# The concentration-compactness principle for the nonlocal anisotropic
$p$-Laplacian of mixed order

Jamil Chaker  Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld,
Germany<EMAIL_ADDRESS>,  Minhyun Kim  Fakultät für
Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany  minhyun.kim@uni-
bielefeld.de and  Marvin Weidner  Fakultät für Mathematik, Universität
Bielefeld, 33615 Bielefeld, Germany<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the existence of minimizers of the Sobolev quotient
for a class of nonlocal operators with an orthotropic structure having
different exponents of integrability and different orders of
differentiability. Our method is based on the concentration-compactness
principle which we extend to this class of operators. One consequence of our
main result is the existence of a nontrivial nonnegative solution to the
corresponding critical problem.

###### Key words and phrases:

concentration-compactness principle, anisotropy, orthotropic structure,
nonlocal operator, Sobolev inequality, quasilinear equation, $p$-Laplacian.

###### 2020 Mathematics Subject Classification:

35R11, 35A01, 49J35, 46E35, 46B50.

Jamil Chaker gratefully acknowledges support by the DFG through CRC 1283.
Minhyun Kim and Marvin Weidner gratefully acknowledge financial support by the
DFG through IRTG 2235.

##  1\. Introduction

In this paper, we study variational problems for the nonlocal anisotropic
$p$-Laplacian of mixed order. For $x\in\mathbb{R}^{n}$, let

(1.1)
$\sum_{i=1}^{n}(-\partial_{ii})^{s_{i}}_{p_{i}}u(x):=\sum_{i=1}^{n}s_{i}(1-s_{i})\int_{\mathbb{R}}\frac{|u(x)-u(x+he_{i})|^{p_{i}-2}(u(x)-u(x+he_{i}))}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h,$

where $s_{1},\dots,s_{n}\in(0,1)$ and $p_{1},\dots,p_{n}>1$. This operator can
be seen as a nonlocal analog of the anisotropic $p$-Laplacian

(1.2)
$\sum_{i=1}^{n}(-\partial_{ii})_{p_{i}}u(x)=-\sum_{i=1}^{n}\frac{\partial}{\partial
x_{i}}\left(\left|\frac{\partial u(x)}{\partial
x_{i}}\right|^{p_{i}-2}\frac{\partial u(x)}{\partial x_{i}}\right).$

Local operators with such an orthotropic structure are well known in the
literature and there are several results related to this type of operators,
see [36, 41, 8, 20, 21, 26, 10, 16, 22, 24, 11] and the references therein. We
can read off (1.1), that the operator under consideration has on the one hand
different exponents of integrability and on the other hand different orders of
differentiability. Considering such operators, it is natural to work on
anisotropic Sobolev spaces. The aim of this paper is to study the existence of
minimizers of the Sobolev quotient for the operator in (1.1). As a consequence
of the existence of a minimizer, we get the existence of a solution to the
corresponding critical problem. One of the main auxiliary results in this
paper is a robust Sobolev-type inequality for the operator
$\sum_{i=1}^{n}(-\partial_{ii})^{s_{i}}_{p_{i}}$.

The approach we use in this paper is based on the concentration-compactness
principle (CCP) that provides an important tool to prove relative compactness
of minimizing sequences. It has been introduced by P.-L. Lions in a series of
papers, see [37, 38, 39, 40]. Before we address known results from the
literature and describe the strategy of our paper, we formulate the main
results and assumptions of the present work.

Given $s_{1},\dots,s_{n}\in(0,1)$ and $p_{1},\dots,p_{n}>1$, we define

$\bar{s}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{s_{i}}\right)^{-1},\quad\overline{sp}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{s_{i}p_{i}}\right)^{-1},\quad\text{and}\quad
p^{\ast}=\frac{n\overline{sp}/\bar{s}}{n-\overline{sp}}.$

######  Assumption 1.

Given $s_{1},\dots,s_{n}\in(0,1)$ and $p_{1},\dots,p_{n}>1$, we assume

$\overline{sp}<n\quad\text{ and }\quad
p_{\max}:=\max\\{p_{1},\dots,p_{n}\\}<p^{\ast}.$

Let $\vec{s}=(s_{1},\dots,s_{n})$ and $\vec{p}=(p_{1},\dots,p_{n})$. We
introduce the homogeneous anisotropic Sobolev space
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ as

$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})=\left\\{u\in
L^{p^{\ast}}(\mathbb{R}^{n}):\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}s_{i}(1-s_{i})\frac{|u(x)-u(x+he_{i})|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h\,\textnormal{d}x<\infty\right\\}$

equipped with the norm
$\|u\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}$, where

$\|u\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}:=\sum_{i=1}^{n}\|D^{s_{i}}_{p_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}:=\sum_{i=1}^{n}\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}s_{i}(1-s_{i})\frac{|u(x)-u(x+he_{i})|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h\,\textnormal{d}x\right)^{1/p_{i}}$

with

$|D^{s_{i}}_{p_{i}}u(x)|^{p_{i}}:=s_{i}(1-s_{i})\int_{\mathbb{R}}\frac{|u(x)-u(x+he_{i})|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h$

for $i=1,\dots,n$. Note that $\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$
becomes a normed space in the light of Theorem 1.1. It is the natural space
for minimizing the functional connected to the Sobolev quotient. The first
main result of this paper is a robust Sobolev-type inequality.

######  Theorem 1.1.

Let $s_{1},\dots,s_{n}\in[s_{0},1)$ for some $s_{0}\in(0,1)$ and
$p_{1},\dots,p_{n}>1$ be such that 1 holds. Then there is a constant
$C=C(n,p^{\ast},p^{\ast}-p_{\max},s_{0})>0$ such that for every
$u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$

$\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq
C\|u\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}.$

Theorem 1.1 is robust in the sense that the appearing constant depends only on
a lower bound $s_{0}\leq s_{1},\dots,s_{n}$. This allows us to recover the
local anisotropic Sobolev inequality (see [47, 44, 45, 35])

(1.3) $\|u\|_{L^{q}(\mathbb{R}^{n})}\leq
C\sum_{i=1}^{n}\|D_{i}u\|_{L^{p_{i}}(\mathbb{R}^{n})},$

where

(1.4)
$p_{i}>1,\quad\bar{p}=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{p_{i}}\right)^{-1},\quad\bar{p}<n,\quad
q=\frac{n\bar{p}}{n-\bar{p}},\quad\text{and}\quad p_{\max}<q.$

Moreover, our result provides an alternative proof of (1.3).

Our second main result establishes the existence of nonnegative minimizers of
the Sobolev quotient.

######  Theorem 1.2.

Assume $\vec{s}=(s_{1},\dots,s_{n})$ and $\vec{p}=(p_{1},\dots,p_{n})$ satisfy
1. Then there exists a nonnegative minimizer
$u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ of

(1.5)
$S:=\inf_{u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n}),\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1}\sum_{i=1}^{n}\frac{1}{p_{i}}\|D^{s_{i}}_{p_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}.$

A consequence of the foregoing result is the existence of a nontrivial,
nonnegative solution to the corresponding critical problem.

######  Corollary 1.3.

Assume $\vec{s}=(s_{1},\dots,s_{n})$ and $\vec{p}=(p_{1},\dots,p_{n})$ satisfy
1. Then there exists a nontrivial nonnegative solution
$u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ of

(1.6)
$\sum_{i=1}^{n}(-\partial_{ii})^{s_{i}}_{p_{i}}u=|u|^{p^{\ast}-2}u\quad\text{in}~{}\mathbb{R}^{n}.$

We first comment on known results from the literature before we explain the
novelty of our results. For $s\in(0,1)$ and $p\in(1,\infty)$, we define the
normalized Gagliardo–Slobodeckiĭ seminorm as

$[u]_{W^{s,p}(\mathbb{R}^{n})}:=\left(s(1-s)\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,\textnormal{d}x\,\textnormal{d}y\right)^{1/p}$

and for $s=1$ and $p\in(1,\infty)$, we define

$[u]_{W^{1,p}(\mathbb{R}^{n})}:=\|\nabla u\|_{L^{p}(\mathbb{R}^{n})}.$

The homogeneous Sobolev space is then given as the completion of
$C_{c}^{\infty}(\mathbb{R}^{n})$ with respect to the
$W^{s,p}(\mathbb{R}^{n})$-seminorm. In the subconformal case $sp<n$, it can be
represented as the function space

$\mathcal{D}^{s,p}(\mathbb{R}^{n})=\\{u\in
L^{p^{\ast}}(\mathbb{R}^{n}):[u]_{W^{s,p}(\mathbb{R}^{n})}<\infty\\},$

where $p^{\ast}=np/(n-sp)$. For details on homogeneous Sobolev spaces, we
refer the reader to [12] and the references therein.

