# A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space \({{\mathbb {H}}}^N\)

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## Abstract

We prove a subset of inequalities of Caffarelli–Kohn–Nirenberg type in the hyperbolic space \({{\mathbb {H}}^N}, N\ge 2\), based on invariance with respect to a certain nonlinear scaling group, and study existence of corresponding minimizers. Earlier results concerning the Moser–Trudinger inequality are now interpreted in terms of CKN inequalities on the Poincaré disk.

## Keywords

Scale invariance CKN inequalities Concentration compactness Weak convergence Hyperbolic space Poincaré ball Hardy inequalities## Mathematics Subject Classification

35J20 35J61 35J75 58J05 46B50## 1 Introduction

In this paper, we study inequalities that define the embedding for the (homogeneous) Sobolev space \(H^{1}({\mathbb {H}}^N)\) of the hyperbolic space \({\mathbb {H}}^N, N\ge 2\), into functional spaces of Lebesgue and Lorentz type, including their radial counterpart, with the radiality understood in terms of the Poincaré ball model. The space \(H^{1}({\mathbb {H}}^N)\) is defined as the completion of \(C_c^\infty ({{\mathbb {H}}^N})\) with respect to the quadratic form of the Laplace–Beltrami operator on \({{\mathbb {H}}^N}\) (see below for details). Similar inequalities in the Euclidean case are a subset of the celebrated Caffarelli–Kohn–Nirenberg (CKN for short) inequalities (see [17], Theorem 4.1, or [10]) and we refer the reader to the paper of Dolbeault, Esteban and Loss [15] for recent results concerning sharp estimates of constants and radiality of minimizers). CKN inequalities are distinguished by various optimality properties, including scaling invariance.

### 1.1 Hyperbolic scaling invariance in the two-dimensional case

*N*-dimensional ball. The inequalities for general values \(p\ge N\) are derived, without losing scaling invariance, from the corresponding inequalities for \(p=N\) and \(p=\infty \) by means of Hölder inequality. For the case \(N=2\), which is considered in the present paper, we have the Leray inequality ([18]) for the case \(p=2\) and the pointwise estimate for radial functions (which also implies the Trudinger inequality for general functions) for the case \(p=\infty \). Leray inequality

### 1.2 Nonlinear scalings for Laplace–Beltrami operators by levels of fundamental solution

Similarly to the original CKN inequalities, which are invariant (up to a normalization factor) with respect to linear scalings \( \{u(x)\mapsto u(tx)\}_{t>0}\), their counterparts in [3], in restriction to radial functions, are invariant up to normalization with respect to nonlinear scalings \( \{u(r)\mapsto u(r^s)\}_{s>0}\). In this paper, we show that this transformation is a particular case of the scaling transformation \(r\mapsto G^{-1}(\lambda G(r))\) where *G*(*r*) is the radial fundamental solution (which can be obviously taken here up to an arbitrary scalar multiple) of the Poisson equation for the hyperbolic Laplace–Beltrami operator. In particular, \(G(r)=\log \frac{1}{r}\) for the Dirichlet Laplacian on the unit disk, and thus, in the Poincaré disk coordinates, also for the Laplace–Beltrami operator on \({{\mathbb {H}}}^2\). The fundamental solution \(G(r)=\frac{1}{r^{N-2}}\) plays a similar role in the Euclidean case for \(N\ge 3\): with this choice of *G* the formula \(r\mapsto G^{-1}(\lambda G(r))\) means multiplication of *r* by a power of \(\lambda \).

The role of fundamental solution in the scaling invariance and furthermore, the role of its square root as a generalized ground state, which we discuss below, is partly motivated by the paper of Adimurthi and Sekar [1].

