = (p -1)/p ≥ 1/2, or m = 0. Define

W := W p-1 p = p -1 p p ∇G v 1 p |m + M -2G| p-2 2(p -2)v 1 + (M -m) 2 . (4.8) Then v := v p-1 p = [(G -m)(M -G)] p-1 p
is a positive solution (resp., supersolution) of Q -W (w) = 0 in Ω, and the following L p -Hardy type inequality holds:

Ω |∇ϕ| p dν ≥ Ω W |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω). (4.9)
Proof. Let 0 ≤ α ≤ 1. By our assumption, G ∈ C 1,β loc (Ω) for some β ∈ (0, 1]. Moreover, the function f (s) = [(s -m)(M -s)] α belongs to C 2 (0, γ) . Consequently, one may apply Lemma 2.10 with G and f to obtain that in the weak sense,

-∆ p (v α ) = (≥) -(p -1)|f ′ (G)| p-2 |∇G| p f ′′ (G) = W α v p-1 α in Ω.
Therefore, v α = f (G) is a positive (super)solution of the equation Q -Wα (w) = 0 in Ω, and the Allegretto-Piepenbrink type theorem (Theorem 2.1) implies

Ω |∇ϕ| p dν ≥ Ω W α |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).
In particular, for α = (p -1)/p we have (4.9).

Remark 4.9. Let Ω 1 ⋐ Ω 2 ⊂ R n be two open sets. Suppose that Ω := Ω 2 \ Ω 1 is a C 1,β -bounded annular-type domain such that ∂Ω is the union of Γ 1 = ∂Ω 1 , and Γ 2 = ∂Ω 2 . Let G be the solution of the Dirichlet problem

     -∆ p (u) = 0 in Ω, u = m on Γ 1 , u = M on Γ 2 ,
where 0 ≤ m < M . Then G satisfies the assumptions of Proposition 4.8. Moreover, if p > n, Ω is a C 1,β -bounded domain with 0 < β ≤ 1, and G := G Ω (•, 0) is the positive minimal p-Green function of the operator -∆ p in Ω with a pole at 0. Then G satisfies the assumptions of Proposition 4.8 in Ω ⋆ , with m := lim x→∂Ω G(x) = 0, and M := lim x→0 G(x). Remark 4.11. A new phenomenon appears in Proposition 4.8: if p = 2, then the weight W α necessarily vanishes in Ω. Indeed, W α = 0 on the set

x ∈ Ω | G(x) = m + M 2 .

Proof of theorems 1.5

The present section is devoted to the proof of the main result of the paper, namely Theorem 1.5, that deals with the case V = 0, and claims the optimality of the supersolution construction for the p-Laplacian in Ω ⋆ . We divide the proofs into three parts: the criticality of Q -W , the optimality of the constant near infinity and zero, and finally the null-criticality of Q -W . 5.1. Criticality. In the present subsection, we prove the criticality of Q -W . We divide the proof into two parts, according to which of the assumptions (1.6), (1.7) is satisfied. We start by showing the criticality of Q -W if either (1.6) or (1.7) with γ = 0 is satisfied. This is a consequence of the following proposition: (5.1)

Then the functional Q -W is critical in Ω ⋆ . Proof. Let v := G p-1
p . Proposition 4.5 implies that v is a positive solution of the equation -∆ p (w) -W |w| p-2 w = 0 in Ω ⋆ . We construct a nullsequence for the functional Q -W in a similar fashion as in the proof of [START_REF] Poliakovsky | Uniqueness of positive solutions for singular problems involving the p-Laplacian[END_REF]Theorem 1.3]. Let

ϕ n (t) :=                      0 0 ≤ t ≤ 1 n 2 , 2 + log t log n 1 n 2 ≤ t ≤ 1 n , 1 1 n ≤ t ≤ n, 2 - log t log n n ≤ t ≤ n 2 , 0 t ≥ n 2 .
Set w n := ϕ n (v), and consider the sequence {vw n } n∈N .

Claim: {vw n } is a null-sequence for the functional

Q -W . Set B := {x ∈ Ω ⋆ | 1 < v < 2}, then B is compact in Ω ⋆ . By Lemma 2.8 we have Q -W (vw) ≍ Q sim (w),
where Q sim is the simplified energy for the functional Q -W (see (2.4)). Thus, we need to prove that

lim n→∞ Q sim (w n ) B (vw n ) p dν = 0. (5.2) Set X n := X(w n ) = Ω ⋆ v p |∇w n | p dν, and Y n := Y (w n ) = Ω ⋆ w p n |∇v| p dν.
Using the coarea formula (3.1), we obtain

X n = c 1 Ω ⋆ v p |ϕ ′ n (v)| p |∇v| p dν = c ∞ 0 (t|ϕ ′ n (t)|) p dt t = c 1 log n p 1 n 1 n 2 dt t + n 2 n dt t = 2c 1 log n p-1 .
Using again (3.1), we get

Y n = Ω ⋆ w p n |∇v| p dν = c ∞ 0 |ϕ n (t)| p dt t ≍ n 1 n dt t ≍ log n.
On the other hand, we clearly have