The existence of a minimizer of the Sobolev quotient

(1.7)
$\inf_{u\in\mathcal{D}^{s,p}(\mathbb{R}^{n})\setminus\\{0\\}}\frac{[u]_{W^{s,p}(\mathbb{R}^{n})}^{p}}{\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p}}$

is an important problem in the analysis of variational problems. One reason
for this is that minimizers of (1.7) satisfy the quasilinear equation
involving the critical exponent

(1.8) $(-\Delta)^{s}_{p}u=|u|^{p^{\ast}-2}u\quad\text{in}~{}\mathbb{R}^{n}.$

For $s=1$ and $p=2$, this equation (1.8) is related to the Yamabe problem,
which addresses a question on the scalar curvature of Riemannian manifolds.
For $s=1$, the problem of existence of minimizers to (1.7) in the case
$p\in(1,n)$ was completely answered in [4, 46]. Not only the existence of
minimizers to (1.7) is proved, but also their explicit form is shown to be (up
to translation and scaling)

$u(x)=\left(1+|x|^{\frac{p}{p-1}}\right)^{\frac{p-n}{p}}.$

Let us now comment on results in the fractional case $s\in(0,1)$. In the
linear case $p=2$, the authors in [17] obtain sharp constants for Sobolev
inequalities. They prove the existence of minimizers and prove that they are
(again up to translation and scaling) of the form

(1.9) $u(x)=\left(1+|x|^{\frac{p}{p-1}}\right)^{\frac{sp-n}{p}}.$

In the case $p\in(1,n/s)$, the existence of minimizers is proved in [13].
Furthermore, it is conjectured that minimizers are of the form (1.9), see [13,
Equation (1.9)]. The authors provide the asymptotic behavior of minimizers,
but their explicit form remains still open.

The aim of this paper is to show the existence of nonnegative minimizers to
(1.5) and as a consequence prove the existence of nontrivial nonnegative
solutions to the corresponding critical problem (1.6). The results in this
paper can be seen as a combination of the results in [23] and [9]. Our
approach is based on an appropriate adjustment of the concentration-
compactness principle. Let us briefly discuss the results in [23] and [9]
before we address results concerning the concentration-compactness principle.
In [23], the authors study the anisotropic $p$-Laplacian (1.2). They prove
that nonnegative minimizers of

$\inf_{u\in\mathcal{D}^{1,\vec{p}}(\mathbb{R}^{n}),\|u\|_{L^{q}(\mathbb{R}^{n})}=1}\sum_{i=1}^{n}\frac{1}{p_{i}}\|D_{i}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}},$

exist, where $\vec{p}=(p_{1},\dots,p_{n})$ and $q$ are given as in (1.4), and

$\mathcal{D}^{1,\vec{p}}(\mathbb{R}^{n})=\\{u\in
L^{q}(\mathbb{R}^{n}):|D_{i}u|\in L^{p_{i}}(\mathbb{R}^{n})\\}.$

The main tool in [23] is an adaption of the concentration-compactness
principle. The existence of a nonnegative minimizer implies the existence of a
nonnegative solution to the corresponding anisotropic critical problem.
In [9], the authors extend the concentration-compactness principle for the
fractional
$p$-Laplacian in unbounded domains. Using the concentration-compactness
principle, they provide sufficient conditions for the existence of a
nontrivial solution to the generalized fractional Brezis–Nirenberg problem.
Combining ideas from the anisotropic local case as in [23] and from the
nonlocal case as in [9] allows us to prove the concentration-compactness
principle for the anisotropic nonlocal operator in (1.1) and the existence of
solutions to the corresponding critical problem.

######  Remark 1.4.

We would like to emphasize that our method of proof — up to straightforward
modifications — carries over to certain anisotropic operators with a possibly
partially orthotropic structure, as for example

(1.10)
$\left(-\partial_{11}-\partial_{22}\right)^{s}_{p}+(-\partial_{33})^{t}_{q},$

where $n=3$, $p,q>1$, $s,t\in(0,1)$. In case $\vec{s}=(s,s,t),\vec{p}=(p,p,q)$
satisfy 1, one can prove analogues of Theorem 1.1, Theorem 1.2 and Corollary
1.3 for the operator in (1.10).

We would like to mention that the classification of nonnegative minimizers
(1.5) remains an open problem. To the best of our knowledge, even in the
anisotropic local case such as (1.2), the shape of minimizers is not known. We
do not even know whether such classification is possible in the anisotropic
case.

Let us now refer to results on the concentration-compactness principle in the
literature. As already mentioned, the concentration-compactness principle
(CCP) has been introduced by P.-L. Lions in a series of papers. In [39] the
CCP was introduced for bounded domains. It has been extended to unbounded
domains by Chabrowski in [14] (see also [7, 5]). For the fractional
$p$-Laplacian on bounded domains, the CCP was established in the linear case
$p=2$ in [43] and for general $1<p<n/s$ in [42]. The CCP for the fractional
$p$-Laplacian was extended to unbounded domains in [9].

We briefly refer to further results addressing the CCP. In [18], the authors
obtain a CCP in the linear case $p=2$ for a class of stable processes in
$\mathbb{R}^{n}$. Concerning the $p$-Laplacian resp. the fractional
$p$-Laplacian, we refer the reader to [29, 13]. For results with regard to the
fractional $(p,q)$-Laplacian, see [6, 1, 3] and the references therein. The
CCP for local operators with variable exponents is studied in [2, 25, 27, 28,
34, 33] and the nonlocal case with variable exponents can be found in [32].

The article is organized as follows. In Section 2, we prove an anisotropic
fractional Sobolev inequality and a compact embedding theorem. Section 3
establishes the CCP and is divided into three subsections. In Section 3.1, we
study the decay of cutoff functions, while Section 3.2 contains the proof of
reverse Hölder inequalities. Finally, in Section 3.3 we prove the main results
of this paper.

##  2\. Sobolev inequality and compact embedding

In this section, we prove an anisotropic fractional Sobolev inequality and a
compact embedding theorem. Even as a self-standing result the anisotropic
fractional Sobolev inequality is very interesting. On the one hand, it allows
for different orders of integrability $p_{1},\dots,p_{n}>1$ and at the same
time also different orders of differentiability $s_{1},\dots,s_{n}\in(0,1)$.
Hence, this Sobolev inequality covers plenty of anisotropic nonlocal
operators, such as $-(-\partial_{11})^{s_{1}}-\dots-(-\partial_{nn})^{s_{n}}$
or fractional orthotropic $p$-Laplacians (see [15]). By comparability of
energy forms, one can deduce the Sobolev inequality also for isotropic
operators such as the fractional Laplacian $-(-\Delta)^{s}$. On the other
hand, the Sobolev inequality is robust in the sense that the appearing
constant depends only on a lower bound of $s_{1},\cdots,s_{n}$. This allows us
to recover local isotropic and anisotropic Sobolev inequalities such as
Troisi’s inequality, see (1.3). Therefore, Theorem 1.1 not only extends (1.3)
to the nonlocal setting, but also provides another proof of it.
The compact embedding
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})\subset\subset
L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})$ for every $1\leq p<p^{\ast}$ is a
consequence of the anisotropic fractional Sobolev inequality and direct
computations.

The proof of Theorem 1.1 is based on the boundedness of maximal and sharp
maximal operators. For this purpose, we define a class of rectangles taking
the anisotropy into account. For $i=1,\dots,n$, set

(2.1)
$m_{i}=\frac{s_{\max}}{s_{i}}\left(\frac{1}{p_{i}}-\frac{1}{p^{\ast}}\right)\left(\frac{1}{p_{\max}}-\frac{1}{p^{\ast}}\right)^{-1}.$

Note that $m_{i}\geq 1$ by the assumption $p_{\max}<p^{\ast}$. For
$x\in\mathbb{R}^{n}$ and $\rho>0$, we define

(2.2)
$M_{\rho}(x)=\mathop{\mathchoice{\raise-2.20001pt\hbox{\huge$\times$}}{\raise-0.50003pt\hbox{\Large$\times$}}{\hbox{\large$\times$}}{\times}}_{i=1}^{n}\left(x_{i}-\rho^{m_{i}},x_{i}+\rho^{m_{i}}\right).$

The sets $M_{\rho}(x)$ are balls with respect to the metric $d_{M}$ defined by

$d_{M}(x,y)=\sup_{i=1,\dots,n}|x_{i}-y_{i}|^{1/m_{i}},\quad
x,y\in\mathbb{R}^{n}.$

Note that $d_{M}$ is a metric since $m_{i}\geq 1$. An important ingredient in
the proof of Theorem 1.1, is the fact that $(\mathbb{R}^{n},d_{M})$ equipped
with the Lebesgue measure is a doubling metric measure space. Since

$|M_{2\rho}|=2^{n}(2\rho)^{\sum_{i=1}^{n}m_{i}}=2^{\sum_{i=1}^{n}m_{i}}|M_{\rho}|=2^{s_{\max}/(\frac{1}{p_{\max}}-\frac{1}{p^{\ast}})}|M_{\rho}|\leq
2^{\frac{(p^{\ast})^{2}}{p^{\ast}-p_{\max}}}|M_{\rho}|,$

the doubling constant depends on $p^{\ast}$ and $p^{\ast}-p_{\max}$.

