# On the existence of a second positive solution to mixed local-nonlocal
concave-convex critical problems

Stefano Biagi  and  Eugenio Vecchi  Dipartimento di Matematica
Politecnico di Milano
Via Bonardi 9, 20133 Milano, Italy<EMAIL_ADDRESS>Dipartimento di
Matematica
Università di Bologna
Piazza di Porta San Donato 5, 40126 Bologna , Italy<EMAIL_ADDRESS>
###### Abstract.

We prove the existence of a second positive weak solution for mixed local-
nonlocal critical semilinear elliptic problems with a sublinear perturbation
in the spirit of [1].

###### Key words and phrases:

Mixed local-nonlocal operators, critical problems

###### 2020 Mathematics Subject Classification:

35J75, 35M12, 35B33

The Authors are member of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM), and 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

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{n}$ be an open and bounded set with smooth
enough boundary $\partial\Omega$. We consider the following mixed local-
nonlocal perturbed critical semilinear elliptic problem:

($\mathrm{P}_{\varepsilon}$)
$\left\\{\begin{array}[]{rl}\mathcal{L}_{\varepsilon}u=\lambda\,u^{p}+u^{2^{\ast}-1}&\textrm{in
}\Omega,\\\ u>0&\textrm{in }\Omega,\\\ u=0&\textrm{in
}\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

where $\lambda>0$ is a positive real parameter, $p\in(0,1)$ and

$\mathcal{L}_{\varepsilon}:=-\Delta+\varepsilon\,(-\Delta)^{s},\qquad
s\in(0,1),\quad\varepsilon\in(0,1].$

Along the paper it will sometimes be useful to denote the above problem as
($\mathrm{P}_{\varepsilon}$)λ to make it clear the choice of the parameter.
Here, $(-\Delta)^{s}$ with $s\in(0,1)$ denotes the fractional Laplacian which
acts in smooth enough functions as

$(-\Delta)^{s}u(x)=2\,\mathrm{P.V.}\int_{\mathbb{R}^{n}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=2\,\lim_{\delta\to
0^{+}}\int_{\\{|x-y|\geq\delta\\}}\frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy.$

We neglect the normalization constant $C_{n,s}>0$ usually appearing in front
of the integral because we are not interested in asymptotics as $s\to 1^{-}$.

The above boundary value problem falls in the framework of the so called
concave-convex problems whose model and most famous example is the one
considered in the paper [1]:

(ABC) $\left\\{\begin{array}[]{rl}-\Delta
u=\lambda\,u^{p}+u^{2^{\ast}-1}&\textrm{in }\Omega,\\\ u>0&\textrm{in
}\Omega,\\\ u=0&\textrm{on }\partial\Omega.\end{array}\right.$

Without any aim of completeness, we mention that (ABC) has been generalized to
various directions, either allowing for different operators either considering
different boundary conditions, see e.g. [32, 24, 22, 7].

Before entering into the details related to ($\mathrm{P}_{\varepsilon}$)λ, let
us spend a few more words on $\mathcal{L}_{\varepsilon}$. The operator
$\mathcal{L}_{1}$ (i.e. with $\varepsilon=1$) is a special instance of a wide
more general class of operators whose study began in the ’60s, see [14] and
[21] for generalizations, in connection with the validity of a maximum
principle. On the other hand, the operator $\mathcal{L}_{1}$ can be seen as
the infinitesimal generator of a stochastic process obtained as a
superposition of a Brownian motion and a Lévy flight, and hence there is a
vast literature which establishes several regularity properties _adopting
probabilistic techniques_ , see e.g. [23] and the references therein.
More recently, the study of regularity properties related to this operator
(and its quasilinear generalizations) has seen an increasing interest, mainly
adopting more analytical and PDEs approaches, see, e.g., [8, 20, 27, 28, 30,
31, 37, 4, 38]. It is worth mentioning that the operator
$\mathcal{L}_{\varepsilon}$ (usually with $\varepsilon=1$) seems to be of
interest in biological applications, see, e.g. [29] and the references
therein.
The operator $\mathcal{L}_{1}$ has a also variational nature and it is
associated to the energy

$E(u):=\int_{\Omega}\dfrac{|\nabla
u|^{2}}{2}\,dx+\dfrac{1}{2}\,\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy.$

defined on a suitable space functions for which we refer to Section 2. It is
clear that

$E(\lambda u)=\lambda^{2}E(u),\quad\textrm{for every }\lambda\in\mathbb{R},$

but there is a lack of scaling invariance, namely,

$E(u_{t})=\dfrac{1}{2}\int_{\Omega}|\nabla
u|^{2}+\dfrac{t^{2s-2}}{2}\iint_{\mathbb{R}^{2n}}\dfrac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy\quad\textrm{for
every }t>0,$

where $u_{t}(x):=t^{(n-2)/2}u(x)$, which typically preserves the
$L^{2}$-gradient norm.

Among other aspects, this phenomenon is responsible of the following fact: the
best Sobolev constant in the natural mixed Sobolev inequality is never
achieved and it coincides with the one coming from the purely local one. We
notice that something similar happens when dealing with mixed Hardy-type
inequalities, see [11]. An interesting consequence of the absence of mixed
Aubin-Talenti functions naturally affects PDEs with critical term. Indeed,
when dealing with critical variational problems for $-\Delta$, it is well
known that a major role is played by the best Sobolev constant, for example
being a threshold of validity of Palais-Smale condition, and this is usually
achieved by testing the critical equation with the Aubin-Talenti functions. An
analogous procedure can be followed in the mixed setting as well, but the lack
of invariance previously mentioned can create troubles, see e.g [10, 13, 25].
We stress that something similar may happen with non-homogeneous operators
like the $(p,q)$-Laplacian, see [33].

Following the approach recently used in [13] for the case of mild singular and
critical mixed problems, we are now interested in the following classical
problem:

Find values of the parameter $\lambda$ for which ($\mathrm{P}_{\varepsilon}$)λ
admits one or more positive weak solutions.

We refer to Definition 2.2 for the precise definition of _weak solution_ of
($\mathrm{P}_{\varepsilon}$)λ. Differently from [10] and [13], here we are in
presence of a sublinear perturbation ($\lambda\,u^{p}$ with $p\in(0,1)$) of
the critical term ($u^{2^{\ast}-1}$).

For $\varepsilon=1$, a first step towards an answer has been recently made in
[2, Theorem 1.2] and [36, Theorem 1.1] where the following has been proved:

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{n}$ _(_ with $n\geq 3$_)_ be a bounded open set
with smooth enough boundary, and let $p\in(0,1)$. Then, there exists
$\Lambda>0$ such that

  * a)

problem $(\mathrm{P}_{1})_{\lambda}$ admits at least one weak solution for
every $0<\lambda\leq\Lambda$;

  * b)

problem $(\mathrm{P}_{1})_{\lambda}$ does not admit weak solutions for every
$\lambda>\Lambda$.

Moreover, for $\lambda\in(0,\Lambda)$, the solution is minimal and increasing
w.r.t. to $\lambda$.

We stress that the the above results actually holds in the supercritical case
as well. For sake of completeness, we recall that the existence of the first
positive solution in [2] is obtained by means of a sub/supersolution scheme as
in [1]. We also note that, if we fix $\varepsilon\in(0,1]$, the above result
holds for a threshold $\Lambda_{\varepsilon}$ now depending on $\varepsilon$.
We also mention that concave-convex problems in the mixed local-nonlocal
setting has been previously studied in [26]: there, the leading operator is a
combination of $p$-Laplacian and fractional $p$-Laplacian and the existence of
weak solutions is proved under the assumption $p>n$, being the interest of the
authors to study the limiting problem as $p\to+\infty$.

Our first result is somehow complementary to Theorem 1.1 stated above and it
is the counterpart of [1, Theorem 2.2] in the mixed local-nonlocal setting.

######  Theorem 1.2.

Let $\varepsilon\in(0,1]$ be fixed. Then, there exists a constant
$M_{\varepsilon}>0$ such that, for every
$\lambda\in(0,\Lambda_{\varepsilon})$, problem ($\mathrm{P}_{\varepsilon}$)λ,ε
has at most one solution $u_{\lambda}$ with

$\|u_{\lambda,\varepsilon}\|_{L^{\infty}(\mathbb{R}^{n})}\leq
M_{\varepsilon}.$

Note that the if the unique solution of Theorem 1.2 exists, it has to be the
minimal one found in Theorem 1.1.

Following the seminal paper [1], it is natural to wonder whether a second
positive solution exists or not for all $\lambda\in(0,\Lambda)$. A similar
question has been answered in [36] for sublinear perturbations of a
subcritical nonlinearity, so leaving the critical case unsolved.

To better understand the difficulties one has to face in the case of mixed
operators like $\mathcal{L}_{\varepsilon}$, let us briefly recall the method
employed in [1] to find a second positive solution of (ABC):

  * i)

show that the first solution found is a minimizer of the functional naturally
associated with (ABC) in $C^{1}$-topology;

  * ii)

thanks to the famous result by Brezis and Nirenberg [18], one inherits that
the first solution is actually a minimizer in the $H^{1}$-topology;

  * iii)

prove the validity of the Palais-Smale condition under a certain value
proportional to the best Sobolev constant, here making use of the Aubin-
Talenti functions;

  * iv)

find a second positive solution of mountain-pass-type.