### 1.3 Square root of fundamental solution as a ground state: case \(p=2\)

It is well known that the Hardy inequality (2) has no minimizer, but any sequence \((u_n)\), that minimizes the quadratic form *Q*(*u*) in (2) under a constraint \(\int _Ku=1\), where \(K\subset {\mathbb {R}}^N\setminus \{0\}\) is an open relatively compact set, converges in \(H^1_{\mathrm {loc}}({\mathbb {R}}^N\setminus \{0\})\) to the unique (up to a constant multiple) positive solution \(\sqrt{G(r)}\), with \(G(r)=\frac{1}{r^{N-2}}\), of the corresponding Euler–Lagrange equation, called the generalized ground state, or virtual bound state. By the ground state alternative of [22], Theorem 1.5 (see also [23], Theorem 1.6), existence of the virtual bound state implies that there is no nonzero nonnegative measurable function *W* such that \(Q(u)\ge \int Wu^2\), i.e., the Hardy potential is optimal. A general result in [14] states that, under general conditions on the elliptic operator, the square root of the *positive minimal Green function* is always a generalized ground state. For the sake of consistency of the paper, instead of applying definitions and quoting the exact statement from [14], we give a short direct proof that \(\sqrt{G}\) is a generalized ground state in our case. This not only provides the best constant in the hyperbolic counterpart of the Hardy inequality, but also assures that the potential in it cannot be improved.

### 1.4 Non-radial case with \(p\in (2, \frac{2N}{N-2}]\)

*u*, relative to the Lebesgue measure on

*B*, derived from (4) and the Polia–Szegö inequality (since the latter holds also with respect to rearrangements relative to the Riemannian measure on \({{\mathbb {H}}}^2\) ([6]), inequality (6) holds also when the rearrangement \(u^\#\) is defined relative to that measure). Note that the left-hand side is stronger than \(\sup _{0<r<1}\frac{u^\#(r)}{\sqrt{\log \frac{e}{r}}}\), which is a quasinorm on the standard Zygmund scale, and is known (see, e.g., [7]) to be equivalent to the Orlicz norm of the functional of critical growth \(\int _B e^{a u^2}dx\) for the Sobolev space \(H^{1}_0(B)\).

When \(N\ge 3\) and \(p\in [2,2^*]\), embeddings of the space \(\dot{H}^1({\mathbb {R}}^N)\) (the completion of \(C_c^\infty ({\mathbb {R}}^N)\) in the norm \(\Vert \nabla \cdot \Vert _2\)) into weighted \(L^p\)-spaces in the non-radial case of CKN inequalities follow from those in the radial case by means of standard rearrangement argument, namely the Polia–Szegö inequality and the Hardy–Littlewood inequality. The latter, however, applies only when the weight in the Lebesgue integral is non-increasing, which is the case only if \(p\le 2^*\). Following the similar approach in the hyperbolic case, we do not have the decresing weight, but instead, for \(N\ge 3, p\in [2,2^*]\), instead of embeddings of \(H^{1}({\mathbb {H}}^N)\) into weighted \(L^p\)-spaces, at embeddings into certain rearrangement-invariant quasi-Banach spaces, which we then identify as intersections of Lorentz spaces \(L^{2,p}({{\mathbb {H}}^N})\cap L^{2^*,p}({{\mathbb {H}}^N})\) (in the case \(N=2\) we have an intersection of spaces of Zygmund–Lorentz type). These intersections are strictly smaller than \(L^p\), and thus these embeddings refine the embedding of \(H^{1}({\mathbb {H}}^N)\) into \(L^p({{\mathbb {H}}^N})\) from [19] (Note that in the Euclidean case there are no embeddings \(\dot{H}^1({\mathbb {R}}^N)\hookrightarrow L^p({\mathbb {R}}^N)\) for \(p\ne 2^*\).)

Scale-invariant inequalities of the present paper follow several other previously established inequalities of Sobolev type on the hyperbolic space. In particular, we would like to mention the Poincaré-Sobolev inequality of Mancini and Sandeep ([19], (1.2), which, as they have shown, by writing it in the half-space coordinates, follows from the Sobolev–Hardy–Mazy’a inequality, which is in turn equivalent to a subset of the original CKN inequalities by means of the ground state transform, also known as Picone identity); as well as related inequalities in [8] and [9]. Inequalities with weight play an important role in the study of Hénon-type equations in hyperbolic space, and a few such embeddings have been developed in [11] and [16]. The scale-invariant inequalities that we prove are significantly sharper than some of those found in literature. In particular, (30) is stronger than (1.1) in [19], while the weight in the embedding in [11], Lemma 2, case \(\alpha =0\), which uses hyperbolic distance from the origin \(d(x)=\log \frac{1+r}{1-r}\), behaves as a positive power of *r* at the origin and as a negative power of \(|\log (1-r)|\) at \(r =1\), while the weight (12) in our embeddings (31) and (26) has the power singularity both at the origin and at \(r=1\), see (19)–(20).