Let us recall the Hardy–Littlewood maximal function and sharp maximal
function: for $u\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$,

${\bf
M}u(x)=\sup_{\rho>0}\fint_{M_{\rho}(x)}u(y)\,\textnormal{d}y\quad\text{and}\quad{\bf
M}^{\sharp}u(x)=\sup_{\rho>0}\fint_{M_{\rho}(x)}|u(y)-(u)_{M_{\rho}(x)}|\,\textnormal{d}y,$

where $(u)_{M_{\rho}(x)}=\fint_{M_{\rho}(x)}u(z)\,\textnormal{d}z$. It is well
known that the maximal inequality and sharp maximal inequality hold true on
doubling metric spaces, see [31, 30, 15].

###### Proof of Theorem 1.1.

Let $u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$, then the maximal
function ${\bf M}u$ and sharp maximal function ${\bf M}^{\sharp}u$ are well
defined. Following the proof of [15, Theorem 2.4], one obtains

(2.3)
$\fint_{M_{\rho}(x)}|u(y)-(u)_{M_{\rho}(x)}|\,\textnormal{d}y\leq\sum_{i=1}^{n}\rho^{s_{i}m_{i}}\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y,$

where the function $F_{i}$ is defined by

$F_{i}(y):=\sup_{\rho>0}\left(2\rho^{-s_{i}m_{i}}\fint_{-2\rho^{m_{i}}}^{2\rho^{m_{i}}}|u(y)-u(y+he_{i})|\,\mathrm{d}h\right).$

By Hölder’s inequality, we have

(2.4)
$\begin{split}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p^{\ast}}&\leq\left(\fint_{M_{\rho}(x)}F_{i}^{p_{i}}(y)\,\mathrm{d}y\right)^{\frac{{p^{\ast}}-p_{i}}{p_{i}}}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p_{i}}\\\
&\leq|M_{\rho}|^{-\frac{{p^{\ast}}-p_{i}}{p_{i}}}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{{p^{\ast}}-p_{i}}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p_{i}}.\end{split}$

Thus, it follows from (2.3) and (2.4) that

$\begin{split}&\left(\fint_{M_{\rho}(x)}|u(y)-(u)_{M_{\rho}(x)}|\,\mathrm{d}y\right)^{p^{\ast}}\\\
&\leq\sum_{i=1}^{n}n^{{p^{\ast}}-1}\rho^{{p^{\ast}}s_{i}m_{i}}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p^{\ast}}\\\
&\leq\sum_{i=1}^{n}n^{{p^{\ast}}-1}\rho^{{p^{\ast}}s_{i}m_{i}}|M_{\rho}|^{-\frac{{p^{\ast}}-p_{i}}{p_{i}}}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{{p^{\ast}}-p_{i}}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p_{i}}.\end{split}$

Note that for each $i$,

$p^{\ast}s_{i}m_{i}-\frac{p^{\ast}-p_{i}}{p_{i}}\sum_{j=1}^{n}m_{j}=0.$

Therefore, we arrive at

$\left(\fint_{M_{\rho}(x)}|u(y)-(u)_{M_{\rho}(x)}|\,\mathrm{d}y\right)^{p^{\ast}}\leq\sum_{i=1}^{n}\frac{n^{{p^{\ast}}-1}}{2^{n\frac{{p^{\ast}}-p_{i}}{p_{i}}}}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{{p^{\ast}}-p_{i}}\left(\fint_{M_{\rho}(x)}F_{i}(y)\,\mathrm{d}y\right)^{p_{i}}.$

We take the supremum over $\rho>0$ and use the estimate
$2^{n\frac{p^{\ast}-p_{i}}{p_{i}}}\geq 1$, then

$\left({\bf M}^{\sharp}u(x)\right)^{p^{\ast}}\leq
n^{p^{\ast}-1}\sum_{i=1}^{n}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{{p^{\ast}}-p_{i}}({\bf
M}F_{i}(x))^{p_{i}}.$

By boundedness of the sharp maximal operator in $L^{p^{\ast}}(\mathbb{R}^{n})$
and the maximal operator in $L^{p_{i}}(\mathbb{R}^{n})$,

$\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}\leq C\|{\bf
M}^{\sharp}u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}\leq
C\sum_{i=1}^{n}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{{p^{\ast}}-p_{i}}\|{\bf
M}F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}\leq
C\sum_{i=1}^{n}\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p^{\ast}}$

for some $C>0$. The constant $C$ depends on the doubling constant and the
number of intersections of anisotropic dyadic rectangles as in [15, Appendix
B]. Therefore, $C=C(n,p^{\ast},p^{\ast}-p_{\max},s_{0})$.

It only remains to show that there is a
$C=C(n,p^{\ast},p^{\ast}-p_{\max},s_{0})>0$ such that

$\|F_{i}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}\leq
Cs_{i}(1-s_{i})\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\frac{|u(x)-u(x+he_{i})|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\mathrm{d}h\,\mathrm{d}x$

for each $i=1,\dots,n$, which follows exactly as in [15, Lemma 2.3 and Theorem
2.4]. ∎

######  Theorem 2.1 (Compact Embedding).

Let $s_{1},\dots,s_{n}\in[s_{0},1)$ for some $s_{0}\in(0,1)$ and
$p_{1},\dots,p_{n}>1$ be such that 1 holds. Then
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})\subset\subset
L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})$ for every $1\leq p<p^{\ast}$.

###### Proof.

We follow the proof in [19, Theorem 4.54].
Let $\mathcal{B}\subset\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ be
bounded and $\Omega\subset\mathbb{R}^{n}$ be an open, bounded set. By Theorem
1.1 we know that $\mathcal{B}$ is bounded in $L^{p^{\ast}}(\mathbb{R}^{n})$.
Therefore, by a standard Hölder interpolation argument, it is sufficient to
prove that $\mathcal{B}$ is precompact in $L^{1}(\Omega)$ in order to deduce
the desired result. We will show that there is $C>0$ such that for every
$r\in\mathbb{R}$, $i\in\\{1,\dots,n\\}$, $u\in\mathcal{B}$ it holds:

(2.5) $\int_{\Omega_{r}}|u(x+re_{i})-u(x)|\,\textnormal{d}x\leq
C|r|^{s_{i}}|\Omega|^{(p_{i}-1)/p_{i}}\|D^{s_{i}}_{p_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})},$

where
$\Omega_{r}=\\{x\in\mathbb{R}^{n}:\operatorname{dist}(x,\partial\Omega)>|r|\\}$.
By the triangle inequality, we have

$\begin{split}&2|r|\int_{\Omega_{r}}|u(x+re_{i})-u(x)|\,\textnormal{d}x\\\
&\leq\int_{\Omega_{r}}\int_{(-|r|,|r|)}|u(x+re_{i})-u(x+te_{i})|+|u(x+te_{i})-u(x)|\,\textnormal{d}t\,\textnormal{d}x\\\
&=\int_{\Omega_{r}}\int_{(-|r|,|r|)}\frac{|u(x+re_{i})-u(x+te_{i})|}{|t-r|^{\frac{1+s_{i}p_{i}}{p_{i}}}}|t-r|^{\frac{1+s_{i}p_{i}}{p_{i}}}\,\textnormal{d}t\,\textnormal{d}x\\\
&\qquad\qquad+\int_{\Omega_{r}}\int_{(-|r|,|r|)}\frac{|u(x+te_{i})-u(x)|}{|t|^{\frac{1+s_{i}p_{i}}{p_{i}}}}|t|^{\frac{1+s_{i}p_{i}}{p_{i}}}\,\textnormal{d}t\,\textnormal{d}x.\end{split}$

Both expressions on the right-hand side can be estimated from above in a
similar way. We demonstrate how to treat the second expression. By Hölder’s
inequality

$\begin{split}\int_{\Omega_{r}}&\int_{(-|r|,|r|)}\frac{|u(x+te_{i})-u(x)|}{|t|^{\frac{1+s_{i}p_{i}}{p_{i}}}}|t|^{\frac{1+s_{i}p_{i}}{p_{i}}}\,\textnormal{d}t\,\textnormal{d}x\\\
&\leq\left(\int_{\Omega_{r}}\int_{(-|r|,|r|)}\frac{|u(x+te_{i})-u(x)|^{p_{i}}}{|t|^{1+s_{i}p_{i}}}\,\textnormal{d}t\,\textnormal{d}x\right)^{1/p_{i}}\left(\int_{\Omega_{r}}\int_{(-|r|,|r|)}|t|^{\frac{1+s_{i}p_{i}}{p_{i}-1}}\,\textnormal{d}t\,\textnormal{d}x\right)^{\frac{p_{i}-1}{p_{i}}}\\\
&\leq
C|r|^{1+s_{i}}|\Omega|^{1-1/p_{i}}\|D^{s_{i}}_{p_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}\end{split}$

for some $C>0$. This proves (2.5). Now let $h\in B_{1}(0)$. Then

$\begin{split}&\int_{\Omega_{2|h|}}|u(x+h)-u(x)|\,\textnormal{d}x\\\
&\leq\sum_{i=1}^{n}\int_{\Omega_{2|h|}}|u(x+h_{1}e_{1}+\cdots+h_{i}e_{i})-u(x+h_{1}e_{1}+\cdots+h_{i-1}e_{i-1})|\,\textnormal{d}x\\\
&=\sum_{i=1}^{n}\int_{h_{1}e_{1}+\cdots+h_{i-1}e_{i-1}+\Omega_{2|h|}}|u(x+h_{i}e_{i})-u(x)|\,\textnormal{d}x.\end{split}$