B (vw n ) p dν ≍ 1.
Recall that by (2.5), the simplified energy can be estimated from above by

Q sim (w n ) ≤ C      X n if 1 < p ≤ 2, X n + Xn Yn 2/p Y n if p > 2.
Therefore, lim n→∞ Q sim (w n ) = 0, and (5.2) is proved. Thus, {vw n : n ∈ N} is a null-sequence for the functional Q -W , and

Q -W is critical in Ω ⋆ .
Next, we prove the criticality of Q -W if assumption (1.7) with γ > 0 is satisfied: (5.3)

Then the functional Q -W is critical in Ω ⋆ .
Proof. The proof follows closely the proof of Proposition 5.1. Assume for simplicity that γ = G(0

) = 1. Recall that v := [G(1 -G)]
p-1 p . Proposition 4.8 implies that v is a positive solution of the equation -∆ p (w) -W |w| p-2 w = 0 in Ω ⋆ . We construct a null-sequence for the functional Q -W . This time, let

ϕ n (t) :=          0 0 ≤ t ≤ 1 n 2 , 2 + log t log n 1 n 2 ≤ t ≤ 1 n , 1 1 n ≤ t,
and consider the sequence {w n = ϕ n (v)} n∈N . By hypothesis, v(0) = 0 and lim x→ ∞ v(x) = 0. Therefore, for every n ∈ N, w n is compactly supported in Ω ⋆ .

Claim: The sequence {vw n } n∈N is a null-sequence for the functional

Q -W . Set B := {x ∈ Ω ⋆ | 1 4 < v < 3 4 }, then B is compact in Ω ⋆ .
As in the proof of Proposition 5.1, we set

X n := X(w n ) = Ω ⋆ v p |∇w n | p dν, and Y n := Y (w n ) = Ω ⋆ w p n |∇v| p dν. Let f (s) := [s(1 -s)] p-1
p . Using the coarea formula (3.2), we obtain

X n = Ω ⋆ v p |∇v| p |ϕ ′ n (v)| p dν = C Ω ⋆ [G(1 -G)] p-2 |1 -2G| p |ϕ ′ n • f (G)| p |∇G| p dν = C (log n) p f (t)∈[1/n 2 ,1/n] |1 -2t| p t(1 -t) dt ≍ 1 (log n) p-1 .
Using again the coarea formula (3.2), we get

Y n = Ω ⋆ (ϕ n (v)) p |∇v| p dν = 1 0 ϕ n (f (t)) p |1 -2t| p t(1 -t) dt ≍ log n.
In light of (2.5) we have lim n→∞ Q sim (w n ) = 0. On the other hand, we clearly have

B (vw n ) p dν ≍ 1.
Hence, {vw n } n∈N is a null-sequence for the functional Q -W .

Optimality of the constant near infinity and zero.

In the present subsection we prove the optimality of the constant C p := p-1 p p near the ends of Ω ⋆ . As in the previous subsection, we split the proof into two parts. (5.4)

Then the constant λ = C p in the Hardy inequality

Ω ⋆ |∇ϕ| p dν ≥ λ Ω ⋆ ∇G G p |ϕ| p dν (5.5)
is also the best constant for functions ϕ compactly supported either in a fixed punctured neighborhood of the origin, or in a fixed neighborhood of infinity in Ω.

Proof. We assume that 1 < p ≤ n, and present the proof of the optimality at infinity, the other cases are proved similarly. We proceed by a contradiction. Suppose that there exists a positive constant λ and a compact set K ⋐ Ω containing zero such that

Ω\K |∇ψ| p -W |ψ| p dν ≥ λ Ω\K W |ψ| p dν ∀ψ ∈ C ∞ 0 (Ω \ K). (5.6)
We apply inequality (5.6) to ψ = vϕ, where v := G (p-1)/p is a positive solution of Q -W (w) = 0, and ϕ ∈ C ∞ 0 (Ω \ K). Now, use Lemma 2.8 and (5.6) to obtain that for some positive constant β we have

βY (ϕ) ≤      X(ϕ) if 1 < p ≤ 2, X(ϕ) + X(ϕ) Y (ϕ) 2 p Y (ϕ) if p > 2, ∀ϕ ∈ C ∞ 0 (Ω \ K),
(5.7) where we recall that X(ϕ) := Ω ⋆ v p |∇ϕ| p dν and Y (ϕ) := Ω ⋆ ϕ p |∇v| p dν = Ω ⋆ v p ϕ p W dν. In the case p > 2, using the fact that for every ε > 0, there is a constant C > 0 such that for every t > 0, t + t 2/p ≤ Ct + ε, we have that

X(ϕ) + X(ϕ) Y (ϕ) 2 p Y (ϕ) ≤ CX(ϕ) + εY (ϕ).
Taking ε < β, we get by (5.7) that for any 1 < p < ∞, there is a constant

C > 0 such that CY (ϕ) ≤ X(ϕ) ∀ϕ ∈ C ∞ 0 (Ω \ K).
(5.8)

Assume without loss of generality that {v ≤ 1} ⊂ Ω \ K. Using the coarea formula (3.1), and applying inequality (5.8) to ϕ = φ(v), where φ ∈ C ∞ 0 (0, 1) we get that

1 0 |φ(t)| p dt t ≤ C 1 0 (t|φ ′ (t)|) p dt t ∀φ ∈ C ∞ 0 (0, 1) . (5.9)
But by [11, Theorem 1 of Sec. 1.3.2], this inequality cannot hold.

