# On mixed local-nonlocal problems with Hardy potential

Stefano Biagi  ,  Francesco Esposito  ,  Luigi Montoro  and  Eugenio Vecchi
Politecnico di Milano - Dipartimento di Matematica
Via Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>Dipartimento di
Matematica e Informatica, Università della Calabria
Ponte Pietro Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy
<EMAIL_ADDRESS>Dipartimento di Matematica e Informatica,
Università della Calabria
Ponte Pietro Bucci 31B, 87036 Arcavacata di Rende, Cosenza, Italy
<EMAIL_ADDRESS>Dipartimento di Matematica, Università di Bologna
Piazza di Porta San Donato 5, 40126 Bologna, Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the effect of the Hardy potential on existence,
uniqueness and optimal summability of solutions of the mixed local-nonlocal
elliptic problem

$-\Delta u+(-\Delta)^{s}u-\gamma\frac{u}{|x|^{2}}=f\text{ in }\Omega,\
u=0\text{ in }{\mathbb{R}}^{n}\setminus\Omega,$

where $\Omega$ is a bounded domain in ${\mathbb{R}}^{n}$ containing the origin
and $\gamma>0$. In particular, we will discuss the existence, non-existence
and uniqueness of solutions in terms of the summability of $f$ and of the
value of the parameter $\gamma$.

###### Key words and phrases:

Mixed local-nonlocal PDEs, Hardy potential, existence and regularity of
solutions

###### 2020 Mathematics Subject Classification:

Primary 35J75, 35A01, 35B65; Secondary 35M10, 35J15

The authors are members of INdAM. S.Biagi and E.Vecchi are partially supported
by the PRIN 2022 project 2022R537CS _$NO^{3}$ \- Nodal Optimization, Nonlinear
elliptic equations, Nonlocal geometric problems, with a focus on regularity_,
founded by the European Union - Next Generation EU. F. Esposito and L. Montoro
are partially supported by the PRIN 2022 project P2022YFAJH Linear and
Nonlinear PDE’s: New directions and Applications. F. Esposito and E. Vecchi
are partially supported by the INdAM-GNAMPA Project _Esistenza, unicità e
regolarità di soluzioni per problemi singolari_.

##  1\. Introduction

Elliptic and parabolic PDEs with the Hardy potential $|x|^{-2}$, where
$x\in\mathbb{R}^{n}$ with $n\geq 3$, are nowadays a classical topic of
investigation. We refer to the monograph [23] for a detailed description of
the mathematical and physical reasons that led to the study of such PDEs as
well as most of the known result both in the model local and nonlocal case,
namely when the leading operator is either $-\Delta$ or $(-\Delta)^{s}$ with
$s\in(0,1)$. In the latter case, the interesting Hardy potential is given by
$|x|^{-2s}$. Our aim here is to start the study of problems of the form

$\left\\{\begin{array}[]{rl}\mathcal{L}u-\gamma\,\dfrac{u}{|x|^{2}}=f(x)&\textrm{in
}\Omega,\\\ u=0&\textrm{in }\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{n}$ is an open and bounded set with smooth
enough boundary $\partial\Omega$, $0\in\Omega$, $f$ belongs to some suitable
$L^{m}(\Omega)$ and $\gamma>0$. It will be soon clarified that the choice of
the local Hardy potential is the most proper in this case, and this will
immediately provide a natural upper bound for $\gamma$. Finally, we introduce
the operator

$\mathcal{L}:=-\Delta+(-\Delta)^{s},$

where $(-\Delta)^{s}$ denotes the fractional Laplacian of order $s\in(0,1)$,
i.e.,

$(-\Delta)^{s}u=C_{n,s}\,\lim_{\varepsilon\to
0}\int_{\\{|x-y|\geq\varepsilon\\}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy,$

and $C_{n,s}>0$ is a suitable normalization constant (see, precisely, Remark
2.1).

The above mixed local-nonlocal operator is a particular instance of a general
class of operators firstly studied in the 60’s [11, 12] in connection with the
validity of maximum principle and more recently from a probabilistic point of
view, see, e.g., [13] and the references therein. Recently there has been a
renewed interest in problems presenting both a local and nonlocal nature,
mainly due to their interest in the applications see, e.g., [16]. The most
common thread of the more recent contributions concerns the development of a
regularity theory making mainly use of purely analytical techniques, see,
e.g., [5, 14, 19, 20, 25, 26] and the references therein, so to be able to
treat PDEs where the leading operator is $\mathcal{L}$.
As already announced, our interest in this paper is to start the study of
mixed local-nonlocal PDEs with a singular potential. We just mention that
other singular PDEs driven by $\mathcal{L}$ have been considered so far, see
e.g. [4, 18, 9]. Coming back to our issues and in order to clarify the choice
of the purely local Hardy potential, the starting point is the Hardy
inequality, originally proved in [21] in the one dimensional case and
subsequently extended in various directions:

(1.1)
$C\,\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx,\quad u\in C^{\infty}_{0}(\mathbb{R}^{n}),$

where $C>0$. As it is customary when dealing with functional inequalities, it
is interesting to detect whether there is an optimal constant for which (1.1)
holds and whether this is ever achieved. These questions are nowadays
classical for the inequality (1.1): the best constant is purely dimensional,
it is given by

$\Lambda_{n}:=\dfrac{(n-2)^{2}}{4},$

and it is never achieved. Similarly, there exists an optimal positive constant
$\overline{\Lambda}_{n,s}$, depending only on $n$ and $s$ and never achieved,
such that

(1.2)
$\overline{\Lambda}_{n,s}\,\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2s}}\,dx\leq\frac{C_{n,s}}{2}[u]^{2}_{s},\quad
u\in C^{\infty}_{0}(\mathbb{R}^{n}),$

where

$[u]^{2}_{s}:=\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy,$

denotes the Gagliardo seminorm of $u$ (of order $s\in(0,1)$).
The inequalities (1.1) with $C=\Lambda_{n}$ and (1.2) are the starting point
of our mixed local-nonlocal analysis. Clearly, the following trivially hold

$\Lambda_{n}\,\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx+\frac{C_{n,s}}{2}[u]^{2}_{s},\quad u\in
C^{\infty}_{0}(\mathbb{R}^{n}),$

and

$\overline{\Lambda}_{n,s}\,\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2s}}\,dx\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx+\frac{C_{n,s}}{2}[u]^{2}_{s},\quad u\in
C^{\infty}_{0}(\mathbb{R}^{n}).$

As probably expected, the best Hardy potential (in a sense to be specified
later on, see after Proposition 2.6) to be considered is the one coming from
the local Hardy inequality. This leads to the following natural question: does
the minimization problem

(1.3) $\Lambda_{s,n}:=\inf_{u\in
C^{\infty}_{0}(\mathbb{R}^{n})\setminus\\{0\\}}\dfrac{\rho(u)^{2}}{\mathcal{H}(u)}$

admit a minimizer? Here, we have introduced the shorthand notation

$\rho(u)^{2}:=\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{n})}+\frac{C_{n,s}}{2}[u]^{2}_{s}\quad\mathrm{and}\quad\mathcal{H}(u):=\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx.$

Our first result answers to the above question.

######  Theorem 1.1.

The number defined by the minimization problem (1.3) coincides with the
optimal local Hardy constant $\Lambda_{n}$ and it is never achieved.

Having in mind applications to mixed local-nonlocal PDEs, we are also
interested in studying Hardy-type inequalities in open and bounded sets
$\Omega\subset\mathbb{R}^{n}$. The second main result of the present paper is
the following

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{n}$ be an open and bounded set such that
$0\in\Omega$. Then

$\Lambda_{s,n}(\Omega):=\inf\left\\{\rho(u)^{2}:u\in
C^{\infty}_{0}(\Omega)\,\textrm{ s.t. }\,\mathcal{H}(u)=1\right\\},$

is independent of $\Omega$, coincides with $\Lambda_{n}$ and it is never
achieved.

In light of similar results proved for mixed local-nonlocal Sobolev
inequalities (see [7]), the above fact is not so surprising and it is
motivated by the lack of scaling invariance of the functional $\rho(u)$.

The aforementioned results open the way to the study of mixed local-nonlocal
PDEs in presence of the classical Hardy potential, namely

(1.4) $\left\\{\begin{array}[]{rl}-\Delta
u+(-\Delta)^{s}u-\gamma\dfrac{u}{|x|^{2}}=f&\mathrm{in}\,\Omega,\\\
u=0&\mathrm{in}\,\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{n}$ is an open and bounded set with smooth
enough boundary and such that $0\in\Omega$, $f\in L^{m}(\Omega)$ for some
$m\geq 1$ and $\gamma\in(0,\Lambda_{n})$.
The boundary condition is the purely nonlocal one and it is quite natural if
one is interested in Dirichlet-type boundary conditions, see, e.g., [5] for
examples in the case $\gamma=0$.
Our aim is to study the regularity of the solutions of (1.4) in terms of the
summability of the source term $f$: this forces to consider different notions
of solutions and it is nowadays quite classical. This line of research is
quite old in the case of variational local elliptic operators as well as
purely nonlocal ones. We refer to Subsection 2.2 for a detailed review of the
known results in the case of $-\Delta$, of $(-\Delta)^{s}$ and $\mathcal{L}$.
Having this background in mind, we state our first result which provides
existence, optimal solvability and gain of integrability of solutions which
can be considered of energy-type since $f\in L^{m}(\Omega)$ with
$m\geq\tfrac{2n}{n+2}$.

######  Remark 1.3.

Here, and in all the paper, we assume the positivity of the nonlinearity $f$,
in the sense that $f\gneq 0$ in $\Omega$, i.e. $f$ is not identically zero.
This assumption is not a restriction, since our problem is linear.

######  Theorem 1.4.

Let $f\in L^{m}(\Omega)$ with $m\geq(2^{*})^{\prime}=\tfrac{2n}{n+2}$. Then,
problem (1.4) admits a unique weak solution
$u_{f}\in\mathcal{X}^{1,2}(\Omega)$ (see Section 2 for the relevant
definition), in the following sense:

(1.5) $\begin{split}\int_{{\mathbb{R}}^{n}}\nabla u_{f}\cdot\nabla
v\,dx\,&+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(u_{f}(x)-u_{f}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy-\gamma\int_{\mathbb{R}^{n}}\dfrac{u_{f}v}{|x|^{2}}\,dx\\\
&=\int_{\Omega}fv\,dx,\quad\textrm{for every
}v\in\mathcal{X}^{1,2}(\Omega),\end{split}$

Moreover, if $f\in L^{m}(\Omega)$ with $\tfrac{2n}{n+2}\leq m<\tfrac{n}{2}$ is
a positive function and

$0<\gamma<\gamma(m):=\dfrac{n(m-1)(n-2m)}{m^{2}},$

then the unique weak solution $u_{f}\in\mathcal{X}^{1,2}(\Omega)$ of problem
(1.4) is such that

$u_{f}\in L^{m^{**}}(\Omega),\quad\textrm{where }m^{**}:=\tfrac{nm}{n-2m}.$

The next result deals with a source term $f\in L^{m}(\Omega)$ with
$1<m<\tfrac{2n}{n+2}$. In this case variational arguments are forbidden but
one can rely on the technique leading to approximated solutions by means of
suitable truncation arguments.

######  Theorem 1.5.

Let $f\in L^{m}(\Omega)$ with $1<m<\tfrac{2n}{n+2}$ be a positive function. If

$0<\gamma<\gamma(m):=\dfrac{n(m-1)(n-2m)}{m^{2}},$

then there exists a unique solution $u_{f}\in W_{0}^{1,m^{*}}(\Omega)$ of
(1.4), where $m^{*}:=\tfrac{nm}{n-m}$, in the following sense,
$u_{f}/|x|^{2}\in L^{1}(\Omega)$ and it holds:

(1.6) $\begin{split}\int_{{\mathbb{R}}^{n}}\nabla u_{f}\cdot\nabla
v\,dx\,&+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(u_{f}(x)-u_{f}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy-\gamma\int_{\mathbb{R}^{n}}\dfrac{u_{f}v}{|x|^{2}}\,dx\\\
&=\int_{\Omega}fv\,dx,\quad\textrm{for every }v\in
W_{0}^{1,(m^{*})^{\prime}}(\Omega)\cap L^{\infty}(\Omega),\end{split}$

where $(m^{*})^{\prime}=(m^{*}-1)/m^{*}$.

The last result concerns the case of $f\in L^{1}(\Omega)$, where we cannot
ensure (in general) that
$f\in\mathbb{X}=\left(\mathcal{X}^{1,2}(\Omega)\right)^{\prime}$, and thus
variational arguments are once again forbidden. As for the local case [2, 23],
we are able to provide an optimal solvability criteria. Nevertheless, due to
the nature of $\mathcal{L}$, which makes the task of finding explicit
solutions quite hard, our condition is somehow implicit. In order to give the
existence result, we need to consider solutions of an auxiliary problem. Let
$g\in L^{\infty}(\Omega)$ and let $w\in\mathcal{X}^{1,2}(\Omega)$ be the
unique weak solution of

(1.7) $\left\\{\begin{array}[]{rl}-\Delta
w+(-\Delta)^{s}w=g&\mathrm{in}\,\Omega,\\\
w=0&\mathrm{in}\,\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

in the sense of (1.5), with $\gamma=0$. We note that the unique solution $w$
belongs to $L^{\infty}(\Omega)$. Now, we introduce the notion of solution in
this setting, adapting a classical concept of duality.

