Optimal Hardy-weights for the $(p,A)$-Laplacian with a potential term

We construct new optimal $L^p$ Hardy-type inequalities for elliptic Schr\"odinger-type operators


Introduction
For any ξ ∈ R n and a positive definite matrix A ∈ R n×n , let |ξ| A := Aξ, ξ , where ·, · denotes the Euclidean inner product on R n . Consider a second order half-linear operator of the form Q p,A,V (u) := −div |∇u| p−2 A A∇u + V |u| p−2 u defined in a domain Ω ⊂ R n , and assume that Q p,A,V admits a positive solution in Ω. We are interested to find an optimal weight function W 0 (see Definition 2.29) such that the equation Q p,A,V −W (u) = 0 admits a positive solution in Ω. Equivalently [19,Theorem 4.3], we are interested to find an optimal weight function W 0 such that the following Hardy-type inequality is satisfied: In some definite sense, an optimal weight W 0 is "as large as possible" nonnegative function such that (1.1) is satisfied for all nonnegative φ ∈ C ∞ 0 (Ω). The search for Hardy-type inequalities with optimal weight function W was originally proposed by Agmon, who raised this problem in connection with his theory of exponential decay of Schrödinger eigenfunctions [1, p. 6]. In the past four decades, the problem of improving Hardy-type inequalities has engaged many authors. In particular, Hardytype inequalities were established for a vast class of operators (e.g., elliptic operators, Schrödinger operators on graphs, fractional differential equations) with different types of boundary conditions, see [2,3,4,6,8,9,10,14,22]. In [9], Devyver and Pinchover studied the problem of optimal weights for the operator Q p,A,V . However, they managed to find optimal weights only in the case where A is the identity matrix and V = 0. They proved (under certain assumptions ) that the p-Laplace operator, −div (|∇u| p−2 ∇u), admits an optimal Hardy-weight. More specifically, it is proved that if 1 < p ≤ n, then W = p−1 p p ∇G G p an optimal Hardy-weight, where G is the associated positive minimal Green function with singularity at 0. For p > n, several cases should be considered, depending on the behavior of a positive p-harmonic function with singularity at 0.
In the present paper we make a nontrivial progress towards the study of (1.1) in the case where A is not necessarily the identity matrix, and V is a slowly growing potential function. Our main result reads as follows.
Theorem 1.1. Let Ω ⊂ R n be a domain and x 0 ∈ Ω. Let Q p,A,V be a subcritical operator in Ω satisfying Assumptions 2.8 in Ω. Suppose that Q p,A,V admits a (nonnegative) Green potential, G ϕ (x), in Ω (see Definition 2.22) satisfying where ∞ denotes the ideal point in the one-point compactification of Ω. Then the operator Q p,A,V /cp admits an optimal Hardy-weight in Ω, where c p = p/(p − 1) p−1 .
As a corollary of the proof of Theorem 1.1 we obtain the following result.
where ∞ denotes the ideal point in the one-point compactification of Ω. Then the operator Q p,A,V /cp admits an optimal Hardy-weight in Ω, where c p = p/(p − 1) p−1 .
The paper is organized as follows. In Section 2, we introduce the necessary notation and recall some previously obtained results needed in the present paper. We proceed in Section 3, with proving essential results needed for the proof of Theorem 1.1, and then we prove Theorem 1.1 and Corollary 1.2.

Preliminaries
Let Ω ⊂ R n be a domain, and let 1 < p < ∞. Throughout the paper we use the following notation and conventions: • For any R > 0 and x ∈ R n , we denote by B R (x) the open ball of radius R centered at x, and B + open in Ω, the set Ω 1 is compact, and Ω 1 ⊂ Ω 2 .
• C refers to a positive constant which may vary from line to line.
• Let g 1 , g 2 be two positive functions defined in Ω. We use the notation g 1 ≍ g 2 in Ω if there exists a positive constant C such that for all x ∈ Ω.
• Let g 1 , g 2 be two positive functions defined in Ω, and let x 0 ∈ Ω. We use the notation g 1 ∼ g 2 near x 0 if there exists a positive constant C such that • The gradient of a function f will be denoted either by ∇f or Df .
• χ B denotes the characteristic function of a set B ⊂ R n .
• For a real valued function W , we write W 0 in Ω if W ≥ 0 in Ω and sup Ω W > 0.
• For a real valued function u and 1 < p < ∞, I p (u) := |u| p−2 u. • ∞ denotes the ideal point in the one-point compactification of Ω.
• supp(u) denotes the support of the function u.
• H l , 1 ≤ l ≤ n, denotes the l-dimensional Hausdorff measure on R n .
2.1. Gauss-Green formula. We continue with several definitions and results concerning the Gauss-Green theorem [7]. (1) We denote by M(D) the space of all signed Radon measures µ on D such that Definition 2.2 (cf. [7] and [11,Section 5] Denote by BV(D) the space of all functions f ∈ L 1 (D) having bounded variation. . Let E ⋐ R n and let 0 ≤ f ∈ BV(E)∩C 1 (E).
We proceed with the following Gauss-Green theorem of divergence measure fields over sets of finite perimeter (see [ where n is a classical outer unit normal to ∂E which is defined H n−1 -a.e. on ∂E.