Since $h_{1}e_{1}+\dots+h_{i-1}e_{i-1}+\Omega_{2|h|}\subset\Omega_{h_{i}}$ for
every $i\in\\{1,\dots,n\\}$, we have

$\begin{split}\int_{\Omega_{2|h|}}|u(x+h)-u(x)|\,\textnormal{d}x&\leq\sum_{i=1}^{n}\int_{\Omega_{h_{i}}}|u(x+h_{i}e_{i})-u(x)|\,\textnormal{d}x\\\
&\leq
C|h|^{s_{0}}\max\\{1,|\Omega|^{1-1/p_{\max}}\\}\|u\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}.\end{split}$

for some $C>0$. By boundedness of $\mathcal{B}$ in
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$, the right-hand side vanishes
uniformly as $h\searrow 0$. Since $\mathcal{B}$ is also bounded in
$L^{1}(\Omega)$, we can deduce precompactness of $\mathcal{B}$ in
$L^{1}(\Omega)$ from the Riesz–Fréchet–Kolmogorov theorem (c.f. [19, Theorem
1.95]). ∎

##  3\. Concentration-compactness principle

In this section we present the concentration-compactness principle for solving
the minimization problem (1.5). Throughout this section, $\vec{s}$ and
$\vec{p}$ always satisfy 1.

Recall that

$S=\inf_{u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n}),\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1}\sum_{i=1}^{n}\frac{1}{p_{i}}\|D^{s_{i}}_{p_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}},$

which is positive by Theorem 1.1. Furthermore, by definition of $S$, we have
the following lemma.

######  Lemma 3.1.

For every $u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ with
$\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq 1$, it holds that

$S\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}\leq\sum_{i=1}^{n}\frac{1}{p_{i}}\|D_{p_{i}}^{s_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}.$

###### Proof.

Let $u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ be such that $u\neq 0$
and $\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq 1$. Let
$v=u/\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}$, then
$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1$. Thus, by the definition of $S$,

$S\leq\sum_{i=1}^{n}\frac{1}{p_{i}}\|D_{p_{i}}^{s_{i}}v\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}=\sum_{i=1}^{n}\frac{1}{p_{i}}\frac{\|D_{p_{i}}^{s_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}}{\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{i}}}\leq\sum_{i=1}^{n}\frac{1}{p_{i}}\frac{\|D_{p_{i}}^{s_{i}}u\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}}{\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}}.$

∎

The following scaling property is a consequence of a straightforward
computation.

######  Lemma 3.2.

For $i=1,\dots,n$, let $m_{i}$ be the exponent defined by (2.1). For every
$y\in\mathbb{R}^{n}$, $\rho>0$, and
$u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$, the function $v$ defined
by

(3.1)
$v(x)=\rho^{s_{\max}p_{\max}/(p^{\ast}-p_{\max})}u(\rho^{m_{1}}x_{1}+y_{1},\cdots,\rho^{m_{n}}x_{n}+y_{n})$

satisfies
$\|v\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}=\|u\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}$
and
$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=\|u\|_{L^{p^{\ast}}(\mathbb{R}^{n})}$.

To prove the existence of a minimizer of $S$, let us take a minimizing
sequence $(u_{k})\subset\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ for $S$
satisfying $\|u_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1$ for every
$k\in\mathbb{N}$. We define the Lévy concentration function

$Q_{k}(\rho)=\sup_{y\in\mathbb{R}^{n}}\int_{M_{\rho}(y)}|u_{k}|^{p^{\ast}}\,\textnormal{d}x,\quad\rho>0,$

where $M_{\rho}(y)$ is the rectangle defined by (2.2). For each
$k\in\mathbb{N}$, there exists $\rho_{k}>0$ such that $Q_{k}(\rho_{k})=1/2$.
Moreover, there exists $y_{k}\in\mathbb{R}^{n}$ such that

$Q_{k}(\rho_{k})=\int_{M_{\rho_{k}}(y_{k})}|u_{k}|^{p^{\ast}}\,\textnormal{d}x=\frac{1}{2}.$

By Lemma 3.2, the function $v_{k}$ defined by (3.1) with $\rho=\rho_{k}$
satisfies

(3.2)
$\int_{M_{1}(0)}|v_{k}|^{p^{\ast}}\,\mathrm{d}x=\frac{1}{2}=\sup_{y\in\mathbb{R}^{n}}\int_{M_{1}(y)}|v_{k}|^{p^{\ast}}\,\mathrm{d}x.$

Furthermore, the sequence $(v_{k})$ is also a minimizing sequence for $S$. By
the Banach–Alaoglu theorem and Theorem 1.1, we have

(3.3) $v_{k}\rightharpoonup
v~{}\text{in}~{}\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})~{}\text{and}~{}L^{p^{\ast}}(\mathbb{R}^{n})$

for some $v\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$, up to
subsequences. Moreover, we have by Theorem 2.1 that

(3.4) $v_{k}\to v~{}\text{a.e. in}~{}\mathbb{R}^{n}$

up to subsequences. By the lower semicontinuity of norms
$\|\cdot\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}$ and
$\|\cdot\|_{L^{p^{\ast}}(\mathbb{R}^{n})}$, we have

(3.5)
$\|v\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}\leq\liminf_{k\to\infty}\|v_{k}\|_{\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})}=S\quad\text{and}\quad\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}\leq\liminf_{k\to\infty}\|v_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1.$

Therefore, it is enough to prove the following theorem to find a minimizer for
$S$.

######  Theorem 3.3.

Let $v:\mathbb{R}^{n}\to\mathbb{R}$ be the defined as above. Then

$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1.$

This is the main theorem of this section. We will provide the proof at the
beginning of Section 3.3.

###  3.1. Decay of cutoff functions

Let us make a few comments on the significance of cutoff functions in this
article. Unlike in the local case, the support of a function is not preserved
by the application of a nonlocal operator. This basic fact causes some
problems since the compact embedding
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})\subset\subset
L^{p}_{\mathrm{loc}}(\mathbb{R}^{n})$, $1\leq p<p^{\ast}$, (c.f. Theorem 2.1)
cannot be used directly. In [9], the authors solve this issue by proving a
compact embedding into weighted spaces of the form
$L^{p}(|D^{s}\phi(x)|^{p}\textnormal{d}x)$, where $\phi\in
C_{c}^{\infty}(\mathbb{R}^{n})$ is a cutoff function. The main observation is
that the fractional $(s,p)$-gradient of a compactly supported function decays
fast enough for such result to hold true, despite having full support.

The following lemmas discuss the adaption of this argument to the class of
anisotropic nonlocal operators considered in this article.

For $x\in\mathbb{R}^{n}$, $\rho>0$, and $i\in\\{1,\dots,n\\}$, we introduce
the notation

$M_{\rho}^{(i)}(x)=\mathop{\mathchoice{\raise-2.20001pt\hbox{\huge$\times$}}{\raise-0.50003pt\hbox{\Large$\times$}}{\hbox{\large$\times$}}{\times}}_{j=1}^{i-1}\left(x_{j}-\rho^{m_{j}},x_{j}+\rho^{m_{j}}\right)\times\mathbb{R}\times\mathop{\mathchoice{\raise-2.20001pt\hbox{\huge$\times$}}{\raise-0.50003pt\hbox{\Large$\times$}}{\hbox{\large$\times$}}{\times}}_{j=i+1}^{n}\left(x_{j}-\rho^{m_{j}},x_{j}+\rho^{m_{j}}\right).$

######  Lemma 3.4.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that
$\operatorname{supp}(\phi)\subset M_{1}(0)$. Then

(3.6) $|D^{s_{i}}_{p_{i}}\phi(x)|^{p_{i}}\leq
C\min(1,|x_{i}|^{-1-s_{i}p_{i}}),\quad i=1,\dots,n,$

where $C=C(n,s_{i},p_{i},\|\phi\|_{\infty},\|\partial_{i}\phi\|_{\infty})>0$.
Moreover, $\operatorname{supp}(|D^{s_{i}}_{p_{i}}\phi|)\subset
M_{1}^{(i)}(0)$.