######  Definition 1.6.

Let $f\in L^{1}(\Omega)$, and let $u\in L^{1}(\Omega)$. We say that $u$ is a
duality solution of (1.4), if the following conditions hold:

  1. (1)

$u\equiv 0$ a.e. in ${\mathbb{R}}^{n}\setminus\Omega$ and $u/|x|^{2}\in
L^{1}(\Omega)$;

  2. (2)

for every $g\in L^{\infty}(\Omega)$ and $w$ solution to (1.7), we have

$\begin{split}\int_{\Omega}gu\,dx=\gamma\int_{\Omega}\dfrac{u}{|x|^{2}}w\,dx+\int_{\Omega}fw\,dx.\end{split}$

######  Remark 1.7.

We notice that any function $u$ which solves (1.4) in the sense of Theorems
1.4 and 1.5 is also duality solutions (in the sense of Definition 1.6).

With Definition 1.6 at hand, we can now state the _optimal-solvability result_
for problem (1.4). Before doing this, we first fix a notation: _we denote by
$\Phi_{\Omega}$ the unique weak solution_ in $\mathcal{X}^{1,2}(\Omega)$ of
the following problem

$\begin{cases}\displaystyle\mathcal{L}u=\gamma\frac{u}{|x|^{2}}+1&\text{in
$\Omega$},\\\ u=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$},\end{cases}$

We explicitly observe that, since the constant function $f\equiv 1$ is
positive and bounded on $\Omega$, the existence and uniqueness of
$\Phi_{\Omega}$ follow from Theorem 1.4; moreover, we also have

$\text{$\Phi_{\Omega}>0$ a.e.\,in $\Omega$}.$

######  Theorem 1.8.

Let $f\in L^{1}(\Omega),\,\text{$f\geq 0$ a.e.\,in $\Omega$}$. Then, there
exists a positive duality solution $u\in L^{1}(\Omega)$ of problem (1.4) in
the sense of Definition 1.6 if and only if

(1.8) $\int_{\Omega}f\Phi_{\Omega}\,dx<\infty.$

In this case, one can also prove that $u$ satisfies the following properties:

  * 1)

for every $k\in\mathbb{N}$, one has $T_{k}(u)\in\mathcal{X}^{1,2}(\Omega)$
(see (2.11) for the definition of $T_{k}$);

  * 2)

$u|_{\Omega}$ is uniformly bounded in $W_{0}^{1,p}(\Omega)$ for every
$p<\frac{n}{n-1}$.

As a final comment, let us stress that either in the purely local and purely
nonlocal case the analogous results have been proved to be sharp. This is at
the present stage still an open problem for us, even if we believe that the
presence of a leading operator like $-\Delta$ should be enough to inherit the
same sharpness of the local case.

The paper is organized as follows: in Section 2 we firstly describe the
natural functional setting associated with $\mathcal{L}$ (see Subsection 2.1),
we then review all the regularity results connected with our problem (see
Subsection 2.2) and then we prove both Theorem 1.1 and Theorem 1.2 (see
Subsection 2.3). In Section 3 we move to the study of (1.4) proving Theorem
1.4, Theorem 1.5 and Theorem 1.8.

##  2\. The functional setting and Preliminary results

The aim of this section is to introduce the adequate functional setting for
the study of the mixed operator $\mathcal{L}=-\Delta+(-\Delta)^{s}$, and to
collect some known results concerning the $\mathcal{L}$-Dirichlet problem

($\mathrm{D}_{f}$) $\begin{cases}\mathcal{L}u=f&\text{in $\Omega$},\\\ u\equiv
0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$}\end{cases}$

where $\Omega\subseteq{\mathbb{R}}^{n}$ is a bounded domain and $f\in
L^{m}(\Omega)$ for some $m\geq 1$. We also study mixed Hardy type inequality,
by proving Theorem 1.1 and 1.2.

###  2.1. The functional setting

Let $s\in(0,1)$ be fixed, and let
$\varnothing\neq\Omega\subseteq{\mathbb{R}}^{n}$ (with $n\geq 3$) be an
_arbitrary_ open set, not necessarily bounded. We define the function space
$\mathcal{X}^{1,2}(\Omega)$ as the completion of $C_{0}^{\infty}(\Omega)$ with
respect to the _mixed global norm_

$\rho(u):=\left(\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\frac{C_{n,s}}{2}[u]^{2}_{s}\right)^{1/2},\qquad
u\in C_{0}^{\infty}(\Omega),$

where $[u]_{s}$ denotes the so-called _Gagliardo seminorm_ of $u$ (of order
$s$), that is,

$[u]_{s}:=\bigg{(}\iint_{{\mathbb{R}}^{2n}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy\bigg{)}^{1/2}.$

######  Remark 2.1.

A couple of observations concerning the space $\mathcal{X}^{1,2}(\Omega)$ are
in order.

  1. (1)

The norm $\rho(\cdot)$ is induced by the scalar product (or bilinear form)

$\mathcal{B}(u,v):=\int_{{\mathbb{R}}^{n}}\nabla u\cdot\nabla
v\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy,$

where $\cdot$ denotes the usual scalar product in the Euclidean space
${\mathbb{R}}^{n}$, while

$C_{n,s}:=2^{2s-1}\pi^{-\frac{n}{2}}\Gamma((n+2s)/2)/|\Gamma(-s)|;$

thus, $\mathcal{X}^{1,2}(\Omega)$ is endowed with a structure of a (real)
_Hilbert space_.

  2. (2)

Even if a function $u\in C_{0}^{\infty}(\Omega)$ _identically vanishes outside
$\Omega$_, it is often still convenient to consider in the definition of
$\rho(\cdot)$ the $L^{2}$-norm of $\nabla u$ _on the whole of
${\mathbb{R}}^{n}$_, rather than restricted to $\Omega$ (though of course the
result would be the same): this is to stress that the elements in
$\mathcal{X}^{1,2}(\Omega)$ are functions defined _on the entire space
${\mathbb{R}}^{n}$_ and not only on $\Omega$ (and this is consistent with the
nonlocal nature of the operator $\mathcal{L}$). The benefit of having this
global functional setting is that these functions can be _globally
approximated on ${\mathbb{R}}^{n}$_ (with respect to the norm $\rho(\cdot)$)
by smooth functions with compact support in $\Omega$.

In particular, when $\Omega\neq{\mathbb{R}}^{n}$, we will see that this
_global_ definition of $\rho(\cdot)$ implies that the functions in
$\mathcal{X}^{1,2}(\Omega)$ naturally satisfy the nonlocal Dirichlet condition

(2.1) $\text{$u\equiv 0$ a.e.\,in ${\mathbb{R}}^{n}\setminus\Omega$ for every
$u\in\mathcal{X}^{1,2}(\Omega)$}.$

In order to better understand the _nature_ of the space
$\mathcal{X}^{1,2}(\Omega)$ (and to recognize the validity of (2.1)), we
distinguish two cases.

(i) If _$\Omega$ is bounded_. In this case we first recall the following
inequality, which expresses the _continuous embedding_ of
$H^{1}({\mathbb{R}}^{n})$ into $H^{s}({\mathbb{R}}^{n})$ (see, e.g., [15,
Proposition 2.2]): there exists a constant $\mathbf{c}=\mathbf{c}(n,s)>0$ such
that, for every $u\in C_{0}^{\infty}(\Omega)$, one has

(2.2)
$[u]^{2}_{s}\leq\mathbf{c}(n,s)\|u\|_{H^{1}({\mathbb{R}}^{n})}^{2}=\mathbf{c}(n,s)\big{(}\|u\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{n})}\big{)}.$

This, together with the classical _Poincaré inequality_ , implies that
$\rho(\cdot)$ and the full $H^{1}$-norm in ${\mathbb{R}}^{n}$ are _actually
equivalent_ on the space $C^{\infty}_{0}(\Omega)$, and hence

$\displaystyle\mathcal{X}^{1,2}(\Omega)$
$\displaystyle=\overline{C_{0}^{\infty}(\Omega)}^{\,\,\|\cdot\|_{H^{1}({\mathbb{R}}^{n})}}$
$\displaystyle=\\{u\in H^{1}({\mathbb{R}}^{n}):\,\text{$u|_{\Omega}\in
H_{0}^{1}(\Omega)$ and $u\equiv 0$ a.e.\,in
${\mathbb{R}}^{n}\setminus\Omega$}\\}.$

(ii) If _$\Omega$ is unbounded_. In this case, even if the _embedding
inequality_ (2.2) is still satisfied, the Poincaré inequality _does not hold_
; hence, the norm $\rho(\cdot)$ is no more equivalent to the full $H^{1}$-norm
in ${\mathbb{R}}^{n}$, and $\mathcal{X}^{1,2}(\Omega)$ _is not a subspace of
$H^{1}({\mathbb{R}}^{n})$_.

On the other hand, by the classical _Sobolev inequality_ , there exists a
constant $\mathcal{S}=\mathcal{S}_{n}>0$, _independent of the open set
$\Omega$_, such that

(2.3) $\mathcal{S}_{n}\|u\|^{2}_{L^{2^{*}}({\mathbb{R}}^{n})}\leq\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{n})}\leq\rho(u)^{2}\qquad\text{for every $u\in
C_{0}^{\infty}(\Omega)$}.$

As a consequence, we deduce that $\mathcal{X}^{1,2}(\Omega)\hookrightarrow
L^{2^{*}}({\mathbb{R}}^{n}).$

######  Remark 2.2.

We explicitly notice that, since the _mixed Sobolev-type inequality_ (2.3)
holds _for every open set $\Omega\subseteq{\mathbb{R}}^{n}$_ (bounded or not),
we always have

(2.4) $\mathcal{X}^{1,2}(\Omega)\hookrightarrow L^{2^{*}}({\mathbb{R}}^{n}).$

Furthermore, by exploiting the density of $C_{0}^{\infty}(\Omega)$ in
$\mathcal{X}^{1,2}(\Omega)$, we can extend inequality (2.3) to _every function
$u\in\mathcal{X}^{1,2}(\Omega)$_, thereby obtaining

$\mathcal{S}_{n}\|u\|_{L^{2^{*}}({\mathbb{R}}^{n})}^{2}\leq\rho(u)^{2}=\|\nabla
u\|_{L^{2}({\mathbb{R}}^{n})}^{2}+\frac{C_{n,s}}{2}[u]^{2}_{s}\quad\text{for
every $u\in\mathcal{X}^{1,2}(\Omega)$}.$

Due to its relevance in the sequel, we also introduce a distinguished notation
for the cone of the non-negative functions in $\mathcal{X}^{1,2}(\Omega)$: we
set

$\mathcal{X}^{1,2}_{+}(\Omega)=\big{\\{}u\in\mathcal{X}^{1,2}(\Omega):\,\text{$u\geq
0$ a.e.\,in $\Omega$}\big{\\}}.$

###  2.2. The $\mathcal{L}$-Dirichlet problem

Now we have introduced the space $\mathcal{X}^{1,2}(\Omega)$, we present some
known results concerning _existence and optimal regularity of solutions_ for
the $\mathcal{L}$-Dirichlet problem ($\mathrm{D}_{f}$). To this end, we
distinguish two different cases.

$\displaystyle\mathrm{i)}$ $\displaystyle\,\,\text{$f\in L^{m}(\Omega)$ for
some $m\geq(2^{*})^{\prime}=\frac{2n}{n+2}$};$ $\displaystyle\mathrm{ii)}$
$\displaystyle\,\,\text{$f\in L^{m}(\Omega)$ for some
$1<m<(2^{*})^{\prime}$}.$

Case i) - The variational case. In this case we observe that, by combining the
_continuous embedding_ (2.4) of $\mathcal{X}^{1,2}(\Omega)$ with the Hölder
inequality, we have

$\int_{\Omega}|fv|\,dx<+\infty\quad\forall\,\,v\in\mathcal{X}^{1,2}(\Omega),$

and thus $f\in\mathbb{X}=(\mathcal{X}^{1,2}(\Omega))^{\prime}$; in view of
this fact, _existence and uniqueness_ of solutions for problem
($\mathrm{D}_{f}$) easily follow from the Lax - Milgram Theorem. More
precisely, we present the following theorem, which collects most of the
published results regarding the existence, uniqueness and regularity of
solutions to problem ($\mathrm{D}_{f}$).

######  Theorem 2.3.

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be a bounded open set with smooth
boundary, and let $f\in L^{m}(\Omega)$ for some $m\geq(2^{*})^{\prime}$. Then,
_there exists a unique weak solution_ $u_{f}\in\mathcal{X}^{1,2}(\Omega)$ of
problem ($\mathrm{D}_{f}$), in the following sense:

(2.5)
$\mathcal{B}(u_{f},v)=\int_{\Omega}fv\,dx\qquad\forall\,\,v\in\mathcal{X}^{1,2}(\Omega).$

Furthermore, the following properties hold.

  * i)

_(_ see [5, Thm. 4.7]_)_ If $f\in L^{m}(\Omega)$ with $m>n/2$, then $u_{f}\in
L^{\infty}({\mathbb{R}}^{n})$.