Local Morrey spaces.
In the present subsection we introduce a certain class of Morrey spaces that depend on the index p, where 1 < p < ∞. We write f ∈ M q loc (Ω) if for any ω ⋐ Ω we have f M q (ω) < ∞.
Next, we define a special local Morrey space M q loc (p; Ω) which depends on the values of the exponent p.
For the regularity theory of equations with coefficients in Morrey spaces we refer the reader to [18,19].
We associate to any domain Ω ⊂ R n an exhaustion, i.e. a sequence of smooth, precom- 2.3. Criticality theory for Q p,A,V . Let 1 < p < ∞, and consider the operator defined on a domain Ω ⊂ R n , n ≥ 2, where ∆ p,A := div (|∇u| p−2 A A∇u) and I p (u) := |u| p−2 u. Unless otherwise stated, we always assume that the matrix A and the potential function V satisfy the following regularity assumptions: is a symmetric positive definite matrix which is locally uniformly elliptic, that is, for any compact K ⋐ Ω there exists Θ K > 0 such that ∀ξ ∈ R n and ∀x ∈ K.
• V ∈ M q loc (p; Ω) is a real valued function. The associated energy functional for the operator Q p,A,V in Ω is defined by It should be noted that the above definition makes sense due to the following Morrey-Adams Theorem (see for example, [19,Theorem 2.4] and references therein). Theorem 2.10 (Morrey-Adams theorem). Let ω ⋐ R n and V ∈ M q (p; ω).
(1) There exists a constant C = C(n, p, q) > 0 such that for any δ > 0 (2) For any ω ′ ⋐ ω with Lipschitz boundary, there exists a positive constant C = C(n, p, q, ω ′ , ω, δ, V M q (p;ω) ) and δ 0 such that for 0 < δ ≤ δ 0 We denote the set of all positive solutions (resp., supersolutions) of Q p,A,V (u) = 0 in Ω by C Q p,A,V (Ω) (resp., K Q p,A,V (Ω)). We say that the operator It is well known that under Assumptions 2.8 any positive solution of the equation Q p,A,V (u) = 0 in Ω belongs to C 1,α (Ω) (see for example [ with a nonzero nonnegative u which is called a principal eigenfunction. Proposition 2.14 ([19, Theorem 3.9]). Let Ω ⊂ R n be a bounded Lipschitz domain, and assume that A is a uniformly elliptic, bounded matrix in Ω, and V ∈ M q (p; Ω). Then, the operator Q p,A,V admits a unique principal eigenvalue λ 1 (Ω). Moreover, λ 1 is simple and its principal eigenfunction is the minimizer of the Rayleigh-Ritz variational problem .
The following well-known Allegretto-Piepenbrink theorem (in short, the AP theorem) connects between the nonnegativity of Q p,A,V and the nonnegativity of its associated energy functional Q Ω p,A,V [19, Theorem 4.3]. Theorem 2.15 (AP theorem). The following assertions are equivalent. ( We say that Q p,A,V satisfies the strong maximum principle in ω if for any u ∈ W 1,p (ω) satisfying Q p,A,V (u) ≥ 0 in ω and u ≥ 0 on ∂ω, either u = 0, or u > 0 in ω.
If u ∈ MG A,V,Ω;∅ , then u is called an Agmon ground state of Q p,A,V in Ω.  The next lemma shows that the energy functional Q Ω p,A,V is equivalent to a simplified energy that does not explicitly depend on V and contains only nonnegative terms.
2.4. Optimal Hardy-weights. Let ∞ denote the ideal point in the one-point compactification of Ω. Let us define the notion of an optimal Hardy-weight for the operator Q p,A,V .
Definition 2.29 ( [9]). Suppose that Q p,A,V is subcritical in Ω. We say that 0 W is an optimal Hardy-weight of Q p,A,V in Ω if the following two assertions are satisfied: (1) Criticality: Q p,A,V −W is critical in Ω.
Remark 2.30. Let us discuss Definition 2.29. Suppose that Q p,A,V is subcritical in a domain Ω containing x 0 , and let x 0 ∈ K ⋐ Ω. Then, for any 0 W ∈ C ∞ 0 (Ω) there exists τ > 0 such that Q p,A,V −τ W is critical in Ω (see for example [21,Proposition 4.4] and [19]). On the other hand, the ground state of Q p,A,V −τ W , φ, satisfies Therefore, there are infinity many weight functions 0 W ∈ C ∞ 0 (Ω) such that Q p,A,V −W is critical in Ω, obviously, for such a weight W , the operator Q p,A,V −W is not null-critical with respect to W . Definition 2.31. We say that a Hardy-weight W is optimal at infinity in Ω if for any Remark 2.32. The definition of an optimal Hardy-weight in [8] includes the requirement that W should be optimal at infinity. But, it is proved in [15] that if Q − W is null-critical with respect to W in Ω, then Q − W is optimal at infinity. The same proof applies under the assumptions considered in the present paper, hence, in Definition 2.29 we avoid the requirement of optimality at infinity.
The following coarea formula is a direct consequence of [9, Proposition 3.1].
The following theorem is proved in [9] for the case A = 1. However, it can be easily checked that the validity of Lemma 2.33 for a general matrix A satisfying Assumptions 2.8, gives rise to the following theorem.
Then the following Hardy-type inequality holds in Ω * : and W is an optimal Hardy-weight of −∆ p,A in Ω * . Moreover, up to a multiplicative constant, v is the ground state of −∆ p,A − W I p in Ω * .
The following simple observation concerns the existence of optimal Hardy-weights for a 'small perturbation' of an operator with a given optimal Hardy-weight.
Proof. Consider the function W+V 1 . Then, Q p,A,V+V 1 −(W+V 1 )I p = Q p,A,V −W I p is a critical operator in Ω * .
Obviously, W + V 1 0, and the ground state ψ of Q p,A,V − W I p in Ω * is the ground state of Q p,A,V +V 1 − (W + V 1 )I p in Ω * . Moreover, implying that Q p,A,V +V 1 −(W +V 1 ) is null-critical in Ω * with respect to W + V 1 . In particular, W + V 1 is an optimal Hardy-weight of Q p,A,V +V 1 in Ω * .