Despite i) can still be proved in our case, the above scheme finds a first
obstacle once a result like the one in [18] is needed. To this aim, and thanks
to the recent regularity results proved in [3] and [38], we establish the
following

######  Theorem 1.3.

Let $\Phi:\mathcal{X}^{1,2}(\Omega)\to\mathbb{R}$ defined as

(1.1) $\Phi(u):=\dfrac{1}{2}\,\rho(u)^{2}-\int_{\Omega}F(x,u)\,dx,$

where $F(x,u):=\int_{0}^{u}f(x,s)\,ds$ and with $f$ satisfying that

(1.2) $|f(x,u)|\leq C_{f}\,(1+|u|^{2^{*}-1}).$

Let $u_{0}\in\mathcal{X}^{1,2}(\Omega)$ be a local minimizer of $\Phi$ in the
$C^{1}$-topology. Then, $u_{0}$ is a local minimizer in the
$\mathcal{X}^{1,2}$-topology.

Theorem 1.3 allows to achieve ii) as well, but a new difficulty arises once
iii) has to be faced, the main reason being the lack of scaling invariance,
and its consequences on the mixed Sobolev inequality, previously discussed. A
similar issue occurred in the recent [13] when dealing with mild singular
problems, and it is the main reason to consider $\mathcal{L}_{\varepsilon}$
instead of $\mathcal{L}_{1}$ because, roughly speaking, the presence of the
parameter $\varepsilon$ allows us to adjust the lack of scaling invariance, at
least for small enough $\varepsilon$. In this way, we are able to fully follow
the above scheme proving the following

######  Theorem 1.4.

Let $\Omega\subset\mathbb{R}^{n}$ _(_ with $n\geq 3$_)_ be a bounded open set
with smooth enough boundary, and let $p\in(0,1)$ be fixed.

Then, there exist $\lambda_{\star}>0$ and $\varepsilon_{0}\in(0,1)$ such that
problem ($\mathrm{P}_{\varepsilon}$)λ admits a second positive solutions for
every $\varepsilon\in(0,\varepsilon_{0})$ and for every
$\lambda\in(0,\lambda_{\star})$.

As it clearly appears from the statement, there is a first price to pay once
considering $\mathcal{L}_{\varepsilon}$, because the existence is not proved
for every $\lambda$. Less evident is the second fee we have to pay. Let us
briefly describe it: the first solution found in Theorem 1.1 can still be
found once $\mathcal{L}_{1}$ is replaced with $\mathcal{L}_{\varepsilon}$ but
now it implicitly depends on $\varepsilon$ and this forces to carefully keep
track of the dependence on $\varepsilon$ of many ingredients like
$L^{\infty}$-bounds and so on.
As in [13], the key result to be proved is Lemma 5.2 and we want to explicit
mention that a careful inspection of its proof shows that a better result can
be obtained, this time with big restrictions on both the dimension $n$ and the
fractional parameter $s$:

######  Corollary 1.5.

Let $n=3$ and $s\in\left(0,\tfrac{1}{2}\right)$. Then problem
($\mathrm{P}_{\varepsilon}$)λ admits at least two positive weak solutions for
every $\lambda\in(0,\Lambda)$.

Finally, once the existence of (at least) two solutions of problem
($\mathrm{P}_{\varepsilon}$)λ has been established (at least for $\varepsilon$
small enough), by exploiting Theorem 1.2 and by proceeding essentially as in
[1] we can prove the following qualitative result.

######  Theorem 1.6.

Let $\varepsilon\in(0,1]$ be fixed, and assume that $\Omega$ is _star-shaped_.
Then,

$\text{$\|v_{\lambda,\varepsilon}\|_{L^{\infty}(\Omega)}\to+\infty$ as
$\lambda\to 0^{+}$},$

where $v_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ is _any weak
solution of_ problem ($\mathrm{P}_{\varepsilon}$)λ distinct from its minimal
solution $u_{\lambda,\varepsilon}$ _(_ see Theorem 1.1 and Lemma 2.6 _)_.

_Plan of the paper:_ The paper is organized as follows:

  * •

In Section 2 we collect all the relevant notation, definitions and preliminary
results needed for the proof of our main results.

  * •

In Section 3 we briefly study the _purely sublinear_ counterpart of problem
($\mathrm{P}_{\varepsilon}$)λ (starting from the results proved in [12]); in
particular, we establish some _uniform bounds_ of the unique solution
$w_{\lambda,\varepsilon}$ of this problem which will be used subsequent
sections, and we establish Theorem 1.2.

  * •

In Section 4 we prove Theorem 1.3.

  * •

In Section 5 we prove Theorem 1.4.

  * •

Finally, in Section 6 we give the proof of Theorem 1.6.

##  2\. Preliminaries

In this section we collect some preliminary definitions and results which will
be used throughout the rest of the paper. First of all, we review some basic
properties of the fractional Sobolev spaces, and we properly introduce the
adequate functional setting for the study of mixed local-nonlocal operators;
we then give the precise definition of _weak sub/supersolution_ of problem
($\mathrm{P}_{\varepsilon}$)λ, and we establish some qualitative properties of
the solutions of this problem (provided they exist). Finally, we spend a few
words the applicability of Theorem 1.1 (established in the case when
$\varepsilon=1$) to our $\varepsilon$-dependent operator
$\mathcal{L}_{\varepsilon}$.

i) Sobolev spaces of fractional order. We begin this section by collecting a
few basic facts fractional Sobolev spaces, which are naturally related to the
fractional Laplacian $(-\Delta)^{s}$; we refer to [34] for a thorough
introduction to this topic.

Let $\varnothing\neq\mathcal{O}\subseteq\mathbb{R}^{n}$ be an arbitrary open
set. The _fractional Sobolev space_ $H^{s}(\mathcal{O})$ (of order
$s\in(0,1)$) is the subset of $L^{2}(\mathcal{O})$ defined as follows

$H^{s}(\mathcal{O}):=\Big{\\{}u\in
L^{2}(\mathcal{O}):\,[u]^{2}_{s,\mathcal{O}}=\iint_{\mathcal{O}\times\mathcal{O}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{N+2s}}\,dx\,dy<+\infty\Big{\\}}.$

We then list the few basic properties of $H^{s}(\mathcal{O})$ we will exploit
in this paper.

  * a)

$H^{s}(\mathcal{O})$ is a real Hilbert space, with the scalar product

$\langle
u,v\rangle_{s,\,\mathcal{O}}:=\iint_{\mathcal{O}\times\mathcal{O}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2s}}\,dx\,dy\qquad(u,v\in
H^{s}(\mathcal{O})).$

  * b)

$C_{0}^{\infty}(\mathcal{O})$ is a _linear subspace of $H^{s}(\mathcal{O})$_;
in addition, in the particular case when $\mathcal{O}=\mathbb{R}^{n}$, we have
that $C_{0}^{\infty}(\mathbb{R}^{n})$ is _dense_ in $H^{s}(\mathbb{R}^{n})$.

  * c)

If $\mathcal{O}=\mathbb{R}^{n}$ or if $\mathcal{O}$ has _bounded boundary
$\partial\mathcal{O}\in C^{0,1}$_, we have the _continuous embedding_
$H^{1}(\mathcal{O})\hookrightarrow H^{s}(\mathcal{O})$, that is, there exists
$\mathbf{c}=\mathbf{c}(n,s)>0$ s.t.

(2.1)
$\iint_{\mathcal{O}\times\mathcal{O}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy\leq\mathbf{c}\,\|u\|_{H^{1}(\mathcal{O})}^{2}\quad\text{for
every $u\in H^{1}(\mathcal{O})$}.$

In particular, if $\mathcal{O}\subseteq\mathbb{R}^{n}$ is a _bounded open set_
(with no regularity assumptions on $\partial\mathcal{O}$) and if $u\in
H_{0}^{1}(\mathcal{O})$, setting $\hat{u}=u\cdot\mathbf{1}_{\mathcal{O}}\in
H^{1}(\mathbb{R}^{n})$ we have

(2.2)
$\iint_{\mathbb{R}^{2n}}\frac{|\hat{u}(x)-\hat{u}(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy\leq\beta\,\int_{\mathcal{O}}|\nabla
u|^{2}\,dx,$

where $\beta>0$ is a suitable constant depending on $n,s$ and on $|\Omega|$.
Here and throughout, $|\cdot|$ denotes the $n$-dimensional Lebesgue measure.

ii) The space $\mathcal{X}^{1,2}(\Omega)$. Now we have briefly recalled some
basic facts regarding fractional Sobolev spaces, we are in a position to
introduce the adequate functional setting for the study of mixed local-
nonlocal operators.