  * ii)

_(_ see [4, Thm. 2.3]_)_ If $(2^{*})^{\prime}\leq m\leq n/2$, then $u_{f}\in
L^{m^{**}}(\Omega)$, where

$m^{**}=\frac{nm}{n-2m}.$

  * iii)

_(_ see [3, Thm. 1.1]_)_ If $f\in L^{m}(\Omega)$ with $m>n$, then

$\text{$u_{f}\in C^{1,\theta}(\overline{\Omega})$ for some $\theta\in(0,1)$}.$

  * iv)

_(_ see [6, Thm. B.1]_)_ If $f\in C^{\alpha}(\overline{\Omega})$ for some
$\alpha\in(0,1)$, and if $2s+\alpha<1$ _(_ hence, in particular, $0<s<1/2$_)_
, then $u_{f}\in C^{2,\alpha}(\overline{\Omega})$.

  * v)

_(_ see, e.g., [5, Thm. 1.2] and [8, Cor. 3.3 and Rem. 3.4] _)_ If $f\geq 0$
a.e. in $\Omega$, then we have $u_{f}\geq 0$ a.e. in $\Omega$; moreover,
either $u_{f}\equiv 0$ or $u_{f}>0$ in $\Omega$.

Case ii) - The nonvariational case. In this case, the inclusion
$L^{m}(\Omega)\subseteq\mathbb{X}=(\mathcal{X}^{1,2}(\Omega))^{\prime}$ _does
not hold_ , and we cannot directly apply the Lax-Milgram Theorem to study the
existence of solutions for problem ($\mathrm{D}_{f}$). As in the purely local
setting, _existence and improved regularity_ are proved at the same time by an
approximation argument.

######  Theorem 2.4 (See [4, Theorem 2.1]).

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be a bounded open set with smooth
boundary, and let $f\in L^{m}(\Omega)$ for some $1<m<(2^{*})^{\prime}$, $f\geq
0$ a.e. in $\Omega$. Then, there exists a positive solution $u_{f}$ of problem
($\mathrm{D}_{f}$), in the following sense

  * a)

$u_{f}\in W_{0}^{1,m^{*}}(\Omega)$, with $m^{*}=nm/(n-m)$;

  * b)

for every $v\in W_{0}^{1,(m^{*})^{\prime}}(\Omega)\cap L^{m^{\prime}}(\Omega)$
with _compact support_ , we have

$\mathcal{B}(u_{f},v)=\int_{\Omega}fv\,dx.$

######  Remark 2.5 (A comparison with the local/nonlocal case).

It is interesting to notice that, on account of Theorems 2.3-2.4, the
_improved integrability_ for the solutions of ($\mathrm{D}_{f}$) is generated
only by $-\Delta$, thus reflecting the merely perturbative role of
$(-\Delta)^{s}$, at least when working on bounded sets. More precisely, let
$u_{f}$ be the solution (to be specified, depending on the summability of $f$)
of

$\begin{cases}-\Delta u=f\in L^{m}(\Omega)&\text{in $\Omega$}\\\ u=0&\text{in
$\partial\Omega$}.\end{cases}.$

Then, in [24] where more general operators are considered, Stampacchia showed
that

  * i)

if $1<m<(2^{*})^{\prime}$ then $u_{f}\in W^{1,m^{*}}_{0}(\Omega)$;

  * ii)

if $(2^{*})^{\prime}\leq m\leq n/2$ then $u_{f}\in H^{1}_{0}(\Omega)\cap
L^{m^{**}}(\Omega)$;

  * iii)

if $m>n/2$ then $u_{f}\in H^{1}_{0}(\Omega)\cap L^{\infty}(\Omega)$.

A similar results holds true in the purely nonlocal framework, that is, for
the solution $u_{f}$ of the following _non-local Dirichlet problem_

$\begin{cases}(-\Delta)^{s}u=f\in L^{m}(\Omega)&\text{in $\Omega$}\\\
u=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$}\end{cases}.$

In this case, setting

$m^{*}_{s}:=\dfrac{nm}{n-ms}\quad\textrm{and}\quad
m^{**}_{s}:=\dfrac{nm}{n-2ms},$

we have that (see [22, Theorem 13, Theorem 16, Theorem 17, Theorem 24])

  * a)

if $1<m<{2n}/(n+2s)$ then $u_{f}\in L^{m^{**}_{s}}(\Omega)$;

  * b)

if $2n/(n+2s)\leq m\leq n/(2s)$ then $u_{f}\in H^{s}_{0}(\Omega)\cap
L^{m^{**}_{s}}(\Omega)$;

  * c)

if $m>n/(2s)$ then $u\in H^{s}_{0}(\Omega)\cap L^{\infty}(\Omega)$.

###  2.3. Mixed Hardy-type inequalities

Now, we focus our attention on the mixed Hardy inequalities. In order to do
this, we need to introduce some notations and recall some well known results.
Let $n\geq 3$. The classical Hardy inequality states that

$\frac{(n-2)^{2}}{4}\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx,\quad u\in C^{\infty}_{0}(\mathbb{R}^{n}),$

and it is known that this constant $\Lambda_{n}=(n-2)^{2}/4$ is never
achieved, see e.g. [23]. Clearly,

(2.6)
$\Lambda_{n}\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx\leq\rho(u)^{2},\quad u\in C^{\infty}_{0}(\mathbb{R}^{n}).$

In particular, due to the mixed nature of the norm $\rho(u)$, it holds

(2.7)
$\overline{\Lambda}_{n,s}\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2s}}\,dx\leq\rho(u)^{2},\quad
u\in C^{\infty}_{0}(\mathbb{R}^{n}),$

as well, where $\overline{\Lambda}_{n,s}$ denotes the optimal constant for the
fractional Hardy inequality, see e.g. [23, Chapter 9]. Combining (2.6) and
(2.7), we get the following

######  Proposition 2.6.

For every $p\in[2s,2]$, there exists a positive constant $C>0$, depending only
on $n$ and $s$, such that

(2.8) $C\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{p}}\,dx\leq\rho(u)^{2},\quad
u\in C^{\infty}_{0}(\mathbb{R}^{n}),$

###### Proof.

Let $p\in[2s,2]$ be arbitrarily fixed. Owing to (2.6)-(2.7), we have

$\displaystyle\int_{{\mathbb{R}}^{n}}\frac{u^{2}}{|x|^{p}}\,dx$
$\displaystyle=\int_{\\{|x|\leq
1\\}}\frac{u^{2}}{|x|^{p}}\,dx+\int_{\\{|x|>1\\}}\frac{u^{2}}{|x|^{p}}\,dx$
$\displaystyle\leq\int_{\\{|x|\leq
1\\}}\frac{u^{2}}{|x|^{2}}\,dx+\int_{\\{|x|>1\\}}\frac{u^{2}}{|x|^{2s}}\,dx$
$\displaystyle\leq\Big{(}\frac{1}{\Lambda_{n}}+\frac{1}{\overline{\Lambda}_{n,s}}\Big{)}\rho(u)^{2}\quad\forall\,\,u\in
C_{0}^{\infty}({\mathbb{R}}^{n}),$

and this is precisely the desired (2.8). ∎

We notice that for $p=2$ we can take $C=\Lambda_{n}$, while for $p=2s$ we can
take $C=\overline{\Lambda}_{n,s}$; however, due to the monotonicity of
$|x|^{-p}$ it seems that the most interesting case is the one for $p=2$. Let
us focus for a moment to the case $p=2$, and let us introduce the shorthand
notation

$\mathcal{H}(u):=\int_{\mathbb{R}^{n}}\dfrac{u^{2}}{|x|^{2}}\,dx.$

It is natural to wonder whether the minimization problem

$\Lambda_{s,n}:=\inf_{u\in
C^{\infty}_{0}(\mathbb{R}^{n})\setminus\\{0\\}}\dfrac{\rho(u)^{2}}{\mathcal{H}(u)}$

admits a minimizer, that is whether $\Lambda_{s,n}$ is achieved or not. The
answer is given by Theorem 1.1 which we will prove right below:

###### Proof of Theorem 1.1.

Let $u\in C^{\infty}_{0}(\mathbb{R}^{n})$. By the very definition of
$\rho(u)$, we have that

$\Lambda_{n}\leq\dfrac{\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{n})}}{\mathcal{H}(u)}\leq\dfrac{\rho(u)^{2}}{\mathcal{H}(u)},$

and therefore, taking the infimum,

(2.9) $\Lambda_{n}\leq\Lambda_{s,n}.$

On the other hand, if $u\in C^{\infty}_{0}(\mathbb{R}^{n})$, we have that (for
$\lambda>0$),

$u_{\lambda}(x):=\lambda^{\tfrac{n-2}{2}}u(\lambda x)\in
C^{\infty}_{0}(\mathbb{R}^{n}),$

and therefore it can be used as a test function, finding

$\displaystyle\Lambda_{s,n}$
$\displaystyle\leq\dfrac{\rho(u_{\lambda})^{2}}{\mathcal{H}(u_{\lambda})}=\dfrac{\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{n})}+\lambda^{2s-2}\frac{C_{n,s}}{2}[u]^{2}_{s}}{\mathcal{H}(u)}$
$\displaystyle=\dfrac{\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{n})}}{\mathcal{H}(u)}+\lambda^{2s-2}\frac{C_{n,s}}{2}\dfrac{[u]^{2}_{s}}{\mathcal{H}(u)}\to\dfrac{\|\nabla
u\|^{2}_{L^{2}(\mathbb{R}^{n})}}{\mathcal{H}(u)},\quad\textrm{as
}\lambda\to+\infty.$

We can then conclude that

$\Lambda_{s,n}\leq\Lambda_{n},$

which combined with (2.9) closes the proof. ∎

Let us now move to the case of bounded sets. Firstly, let us define the set

$\mathcal{M}(\Omega):=\left\\{u\in
C^{\infty}_{0}(\Omega):\mathcal{H}(u)=1\right\\},$

and the constants

$\Lambda_{s,n}(\Omega):=\inf_{u\in
C^{\infty}_{0}(\Omega)\setminus\\{0\\}}\dfrac{\rho(u)^{2}}{\mathcal{H}(u)}=\inf\left\\{\rho(u)^{2}:u\in
C^{\infty}_{0}(\Omega)\cap\mathcal{M}(\Omega)\right\\}.$

We now prove the following proposition leading to the demonstration of Theorem
1.2.

######  Proposition 2.7.

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open and bounded set such that
$0\in\Omega$. Then

(2.10) $\Lambda_{s,n}(\Omega)=\Lambda_{n}.$

###### Proof.

Since $\rho(u)\geq\|\nabla u\|_{L^{2}(\Omega)}$ for every $u\in
C^{\infty}_{0}(\Omega)$, it holds that

$\Lambda_{s,n}(\Omega)\geq\Lambda_{n}.$

To prove the reverse inequality, we consider $r>0$ be such that
$B_{r}(0)\subseteq\Omega$. We now observe that, given any $u\in
C_{0}^{\infty}({\mathbb{R}}^{n})\cap\mathcal{M}({\mathbb{R}}^{n})$, there
exists $k_{0}=k_{0}(u)\in{\mathbb{N}}$ such that

$\mathrm{supp}(u)\subseteq B_{kr}(0)\quad\text{for every $k\geq k_{0}$};$

as a consequence, setting $u_{k}(x):=k^{\frac{n-2}{2}}u(kx)$ (for $k\geq
k_{0}$), we readily see that

$\mathrm{supp}(u_{k})\subseteq
B_{r}(0)\subseteq\Omega\quad\text{and}\quad\mathcal{H}(u_{k})=1.$

Taking into account the definition of $\Lambda_{s,n}(\Omega)$, we find that,
for every $k\geq k_{0}$,

$\displaystyle\Lambda_{s,n}(\Omega)$
$\displaystyle\leq\rho(u_{k})^{2}=\|\nabla
u_{k}\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\frac{C_{n,s}}{2}[u_{k}]^{2}_{s}=\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{n})}+k^{2s-2}\frac{C_{n,s}}{2}[u]^{2}_{s}.$

From this, letting $k\to\infty$ (and recalling that $s<1$), we obtain

$\Lambda_{s,n}(\Omega)\leq\|\nabla u\|^{2}_{L^{2}({\mathbb{R}}^{n})}.$

By the arbitrariness of $u\in
C^{\infty}_{0}(\Omega)\cap\mathcal{M}({\mathbb{R}}^{n})$ and the fact that
$\Lambda_{n}$ _is independent of the open set $\Omega$_, we finally infer that

$\Lambda_{s,n}(\Omega)\leq\inf\big{\\{}\|\nabla
u\|^{2}_{L^{2}({\mathbb{R}}^{n})}:\,C^{\infty}_{0}(\Omega)\cap\mathcal{M}({\mathbb{R}}^{n})\big{\\}}=\Lambda_{n},$

and hence $\Lambda_{s,n}(\Omega)=\Lambda_{n}$. ∎

######  Proposition 2.8.

Let $\Omega\subseteq\mathbb{R}^{n}$ be an open set (not necessarily bounded).
Then, $\Lambda_{s,n}(\Omega)$ is never achieved.

###### Proof.

Arguing by contradiction, let us suppose that there exists a function
$u_{0}\in\mathcal{X}^{1,2}(\Omega)$ such that $\mathcal{H}(u_{0})=1$ (hence,
$u_{0}\not\equiv 0$) and

$\rho(u_{0})^{2}=\|\nabla
u_{0}\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\frac{C_{n,s}}{2}[u_{0}]^{2}_{s}=\Lambda_{n}.$

Since $\mathcal{X}^{1,2}(\Omega)\subseteq\mathcal{D}^{1,2}(\Omega)$, we can
then infer that

$\Lambda_{n}\leq\|\nabla u_{0}\|^{2}_{L^{2}(\Omega)}\leq\|\nabla
u_{0}\|^{2}_{L^{2}({\mathbb{R}}^{n})}+\frac{C_{n,s}}{2}[u_{0}]^{2}_{s}=\rho(u_{0})^{2}=\Lambda_{n},$

from which we derive that $[u_{0}]_{s}=0$. As a consequence, the function
$u_{0}$ must be _constant in ${\mathbb{R}}^{n}$_, but this is contradiction
with the fact that $\mathcal{H}(u_{0})=1$. ∎

We are therefore ready to prove Theorem 1.2.