Alternatively, an easy way to see that (5.9) does not hold is to define a sequence {φ ε } of compactly supported Lipschitz continuous functions in (0, 1) of the form

φ ε (t) :=                t ε| log ε| γ t ∈ (0, ε), 1 | log t| γ t ∈ (ε, 1 2 ), ψ(t) t ∈ ( 1 2 , 1),
where ψ is a smooth function, independent of ε such that ψ(1) = 0, and γ > 0 will be determined later. Apply inequality (5.9) to φ ε to get

1 2 ε |φ ε (t)| p dt t ≤ C 1 ε| log ε| γ ε 0 t p dt t + 1 ε (t|φ ′ ε (t)|) p dt t .
(5.10)

Since p > 1, lim ε→0 1 ε| log ε| γ ε 0 t p dt t = lim ε→0 ε p-1 | log ε| γ = 0, therefore, letting ε → 0 in (5.10), we get 1 2 0 1 | log t| γ p dt t ≤ C 1 2 0 1 | log t| γ+1 p dt t + 1 1 2 (tψ ′ (t)) p dt t .
The right-hand side is finite for every positive value of γ, since p(γ + 1) > 1. left-hand side, on the contrary, is finite if and only if pγ > 1. Thus, taking γ such that pγ ≤ 1, we get a contradiction. As a consequence, inequality (5.9) cannot hold. (5.11)

Denote V := ∇G G(γ -G) p |γ -2G| p-2 2(p -2)G(γ -G) + γ 2 .
Then in the Hardy inequality

Ω ⋆ |∇ϕ| p dν ≥ λ Ω ⋆ V |ϕ| p dν (5.12)
the constant λ = C p is also the best constant for functions compactly supported either in a fixed punctured neighborhood of the origin, or in a fixed neighborhood of infinity in Ω.

Proof. We prove the optimality of the constant C p at infinity, the proof of the optimality at zero is similar (by replacing G with (γ -G)). Note that W = C p V . Assume by contradiction that C p is not optimal at infinity, then there is a positive constant λ and a compact subset K of Ω containing 0, such that

Ω\K |∇ψ| p -W |ψ| p dν ≥ λ Ω\K W |ψ| p dν ∀ψ ∈ C ∞ 0 (Ω \ K). (5.13)
Since by our assumption lim x→ ∞ G(x) = 0, we have

W ∼ x→ ∞ p -1 p p ∇G G p .
Therefore, by enlarging K, we may assume that the following inequality is satisfied, for some µ > 0:

Ω\K |∇ψ| p -W |ψ| p dν ≥ µ Ω\K ∇G G p |ψ| p dν ∀ψ ∈ C ∞ 0 (Ω \ K). (5.14)
We apply this inequality to ψ = ϕv,

where v = [G(γ -G)] p-1 p is a positive solution of Q -W (w) = 0, and ϕ ∈ C ∞ 0 (Ω \ K). Define ṽ := G p-1
p , and notice that at infinity, v ∼ x→ ∞ ṽ, and

∇v v p ∼ x→ ∞ p -1 p p ∇G G p = ∇ṽ ṽ p .
Therefore, from Lemma 2.8, (2.5) with p > 2, and (5.14), one gets that for some positive constant β,

β Ỹ (ϕ) ≤ X(ϕ) + X(ϕ) Ỹ (ϕ) 2 p Ỹ (ϕ), ∀ϕ ∈ C ∞ 0 (Ω \ K), (5.15)
where X(ϕ) := Ω ⋆ ṽp |∇ϕ| p dν and Ỹ (ϕ) := Ω ⋆ ϕ p |∇ṽ| p dν. We are back to inequality (5.7) of the proof of Proposition 5.3, where we have shown that such an inequality cannot hold. Consequently, (5.13) does not hold, and the constant p-1 p p in the inequality (5.12) is optimal at infinity. 5.3. Null-criticality. The null-criticality of the operators Q -W in Ω ⋆ follows from our coarea formula (3.1). First, we have: Proposition 5.5. Assume that in Theorem 1.5 the positive p-harmonic function G satisfies

     lim x→0 G = ∞ and lim x→ ∞G = 0 if 1 < p ≤ n, lim x→0 G = 0 and lim x→ ∞G = ∞ if p > n.
(5.16)

Then the functional Q -W is null-critical at 0 and at infinity in Ω.

