SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS

ROBERTO MONTI

Abstract. We study symmetry, existence and uniqueness properties of extremal functions for the weighted Sobolev inequality

(∫_{ℝ^{n}} |u|^{\frac{2Q}{Q−2}} dxdy)^{\frac{Q−2}{Q}} ≤ c_{m,k,α} ∫_{ℝ^{n}} \{|∇_{x}u|^{2} + (α + 1)^{2}|x|^{2α}|∇_{y}u|^{2}\}dxdy,

where x ∈ ℝ^{m}, y ∈ ℝ^{k} with m, k ≥ 1 and n = m + k, α > 0 and Q = m + k(α + 1).

1. Introduction

Let x ∈ ℝ^{m}, y ∈ ℝ^{k} and n = m + k, with m, k ≥ 1, introduce the weight function w(x) = (α + 1)|x|^{α}, where α > 0 is a positive real number, and consider the “homogeneous dimension” Q = m + k(α + 1). We study the weighted Sobolev inequality

(∫_{ℝ^{n}} |u|\frac{2Q}{Q−2} dxdy)^{\frac{Q−2}{Q}} ≤ c_{m,k,α} ∫_{ℝ^{n}} \{|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}\}dxdy  (1.1)

for functions u in an appropriate Sobolev space. Here, ∇_{x}u and ∇_{y}u denote the gradients of u in the variables x and y, respectively. We are interested in symmetry, existence and uniqueness properties of extremal functions for (1.1), i.e. functions achieving the best constant c_{m,k,α} > 0. This paper continues the research initiated in [MM2], where D. Morbidelli and the author studied symmetry and uniqueness properties of non negative, entire solutions to the critical semilinear equation

Δ_{x}u + w^{2}Δ_{y}u = −u^{\frac{Q+2}{Q−2}}  in ℝ^{n}.  (1.2)

This is the Euler equation associated with (1.1). When α = 0, we have the Yamabe equation and positive, entire solutions are exactly the extremal functions for the Sobolev inequality (1.1) with Q = n and w = 1. For α > 0, the inequality is known as Sobolev inequality for the Grushin operator and it can be proved by general techniques in doubling metric spaces with Poincaré inequality. A general reference to the topic is [HK] (see also [FGW] and [GN]).

For α = 0, extremal functions are radially symmetric and decreasing about some point in ℝ^{n}. This is no longer true for α > 0, because inequality (1.1) is not invariant for the full group of Euclidean translations and orthogonal transformations. Let us introduce the following notion of symmetry. We say that a non negative function u : ℝ^{n} → [0, +∞) belongs to the class S if: i) there is a function v : ℝ^{m} → [0, +∞) such that

Date: 1.4.2005.

u(x, y) = |x|^{\frac{2−Q}{2}} v (\frac{(1 − |x|^{α+1})^{2} + |y|^{2}}{4|x|^{α+1}} \frac{x}{|x|}) , x ≠ 0;  (1.3)

ii) the function t ↦ t^{\frac{Q−2}{2}} u(tζ, 0) is non decreasing on [0, 1] for all ζ ∈ 𝕊^{m−1}, i.e. ζ ∈

ℝ^{m} and |ζ| = 1. In [MM2], any positive function u ∈ C^{2}(ℝ^{n}) solving equation (1.2) is proved to be of the form (1.3), up to translation in the variable y and scaling u_{λ}(x, y) = λ^{\frac{Q−2}{2}} u(λx, λ^{α+1}y) for some positive number λ. The proof relies on the Kelvin transform for Grushin operators and on the moving spheres method. In this work, we take a different point of view and we show by a rearrangement argument that the family of competing functions for inequality (1.1) with sharp constant can be reduced to the class S.

Let D^{1}(ℝ^{n}) be the Sobolev space obtained as completion of C_{0}^{∞}(ℝ^{n}) with respect to the norm

∥u∥_{D^{1}(ℝ^{n})} = (∫_{ℝ^{n}} {|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}}dxdy)^{1/2} .

The space D^{1}(ℝ^{n}) is the natural framework for inequality (1.1). It can be equivalently defined as the space of functions u ∈ L^{\frac{2Q}{Q−2}} (ℝ^{n}) with weak derivatives satisfying ∥u∥_{D^{1}(ℝ^{n})} < +∞. Any function in C_{0}^{∞}(ℝ^{n}) can be approximated uniformly and in D^{1}(ℝ^{n}) by compactly supported, piecewise affine functions u satisfying the additional condition

|∇_{y}u| ≠ 0 on the support of u.  (1.4)

We denote by A the family of all such functions. The set A is dense in D^{1}(ℝ^{n}) and it will sufficient to work in this class.

Theorem 1. Let m, k ∈ ℕ, α ≥ 0 and Q > 2. For any non negative function u ∈ A there exists a non negative function u^{∗} ∈ D^{1}(ℝ^{n}) in the class S such that

∫_{ℝ^{n}} {|∇_{x}u^{∗}|^{2} + w^{2}|∇_{y}u^{∗}|^{2}}dxdy ≤ ∫_{ℝ^{n}} {|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}}dxdy.  (1.5)

Moreover,

∫_{ℝ^{n}} u^{∗}^{\frac{2Q}{Q−2}} dxdy = ∫_{ℝ^{n}} u^{\frac{2Q}{Q−2}} dxdy.  (1.6)

The proof of Theorem 1 relies on a rearrangement argument in the hyperbolic space. We decompose ℝ^{n} as 𝕊^{m−1} × ℍ^{k+1}, where 𝕊^{m−1} is the unit Euclidean sphere in ℝ^{m} and ℍ^{k+1} is the (k + 1)-dimensional hyperbolic space. After a suitable functional change, functions are rearranged in each slice ℍ^{k+1} with respect to hyperbolic volume and distance. The rearranged function is radially symmetric in each slice ℍ^{k+1} about some fixed origin. Inverting the functional change, we get a rearranged function in the class S. The procedure is introduced in Section 2, and Theorem 1 is proved in

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 3

Section 3. Beckner recognized in [B] the connection between Grushin operators and hyperbolic symmetry: in the case of the plane (m = k = 1 and also α = 1) he was able to determine extremal functions, controlling inequality (1.1) by the sharp Sobolev inequality on the sphere.

As functions in the class S are determined by their values in the “disk” {(x, 0) ∈ ℝ^{m} × ℝ^{k} : |x| ≤ 1}, inequality (1.1) is related to a Sobolev-Poincaré inequality in the unit ball B = {x ∈ ℝ^{m} |x| < 1}. Let us introduce the weights

p(x) = (1 − |x|^{2(α+1)})^{k},  q(x) = (Q − m)(Q − 2)(1 − |x|^{2(α+1)})^{k−1}|x|^{2α}. (1.7)

Both p and q are radial and non negative on B. We omit reference to the relevant parameters m, k, α.

Theorem 2. Let m, k ∈ ℕ, α ≥ 0, Q > 2 and p, q as in (1.7). Any non negative function v ∈ C^{1}(B̄) satisfies the inequality

(∫_{B} v^{\frac{2Q}{Q−2}} p dx)^{\frac{Q−2}{Q}} ≤ c′_{m,k,α} ∫_{B} \{|∇_{x}v|^{2}p + v^{2}q\} dx  (1.8)

with the sharp constant c′_{m,k,α} = (ω_{k}2^{−k})^{\frac{2}{Q}} c_{m,k,α}. Here, ω_{k} is the area of the unit sphere in ℝ^{k+1} and c_{m,k,α} is the sharp constant in (1.1).

The vanishing of p and q at the boundary ∂B is, even for α = 0, the main difference between inequality (1.8) and the standard Sobolev-Poincaré inequality in the ball. Its proof relies on the fact that a function in C^{1}(B̄) can be extended to a function in ℝ^{n} of the form (1.3) satisfying the Sobolev inequality (1.1). The reduction procedure from ℝ^{n} to B, which produces the weights p and q, is performed in Theorem 3.2.

The Euler equation for (1.8) is

div_{x}(p∇_{x}v) − qv = −pv^{\frac{Q+2}{Q−2}} , |x| < 1.

This equation, which can also be obtained from (1.2) assuming u of the form (1.3), was studied in [MM2]. We believe that the positive solution v of this equation with mixed boundary condition

\frac{∂v}{∂ν} + (\frac{Q}{2} − 1)v = 0, |x| = 1,

is unique. Equivalently, we believe that extremal functions for inequality (1.8) are radial in the variable x. We are able to prove this conjecture for m = 1. In this case, the inequality is onedimensional

(∫_{−1}^{+1} v^{\frac{2Q}{Q−2}} p dx)^{\frac{Q−2}{Q}} ≤ c′_{1,k,α} ∫_{−1}^{+1} \{v′^{2}p + v^{2}q\} dx,  (1.9)

where now Q = 1 + k(α + 1). We prove that (1.9) is optimized by even functions and we establish an existence result of extremals. As a corollary we have the following

Theorem 3. Let m = 1, k ∈ ℕ and α > 0. There exist extremal functions in D^{1}(ℝ^{n}) for inequality (1.1), they are even in x and belong to the class S (up to translation in y and scaling).

4 ROBERTO MONTI

Existence of extremals can be proved elementarly, because the loss of compactness for (1.1) is removed in the class S (see Theorem 4.2). The symmetry property u(−x, y) = u(x, y) is the main point. The proof is based on a rearrangement argument for (1.9) combined with a uniqueness result for the corresponding Euler equation, which is proved in [MM2] (see Theorem 4.5). The premise for Theorem 3 is a complicate functional change discussed at the beginning of Section 4 (see Proposition 4.1).

In the plane m = k = 1, we have uniqueness of extremals. In this case, a positive extremal of (1.9) solves the problem (see Lemma 4.4)

\{ \begin{matrix}(pv′)′ − qv = −pv^{\frac{Q+2}{Q−2}} & in (−1, 1) \\ 2v′(x) + (Q − 2)xv(x) = 0 & x = ±1. \end{matrix}  (1.10)

Here it is Q = α+2. By Theorem 6 in [MM2], the positive solution to problem (1.10) is unique. In the case α = 1, the solution can be expressed in closed form and it is

v(x) = \frac{1}{(1 + x^{2})^{1/2}} .  (1.11)

Then we have the following

Theorem 4. Let m = k = 1 and α > 0. Extremal functions for inequality (1.1) are unique, up to translation in y, scaling and multiplication by a non zero constant. In particular, for α = 1 the extremal is

u(x, y) = \frac{1}{((1 + x^{2})^{2} + y^{2})^{1/4}} .  (1.12)

Beckner’s function (1.12) is obtained from (1.11) and (1.3). In the case of the plane, the sharp Sobolev inequality for the L^{1} norm of the Grushin gradient was established in [MM1] as a corollary of a sharp isoperimetric inequality. Grushin isoperimetric sets and level sets of extremal functions (1.12) are different.