Let then $\Omega\subseteq\mathbb{R}^{n}$ be a bounded open set Lipschitz
boundary $\partial\Omega$. We define the space $\mathcal{X}^{1,2}(\Omega)$ as
the completion of $C_{0}^{\infty}(\Omega)$ with respect to the _global norm_

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

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

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

Since this norm $\rho$ is induced by the scalar product

$\mathcal{B}_{\rho}(u,v):=\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla
v\,dx+\langle u,v\rangle_{s,\mathbb{R}^{n}}$

(where $\cdot$ denotes the usual scalar product in $\mathbb{R}^{n}$), the
space $\mathcal{X}^{1,2}(\Omega)$ is a real _Hilbert space_ ; most
importantly, since $\Omega$ is bounded and $\partial\Omega$ is Lipschitz, by
combining the above (2.1) with the classical Poincaré inequality we infer that

$\vartheta^{-1}\|u\|_{H^{1}(\mathbb{R}^{n})}\leq\rho(u)\leq\vartheta\|u\|_{H^{1}(\mathbb{R}^{n})}\qquad\text{for
every $u\in C_{0}^{\infty}(\Omega)$},$

where $\vartheta>1$ is a suitable constant depending on $n,s$ and on
$|\Omega|$. Thus, $\rho(\cdot)$ and the full $H^{1}$-norm in $\mathbb{R}^{n}$
are _actually equivalent_ on the space $C^{\infty}_{0}(\Omega)$, so that

(2.3)
$\begin{split}\mathcal{X}^{1,2}(\Omega)&=\overline{C_{0}^{\infty}(\Omega)}^{\,\,\|\cdot\|_{H^{1}(\mathbb{R}^{n})}}\\\
&=\\{u\in H^{1}(\mathbb{R}^{n}):\,\text{$u|_{\Omega}\in H_{0}^{1}(\Omega)$ and
$u\equiv 0$ a.e.\,in $\mathbb{R}^{n}\setminus\Omega$}\\}.\end{split}$

We explicitly observe that, on account of (2.3), the functions in
$\mathcal{X}^{1,2}(\Omega)$ naturally satisfy the nonlocal Dirichlet condition
prescribed in problem ($\mathrm{P}_{\varepsilon}$)λ, that is,

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

######  Remark 2.1 (Properties of the space $\mathcal{X}^{1,2}(\Omega)$).

For a future reference, we list in this remark some properties of the function
space $\mathcal{X}^{1,2}(\Omega)$ which will be repeatedly exploited in the
rest of the paper.

  1. 1)

Since both $H^{1}(\mathbb{R}^{n})$ and $H_{0}^{1}(\Omega)$ are _closed_ under
the maximum/minimum operation, it is readily seen that

$u_{\pm}\in\mathcal{X}^{1,2}(\Omega)\quad\text{for every
$u\in\mathcal{X}^{1,2}(\Omega)$},$

where $u_{+}=\max\\{u,0\\}$ and $u_{-}=\max\\{-u,0\\}$.

  2. 2)

Since we are assuming that $\partial\Omega$ is smooth, from (2.3) we see that
a function $u\in H^{1}(\mathbb{R}^{n})\cap C(\overline{\Omega})$ belongs to
the space $\mathcal{X}^{1,2}(\Omega)$ _if and only if_

$\text{$u\equiv 0$ pointwise on $\mathbb{R}^{n}\setminus\Omega$}.$

  3. 3)

On account of (2.2), for every $u\in\mathcal{X}^{1,2}(\Omega)$ we have

(2.4)
$[u]_{s,\mathbb{R}^{n}}^{2}=\iint_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2s}}\,dx\,dy\leq\beta\int_{\Omega}|\nabla
u|^{2}\,dx.$

As a consequence, the norm $\rho$ is _globally equivalent_ on
$\mathcal{X}^{1,2}(\Omega)$ to the $H_{0}^{1}$-norm: in fact, by (2.4) there
exists a constant $\Theta>0$ such that

(2.5) $\||\nabla u|\|_{L^{2}(\Omega)}\leq\rho(u)\leq\Theta\||\nabla
u|\|_{L^{2}(\Omega)}\quad\text{for every $u\in\mathcal{X}^{1,2}(\Omega)$}.$

  4. 4)

By the (local) Sobolev inequality, for every $u\in\mathcal{X}^{1,2}(\Omega)$
we have

$\displaystyle S_{n}\|u\|_{L^{2^{*}}(\Omega)}^{2}$
$\displaystyle=\|u\|_{L^{2^{*}}(\mathbb{R}^{n})}^{2}\leq\int_{\mathbb{R}^{n}}|\nabla
u|^{2}\,dx\leq\rho(u)^{2}.$

This, together with Hölder’s inequality (recall that $\Omega$ is _bounded_),
proves the _continuous embedding_
$\text{$\mathcal{X}^{1,2}(\Omega)\hookrightarrow L^{q}(\Omega)$ for every
$1\leq q\leq 2^{*}$}.$

  5. 5)

By combining (2.5) with the _compact embedding_ of
$H_{0}^{1}(\Omega)\hookrightarrow L^{q}(\Omega)$ (holding true for every
$1\leq q<2^{*}$), we derive that also the embedding

$\text{$\mathcal{X}^{1,2}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact for
every $1\leq q<2^{*}$}.$

As a consequence, if $\\{u_{k}\\}_{k}$ is a bounded sequence in
$\mathcal{X}^{1,2}(\Omega)$, it is possible to find a (unique) function
$u\in\mathcal{X}^{1,2}(\Omega)$ such that (up to a sub-sequence)

     * a)

$u_{n}\to u$ weakly in $\mathcal{X}^{1,2}(\Omega)$;

     * b)

$u_{n}\to u$ _strongly_ in $L^{q}(\Omega)$ for every $1\leq q<2^{*}$;

     * c)

$u_{n}\to u$ pointwise a.e. in $\Omega$.

Clearly, since both $u_{n}$ (for all $n\geq 1$) and $u$ _identically vanish_
out of $\Omega$, see (2.3), we can replace $\Omega$ with $\mathbb{R}^{n}$ in
the above assertions b)-c).

We will exploit these properties without any further comment.

We now observe that, since the leading operator of
($\mathrm{P}_{\varepsilon}$)λ is given by the $\varepsilon$-dependent operator
$\mathcal{L}_{\varepsilon}=-\Delta+\varepsilon(-\Delta)^{s}$, it follows that
the bilinear form naturally associated with $\mathcal{L}_{\varepsilon}$ is the
following

$\mathcal{B}_{\varepsilon}(u,v)=\int_{\mathbb{R}^{n}}\nabla u\cdot\nabla
v\,dx+\varepsilon\,\langle u,v\rangle_{s,\mathbb{R}^{n}};$

in its turn, this form $\mathcal{B}_{\varepsilon}$ induces the
$\varepsilon$-dependent quadratic form

$\rho_{\varepsilon}(u)=\||\nabla
u|\|^{2}_{L^{2}(\mathbb{R}^{n})}+\varepsilon\,[u]^{2}_{s}\qquad(\text{for
$u\in\mathcal{X}^{1,2}(\Omega)$}).$

While in this perspective it should seem more _natural_ to use the norm
$\rho_{\varepsilon}$ in place of $\rho$ on the space
$\mathcal{X}^{1,2}(\Omega)$, it is readily seen that these two norms are
indeed _equivalent on $\mathcal{X}^{1,2}(\Omega)$_ (and equivalent to the
$H_{0}^{1}$-norm), _uniformly with respect to $\varepsilon$_: in fact, taking
into account (2.5) (and since $0<\varepsilon\leq 1$), we have

(2.6) $\||\nabla
u|\|_{H^{1}_{0}(\Omega)}\leq\rho_{\varepsilon}(u)\leq\rho(u)\leq\Theta\||\nabla
u|\|_{H^{1}_{0}(\Omega)}$

for some $\Theta>0$ only depending on $n,s$. On account of (2.6), we can
indifferently use $\rho_{\varepsilon}(\cdot),\,\rho(\cdot)$ and the
$H_{0}^{1}$-norm to define the topology of the space
$\mathcal{X}^{1,2}(\Omega)$, and this choice does not produce any dependence
on $\varepsilon\in(0,1]$.

Notation. We conclude this second part of the section with a short list of
notation, which will be used in the sequel; here, as usual,
$\varepsilon\in(0,1]$ is a fixed parameter.

1) Given any open set $\mathcal{O}\subseteq\mathbb{R}^{n}$ (not necessarily
bounded), we set

$\begin{split}\mathrm{a)}&\,\,\mathcal{B}_{\varepsilon,\mathcal{O}}(u,v)=\int_{\mathcal{O}}\nabla
u\cdot\nabla v\,dx+\varepsilon\,\langle
u,v\rangle_{s,\mathbb{R}^{n}}\,\,(\text{for
$u,v\in\mathcal{X}^{1,2}(\Omega)$});\\\
\mathrm{b)}&\,\,\mathcal{Q}_{\varepsilon,\mathcal{O}}(u)=\mathcal{B}_{\rho_{\varepsilon},\mathcal{O}}(u,u)\,\,(\text{for
$u\in\mathcal{X}^{1,2}(\Omega)$}).\end{split}$

Since $\mathcal{X}^{1,2}(\Omega)\subseteq H^{1}(\mathbb{R}^{n})$ (see (2.3)),
the above forms $\mathcal{B}_{\varepsilon,\mathcal{O}}$ and
$\mathcal{Q}_{\varepsilon,\mathcal{O}}$ are well-defined; moreover, again by
taking into account (2.3) we have

  * •

$\mathcal{B}_{\varepsilon,\Omega}(u,v)=\mathcal{B}_{\varepsilon,\mathbb{R}^{n}}(u,v)\equiv\mathcal{B}_{\varepsilon}(u,v)$
for all $u,v\in\mathcal{X}^{1,2}(\Omega)$;

  * •

$\mathcal{Q}_{\varepsilon,\Omega}(u)=\mathcal{Q}_{\varepsilon,\mathbb{R}^{n}}(u)\equiv\rho_{\varepsilon}(u)$
for all $u\in\mathcal{X}^{1,2}(\Omega)$.