Proof. Let v := G p-1 p , and denote also u := G. A minimizer of the variational problem (1.5) is necessarily a positive solution of the equation Q -W = 0 in Ω * . Since Q -W is critical, a minimizer in D 1,p (Ω) should be the ground state v. We claim that for any neighborhood O of 0, the ground state v belongs neither to D 1,p (Ω \ Ō) nor to D 1,p (O \ {0}). Indeed, the coarea formula (3.1) implies that

{t -<u(x)<t + } |∇v| p dν = c 1 t + t - dt t -→ t ± →ε ± ∞,
with ε + = ∞ and ε -= 0. Thus, the claim is proved.

The corresponding result, under assumption (1.7) with γ > 0, reads as follows Proposition 5.6. Assume that in Theorem 1.5 p > n, and the positive p-harmonic function G satisfies

lim x→0 G = γ > 0 and lim x→ ∞ G = 0.
(5.17)

Then the functional Q -W is null-critical at 0 and at infinity in Ω.

Proof. The proof is similar to the proof of Proposition 5.5. Indeed, recall that

v := [G(γ -G)] p-1 p . Let ε + = γ and ε -= 0. It is enough to prove that lim t ± →ε ± {t -<G<t + } |∇v| p dν = ∞.
We prove it when t -→ 0, the other case is similar, (replace G with (γ -G)).

Define ṽ = G p-1 p . At infinity in Ω, we have v ∼ x→ ∞ ṽ, and

∇v v p ∼ x→ ∞ p -1 p p ∇G G p = ∇ṽ ṽ p .
Therefore, by the coarea formula (3.1), one has as t -→ 0,

{t -<G<γ/2} |∇v| p dν ∼ {t -<G<γ/2} |∇ṽ| p dν = γ 2 t - dt t ,
and consequently, lim

t -→ε -{t -<G<γ/2} |∇v| p dν = ∞.
We conclude the present section with a corollary concerning Caccioppoli inequality. Recall the logarithmic Caccioppoli inequality (4.5) which holds in particular in Ω ⋆ :

Ω ⋆ |∇ϕ| p dν ≥ µ Ω ⋆ ∇v v p |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω ⋆ ), (5.18)
where v is any positive p-superharmonic functions in Ω ⋆ , and µ ≥ C P = p-1 p p

. By the results of [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] it follows that in the linear case (where p = 2) the constant C 2 = 1/4 in (5.18) is optimal. Now, Theorem 1.5 clearly implies the optimality of the constant C p also for any 1 < p ≤ n. More precisely, we have.

Corollary 5.7. Assume that 1 < p ≤ n, and suppose that Ω is a C 1,αdomain of a noncompact Riemannian manifold M (where α ∈ (0, 1]), and -∆ p is subcritical in M . Let G M be the positive minimal Green function, and assume that lim x→ ∞ G M (x, 0) = 0.

Then the best constant in the logarithmic Caccioppoli inequality (5.18) equals to p-1 p p .

Optimal weights for annular and exterior domains

In the present section we extend our main result (Theorem 1.5), obtained for punctured domains, to two additional types of domains: annular-type domains and exterior-type domains. As in the case of punctured domains, we view these two types of domains as manifolds with two ends. In particular, Definition 1.2 of optimal Hardy-type weight (which was given for a punctured domain) is extended naturally to handle annular-type and exteriortype domains.

We assume that the given positive p-harmonic function admits limits at the two ends (one limit might be infinity). We use the supersolution constructions obtained in propositions 4.5 and 4.8, and the techniques used in the proof of Theorem 1.5 to obtain optimal Hardy-weights for these cases. We omit the proofs since they differ only slightly from the proof of Theorem 1.5. Theorem 6.1. Let Ω be a C 1,α domain for some α > 0. Let U ⋐ Ω be an open C 1,α subdomain of Ω, and consider Ω := Ω \ U . Denote by ∞ the infinity in Ω, and assume that -∆ p admits a positive p-harmonic function G in Ω satisfying the following conditions

lim x→∂U G(x) = γ 1 , lim x→ ∞ G(x) = γ 2 , (6.1)
where γ 1 = γ 2 , and

0 ≤ γ 1 , γ 2 ≤ ∞. Denote m := min{γ 1 , γ 2 }, M := max{γ 1 , γ 2 }.
Define positive functions v 1 and v, and a nonnegative weight W on Ω as follows:

(a) If M < ∞, assume further that either m = 0 or p ≥ 2, and let

v 1 := (G -m)(M -G), v := v (p-1)/p 1 = [(G -m)(M -G)] (p-1)/p , and
W := p -1 p p ∇G v 1 p |m + M -2G| p-2 2(p -2)v 1 + (M -m) 2 , (6.2) (b) If M = ∞, define v 1 := (G -m), v := v (p-1)/p 1 = (G -m) (p-1)/p , and
W := p -1 p p ∇G v 1 p . (6.3)
Then the following Hardy-type inequality holds true

Ω |∇ϕ| p dν ≥ Ω W |ϕ| p dν ∀ϕ ∈ C ∞ 0 ( Ω), (6.4)
and W is an optimal Hardy-weight for -∆ p in Ω.

Moreover, up to a multiplicative constant, v is the unique positive supersolution of the equation Q -W (w) = 0 in Ω.