The paper is organized as follows. In Sect. 2 we recall definitions related to the hyperbolic space. In Sect. 3 state the main results. In Sect. 4 we prove the inequalities of CKN type. In Sect. 5 we prove existence of minimizers in the hyperbolic CKN inequalities and study related compactness issues. A refined analysis of concentration compactness phenomena, in the form of profile decomposition, is given for sequences of radial functions. In Appendix, we provide cross-references between the results of this paper and the results in [3] for the case \(N=2\), one of them being representation of the Moser–Trudinger inequality as a two-dimensional case of the Sobolev embedding for the hyperbolic space.

## 2 Preliminaries

### 2.1 Poincaré ball

*B*in \({\mathbb {R}}^N\) centered at the origin and equipped with the metric

*r*refers to \(\sqrt{\sum _{i=1}^Nx_i^2}\), the Euclidean distance of a point \(x\in B\) from the origin.

Notation \(\Vert u\Vert _p\) will refer to the \(L^p({{\mathbb {H}}^N},{\mathrm d}V_{{\mathbb {H}}})\)-norms. Norms with weight *W* relative to the measure on \({{\mathbb {H}}^N}\) for the spaces \(L^p({{\mathbb {H}}^N},W{\mathrm d}V_{{\mathbb {H}}})\), \(1\le p<\infty \), be denoted as \(\Vert u\Vert _{p,W}\). Reference to the weight *W* when \(W=1\) will be omitted from notation in some instances. Notation \(\Vert u\Vert _{\infty ,W}\) will refer to the supremum norm for the product |*u*(*x*)*W*(*x*)|, and the corresponding space will be denoted as \(L^\infty ({{\mathbb {H}}^N},W)\).

The Sobolev space \(H^{1}({\mathbb {H}}^N)\) is defined as a completion of \(C_c^\infty \) in the norm defined by the quadratic form above. By \(H^{1}_r({\mathbb {H}}^N)\) we will denote the subspace of radially symmetric functions of \(H^{1}({\mathbb {H}}^N)\) (which is the same as functions in \({{\mathbb {H}}^N}\) which are radial with respect to the hyperbolic distance from 0.). We will denote \(u \in H^{1}_r({\mathbb {H}}^N)\) by its radial representative \(u:[0,1)\rightarrow {\mathbb {R}}.\)

We will denote by \(\omega _{N-1}\) the surface measure of the unit sphere \(S^{N-1} \subset \mathbb {{\mathbb {R}}}^N\).

### 2.2 Scaling by fundamental solution

*G*on \({\mathbb {H}}^N\) is \((0,\infty )\) we define the following multiplicative transformation group on \(H^{1}_r({\mathbb {H}}^N)\) by means of the change of radial variable in the Poincaré ball coordinates:

*G*the fundamental solution \(\frac{C(N)}{r^{N-2}}\) of the Laplace operator in \({\mathbb {R}}^N\). Action of the linear scaling on functions on \({\mathbb {R}}^N\), under suitable normalizations, preserves the right- and the left-hand sides in the original CKN inequalities including the quadratic form of the Laplace operator. Transformation (9) similarly preserves the quadratic form of the Laplace–Beltrami operator, if only in restriction to radial functions, and, furthermore, one can show by elementary computations based on change of variable under the integral that every radial diffeomorphism with this property is necessarily of the form (9).