2) Given any _bounded_ open set $\mathcal{O}\subseteq\mathbb{R}^{n}$, we set

$\|u\|_{H_{0}^{1}(\mathcal{O})}:=\||\nabla
u|\|_{L^{2}(\mathcal{O})}=\Big{(}\int_{\mathcal{O}}|\nabla
u|^{2}\,dx\Big{)}^{1/2}.$

iii) Weak sub/supersolutions of problem ($\mathrm{P}_{\varepsilon}$)λ. Thanks
to all the preliminaries reviewed so far, we are ready to provide the precise
definition of _weak sub/supersolutions_ of problem
($\mathrm{P}_{\varepsilon}$)λ. Actually, for a reason which will be clear
later on, we consider the more general problem

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

where $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is an arbitrary Carathéodory
function satisfying the growth condition (1.2), that is, there exists a
constant $C>0$ such that

$|f(x,t)|\leq C(1+|t|^{2^{*}-1})\quad\text{\emph{for a.e.\,$x\in\Omega$ and
every $t\in\mathbb{R}$}}.$

Clearly, problem ($\mathrm{P}_{\varepsilon}$)λ is of the form (2.7), with the
choice $f(x,t)=\lambda t^{p}+t^{2^{*}-1}$.

######  Definition 2.2.

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function
satisfying the growth condition (1.2). We say that a function
$u\in\mathcal{X}^{1,2}(\Omega)$ is

  * a)

a _weak subsolution_ (resp.  _supersolution_) of problem (2.7) if we have

(2.8)
$\begin{gathered}\mathcal{B}_{\rho}(u,\varphi)\leq\,[\text{resp.}\,\geq]\,\,\int_{\Omega}f(x,u)\varphi\,dx\\\
\text{for every test function $\varphi\in
C_{0}^{\infty}(\Omega)$,\,$\varphi\geq 0$ in $\Omega$}.$$\end{gathered}$

  * b)

a _weak solution_ of problem (2.7) if $u$ is both a weak subsolution and a
weak supersolution of the same problem;

  * c)

a _weak subsolution_ (resp.  _supersolution, solution_) of problem

(2.9) $\begin{cases}\mathcal{L}_{\varepsilon}u=f(x,u)&\text{in $\Omega$}\\\
u>0&\text{in $\Omega$}\\\ u=0&\text{in
$\mathbb{R}^{n}\setminus\Omega$}\end{cases}$

if $u$ is a weak subsolution (resp. supersolution, solution) of problem (2.7)
in the sense already specified, further satisfying

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

Now we have introduced Definition 2.2, we prove a regularity result for
solutions of problem (2.7) which will be used in the sequel.

######  Theorem 2.3.

Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function
satisfying the growth condition (1.2), and suppose that there exists a weak
solution $u_{0}\in\mathcal{X}^{1,2}(\Omega)$ of problem (2.7) _(_ in the sense
of Definition 2.2 _)_. Then, we have

(2.10) $\text{$u\in C^{1,\alpha}(\overline{\Omega})$ for some
$\alpha\in(0,1)$}.$

###### Proof.

First of all we observe that, since the function $f$ satisfies the growth
condition (1.2), we can apply [38, Theorem 1.1], ensuring that $u_{0}\in
L^{\infty}(\Omega)$; from this, again by exploiting condition (1.2), we derive
that

(2.11) $g(x):=f(x,u(x))\in L^{\infty}(\Omega).$

On account of (2.11), and since $\Omega$ has smooth boundary, we can then
invoke the _global regularity result_ for the weak solutions of the
_$\mathcal{L}_{\varepsilon}$ \- Dirichlet problem_

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

proved in [3, Theorem 1.1], from which we derive (2.10). This ends the proof.
∎

To make the paper self-contained, we finally state a strong maximum principle
and Hopf Lemma which is a mere combination of results already appearing in the
literature, see [3, 8, 12].

######  Theorem 2.4.

Let $u\in\mathcal{X}^{1,2}(\Omega)\cap C^{1}(\overline{\Omega})$ be a weak
supersolution of

(2.12) $\begin{cases}\mathcal{L}_{\varepsilon}u=0&\text{in $\Omega$},\\\
u=0&\text{in $\mathbb{R}^{n}\setminus\Omega$}.\end{cases}$

We also assume that $u\not\equiv 0$ in $\Omega$. Then

  * i)

$u>0$ pointwise in $\Omega$ and $u=0$ pointwise in $\partial\Omega$;

  * ii)

$\partial_{\nu}<0$ pointwise on $\partial\Omega$ _(_ where $\nu$ is the outer
exterior normal at $\partial\Omega$_)_.

###### Proof.

We first observe that, since $u\in\mathcal{X}^{1,2}(\Omega)\cap
C^{1}(\overline{\Omega})$, by Remark 2.1 we have

(2.13) $\text{$u\in H^{1}(\mathbb{R}^{n})$ and $u\equiv 0$ pointwise in
$\mathbb{R}^{n}\setminus\Omega$};$

moreover, since $u$ is a weak supersolution of (2.12), we also have

(2.14) $\mathcal{B}_{\varepsilon}(u,\varphi)\geq 0\quad\forall\,\,\varphi\in
C_{0}^{\infty}(\Omega),\,\text{$\varphi\geq 0$ in $\Omega$}.$

Gathering (2.13)-(2.14), we are then entitled to apply the Weak Maximum
Principle for $\mathcal{L}_{\varepsilon}$ in [8, Theorem 1.2], ensuring that
$u\geq 0$ pointwise in $\mathbb{R}^{n}$; as a consequence, since $u\geq
0,\,u\not\equiv 0$ in $\Omega$ and since $u\equiv 0$ on
$\mathbb{R}^{n}\setminus\Omega$ (see (2.13)), by combining the Strong Maximum
Principle in [12, Theorem 3.1] (applied here with $f\equiv 0$) and the Hopf
lemma in [3, Theorem 1.2] (remind that $u\in C^{1}(\overline{\Omega})$), we
conclude that

$\text{$u>0$ pointwise in
$\Omega$}\quad\text{and}\quad\text{$\partial_{\nu}u<0$ on $\partial\Omega$}.$

This ends the proof. ∎

######  Lemma 2.5.

Let $n>3$ and $h=h(x)\in L^{p}(\Omega)$ with $p>\tfrac{n}{2}$. Assume that
there exists a weak solution $u\in\mathcal{X}^{1,2}(\Omega)$ of the Dirichlet
problem

$\left\\{\begin{array}[]{rl}\mathcal{L}_{\varepsilon}u=h&\textrm{in
}\Omega,\\\ u=0&\textrm{in }\mathbb{R}^{n}\setminus\Omega.\end{array}\right.$

Then $u\in L^{\infty}(\mathbb{R}^{n})$ and

$\|u\|_{L^{\infty}(\mathbb{R}^{n})}\leq C\,\|h\|_{L^{p}(\Omega)},$

for some positive constant $C>0$ independent of $\varepsilon$.

###### Proof.

It is enough to closely follow the proof of [8, Theorem 4.7] in the case
$\varepsilon=1$. Take $\delta>0$ to be chosen later on and define the
functions

$\tilde{u}:=\dfrac{\delta\,u}{K}\quad\textrm{and
}\quad\tilde{v}:=\dfrac{\delta\,h}{K},$

where
$K=K(u,h,\Omega,p,n):=\|u\|_{L^{2^{\ast}}(\Omega)}+\|h\|_{L^{p}(\Omega)}$. One
can now perform the same computations made in [8, Theorem 4.7], getting rid of
the nonlocal part thanks to [8, Equation (4.47)], reaching the following:

(2.15)
$u(x)\leq\dfrac{K}{\delta}=\dfrac{\|u\|_{L^{2^{\ast}}(\Omega)}+\|h\|_{L^{p}(\Omega)}}{\delta},\quad\textrm{for
every }x\in\Omega,$

where $\delta>0$ has been chosen conveniently small but independently of
$\varepsilon$. Now, by Sobolev and Hölder inequalities, and testing the
equation with $u$ itself, we get

$\mathcal{S}_{n}\,\|u\|^{2}_{L^{2^{\ast}}(\Omega)}\leq|\Omega|^{1/q}\,\|u\|_{L^{2^{\ast}}(\Omega)}\,\|h\|_{L^{p}(\Omega)},$

where $\tfrac{1}{2^{\ast}}+\tfrac{1}{p}+\tfrac{1}{q}=1$ are the Hölder
exponents. Therefore,

$\|u\|_{L^{2^{\ast}}(\Omega)}\leq\dfrac{|\Omega|^{1/q}}{\mathcal{S}_{n}}\,\|h\|_{L^{p}(\Omega)},$

which combined with (2.15) gives the desired conclusion. ∎

iv) First positive solution to ($\mathrm{P}_{\varepsilon}$)λ. We conclude this
section of preliminaries by spending a few words on the applicability of
Theorem 1.1 (which is proved in the case when $\varepsilon=1$) to our operator
$\mathcal{L}_{\varepsilon}$.