Optimal L p Rellich-type inequalities

Throughout the present section we consider a linear operator P . In [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF] we proved the following L 2 -Rellich-type inequality.

P := -div(A(x)∇•) + V (x),
and let v 0 and v 1 be two linearly independent positive solutions of the equa-

tion P u = 0 in Ω. Let W := 1 4 ∇ log v 0 v 1 2 A
be the Hardy-weight obtained by the supersolution construction with a pair (v 0 , v 1 ) (see (4.2)). Suppose that W is strictly positive, and fix 0 ≤ λ ≤ 1. Then (a) For a fixed 0 ≤ α < 1 and all ϕ ∈ C ∞ 0 (Ω) the following Rellich-type inequality holds true

Ω |P ϕ| 2 W (x) v 0 v 1 α dν ≥ λ 1 -α 2 2 Ω |ϕ| 2 W (x) v 0 v 1 α dν. (7.1) (b) If P -W is critical in Ω, then λ = 1 is the best constant in (7.1).
We are interested in generalizing Lemma 7.1, and prove L p -Rellich-type inequalities for the operator P . Our result hinges on the following L p -Rellich-type inequality of E.B. Davies and A.M. Hinz: Theorem 7.2 ([7, Theorem 4]). Let Ω be a domain in a Riemannian manifold of dimension n ≥ 2, and let 1 ≤ p < ∞. If 0 < v ∈ C(Ω) with -∆v > 0 and -∆(v δ ) ≥ 0 for some δ > 1, then

Ω v p |∆v| p-1 |∆ϕ| p dν ≥ [(p -1)δ + 1] p p 2p Ω |∆v||ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).
If P = -div(A(x)∇•) (i.e., V = 0), Theorem 7.2 implies the following L p -Rellich-type inequality: Theorem 7.3. Let P := -div(A∇•) be a subcritical operator in Ω, and let v 0 be a positive (super)solution of the equation P u = 0 in Ω and v 1 := 1.

Let W := 1 4 |∇ log v 0 | 2
A be the Hardy-weight obtained by the supersolution construction with a pair (v 0 , v 1 ), and suppose that W > 0. Then for every α ∈ (0, 1) and 1 ≤ p < ∞ the following Rellich-type inequality holds:

Ω |P ϕ| p W p-1 (v 0 ) α dν ≥ 4 p (1 -α) p (p -1 + α) p p 2p Ω |ϕ| p W (v 0 ) α dν ∀ϕ ∈ C ∞ 0 (Ω). (7.2)
Proof. Apply Theorem 7.2, with v := (v 0 ) α , and δ = 1/α. Since -∆v ≥ 4α(1 -α)W v > 0, and -∆v δ ≥ 0, we obtain (7.2).

Using the ground state transform with a positive solution v 1 , Theorem 7.3 implies: Theorem 7.4. Let P := -div(A∇•)+ V be a subcritical linear Schrödingertype operator in Ω, and let v 0 and, v 1 be two positive solutions of the equation P u = 0 in Ω. Let W := 1 4 |∇ log (v 0 /v 1 )| 2 A be the Hardy-weight obtained by the supersolution construction with a pair (v 0 , v 1 ), and suppose that W > 0. Then for every α ∈ (0, 1) and 1 ≤ p < ∞ the following L p -Rellich-type inequality holds:

Ω |P ϕ| p W p-1 v 0 v 1 α v 2-p 1 dν ≥ 4 p (1 -α) p (p -1 + α) p p 2p Ω |ϕ| p W v 0 v 1 α v 2-p 1 dν (7.
3) for all ϕ ∈ C ∞ 0 (Ω). Remark 7.5. In the case p = 2, we recover the best constant 1 -α 2 2 obtained in Lemma 7.1. We note that for p = 2, the constant of the L p -Rellich-type inequalities (7.2) and (7.3) is optimal at least in the classical case, where Ω = R n \ {0}, P = -∆, v 0 = |x| 2-n and v 1 = 1. The optimality of the constant in this case follows from the remark in [7, page 521].

The supersolution construction for Q V

In the present section we study the supersolution construction for operators Q V of the form (1.3) under the assumption that (roughly speaking) the supersolutions v j have the same level sets. In Appendix A we present a proof of the particular case of radially symmetric potentials.

The following result generalizes Lemma 4.1 for p = 2.

Theorem 8.1. Let v j , j = 0, 1, be two positive, linearly independent, C 2 -(super)solutions of the equation Q V j (u) = 0 in Ω. Assume that ∇v 0 does not vanish in Ω, and that

v 1 = ϕ 1 (v 0 ) for some C 2 -function ϕ 1 such that ϕ ′ 1 (u) = 0. For 0 ≤ α ≤ 1, define the function v α := v α 1 v 1-α 0 ,
and let

V α := (1 -α)V 0 |∇ log v 0 | 2-p + αV 1 |∇ log v 1 | 2-p |∇ log v α | p-2 , W α := α(1 -α)(p -1) ∇ log v 0 v 1 2 |∇ log v α | p-2 ,
Then v α is a positive (super)solution of the equation

Q Vα-Wα (u) = 0 in Ω, (8.1)
and the following improved inequality holds

Q Vα (ϕ) ≥ Ω W α |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).
Remark 8.2. If both v 0 and v 1 do not admit critical points, then the condition v 1 = ϕ 1 (v 0 ) is equivalent to the fact that ∇v 0 and ∇v 1 are collinear at every point, and also to the fact that the level sets of v 0 and v 1 coincide, that is, for every t 0 > 0, there is

t 1 > 0 such that {x ∈ Ω | v 0 (x) = t 0 } = {x ∈ Ω | v 1 (x) = t 1 },
and vice versa. A particular case appears when v j are radially symmetric positive supersolutions (see Appendix A).