###### Proof of Theorem 1.2.

It is enough to combine Proposition 2.7 with Proposition 2.8. ∎

We end this section with the following lemma, showing a connection between the
Hardy-type inequality (2.8) and the existence of solutions for the variational
inequality

$\mathcal{L}u\geq\gamma\frac{u}{|x|^{2}}\quad\text{in $\Omega$}.$

In order to state and prove such a result, we first fix a notation: given any
$k\in\mathbb{N}$, we set

(2.11)
$T_{k}(t):=\max\\{\min\\{k,t\\},-k\\}=\left\\{\begin{array}[]{lr}t,&|t|\leq
k,\\\ k\dfrac{t}{|t|},&|t|>k.\end{array}\right.$

######  Lemma 2.9.

Let $\gamma>0$ be fixed, and let $u\in\mathcal{X}^{1,2}(\Omega)$, $u>0$, be
such that

$\mathcal{B}(u,v)\geq\gamma\int_{\Omega}\frac{uv}{|x|^{2}}\,dx\qquad\text{for
all $v\in\mathcal{X}^{1,2}_{+}(\Omega)$}.$

Then, we have

$\rho(v)^{2}\geq\gamma\int_{\Omega}\frac{v^{2}}{|x|^{2}}\,dx\quad\text{for all
$v\in\mathcal{X}^{1,2}(\Omega)$}.$

###### Proof.

This proof is inspired by [17, Lem. B.1]. First of all, let us consider the
following truncated problem given by

(2.12) $\begin{cases}\mathcal{L}w=T_{1}\left(\frac{u}{|x|^{2}}\right)&\text{in
$\Omega$}\\\ w=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$}.\end{cases}$

Thanks to Theorem 2.5, we have that problem (2.12) admits a unique positive
weak solution $w\in\mathcal{X}^{1,2}(\Omega)$. Hence, noticing that

$\mathcal{L}u\geq\gamma\frac{u}{|x|^{2}}\geq\gamma
T_{1}\left(\frac{u}{|x|^{2}}\right)=\mathcal{L}w\quad\text{in $\Omega$},$

and since the operator $\mathcal{L}$ is linear, we easily deduce that

$\begin{cases}\mathcal{L}(w-u)\leq 0&\text{in $\Omega$}\\\ w-u=0&\text{in
${\mathbb{R}}^{n}\setminus\Omega$}.\end{cases}$

As a consequence, since $v=w-u\in\mathcal{X}^{1,2}(\Omega)\subseteq
H^{1}({\mathbb{R}}^{n})$, we are entitled to apply the Weak Maximum Principle
in [5, Thm. 1.2], obtaining

$u\geq w\qquad\text{in }{\mathbb{R}}^{n}.$

In particular, since $w\in C^{1,\alpha}(\overline{\Omega})$ (see again Theorem
2.5), we deduce that for every compact set $\mathcal{K}\subset\Omega$ there
exists a constant $\vartheta>0$ such that

(2.13) $u\geq w\geq\vartheta>0\qquad\text{in }\mathcal{K},$

and this ensures that $u^{-1}\in L^{\infty}_{\mathrm{loc}}(\Omega)$.

Now we have established (2.13), we can easily complete the proof of the lemma.
Indeed, let $\varphi\in C_{0}^{\infty}(\Omega)$ be arbitrarily fixed, and let

$\psi=u^{-1}\varphi^{2}.$

By (2.13), we have $\psi\in L^{\infty}(\Omega)$; moreover,

$(u(x)-u(y))(\psi(x)-\psi(y))\leq|\varphi(x)-\varphi(y)|^{2}.$

Indeed, by the very definition of $\psi$ we have

(2.14)
$\begin{split}(u(x)-u(y))&(\psi(x)-\psi(y))=\varphi(x)^{2}+\varphi(y)^{2}-u(x)\frac{\varphi^{2}(y)}{u(y)}-u(y)\frac{\varphi^{2}(x)}{u(x)}\\\
&=(\varphi(x)-\varphi(y))^{2}+2\varphi(x)\varphi(y)-u(x)\frac{\varphi^{2}(y)}{u(y)}-u(y)\frac{\varphi^{2}(x)}{u(x)}\\\
&=(\varphi(x)-\varphi(y))^{2}+2\varphi(x)\varphi(y)-u(x)u(y)\frac{\varphi^{2}(y)}{u^{2}(y)}-u(x)u(y)\frac{\varphi^{2}(x)}{u^{2}(x)}\\\
&=(\varphi(x)-\varphi(y))^{2}-u(x)u(y)\Big{\\{}\frac{\varphi^{2}(x)}{u^{2}(x)}+\frac{\varphi^{2}(y)}{u^{2}(y)}-2\frac{\varphi(x)}{u(x)}\frac{\varphi(y)}{u(y)}\Big{\\}}\\\
&=(\varphi(x)-\varphi(y))^{2}-u(x)u(y)\Big{(}\frac{\varphi(x)}{u(x)}-\frac{\varphi(y)}{u(y)}\Big{)}^{2}\\\
&\leq(\varphi(x)-\varphi(y))^{2},\end{split}$

where we have also used the fact that $u>0$ a.e. in $\Omega$.

From now on we argue as in [17]. In order to prove the result we take $\psi$
as test function in

$\int_{{\mathbb{R}}^{n}}\nabla
u\cdot\nabla\psi\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(u(x)-u(y))(\psi(x)-\psi(y))}{|x-y|^{n+2s}}\,dxdy\geq\gamma\int_{\Omega}\frac{u\psi}{|x|^{2}}\,dx.$

Noticing that $\nabla
u\cdot\nabla\psi=|\nabla\varphi|^{2}-u^{2}|\nabla(u^{-1}\varphi)|^{2}\leq|\nabla\varphi|^{2}$,
and using (2.14), we deduce

$\int_{{\mathbb{R}}^{n}}|\nabla\varphi|^{2}\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(\varphi(x)-\varphi(y))^{2}}{|x-y|^{n+2s}}\,dxdy\geq\gamma\int_{\Omega}\frac{\varphi^{2}}{|x|^{2}}\,dx.$

∎

##  3\. The $\mathcal{L}$-Dirichlet problem involving the Hardy potential

Taking into account all the results recalled and/or established so far, we now
aim to study _existence and improved-integrability_ of the solutions for the
following $\mathcal{L}$-Dirichlet problem involving the Hardy potential, that
is,

($\mathrm{D}_{\gamma,f}$)
${}\left\\{\begin{array}[]{rl}\mathcal{L}u-\gamma\dfrac{u}{|x|^{2}}=f&\textrm{in
}\Omega,\\\ u=0&\textrm{in }\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\Omega\subset\mathbb{R}^{n}$ is an open, regular and bounded set with
$0\in\Omega$, and

$\text{$f\in L^{m}(\Omega)$ for some $m\geq
1$}\quad\text{and}\quad\gamma\in\left(0,\Lambda_{n}\right).$

To this end, motivated by the discussion in Section 2, we distinguish the
following three cases, according to the values of $m$.

$\displaystyle\mathrm{i)}$ $\displaystyle\,\,\text{$f\in L^{m}(\Omega)$ for
some $m\geq(2^{*})^{\prime}=\frac{2n}{n+2}$};$ $\displaystyle\mathrm{ii)}$
$\displaystyle\,\,\text{$f\in L^{m}(\Omega)$ for some
$1<m<(2^{*})^{\prime}$};$ $\displaystyle\mathrm{iii)}$ $\displaystyle\,\,f\in
L^{1}(\Omega).$

######  Remark 3.1.

We explicitly stress that the restriction $0<\gamma<\Lambda_{n}$, generally,
is motivated by the results in the previous section. Indeed, if we assume that
there exists a _nonnegative weak solution_ of problem
($\mathrm{D}_{\gamma,f}$) for some $\lambda>0$ and some $f\geq 0$, say
$u_{f}\in\mathcal{X}^{1,2}_{+}(\Omega)$, then

$\mathcal{B}(u_{f},v)=\gamma\int_{\Omega}\frac{u_{f}v}{|x|^{2}}\,dx+\int_{\Omega}fv\,dx\geq\gamma\int_{\Omega}\frac{u_{f}v}{|x|^{2}}\,dx\quad\text{for
all $v\in\mathcal{X}^{1,2}_{+}(\Omega)$};$

this, together with Lemma 2.9 and Theorem 1.1, implies that

$\Lambda_{n}=\inf_{u\in
C^{\infty}_{0}(\Omega)\setminus\\{0\\}}\dfrac{\rho(u)^{2}}{\mathcal{H}(u)}\geq\gamma.$

Hence, the last fact motivates nonexistence of positive energy solution of
problem ($\mathrm{D}_{\gamma,f}$) in the case $\gamma>\Lambda_{n}$ and $f$
sufficiently regular.

On the other hand, if we consider the case $\gamma=\Lambda_{n}$ in
($\mathrm{D}_{\gamma,f}$), there is a different approach that can be borrowed,
as pointed out in [23]. Thanks to a more general version of the Hardy
inequality, the improved Hardy inequality (see, e.g., [27]), one can define an
Hilbert space $\mathcal{H}(\Omega)$ larger than $\mathcal{X}^{1,2}(\Omega)$
where it is possible to find a unique positive solution
$u_{f}\in\mathcal{H}(\Omega)$ to ($\mathrm{D}_{\gamma,f}$) if $f$ belongs to
the dual space $\mathcal{H}^{\prime}(\Omega)$. We plan to come back to
possible extensions of such result in a future work.

###  3.1. Case i) - $f\in L^{m}(\Omega)$ with $m\geq(2^{*})^{\prime}$.

As already pointed out in Section 2, in this case we have
$f\in\mathbb{X}=(\mathcal{X}^{1,2}(\Omega))^{\prime}$; as a consequence, we
can investigate existence and uniqueness of solutions for ($\mathrm{D}_{f}$)
using the Lax-Milgram Theorem, obtaining the following result.

######  Theorem 3.2.

Let $f\in L^{m}(\Omega)$ with $m\geq(2^{*})^{\prime}$. Then, problem
($\mathrm{D}_{\gamma,f}$) admits a unique weak solution
$u_{f}\in\mathcal{X}^{1,2}(\Omega)$, in the following sense:

(3.1)
$\mathcal{B}(u,v)-\gamma\int_{\mathbb{R}^{n}}\dfrac{uv}{|x|^{2}}\,dx=\int_{\Omega}fv\,dx,\quad\textrm{for
every }v\in\mathcal{X}^{1,2}(\Omega).$

In particular, if $f\equiv 0$, the unique weak solution is the trivial one;
while, if $f\geq 0$ a.e. in $\Omega$, then $u_{f}\geq 0$ a.e. in $\Omega$.

Furthermore, if $f>0$ a.e. in $\Omega$, then $u_{f}>0$ a.e. in $\Omega$.

###### Proof.

The first assertion follows from Lax-Milgram Theorem: in fact, it suffices to
notice that the bilinear form defined as

$\mathcal{B}(u,v)-\gamma\int_{\mathbb{R}^{n}}\dfrac{uv}{|x|^{2}}\,dx$

is coercive and continuous on the Hilbert space $\mathcal{X}^{1,2}(\Omega)$.
The second assertion easily follows noticing that $u\equiv 0$ is a weak
solution to ($\mathrm{D}_{\gamma,f}$) when $f\equiv 0$.