If radiality in x is assumed, existence, symmetry and uniqueness results can be obtained also for m > 1. Precisely, let R be the family of functions u : ℝ^{m} × ℝ^{k} → [0, +∞) such that u(x, y) = v(|x|, y) for some v : [0, +∞) × ℝ^{k} → [0, +∞) and define c_{m,k,α}^{R} > 0 by

\frac{1}{c_{m,k,α}^{R}} = inf \{∫_{ℝ^{n}}\{|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}\}dxdy : u ∈ R ∩ D^{1}(ℝ^{n}), ∫_{ℝ^{n}} |u|^{\frac{2Q}{Q−2}} dxdy = 1\} .

By Theorem 1 and Lions’s concentration-compactness principle [L], the constant c_{m,k,α}^{R} > 0 is achieved by functions belonging to the class S (up to translation in y and scaling). In the case α = 1, this result is linked to the sharp Sobolev inequalities in groups of Heisenberg type proved by Garofalo and Vassiliev in [GV].

A short overview of the paper is in order. In Section 2, we introduce the main notion of rearrangement and we describe its basic properties. In Section 3, we prove Theorems 1 and 2. Section 4 is devoted to the proof of Theorem 3.

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 5

It is a pleasure to acknowledge the debt I owe Daniele Morbidelli. The problems and the ideas contained in this paper originated in our discussions and in our common research activity.

2. Hyperbolic type rearrangement

Let u : ℝ^{n} → [0, +∞) be a non negative function and associate with it the new function U : ℝ^{n} → [0 + ∞) of the variables ξ ∈ ℝ^{m} and η ∈ ℝ^{k}

U(ξ, η) = |ξ|^{\frac{Q−2}{2(α+1)}} u(|ξ|^{− \frac{α}{α+1}} ξ, η).  (2.1)

If u is of the form (1.3) and U is given by (2.1) then

U(ξ, η) = Φ (\frac{(1 − |ξ|)^{2} + |η|^{2}}{4|ξ|} \frac{ξ}{|ξ|}) .  (2.2)

For any fixed direction ξ/|ξ| ∈ 𝕊^{m−1}, a function U of the form (2.2) is radially symmetric for the hyperbolic metric in the (k + 1)-dimensional halfspace “spanned” by ξ/|ξ| and η.

Let ℍ^{k+1} = ℝ^{+} × ℝ^{k} be the (k + 1)-dimensional hyperbolic space with metric r^{−2}(dr^{2} + |dη|^{2}), r > 0 and η ∈ ℝ^{k}. The Riemannian volume μ and the Riemannian hypersurface measure ν are respectively

μ = \frac{1}{r^{k+1}} L^{k+1} and ν = \frac{1}{r^{k}} H^{k},

where L^{k+1} is the Lebesgue measure in ℝ^{k+1} and H^{k} is the k-dimensional Hausdorff measure in ℝ^{k+1}. If A ⊂ ℍ^{k+1} is an open set and B is a metric ball in ℍ^{k+1}, then the hyperbolic isoperimetric inequality states that

μ(B) = μ(A) ⇒ ν(∂B) ≤ ν(∂A),  (2.3)

with equality only if A is a ball (see [BZ, §10.2]).

A ball B_{s} centered at (1, 0) ∈ ℍ^{k+1} having radius s = log (\frac{1+ϱ}{1−ϱ}), ϱ ∈ (0, 1), is the set of points (r, η) ∈ ℍ^{k+1} satisfying

\frac{(1 − r)^{2} + |η|^{2}}{4r} < \frac{ϱ^{2}}{1 − ϱ^{2}} .  (2.4)

The measure of B_{s} is

μ(B_{s}) = ω_{k}2^{k+1} ∫_{0}^{ϱ} \frac{r^{k}}{(1 − r^{2})^{k+1}} dr,  (2.5)

where ω_{k} is the surface measure of the unit Euclidean sphere in ℝ^{k+1}.

Formulae (2.4) and (2.5) can be easily proved in the ball-model of the hyperbolic space. Endow 𝔹^{k+1} = {z ∈ ℝ^{k+1} : |z| < 1} with the metric 4(1−|z|^{2})^{−2}|dz|^{2} and with the Riemannian volume 2^{k+1}(1 − |z|^{2})^{−k−1}dz. The Möbius map S : ℍ^{k+1} → 𝔹^{k+1}

S(r, η) = (\frac{1 − r^{2} − |η|^{2}}{(1 + r)^{2} + |η|^{2}} , \frac{−2η}{(1 + r)^{2} + |η|^{2}} ) .  (2.6)

6 ROBERTO MONTI

is an isometry and it maps the ball (2.4) onto the ball {|z| < ϱ}. Formula (2.5) easily follows.

Now we construct a function u^{∗} ∈ S starting from a non negative function u ∈ C_{0}(ℝ^{n}). Let U be the function defined in (2.1) and for any ζ ∈ 𝕊^{m−1} let U_{ζ}(r, η) = U(rζ, η). Analogously, for any t ≥ 0 we have the superlevel sets Λ_{t} = {U > t} ⊂ ℝ^{n} and Λ_{ζ,t} = {U_{ζ} > t} ⊂ ℍ^{k+1}. For fixed ζ, the distribution function φ_{ζ}(t) = μ(Λ_{ζ,t}) is decreasing in t and φ_{ζ}(t) = φ_{ζ}(t + 0). Its “inverse” ψ_{ζ} : [0, μ(Λ_{ζ,0})) → [0, +∞) is defined by ψ_{ζ}(s) = inf{t > 0 : φ_{ζ}(t) ≤ s}. The rearranged function U_{ζ}^{∗} is by definition radially symmetric about the point (1, 0) ∈ ℍ^{k+1} and with level sets Λ_{ζ,t}^{∗} satisfying the condition

μ(Λ_{ζ,t}^{∗}) = μ(Λ_{ζ,t}), t > 0.  (2.7)

Precisely, we let

U_{ζ}^{∗}(r, η) = ψ_{ζ}(μ(B_{d(r,η)})),  (2.8)

where

d(r, η) = log \frac{1 + ϱ}{1 − ϱ},  with ϱ^{2} = \frac{(1 − r)^{2} + |η|^{2}}{(1 + r)^{2} + |η|^{2}} ,  (2.9)

is the hyperbolic distance of (r, η) from (1, 0). By Cavalieri principle, for any real number p > 0 condition (2.7) implies the identity

∫_{ℍ^{k+1}} U_{ζ}^{∗}^{p}dμ = ∫_{0}^{+∞} μ(Λ_{ζ,t^{1/p}}^{∗})dt = ∫_{0}^{+∞} μ(Λ_{ζ,t^{1/p}})dt = ∫_{ℍ^{k+1}} U_{ζ}^{p}dμ.  (2.10)

Letting U^{∗}(rζ, η) = U_{ζ}^{∗}(r, η), we can finally define u^{∗} by

U^{∗}(ξ, η) = |ξ|^{\frac{Q−2}{2(α+1)}} u^{∗}(|ξ|^{− \frac{α}{α+1}} ξ, η),  (2.11)

as in (2.1). We call u^{∗} the rearrangement of u. The function u^{∗} is always in the class S, but it has not necessarily compact support even if u itself has compact support.

Now we discuss the properties of u^{∗}, assuming u in A, the family of piecewise affine functions with compact support satisfying (1.4). In this case, the level sets ∂Λ_{ζ,t} = {U_{ζ} = t} are negligible and (ζ, t) ↦ φ_{ζ}(t) is therefore continuous. Moreover, t ↦ φ_{ζ}(t) is strictly decreasing. We shall equivalently write φ(ζ, t) = φ_{ζ}(t) = φ_{t}(ζ), according to the relevant variable.

Throughout the paper we use the following notation. ∇_{ξ}U and ∇_{η}U denote the gradients of a function U in ℝ^{n} in the variables ξ and η, respectively, ∇_{R}U = ⟨∇_{ξ}U, ξ/|ξ|⟩ is the radial part of ∇_{ξ}U and ∇_{T} U = ∇_{ξ}U − ∇_{R}U ξ/|ξ| is the tangential part of ∇_{ξ}U. We also denote by ∇_{H}U = |ξ|(∇_{R}U, ∇_{η}U) the hyperbolic component of ∇U. Finally, if W : 𝕊^{m−1} → ℝ is differentiable at ζ ∈ 𝕊^{m−1} and v ∈ T_{ζ}𝕊^{m−1} is a tangent vector, we define the tangential derivative in the direction v

∇_{v}W (ζ) = lim_{ε→0} \frac{W (γ(ε)) − W (ζ)}{ε},

where γ : (−δ, δ) → 𝕊^{m−1} is a smooth curve such that γ(0) = ζ and γ̇(0) = v.

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 7

The next propositions describe some properties of the rearrangement u^{∗} and of the distribution function μ(Λ_{ζ,t}). The proof is a standard application of coarea and area formulas and it is sketched in the Appendix at the end of the paper.

Proposition 2.1. Let u ∈ A be non negative, U as in (2.1) and Λ_{ζ,t} = {U_{ζ} > t}. The function φ(ζ, t) = μ(Λ_{ζ,t}) is differentiable at almost every (ζ, t) ∈ 𝕊^{m−1} ×ℝ^{+} and moreover

\frac{∂φ_{ζ}(t)}{∂t} = − ∫_{∂Λ_{ζ,t}} \frac{1}{|∇_{H}U_{ζ}|}dν,  (2.12)

and, for v ∈ T_{ζ}𝕊^{m−1} (with m ≥ 2),

∇_{v}φ_{t}(ζ) = ∫_{∂Λ_{ζ,t}} \frac{∇_{v}U_{ζ}}{|∇_{H}U_{ζ}|}dν,  (2.13)

where ∇_{v}U_{ζ} denotes the tangential derivative of U_{ζ} in the variable ζ and direction v.

Proposition 2.2. If u ∈ A is non negative, then the rearrangement u^{∗} is continuous in ℝ^{n} minus the set {x = 0} and locally Lipschitz continuous in ℝ^{n} minus the set {x = 0} ∪ {|x| = 1, y = 0}.

In the next proposition we show that the function u^{∗} is actually continuous on the whole ℝ^{n}. The proof also provides the profile of extremals for (1.1) on the subspace x = 0.

Proposition 2.3. If u ∈ A is a non negative function, then u^{∗} can be continuously extended to ℝ^{n}. Moreover,

u^{∗}(0, y) = C(m, k, α) (∫_{Ω_{0}} s^{−k(α+1)−1}dsdy)^{\frac{Q−2}{2k(α+1)}} (1 + |y|^{2})^{\frac{2−Q}{2(α+1)}} ,  (2.14)

where Ω_{0} = \{(s, y) ∈ ℝ^{+} × ℝ^{k} : s^{\frac{Q−2}{2}} u(0, y) > 1\} and C(m, k, α) > 0 depends on m, k, α.