When \(N=2\), the Laplace operator does not have a positive fundamental solution on the whole on \({\mathbb {R}}^2\), but the same construction on the unit disk *B*, using the fundamental solution \(G(r)=\frac{1}{2\pi }\log \frac{1}{r}\), defines an automorphism \(\rho _t(r)=r^t\) of *B*, whose action preserves, up to a normalization factor, the quadratic form of the Laplacian on *B* evaluated on radial functions (see [2, 3]). There also exists a family of maps, which we write in the notation of a complex variable as \(z\mapsto z^m, m\in {\mathbb {N}}\), whose action preserves, up to normalization, the quadratic form of the Laplacian on *B* for general functions as well (see [5]). This case is appended to the present paper, via the Poincaré ball model, as the case of \({{\mathbb {H}}}^2\).

Indeed, for general *N*, we also have

### Proposition 2.1

### Proof

*u*is radial. Let \(\rho \) be a general increasing \(C^1\)-function that maps [0, 1] bijectively onto itself, then changing the variable as \(r=\rho (t)\) and \(v(t)= u(r)\) we get

*G*is a diffeomorphism between (0, 1) and \((0,\infty )\), this defines the function \(\rho \) as in (9) with the required isometric property, and, by necessity, it is the only radial function with this property. \(\square \)

### Proposition 2.2

### Proof

*t*,

The following statement also follows by direct computation.

*G*and \(V_p\).

### 2.3 Lorentz spaces involved in the estimates

*u*. We recall that \(L^{p,p}\) coincides with the Lebesgue space \(L^p\).

*x*|, centered at the origin.

### Theorem 2.3

### Proof

*C*such that for every measurable functions

*u*and

*v*one has

*f*/

*g*has a positive limit at that point. Omitting multiplicative constants, we have \(t=V_{{\mathbb {H}}}(B_r)\sim r^N\) at zero and \(t\sim (1-r)^{1-N}\) at infinity, so that \(r\sim t^{1/N}\) near zero and \(1-r\sim t^{-\frac{1}{N-1}}\) near infinity. Assume first that \(N\ge 3\). From (19) we have \({\hat{V}}_q(t)\sim r^{-N(1-q/2^*)}\sim t^{-(1-q/2^*)}\) near zero. From (20) we have \({\hat{V}}_q(t)\sim (1-r)^{-\frac{(N-1)(q-2)}{2}}\sim t^{\frac{(q-2)}{2}}\) near infinity. Both expressions satisfy doubling property (25). If \(N=2\), we have by (18) \({\hat{V}}_q(t)=\frac{(1-r^2)^2}{r^2(\log \frac{1}{r})^\frac{q+2}{2}}\). This gives us \({\hat{V}}_q(t)\sim \frac{1}{t(\log |t|)^\frac{q+2}{2}}\) near zero and \({\hat{V}}_q(t)\sim \frac{1}{t^2(\log t)^\frac{q+2}{2}}\) near infinity. Since the function \(1/\log |t|\) easily verifies doubling property (25), so does \({\hat{V}}_q(t)\). We conclude that (23) defines a quasinorm, from which it is immediate that condition (23) defines a linear space.

2. Calculations in the previous step for \(N\ge 3\) give that \({\hat{V}}_q(t) \sim t^{q/2^*-1}\) at zero and is \({\hat{V}}_q(t) \sim t^{q/2-1}\) at infinity. The exponents in the asymptotics are the same as in the weights for the Lorentz spaces \(L^{2^*,q}\) and \(L^{2,q}\), respectively. When \(2\le q\le 2^*\), the quantity \(t^{q/2^*-1}\) dominates \(t^{q/2-1}\) at zero, while \(t^{q/2-1}\) dominates \(t^{q/2^*-1}\) at infinity, and therefore \(C_1(t^{q/2^*-1}+t^{q/2-1})\le {\hat{V}}_q(t)\le C_2(t^{q/2^*-1}+t^{q/2-1})\). This proves the second assertion of the theorem.