Proof. Fix x ∈ Ω and set u := v 0 (x). By Lemma 2.10 we have

Q V 1 (v 1 ) = -∆ p ϕ 1 (v 0 ) + V 1 (ϕ 1 (v 0 )) p-1 = |ϕ ′ 1 (u)| p-2 |∇v 0 | p -(p-1)ϕ ′′ 1 (u)- ∆ p (v 0 ) |∇v 0 | p ϕ ′ 1 (u)+V 1 |(log ϕ 1 ) ′ (u)| 2-p |∇v 0 | p ϕ 1 (u) (8.2)
in the weak sense. On the other hand, with the identity map ϕ 0 (t) := t on R + we have at x

Q V 0 (v 0 ) = -∆ p (ϕ 0 (v 0 ) + V 0 (ϕ 0 (v 0 )) p-1 = |ϕ ′ 0 (u)| p-2 |∇v 0 | p -(p-1)ϕ ′′ 0 (u)- ∆ p (v 0 ) |∇v 0 | p ϕ ′ 0 (u)+V 0 | (log ϕ 0 ) ′ (u)| 2-p |∇v 0 | p ϕ 0 (u) . (8.3)
Therefore, for j = 0, 1, ϕ j (u) satisfies at the point u the following linear ordinary differential inequality

-(p -1)ϕ ′′ j (u) - ∆ p (v 0 ) |∇v 0 | p ϕ ′ j (u) + V j | (log ϕ j ) ′ (u)| 2-p |∇v 0 | p ϕ j (u) = (≥) 0.
Denote ϕ α (u) := ϕ 0 (u) 1-α ϕ 1 (u) α , and apply the one-dimensional version of Corollary 4.2. We obtain the following linear differential inequality at u

-(p -1)ϕ ′′ α (u) - ∆ p (v 0 ) |∇v 0 | p ϕ ′ α (u) + (1 -α)V 0 | (log ϕ 0 ) ′ (u)| 2-p |∇v 0 | p ϕ α (u)+ αV 1 (log ϕ 1 ) ′ (u) 2-p |∇v 0 | p ϕ α (u)-(p-1)α(1-α) log ϕ 0 (u) ϕ 1 (u) ′ 2 ϕ α (u) = (≥) 0. (8.4)
In view of Lemma 2.10 we have

-∆ p (ϕ α ) = |ϕ ′ α | p-2 |∇v 0 | p -(p -1)ϕ ′′ α - ∆ p (v 0 ) |∇v 0 | p ϕ ′ α .
On the other hand,

| (log ϕ j ) ′ | 2-p | (log ϕ α ) ′ | p-2 = |∇ log v j | 2-p |∇ log v α | p-2 j = 0, 1, log ϕ 0 ϕ 1 ′ 2 | (log ϕ α ) ′ | p-2 |∇v 0 | p = ∇ log v 0 v 1 2 |∇ log v α | p-2 .
Hence, (8.4) implies the result of the theorem.

Remark 8.3. In particular, let V = 0 and v 0 = G be the p-Laplacian's Green function with a pole at 0 ∈ Ω, and v 1 = 1. Then V α = 0, and a computation shows that W α = (p-1)α p-1 (1-α) ∇G G p (cf. Proposition 4.5).

Corollary 8.4. Assume that p > n, V = 0, and -∆ p is subcritical in Ω.

Let G be (up to a constant) the p-Green function with a pole at 0 ∈ Ω.

Suppose that lim

x→0 G(x) = γ > 0 and lim x→ ∞ G(x) = 0. For 0 ≤ α ≤ 1, let v α := G 1-α (γ-G) α , W α := α(1-α)(p-1) γ(1-α)-G p-2 ∇G G(γ -G) p (8.5)
Then the following improved Hardy inequality holds in Ω ⋆ :

Ω ⋆ |∇ϕ| p dν ≥ Ω ⋆ W α |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω ⋆ ). (8.6)
Moreover, for any 0 ≤ α ≤ 1 the operator Q -Wα is subcritical in Ω.