Proof. We claim that for any ε > 0 there is a δ > 0 such that for |x| ≤ δ and y ∈ ℝ^{k} we have |u^{∗}(x, y) − u^{∗}(0, y)| ≤ ε with u^{∗}(0, y) given by (2.14). If (|x|^{α}x, y) ∈ ∂Λ_{t}^{∗} = ∂{U^{∗} > t} for some t > 0 then

u^{∗}(x, y) = |x|^{\frac{2−Q}{2}} U^{∗}(|x|^{α}x, y) = |x|^{\frac{2−Q}{2}} t.  (2.15)

For ζ ∈ 𝕊^{m−1} let Λ_{ζ,t} = {U_{ζ} > t}. As in (2.7) we have μ(Λ_{ζ,t}) = μ(Λ_{ζ,t}^{∗}), where Λ_{ζ,t}^{∗} is a ball B_{s} centered at (1, 0) with radius s = log \frac{1−ϱ}{1+ϱ}, which, by (2.5), is determined by

\begin{aligned}ω_{k}2^{k+1} ∫_{0}^{ϱ} \frac{r^{k}}{(1 − r^{2})^{k+1}} dr & = μ(Λ_{ζ,t}) = ∫_{U_{ζ}>t} r^{−k−1}drdη \\ & = (α + 1)t^{− \frac{2k(α+1)}{Q−2}} ∫_{Ω_{ζ,t}} s^{−k(α+1)−1}dsdy \\ & = t^{− \frac{2k(α+1)}{Q−2}} λ(ζ, t), \end{aligned}  (2.16)

where Ω_{ζ,t} = \{(s, y) ∈ ℝ^{+} × ℝ^{k} : s^{\frac{Q−2}{2}} u(t^{\frac{2}{Q−2}} sζ, y) > 1\} and λ(ζ, t) is defined by the last equality. In order to get the second equality in (2.16), we used (2.1) and we performed the change of variable rt^{\frac{2(α+1)}{Q−2}} = s^{α+1}. Notice that if |x| → 0 then t → 0 and that

lim_{t→0} λ(ζ, t) = (α + 1) ∫_{Ω_{0}} s^{−k(α+1)−1}dsdy.  (2.17)

The convergence in (2.17) is uniform in ζ. In order to prove this fact, we note that

lim_{s_{0}→+∞} ∫_{Ω_{ζ,t}∩{s>s_{0}}} s^{−k(α+1)−1}dsdy = 0

uniformly in ζ. Moreover, u is uniformly continuous and then for any σ > 0 and s_{0} > 0 there exists t_{0} = t_{0}(σ, s_{0}) > 0 such that

s^{\frac{Q−2}{2}} u(0, y) > 1 + σ ⇒ s^{\frac{Q−2}{2}} u(t^{\frac{2}{Q−2}} sζ, y) > 1 ⇒ s^{\frac{Q−2}{2}} u(0, y) > 1 − σ

for all s ∈ (0, s_{0}), t ∈ (0, t_{0}) and ζ ∈ 𝕊^{m−1}. The uniform convergence easily follows. Now, from (2.15) and (2.16) we get

u^{∗}(x, y) = (\frac{ω_{k}2^{k+1}|x|^{k(α+1)}}{λ(x/|x|, t)} ∫_{0}^{ϱ} \frac{r^{k}}{(1 − r^{2})^{k+1}} dr)^{\frac{2−Q}{2k(α+1)}} ,

where t and ϱ are related through (2.16) and moreover, by (2.9) with (|x|^{α}x, y) ∈ ∂Λ_{t}^{∗},

ϱ^{2} = \frac{(1 − |x|^{α+1})^{2} + |y|^{2}}{(1 + |x|^{α+1})^{2} + |y|^{2}} .

By Hôpital’s rule

lim_{|x|→0} |x|^{k(α+1)} ∫_{0}^{ϱ} \frac{r^{k}}{(1 − r^{2})^{k+1}} dr = lim_{|x|→0} −\frac{|x|^{k(α+1)+1}ϱ^{k}ϱ′}{k(α + 1)(1 − ϱ^{2})^{k+1}} ,

where ϱ′ stands for the derivative of ϱ with respect to |x|. Using

2ϱϱ′ = (ϱ^{2})′ = \frac{−4(α + 1)|x|^{α}(1 + |y|^{2} − |x|^{2(α+1)})}{((1 + |x|^{α+1})^{2} + |y|^{2})^{2}},

we find

−\frac{|x|^{k(α+1)+1}ϱ^{k}ϱ′}{k(α + 1)(1 − ϱ^{2})^{k+1}} = \frac{2}{4^{k}k} (1 + |y|^{2} − |x|^{2(α+1)})((1 + |x|^{α+1})^{2} + |y|^{2})^{k−1},

which converges to \frac{2}{4^{k}k}(1 + |y|^{2})^{k} as |x| → 0, uniformly for y ∈ ℝ^{k}. The claim follows.

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 9

3. Hyperbolic symmetry for competing functions

Theorem 3.1. Let u ∈ A be a non negative function and let u^{∗} be its rearrangement. Then u^{∗} ∈ D^{1}(ℝ^{n}) and relations (1.5) and (1.6) holds.

Proof. The statement u^{∗} ∈ D^{1}(ℝ^{n}) follows from Propositions 2.2 and 2.3, as soon as we prove (1.5) and (1.6).

Let U be the function related to u as in (2.1). By a short computation, we find

\begin{aligned}|∇_{x}u|^{2} & = |x|^{−Q}\{|x|^{2(α+1)}((α + 1)^{2}|∇_{R}U|^{2} + |∇_{T} U|^{2}) + (1 − \frac{Q}{2})^{2}U^{2} \\ & + (α + 1)(2 − Q)|x|^{α+1}U∇_{R}U\}, \\ |∇_{y}u|^{2} & = |x|^{2−Q}|∇_{η}U|^{2}. \end{aligned}  (3.1)

Here, ∇_{x}u and ∇_{y}u are evaluated at (x, y), while U, ∇_{R}U, ∇_{T} U and ∇_{η}U are evaluated at (ξ, η) = (|x|^{α}x, y).

For any t > 0 let Λ_{t} = {U > t} and Ω_{t} = Ψ(Λ_{t}), where

Ψ(ξ, η) = (|ξ|^{− \frac{α}{α+1}} ξ, η).  (3.2)

Using (3.1), we get

\begin{aligned}E_{t} : & = ∫_{Ω_{t}} \{|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}\} dxdy \\ & = ∫_{Ω_{t}} \{(α + 1)^{2}|x|^{2(α+1)}(|∇_{R}U|^{2} + |∇_{η}U|^{2}) + |x|^{2(α+1)}|∇_{T} U|^{2} \\ & + (α + 1)(2 − Q)|x|^{α+1}U∇_{R}U + (1 − \frac{Q}{2})^{2}U^{2}\}\frac{dxdy}{|x|^{Q}} \\ & = ∫_{Λ_{t}} \{δ|∇_{H}U|^{2} + \frac{|ξ|^{2}}{δ} |∇_{T} U|^{2} + \frac{2γ|ξ|}{k}U∇_{R}U + βU^{2}\}\frac{dξdη}{|ξ|^{n}} , \end{aligned}  (3.3)

where we set

β = \frac{(2 − Q)^{2}}{4(α + 1)} , γ = \frac{k}{2}(2 − Q), δ = α + 1.  (3.4)

We got the last equality in (3.3) by the change of variable (x, y) = Ψ(ξ, η), which has Jacobian determinant | det JΨ| = (a + 1)^{−1}|ξ|^{− \frac{mα}{α+1}} .

Let U_{ζ}(r, η) = U(rζ, η), with ζ ∈ 𝕊^{m−1} and r > 0, and Λ_{ζ,t} = {U_{ζ} > t} ⊂ ℍ^{k+1}. By Fubini-Tonelli theorem, we split the last integral in (3.3) into a double integral in ℝ^{k} and ℝ^{m}. Then, introducing polar coordinates in ℝ^{m}, we obtain

E_{t} = ∫_{𝕊^{m−1}} ∫_{Λ_{ζ,t}} \{δ|∇_{H}U_{ζ}|^{2} + \frac{r^{2}}{δ} |∇_{T} U_{ζ}|^{2} + \frac{2γ}{k} rU_{ζ}∇_{R}U_{ζ} + βU_{ζ}^{2}\}\frac{drdη}{r^{k+1}} dH^{m−1}, (3.5)

where ∇_{T} U_{ζ}(r, η) = ∇_{T} U(rζ, η) and ∇_{R}U_{ζ}(r, η) = ∇_{R}U(rζ, η) = \frac{∂}{∂r}U_{ζ}(r, η). Note in the inner integral the hyperbolic measure dμ = \frac{drdη}{r^{k+1}} .

We have to estimate four terms. Let U_{ζ}^{∗} be the rearrangement of U_{ζ} as in (2.8) and let Λ_{ζ,t}^{∗} = {U_{ζ}^{∗} > t}. We claim that for all ζ ∈ 𝕊^{m−1} and for all t > 0 we have

∫_{Λ_{ζ,t}^{∗}} U_{ζ}^{∗}^{2}dμ = ∫_{Λ_{ζ,t}} U_{ζ}^{2}dμ,  (3.6)

∫_{Λ_{ζ,t}^{∗}} rU_{ζ}^{∗} \frac{∂U_{ζ}^{∗}}{∂r} dμ = ∫_{Λ_{ζ,t}} rU_{ζ} \frac{∂U_{ζ}}{∂r} dμ,  (3.7)

∫_{Λ_{ζ,t}^{∗}} r^{2}|∇_{T} U_{ζ}^{∗}|^{2}dμ ≤ ∫_{Λ_{ζ,t}} r^{2}|∇_{T} U_{ζ}|^{2}dμ,  (3.8)

∫_{Λ_{ζ,t}^{∗}} |∇_{H}U_{ζ}^{∗}|^{2}dμ ≤ ∫_{Λ_{ζ,t}} |∇_{H}U_{ζ}|^{2}dμ.  (3.9)

Relations (3.6)–(3.9) imply

E_{t} ≥ ∫_{Ω_{t}^{∗}} {|∇_{x}u^{∗}|^{2} + w^{2}|∇_{y}u^{∗}|^{2}} dxdy,

where u^{∗} is the function defined in (2.11) and Ω_{t}^{∗} = Ψ(Λ_{t}^{∗}). Our claim (1.5) then follows by monotone convergence letting t → 0.

The proof of identity (3.6) is as in line (2.10). We prove (3.7). This is the part of the proof where we need to work with superlevel sets Λ_{t} with t > 0. For any η ∈ ℝ^{k} let Λ_{ζ,η,t} = {r > 0 : (r, η) ∈ Λ_{ζ,t}}. Then

∫_{Λ_{ζ,t}} rU_{ζ} \frac{∂U_{ζ}}{∂r} dμ = \frac{1}{2} ∫_{Λ_{ζ,t}} \frac{∂}{∂r}(U_{ζ}^{2})\frac{drdη}{r^{k}} = \frac{1}{2} ∫_{ℝ^{k}} ∫_{Λ_{ζ,η,t}} \frac{∂}{∂r}(U_{ζ}^{2})\frac{dr}{r^{k}} dη.