In the case $\varepsilon=1$, the existence of a first positive solution is
proved in [2] by means of a classical sub and supersolution scheme that can be
performed in the same way even replacing $\mathcal{L}_{1}$ with
$\mathcal{L}_{\varepsilon}$. In particular, one can define

(2.16)
$\Lambda_{\varepsilon}:=\sup\left\\{\lambda>0:\eqref{eq:Problem}_{\lambda}\textrm{
admits a weak solution }\right\\}.$

One then has to show what follows:

  * a)

$\Lambda_{\varepsilon}$ is well defined and $\Lambda_{\varepsilon}<+\infty$;

  * b)

problem ($\mathrm{P}_{\varepsilon}$)λ admits a weak solution for every
$0<\lambda\leq\Lambda_{\varepsilon}$.

For every fixed $0<\varepsilon\leq 1$, assertions a) - b) can be proved
exactly as in [2]. In particular, following the proof of [2, Theorem 1.1],
where the authors used a recent result in critical point theory proved in
[35], there exists $\mu_{\sharp,\varepsilon}>0$ (a priori depending on
$\varepsilon$ in a not explicit way) such that a first solution of
($\mathrm{P}_{\varepsilon}$)λ (for $0<\lambda<\mu_{\sharp,\varepsilon}$) is
obtained as a critical point over the set

$M=\left\\{v\in\mathcal{X}^{1,2}(\Omega):\|v\|_{L^{\infty}(\Omega)}\leq
r,u\geq 0\textrm{ a.e. in }\Omega\right\\}.$

This is the reason why we need the following

######  Lemma 2.6.

There exists $\mu_{\sharp}>0$ independent of $\varepsilon$ such that
($\mathrm{P}_{\varepsilon}$)λ possesses a minimal weak solution
$\bar{u}_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)\cap
L^{\infty}(\Omega)$ for every $0<\lambda\leq\mu_{\sharp}$ and every
$\varepsilon\in(0,1)$, which is increasing w.r.t. $\lambda$ and further
satisfying that

(2.17) $\|\bar{u}_{\lambda,\varepsilon}\|_{L^{\infty}(\Omega)}\ \leq C,$

where $C>0$ is independent of $\varepsilon$.

###### Proof.

As already discussed, Theorem 1.1 applies to our problem
($\mathrm{P}_{\varepsilon}$)λ, the role of $\varepsilon$ being immaterial; as
a consequence, there exists $\Lambda_{\varepsilon}>0$ such that problem
($\mathrm{P}_{\varepsilon}$)λ possesses a _minimal weak solution_

$\bar{u}_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$

for every $0<\lambda\leq\Lambda_{\varepsilon}$; moreover, if
$0<\lambda_{1}<\lambda_{2}<\Lambda_{\varepsilon}$, we have

$\text{$\bar{u}_{\lambda_{1},\varepsilon}\leq\bar{u}_{\lambda_{2},\varepsilon}$
a.e.\,in $\Omega$}.$

We claim that there exists a constant $\lambda_{\sharp}>0$, independent of
$\varepsilon\in(0,1)$, such that

(2.18) $\Lambda_{\varepsilon}\geq\lambda_{\sharp}.$

Indeed, let $\varepsilon\in(0,1)$ be fixed. By arguing as in the proof of [2,
Theorem 1.2] we see that there exists
$\Lambda_{\varepsilon}\geq\mu_{\sharp,\varepsilon}$, where
$\mu_{\sharp,\varepsilon}>0$ satisfies the following property (see, precisely,
the proof of [2, Lemma 3.9]):

$\begin{gathered}(\star)\qquad\quad\text{for every
$0<\lambda<\lambda_{\sharp,\varepsilon}$ there exist $0<r_{1}<r_{2}$ such
that}\\\ \text{$C_{\varepsilon}(r^{{2^{*}}-2}+\lambda r^{p-1})\leq 1$ for all
$r\in[r_{1},r_{2}]$};\end{gathered}$

here, $C_{\varepsilon}>0$ is a positive constant such that, for every $h\in
L^{p}(\Omega)$ (with $p>n/2$) and every weak solution
$u\in\mathcal{X}^{1,2}(\Omega)$ of the problem

(2.19) $\begin{cases}\mathcal{L}_{\varepsilon}u=h&\text{in $\Omega$}\\\
u=0&\text{in $\mathbb{R}^{n}\setminus\Omega$}\end{cases}$

we have the _a - priori estimate_

(2.20) $\|u\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C_{\varepsilon}\|h\|_{L^{p}(\Omega)}$

(the existence of such a constant follows from [8, Theorem 4.7]). Hence, the
lower bound $\lambda_{\sharp,\varepsilon}$ depends on $\varepsilon$ _only
through the constant $C_{\varepsilon}$ in estimate (2.20)_. On the other hand,
by Lemma 2.5, the constant $C_{\varepsilon}$ can be chosen _independently of
$\varepsilon$_: this means, precisely, that there exists $C_{0}>0$ such that

$\|u\|_{L^{\infty}(\mathbb{R}^{n})}\leq C_{0}\|h\|_{L^{p}(\Omega)}$

for every $h\in L^{p}(\Omega)$ (with $p>n/2$), every weak solution
$u\in\mathcal{X}^{1,2}(\Omega)$ of (2.19) _and every $\varepsilon\in(0,1]$_.
Hence, if we choose $\mu_{\sharp}>0$ in such a way that $(\star)$ holds with
$C_{0}$ in place of $C_{\varepsilon}$ (that is, if we choose the _very same
constant considered in[2]_ and corresponding to the case $\varepsilon=1$), we
conclude that (2.18) is satisfied.

We now turn to prove (2.17). To this end we notice that, by proceeding exactly
as in the proof of [2, Theorem 1.1] (and by recalling that our constant
$\mu_{\sharp}$ is precisely the one considered in [2] and corresponding to the
case $\varepsilon=1$), there exists a solution
$w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ of problem
($\mathrm{P}_{\varepsilon}$)λ satisfying the a-priori estimate

$\|w_{\lambda,\varepsilon}\|_{L^{\infty}(\Omega)}\leq r,$

where $r\in[r_{1},r_{2}]$ is arbitrarily chosen, and $r_{1}<r_{2}$ are as in
$(\star)$ (and they depend on the fixed $\lambda$); thus, by the minimality
property of $\bar{u}_{\lambda,\varepsilon}$ we infer that

$0\leq\bar{u}_{\lambda,\varepsilon}\leq w_{\lambda,\varepsilon}\leq
r\quad\text{a.e.\,in $\Omega$},$

and this readily gives the desired (2.17). ∎

######  Remark 2.7.

Arguing exactly as in [2, Proof of Theorem 1.2] one can also realize that
there exists $\Lambda_{\star}>0$ (indicated there as $\tilde{\lambda}$),
depending on the first Dirichlet eigenvalue of $\mathcal{L}_{1}$ (see e.g.
[9]) but independent of $\varepsilon$, such that

$\Lambda_{\varepsilon}\leq\Lambda_{\star}.$

##  3\. Purely sublinear problems and the proof of Theorem 1.2

In this short section we investigate the _purely sublinear_ problem

(3.1)
$\left\\{\begin{array}[]{rl}\mathcal{L}_{\varepsilon}u=\lambda\,u^{p}&\textrm{in
}\Omega,\\\ u>0&\textrm{in }\Omega,\\\ u=0&\textrm{in
}\mathbb{R}^{n}\setminus\Omega,\end{array}\right.$

To be more precise, we establish/review some results which will be used (as
fundamental tools) in the proof of Theorem 1.4, and we demonstrate our first
main result, namely Theorem 1.2.

To begin with, we prove the following proposition.

######  Proposition 3.1.

Let $\varepsilon\in(0,1]$ be fixed. Moreover, let $\lambda>0$ and $p\in(0,1)$.

Then, the following assertions hold.

  * i)

There exists a _unique weak solution_
$w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ of problem (3.1), which
is the unique _global minimizer of the functional_

(3.2)
$J_{\lambda,\varepsilon}(u)=\frac{1}{2}\rho_{\varepsilon}(u)^{2}-\frac{\lambda}{p+1}\int_{\Omega}|u|^{p+1}\,dx.$

Moreover, $J_{\lambda,\varepsilon}(w_{\lambda,\varepsilon})<0$.

  * ii)

$w_{\lambda,\varepsilon}\in C^{1,\alpha}(\overline{\Omega})$ for every
$\alpha\in(0,1)$, and $\partial_{\nu}w_{\lambda,\varepsilon}<0$ on
$\partial\Omega$.

  * iii)

There exists $\varepsilon_{0}>0$ with the following property: _given any ball
$B_{R}(x_{0})\subseteq\Omega$ and any $0<r\leq\min\\{1,R\\}$, for every
$0<\varepsilon\leq\varepsilon_{0}$ we have_

$w_{\lambda,\varepsilon}\geq c_{2}\quad\text{a.e.\,on $B_{r}(x_{0})$},$

for some constant $c_{2}>0$ independent of $\varepsilon$.

We explicitly stress that, since problem (3.1) is of the form (2.7) (with
$f=\lambda t^{p}$), the definition of weak solution of (3.1) is given in
Definition 2.2.