###### Proof.

We first observe that for given $j\in\\{1,\dots,n\\}$ and $x\in\mathbb{R}^{n}$
with $|x_{j}|>1$, it holds that $\phi(x+e_{i}h)=0$ for every $h\in\mathbb{R}$
and $i\neq j$. It becomes evident that
$\operatorname{supp}(|D^{s_{i}}_{p_{i}}\phi|)\subset M_{1}^{(i)}(0)$ for every
$i\in\\{1,\dots,n\\}$.
For $x\in\mathbb{R}^{n}$ we compute

$s_{i}(1-s_{i})\int_{\mathbb{R}}\frac{|\phi(x)-\phi(x+e_{i}h)|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h\leq
C(p_{i},n)\left(\|\phi\|_{\infty}^{p_{i}}+\|\partial_{i}\phi\|_{\infty}^{p_{i}}\right).$

If $|x_{i}|>2$ and $|x_{i}+h|<1$, then $|h|>|x_{i}|/2$. Therefore, we estimate
for $x\in\mathbb{R}^{n}$ with $|x_{i}|>2$:

$\begin{split}s_{i}(1-s_{i})\int_{\mathbb{R}}\frac{|\phi(x)-\phi(x+e_{i}h)|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h&=s_{i}(1-s_{i})\int_{|x_{i}+h|<1}\frac{|\phi(x+e_{i}h)|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,\textnormal{d}h\\\
&\leq C\frac{\|\phi\|_{\infty}^{p_{i}}}{|x_{i}|^{1+s_{i}p_{i}}}.\end{split}$

Assertion (3.6) follows by combining the aforementioned estimates. ∎

Given $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$, $\varepsilon>0$, and
$\xi\in\mathbb{R}^{n}$, we introduce the scaled cutoff function
$\phi_{\varepsilon,\xi}$ defined by

(3.7) $\phi_{\varepsilon,\xi}(x)=\phi(x^{(\varepsilon,\xi)}),$

where
$x^{(\varepsilon,\xi)}:=((x_{1}-\xi_{1})\varepsilon^{-m_{1}},\dots,(x_{n}-\xi_{n})\varepsilon^{-m_{n}})$
and $m_{i}$ is given by (2.1).

One easily computes:

$|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,0}(x)|^{p_{i}}=\varepsilon^{-m_{i}s_{i}p_{i}}|D^{s_{i}}_{p_{i}}\phi(x^{(\varepsilon,0)})|^{p_{i}}.$

Consequently, we have the following corollary of Lemma 3.4:

######  Corollary 3.5.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that
$\operatorname{supp}(\phi)\subset M_{1}(0)$. Let $\varepsilon>0$ and
$\xi\in\mathbb{R}^{n}$. Then

$|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,\xi}(x)|^{p_{i}}\leq
C\min(\varepsilon^{-m_{i}s_{i}p_{i}},\varepsilon^{m_{i}}|x_{i}-\xi_{i}|^{-1-s_{i}p_{i}}),\quad
i=1,\dots,n,$

where $C=C(n,s_{i},p_{i},\|\phi\|_{\infty},\|\partial_{i}\phi\|_{\infty})>0$.
Moreover,
$\operatorname{supp}(|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,\xi}|)\subset
M_{\varepsilon}^{(i)}(\xi)$.

The following lemma makes use of the compact embedding Theorem 2.1 and the
decay estimates. It follows in the spirit of [9, Lemma 2.4] and can be
interpreted as a compact embedding of
$\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ into the weighted space
$L^{p_{i}}(|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,\xi}|^{p_{i}}\,\textnormal{d}x)$
for every $i\in\\{1,\dots,n\\}$. Let us recall the function $v_{k}$
constructed in Section 3 and define

$w_{k}:=v_{k}-v.$

######  Lemma 3.6.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that
$\operatorname{supp}(\phi)\subset M_{1}(0)$. Let $\xi\in\mathbb{R}^{n}$ and
$\varepsilon>0$. Then for every $i\in\\{1,\dots,n\\}$,

$\int_{\mathbb{R}^{n}}|D_{p_{i}}^{s_{i}}(\phi_{\varepsilon,\xi})w_{k}|^{p_{i}}\,\textnormal{d}x\to
0,$

as $k\to\infty$.

###### Proof.

For $R>0$ to be chosen later, we decompose

$\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi})w_{k}|^{p_{i}}\,\textnormal{d}x=\int_{|x_{i}-\xi_{i}|<R}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi})w_{k}|^{p_{i}}\,\textnormal{d}x+\int_{|x_{i}-\xi_{i}|>R}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi})w_{k}|^{p_{i}}\,\textnormal{d}x=:I+II.$

In order to bound $II$, we apply Hölder’s inequality, Lemma 3.2, and Corollary
3.5:

$\begin{split}II&\leq
C\left(\|v_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{i}/p^{\ast}}+\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{i}/p^{\ast}}\right)\left(\int_{|x_{i}-\xi_{i}|>R}|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,\xi}|^{p_{i}\left(\frac{p^{\ast}}{p^{\ast}-p_{i}}\right)}\,\textnormal{d}x\right)^{\frac{p^{\ast}-p_{i}}{p^{\ast}}}\\\
&\leq
C\left(\int_{M_{\varepsilon}^{(i)}(\xi)\cap\\{|x_{i}-\xi_{i}|>R\\}}|x_{i}-\xi_{i}|^{(-1-s_{i}p_{i})\left(\frac{p^{\ast}}{p^{\ast}-p_{i}}\right)}\,\textnormal{d}x\right)^{\frac{p^{\ast}-p_{i}}{p^{\ast}}}\\\
&\leq
C\left(\int_{|x_{i}|>R}|x_{i}|^{(-1-s_{i}p_{i})\left(\frac{p^{\ast}}{p^{\ast}-p_{i}}\right)}\,\textnormal{d}x_{i}\right)^{\frac{p^{\ast}-p_{i}}{p^{\ast}}}\\\
&\leq CR^{\gamma},\end{split}$

where $C>0$ is independent of $k$, and
$\gamma=\left((-1-s_{i}p_{i})\left(\frac{p^{\ast}}{p^{\ast}-p_{i}}\right)+1\right)\frac{p^{\ast}-p_{i}}{p^{\ast}}=-p_{i}(s_{i}+1/p^{\ast})<0$.
Thus, $II\to 0$ as $R\to\infty$ uniformly in $k$.
For $I$, we observe that the integration takes place on the compact set

$\Omega:=M_{\varepsilon}^{(i)}(\xi)\cap\\{|x_{i}-\xi_{i}|<R\\}\subset\mathbb{R}^{n}$

due to Corollary 3.5. From Theorem 2.1, we know that
$\|w_{k}\|_{L^{p_{i}}(\Omega)}\to 0$ as $k\to\infty$. Hence, it follows that
$I\to 0$ as $k\to\infty$ for any $R>0$ upon recalling that
$|D^{s_{i}}_{p_{i}}\phi_{\varepsilon,\xi}|$ is bounded due to Corollary 3.5. ∎

We end this section by another convergence result for sequences of scaled
cutoff functions $(\phi_{\varepsilon,\xi})_{\varepsilon}$ given
$\xi\in\mathbb{R}^{n}$. Let $v$ be the function given in Theorem 3.3.

######  Lemma 3.7.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that
$\operatorname{supp}(\phi)\subset M_{1}(0)$. Let $\xi\in\mathbb{R}^{n}$ and
$\varepsilon>0$. Then for every $i\in\\{1,\dots,n\\}$,

$\int_{\mathbb{R}^{n}}|D_{p_{i}}^{s_{i}}(\phi_{\varepsilon,\xi})v|^{p_{i}}\,\textnormal{d}x\to
0,$

as $\varepsilon\to 0$.

###### Proof.

For $\varepsilon>0$ we estimate, using Corollary 3.5,

$\begin{split}\int_{\mathbb{R}^{n}}|D_{p_{i}}^{s_{i}}(\phi_{\varepsilon,\xi})v|^{p_{i}}\,\textnormal{d}x&\leq\varepsilon^{-m_{i}s_{i}p_{i}}\int_{M_{\varepsilon}(\xi)}|v(x)|^{p_{i}}\,\textnormal{d}x+\varepsilon^{m_{i}}\int_{M_{\varepsilon}^{(i)}(\xi)\cap\\{|x_{i}-\xi_{i}|>\varepsilon^{m_{i}}\\}}\frac{|v(x)|^{p_{i}}}{|x_{i}-\xi_{i}|^{1+s_{i}p_{i}}}\,\textnormal{d}x\\\
&=:I+II.\end{split}$

The first summand can be bounded from above by Hölder’s inequality:

$I\leq\varepsilon^{-m_{i}s_{i}p_{i}}\left(\int_{M_{\varepsilon}(\xi)}|v(x)|^{p^{\ast}}\,\textnormal{d}x\right)^{p_{i}/p^{\ast}}|M_{\varepsilon}(\xi)|^{\frac{p^{\ast}-p_{i}}{p^{\ast}}}\leq
C\|v\|_{L^{p^{\ast}}(M_{\varepsilon}(\xi))}^{p_{i}}$

for some $C>0$, where we used that
$m_{i}s_{i}p_{i}=\left(1-\frac{p_{i}}{p^{\ast}}\right)\sum_{j=1}^{n}m_{j}$.
Since $v\in L^{p^{\ast}}(\mathbb{R}^{n})$ it is clear that $I\to 0$ as
$\varepsilon\searrow 0$.
For $II$, we decompose the domain of integration as follows:

$\begin{split}II&=\sum_{l=0}^{\infty}\varepsilon^{m_{i}}\int_{M^{(i)}_{\varepsilon}(\xi)\cap\\{(2^{l+1}\varepsilon)^{m_{i}}>|x_{i}-\xi_{i}|>(2^{l}\varepsilon)^{m_{i}}\\}}\frac{|v(x)|^{p_{i}}}{|x_{i}-\xi_{i}|^{1+s_{i}p_{i}}}\,\textnormal{d}x\\\
&\leq\sum_{l=0}^{\infty}\varepsilon^{-m_{i}s_{i}p_{i}}2^{-lm_{i}(1+s_{i}p_{i})}\int_{M_{2^{l+1}\varepsilon}(\xi)}|v(x)|^{p_{i}}\,\textnormal{d}x.\end{split}$

By Hölder’s inequality, we deduce that

$II\leq
c\sum_{l=0}^{\infty}2^{-lm_{i}}\left(\int_{M_{2^{l+1}\varepsilon}(\xi)}|v(x)|^{p^{\ast}}\,\textnormal{d}x\right)^{p_{i}/p^{\ast}}.$

Note that for every $\delta>0$, we can find $l_{0}\in\mathbb{N}$ such that
$\sum_{l=l_{0}+1}^{\infty}2^{-lm_{i}}<\delta$. Therefore

(3.8)
$II\leq\delta\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{i}}+C\|v\|_{L^{p^{\ast}}\left(M_{2^{l_{0}+1}\varepsilon}(\xi)\right)}^{p_{i}},$

where $C=C(l_{0})>0$. Since (3.8) holds true for every $\delta>0$, also $II\to
0$ as $\varepsilon\searrow 0$. This implies the desired result. ∎

###  3.2. Reverse Hölder inequalities

In this subsection, we prove reverse Hölder inequalities for the “components”
of limiting measures of the minimizing sequence that spread or concentrate.
Recall the sequence $(v_{k})$ constructed in Section 3. We may take a further
subsequence of $(v_{k})$ so that

$\displaystyle|D^{s_{i}}_{p_{i}}(v_{k}-v)|^{p_{i}}\,\mathrm{d}x$
$\displaystyle\overset{v}{\rightharpoonup}\mu_{i},$
$\displaystyle|v_{k}-v|^{p^{\ast}}\,\mathrm{d}x$
$\displaystyle\overset{v}{\rightharpoonup}\nu,$
$\displaystyle\sum_{i=1}^{n}\frac{1}{p_{i}}|D_{p_{i}}^{s_{i}}v_{k}|^{p_{i}}\,\textnormal{d}x$
$\displaystyle\overset{v}{\rightharpoonup}\widetilde{\mu}$

for some positive bounded Borel measures $\mu_{i}$, $\nu$, and
$\widetilde{\mu}$, where $\overset{v}{\rightharpoonup}$ denotes vague
convergence. Moreover, we define $\mu=\sum_{i=1}^{n}\frac{1}{p_{i}}\mu_{i}$.
These quantities contain information on the concentration of $(v_{k})$. We
also define the limits

$\begin{split}&\mu_{\infty}=\lim_{R\to\infty}\limsup_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}\setminus
M_{R}(0)}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}\,\mathrm{d}x,\\\
&\nu_{\infty}=\lim_{R\to\infty}\limsup_{k\to\infty}\int_{\mathbb{R}^{n}\setminus
M_{R}(0)}|v_{k}|^{p^{\ast}}\,\mathrm{d}x,\end{split}$

which encode information on the spreading of $(v_{k})$.

In the following, we will make use of the estimate

(3.9)
$\|D^{s_{i}}_{p_{i}}(fg)\|_{L^{p_{i}}(\mathbb{R}^{n})}\leq\|D^{s_{i}}_{p_{i}}(f)g\|_{L^{p_{i}}(\mathbb{R}^{n})}+\|D^{s_{i}}_{p_{i}}(g)f\|_{L^{p_{i}}(\mathbb{R}^{n})},$

which holds true for all functions $f,g:\mathbb{R}^{n}\to\mathbb{R}$, whenever
the aforementioned quantities are finite. This inequality is an immediate
consequence of Minkowski’s inequality.

Our first result provides reverse Hölder inequalities for $\mu_{\infty}$ and
$\nu_{\infty}$.

######  Lemma 3.8.

For $\mu_{\infty}$ and $\nu_{\infty}$ as above, the following is true:

$S\nu_{\infty}^{p_{\max}/p^{\ast}}\leq\mu_{\infty}.$

###### Proof.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that $0\leq\phi\leq 1$,
$\phi\equiv 1$ in $M_{1}(0)$ and $\phi\equiv 0$ in $\mathbb{R}^{n}\setminus
M_{2}(0)$. For $R>0$, we define $\psi_{R}:=1-\phi_{R,0}$, where $\phi_{R,0}$
is defined as in (3.7). Then, $\psi_{R}\equiv 1$ in $\mathbb{R}^{n}\setminus
M_{2R}(0)$ and $\psi_{R}\equiv 0$ in $M_{R}(0)$. We first prove that

(3.10) $\displaystyle\nu_{\infty}$
$\displaystyle=\lim_{R\to\infty}\limsup_{k\to\infty}\int_{\mathbb{R}^{n}}|w_{k}|^{p^{\ast}}\psi_{R}^{p^{\ast}}\,\mathrm{d}x\quad\text{and}$
(3.11) $\displaystyle\mu_{\infty}$
$\displaystyle=\lim_{R\to\infty}\limsup_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}w_{k}|^{p_{i}}\psi_{R}^{p_{i}}\,\mathrm{d}x.$

For (3.10) we observe that

$\int_{\mathbb{R}^{n}\setminus
M_{2R}(0)}|v_{k}|^{p^{\ast}}\,\textnormal{d}x\leq\int_{\mathbb{R}^{n}}|v_{k}|^{p^{\ast}}\psi_{R}^{p^{\ast}}\,\textnormal{d}x\leq\int_{\mathbb{R}^{n}\setminus
M_{R}(0)}|v_{k}|^{p^{\ast}}\,\textnormal{d}x,$

which implies that

(3.12)
$\nu_{\infty}=\lim_{R\to\infty}\limsup_{k\to\infty}\int_{\mathbb{R}^{n}}|v_{k}|^{p^{\ast}}\psi_{R}^{p^{\ast}}\,\mathrm{d}x.$

Thus, the identity (3.10) follows now upon the observation that

$\lim_{R\to\infty}\int_{\mathbb{R}^{n}}|v|^{p^{\ast}}\psi_{R}^{p^{\ast}}\,\textnormal{d}x=\lim_{R\to\infty}\int_{\mathbb{R}^{n}\setminus
M_{R}(0)}|v|^{p^{\ast}}\psi_{R}^{p^{\ast}}\,\textnormal{d}x=0,$

since $v\in L^{p^{\ast}}(\mathbb{R}^{n})$. The proof of (3.11) is analogous.

Let us next prove that $S\nu_{\infty}^{p_{\max}/p^{\ast}}\leq\mu_{\infty}$. By
Brezis–Lieb’s lemma and (3.4) we have

$\lim_{k\to\infty}\|w_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}=\lim_{k\to\infty}\|v_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}-\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}<1.$

It follows from Lemma 3.1 that for large $k$

$S\|\psi_{R}w_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}\leq\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\psi_{R}w_{k})|^{p_{i}}\,\mathrm{d}x.$

By Lemma 3.6, we have

$\lim_{k\to\infty}\int_{\mathbb{R}^{n}}|w_{k}|^{p_{i}}|D^{s_{i}}_{p_{i}}\psi_{R}|^{p_{i}}\,\mathrm{d}x=\lim_{k\to\infty}\int_{\mathbb{R}^{n}}|w_{k}|^{p_{i}}|D^{s_{i}}_{p_{i}}\phi_{R,0}|^{p_{i}}\,\mathrm{d}x=0.$

Therefore, by (3.9),

$S\left(\lim_{R\to\infty}\limsup_{k\to\infty}\|\psi_{R}w_{k}\|_{p^{\ast}}^{p_{\max}}\right)\leq\lim_{R\to\infty}\limsup_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}w_{k}|^{p_{i}}\psi_{R}^{p_{i}}\,\mathrm{d}x,$

and the desired result follows from (3.10) and (3.11). ∎

The following lemma provides the reverse Hölder inequalities for $\mu_{i}$ and
$\nu$.