The open set Λ_{ζ,η,t} ⊂ ℝ^{+} is an at most countable disjoint union of open intervals

Λ_{ζ,η,t} = ⋃_{h=1}^{+∞}(a_{h}, b_{h}), 0 < a_{h} ≤ b_{h} ≤ a_{h+1}.

(The numbers a_{h}, b_{h} depend on ζ, η, t.) Then, integrating by parts, we obtain

∫_{Λ_{ζ,η,t}} \frac{∂}{∂r}(U_{ζ}^{2})\frac{dr}{r^{k}} = ∑_{h=1}^{+∞} ∫_{a_{h}}^{b_{h}} \frac{∂}{∂r}(U_{ζ}^{2})\frac{dr}{r^{k}}

= ∑_{h=1}^{+∞} (\frac{U_{ζ}(b_{h}, η)^{2}}{b_{h}^{k}} − \frac{U_{ζ}(a_{h}, η)^{2}}{a_{h}^{k}} + k ∫_{a_{h}}^{b_{h}} U_{ζ}^{2} \frac{dr}{r^{k+1}} )

= k ∫_{Λ_{ζ,η,t}} U_{ζ}^{2} \frac{dr}{r^{k+1}} + t^{2} ∑_{h=1}^{+∞} (\frac{1}{b_{h}^{k}} − \frac{1}{a_{h}^{k}} )

= k ∫_{Λ_{ζ,η,t}} (U_{ζ}^{2} − t^{2})\frac{dr}{r^{k+1}} .

Hence, integrating in η and using (2.7) and (3.6), we get

∫_{Λ_{ζ,t}} \frac{∂}{∂r}(U_{ζ}^{2})dμ = k ∫_{Λ_{ζ,t}} (U_{ζ}^{2} − t^{2})dμ = k ∫_{Λ_{ζ,t}^{∗}} (U_{ζ}^{∗}^{2} − t^{2})dμ = ∫_{Λ_{ζ,t}^{∗}} \frac{∂}{∂r}(U_{ζ}^{∗}^{2})dμ. (3.10)

This concludes the proof of (3.7).

We prove (3.8) (we need this part of the argument only if m ≥ 2). First of all notice that Proposition 2.1 applies also to U_{ζ}^{∗}. From (2.12), we have for a.e. t > 0

∫_{∂Λ_{ζ,t}} \frac{1}{|∇_{H}U_{ζ}|}dν = ∫_{∂Λ_{ζ,t}^{∗}} \frac{1}{|∇_{H}U_{ζ}^{∗}|}dν.  (3.11)

From (2.13), we have for a.e. ζ ∈ 𝕊^{m−1} and for every tangential direction v ∈ T_{ζ}𝕊^{m−1}

∫_{∂Λ_{ζ,t}} \frac{∇_{v}U_{ζ}}{|∇_{H}U_{ζ}|}dν = ∫_{∂Λ_{ζ,t}^{∗}} \frac{∇_{v}U_{ζ}^{∗}}{|∇_{H}U_{ζ}^{∗}|}dν.  (3.12)

By the coarea formula, as in (4.1), we have

∫_{Λ_{ζ,t}} r^{2}|∇_{T} U_{ζ}|^{2}dμ = ∫_{t}^{+∞}∫_{∂Λ_{ζ,s}} \frac{r^{2}|∇_{T} U_{ζ}|^{2}}{|∇_{H}U_{ζ}|} dν ds.  (3.13)

Let v_{1}, ..., v_{m−1} ∈ T_{ζ}𝕊^{m−1} be an orthonormal basis. Then

r^{2}|∇_{T} U_{ζ}|^{2} = ∑_{i=1}^{m−1} |∇_{v_{i}}U_{ζ}|^{2},  (3.14)

where ∇_{v_{i}}U_{ζ} is the tangential derivative of ζ ↦ U_{ζ} in the direction v_{i}.

By Hölder inequality, (3.11) and (3.12) we have

∫_{∂Λ_{ζ,s}} \frac{|∇_{v_{i}}U_{ζ}|^{2}}{|∇_{H}U_{ζ}|} dν ≥ ( ∫_{∂Λ_{ζ,s}} \frac{∇_{v_{i}}U_{ζ}}{|∇_{H}U_{ζ}|}dν)^{2}( ∫_{∂Λ_{ζ,s}} \frac{1}{|∇_{H}U_{ζ}|}dν)^{−1}

= ( ∫_{∂Λ_{ζ,s}^{∗}} \frac{∇_{v_{i}}U_{ζ}^{∗}}{|∇_{H}U_{ζ}^{∗}|}dν)^{2}( ∫_{∂Λ_{ζ,s}^{∗}} \frac{1}{|∇_{H}U_{ζ}^{∗}|}dν)^{−1} = \frac{|∇_{v_{i}}U_{ζ}^{∗}|^{2}}{|∇_{H}U_{ζ}^{∗}|} ν(∂Λ_{ζ,s}^{∗}).

The last equality follows from the fact that ∇_{v_{i}}U_{ζ}^{∗} and |∇_{H}U_{ζ}^{∗}| are constant on ∂Λ_{ζ,s}^{∗}. Indeed, the length of the gradient of a radial function in ℍ^{k+1} is still radial. Precisely, let V : ℍ^{k+1} → ℝ be a function of the form V (r, η) = ψ(d(r, η)) where d stands for the hyperbolic distance from some fixed point and ψ is a differentiable function. Then

|∇_{H}V (r, η)| = |ψ′(d(r, η))∇_{H}d(r, η)| = |ψ′(d(r, η))|,  (3.15)

because |∇_{H}d(r, η)| = 1, and thus |∇_{H}V (r, η)| is radial.

Now (3.8) is a consequence of (3.13), (3.14) and

∫_{∂Λ_{ζ,s}^{∗}} \frac{|∇_{v_{i}}U_{ζ}^{∗}|^{2}}{|∇_{H}U_{ζ}^{∗}|} dν ≤ ∫_{∂Λ_{ζ,s}} \frac{|∇_{v_{i}}U_{ζ}|^{2}}{|∇_{H}U_{ζ}|} dν.

In order to prove (3.9), note that by the coarea formula, analogously to (4.1), we have

∫_{Λ_{ζ,t}} |∇_{H}U_{ζ}|^{2} = ∫_{t}^{+∞}∫_{∂Λ_{ζ,s}} |∇_{H}U_{ζ}|dν ds,  (3.16)

and by Hölder inequality

ν(∂Λ_{ζ,s}) ≤ ( ∫_{∂Λ_{ζ,s}} |∇_{H}U_{ζ}|dν )^{1/2}( ∫_{∂Λ_{ζ,s}} \frac{1}{|∇_{H}U_{ζ}|}dν )^{1/2}.  (3.17)

For μ(Λ_{ζ,s}^{∗}) = μ(Λ_{ζ,s}), the isoperimetric inequality (2.3) implies ν(∂Λ_{ζ,s}^{∗}) ≤ ν(∂Λ_{ζ,s}). Thus, from (3.17) and (3.11) it follows that

∫_{∂Λ_{ζ,s}} |∇_{H}U_{ζ}|dν ≥ ν(∂Λ_{ζ,s}^{∗})^{2}( ∫_{∂Λ_{ζ,s}^{∗}} \frac{1}{|∇_{H}U_{ζ}^{∗}|}dν )^{−1} = ∫_{∂Λ_{ζ,s}^{∗}} |∇_{H}U_{ζ}^{∗}|dν.

In the last equality we used again the fact that, by (3.15), |∇_{H}U_{ζ}^{∗}| is constant on ∂Λ_{ζ,s}^{∗}. Now (3.9) follows from (3.16) and the proof of (1.5) is concluded.

The proof of (1.6) follows from

∫_{ℝ^{n}} u^{\frac{2Q}{Q−2}} dxdy = \frac{1}{α + 1} ∫_{𝕊^{m−1}} ∫_{ℍ^{k+1}} Uζ^{\frac{2Q}{Q−2}} dμ dH^{m−1},

and from (2.10).

We recall that B = {x ∈ ℝ^{m} : |x| < 1}. By Theorem 3.1, competing functions in the class S optimize inequality (1.1). Theorem 2 is a consequence of (1.1) and of the following

Theorem 3.2. Let m, k ∈ ℕ, α ≥ 0, Q = m + k(α + 1) > 2. For any non negative function v ∈ C^{1}(B̄), the function u of the form (1.3) and such that u(x, 0) = v(x), |x| ≤ 1, satisfies the identities

∫_{ℝ^{n}} \{|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}\}dxdy = \frac{ω_{k}}{2^{k}} ∫_{B} \{|∇_{x}v|^{2}p + v^{2}q\}dx,  (3.18)

∫_{ℝ^{n}} u^{\frac{2Q}{Q−2}} dxdy = \frac{ω_{k}}{2^{k}} ∫_{B} v^{\frac{2Q}{Q−2}}p dx.  (3.19)

Here, p and q are the weights (1.7), and ω_{k} stands for the surface measure of the unit Euclidean sphere in ℝ^{k+1}.

Proof. Let U be the function associated with u as in (2.1), U_{ζ}(r, η) = U(rζ, η), Λ_{t} = {U > t}, Λ_{ζ,t} = {U_{ζ} > t} ⊂ ℍ^{k+1}, Ω_{t} = Ψ(Λ_{t}), with t > 0 and ζ ∈ 𝕊^{m−1}. Here, Ψ is the mapping introduced in (3.2). In order to prove (3.18), we begin with the identity

E_{t} : = ∫_{Ω_{t}} {|∇_{x}u|^{2} + w^{2}|∇_{y}u|^{2}} dxdy = ∫_{𝕊^{m−1}} E_{ζ,t} dH^{m−1},  (3.20)

where

E_{ζ,t} = ∫_{Λ_{ζ,t}} \{δ|∇_{H}U_{ζ}|^{2} + \frac{r^{2}}{δ} |∇_{T} U_{ζ}|^{2} + \frac{2γ}{k} rU_{ζ}∇_{R}U_{ζ} + βU_{ζ}^{2}\}dμ.