Finally, we turn to prove $u_{f}\geq 0$ a.e. in $\Omega$ when $f\geq 0$. To
this end we first observe that, since $u_{f}\in\mathcal{X}^{1,2}(\Omega)$, we
have

$v:=(u_{f})^{-}=\max\\{-u_{f},0\\}\in\mathcal{X}^{1,2}(\Omega).$

We are then entitled to use this function $v$ as a test function in (3.1),
obtaining

$\displaystyle 0$
$\displaystyle\leq\int_{\Omega}fv\,dx=\mathcal{B}(u_{f},v)-\gamma\int_{\mathbb{R}^{n}}\dfrac{u_{f}v}{|x|^{2}}\,dx$
$\displaystyle=\int_{\Omega}\nabla u_{f}\cdot\nabla
v\,dx+\iint_{{\mathbb{R}}^{2n}}\frac{(u_{f}(x)-u_{f}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy-\gamma\int_{\mathbb{R}^{n}}\dfrac{u_{f}v}{|x|^{2}}\,dx$
$\displaystyle=-\Big{(}\int_{\Omega}|\nabla
v|^{2}\,dx-\gamma\int_{\Omega}\frac{v^{2}}{|x|^{2}}\,dx\Big{)}+\iint_{{\mathbb{R}}^{2n}}\frac{(u_{f}(x)-u_{f}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dx\,dy$
$\displaystyle(\text{since
$(u_{f}(x)-u_{f}(y))(v(x)-v(y))\leq-|v(x)-v(y)|^{2}$})$
$\displaystyle\leq-\Big{(}\rho(v)^{2}-\gamma\int_{\Omega}\frac{v^{2}}{|x|^{2}}\,dx\Big{)}$
$\displaystyle(\text{using Theorem \ref{thm:BestConstant}})$
$\displaystyle\leq-(\Lambda_{n}-\gamma)\rho(v)^{2}$

From this, since $0<\gamma<\Lambda_{n}$, we conclude that $v=(u_{f})^{-}=0$
a.e. in $\Omega$, and thus

(3.2) $u_{f}\geq 0\quad\textrm{ a.e.\,in }\Omega.$

Now, let us assume $f>0$ a.e. in $\Omega$. To show that $u_{f}>0$ a.e. in
$\Omega$ we now observe that, since $u_{f}$ is a weak solution of
($\mathrm{D}_{\gamma,f}$), we can write

$\mathcal{B}(u_{f},v)=\int_{\Omega}g(x,u_{f})v\,dx\quad\forall\,\,v\in\mathcal{X}^{1,2}(\Omega),$

where $g(x,t)=f(x)+\lambda\,t/|x|^{2}$; thus, since $f>0$ a.e. $\Omega$, we
have

$\text{$g(x,t)\geq 0$ for a.e.\,$x\in\Omega$ and $t\geq 0$}.$

Recalling (3.2), we can apply the Strong Maximum Principle in [8, Cor. 3.3],
showing that $u_{f}>0$ a.e. in $\Omega$ (as $f\not\equiv 0$). ∎

Similarly to the case $\gamma=0$ discussed in Section 2, see Theorem 2.3 \-
ii), also for the unique solution $u_{f}$ of problem ($\mathrm{D}_{\gamma,f}$)
we have an _improved-integrability result_.

######  Theorem 3.3.

Let $f\in L^{m}(\Omega)$ with $\tfrac{2n}{n+2}\leq m<\tfrac{n}{2},$ be a
positive function. If

$0<\gamma<\gamma(m):=\dfrac{n(m-1)(n-2m)}{m^{2}},$

then the unique weak solution $u_{f}\in\mathcal{X}^{1,2}(\Omega)$ of problem
($\mathrm{D}_{\gamma,f}$) _(_ whose existence is guaranteed by Theorem 3.2 _)_
is such that

$u_{f}\in L^{m^{**}}(\Omega),\quad\textrm{where }m^{**}:=\tfrac{nm}{n-2m}$

###### Proof.

As it is customary for such kind of problems, for every $k\in\mathbb{N}$ we
consider the unique weak solution $\varphi_{k}\in\mathcal{X}^{1,2}(\Omega)\cap
L^{\infty}(\mathbb{R}^{n})$ of the regularized problem

(3.3)
$\left\\{\begin{array}[]{rl}\mathcal{L}u=\gamma\dfrac{\varphi_{k-1}}{|x|^{2}+1/k}+f_{k}(x)&\textrm{in
}\Omega,\\\ u=0&\textrm{in }\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\varphi_{0}:=0$ and $f_{k}(x):=T_{k}(f(x))$. Owing to Lemma A.1, we
know that

  * i)

$\varphi_{k}$ is well-defined for every $k\geq 1$, and $\\{\varphi_{k}\\}_{k}$
is non-decreasing;

  * ii)

$\varphi_{k}\to u_{f}$ as $k\to+\infty$ strongly in $L^{p}(\Omega)$ for every
$p\in[1,2^{*})$, where $u_{f}\in\mathcal{X}^{1,2}(\Omega)$ is the unique weak
solution of problem ($\mathrm{D}_{\gamma,f}$) (in the sense of Theorem 3.2);

Let now

(3.4) $\alpha:=\dfrac{m(n-2)}{n-2m},$

and notice the following:

  * a)

$m\geq\tfrac{2n}{n+2}$ is equivalent to $\alpha\geq 2$;

  * b)

denoting by $m^{\prime}=\tfrac{m}{m-1}$ the usual Hölder conjugate exponent of
$m$, one has

$\frac{2^{*}\alpha}{2}=\frac{nm}{n-2m}=m^{**}=(\alpha-1)m^{\prime}.$

Since $\varphi_{k}\in\mathcal{X}^{1,2}_{+}(\Omega)$ for every
$k\in\mathbb{N}$, we can use
$v_{k}:=\varphi_{k}^{\alpha-1}(=|\varphi_{k}|^{\alpha-2}\varphi_{k}),$ as test
function in the variational formulation of (3.3), finding

(3.5)
$\displaystyle\mathcal{B}\left(\varphi_{k},\varphi_{k}^{\alpha-1}\right)$
$\displaystyle=\gamma\int_{\Omega}\dfrac{\varphi_{k-1}\varphi_{k}^{\alpha-1}}{(|x|^{2}+1/k)}\,dx+\int_{\Omega}f_{k}\varphi_{k}^{\alpha-1}\,dx$
$\displaystyle\leq\gamma\int_{\Omega}\dfrac{\varphi_{k}^{\alpha}}{|x|^{2}}\,dx+\int_{\Omega}f_{k}\varphi_{k}^{\alpha-1}\,dx,$

where we used the increasing monotonicity of $\\{\varphi_{k}\\}_{k}$.

Let us now consider the left hand side of (3.5). We have

$\displaystyle\mathcal{B}\left(\varphi_{k},\varphi_{k}^{\alpha-1}\right)$
$\displaystyle\qquad=\int_{\Omega}\nabla\varphi_{k}\cdot\nabla(\varphi_{k}^{\alpha-1})\,dx+\frac{C_{n,s}}{2}\iint_{\mathbb{R}^{2n}}\dfrac{(\varphi_{k}(x)-\varphi_{k}(y))(\varphi_{k}^{\alpha-1}(x)-\varphi_{k}^{\alpha-1}(y))}{|x-y|^{n+2s}}\,dx\,dy$
$\displaystyle\qquad=:(L)+(NL).$

The nonlocal part can be treated as in the proof of [1, Theorem 4.2]. Let us
give the details for sake of completeness: first, exploiting the algebraic
inequality [1, Equation (17)], i.e. for $s_{1},s_{2}\geq 0$ and $a>0$ it holds

$(s_{1}-s_{2})(s_{1}^{a}-s_{2}^{a})\geq\frac{4a}{(a+1)^{2}}\left(s_{1}^{\frac{a+1}{2}}-s_{2}^{\frac{a+1}{2}}\right)^{2},$

we deduce that

$(\varphi_{k}(x)-\varphi_{k}(y))(\varphi_{k}^{\alpha-1}(x)-\varphi_{k}^{\alpha-1}(y))\geq\dfrac{4(\alpha-1)}{\alpha^{2}}\left(\varphi_{k}^{\alpha/2}(x)-\varphi_{k}^{\alpha/2}(y)\right)^{2},$

and therefore

(3.6)
$(NL)\geq\dfrac{4(\alpha-1)}{\alpha^{2}}\left[\varphi_{k}^{\alpha/2}\right]_{s}^{2}.$

In the local part we take advantage of the following identity (see the proof
of [10, Theorem 2.1]),

$\nabla\varphi_{k}\cdot\nabla(\varphi_{k}^{\alpha-1})=\dfrac{4(\alpha-1)}{\alpha^{2}}\left|\nabla\varphi_{k}^{\alpha/2}\right|^{2},$

and therefore

(3.7)
$(L)=\dfrac{4(\alpha-1)}{\alpha^{2}}\int_{\Omega}\left|\nabla\varphi_{k}^{\alpha/2}\right|^{2}\,dx.$

Hence, combining (3.7) and (3.6) we get

(3.8)
$\mathcal{B}\left(\varphi_{k},\varphi_{k}^{\alpha-1}\right)\geq\dfrac{4(\alpha-1)}{\alpha^{2}}\mathcal{B}\left(\varphi_{k}^{\alpha/2},\varphi_{k}^{\alpha/2}\right)=\dfrac{4(\alpha-1)}{\alpha^{2}}\rho\left(\varphi_{k}^{\alpha/2}\right)^{2}.$

We now move to the right hand side of (3.5). By Hölder inequality, we find
that

(3.9)
$\int_{\Omega}f_{k}\varphi_{k}^{\alpha-1}\,dx\leq\|f_{k}\|_{L^{m}(\Omega)}\left(\int_{\Omega}\varphi_{k}^{(\alpha-1)m^{\prime}}\,dx\right)^{1/m^{\prime}},$

where $m^{\prime}$ is the Hölder conjugate exponent of $m$ defined in b). It
remains to deal with the last term: by the mixed Hardy inequality with best
constant, see Theorem 1.1, we get

(3.10)
$\gamma\int_{\Omega}\dfrac{\varphi_{k}^{\alpha}}{|x|^{2}}\,dx=\gamma\int_{\Omega}\left(\dfrac{\varphi_{k}^{\alpha/2}}{|x|}\right)^{2}\,dx\leq\dfrac{\gamma}{\Lambda_{n}}\rho\left(\varphi_{k}^{\alpha/2}\right)^{2}.$

Combining (3.5) with (3.8), (3.9) and (3.10), we get

(3.11)
$\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\rho\left(\varphi_{k}^{\alpha/2}\right)^{2}\leq\|f_{k}\|_{L^{m}(\Omega)}\left(\int_{\Omega}\varphi_{k}^{(\alpha-1)m^{\prime}}\,dx\right)^{1/m^{\prime}}.$

As in the purely local case, in order to have a meaningful a priori estimate
we need to have

$\gamma<\dfrac{4(\alpha-1)\Lambda_{n}}{\alpha^{2}}=\dfrac{n(m-1)(n-2m)}{m^{2}}=\gamma(m),$

as assumed. In order to conclude the proof we can follow the argument in [10].
By (2.3), and recalling b), we get

$\rho\left(\varphi_{k}^{\alpha/2}\right)^{2}\geq\mathcal{S}_{n}\|\varphi_{k}^{\alpha/2}\|_{L^{2^{*}}(\Omega)}^{2}=\mathcal{S}_{n}\left(\int_{\Omega}\varphi_{k}^{(2^{*}\alpha)/2}\,dx\right)^{2/2^{*}}\stackrel{{\scriptstyle
b)}}{{=}}\mathcal{S}_{n}\left(\int_{\Omega}\varphi_{k}^{(\alpha-1)m^{\prime}}\,dx\right)^{2/2^{*}},$

which, combined with (3.11) gives

$\displaystyle\mathcal{S}_{n}$
$\displaystyle\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\left(\int_{\Omega}\varphi_{k}^{(\alpha-1)m^{\prime}}\,dx\right)^{2/2^{*}-1/m^{\prime}}$
$\displaystyle=\mathcal{S}_{n}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\,\|\varphi_{k}\|_{L^{m^{**}}(\Omega)}\leq\|f_{k}\|_{L^{m}(\Omega)},$

and thus

$\dfrac{\mathcal{S}_{n}}{\Lambda_{n}}\,\,(\gamma(m)-\gamma)\,\,\|\varphi_{k}\|_{L^{m^{**}}(\Omega)}\leq\|f_{k}\|_{L^{m}(\Omega)}.$

Now, by using the Fatou Lemma it is sufficient to pass to the limit for
$k\rightarrow+\infty$ and note that $\varphi_{k}$ converges to the solution
$u_{f}$ of problem ($\mathrm{D}_{\gamma,f}$). From this we deduce the thesis.
∎

We are now ready to prove Theorem 1.4.

###### Proof of Theorem 1.4.

It is enough to combine Theorem 3.2 with Theorem 3.3. ∎

###  3.2. Case ii) - $f\in L^{m}(\Omega)$ with $1<m<(2^{*})^{\prime}$

As already observed in Section 2, in this case we cannot ensure that
$f\in\mathbb{X}=(\mathcal{X}^{1,2}(\Omega))^{\prime}$, and thus we _cannot
apply_ the Lax-Milgram Theorem to study existence and uniqueness of solutions
of problem ($\mathrm{D}_{\gamma,f}$). Hence, following the approach adopted in
[10] for the local case (and in [4] for the case $\gamma=0$), we prove
_existence and improved integrability_ at the same time with a truncation
argument.