######  Lemma 3.9.

Let $\phi\in C_{c}^{\infty}(\mathbb{R}^{n})$ be such that
$\operatorname{supp}(\phi)\subset M_{1}(0)$. Then, for every
$\xi\in\mathbb{R}^{n}$ and $\varepsilon>0$,

(3.13)
$\displaystyle\left(\int_{\mathbb{R}^{n}}|\phi_{\varepsilon,\xi}|^{p^{\ast}}\,\textnormal{d}\nu\right)^{1/p^{\ast}}$
$\displaystyle\leq
C\sum_{i=1}^{n}\left(\int_{\mathbb{R}^{n}}|\phi_{\varepsilon,\xi}|^{p_{i}}\,\textnormal{d}\mu_{i}\right)^{1/p_{i}}\quad\text{and}$
(3.14)
$\displaystyle\left(\int_{\mathbb{R}^{n}}|\phi_{\varepsilon,\xi}|^{p^{\ast}}\,\textnormal{d}\nu\right)^{1/p^{\ast}}$
$\displaystyle\leq
C\left(\int_{\mathbb{R}^{n}}|\phi_{\varepsilon,\xi}|^{p_{\max}}\,\textnormal{d}\mu\right)^{1/p_{\max}}$

for some constant $C=C(\|\mu\|,\|\phi\|_{\infty})>0$.

###### Proof.

For simplicity, we write $\phi=\phi_{\varepsilon,\xi}$. First, we prove
(3.13). By definition of $\nu$ we deduce that $\int_{\mathbb{R}^{n}}|\phi
w_{k}|^{p^{\ast}}\,\textnormal{d}x\to\int_{\mathbb{R}^{n}}|\phi|^{p^{\ast}}\,\textnormal{d}\nu$
as $k\to\infty$. By Theorem 1.1 it is sufficient to prove that for
$i=1,\dots,n$:

(3.15) $\limsup\limits_{k\to\infty}\|D^{s_{i}}_{p_{i}}(\phi
w_{k})\|_{L^{p_{i}}(\mathbb{R}^{n})}\leq
C\left(\int_{\mathbb{R}^{n}}|\phi|^{p_{i}}\,\textnormal{d}\mu_{i}\right)^{1/p_{i}}.$

By definition of $\mu_{i}$ it follows that
$\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(w_{k})\phi|^{p_{i}}\,\textnormal{d}x\to\int_{\mathbb{R}^{n}}|\phi|^{p_{i}}\,\textnormal{d}\mu_{i}$,
as $k\to\infty$. Therefore, by (3.9), it remains to show that

(3.16)
$\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi)w_{k}|^{p_{i}}\,\textnormal{d}x\to
0,~{}~{}\text{ as }k\to\infty,$

in order to deduce (3.15). Since (3.16) follows from Lemma 3.6, estimate
(3.13) is proved.
For (3.14) we observe that for every $i\in\\{1,\dots,n\\}$ by Hölder’s
inequality:

$\int_{\mathbb{R}^{n}}|\phi|^{p_{i}}\,\textnormal{d}\mu_{i}\leq
C\int_{\mathbb{R}^{n}}|\phi|^{p_{i}}\,\textnormal{d}\mu\leq
C\left(\|\mu\|\|\phi\|_{\infty}\right)^{1-\frac{p_{i}}{p_{\max}}}\left(\int_{\mathbb{R}^{n}}|\phi|^{p_{\max}}\,\textnormal{d}\mu\right)^{p_{i}/p_{\max}}.$

∎

######  Corollary 3.10.

There exists an at most countable set $\Xi\subset\mathbb{R}^{n}$ and positive
weights $(\nu_{\xi})_{\xi\in\Xi}$, $(\mu_{\xi})_{\xi\in\Xi}$ such that

$\nu=\sum_{\xi\in\Xi}\nu_{\xi}\delta_{\\{\xi\\}}\quad\text{and}\quad\mu\geq\sum_{\xi\in\Xi}\mu_{\xi}\delta_{\\{\xi\\}}.$

###### Proof.

By Lemma 3.9 it becomes clear that $\nu\ll\mu$, and for every
$\xi\in\mathbb{R}^{n}$ and $\varepsilon\in(0,1/2)$,

(3.17) $\nu(M_{\varepsilon}(\xi))\leq
C\mu(M_{\varepsilon}(\xi))^{\frac{p^{\ast}}{p_{\max}}}$

for some constant $C=C(\|\mu\|)>0$. Since $\mu$ is a finite measure, the set
$\Xi$ of atoms is at most countable. This proves already that
$\mu\geq\sum_{\xi\in\Xi}\mu_{\xi}\delta_{\\{\xi\\}}$ and
$\nu\geq\sum_{\xi\in\Xi}\nu_{\xi}\delta_{\\{\xi\\}}$ for two families of
positive weights $(\mu_{\xi})_{\xi\in\Xi}$, $(\nu_{\xi})_{\xi\in\Xi}$ that are
given by $\mu_{\xi}=\lim_{\varepsilon\to 0}\mu(M_{\varepsilon}(\xi))$ and
$\nu_{\xi}=\lim_{\varepsilon\to 0}\nu(M_{\varepsilon}(\xi))$ for every
$\xi\in\Xi$. By $\nu\ll\mu$, it follows that

$\nu(M_{\varepsilon}(\xi))=\int_{M_{\varepsilon}(\xi)}D_{\mu}(x)\,\textnormal{d}\mu(x),$

where

$D_{\mu}(x):=\lim_{r\to 0}\frac{\nu(M_{r}(x))}{\mu(M_{r}(x))}\leq C\lim_{r\to
0}\mu(M_{r}(x))^{\frac{p^{\ast}}{p_{\max}}-1}=\begin{cases}C\mu_{x}^{\frac{p^{\ast}}{p_{\max}}-1},&~{}~{}x\in\Xi,\\\
0,&~{}~{}\text{else},\end{cases}$

and we used (3.17). Consequently,
$\nu=\sum_{\xi\in\Xi}\nu_{\xi}\delta_{\\{\xi\\}}$, as desired. ∎

As a consequence of Lemma 3.9 and Corollary 3.10, we can infer a reverse
Hölder inequality for the atoms of $\nu$ and $\widetilde{\mu}$.

######  Corollary 3.11.

Every $\xi\in\Xi$ is an atomic point of $\widetilde{\mu}$ and it holds

(3.18) $S\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}\leq\widetilde{\mu}(\\{\xi\\}).$

In particular,

$\|\widetilde{\mu}\|\geq
S\left(\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}+\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}\right).$

###### Proof.

In order to prove (3.18), we approximate $\widetilde{\mu}(\\{\xi\\})$ by
cutoff functions $(\phi_{\varepsilon,\xi})_{\varepsilon}$, where $\phi\in
C_{c}^{\infty}(\mathbb{R}^{n})$ with $\operatorname{supp}(\phi)\subset
M_{1}(0)$, $\phi(0)=1$ and $0\leq\phi\leq 1$. By Brezis–Lieb’s lemma and
(3.4), it holds that
$|v_{k}|^{p^{\ast}}\,\textnormal{d}x\rightharpoonup|v|^{p^{\ast}}\,\textnormal{d}x+\nu$,
and consequently:

(3.19) $\lim_{\varepsilon\to
0}\lim_{k\to\infty}\int_{\mathbb{R}^{n}}\phi_{\varepsilon,\xi}^{p^{\ast}}|v_{k}|^{p^{\ast}}\,\textnormal{d}x=\lim_{\varepsilon\to
0}\int_{M_{\varepsilon}(\xi)}\phi_{\varepsilon,\xi}^{p^{\ast}}|v|^{p^{\ast}}\,\textnormal{d}x+\lim_{\varepsilon\to
0}\int_{M_{\varepsilon}(\xi)}\phi_{\varepsilon,\xi}^{p^{\ast}}\,\textnormal{d}\nu\geq\nu_{\xi}.$

On the other hand, by (3.9), Lemma 3.7 and Lemma 3.6:

(3.20) $\displaystyle\limsup_{\varepsilon\to
0}\limsup_{k\to\infty}\left(\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi}v_{k})|^{p_{i}}\,\textnormal{d}x\right)^{1/p_{i}}$
$\displaystyle\leq\limsup_{\varepsilon\to
0}\limsup_{k\to\infty}\Bigg{[}\left(\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi})w_{k}|^{p_{i}}\,\textnormal{d}x\right)^{1/p_{i}}+\left(\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi})v|^{p_{i}}\,\textnormal{d}x\right)^{1/p_{i}}$
$\displaystyle\hskip
113.81102pt+\left(\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}|\phi_{\varepsilon,\xi}|^{p_{i}}\,\textnormal{d}x\right)^{1/p_{i}}\Bigg{]}$
$\displaystyle=\limsup_{\varepsilon\to
0}\limsup_{k\to\infty}\left(\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}|\phi_{\varepsilon,\xi}|^{p_{i}}\,\textnormal{d}x\right)^{1/p_{i}}.$

Consequently, by combining (3.19), (3.20) and applying Lemma 3.1,

$\begin{split}S\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}&\leq\lim_{\varepsilon\to
0}\lim_{k\to\infty}S\left(\int_{\mathbb{R}^{n}}\phi_{\varepsilon,\xi}^{p^{\ast}}|v_{k}|^{p^{\ast}}\,\textnormal{d}x\right)^{\frac{p_{\max}}{p^{\ast}}}\\\
&\leq\limsup_{\varepsilon\to
0}\limsup_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}(\phi_{\varepsilon,\xi}v_{k})|^{p_{i}}\,\textnormal{d}x\\\
&\leq\limsup_{\varepsilon\to
0}\limsup_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\int_{\mathbb{R}^{n}}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}|\phi_{\varepsilon,\xi}|^{p_{i}}\,\textnormal{d}x\\\
&\leq\limsup_{\varepsilon\to 0}\widetilde{\mu}(M_{\varepsilon}(\xi))\\\
&=\widetilde{\mu}(\\{\xi\\}).\end{split}$

Hence, (3.18) holds true. From the observation that
$\sum_{i=1}^{n}\frac{1}{p_{i}}|D^{s_{i}}_{p_{i}}v|^{p_{i}}\textnormal{d}x$ is
orthogonal to the atomic part of $\widetilde{\mu}$, we conclude that

$\|\widetilde{\mu}\|\geq
S\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}+\sum_{i=1}^{n}\frac{1}{p_{i}}\|D_{p_{i}}^{s_{i}}v\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}\geq
S\left(\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}+\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}\right),$

where we also applied Lemma 3.1. This concludes the proof. ∎

###  3.3. Proof of main results

This subsection is devoted to the proofs of the main results Theorem 1.2 and
Corollary 1.3. We first prove Theorem 3.3.