Identity (3.20) is proved in Theorem 3.1, formulae (3.1)–(3.5). The constants β, γ, δ are defined in (3.4). Using the identity

∫_{Λ_{ζ,t}} rU_{ζ}∇_{R}U_{ζ}dμ = \frac{k}{2} ∫_{Λ_{ζ,t}} (U_{ζ}^{2} − t^{2})dμ,

which is proved in (3.10), we also have

E_{ζ,t} = ∫_{Λ_{ζ,t}} \{δ|∇_{H}U_{ζ}|^{2} + \frac{r^{2}}{δ} |∇_{T} U_{ζ}|^{2} + (β + γ)U_{ζ}^{2} − γt^{2}\}dμ.  (3.21)

The function U_{ζ} is radial in the hyperbolic metric about the point (1, 0) ∈ ℍ^{k+1}. In order to exploit this symmetry, it is useful to work with the ball model of the hyperbolic space. Let 𝔹^{k+1} = {z ∈ ℝ^{k+1} : |z| < 1} and let S : ℍ^{k+1} → 𝔹^{k+1} be the isometry defined in (2.6). Notice that S^{−1} = S and then in the coordinates z = (z_{0}, z′) ∈ ℝ^{+} × ℝ^{k} the components (S_{1}, S_{2}) of S^{−1} are

S_{1}(z) = \frac{1 − |z|^{2}}{(1 + z_{0})^{2} + |z′|^{2}} , S_{2}(z) = \frac{−2z′}{(1 + z_{0})^{2} + |z′|^{2}} .

We need the function

σ(r) = \frac{1 − r}{1 + r},  with σ′(r) = −\frac{2}{(1 + r)^{2}} , r ≠ −1.  (3.22)

Note that S(r, 0) = (σ(r), 0).

The function V_{ζ} : 𝔹^{k+1} → ℝ, V_{ζ}(z) = U_{ζ}(S(z)), is radial in the variable z. Moreover, the hyperbolic gradient transforms as

|∇_{H}U_{ζ}(S(z))|^{2} = \frac{(1 − |z|^{2})^{2}}{4} |∇_{z}V_{ζ}(z)|^{2}.  (3.23)

We transform the tangential component r^{2}|∇_{T} U_{ζ}|^{2}. Fix a point (r, η) ∈ ℍ^{k+1} and let r_{0} ∈ (0, 1] be the solution of the equation

\frac{(1 − r_{0})^{2}}{4r_{0}} = \frac{(1 − r)^{2} + |η|^{2}}{4r} .  (3.24)

For U_{ζ} is radial about (1, 0), we have r^{2}|∇_{T} U_{ζ}(r, η)|^{2} = r_{0}^{2}|∇_{T} U_{ζ}(r_{0}, 0)|^{2}. Letting z = S(r, η), by (3.24) we have |z| = σ(r_{0}), which is equivalent to r_{0} = σ(|z|). Then we find U_{ζ}(r_{0}, 0) = V_{ζ}(S(r_{0}, 0)) = V_{ζ}(σ(r_{0}), 0), and hence

r^{2}|∇_{T} U_{ζ}(r, η)|^{2} = r_{0}^{2}|∇_{T} V_{ζ}(σ(r_{0}), 0)|^{2} = σ(|z|)^{2}|∇_{T} V_{ζ}(|z|, 0)|^{2}.  (3.25)

The Möbius map S transforms the measure μ into the measure \frac{2^{k+1}dz}{(1−|z|^{2})^{k+1}} and the set Λ_{ζ,t} onto the set Z_{ζ,t} = {V_{ζ} > t}. Taking into account (3.23) and (3.25), identity (3.21) becomes

E_{ζ,t} =∫_{Z_{ζ,t}}\{δ \frac{(1 − |z|^{2})^{2}}{4} |∇_{z}V_{ζ}|^{2} +\frac{σ(|z|)^{2}}{δ} |∇_{T} V_{ζ}|^{2} + (β + γ)V_{ζ}^{2} − γt^{2}\}\frac{2^{k+1}dz}{(1 − |z|^{2})^{k+1}} . (3.26)

From V_{ζ}(z) = V_{ζ}(|z|, 0) = U_{ζ}(S(|z|, 0)) = U_{ζ}(σ(|z|), 0)) and (3.22), we find

\frac{(1 − |z|^{2})^{2}}{4} |∇_{z}V_{ζ}(z)|^{2} = σ(|z|)^{2}|∇_{R}U_{ζ}(σ(|z|), 0)|^{2}.

Thus, by polar coordinates we obtain from (3.26)

E_{ζ,t} = 2^{k+1}∫_{0}^{r_{ζ,t}}\{σ(r)^{2}(δ|∇_{R}U_{ζ}|^{2} + \frac{1}{δ}|∇_{T} U_{ζ}|^{2}) + (β + γ)U_{ζ}^{2} − γt^{2}\}\frac{ω_{k}r^{k}}{(1 − r^{2})^{k+1}} dr,

where r_{ζ,t} ∈ (0, 1) is a number such that U_{ζ}(σ(r_{ζ,t}), 0) = t and functions inside the integral are evaluated at (σ(r), 0). Now the change of variable ϱ = σ(r) with dϱ = σ′(r)dr furnishes

E_{ζ,t} = \frac{ω_{k}}{2^{k}} ∫_{ϱ_{ζ,t}}^{1}\{ϱ^{2}(δ|∇_{R}U_{ζ}|^{2} + \frac{1}{δ}|∇_{T} U_{ζ}|^{2}) + (β + γ)U_{ζ}^{2} − γt^{2}\}\frac{(1 − ϱ^{2})^{k}}{ϱ^{k+1}} dϱ, (3.27)

where ϱ_{ζ,t} = σ(r_{ζ,t}) and functions inside the integral are evaluated at (ϱ, 0).

In the next step we write (3.27) for the function v. By (3.1) with v(x) = u(x, 0) (recall also the constants in (3.4)) we have

\frac{1}{δ}ϱ^{\frac{Q}{α+1}} |∇_{x}v(ϱ^{\frac{1}{α+1}} ζ)|^{2} = ϱ^{2} (δ|∇_{R}U_{ζ}|^{2} + \frac{1}{δ}|∇_{T} U_{ζ}|^{2}) + βU_{ζ}^{2} + \frac{2γ}{k} ϱU_{ζ}∇_{R}U_{ζ}, (3.28)

where functions in the right hand side are evaluated at (ϱ, 0). Identity (3.27) then becomes

E_{ζ,t} = \frac{ω_{k}}{2^{k}} ∫_{ϱ_{ζ,t}}^{1}\{\frac{1}{δ}ϱ^{\frac{Q}{α+1}} |∇_{x}v(ϱ^{\frac{1}{α+1}} ζ)|^{2} + γ(U_{ζ}^{2} − t^{2}) − \frac{2γ}{k} ϱU_{ζ}∇_{R}U_{ζ}\}\frac{(1 − ϱ^{2})^{k}}{ϱ^{k+1}} dϱ, (3.29)

with U_{ζ} and ∇_{R}U_{ζ} evaluated at (ϱ, 0). Let us consider

\begin{aligned}−\frac{2γ}{k} ∫_{ϱ_{ζ,t}}^{1} U_{ζ}∇_{R}U_{ζ} \frac{(1 − ϱ^{2})^{k}}{ϱ^{k}} dϱ & = −\frac{γ}{k} ∫_{ϱ_{ζ,t}}^{1} ∇_{R}(U_{ζ}^{2} − t^{2})\frac{(1 − ϱ^{2})^{k}}{ϱ^{k}} dϱ \\ & = \frac{γ}{k} ∫_{ϱ_{ζ,t}}^{1}(U_{ζ}^{2} − t^{2})f(ϱ)dϱ, \end{aligned}  (3.30)

with

f(ϱ) = \frac{d}{dϱ} \frac{(1 − ϱ^{2})^{k}}{ϱ^{k}} = −2k \frac{(1 − ϱ^{2})^{k−1}}{ϱ^{k−1}} − k \frac{(1 − ϱ^{2})^{k}}{ϱ^{k+1}} = f_{1}(ϱ) + f_{2}(ϱ).  (3.31)

Integration by parts in (3.30) produces no boundary contribution because U_{ζ}(ϱ_{ζ,t}, 0) = t. The term γ(U_{ζ}^{2} − t^{2}) in (3.29) is cancelled by the part of (3.30) corresponding to f_{2} in (3.31).

We are therefore left with

\begin{aligned}E_{ζ,t} = \frac{ω_{k}}{2^{k}} ∫_{ϱ_{ζ,t}}^{1} & \{\frac{1}{δ}|∇_{x}v(ϱ^{\frac{1}{α+1}} ζ)|^{2}ϱ^{\frac{m}{α+1} −1}(1 − ϱ^{2})^{k} − 2γv(ϱ^{\frac{1}{α+1}} ζ)^{2}ϱ^{\frac{m−2}{α+1} +1}(1 − ϱ^{2})^{k−1}\}dϱ \\ & + \frac{γω_{k}}{2^{k−1}} R(ζ, t), \end{aligned} (3.32)

where

R(ζ, t) = t^{2} ∫_{ϱ_{ζ,t}}^{1} ϱ^{1−k}(1 − ϱ^{2})^{k−1}dϱ.

We claim that R(ζ, t) converges to zero as t → 0, uniformly in ζ ∈ 𝕊^{m−1}. For U_{ζ}(ϱ_{ζ,t}, 0) = t and U(ϱζ, 0) = ϱ^{\frac{Q−2}{2(α+1)}} u(ϱ^{\frac{1}{α+1}} ζ, 0), by (2.1), the claim follows from the boundedness of u and from

lim_{s→0} s^{\frac{Q−2}{α+1}} ∫_{s}^{1} ϱ^{1−k}(1 − ϱ^{2})^{k−1}dϱ = 0.

Letting t tend to zero in (3.32) and performing the change of variable ϱ = r^{α+1} we get (recall (3.4))

E_{ζ,0} := lim_{t→0} E_{ζ,t} = \frac{ω_{k}}{2^{k}} ∫_{0}^{1} \{|∇_{x}v(rζ)|^{2}p(r) + v(rζ)^{2}q(r)\}r^{m−1}dr.

Here, consistently with (1.7), we let p(r) = (1 − r^{2(α+1)})^{k} and q(r) = (Q − m)(Q − 2)(1 − r^{2(α+1)})^{k−1}r^{2α}. Integrating E_{ζ,0} over 𝕊^{m−1} and using the coarea formula (polar coordinates), we complete the proof of (3.18).

The proof of (3.19) is much more easier and it is left to the reader.

4. Existence and even symmetry of extremals

In this section we prove Theorem 3. First, we transform inequality (1.8) into a Sobolev inequality of Riemannian type. Then, in Theorem 4.2, we prove existence of extremals for inequality (1.9) and hence for inequality (1.1), in the case m = 1. Finally, in Theorem 4.5, we show that extremals are even in the variable x. Note that outside the class S there can be no extremals for (1.1), because, by Theorem 4 in [MM2], any positive solution to equation (1.2) must be in S, up to scaling and translation in y. The important technical point in this section is Proposition 4.1, which provides the functional transformation suitable to proving existence and symmetry of extremals. The coefficient functions and the constants appearing in this proposition will play a fundamental role.