3. The last assertion of the theorem follows from the fact that the weight \(t^{\frac{2}{q}-1}+t^{\frac{2^*}{q}-1}\) is bounded away from zero for \(2<q<2^*\), which implies that the \(L^q\)-norm is bounded by the quasinorm of \(L^{2,q}\cap L^{2^*,q}\). \(\square \)

### Remark 2.4

When \(N=2\), an argument similar to the step 2 of the proof above shows the space defined by the quasinorm \(\Vert \cdot \Vert _{q,V_q}, 2\le q \le \infty \), is an intersection of two spaces of Lorentz–Zygmund type. For the sake of brevity, we will denote this space as \({\mathcal {L}}^q({{\mathbb {H}}}^2)\). When \(q=\infty \) these spaces coincide and the quasinorm of \({\mathcal {L}}^\infty ({{\mathbb {H}}}^2)\) is \(\sup _{0<r<1}\frac{u^\#(r)}{\sqrt{\log \frac{1}{r}}}\).

## 3 Main results

The main results of this paper are the inequalities below. They all use weights \(V_p, 2\le p\le \infty \), defined in (12).

### Theorem 3.1

### Remark 3.2

### Remark 3.3

To illustrate optimality of the inequality (26), consider its restriction to \(H^{1}_r({\mathbb {H}}^N)\). If *V*(*r*) is a continuous function and \(\frac{V(r)}{V_p(r)}\rightarrow +\infty \) when \(r\rightarrow 0\) or \(r\rightarrow 1\), then the inequality (26) with \(V_p\) replaced by *V* will be false. Indeed, one can fix any nonzero function \(w\in H^{1}_r({\mathbb {H}}^N)\) and, by changing the radial variable under the integral defining the \(L^p-norm\), easily find that \(\sup _{t>0}\Vert g_tw\Vert _{p,V}= \infty \).

### Theorem 3.4

*W*instead of \(V_2\).

Note that for \(N=2\) this is the classical Leray inequality, [18].

### Remark 3.5

*r*, so that \(V_2(r)\ge V_2(1)\). From the definition of

*G*we have \(G(r)=\frac{2^{N-2}(1-r)^{N-1}}{N-1}(1+o_{r\rightarrow 1}(1))\), and thus

### Theorem 3.6

The case \(p=\infty \) has been already proved by Hasegawa ([16], formula (A.3), Lemma A.1.)

We have the following result concerning minimizers of the above inequality.

### Theorem 3.7

Note that the function (33) is a generalization of the test function used by Moser [21] in the case \(N=2\).

Our further results include Theorem 5.1 on cocompactness of embeddings of \(H^{1}_r({\mathbb {H}}^N)\) relative to the group (10), Theorem 5.3 on structure of unbounded sequences in \(H^{1}_r({\mathbb {H}}^N)\), Theorem 5.4 on compactness of embeddings of the inhomogeneous counterpart of \(H^{1}_r({\mathbb {H}}^N)\) into \(L^p\)-spaces, and an elementary Theorem 5.5 on compactness of embeddings in presence of ”sub-Hardy” potentials.

## 4 Proofs of the inequalities

Let us look in more detail at the weight \(V_2(r)=\left( \frac{f(r)(1-r^2)}{G(r)}\right) ^2\) for \(N\ge 3\). At \(r=1\) the value of \(V_2\) is finite and positive, and near zero \(V_2(r)=\frac{(N-2)^2}{r^2}(1+o_{r \rightarrow 0}(1))\). In particular, \(V_2(r)=\left( \frac{(1+r)^2}{r}\right) ^2\) for \(N=3\). In other words, the weight \(\frac{1}{16}V_2\) that appears in the theorem below is qualitatively similar to the weight \(\frac{(N-2)^2}{4r^2}\) from the usual radial Hardy inequality in \({\mathbb {R}}^N\). We are going to use the following well-known identity (see, e.g., [13]).