###### Proof of Theorem 1.5.

Similarly to the proof of Theorem 3.3, for every $k\in\mathbb{N}$ we consider
the unique weak solution $\varphi_{k}\in\mathcal{X}^{1,2}_{+}(\Omega)\cap
L^{\infty}({\mathbb{R}}^{n})$ of the regularized problem

(3.12)
$\left\\{\begin{array}[]{rl}\mathcal{L}u=\gamma\dfrac{\varphi_{k-1}}{|x|^{2}+1/k}+f_{k}(x)&\textrm{in
}\Omega,\\\ u=0&\textrm{in }\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\varphi_{0}\equiv 0$ and $f_{k}(x):=\min\\{f(x),k\\}$. By arguing
exactly as in the _incipit_ of the proof of Lemma A.1, we easily see that
$\varphi_{k}$ is well-defined for every $k\geq 1$, and the sequence
$\\{\varphi_{k}\\}_{k}$ is non-decreasing. We then define the number
$\alpha\in(1,2)$ as in (3.4), and we use

$v_{k}:=(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1},\qquad\varepsilon>0,$

as test function in the variational formulation of (3.12). Recalling that

$(\varphi_{k}+\varepsilon)(x)-(\varphi_{k}+\varepsilon)(y)=\varphi_{k}(x)-\varphi_{k}(y),$

and making the same computations performed in the proof of Theorem 3.3, we get

(3.13)
$\mathcal{B}\left(\varphi_{k},v_{k}\right)=\mathcal{B}\left(\varphi_{k},(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}\right)\geq\dfrac{4(\alpha-1)}{\alpha^{2}}\rho\left((\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\right)^{2},$

and, using the monotonicity of $\\{\varphi_{k}\\}_{k}$ and the mixed Hardy
inequality, we also find

(3.14) $\displaystyle\gamma$
$\displaystyle\int_{\Omega}\dfrac{\varphi_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]}{|x|^{2}}\,dx+\int_{\Omega}f_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]\,dx$
$\displaystyle\leq\gamma\int_{\Omega}\left(\dfrac{(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}}{|x|}\right)^{2}\,dx$
$\displaystyle\quad+\gamma\int_{\Omega}\dfrac{|\varphi_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]-[(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}]^{2}|}{|x|^{2}}\,dx$
$\displaystyle\quad+\int_{\Omega}f_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]\,dx$
$\displaystyle\leq\dfrac{\gamma}{\Lambda_{n}}\rho\left((\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\right)^{2}$
$\displaystyle\quad+\gamma\int_{\Omega}\dfrac{|\varphi_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]-[(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}]^{2}|}{|x|^{2}}\,dx$
$\displaystyle\quad+\int_{\Omega}f_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]\,dx$

Combining (3.13), (3.14), we finally get

(3.15)
$\begin{split}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)&\rho\left((\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\right)^{2}\leq\int_{\Omega}f_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]\,dx\\\
&+\gamma\int_{\Omega}\dfrac{|\varphi_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]-[(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}]^{2}|}{|x|^{2}}\,dx.\end{split}$

Hence, using (2.3), we get

$\begin{split}\rho\left((\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\right)^{2}&\geq\mathcal{S}_{n}\|(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\|_{L^{2^{*}}(\Omega)}^{2}\\\
&=\mathcal{S}_{n}\left(\int_{\Omega}[(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}]^{2^{*}}\,dx\right)^{2/2^{*}}.\end{split}$

Now, for $k\in{\mathbb{N}}$ fixed, thanks to the Lebesgue’s dominated
convergence theorem we can pass to the limit for $\varepsilon$ that goes to
$0^{+}$, and thus we obtain

(3.16)
$\mathcal{S}_{n}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\left(\int_{\Omega}\varphi_{k}^{(2^{*}\alpha)/2}\,dx\right)^{2/2^{*}}\leq\int_{\Omega}f_{k}\varphi_{k}^{\alpha-1}\,dx.$

Hence, applying Hölder inequality in the right hand side of (3.16), we deduce

$\mathcal{S}_{n}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\left(\int_{\Omega}\varphi_{k}^{(2^{*}\alpha)/2}\,dx\right)^{2/2^{*}}\leq\|f_{k}\|_{L^{m}(\Omega)}\left(\int_{\Omega}\varphi_{k}^{(\alpha-1)m^{\prime}}\,dx\right)^{\frac{1}{m^{\prime}}}.$

Using the definition of $\alpha$ in (3.4) and recalling that
$\tfrac{\alpha}{2}\,2^{*}=\tfrac{nm}{n-2m}=m^{**}=(\alpha-1)m^{\prime}$, we
deduce

(3.17)
$\mathcal{S}_{n}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\|\varphi_{k}\|_{L^{m^{**}}(\Omega)}\leq\|f_{k}\|_{L^{m}(\Omega)}.$

Our aim is to show that the sequence $\\{\varphi_{k}\\}_{k}$ is bounded in
$W^{1,m^{*}}_{0}(\Omega)$. In order to prove this fact, from (3.15), we have

$\begin{split}&\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\left(\frac{\alpha-2}{2}\right)^{2}\int_{\Omega}\frac{|\nabla\varphi_{k}|^{2}}{(\varphi_{k}+\varepsilon)^{2-\alpha}}\,dx\\\
&\qquad\leq\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\rho\left((\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}\right)^{2}\\\
&\qquad\leq\gamma\int_{\Omega}\dfrac{|\varphi_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]-[(\varphi_{k}+\varepsilon)^{\alpha/2}-\varepsilon^{\alpha/2}]^{2}|}{|x|^{2}}\,dx\\\
&\qquad\qquad+\int_{\Omega}f_{k}[(\varphi_{k}+\varepsilon)^{\alpha-1}-\varepsilon^{\alpha-1}]\,dx.\end{split}$

If we fix $\varepsilon=R$ in (3.15), using (3.17), we get

$\begin{split}\int_{\Omega}\frac{|\nabla\varphi_{k}|^{2}}{(\varphi_{k}+R)^{2-\alpha}}\,dx\leq\mathcal{C}(m,n,\gamma,\mathcal{S}_{n},R)=:\mathcal{C}.\end{split}$

Thanks to Hölder inequality with exponents $(2/m^{*},2/(2-m^{*}))$ we get

$\begin{split}\int_{\Omega}|\nabla\varphi_{k}|^{m^{*}}\,dx&\leq\int_{\Omega}\frac{|\nabla\varphi_{k}|^{m^{*}}}{(\varphi_{k}+R)^{\frac{(2-\alpha)m^{*}}{2}}}(\varphi_{k}+R)^{\frac{(2-\alpha)m^{*}}{2}}\,dx\\\
&\leq\mathcal{C}^{\frac{m^{*}}{2}}\left(\int_{\Omega}(\varphi_{k}+R)^{\frac{(2-\alpha)m^{*}}{2-m^{*}}}\,dx\right)^{\frac{2-m^{*}}{2}}.\end{split}$

Finally, we have

$\begin{split}\int_{\Omega}|\nabla\varphi_{k}|^{m^{*}}\,dx\leq\mathcal{C}^{\frac{m^{*}}{2}}\left(\int_{\Omega}(\varphi_{k}+R)^{m^{**}}\,dx\right)^{\frac{2-m^{*}}{2}},\end{split}$

which implies the boundedness of $\\{\varphi_{k}\\}_{k}$ in
$W_{0}^{1,m^{*}}(\Omega)$. Since $m>1$ and $\varphi_{k}$ weakly solves (3.12),
namely

$\begin{split}\int_{{\mathbb{R}}^{n}}\nabla\varphi_{k}\cdot\nabla
v\,dx&+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(\varphi_{k}(x)-\varphi_{k}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dxdy\\\
&=\gamma\int_{\Omega}\dfrac{\varphi_{k-1}}{|x|^{2}+1/k}v\,dx+\int_{\Omega}f_{k}(x)v\,dx,\end{split}$

for every $v\in\mathcal{X}^{1,2}(\Omega)$, we deduce that there exists $u$
such that (up to subsequences)

$\varphi_{k}\rightharpoonup u\text{ in }W^{1,m^{*}}_{0}(\Omega)\text{ and a.e.
in }\Omega,\frac{\varphi_{k}}{|x|^{2}}\rightarrow\frac{u}{|x|^{2}}\text{
strongly in }L^{1}(\Omega).$

We note that, for any $v\in W^{1,(m^{*})^{\prime}}_{0}(\Omega)$ the following

$W^{1,m^{*}}_{0}(\Omega)\ni g\mapsto\int_{{\mathbb{R}}^{n}}\nabla g\cdot\nabla
v\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(g(x)-g(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dxdy$

is a bounded linear functional in $W^{1,m^{*}}_{0}(\Omega)$. Then for every
$v\in W^{1,(m^{*})^{\prime}}_{0}(\Omega)$, we obtain

$\begin{split}&\int_{{\mathbb{R}}^{n}}\nabla\varphi_{k}\cdot\nabla
v\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(\varphi_{k}(x)-\varphi_{k}(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dxdy\\\
&\qquad\longrightarrow\int_{{\mathbb{R}}^{n}}\nabla u\cdot\nabla
v\,dx+\frac{C_{n,s}}{2}\iint_{{\mathbb{R}}^{2n}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+2s}}\,dxdy\text{
as }k\rightarrow+\infty.\end{split}$

Collecting the last information on the operator, the convergence of $u_{k}$
discussed before, and using the Lebesgue’s dominated convergence theorem in
the right hand side of the truncated problem, we can pass to the limit in it,
showing that $u$ is a solution (as defined in (1.6)) belonging to
$W^{1,m^{*}}_{0}(\Omega)$.

Now we want to prove that this solution $u\in W^{1,m^{*}}_{0}(\Omega)$ is
unique. Consider another sequence of functions $\\{g_{k}\\}_{k}$ converging to
$f$ in $L^{m}(\Omega)$. Then let us consider the following regularized problem

$\left\\{\begin{array}[]{rl}\mathcal{L}w_{k}=\gamma\dfrac{w_{k-1}}{|x|^{2}+1/k}+g_{k}(x)&\textrm{in
}\Omega,\\\ w_{k}=0&\textrm{in
}\mathbb{R}^{n}\setminus\Omega.\end{array}\right.$

Hence, as before we can prove that, up to subsequences, $w_{k}$ converges to a
solution $w$ belonging to $W^{1,m^{*}}_{0}(\Omega)$. Hence, arguing on the
function $\varphi_{k}-w_{k}$, we deduce that it solves

$\left\\{\begin{array}[]{rl}\mathcal{L}(\varphi_{k}-w_{k})=\gamma\dfrac{(\varphi_{k-1}-w_{k-1})}{|x|^{2}+1/k}+f_{k}(x)-g_{k}(x)&\textrm{in
}\Omega,\\\ \varphi_{k}-w_{k}=0&\textrm{in
}\mathbb{R}^{n}\setminus\Omega.\end{array}\right.$

Arguing as above, we can derive an analogous estimate to (3.17) for the
difference function

$\mathcal{S}_{n}\left(\dfrac{4(\alpha-1)}{\alpha^{2}}-\dfrac{\gamma}{\Lambda_{n}}\right)\|\varphi_{k}-w_{k}\|_{L^{m^{**}}(\Omega)}\leq\|f_{k}-g_{k}\|_{L^{m}(\Omega)}.$

Passing to the limit for $k$ that goes to $+\infty$, we easily deduce that

$\|u-w\|_{L^{m^{**}}(\Omega)}=0,$

which immediately implies the uniqueness. ∎

###  3.3. Case iii) - $f\in L^{1}(\Omega)$

Taking into account all the results established so far, we end this section by
proving an _optimal condition_ for the existence of a positive duality
solution of problem ($\mathrm{D}_{\gamma,f}$) when $\beta\in(0,\Lambda_{n})$
and $f$ is just a _positive $L^{1}$ function_ in the sense of Definition 1.6.

###### Proof of Theorem 1.8.

(_Necessity of ( 1.8)_) We first assume that there exists a positive duality
solution $u$ of problem ($\mathrm{D}_{\gamma,f}$), and we show that the
‘integrability condition’ (1.8) is fulfilled by datum $f$.

To this end (and similarly to the proof of Theorems 3.3 \- 1.5), for every
$k\geq 1$ we consider the unique weak solution
$\phi_{k}\in\mathcal{X}_{+}^{1,2}(\Omega)\cap L^{\infty}({\mathbb{R}}^{n})$ of
the problem

(3.18) $\begin{cases}\mathcal{L}u=g_{k}&\text{in $\Omega$},\\\ u=0&\text{in
${\mathbb{R}}^{n}\setminus\Omega$},\end{cases}$

where $\phi_{0}\equiv 0$ and, to simplify the notation, we have set

$g_{k}(x)=\gamma\frac{\hat{\varphi}_{k-1}}{|x|^{2}+1/k}+1.$

Notice that, owing to Lemma A.1 (with $f\equiv 1$), we have

  * a)

$\phi_{k}$ is well-defined for every $k\geq 1$, and the sequence
$\\{\phi_{k}\\}_{k}$ is non-decreasing;

  * b)

$\phi_{k}\to\Phi_{\Omega}$ as $k\to+\infty$ in $L^{p}(\Omega)$, for every
$1\leq p<2^{*}$.

Now, choosing $w:=\phi_{k}$ and $g:=g_{k}$ in Definition 1.6 (notice that this
is legitimate, since we have $g_{k}\in L^{\infty}(\Omega)$ and
$\phi_{k}\in\mathcal{X}_{+}^{1,2}(\Omega)\cap L^{\infty}({\mathbb{R}}^{n})$
solves (3.18)), we get

(3.19)
$\begin{split}\int_{\Omega}f\phi_{k}\,dx&=\int_{\Omega}u\left(g_{k}-\gamma\frac{\phi_{k}}{|x|^{2}}\right)dx\\\
&(\text{by definition of $g_{k}$})\\\
&=\int_{\Omega}u\left(1+\gamma\frac{\phi_{k-1}}{|x|^{2}+1/k}-\gamma\frac{\phi_{k}}{|x|^{2}}\right)dx\\\
&(\text{by monotonicity of $\hat{\varphi}_{k}$})\\\
&\leq\int_{\Omega}u\left[1+\gamma\phi_{k}\left(\frac{1}{|x|^{2}+1/k}-\frac{1}{|x|^{2}}\right)\right]dx\\\
&(\text{since $\phi_{k}$ is positive})\\\
&=\int_{\Omega}u\,dx=\|u\|_{L^{1}(\Omega)}<\infty.\end{split}$

From this, by passing to the limit as $k\to\infty$ in (3.19) with the aid of
the Monotone Convergence Theorem (and recalling b)), we immediately obtain the
desired (1.8).

_Proof_. (_Sufficiency of ( 1.8)_). We now assume that $f$ satisfies condition
(1.8), and we prove that there exists a solution $u$ of problem
($\mathrm{D}_{\gamma,f}$), further satisfying properties 1) - 2).