###### Proof of Theorem 3.3.

Let us assume by contradiction that $\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}<1$.
First, we have that

(3.21) $\displaystyle
1=\frac{1}{S}\left(\|\widetilde{\mu}\|+\mu_{\infty}\right)$
$\displaystyle\geq\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}+\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}+\nu_{\infty}^{\frac{p_{\max}}{p^{\ast}}}$
$\displaystyle\geq\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}+\|\nu\|^{\frac{p_{\max}}{p^{\ast}}}+\nu_{\infty}^{\frac{p_{\max}}{p^{\ast}}},$

where the second step follows from Lemma 3.8 and Corollary 3.11. In order to
prove the first equality in (3.21) we define
$\psi_{R}:\mathbb{R}^{n}\to\mathbb{R}$ as in the proof of Lemma 3.8 and
compute

$\begin{split}S&=\lim_{k\to\infty}\sum_{i=1}^{n}\frac{1}{p_{i}}\|D_{p_{i}}^{s_{i}}v_{k}\|_{L^{p_{i}}(\mathbb{R}^{n})}^{p_{i}}\\\
&=\lim_{R\to\infty}\lim_{k\to\infty}\int_{\mathbb{R}^{n}}\sum_{i=1}^{n}\frac{1}{p_{i}}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}(1-\psi_{R}^{p_{i}})\,\textnormal{d}x+\lim_{R\to\infty}\limsup_{k\to\infty}\int_{\mathbb{R}^{n}}\sum_{i=1}^{n}\frac{1}{p_{i}}|D^{s_{i}}_{p_{i}}v_{k}|^{p_{i}}\psi_{R}^{p_{i}}\,\textnormal{d}x\\\
&=\lim_{R\to\infty}\int_{\mathbb{R}^{n}}(1-\psi_{R}^{p_{i}})\,\textnormal{d}\widetilde{\mu}+\lim_{R\to\infty}\limsup_{k\to\infty}\int_{\mathbb{R}^{n}}\sum_{i=1}^{n}\frac{1}{p_{i}}|D^{s_{i}}_{p_{i}}w_{k}|^{p_{i}}\psi_{R}^{p_{i}}\,\textnormal{d}x\\\
&=\|\widetilde{\mu}\|+\mu_{\infty},\end{split}$

where we used (3.11). This proves (3.21). On the other hand, it holds that

(3.22)
$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}+\|\nu\|^{\frac{p_{\max}}{p^{\ast}}}+\nu_{\infty}^{\frac{p_{\max}}{p^{\ast}}}\geq
1.$

This follows from the observation

(3.23)
$\begin{split}1&=\lim_{k\to\infty}\|v_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}=\lim_{R\to\infty}\lim_{k\to\infty}\int_{\mathbb{R}^{n}}(1-\psi_{R}^{p^{\ast}})|v_{k}|^{p^{\ast}}\,\textnormal{d}x+\lim_{R\to\infty}\lim_{k\to\infty}\int_{\mathbb{R}^{n}}\psi_{R}^{p^{\ast}}|v_{k}|^{p^{\ast}}\,\textnormal{d}x\\\
&=\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}}+\|\nu\|+\nu_{\infty},\end{split}$

where we used that
$|v_{k}|^{p^{\ast}}\,\textnormal{d}x\rightharpoonup|v|^{p^{\ast}}\,\textnormal{d}x+\nu$
as a consequence of Brezis–Lieb’s lemma and (3.4), and (3.12). In order to
derive (3.22), we take (3.23) to the power $p_{\max}/p^{\ast}<1$. Combining
(3.21) and (3.22) yields that

$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p_{\max}}+\|\nu\|^{\frac{p_{\max}}{p^{\ast}}}+\nu_{\infty}^{\frac{p_{\max}}{p^{\ast}}}=1.$

Consequently,
$\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}^{p^{\ast}},\|\nu\|,\nu_{\infty}\in\\{0,1\\}$.
By assumption, we can now deduce that $v=0$. Moreover, by (3.2) we know that
$\nu_{\infty}\leq 1/2$ and therefore it must be that

$1=\|\nu\|=\sum_{\xi\in\Xi}\nu_{\xi},$

where we applied Corollary 3.10 in the last step. From (3.21), we deduce that
$1\geq\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}$. It follows that

$1=\left(\sum_{\xi\in\Xi}\nu_{\xi}\right)^{\frac{p_{\max}}{p^{\ast}}}\leq\sum_{\xi\in\Xi}\nu_{\xi}^{\frac{p_{\max}}{p^{\ast}}}\leq
1,$

and therefore all inequalities in the above line can be replaced by equality
signs. Thus, $\nu_{\xi}\in\\{0,1\\}$ for every $\xi\in\Xi$ and therefore,
there exists exactly one $\xi_{0}\in\Xi$ such that $\nu_{\xi_{0}}=1$ and
$\nu_{\xi}=0$ for every $\xi\in\Xi\setminus\\{\xi_{0}\\}$. This leads to a
contradiction since by (3.2):

$\frac{1}{2}=\sup_{y\in\mathbb{R}^{n}}\int_{M_{1}(y)}|v_{k}|^{p^{\ast}}\,\textnormal{d}x\geq\sup_{y\in\mathbb{R}^{n}}\int_{M_{1}(\xi_{0})}|v_{k}|^{p^{\ast}}\,\textnormal{d}x\to\|\nu\|=1.$

Therefore, it must hold that $\|v\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1$, as
desired. ∎

We are now ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

As explained in the beginning of Section 3, let us take a minimizing sequence
$(u_{k})\subset\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ for $S$
satisfying $\|u_{k}\|_{L^{p^{\ast}}(\mathbb{R}^{n})}=1$ for every
$k\in\mathbb{N}$. We may assume that $u_{k}$ is nonnegative because
$(|u_{k}|)$ is also a minimizing sequence for $S$. Indeed,
$\|D^{s_{i}}_{p_{i}}|u_{k}|\|_{L^{p_{i}}(\mathbb{R}^{n})}\leq\|D^{s_{i}}_{p_{i}}u_{k}\|_{L^{p_{i}}(\mathbb{R}^{n})}$
follows from the inequality

$||u(x)|-|u(x+he_{i})||\leq|u(x)-u(x+he_{i})|.$

Let us now consider the rescaled function $v_{k}$ as in the beginning of
Section 3, which satisfies (3.2), (3.3), and (3.4). Then, it follows from
(3.5) and Theorem 3.3 that $v$ is a nontrivial minimizer for $S$. Moreover,
since $v_{k}$ is nonnegative a.e. in $\mathbb{R}^{n}$, so is $v$. ∎

Let us next prove Corollary 1.3.

###### Proof of Corollary 1.3.

By the standard Lagrange multiplier rule, any minimizer
$u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ of (1.5) satisfies

$\sum_{i=1}^{n}s_{i}(1-s_{i})\int_{\mathbb{R}}\frac{|u(x)-u(x+he_{i})|^{p_{i}-2}(u(x)-u(x+he_{i}))}{|h|^{1+s_{i}p_{i}}}\,\mathrm{d}h=S|u|^{p^{\ast}-2}u\quad\text{in}~{}\mathbb{R}^{n}.$

Thus, the function $v$ defined by
$v(x)=u(S^{-\frac{1}{s_{1}p_{1}}}x_{1},\dots,S^{-\frac{1}{s_{n}p_{n}}}x_{n})$
solves (1.6). ∎

By rescaling a solution of (1.6), we get infinitely many solutions.

######  Corollary 3.12.

If $u\in\mathcal{D}^{\vec{s},\vec{p}}(\mathbb{R}^{n})$ solves (1.6), then the
function $v$ defined by (3.1) also solves (1.6).

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