### Lemma 4.1

*V*be a continuous function on \(\Omega \). Let \(A(x), x\in \Omega \), be a symmetric real-valued positive matrix with continuous coefficients. If \(v\in C^2(\Omega )\) is a positive solution of the equation (understood in the sense of weak derivatives) \(-\nabla \cdot A(x)\nabla v(x)=V(x)v(x), x\in \Omega \), then the following identity holds for any \(u\in C_c^\infty (\Omega )\):

### Proof of Theorem 3.4

*null sequence*(Definition 1.1, [22]), it admits no spectral gap. Absence of spectral gap (Definition 1.2, [22]) is exactly the second assertion of our theorem. A null sequence is a sequence \((v_n)\) which converges to a positive solution

*v*of \(Q'(v)=0\) uniformly on compact subsets, and satisfies \(Q(v_n)\rightarrow 0\). Thus it suffices to construct a null sequence. Let, \(r_1=G^{-1}(1)\), and let

*G*at 0 and 1 provided at the end of Sect. 2.2 that yield \(G(r)^{-2\epsilon _n(r)}\rightarrow 0\) as \(r\rightarrow 0\) or \(r\rightarrow 1\). \(\square \)

### Proof of Theorem 3.6

The exact constant \(c_r(N,2)\) is realized on the normalized sequence \(v_n\) from (37). An easy computation shows that \(\Vert \nabla _{{\mathbb {H}}}v_n\Vert _2\rightarrow \infty \) and thus \(Q(w_n)\rightarrow 0\) where \(w_n=v_n/\Vert \nabla _{{\mathbb {H}}}v_n\Vert _2\). This immediately implies that \(\frac{1}{16}\int _{{\mathbb {H}}^N}|w_n|^2V_2{\mathrm d}V_{{\mathbb {H}}}=\Vert \nabla _{{\mathbb {H}}}v_n\Vert _2^2-Q(w_n)\rightarrow 1\).

### Proof of Theorem 3.1

Inequality (26) follows from (31) once we take into account the Polya-Szegö inequality for rearrangements in the hyperbolic space, \(\Vert \nabla _{{\mathbb {H}}}u^\#\Vert _2\le \Vert \nabla _{{\mathbb {H}}}u\Vert _2\) (see [6]). Embedding of \(H^{1}({\mathbb {H}}^N)\) into \(L^{2,p}\cap L^{2^*,p}\) follows from Theorem 2.3. \(\square \)

## 5 Cocompactness, profile decomposition, and minimizers

In this section we follow the framework of [3] (which, implicitly, studied the case \(N=2\) of the present paper).

We recall that an embedding of a Hilbert space *H* into a Banach space *Y* is called *cocompact relative to a group of unitary operators* *D* if any sequence \((u_n)\subset H\;D\)-weakly convergent to zero (i.e., is such that for any \((g_n)\subset D, g_nu_n\rightharpoonup 0\)), converges in the norm of *Y*.

### Theorem 5.1

Let \(N\ge 2\). For any \(p\in (2, \infty ]\), the embedding \(H^{1}_r({\mathbb {H}}^N)\hookrightarrow L^p({\mathbb {H}}^N;V_p{\mathrm d}V_{{\mathbb {H}}})\) is cocompact relative to the group \(D_0\).

The argument is an elementary generalization of the proof for the case \(N=2\) in [3], Lemma 3.3, Lemma 3.4 and their interpretation in Remark 3.5. The main step in the proof of the theorem is the case \(p=\infty \) which is a trivial generalization of [3], Lemmas 3.3.

### Proposition 5.2

The embedding \(H^{1}_r({\mathbb {H}}^N)\hookrightarrow L^\infty ({{\mathbb {H}}^N},V_\infty )\) is cocompact relative to the group (10).

### Proof

We can now prove Theorem 5.1.

### Proof

As a consequence of Theorem 5.1, we have the following structural result for general bounded sequences in \(H^{1}_r({\mathbb {H}}^N)\).

### Theorem 5.3

*k*.

### Proof

This theorem is just a particular case of the theorem on profile decompositions in general Hilbert space, Theorem 3.1 in [27], for \(H^{1}({\mathbb {H}}^N)\) equipped with the group *D* (the unconditional convergence has been explicitly stated only later, in a more general result, Theorem 5.5 (see Definition 2.5) in [24]). For more details about application of the general theorem to the group *D*, we refer the reader to [3]. \(\square \)

We can now prove Theorem 3.7.