To this end, for every fixed $k\geq 1$ we consider once again the unique
variational solution $\varphi_{k}\in\mathcal{X}_{+}^{1,2}(\Omega)\cap
L^{\infty}({\mathbb{R}}^{n})$ of the approximated problem

(3.20)
$\begin{cases}\mathcal{L}u=\gamma\frac{\varphi_{k-1}}{|x|^{2}+1/k}+f_{k}&\text{in
$\Omega$},\\\ u=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$},\end{cases}$

where $\varphi_{0}\equiv 0$ and $f_{k}=T_{k}(f)$ (here, $T_{k}(\cdot)$ is the
truncation operator defined in (2.11)). By arguing as in the _incipit_ of the
proof of Lemma A.1, we see that $\varphi_{k}$ is well-defined for every $k\geq
1$, and the sequence $\\{\varphi_{k}\\}_{k}$ is non-decreasing; hence, we can
define

$u(x)=\lim_{n\to\infty}\varphi_{k}(x)=\sup\\{\varphi_{k}(x):\,n\in\mathbb{N}\\}\geq
0.$

We then prove that this $u$ is a duality solution of
($\mathrm{D}_{\gamma,f}$), further satisfying 1) - 2).

a)  _$u$ is a duality solution of ($\mathrm{D}_{\gamma,f}$)._ First of all, we
show that $u\in L^{1}(\Omega)$. To this end it suffices to observe that, using
$\varphi_{k}$ as a test function for the (variational) equation solved by
$\Phi_{\Omega}$ and viceversa, see (3.1), from the assumed condition (1.8) we
have

(3.21)
$\begin{split}\int_{\Omega}\varphi_{k}\,dx&=-\gamma\int_{\Omega}\frac{\Phi_{\Omega}\varphi_{k}}{|x|^{2}}\,dx+\mathcal{B}(\Phi_{\Omega},\varphi_{k})\\\
&(\text{since $\varphi_{k}$ solves \eqref{eq:PbApproxBis}})\\\
&=\gamma\int_{\Omega}\Phi_{\Omega}\Big{(}\frac{\varphi_{k-1}}{|x|^{2}+1/k}-\frac{\varphi_{k}}{|x|^{2}}\Big{)}dx+\int_{\Omega}f_{k}\Phi_{\Omega}\,dx\\\
&\leq\int_{\Omega}f_{k}\Phi_{\Omega}\,dx\leq\int_{\Omega}f\Phi_{\Omega}\,dx=\mathbf{c}<\infty;\end{split}$

this, together with the Monotone Convergence theorem, proves at once that
$u\in L^{1}(\Omega)$ (in particular, $u<\infty$ a.e. in $\Omega$), and
$\varphi_{k}\to u$ in $L^{1}(\Omega)$.

Taking into account Definition 1.6, we now turn to prove that
${u}/{|x|^{2}}\in L^{1}(\Omega).$ To this end, let
$\phi_{1}\in\mathcal{X}^{1,2}_{+}(\Omega)\cap L^{\infty}(\Omega)$ be as
before, that is, $\phi_{1}$ is the unique solution of

$\begin{cases}\mathcal{L}u=1&\text{in $\Omega$}\\\ u=0&\text{in
${\mathbb{R}}^{n}\setminus\Omega$}.\end{cases}$

Moreover, let $r>0$ be such that $B_{r}(0)\Subset\Omega$. Owing to the Weak
Harnack inequality in [19, Theorem 8.1], we can find a constant $C>0$,
depending on $\phi_{1}$ and on $r$, such that

$\text{$\phi_{1}\geq C$ a.e.\,in $B_{r}(0)$};$

as a consequence, using $\varphi_{k}$ as a test function for the equation
solved by $\phi_{1}$ and viceversa, from the above (3.21) we obtain the
following estimate

(3.22)
$\begin{split}\int_{\Omega}\frac{\varphi_{k-1}}{|x|^{2}+1/k}\,dx&\leq\frac{1}{C}\int_{B_{r}(0)}\frac{\phi_{1}\varphi_{k-1}}{|x|^{2}+1/k}\,dx+\frac{1}{r^{2}}\int_{\Omega\setminus
B_{r}(0)}\varphi_{k-1}\,dx\\\ &(\text{since $\phi_{1},\,\varphi_{k-1}\geq 0$
a.e.\,in $\Omega$})\\\
&\leq\frac{1}{C}\int_{\Omega}\frac{\phi_{1}\varphi_{k-1}}{|x|^{2}+1/k}\,dx+\frac{1}{r^{2}}\int_{\Omega}\varphi_{k-1}\,dx\\\
&(\text{using $\phi_{1}$ as a test function for the equation solved by
$\varphi_{k}$})\\\
&\leq\frac{1}{C}\mathcal{B}(\varphi_{k},\phi_{1})+\frac{1}{r^{2}}\int_{\Omega}\varphi_{k-1}\,dx\\\
&(\text{using $\varphi_{k}$ as a test function for the equation solved by
$\phi_{1}$})\\\
&=\frac{1}{C}\int_{\Omega}\varphi_{k}\,dx+\frac{1}{r^{2}}\int_{\Omega}\varphi_{k-1}\,dx\leq\mathbf{c}<\infty,\end{split}$

where $\mathbf{c}>0$ is a constant independent of $k$. From this, again by the
Monotone Convergence Theorem we infer that

$\frac{u}{|x|^{2}}\in
L^{1}(\Omega)\quad\text{and}\quad\frac{\varphi_{k-1}}{|x|^{2}+1/k}\to\frac{u}{|x|^{2}}\,\,\text{in
$L^{1}(\Omega)$}.$

Gathering all these facts, we can easily prove that $u$ is a duality solution
of problem ($\mathrm{D}_{\gamma,f}$) in the sense of Definition 1.6: in fact,
we already know that $u,\,u/|x|^{2}\in L^{1}(\Omega)$; moreover, since
$\varphi_{k}$ solves (3.20), we get

$\int_{\Omega}\mathcal{B}(\varphi_{k},\psi)\,dx=\int_{\Omega}\gamma\frac{\varphi_{k-1}\psi}{|x|^{2}+1/k}dx+\int_{\Omega}f_{k}\psi\,dx\quad\forall\,\,\psi\in\mathcal{X}^{1,2}(\Omega)\cap
L^{\infty}(\Omega).$

If, in particular, $\psi$ is a solution of (1.7), we have

$\int_{\Omega}g\varphi_{k}=\int_{\Omega}\gamma\frac{\varphi_{k-1}\psi}{|x|^{2}+1/k}dx+\int_{\Omega}f_{k}\psi\,dx.$

Then, letting $k\to\infty$ (and observing that $f_{k}\to f$ in
$L^{1}(\Omega)$), by the above considerations we conclude that $u$ is a
solution of ($\mathrm{D}_{\gamma,f}$).

b)  _$u$ satisfies _1) - 2)__. We first prove that validity of property 1). To
this end, we arbitrarily fix $j,k\in\mathbb{N}$ and we use
$T_{k}(\varphi_{j})$ as a test function for the equation solved by
$\varphi_{j}$: taking into account the computation in (3.22), we get

$\displaystyle\int_{\Omega}|\nabla T_{k}(\varphi_{j})|^{2}\,dx$
$\displaystyle\leq\mathcal{B}(\varphi_{j},T_{k}(\varphi_{j}))=\gamma\int_{\Omega}\frac{\varphi_{j-1}\,T_{k}(\varphi_{j})}{|x|^{2}+1/j}\,dx+\int_{\Omega}f_{j}\,T_{k}(\varphi_{j})\,dx$
$\displaystyle(\text{since $\varphi_{j-1},\,f_{j}\geq 0$ and $0\leq
T_{k}(\varphi_{k})\leq k$})$ $\displaystyle\leq
k\Big{(}\gamma\int_{\Omega}\frac{\varphi_{j-1}}{|x|^{2}+1/j}\,dx+\int_{\Omega}f\,dx\Big{)}$
$\displaystyle(\text{using \eqref{eq:uoverxL1}, and since $f\in
L^{1}(\Omega)$})$ $\displaystyle\leq\mathbf{c}_{k}<\infty,$

and this shows that the sequence $\\{T_{k}(\varphi_{j})\\}_{j}$ is bounded in
$\mathcal{X}^{1,2}(\Omega)$. Then, by the Sobolev Embedding Theorem there
exists a function $g\in\mathcal{X}^{1,2}(\Omega)$ such that (up to a
subsequence)

i) $T_{k}(\varphi_{j})\rightharpoonup g$ in $\mathcal{X}^{1,2}(\Omega)$ as
$j\to\infty$;

ii) $T_{k}(\varphi_{j})\to g$ strongly in $L^{2}(\Omega)$ and pointwise a.e.
in $\Omega$.

On the other hand, since we also have $T_{k}(\varphi_{j})\to T_{k}(u)$
pointiwse a.e. in $\Omega$ as $j\to\infty$ (recall that, by definition, $u$ is
the pointwise limit of $\varphi_{j}$), we conclude that

$g=T_{k}(u),$

and this proves that $T_{k}(u)\in\mathcal{X}^{1,2}(\Omega)$ for every
$k\in\mathbb{N}$, as desired.

We then turn to prove the validity of property 2). To this end, we fix
$\beta<1/2$ (to be conveniently chosen later on), and we use the function
$w_{k}=(1+\varphi_{k})^{2\beta-1}-1$ as a test function for the equation
solved by $\varphi_{k}$ (see, precisely, (3.20)): this gives

(3.23)
$\mathcal{B}(\varphi_{k},w_{k})=\gamma\int_{\Omega}\frac{\varphi_{k-1}w_{k}}{|x|^{2}+1/k}\,dx+\int_{\Omega}f_{k}w_{k}\,dx.$

We now observe that, since $\beta<1/2$, we have $|w_{k}|\leq 1$; as a
consequence, for the second term of the right hand side of the above (3.23),
we have

$\int_{\Omega}f_{k}w_{k}\,dx\leq\int_{\Omega}f_{k}\,dx\leq\|f\|_{L^{1}(\Omega)}.$

For the second term, thanks to (3.22), we have

$\gamma\int_{\Omega}\frac{\varphi_{k-1}w_{k}}{|x|^{2}+1/k}\,dx\leq\gamma\int_{\Omega}\frac{\varphi_{k-1}}{|x|^{2}+1/k}\,dx\leq\mathbf{c}.$

Combining this last two information we deduce

$\|f\|_{L^{1}(\Omega)}+\mathbf{c}\geq\mathcal{B}(\varphi_{k},w_{k}).$

Using the same inequality of (3.13) with $\varepsilon=1$, we deduce

$\|f\|_{L^{1}(\Omega)}+\mathbf{c}\geq\mathcal{B}(\varphi_{k},w_{k})\geq\frac{2\beta-1}{\beta^{2}}\rho\left((1+\varphi_{k})^{\beta}-1\right)^{2}\geq\frac{2\beta-1}{\beta^{2}}\int_{\Omega}|\nabla(1+\varphi_{k})^{\beta}|^{2}\,dx.$

Hence, from the previous estimate we have

(3.24)
$\int_{\Omega}\frac{|\nabla\varphi_{k}|^{2}}{(1+\varphi_{k})^{2(1-\beta)}}\,dx\leq\frac{\|f\|_{L^{1}(\Omega)}+\mathbf{c}}{2\beta-1}.$

On the other hand we can write

$\int_{\Omega}|\nabla\varphi_{k}|^{p}\,dx=\int_{\Omega}\frac{|\nabla\varphi_{k}|^{p}}{(1+\varphi_{k})^{2(1-\beta)\frac{p}{2}}}(1+\varphi_{k})^{2(1-\beta)\frac{p}{2}}\,dx.$

Applying the Hölder inequality with exponents $(2/p,2/(2-p))$ on the right
hand side, and the Sobolev inequality on the left hand side, we obtain

$\displaystyle\mathcal{S}_{n}^{p}\left(\int_{\Omega}\varphi_{k}^{p^{*}}\,dx\right)^{\frac{p}{p^{*}}}$
$\displaystyle\leq\int_{\Omega}|\nabla\varphi_{k}|^{p}\,dx$
$\displaystyle\leq\left(\int_{\Omega}\frac{|\nabla\varphi_{k}|^{2}}{(1+\varphi_{k})^{2(1-\beta)}}\,dx\right)^{\frac{p}{2}}\left(\int_{\Omega}(1+\varphi_{k})^{\frac{2p(1-\beta)}{2-p}}\,dx\right)^{1-\frac{p}{2}}.$

Thanks to (3.24), we get

(3.25)
$\mathcal{S}_{n}^{p}\left(\int_{\Omega}\varphi_{k}^{p^{*}}\,dx\right)^{\frac{p}{p^{*}}}\leq\int_{\Omega}|\nabla\varphi_{k}|^{p}\,dx\leq
C|\Omega|^{1-\frac{p}{2}}+C\left(\int_{\Omega}\varphi_{k}^{\frac{2p(1-\beta)}{2-p}}\,dx\right)^{1-\frac{p}{2}},$

where $C=\left[(\|f\|_{L^{1}(\Omega)}+\mathbf{c})/(2\beta-1)\right]^{p/2}$,
and it is independent on $k$.