###### Proof.

We prove separately the three assertions.

i) To begin with, by using [12, Theorem 1.2] with the choice $f(t)=\lambda
t^{p}$ (notice that this theorem certainly applies to the operator
$\mathcal{L}_{\varepsilon}$, the role of $\varepsilon$ being immaterial) we
immediately get the existence of a _unique weak solution_

$w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)\cap L^{\infty}(\Omega)$

of problem (3.1); moreover, owing to [12, Proposition 6.2] (which also can be
applied to the $\varepsilon$ \- dependent operator
$\mathcal{L}_{\varepsilon}$), we know that this $w_{\lambda,\varepsilon}$
satisfies

(3.3)
$J_{\lambda,\varepsilon}(w_{\lambda,\varepsilon})=\min_{\varphi\in\mathcal{X}^{1,2}(\Omega)}J_{\lambda,\varepsilon}(\varphi)\quad\text{and}\quad
J_{\lambda,\varepsilon}(w_{\lambda,\varepsilon})<0.$

ii) Since $f(t)=\lambda t^{p}$ clearly satisfies the growth condition (1.2)
(with $C_{f}=1$), the regularity up the boundary of $w_{\lambda,\varepsilon}$
follows from Theorem 2.3; this, jointly with the fact that
$w_{\lambda,\varepsilon}>0$ in $\Omega$, allows us to apply 2.4, obtaining

$\text{$\partial_{\nu}w_{\lambda,\varepsilon}<0$ pointwise on
$\partial\Omega$}.$

iii) The proof of this assertion is split into three steps, and it is very
similar to that of [13, Proposition 3.7]; we present it in detail for the sake
of completeness.

Step 1). In this first step, we prove that

(3.4)
$\|w_{\lambda,\varepsilon}\|_{H^{1}_{0}(\Omega)}\leq\lambda^{\frac{1}{1-p}}\,c\quad\text{for
all $\varepsilon\in(0,1]$}.$

for some constant $c>0$ only depending on $\Omega$ and on $p$.

Indeed, since $w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ is a weak
solution of problem (3.1) (in the sense of Definition 2.2), by choosing
$w_{\lambda,\varepsilon}$ as a test function in (2.8), we get

$\displaystyle\|w_{\lambda,\varepsilon}\|_{H^{1}_{0}(\Omega)}^{2}$
$\displaystyle\leq\rho_{\varepsilon}(w_{\lambda,\varepsilon})^{2}=\frac{\lambda}{p+1}\int_{\Omega}w_{\lambda,\varepsilon}^{p+1}\,dx$
$\displaystyle(\text{using H\"{o}lder's and Poincar\'{e} inequality})$
$\displaystyle\leq\frac{\lambda|\Omega|^{1-\frac{p+1}{2}}}{p+1}\|w_{\lambda,\varepsilon}\|^{p+1}\leq\lambda\,c(\Omega,p)\|w_{\lambda,\varepsilon}\|^{p+1}_{H^{1}_{0}(\Omega)},$

where $c(\Omega,p)>0$ is a suitable constant only depending on $\Omega$ and on
$p$. From this, since $w_{\lambda,\varepsilon}\not\equiv 0$ (and since
$0<p<1$), we immediately derive the claimed (3.4).

Step 2). In this second step we prove that, as $\varepsilon\to 0^{+}$, we have

  * a)

$w_{\lambda,\varepsilon}\to w_{\lambda}$ weakly in
$\mathcal{X}^{1,2}(\Omega)$;

  * b)

$w_{\lambda,\varepsilon}\to w_{\lambda}$ strongly in $L^{m}(\Omega)$ for every
$1\leq m<2^{*}$;

where $w_{\lambda}\in H^{1}_{0}(\Omega)$ is the unique solution of the purely
local problem

(3.5) $\begin{cases}-\Delta u=\lambda u^{p}&\textrm{in $\Omega$},\\\
u>0&\textrm{in $\Omega$},\\\ u=0&\textrm{in
$\mathbb{R}^{n}\setminus\Omega$},\end{cases}$

To this end, we let $\\{\varepsilon_{j}\\}_{j}\subseteq(0,1]$ be any sequence
converging to $0$ as $j\to+\infty$, and we arbitrarily choose a subsequence
$\\{\varepsilon_{j_{k}}\\}_{k}$ of $\\{\varepsilon_{j}\\}_{j}$. On account of
(3.4), and using Remark 2.1 \- 5), we know that there exists some function
$\bar{w}\in\mathcal{X}^{1,2}(\Omega)$ such that, as $k\to+\infty$ and up to
possibly choosing a further subsequence,

  * •

$w_{\lambda,\varepsilon_{j_{k}}}\to\bar{w}$ weakly in
$\mathcal{X}^{1,2}(\Omega)$;

  * •

$w_{\lambda,\varepsilon_{j_{k}}}\to\bar{w}$ strongly in $L^{m}(\Omega)$ for
every $1\leq m<2^{*}$;

from this, since $w_{\lambda,\varepsilon}$ is a global minimizer for
$J_{\lambda,\varepsilon}$ (see (3.3)), we get

$\displaystyle J_{\lambda}(\bar{w})$
$\displaystyle=\frac{1}{2}\|\bar{w}\|^{2}_{H^{1}_{0}(\Omega)}-\frac{\lambda}{p+1}\int_{\Omega}|\bar{w}|^{p+1}\,dx$
$\displaystyle\leq\liminf_{k\to+\infty}J_{\lambda,\varepsilon_{j_{k}}}(w_{\lambda,\varepsilon_{j_{k}}})$
$\displaystyle\leq\liminf_{k\to+\infty}J_{\lambda,\varepsilon_{j_{k}}}(\varphi)=J_{\lambda}(\varphi)\quad\text{for
every $\varphi\in H_{0}^{1}(\Omega)$},$

and thus $\bar{w}$ is a global minimizer for the functional $J_{\lambda}$
naturally associated with the purely local, singular problem (3.5). As a
consequence, since $J_{\lambda}$ as a unique global minimizer which is the
unique solution $w_{\lambda}$ of (3.5) (that is, assertion i) also holds in
the purely local case, see [19]), we derive that

$\bar{w}=w_{\lambda}.$

Due to arbitrariness of the sequence $\\{\varepsilon_{j}\\}_{j}\subseteq(0,1]$
of its subsequence $\\{\varepsilon_{j_{k}}\\}_{k}$, and since the limit _is
always the same_ , we conclude the validity of a) - b).

Step 3) In this last step, we complete the proof of the assertion. First of
all we observe that, since $w_{\lambda,\varepsilon}$ is a weak solution of
problem (3.1), we readily derive that $w_{\lambda,\varepsilon}$ is a also
_weak supersolution_ (in the sense of Definition 2.2) of
$\mathcal{L}_{\varepsilon}u=0$ in $\Omega$; thus, by [13, Proposition 3.3]
(and since $w_{\lambda,\varepsilon}\geq 0$ a.e. in $\mathbb{R}^{n}$) we get

(3.6) $w_{\lambda,\varepsilon}\geq
c\,\left(\fint_{B_{r}(x_{0})}w_{\lambda,\varepsilon}^{Q}\,dx\right)^{1/Q}\quad\text{a.e.\,on
$B_{r}(x_{0})$},$

where $Q,\,c>0$ do not depend on $\varepsilon$. On the other hand, since by
Step 2) we know that $w_{\lambda,\varepsilon}\to w_{\lambda}$ as
$\varepsilon\to 0$ strongly in $L^{m}(\Omega)$ for every $1\leq m<2^{*}$, we
have

(3.7) $\lim_{\varepsilon\to
0^{+}}\fint_{B_{r}(x_{0})}w_{\lambda,\varepsilon}^{Q}\,dx=\fint_{B_{r}(x_{0})}w_{\lambda}^{Q}\,dx.$

Gathering (3.6)-(3.7), and recalling that $w_{\lambda}$ is the unique solution
of (3.5) (hence, $w_{\lambda}>0$ a.e. in $\Omega$), we can then fined some
$\varepsilon_{0}>0$ such that

$w_{\lambda,\varepsilon}\geq\frac{c}{2}\left(\fint_{B_{r}(x_{0})}w_{\lambda}^{Q}\,dx\right)^{1/Q}>0$

a.e. in $B_{r}(x_{0})$ and _for every $0<\varepsilon\leq\varepsilon_{0}$_.
This ends the proof. ∎

We now turn to prove our first main result, namely Theorem 1.4. Before doing
this, we establish the following auxiliary lemma.