Proof. By our assumption, γ -G is a positive p-harmonic function in Ω ⋆ . Apply Theorem 8.1 with v 0 = G and v 1 = γ -G to obtain (8.6). Assume to the contrary that Q -Wα is critical in Ω ⋆ . Two cases should be considered: either α < (p -1)/p, or 1 -α < (p -1)/p. Let us assume for example α < (p -1)/p, the other case being similar (exchanging the roles of zero and infinity). Then v p-1 p is a positive supersolution of Q -Wα in a neighborhood of zero, and v α is a positive solution of Q -Wα of minimal growth in Ω ⋆ . Therefore, there exists C > 0 such that v α ≤ Cv p-1 p in a neighborhood of zero. But since α < (p -1)/p, this is impossible, and we get a contradiction. Remark 8.5. A priori it is clear that for W α (given by (8.5)) to be optimal at the origin, it is needed that α = (p -1)/p, but for the constant to be optimal at ∞, we must choose α = 1/p, and thus v α cannot be a ground state (if p = 2). Thus, in the nontrivial cases (v j = constant), the supersolution construction of the form v α = v Then under the conditions of Theorem 1.5, W is an optimal Hardy-weight for -∆ p , and ψ(G) is the ground state of the critical operator Q -W in Ω ⋆ . Note that nevertheless, W = 0 on the set {x ∈ Ω ⋆ | G(x) = γ/2}.

It turns out that if V j both have the same definite sign, then one can find potentials V α ≥ V α (with the same definite sign) which does not depend on v j , such that the corresponding Hardy inequality is satisfied with the same Hardy-weight W α . We have Corollary 8.6. Let Ω, V j , v j (where j = 0, 1), v α , and W α be as in Theorem 8.1 (or as in Theorem A.1). Suppose further that V j ≥ 0 if 1 < p ≤ 2 (resp., V j ≤ 0 if p ≥ 2), where j = 0, 1. Define

V α := ± (1 -α)|V 0 | 1/(p-1) + α|V 1 | 1/(p-1) p-1 ,
where one should take the minus sign if V j ≤ 0. Then v α is a positive (super)solution of the equation

Q Vα-Wα (u) = 0 in Ω, (8.8)
and the following improved inequality holds If and |V 0 | + |V 1 | = 0, and p = 2, then the strict convexity (resp., concavity) of f implies that v α is a positive supersolution of Q Vα-Wα (u) = 0 which is not a solution, and therefore by Lemma 2.6, the corresponding improved functional Q Vα-Wα is subcritical in Ω. Remark 8.7. 1. Suppose that V 0 = V 1 = 0 and V 0 has a definite sign, then V α = V 0 . By Corollary 8.6, the operator Q V 0 -Wα is subcritical in Ω if p = 2. This is in contrast with the linear case where p = 2. Indeed, if v 0 is the Green function of the operator P u := -div A(x)∇ • + V (x) in Ω with a pole 0, and if v 1 is a positive solution satisfying lim x→ ∞ v 0 (x)

Q Vα (ϕ) ≥ Ω W α |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).
v 1 (x) = 0, then P -W 1/2 = P -1 4 ∇ log v 0 v 1 2
is critical in Ω ⋆ (see [START_REF] Devyver | Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon[END_REF]Theorem 2.2]). 2. In general, it is not clear how to optimize in α the potentials W α in the case V 0 = V 1 = 0, and V 0 has a definite sign (so, V α = V 0 ). But if we take v 0 = 1 (so, V ≥ 0 and 1 < p ≤ 2), and v 1 = v is a positive supersolution of the equation Q V 0 (u) = 0, then

W α = α p-1 (1 -α)(p -1) ∇v v p ,
and by optimizing α one obtains

Q V (ϕ) ≥ p -1 p p Ω |∇v| v p |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω), (8.9)
which in particular reproves (2.12) in [START_REF] Adimurthi | Role of the fundamental solution in Hardy-Sobolev-type inequalities[END_REF] if A is the identity matrix. Remark A.2. If 0 ∈ Ω, Ω is a radially symmetric domain, V 0 = V 1 is a radially symmetric potential, and Q V 0 is subcritical in Ω, then one can apply Theorem A.1 in Ω ⋆ = Ω \ {0} with v 0 equals to the corresponding (unique) p -Green function of Q V 0 with a pole at the origin, and v 1 a global radial positive supersolution of the equation Q V 0 (u) = 0 in Ω.

Definition 2 . 2 .

 22 We say that a sequence {ϕ k } k∈N of nonnegative functions belonging to C ∞ 0 (Ω) is a null-sequence for Q in Ω if there exists an open set B ⋐ Ω such that lim k→∞ Q(ϕ k ) = lim k→∞ Ω |∇ϕ k | p + V |ϕ k | p dν = 0, and B |ϕ k | p dν ≍ 1.

Remark 4 . 4 .

 44 Lemma 4.3 does not provide us with an explicit nonzero Hardy-weight W α for Q α , although the subcriticality of Q α ensures the existence of a strictly positive weight.

Example 4 . 7 .

 47 Consider Proposition 4.5 in the particular case Ω = R n \{0}, p = n, and G(x) = |x| p-n p-1 . Then (4.5) clearly implies the classical Hardy inequality (with the best constant):

Remark 4 . 10 .