We introduce in M = {ξ ∈ ℝ^{m} : |ξ| < τ} a Riemannian structure with “axial symmetry”. The number τ > 1 will be fixed later. For v ∈ C^{1}(M) let

|∇_{M}v|^{2} = |∇_{R}v|^{2} + \frac{1}{κ^{2}} |∇_{T} v|^{2},  (4.1)

where ∇_{R}v and ∇_{T} v are the radial and tangential components of the Euclidean gradient of v and κ is a positive function. Let σ be the measure on M

σ = κ^{m−1}L^{m},  (4.2)

where L^{m} is Lebesgue measure in M.

Now consider the unit ball B = {x ∈ ℝ^{m} : |x| < 1}. Given u ∈ C^{1}(B̄), define the function v ∈ C^{1}(M) by letting

u(x) = φ(|x|) v (ψ(|x|)\frac{x}{|x|}) ,  (4.3)

where φ, ψ : [0, 1) → [0, +∞) are the functions

φ(t) = (1 − t^{2(α+1)})^{− \frac{Q−2}{2(α+1)}} and ψ(t) = ∫_{0}^{t}(1 − s^{2(α+1)})^{− \frac{1}{α+1}} ds.  (4.4)

The function ψ can be continuously extended to [0, 1] and it is strictly increasing on this interval. Now we fix τ := ψ(1) > 1 and we fix the function κ by letting

h = |x|\frac{ψ′}{ψ} and κ(ξ) = h(ψ^{−1}(|ξ|)), |ξ| < τ.  (4.5)

We are ready to transform the left and right hand side of inequality (1.8). We draw the reader’s attention to the function G and to the constant ϑ in line (4.7) below. The qualitative behaviour of G and the sign of ϑ will play an important role.

Proposition 4.1. Let m, k ≥ 1 and α > 0. Given a non negative function u ∈ C^{1}(B̄), let v be defined as in (4.3). Then we have the identities

∫_{B}{|∇_{x}u|^{2}p + u^{2}q} dx = ε_{k}∫_{∂B} u^{2} dH^{m−1} + ∫_{M} {|∇_{M}v|^{2} + F v^{2}}dσ,  (4.6)

where ε_{k} = Q − 2 if k = 1 and ε_{k} = 0 otherwise, and F (ξ) = G(ψ^{−1}(|ξ|)) with

G(|x|) = \frac{ϑ|x|^{2α}}{(1 − |x|^{2(α+1)})^{\frac{2α}{α+1}}} ,  ϑ = (Q − 2)(k(α + 1) − 2α − m);  (4.7)

and

∫_{B} u^{\frac{2Q}{Q−2}} p dx = ∫_{M} v^{\frac{2Q}{Q−2}} dσ.  (4.8)

Proof. If u and v are related as in (4.3), then by a short computation we find

|∇_{x}u|^{2} = φ^{2}ψ′^{2}|∇_{R}v|^{2} + \frac{1}{|x|^{2}} φ^{2}ψ^{2}|∇_{T} v|^{2} + 2φφ′ψ′v∇_{R}v + φ′^{2}v^{2},  (4.9)

with v, ∇_{R}v, ∇_{T} v evaluated at ψ(|x|)x/|x|.

We explain how φ, ψ and h can be determined. In order that the first two terms in the right hand sides of (4.9) have the form (4.1), we require

\frac{1}{|x|^{2}} φ^{2}ψ^{2} = \frac{1}{h^{2}} φ^{2}ψ′^{2}.  (4.10)

Expression (4.9) is multiplied by p, added to u^{2}q and finally integrated over the unit ball B. In the integral, we have to perform the change of variable ξ = ψ(|x|)x/|x| = Φ(x), which has Jacobian determinant

| det JΦ| = \frac{ψ^{m−1}ψ′}{|x|^{m−1}} .  (4.11)

In order to get the measure (4.2) after this change of variable, we also require

\frac{φ^{2}ψ′^{2}p|x|^{m−1}}{ψ^{m−1}ψ′} = h^{m−1}.  (4.12)

Analogously, the condition producing the measure (4.2) after the change of variable ξ = Φ(x) in (4.8) is

\frac{φ^{2^{∗}}p|x|^{m−1}}{ψ^{m−1}ψ′} = h^{m−1}.  (4.13)

Here and in the following we let 2^{∗} = \frac{2Q}{Q−2}.

Now, (4.10), (4.12) and (4.13) determine φ, ψ and h. Indeed, comparing (4.12) and (4.13) we find

φ = ψ′^{\frac{2}{2^{∗}−2}} .  (4.14)

From (4.10) we get

h = |x|\frac{ψ′}{ψ} .  (4.15)

Note that, with such an h, the Jacobian determinant (4.11) is

| det JΦ| = ψ′^{m}h^{1−m}.  (4.16)

Replacing (4.15) and (4.14) into (4.12), we determine

ψ′(|x|) = p^{−(2−m+4/(2^{∗}−2))^{−1}} = p^{− \frac{1}{k(α+1)}} = (1 − |x|^{2(α+1)})^{− \frac{1}{α+1}} .  (4.17)

With ψ(0) = 0, ψ is also determined. Note that

τ = ψ(1) = ∫_{0}^{1} (1 − t^{2(α+1)})^{− \frac{1}{α+1}} dt  (4.18)

is finite for any α > 0. From (4.14), we have

φ(|x|) = p^{− \frac{Q−2}{2k(α+1)}} = (1 − |x|^{2(α+1)})^{− \frac{Q−2}{2(α+1)}} .  (4.19)

Finally, h is determined by (4.15). It is not difficult to check that h(0) = 1. By (4.9), (4.10), (4.12) and (4.15) we obtain

\begin{matrix}E := ∫_{B} \{|∇_{x}u|^{2}p + u^{2}q\} dx = ∫_{B} \{ & (|∇_{R}v|^{2} + h^{−2}|∇_{T} v|^{2})ψ′^{m} \\ + pφφ′∇_{R}(v^{2}) + v^{2}(φ′^{2}p + φ^{2}q)\} dx, \end{matrix} (4.20)

where ∇_{R}(v^{2}) = 2ψ′v∇_{R}v and v, ∇_{R}v are evaluated at ψ(|x|)x/|x|. We are going to integrate by parts the term with ∇_{R}(v^{2})

\begin{aligned}∫_{B} pφφ′∇_{R}(v^{2})dx & = ∫_{𝕊^{m−1}} ∫_{0}^{1} pφφ′t^{m−1}∇_{R}(v^{2})dt dH^{m−1} \\ & = ∫_{𝕊^{m−1}} L(x) dH^{m−1} − ∫_{B} ((pφφ′)′ + (m − 1)\frac{pφφ′}{|x|} )v^{2}dx, \end{aligned} (4.21)

L(x) = lim_{t→1} p(t)φ(t)φ′(t)v(ψ(t)x)^{2}, |x| = 1.  (4.22)

Letting λ(t) = (1 − t^{2(α+1)}), t = |x|, we have

p = λ^{k}, q = (Q − m)(Q − 2)t^{2α}λ^{k−1}, φ = λ^{b}, with b = −\frac{Q − 2}{2(α + 1)}. (4.23)

Hence

λ′ = −2(α + 1)t^{2α+1}, p′ = kλ^{k−1}λ′, φ′ = bλ^{b−1}λ′,  (4.24)

and

φφ′p = bλ^{2b+k−1}λ′, (φφ′p)′ = bλ^{2b+k−2}((2b + k − 1)λ′^{2} + λλ′′).  (4.25)

Using (4.3) and (4.23)–(4.24), we can compute L(x) in (4.22) for |x| = 1

L(x) = (Q − 2) lim_{t→1} λ(t)^{k−1}u(tx)^{2} = \{ \begin{matrix}(Q − 2)u(x)^{2} & for k = 1, \\ 0 & for k > 1. \end{matrix}  (4.26)

Taking into account (4.21), the coefficient of v^{2} in (4.20) is

P := φ′^{2}p + φ^{2}q − (pφφ′)′ − (m − 1)pφφ′/t.

Here and in the following t = |x|. By (4.23)–(4.25) we find

P = bλ^{k+2b−2} {(1 − b − k)λ′^{2} − (2(α + 1)(Q − m)t^{2α} + λ′′ + (m − 1)λ′/t) λ}
= 2(α + 1)bt^{2α}λ^{k+2b−2} {2(α + 1)(1 − b − k)t^{2(α+1)}
+(2α + m − k(α + 1))(1 − t^{2(α+1)})} .

By the definition of b in line (4.23), it can be checked that 2(α + 1)(1 − b − k) = 2α + m − k(α + 1) and we have

P = \frac{ϑ|x|^{2α}}{(1 − |x|^{2(α+1)})^{\frac{m+2α}{α+1}}} ,  with ϑ = (Q − 2)(k(α + 1) − 2α − m). (4.27)

The function

G = \frac{P}{ψ′^{m}} = \frac{ϑ|x|^{2α}}{(1 − |x|^{2(α+1)})^{\frac{2α}{α+1}}}

is the one in (4.7). Now, by (4.21) and (4.26), (4.20) becomes

E = ε_{k} ∫_{|x|=1} u^{2}dH^{m−1} + ∫_{B} {|∇_{R}v|^{2} + h^{−2}|∇_{T} v|^{2} + Gv^{2}} ψ′^{m}dx,

and after the change of variable ξ = Φ(x) with Jacobian determinant (4.16), and with κ and F as in the statement of the proposition, we get (4.6).

Finally, (4.8) easily follows from (4.3), (4.13) and (4.11).

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 19

We begin the proof of Theorem 3. Now let m = 1. Our first task is to prove existence of extremal functions for inequality (1.9). We introduce the following norm of a function u ∈ C^{1}([−1, 1])

∥u∥ = (∫_{−1}^{+1} {|u′|^{2}p + |u|^{2}q} dx)^{1/2} ,

where p and q are the functions (1.7) with m = 1. Clearly, the norm ∥ · ∥ origins from a scalar product. Denote by H^{1} the Hilbert space obtained completing C^{1}([−1, 1]) in the norm ∥ · ∥. Vectors in this Sobolev space are measurable functions u with measurable distributional derivative u′ satisfying ∥u∥ < +∞.

Consider the following minimum problem

\frac{1}{c′_{1,k,α}} = inf \{∥u∥^{2} : u ∈ H^{1}, ∫_{−1}^{+1} |u|^{\frac{2Q}{Q−2}} p dx = 1\} .  (4.28)

Theorem 4.2. The infimum (4.28) is attained in H^{1}.