######  Lemma 3.2.

Let $0<\varepsilon\leq 1,\,\lambda>0$ be fixed, and let
$w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ be the unique solution
of problem (3.1) _(_ whose existence and uniqueness is guaranteed by
Proposition 3.1 _)_. Then there exists a positive constant
$\beta=\beta_{\lambda,\varepsilon}>0$ such that

(3.8) $\rho_{\varepsilon}(\varphi)^{2}-\lambda
p\int_{\Omega}w_{\lambda,\varepsilon}^{p-1}\varphi\,dx\geq\beta_{\lambda,\varepsilon}\,\int_{\Omega}\varphi^{2}\,dx,\quad\textrm{for
all }\varphi\in\mathcal{X}^{1,2}(\Omega).$

###### Proof.

We first remind that, on account of Proposition 3.1 \- i), the function
$w_{\lambda,\varepsilon}$ is the unique, global minimizer of the functional
$J_{\lambda,\varepsilon}$ in (3.2), that is,

$J_{\lambda,\varepsilon}(w_{\lambda,\varepsilon})=\min_{u\in\mathcal{X}^{1,2}(\Omega)}J_{\lambda,\varepsilon}(u);$

this, together with the fact that $w_{\lambda,\varepsilon}\in
C^{1}(\overline{\Omega})$, see Proposition 3.1 \- ii), ensures that the
_second variation_ of the functional $J_{\lambda,\varepsilon}$ at
$w_{\lambda,\varepsilon}$ is non-negative, namely

(3.9)
$\rho_{\varepsilon}(\varphi)^{2}-p\lambda\int_{\Omega}w_{\lambda,\varepsilon}^{p-1}\varphi^{2}\,dx\geq
0\quad\text{for every $\varphi\in\mathcal{X}^{1,2}(\Omega)$}.$

As a consequence, we get

(3.10)
$\begin{split}&\lambda_{1}(\mathcal{L}_{\varepsilon}-\lambda\,p\,w_{\lambda,\varepsilon}^{p-1})\\\
&\qquad=\inf\Big{\\{}\rho_{\varepsilon}(\varphi)^{2}-\lambda\,p\,\int_{\Omega}w_{\lambda,\varepsilon}^{p-1}\varphi^{2}\,dx:\,\varphi\in\mathcal{X}^{1,2}(\Omega),\,\|\varphi\|_{L^{2}(\Omega)}=1\Big{\\}}\geq
0.\end{split}$

With (3.10) at hand, to prove (3.8) it then suffices to show that

(3.11)
$\lambda_{1}(\mathcal{L}_{\varepsilon}-\lambda\,pw_{\lambda,\varepsilon}^{p-1})>0;$

in fact, if (3.11) holds, the desired (3.8) follows with the choice

$\beta_{\lambda,\varepsilon}=\lambda_{1}(\mathcal{L}_{\varepsilon}-\lambda\,pw_{\lambda,\varepsilon}^{p-1}).$

Hence, we turn to prove (3.11). To this end, we argue by contradiction
assuming that (3.11) is not true; thus, by (3.10) we necessarily have that

$\lambda_{1}(\mathcal{L}_{\varepsilon}-\lambda\,pw_{\lambda,\varepsilon}^{p-1})=0.$

Taking into account Proposition 3.1 \- ii), and reminding that
$w_{\lambda,\varepsilon}>0$ _pointwise in $\Omega$_ (as
$w_{\lambda,\varepsilon}\in C^{1}(\overline{\Omega})$), we can find some
$\delta=\delta_{\lambda,\varepsilon}>0$ such that

$w_{\lambda,\varepsilon}(x)\geq\delta_{\lambda,\varepsilon}\,\mathrm{dist}(x,\partial\Omega)\quad\text{for
every $x\in\overline{\Omega}$};$

from this, by arguing exactly as in [1, Remark 3.1], we infer the existence of
a _non-negative eigenfunction_ associate with
$\lambda_{1}(\mathcal{L}_{\varepsilon}-a(x))$; this means, precisely, that
there exists a function
$\psi_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega),\,\psi_{\lambda,\varepsilon}\gneqq
0$, such that

$\mathcal{B}_{\varepsilon}(\psi_{\lambda,\varepsilon},\varphi)-\lambda\,p\int_{\Omega}w_{\lambda,\varepsilon}^{p-1}\psi_{\lambda,\varepsilon}\,dx=0\quad\text{for
every $\varphi\in\mathcal{X}^{1,2}(\Omega)$}.$

Now, choosing $\varphi=w_{\lambda,\varepsilon}$ in the above identity, we get

(3.12)
$\mathcal{B}_{\varepsilon}(\psi_{\lambda,\varepsilon},w_{\lambda,\varepsilon})=\lambda\,p\int_{\Omega}w_{\lambda,\varepsilon}^{p}\psi_{\lambda,\varepsilon}\,dx;$

on the other hand, if we choose $\varphi=\psi_{\lambda,\varepsilon}$ as a test
function for the equation solved by $w_{\lambda,\varepsilon}$ appearing in
problem (3.1) (see Definition 2.2), we have

(3.13)
$\mathcal{B}_{\varepsilon}(\psi_{\lambda,\varepsilon},w_{\lambda,\varepsilon})=\lambda\int_{\Omega}w_{\lambda,\varepsilon}^{p}\psi_{\lambda,\varepsilon}\,dx.$

Gathering (3.13) - (3.12), we then conclude that

$\lambda\int_{\Omega}w_{\lambda,\varepsilon}^{p}\psi_{\lambda,\varepsilon}\,dx=\lambda\,p\int_{\Omega}w_{\lambda,\varepsilon}^{p}\psi_{\lambda,\varepsilon}\,dx$

which is a contradiction, since $p\in(0,1)$ (also recall that
$w_{\lambda,\varepsilon}>0$ pointwise in $\Omega$, and that
$\psi_{\lambda,\varepsilon}\gneqq 0$). This ends the proof. ∎

With Lemma 3.2 at hand, we can provide the

###### Proof of Theorem 1.2.

Let $\varepsilon\in(0,1]$ be fixed, and let
$\beta_{\varepsilon}=\beta_{1,\varepsilon}>0$ be as in Lemma 3.2, with
$\lambda=1$. Moreover, let $M_{\varepsilon}>0$ be such that

(3.14) $p\,M_{\varepsilon}^{2^{\ast}-1}<\beta_{\varepsilon}.$

We now prove that, _with this choice of $M_{\varepsilon}$_, assertion 1) is
satisfied.

To this end, we argue by contradiction assuming that, for some fixed
$\lambda\in(0,\Lambda_{\varepsilon})$, there exists a second solution
$v_{\lambda,\varepsilon}$ of problem ($\mathrm{P}_{\varepsilon}$)λ such that

(3.15) $\|v_{\lambda,\varepsilon}\|_{L^{\infty}(\mathbb{R}^{n})}\leq
M_{\varepsilon}.$

By Theorem 1.1 (which can be applied to $\mathcal{L}_{\varepsilon}$, see Lemma
2.6), we know that there exists a _minimal solution $u_{\lambda,\varepsilon}$_
of problem ($\mathrm{P}_{\varepsilon}$)λ; hence, we have

$v_{\lambda,\varepsilon}=u_{\lambda,\varepsilon}+g_{\varepsilon},$

for some function
$g_{\varepsilon}\in\mathcal{X}^{1,2}(\Omega),\,g_{\varepsilon}\geq 0$ in
$\Omega$.

We now define the auxiliary function
$\zeta_{\varepsilon}:\mathbb{R}^{n}\to\mathbb{R}$ as

$\zeta_{\varepsilon}(x):=\lambda^{1/(1-p)}\,w_{1,\varepsilon}(x),$

where $w_{1,\varepsilon}$ is the unique solution of the purely sublinear
problem (3.1) with $\lambda=1$. A direct computation shows that $\zeta$ weakly
solves

(3.16)
$\mathcal{L}_{\varepsilon}\zeta_{\varepsilon}=\lambda\,\zeta_{\varepsilon}^{p}\quad\textrm{in
}\Omega.$

On the other hand, since $u_{\lambda,\varepsilon}$ is a weak solution of
problem ($\mathrm{P}_{\varepsilon}$)λ (hence, in particular,
$u_{\lambda,\varepsilon}>0$ a.e. in $\Omega$), this function is a _weak
supersolution_ of problem (3.1) (in the sense of Definition 2.2 with
$f(t)=\lambda t^{p}$); as a consequence, since we have already observed that
$\zeta$ solves (3.1), we can use [2, Lemma 3.11] (which can certainly be
applied to the operator $\mathcal{L}_{\varepsilon}$, the role of $\varepsilon$
being immaterial), obtaining

(3.17) $\text{$\zeta_{\varepsilon}=\lambda^{1/(1-p)}w_{1,\varepsilon}\leq
u_{\lambda,\varepsilon}$ in $\mathbb{R}^{n}$}.$