 410 If in Proposition 4.8 the supersolution G is unbounded and satisfies G > m in Ω, then one should simply consider the supersolution construction with v α := (G -m) α with 0 ≤ α ≤ 1 to obtain the Hardy-type inequality Ω |∇ϕ| p dν ≥ p -

Proposition 5 . 1 .G

 51 Assume that in Theorem 1.5 the positive p-harmonic function G satisfies = 0 and lim x→ ∞G = ∞ if p > n.

Proposition 5 . 2 .

 52 Assume that in Theorem 1.5 p > n, and the positive p-harmonic function G satisfies lim x→0 G = γ > 0 and lim x→ ∞ G = 0.

Proposition 5 . 3 .G

 53 Assume that in Theorem 1.5 the positive p-harmonic function G satisfies = ∞ and lim x→ ∞G = 0 if 1 < p ≤ n, lim x→0 G = 0 and lim x→ ∞G = ∞ if p > n.

Proposition 5 . 4 .

 54 Next, we prove the optimality of the constant C p = p-1 p p near the ends of Ω ⋆ if assumption (1.7) with γ > 0 is satisfied: Assume that in Theorem 1.5 p > n, and the positive p-harmonic function G satisfies lim x→0 G = γ > 0 and lim x→ ∞ G = 0.

Lemma 7 . 1 ([ 6 ,

 716 Corollary 10.3]). Assume that P is a subcritical linear Schrödinger-type operator in Ω of the form

α 1 v 1 -α 0 ,

 10 does not provide us with an optimal Hardy weight. On the other hand, let ψ(G) := [G(γ -G)] (p-1)/p andW := -∆ p (ψ(G)) ψ(G) p-1 = p -1 p p ∇G G(γ -G) p |γ -2G| p-2 2(p -2)G(γ -G) + γ 2 ≥ 0. (8.7)

Moreover, if p = 2 , 1 p- 1 1 p- 1

 21111 and |V 0 | + |V 1 | = 0, then the functional Q Vα-Wα is subcritical in Ω.Proof. Assume that the conditions of Theorem 8.1 are satisfied. Then v α is a positive (super)solution of the equation Q Vα-Wα (u) = 0.We claim that the function (ξ, η) → f (ξ, η) := ξ p-1 η 2-p on R 2 + is convex (resp., concave) if p ≥ 2 (resp., p ≤ 2). Indeed, Hess (f ) = (p -1)(p -2)ξ p-1 η 2-p be easily checked that Hess (f ) is nonnegative (resp., nonpositive) on R 2 + if and only if (p -1)(p -2) ≥ 0 (resp., (p -1)(p -2) ≤ 0). Hence,(1 -α)|V 0 | p-|∇ log v 0 | 2-p + α|V 1 | p-|∇ log v 1 | 2-p ≥ respect. ≤ (1 -α)|V 0 | 1/(p-1) + α|V 1 | 1/(p-1) p-1 (1 -α)∇ log v 0 + α∇ log v 1 2-p .So, V α ≥ V α , and hence v α is a positive supersolution of the equationQ Vα-Wα (u) = 0 in Ω,and we haveQ Vα (ϕ) ≥ Ω W α |ϕ| p dν ∀ϕ ∈ C ∞ 0 (Ω).

Therefore, by Lemma 4 . 1 ,v 1 ′ 2 (

 4112 v α is a positive (super)solution of the linear equationP + (1 -α)V 0 (log v 0 ) ′ 2-p + αV 1 (log v 1 ) ′ 2-p -(p -1)α(1 -α) log v α satisfies the quasilinear differential (in)equality -∆ p (v α )+ (1 -α)V 0 (log v 0 ) ′ 2-p +αV 1 (log v 1 ) ′ 2-p (log v α ) ′ p-2 v p-1 α -(p-1)α(1-α) log v 0 log v α ) ′ p-2 v p-1 α = Q Vα-Wα (v α ) = (≥) 0. (A.5)

Acknowledgments

The authors wish to thank Martin Fraas for valuable discussions. They acknowledge the support of the Israel Science Foundation (grants No. 963/11) founded by the Israel Academy of Sciences and Humanities. B. D. was supported in part by a Technion fellowship.

Appendix A. Radially symmetric potentials

In this Appendix we present a proof of a particular case of Theorem 8.1, where the two positive supersolutions are radially symmetric functions, and in particular, have the same level sets.

Theorem A.1. Assume that for j = 0, 1

where Ω is a domain in R n not containing the origin, and the potentials V j are two radially symmetric potentials. Let v j , j = 0, 1, be two positive, linearly independent, radially symmetric, C 2 -supersolutions of the equation

where r := |x|. Assume further that (v 0 ) ′ , (v 1 ) ′ , and (v α ) ′ do not vanish, and let

Then v α is a positive supersolution of the equation

and the following improved inequality holds

Proof. Assume that v is a radially symmetric C 2 -function, and denote r := |x|, v ′ := dv/dr. Then by Lemma 2.10 the p -Laplacian of v satisfies

in the weak sense. Denote the linear operator

By our assumptions, v j are positive radial (super)solutions of the equation Q V j (u) = 0 in Ω, where j = 0, 1. Hence, P v j + V j (log v j ) ′ 2-p v j = (≥) 0 j = 0, 1.