Proof. Let (u_{h})_{h∈ℕ} be a minimizing sequence for (4.28)

lim_{h→∞} ∥u_{h}∥^{2} = \frac{1}{c′_{1,k,α}}  and ∫_{−1}^{+1} |u_{h}|^{\frac{2Q}{Q−2}} p dx = 1, h ∈ ℕ.  (4.29)

We can assume without loss of generality that u_{h} ∈ C^{1}([−1, 1]) and u_{h}(±1) > 0. Indeed, we can work with non negative functions u_{h} ≥ 0. Moreover, if u_{h}(1) = 0 or u_{h}(−1) = 0 we replace u_{h} with au_{h} + b, where b > 0 is close to zero and a ∈ (0, 1) is close to 1 as we wish and such that the integral constraint is maintained. The error in the norm can be made arbitrarily small. By weak compactness, there is a subsequence – which is still denoted by u_{h} – weakly converging to a function u ∈ H^{1}. The norm is lower semicontinuous for the weak convergence

∥u∥ ≤ lim inf_{h→∞} ∥u_{h}∥.  (4.30)

Possibly extracting a new subsequence, we can also assume that u_{h}(x) → u(x) for almost every x ∈ (−1, 1). If u satisfies the integral constraint

∫_{−1}^{+1} u^{\frac{2Q}{Q−2}} p dx = 1,  (4.31)

i.e. there is no concentration of measure, then from (4.29)–(4.31) it follows that u is a minimizer for (4.28).

In order to prove (4.31), consider for any h ∈ ℕ the function v_{h} ∈ C^{1}(−τ, τ) associated with u_{h} as in (4.3). Note that v_{h} is continuous on [−τ, τ] and v_{h}(±τ) = 0. The number τ > 1 is defined in (4.18). By Proposition 4.1, we have the identities

∫_{−1}^{+1} uh^{\frac{2Q}{Q−2}} p dx = ∫_{−τ}^{+τ} vh^{\frac{2Q}{Q−2}} dξ,  (4.32)

and

∫_{−1}^{+1}{u′_{h}^{2}p + u_{h}^{2}q}dx = ε_{k}{u_{h}(−1)^{2} + u_{h}(1)^{2}} + ∫_{−τ}^{+τ}{v′_{h}^{2} + F v_{h}^{2}}dξ,  (4.33)

where ε_{k} = Q − 2 if k = 1 and ε_{k} = 0 otherwise. F is the function defined in the statement of Proposition 4.1. Examining the constant ϑ in (4.7), we notice that with m = 1 we have

F ≤ 0 for k = 1, F ≥ 0 for k ≥ 2.

We must discuss separately the cases k = 1 and k ≥ 2.

We study the case k = 1. By continuity, for any h ∈ ℕ there is a point ξ_{h} ∈ [−τ, τ] such that v_{h}(ξ_{h}) = max_{[−τ,τ]} v_{h} := M_{h}. Let z_{h} be the piecewise affine function such that z_{h}(±τ) = 0 and z_{h}(ξ_{h}) = M_{h}. Define v̄_{h} = max{v_{h}, z_{h}} and let ū_{h} be associated with v̄_{h} as in (4.3). Note that, with m = k = 1, the functions in (4.17) and (4.19) are respectively ψ′(x) = (1 − x^{2(α+1)})^{− \frac{1}{α+1}} and φ(x) = (1 − x^{2(α+1)})^{− \frac{α}{2(α+1)}} , x ∈ (0, 1). Then, by Hôpital’s rule

lim_{x→1^{−}} \frac{v_{h}(ψ(x))}{ψ(x) − τ} = lim_{x→1^{−}} \frac{φ(x)^{−1}u_{h}(x)}{ψ(x) − τ} = −αu_{h}(1) lim_{x→1^{−}}(1 − x^{2(α+1)})^{− \frac{α}{2(α+1)}} = −∞,

because u_{h}(1) > 0. It follows that v_{h} ≥ z_{h} in some left neighborhood of τ and this proves that ū_{h}(1) = u_{h}(1). The same holds at x = −1.

For the L^{2} norm of v̄′_{h} is less than or equal to the L^{2} norm of v′_{h}, we have

∫_{−τ}^{+τ} {|v̄′_{h}|^{2} + F v̄_{h}^{2}}dξ ≤ ∫_{−τ}^{+τ} {|v′_{h}|^{2} + F v_{h}^{2}}dξ.

Indeed, F ≤ 0 and v̄_{h} ≥ v_{h}. Analogously,

∫_{−τ}^{+τ} v̄h^{\frac{2Q}{Q−2}} dξ ≥ ∫_{−τ}^{+τ} vh^{\frac{2Q}{Q−2}} dξ.

Then, possibly renormalizing each ū_{h} by a positive factor smaller than 1 (in order that the integral constraint be satisfied), the sequence (ū_{h})_{h∈ℕ} is a new minimizing sequence for (4.28) and we can without loss of generality assume u_{h} = ū_{h}, i.e. v_{h} ≥ z_{h}. Thus, letting 2^{∗} = \frac{2Q}{Q−2}, we have

1 = (∫_{−τ}^{+τ} v_{h}^{2^{∗}}dξ)^{1/2^{∗}} ≥ (∫_{−τ}^{+τ} z_{h}^{2^{∗}}dξ)^{1/2^{∗}} ≥ 2^{\frac{1}{2^{∗}} −1}τ^{\frac{1}{2^{∗}}} M_{h},

and then ∥v_{h}∥_{∞} is uniformly bounded. Now, (4.31) easily follows from (4.32) by dominated convergence.

In the case k ≥ 2, we have ε_{k} = 0 and F ≥ 0 in (4.33), with left hand side uniformly bounded in h ∈ ℕ. It follows that the sequence (v_{h})_{h∈ℕ} is bounded in H_{0}^{1}(−τ, τ), the usual Sobolev space of functions vanishing at the boundary and with derivative in L^{2}. Then sup_{h∈ℕ} ∥v_{h}∥_{∞} < +∞ and the proof of (4.31) is concluded as in the case k = 1.

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 21

The extension of a function u ∈ H^{1} to a function on ℝ^{n}, n = 1+k, of the form (1.3) produces a function which is in D^{1}(ℝ^{n}), by Theorem 3.2. In view of the minimization problem, this function can also be assumed to satisfy condition ii) in the definition of the class S, by Theorem 3.1. Then, without loss of generality, competing functions u ∈ H^{1} for problem (4.28) can also be assumed to satisfy the condition

t ↦ t^{\frac{Q−2}{2}} u(±t) is non decreasing for t ∈ [0, 1).  (4.34)

To a minimum u ∈ H^{1} for problem (4.28) there corresponds an extremal u ∈ D^{1}(ℝ^{n}) for inequality (1.1) solving in weak sense equation (1.2). By abuse of notation, we denote the two functions with the same symbol u. By a result due to Serrin, weak solutions to (1.2) are locally bounded. For our purposes, the boundedness near |x| = 1 is enough. We state the next Lemma for general m.

Lemma 4.3. Let m, k ∈ ℕ, α > 0, Q = m+k(α+1) and n = m+k. A non negative weak solution u ∈ D^{1}(ℝ^{n}) of equation (1.2) is in L_{loc}^{∞}(Ω), with Ω = ℝ^{n} \ {x = 0}.

Proof. Letting V = −u^{\frac{4}{Q−2}} , the function u solves the equation

Δ_{x}u + w^{2}Δ_{y}u = V u.

If we show that for some positive ε ∈ (0, 1]

V ∈ Lloc^{\frac{n}{2−ε}} (Ω),  (4.35)

then the claim follows from Theorems 1 and 2 in [Se]. If n ≥ 3, we have u ∈ Lloc^{\frac{2n}{n−2}} (Ω), by the standard Sobolev-Poincaré inequality, and (4.35) follows from

\frac{4n}{(Q − 2)(2 − ε)} ≤ \frac{2n}{n − 2},

which is satisfied for some positive ε, because Q > n for α > 0.

If n = 2, i.e. m = k = 1, we have u ∈ L^{\frac{2Q}{Q−2}} (ℝ^{2}), by the Sobolev inequality (1.1), and (4.35) follows from

\frac{8}{(Q − 2)(2 − ε)} ≤ \frac{2Q}{Q − 2},

which is satisfied for some positive ε, because Q > 2 for α > 0.

The next Lemma describes the regularity and the boundary behaviour of minimizers of (4.28).

Lemma 4.4. Let m = 1, k ∈ ℕ and α > 0. A non negative function u ∈ H^{1} satisfying (4.34) and solving (4.28) is in C^{2}(−1, 1)∩C^{1}([−1, 1]) and moreover 2u′(x)+ (Q − 2)xu(x) = 0 for x = ±1.

Proof. For it is u ∈ H^{1}, u is continuous in (−1, 1) with weak derivative in L_{loc}^{2}(−1, 1). By Lemma 4.3, we also have u ∈ L^{∞}(−1, 1), and then by (4.34) u can be continuously extended to [−1, 1]. By minimality, u solves in weak sense the Euler equation for problem (4.28), and precisely (up to a multiplicative constant which can be assumed to be equal to 1 upon changing the value of the integral constraint)

(pu′)′ − qu = −pu^{\frac{Q+2}{Q−2}} in (−1, 1).  (4.36)

Then u′′ ∈ L_{loc}^{2}(−1, 1) and consequently u ∈ C^{1}(−1, 1) and thus equation (4.36) implies u ∈ C^{2}(−1, 1). Higher regularity of u depends on α > 0. The function (pu′)′ and hence the function pu′ can be continuously extended to [−1, 1]. Il must be

L := lim_{x→1} p(x)u′(x) = 0.

If it were L > 0, then pu′ > L/2 in some left neighborhood of 1 and thus u(1) = +∞, that is not possible. The case L < 0 can be excluded in the same way. Integrating (4.36) we obtain

u′(x) = \frac{1}{p(x)} ∫_{x}^{1} \{pu^{\frac{Q+2}{Q−2}} − qu\}dt,  (4.37)

and by Hôpital’s rule there exists

u′(1) = lim_{x→1} u′(x) = lim_{x→1} \frac{p(x)u(x)^{\frac{Q+2}{Q−2}} − q(x)u(x)}{p′(x)} = −\frac{Q − 2}{2}u(1).  (4.38)

Analogously, u′(−1) = \frac{Q−2}{2}u(−1).

We prove the even symmetry for extremals of (1.1) when m = 1. This finishes the proof of Theorem 3. The proof is different according to the case k = 1 or k ≥ 2. If k ≥ 2 then the function F introduced in Proposition 4.1 satisfies F ≥ 0. This makes possible a rearrangement argument. If k = 1 then F ≤ 0, and we can use a uniqueness result for problem (1.10).

Theorem 4.5. Let m = 1, k ∈ ℕ and α > 0. Any non negative function u ∈ H^{1} solving (4.28) is even.

Proof. We begin with the case k ≥ 2. We claim that for any non negative function u ∈ C^{1}([−1, 1]) there exists an even function u^{∗} ≥ 0 such that

∫_{−1}^{+1} u^{∗}^{\frac{2Q}{Q−2}} pdx = ∫_{−1}^{+1} u^{\frac{2Q}{Q−2}} pdx,  (4.39)

∫_{−1}^{+1} {|u^{∗}′|^{2}p + u^{∗}^{2}q} dx ≤ ∫_{−1}^{+1} {|u′|^{2}p + u^{2}q} dx.  (4.40)

Moreover, if u ≠ u^{∗} the second inequality is strict.