Optimal Hardy-weights for nonpositive potentials
Lemma 2.35 obviously applies when V 1 ≥ 0. The main goal in the current section is to obtain optimal Hardy-weights for a general subcritical operator Q p,A,V in a domain Ω, without assuming V = 0 in Ω. In particular, we prove Theorem 1.1.
In the following lemma we generalize the notion of Green potential for Q p,A,V .
Lemma 3.2. Assume that Q p,A,V is subcritical in Ω, and let 0 ϕ ∈ C ∞ 0 (Ω). Then there exists a positive function G ϕ ∈ W 1,p loc (Ω), such that G ϕ is a positive solution of minimal growth at infinity and satisfies Q p,A,V (G ϕ ) = ϕ in Ω.

Lemma 3.2 and Proposition 3.4 imply:
Corollary 3.6. Assume that Q p,A,cpV is subcritical in Ω. For 0 ϕ ∈ C ∞ 0 (Ω), let G ϕ be a Green potential satisfying Q p,A,cpV (G ϕ ) = ϕ in Ω, and let f (t) = t p−1 p . Then, In particular, f (G ϕ ) is a positive solution of the equation The following lemma is a generalization of Lemma 2.33 to the case V = 0.
Lemma 3.7. Assume that Q p,A,V is subcritical in Ω, and let G ϕ ∈ C 1,α loc (Ω) be a Green potential (with respect to 0 ϕ ∈ C ∞ 0 (Ω)), and assume that Then, there exists 0 < M ϕ < sup Ω G ϕ such that for almost every 0 < t < M ϕ , satisfying there exists C > 0, independent of t, such that The assumptions lim x→∞ G ϕ = 0, and´Ω V G p−1 ϕ dx < 0 imply that for a sufficiently small M ϕ > 0 and 0 < t < M ϕ , Moreover, the assumption supp(ϕ) ⋐ Ω t implies and C does not depend on t. Sard's theorem for C 1,α functions implies that for H n−1 -a.e.
We proceed with the following lemma. ϕ ∈ C ∞ 0 (Ω), and assume that Q p,A,cpV is subcritical in Ω. Let G ϕ ∈ C 1,α loc (Ω) be a Green potential satisfying Q p,A,cpV (G ϕ ) = ϕ in Ω, and assume that Consider the function f (t) = t p−1 p , and let Hence, W is an optimal Hardy-weight for Q p,A,V in Ω.
Proof. Criticality: Let M ϕ be given by Lemma 3.7, and let K ⋐ Ω be a precompact smooth subdomain satisfying supp ϕ ⋐ K, max Ω\K G ϕ < M ϕ and G ϕ < 1 for all x ∈ Ω \ K.
Assume without loss of generality that inf K G ϕ ≥ 1.
By Lemma 3.9 and Remark 3.10, the operator Q p,A,V /cp admits an optimal Hardy-weight in Ω. Corollary 1.2 and the following remark give rise to new optimal Hardy-type inequalities in the smooth case.
Remark 3.11. Let Ω ⊂ R n be a domain and let Q p,A,V be a subcritical operator in Ω satisfying Assumptions 2.8. Assume further that V ≤ 0 in Ω. Then, there exists K ⋐ Ω and x 0 ∈ intK ⋐ Ω, such that the operator Q p,A,V admits a positive solution G(x) in Ω \ {x 0 } satisfying (1.2) in each of the following cases : • A is a constant, symmetric, positive definite matrix; V ∈ L ∞ (Ω); Ω is a bounded C 1,α domain and λ 1 (Ω) > 0 [17]. • A is a constant, symmetric, positive definite matrix; V ∈ C ∞ 0 (R n ); Ω = R n [12,13]. In particular, Theorem 1.1 can be applied in each of the latter cases.