Now, we can choose $\beta$ such that

$\frac{2p(1-\beta)}{2-p}=p^{*},$

and since $\beta<1/2$, we have $p<\frac{n}{n-1}$. Thanks to this choice of
$\beta$, we have

$\mathcal{S}_{n}^{p}\left(\int_{\Omega}\varphi_{k}^{p^{*}}\,dx\right)^{\frac{p}{p^{*}}}\leq
C|\Omega|^{1-\frac{p}{2}}+C\left(\int_{\Omega}\varphi_{k}^{p^{*}}\,dx\right)^{1-\frac{p}{2}},$

thus we can deduce that $\varphi_{k}$ is uniformly bounded in
$L^{p^{*}}(\Omega)$. From (3.25) we also deduce that $\nabla\varphi_{k}$ is
uniformly bounded in $L^{p}(\Omega)$. Combining the last two information, we
get that $\varphi_{k}$ is uniformly bounded in $W^{1,p}_{0}(\Omega)$. Arguing
as before, it is possible to show that up to subsequences
$\varphi_{k}\rightarrow u$ in $W^{1,p}_{0}(\Omega)$, for $p<\frac{n}{n-1}$.
This closes the proof. ∎

##  Appendix A A convergence lemma

The aim of this appendix is to give an auxiliary result which clarify the
convergence of the sequence $\\{u_{k}\\}$ in in the proof of Theorem 1.5.

######  Lemma A.1 (Approximated Dirichlet problems).

Let $\Omega\subseteq{\mathbb{R}}^{n}$ be a bounded open set with smooth enough
boundary $\partial\Omega$, and let $f\in
L^{\frac{2n}{n+2}}(\Omega),\,\text{$f\geq 0$ a.e.\,in $\Omega$}.$ For a fixed
$\gamma\in(0,\Lambda_{n})$, we let $u\in\mathcal{X}^{1,2}_{+}(\Omega)\cap
L^{\infty}({\mathbb{R}}^{n})$ be the unique weak solution of

(A.1) $\begin{cases}\mathcal{L}u=\gamma\frac{u}{|x|^{2}}+f&\text{in
$\Omega$},\\\ u=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$}.\end{cases}$

Setting $\varphi_{0}=0$, for every $k\geq 1$ we then consider the unique weak
solution $\varphi_{k}\in\mathcal{X}^{1,2}_{+}(\Omega)$ of the following
_approximated_ $\mathcal{L}$ \- Dirichlet problem

(A.2)
$\begin{cases}\mathcal{L}u=\gamma\frac{\varphi_{k-1}}{|x|^{2}+1/k}+f_{k}&\text{in
$\Omega$},\\\ u=0&\text{in ${\mathbb{R}}^{n}\setminus\Omega$}.\end{cases}$

where $f_{k}=T_{k}(f)=\min\\{k,f\\}$. Then,

$\text{$\varphi_{k}\to u$ weakly in $\mathcal{X}^{1,2}(\Omega)$ and strongly
in $L^{p}(\Omega)$ for every $1\leq p<2^{*}$}.$

###### Proof.

We begin with a couple of preliminary observations. First of all, the
existence and the uniqueness of the solution
$\varphi_{k}\in\mathcal{X}_{+}^{1,2}(\Omega)\cap L^{\infty}({\mathbb{R}}^{n})$
follow from Theorem 2.3 (and from the recursive structure of problem (A.2)),
since the right-hand side

$g_{k}(x)=\gamma\frac{\varphi_{k-1}}{|x|^{2}+1/k}+f_{k}$

is non-negative and bounded on $\Omega$ (as, by induction, the same is true of
$\varphi_{k-1}$). Furthermore, by using a classical induction argument based
on the Weak Maximum Principle for $\mathcal{L}$ (and since the sequence
$\\{f_{k}\\}_{k}$ is non-decreasing), we have that

$\text{$\\{\varphi_{k}(x)\\}_{k}$ is non-negative and non-decreasing for every
$x\in\Omega$}.$

As a consequence, we can define

$\psi(x)=\lim_{k\to\infty}\varphi_{k}(x)\in(0,+\infty]$ for all $x\in\Omega$.

Now, using $\varphi_{k}$ as a test function for the equation solved by $u$ and
viceversa, we have

$\displaystyle\int_{\Omega}f_{k}\varphi_{k}\,dx$
$\displaystyle\leq\int_{\Omega}f\varphi_{k}\,dx=\mathcal{B}(u,\varphi_{k})-\gamma\int_{\Omega}\frac{u\varphi_{k}}{|x|^{2}}\,dx$
$\displaystyle(\text{since $\varphi_{k}\geq\varphi_{k-1}$ and
$|x|^{2}\leq|x|^{2}+1/k$})$
$\displaystyle\leq\mathcal{B}(\varphi_{k},u)-\gamma\int_{\Omega}\frac{\varphi_{k-1}u}{|x|^{2}+1/k}\,dx$
$\displaystyle=\int_{\Omega}f_{k}u\,dx\leq\int_{\Omega}fu\,dx;$

hence, by the Monotone Convergence Theorem we derive that

$\text{$f\psi\in L^{1}(\Omega)$}.$

On account of this fact, and using $\varphi_{k}$ as a test function for the
equation solved by $\varphi_{k}$ itself, we obtain the following estimate

$\displaystyle\rho(\varphi_{k})^{2}$
$\displaystyle=\mathcal{B}(\varphi_{k},\varphi_{k})=\gamma\int_{\Omega}\frac{\varphi_{k}\varphi_{k-1}}{|x|^{2}+1/k}\,dx+\int_{\Omega}f_{k}\varphi_{k}\,dx$
$\displaystyle(\text{since $\varphi_{k-1}\leq\varphi_{k}\leq\psi$ a.e.\,in
$\Omega$})$
$\displaystyle\leq\gamma\int_{\Omega}\frac{\varphi_{k}^{2}}{|x|^{2}}\,dx+\int_{\Omega}f\psi\,dx$
$\displaystyle(\text{by Hardy's inequality})$
$\displaystyle\leq\frac{\gamma}{\Lambda_{n}}\rho(\varphi_{k})^{2}+\|f\psi\|_{L^{1}(\Omega)}.$

This, together with the fact that $\gamma\in(0,\Lambda_{n})$, shows that the
sequence $\\{\varphi_{k}\\}_{k}$ is _bounded in the Hilbert space
$\mathcal{X}^{1,2}(\Omega)$_; hence, recalling that $\varphi_{k}\to\psi$
pointwise in $\Omega$ and using the Sobolev Embedding Theorem, we deduce that
$\psi\in\mathcal{X}^{1,2}(\Omega)$ and

$\text{$\varphi_{k}\to\psi$ weakly in $\mathcal{X}^{1,2}(\Omega)$ and strongly
in $L^{p}(\Omega)$ for every $1\leq p<2^{*}$}.$

To complete the proof it now suffices to show that
$\psi\in\mathcal{X}^{1,2}(\Omega)$ solves the same equation solved by $u$; by
uniqueness, this will prove that $\psi=u$, as desired. Let then
$\varphi\in\mathcal{X}^{1,2}(\Omega)$ be arbitrarily fixed. Since
$\varphi_{k}$ is a weak solution of (A.2), we can write

(A.3)
$\mathcal{B}(\varphi_{k},\varphi)=\gamma\int_{\Omega}\frac{\varphi_{k}\varphi}{|x|^{2}+1/k}\,dx+\int_{\Omega}f_{k}\varphi\,dx.$

Since $\varphi_{k}\rightharpoonup\psi$ in $\mathcal{X}^{1,2}(\Omega)$, we have

$\mathcal{B}(\varphi_{k},\varphi)\to\mathcal{B}(\psi,\varphi)\quad\text{as
$k\to\infty$}.$

Moreover, since $0\leq\varphi_{k}\leq\psi$, we have

$\displaystyle\Big{|}\frac{\varphi_{k}\varphi}{|x|^{2}+1/k}\Big{|}$
$\displaystyle\leq\frac{\psi|\varphi|}{|x|^{2}}=g(x)\quad\forall\,\,k\geq 1,$

and $g\in L^{1}(\Omega)$ by Hölder’s and Hardy’s inequality (recall that
$\psi,\varphi\in\mathcal{X}^{1,2}(\Omega)$). We are then entitled to apply the
Lebesgue Dominated Convergence theorem, showing that

$\int_{\Omega}\frac{\varphi_{k}\varphi}{|x|^{2}+1/k}\,dx\to\int_{\Omega}\frac{\psi\varphi}{|x|^{2}}\,dx.$

Finally, since $0\leq f_{k}\leq f$, we also have

$\displaystyle|f_{k}\varphi|$ $\displaystyle\leq
f\varphi=h(x)\quad\forall\,\,k\geq 1,$

and $h\in L^{1}(\Omega)$ by Hölder’s inequality (recall that $f\in
L^{\frac{2n}{n+2}}(\Omega)$ and $\varphi\in\mathcal{X}^{1,2}(\Omega)\subset
L^{2^{*}}(\Omega)$). Again by the Lebesgue Dominated Convergence theorem, we
then get

$\int_{\Omega}f_{k}\varphi\,dx\to\int_{\Omega}f\varphi\,dx.$

Gathering all these facts, we can finally pass to the limit as $k\to\infty$ in
the above (A.3), thus deducing that $\psi$ is a weak solution of the same
equation solved by $u$. ∎

## References

  * [1] B. Abdellaoui, M. Medina, I. Peral, A. Primo, The effect of the Hardy potential in some Calderón-Zygmund properties for the fractional Laplacian, J. Differential Equations 260(11), (2016), 8160–8206.
  * [2] B. Abdellaoui, I. Peral, A note on a critical problem with natural growth in the gradient, J. Eur. Math. Soc. (JEMS) 8(2), 2006, 157–170.
  * [3] C.A. Antonini, M. Cozzi, Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type, preprint. https://arxiv.org/abs/2308.06075
  * [4] R. Arora, V. Radulescu, _Combined effects in mixed local-nonlocal stationary problems_ , Proceedings of the Royal Society of Edinburgh: Section A Mathematics. Published online 2023:1-47. doi:10.1017/prm.2023.801.06701
  * [5] S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, Mixed local and nonlocal elliptic operators: regularity and maximum principles, Comm. Partial Differential Equations 47(3) (2022), 585–629.
  * [6] S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Faber-Krahn inequality for mixed local and nonlocal operators, J. Anal. Math. 150(2), (2023), 405–448.
  * [7] S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Brezis-Nirenberg type result for mixed local and nonlocal operators, preprint. https://arxiv.org/abs/2209.07502
  * [8] S. Biagi, D. Mugnai, E. Vecchi, A Brezis–Oswald approach for mixed local and nonlocal operators, Commun. Contemp. Math. 26(2), (2024), 2250057.
  * [9] S. Biagi, E. Vecchi, Multiplicity of positive solutions for mixed local-nonlocal singular critical problems, preprint. https://arxiv.org/abs/2308.09794
  * [10] L. Boccardo, L. Orsina, I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential, Discrete Contin. Dyn. Syst. 16(3), (2006), 513–523.
  * [11] J.-M. Bony, P. Courrège, P. Priouret, Semi-groupes de Feller sur une variété à bord compacte et problèmes aux limites intégro-différentiels du second ordre donnant lieu au principe du maximum, Ann. Inst. Fourier (Grenoble) 18(2), (1968), 369–521.
  * [12] C. Cancelier, Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum, Comm. Partial Differential Equations 11(15), (1986), 1677–1726.
  * [13] Z.-Q. Chen, P. Kim, R. Song, Z. Vondraček, Boundary Harnack principle for $\Delta+\Delta^{\alpha/2}$, Trans. Amer. Math. Soc. 364(8), (2012), 4169–4205.
  * [14] C. De Filippis, G. Mingione, Gradient regularity in mixed local and nonlocal problems, Math. Ann.388, (2024), 261–328.
  * [15] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
  * [16] S. Dipierro, E. Proietti Lippi, E. Valdinoci, (Non)local logistic equations with Neumann conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire (2022), DOI: 10.4171/AIHPC/57 .
  * [17] M. M. Fall, R. Musina, Sharp Nonexistence Results for a Linear Elliptic Inequality Involving Hardy and Leray Potentials, Journal of Inequalities and Applications 2011.
  * [18] P. Garain, On a class of mixed local and nonlocal semilinear elliptic equation with singular nonlinearity, J. Geom. Anal. 33, 212, (2023).
  * [19] P. Garain, J. Kinnunen, On the regularity theory for mixed local and nonlocal quasilinear elliptic equations, Trans. Amer. Math. Soc. 375(8), (2022),5393–5423.
  * [20] P. Garain, E. Lindgren, Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations, Calc. Var. 62, 67, (2023).
  * [21] G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988.
  * [22] T. Leonori, I. Peral, A. Primo, F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. 35(12), (2015), 6031–6068.
  * [23] I. Peral, F. Soria, Elliptic and Parabolic Equations Involving the Hardy-Leray Potential, De Gruyter Series in Nonlinear Analysis and Applications, Volume 38, (2021) Walter de Gruyter GmbH, Berlin/Boston.
  * [24] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15, (1965), 189–258.
  * [25] X. Su, E. Valdinoci, Y. Wei, J. Zhang, Regularity results for solutions of mixed local and nonlocal elliptic equations, Math. Z. 302, (2022), 1855–1878.
  * [26] X. Su, E. Valdinoci, Y. Wei, J. Zhang, On Some Regularity Properties of Mixed Local and Nonlocal Elliptic Equations, preprint. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4617397
  * [27] J.L. Vazquez, E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173, (2000), 103–153.