### Proof

1. Let \((u_n)\) be a minimizing sequence for (32), that is, \(\Vert u_n\Vert _{p,V_p}= 1\) and \(\Vert \nabla _{{\mathbb {H}}}u_n\Vert _{2}^2\rightarrow c_r(p)\). Since both norms in (32) are *D*-invariant, for any sequence \(j_n\in 2^{\mathbb {Z}}, (g_{j_n}u_n)\) is also a minimizing sequence. Without loss of generality, we may assume that \((u_n)\) has a weak limit \(u\ne 0\). Indeed, if \((g_{j_n}u_n)\rightharpoonup 0\) for any \(j_n\in 2^{\mathbb {Z}}\), then by Theorem 5.1 \(u_n\rightarrow 0\) in \(L^p({{\mathbb {H}}^N},V_p{\mathrm d}V_{{\mathbb {H}}})\), which is a contradiction. We may then pass to a subsequence of \((g_{j_n}u_n)\) that has a nonzero weak limit and rename it as \(u_n\).

*u*is a minimizer. Furthermore, by weak convergence and convergence of the norm, we have \(u_n\rightarrow u\) in \(H^{1}_r({\mathbb {H}}^N)\).

*w*is a minimizer for (38), \(\Vert \nabla _{{\mathbb {H}}}w_n\Vert _2\rightarrow \Vert \nabla _{{\mathbb {H}}}w\Vert _2\), and, consequently, \(w_n\rightarrow w\) and \(g_{t_n}u_n\rightarrow w\) in the norm of \(H^{1}_r({\mathbb {H}}^N)\). Furthermore, we have

*w*satisfies the additional constraint in the left-hand side.

Consider now the infimum in the left-hand side of (46). It is necessarily attained on a function which is harmonic on \((0,r_1)\) and on \((r_1,\infty )\), which, by the requirement that its squared gradient is integrable, equals necessarily to the function (33).

We conclude that any minimizing sequence for (38) admits a renamed subsequence and a sequence of positive numbers \((t_n)\) such that \(g_{t_n}u_n\rightarrow w\) in the norm of \(H^{1}_r({\mathbb {H}}^N)\).

Furthermore, if \({\tilde{w}}\) is any minimizer for (38), the constant minimizing sequence \(({\tilde{w}})_{n\in {\mathbb {N}}}\) admits a sequence of positive numbers \((t_n)_{n\in {\mathbb {N}}}\) such that \(g_{t_n}{\tilde{w}}\rightarrow w\) in the \(H^{1}_r({\mathbb {H}}^N)\)-norm. Then necessarily \(t_n\rightarrow t\) with some \(t>0\) and \({\tilde{w}}=g_\frac{1}{t}w\). \(\square \)

As a consequence of the profile decomposition we have the following compactness result. The intersection space below is assumed to have the scalar product that is a sum of scalar products of constituent spaces.

### Theorem 5.4

Note that the exponent in the right-hand side of (47) is positive, so the weight \(W_p\) is bounded away from zero and goes to infinity at \(r=1\).

### Proof

*n*such that \((j^{(n)}_k)_k\) has a subsequence convergent to zero, the corresponding profile \(w^{(n)}\) will be zero. Indeed, taking into account that \(r^2V_2(r)\) is bounded away from zero (since it follows that \(\frac{f(r)(1-r^2)}{G(r)}\ge \frac{C}{r}\) from (16) and (17) for \(N\ge 3\), and from (14) for \(N=2\)), we have

We add now a more elementary compactness result.

### Theorem 5.5

*V*(

*r*) be a measurable function such that, for some \(C>0\) and all \(r>0, |V(r)|\le CV_p(r)\). If

The proof of this statement is entirely similar to that for Theorem 1.1 in [12], is left to the reader. Note that this result is sharp in the sense that if \(V=V_p\), then the embedding is not compact, due to invariance of both norms with respect to the non-compact group *D*.

## Notes

### Acknowledgements

The authors express their sincere gratitude to the referee whose insightful remarks led to a significant improvement of the results in paper.

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