Let v be the function associated with u as in (4.3). Identity (4.6) with m = 1 and k ≥ 2 reads

∫_{−1}^{+1} {u′^{2}p + u^{2}q} dx = ∫_{−τ}^{+τ} {v′^{2} + F v^{2}}dξ,

SOBOLEV INEQUALITIES FOR WEIGHTED GRADIENTS 23

where F is the function introduced Proposition 4.1. Note that F is non negative, because if m = 1 and k ≥ 2 it is ϑ > 0 in (4.7). Moreover, F is even and strictly increasing on [0, τ).

Given a measurable set I ⊂ (−τ, τ), let I^{∗} = (−|I|/2, |I|/2) be the interval centered at the origin with length equal to |I|, the Lebesgue measure of I. Then

∫_{I} F dξ = ∫_{I∩I^{∗}} F dξ + ∫_{I\I^{∗}} F dξ ≤ ∫_{I∩I^{∗}} F dξ + ∫_{I^{∗}\I} F dξ = ∫_{I^{∗}} F dξ, (4.41)

because |I^{∗} \ I| = |I \ I^{∗}| and F on I^{∗} \ I is less or equal than F on I \ I^{∗}. Moreover, inequality in (4.41) is strict as soon as |I \ I^{∗}| > 0, because of the strict monotonicity of F.

Denote by v^{∗} the symmetric decreasing rearrangement of v. The function v^{∗} is defined by the condition on level sets

{v^{∗} > t} = {v > t}^{∗},  t > 0.  (4.42)

Note that, by (4.41) and (4.42), we have

∫_{−τ}^{+τ}F v^{∗}^{2} dξ = ∫_{0}^{+∞}∫_{{v^{∗}>t^{1/2}}} F dξ dt ≤ ∫_{0}^{+∞}∫_{{v>t^{1/2}}} F dξ dt = ∫_{−τ}^{+τ}F v^{2} dξ (4.43)

and the inequality is strict if v ≠ v^{∗}. Moreover, the following well known facts hold

∫_{−τ}^{+τ} |v^{∗}′|^{2}dξ ≤ ∫_{−τ}^{+τ} |v′|^{2}dξ,  ∫_{−τ}^{+τ} v^{∗}^{\frac{2Q}{Q−2}} dξ = ∫_{−τ}^{+τ} v^{\frac{2Q}{Q−2}} dξ.  (4.44)

Let u^{∗} be defined by v^{∗} as in (4.3). Then our claims (4.39)–(4.40) follow from (4.43) and (4.44), as soon as the integration by parts in (4.21) involving v^{∗} and u^{∗} produces no boundary contribution in (4.22), and precisely we need

lim_{x→±1} pφφ′v^{∗}(ψ) = 0,  (4.45)

where φ and ψ are the functions in (4.4) with m = 1. This holds for v (this is (4.26) with k ≥ 2). By the rearrangement property (4.42), for any x ∈ (0, 1) there is an x̄ ∈ (−1, 1) such that |x̄| ∈ [|x|, 1) and v^{∗}(ψ(x)) ≤ v(ψ(x̄)). For the function |pφφ′| is even and increasing on (0, 1), we have |pφφ′v^{∗}(ψ)(x)| ≤ |pφφ′v^{∗}(ψ)(x̄)| and the claim (4.45) is proved.

If k = 1 we argue in the following way. By Lemma 4.4, a minimizer u of problem (4.28) is in C^{2}(−1, 1) ∩ C^{1}([−1, 1]) and solves the problem

\{ \begin{matrix}(pu′)′ − qu + pu^{\frac{Q+2}{Q−2}} = 0 & in (−1, 1) \\ u > 0 & in [−1, 1] \\ 2u′(x) + (Q − 2)xu(x) = 0 & x = ±1. \end{matrix}  (4.46)

By Theorem 6 in [MM2], the solution u ∈ C^{1}([−1, 1]) ∩ C^{2}(−1, 1) of problem (4.46) is unique and thus even. The proof of the theorem is complete.

Appendix

Proof of Proposition 2.1. We begin with (2.12). By the coarea formula (see [F, 3.2.12])

μ(Λ_{ζ,t}) = ∫_{Λ_{ζ,t}} \frac{1}{r^{k+1}} drdη = ∫_{t}^{∞}∫_{∂Λ_{ζ,s}} \frac{1}{|∇_{rη}U_{ζ}|} \frac{dH^{k}}{r^{k+1}} ds = ∫_{t}^{∞}∫_{∂Λ_{ζ,s}} \frac{1}{|∇_{H}U_{ζ}|}dν ds. (4.1)

Here, ∇_{rη}U_{ζ} denotes the gradient of U_{ζ} in the variables r and η. Indeed, note that r|∇_{rη}U_{ζ}| = |∇_{H}U_{ζ}| and r^{−k}H^{k} = ν. Since u ∈ A satisfies (1.4), |∇_{H}U_{ζ}| is bounded from below by a positive constant and from (4.1) it follows that t ↦ φ_{ζ}(t) is locally Lipschitz for t > 0. Upon differentiating (4.1), we get (2.12).

We prove (2.13). Without loss of generality, assume m = 2, ζ = (1, 0) and v = (0, 1). Let ζ_{ϑ} = (cos ϑ, sin ϑ), ϑ ∈ [0, 2π), and write – with abuse of notation – Λ_{ϑ} = Λ_{ζ_{ϑ},t}. Then we have

∇_{v}φ_{t}(ζ) = lim_{ϑ→0} \frac{μ(Λ_{ϑ}) − μ(Λ_{0})}{ϑ}.

Let

\begin{aligned}Ω_{ϑ} & = \{(rζ_{s}, η) ∈ ℝ^{n} : 0 < s < ϑ, (r, η) ∈ Λ_{s}\}, \\ S_{ϑ} & = \{(rζ_{s}, η) ∈ ℝ^{n} : 0 < s < ϑ, (r, η) ∈ ∂Λ_{s}\}, \end{aligned}

and denote by N the exterior unit normal to ∂Ω_{ϑ}. The vector field Y (ξ, η) = |ξ|^{−k−2}(−ξ_{2}, ξ_{1}, 0) is divergence free and ⟨Y, N⟩ = |ξ|^{−k−1} on the part of ∂Ω_{ϑ} corresponding to Λ_{ϑ} and ⟨Y, N⟩ = −|ξ|^{−k−1} on the part of ∂Ω_{ϑ} corresponding to Λ_{0}. The divergence theorem therefore implies

μ(Λ_{ϑ}) − μ(Λ_{0}) = − ∫_{S_{ϑ}}⟨Y, N⟩dH^{k+1},

where H^{k+1} is the (k + 1)-dimensional Hausdorff measure in ℝ^{k+2}. From this formula we see that ϑ ↦ μ(Λ_{ϑ}) is Lipschitz. Then, for general m > 1, ζ ↦ φ_{t}(ζ) is Lipschitz on 𝕊^{m−1} for any t > 0. By Rademacher theorem, φ is differentiable almost everywhere on 𝕊^{m−1} × ℝ^{+}.

We claim that

lim_{ϑ→0} \frac{1}{ϑ} ∫_{S_{ϑ}}⟨Y, N⟩dH^{k+1} = − ∫_{∂Λ_{ζ,t}} \frac{∇_{v}U_{ζ}}{|∇_{H}U_{ζ}|}dν.  (4.2)

First of all, note that on S_{ϑ} we have N = −∇U/|∇U| and thus

⟨Y, N⟩ = −\frac{⟨(−ξ_{2}, ξ_{1}), ∇_{ξ}U⟩}{|ξ|^{k+2}|∇U|}.

It is enough to prove (4.2) in a local form. Consider a (k + 1)-dimensional surface S_{ϑ} parameterized by a Lipschitz function f : [0, ϑ] × [0, 1]^{k} → ℝ^{k+2} of the form f = (r cos s, r sin s, η) where r = r(s, η) is Lipschitz. Let T be the k-dimensional surface parameterized by g(η) = f(0, η), η ∈ [0, 1]^{k}. Applying the area formula to the integral on S_{ϑ} (see [F, 3.2.3]), taking the limit in ϑ, and finally applying the area formula to the integral over T below, we get

lim_{ϑ→0} \frac{1}{ϑ} ∫_{S_{ϑ}}⟨Y, N⟩dH^{k+1} = − ∫_{T} \frac{∇_{v}U_{ζ}}{|∇U|} (\frac{det(Jf^{T} Jf)}{det(Jg^{T} Jg)})^{1/2} |ξ|^{−k−2}dH^{k},

where Jf and Jg are the differential matrices of f and g, and Jf^{T} stands for the transposed matrix. Here, Jf is evaluated on T.

A short computation furnishes

det(Jf^{T} Jf) = |\frac{∂r}{∂s}|^{2} + r^{2} (1 + |\frac{∂r}{∂η} |^{2}) , det(Jg^{T} Jg) = 1 + |\frac{∂r}{∂η} |^{2},

and moreover, differentiating the identity U(r cos s, r sin s, η) = 0, we find

|\frac{∂r}{∂η} |^{2} = \frac{|∇_{η}U|^{2}}{|∇_{R}U|^{2}} , |\frac{∂r}{∂s}|^{2} = r^{2} \frac{|∇_{T} U|^{2}}{|∇_{R}U|^{2}} .

Thus

(\frac{det(Jf^{T} Jf)}{det(Jg^{T} Jg)})^{1/2} = |ξ|^{2} \frac{|∇U|}{|∇_{H}U|}

and the claim (4.2) follows.

Proof of Proposition 2.2. Without loss of generality we assume μ(Λ_{ζ,0}) = +∞ for all ζ ∈ 𝕊^{m−1}. The distribution function φ(ζ, t) = μ(Λ_{ζ,t}) is continuous and strictly decreasing in t. The inverse function ψ_{ζ} : [0, +∞) → (0, max U_{ζ}] is defined by the identity

ψ(ζ, φ(ζ, t)) = t.  (4.3)

We write ψ_{ζ}(s) = ψ(ζ, s). Note that from (4.1) it follows that for any ε > 0 there is a constant C > 0 such that for ε < t_{1} < t_{2} < max U_{ζ} − ε

φ_{ζ}(t_{1}) − φ_{ζ}(t_{2}) ≥ C(t_{2} − t_{1}),

and thus also ψ_{ζ} is locally Lipschitz for s ∈ (0, +∞).

Now, from the Lipschitz continuity of φ, it easily follows that ψ : 𝕊^{m−1} × ℝ^{+} → ℝ is locally Lipschitz continuous. The function (r, ζ, η) ↦ U_{ζ}^{∗}(r, η) = ψ_{ζ}(μ(B_{d(r,η)})) is continuous for r > 0 and is locally Lipschitz continuous for r > 0 and (r, η) ≠ (1, 0). The statement of Proposition 2.2 immediately follows from (2.11).

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Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Belzoni 7, I–35131 Padova, Italy

  E-mail address: monti@math.unipd.it