With all this at hand we can now conclude. First,

$\displaystyle\mathcal{L}_{\varepsilon}v_{\lambda,\varepsilon}$
$\displaystyle=\mathcal{L}_{\varepsilon}(u_{\lambda,\varepsilon}+g_{\varepsilon})$
$\displaystyle=\lambda(u_{\lambda,\varepsilon}+g_{\varepsilon})^{p}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}\quad\mbox{($v_{\lambda,\varepsilon}$
is a solution of \eqref{eq:Problem}${}_{\lambda}$)}$ $\displaystyle\leq\lambda
u_{\lambda,\varepsilon}^{p}+\lambda
p\,u_{\lambda,\varepsilon}^{p-1}g_{\varepsilon}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}\quad\mbox{(by
concavity of $t\mapsto t^{p}$)};$

as a consequence, we have

$\displaystyle\mathcal{L}_{\varepsilon}g_{\varepsilon}$
$\displaystyle\leq\lambda u_{\lambda,\varepsilon}^{p}+\lambda
p\,u_{\lambda,\varepsilon}^{p-1}g_{\varepsilon}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}-\mathcal{L}_{\varepsilon}u_{\lambda,\varepsilon}$
$\displaystyle=\lambda u_{\lambda,\varepsilon}^{p}+\lambda
p\,u_{\lambda,\varepsilon}^{p-1}g_{\varepsilon}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}-\lambda
u_{\lambda,\varepsilon}^{p}-u_{\lambda,\varepsilon}^{2^{\ast}-1}$
$\displaystyle=\lambda
p\,u_{\lambda,\varepsilon}^{p-1}g_{\varepsilon}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}-u_{\lambda,\varepsilon}^{2^{\ast}-1}$
$\displaystyle\leq
p\,w_{1,\varepsilon}^{p-1}g_{\varepsilon}+(u_{\lambda,\varepsilon}+g_{\varepsilon})^{2^{\ast}-1}-u_{\lambda,\varepsilon}^{2^{\ast}-1}\quad\mbox{(by
\eqref{eq:Comparison_ulambda_v}, and since $0<p<1$)}$ $\displaystyle\leq
p\,w_{1,\varepsilon}^{p-1}g_{\varepsilon}+p\,M_{\varepsilon}^{2^{\ast}-1}g_{\varepsilon}\qquad\mbox{(by
\eqref{eq:Stima_w_lambda})}$
$\displaystyle<p\,w_{1,\varepsilon}^{p-1}g_{\varepsilon}+\beta_{\varepsilon}\,g_{\varepsilon}\qquad\qquad\mbox{(by
\eqref{eq:Def_M})}.$

Summing up, the function $g_{\varepsilon}$ satisfies (in the weak sense on
$\Omega$)

$\mathcal{L}_{\varepsilon}g_{\varepsilon}-p\,w_{1,\varepsilon}^{p-1}g_{\varepsilon}<\beta_{\varepsilon}g_{\varepsilon}.$

Testing it with $g$ itself, and using (3.8) (with $\lambda=1$), we get

$\beta_{\varepsilon}\int_{\Omega}g_{\varepsilon}^{2}\,dx\leq\rho_{\varepsilon}(g_{\varepsilon})^{2}-p\int_{\Omega}w_{1,\varepsilon}^{p-1}g_{\varepsilon}\,dx<\beta_{\varepsilon}\int_{\Omega}g_{\varepsilon}^{2}\,dx,$

which is a contradiction. ∎

##  4\. Proof of Theorem 1.3

In this section we show that the famous result [18, Theorem 1] still holds in
the mixed local-nonlocal setting. The proof is a quite simple adaptation of
the original proof by Brezis and Nirenberg and the major role is played by the
recent regularity results established in [3] and [38].

###### Proof of Theorem 1.3.

By assumption, $u_{0}\in\mathcal{X}^{1,2}(\Omega)$ is a local minimizer for
$\Phi$ in the $C^{1}$-topology; this means, precisely, that there exists some
$r>0$ such that

(4.1) $\Phi(u_{0})\leq\Phi(u_{0}+v),\quad\textrm{for all }v\in
C^{1}_{0}(\overline{\Omega})\,\textrm{ s.t. }\|v\|_{C^{1}(\Omega)}\leq r.$

It follows that the Euler-Lagrange equation satisfied by $u_{0}$ is

$\begin{cases}\mathcal{L}_{\varepsilon}v=f(x,v)&\textrm{ in }\Omega,\\\
v=0&\textrm{ in }\mathbb{R}^{n}\setminus\Omega;\end{cases}$

as a consequence, since (by assumption) $f$ satisfies the growth condition
(1.2), from Theorem 2.3 we derive that $u_{0}\in
C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$. In view of this
fact, from now on we may assume without loss of generality that

$u_{0}=0.$

We now argue by contradiction, assuming that $u_{0}$ _is not a local minimizer
for $\Phi$ in the $\mathcal{X}^{1,2}$-topology_; thus, for every $\delta>0$
there exists $v_{\delta}\in\mathcal{X}^{1,2}(\Omega)$ such that

(4.2)
$\rho_{\varepsilon}(v_{\delta})<\delta\quad\text{and}\quad\Phi(v_{\delta})<\Phi(0).$

We explicitly notice that use of $\rho_{\varepsilon}(\cdot)$ to define an open
neighborhood of $u_{0}=0$ in (4.2) is motivated by the fact that, since
$\varepsilon\in(0,1)$, by (2.6) we have

$\|u\|_{H^{1}_{0}(\Omega)}\leq\rho_{\varepsilon}(u)\leq\rho(u)\leq\Theta\|u\|_{H^{1}_{0}(\Omega)}$

for some $\Theta>0$ only depending on $n,s$; as a consequence, the norm
$\rho_{\varepsilon}(\cdot)$ is globally equivalent to $\rho(\cdot)$ (and to
the $H_{0}^{1}$-norm), uniformly w.r.t. $\varepsilon$.

Now, since the closed set
$K_{\delta}=\\{u\in\mathcal{X}^{1,2}(\Omega):\,\rho_{\varepsilon}(u)\leq\delta\\}\subseteq$
is weakly sequentially compact, and since $\Phi$ is weakly lower
semincontinuous and bounded from below (by the growth assumption (1.2) on
$f$), we can find $w_{\delta}\in K_{\delta}$ such that

$\Phi(w_{\delta})=\min_{u\in K_{\delta}}\Phi(u).$

In particular, by (4.2) we get

(4.3) $\Phi(w_{\delta})\leq\Phi(v_{\delta})<\Phi(0)\quad\text{for every
$\delta>0$}.$

We then claim that

(4.4) $\text{$\mathrm{i)}\,\,w_{\delta}\in C_{0}^{1}(\overline{\Omega})$ \quad
and\quad$\mathrm{ii)}\,\,\lim_{\delta\to 0}w_{\delta}=0$ in the
$C^{1}(\overline{\Omega})$-topology}.$

Taking this claim for granted for a moment, we can immediately complete the
proof of the theorem: in fact, owing to (4.4), we see that there exists
$\delta_{0}>0$ such that

$\|w_{\delta}\|_{C^{1}(\overline{\Omega})}\leq r\quad\text{for every
$0<\delta<\delta_{0}$},$

where $r>0$ is as in (4.1); thus, by (4.1) (and since we are assuming
$u_{0}=0$) we infer that $\Phi(0)\leq\Phi(w_{\delta})$ for every
$0<\delta<\delta_{0}$, but this clearly contradicts (4.3).

We then turn to prove (4.4), and we proceed by steps.

Step I). In this first step we prove that there exist $\delta_{0}>0$ and a
constant $c_{0}>0$, _independent of $\delta$_, such that the following
estimate holds:

(4.5) $\|w_{\delta}\|_{L^{\infty}(\Omega)}\leq c_{0}\quad\text{for every
$0<\delta<\delta_{0}$}.$

To this end we first notice that, since $w_{\delta}$ is a global minimizer for
$\Phi$ on $K_{\delta}$, by the Lagrange Multiplier Theorem, there exists
$\mu_{\delta}\in\mathbb{R}$ such that

(4.6)
$\mathcal{B}_{\varepsilon}(w_{\delta},\varphi)-\int_{\Omega}f(x,w_{\delta})\varphi\,dx=\mu_{\delta}\,\mathcal{B}_{\varepsilon}(w_{\delta},\varphi)\quad\textrm{for
all }\varphi\in\mathcal{X}^{1,2}(\Omega).$

If $\rho_{\varepsilon}(v_{\delta})<\delta$ (that is, if
$w_{\delta}\in\mathrm{int}(K_{\delta})$), then $\mu_{\delta}=0$; if, instead,
$\rho_{\varepsilon}(v_{\delta})=\delta$ (that is, if $w_{\delta}\in\partial
K_{\delta}$), by choosing $\varphi=v_{\delta}$ in (4.6) we derive that

$\displaystyle\mu_{\delta}\,\delta^{2}$
$\displaystyle=\mathcal{B}_{\varepsilon}(w_{\delta},w_{\delta})=\mathcal{B}_{\varepsilon}(w_{\delta},w_{\delta})-\int_{\Omega}f(x,w_{\delta})w_{\delta}\,dx$
$\displaystyle=\Phi(w_{\delta})+\int_{\Omega}F(x,w_{\delta})\,dx-\int_{\Omega}f(x,w_{\delta})w_{\delta}\,dx$
$\displaystyle(\text{using \eqref{eq:tocontradictH1C1}, and since
$\Phi(0)=0$})$
$\displaystyle<\int_{\Omega}\big{(}F(x,w_{\delta})\,dx-\int_{\Omega}f(x,w_{\delta})w_{\delta}\big{)}dx\leq
0,$

where we have used the growth condition (1.2). Summing up, we derive that
$\mu_{\delta}\leq 0$ and that $w_{\delta}\in\mathcal{X}^{1,2}(\Omega)$ is a of
the Dirichlet problem

(4.7)
$\begin{cases}\displaystyle\mathcal{L}_{\varepsilon}u=\frac{f(x,u)}{1-\mu_{\delta}}&\text{in
$\Omega$},\\\ u=0&\text{in $\mathbb{R}^{n}\setminus\Omega$}\end{cases}$

(in the sense of Definition 2.2). Now, since $f$ satisfies the growth
condition (1.2) and since $\mu_{\delta}\leq 0$, it is readily seen that the
function

(4.8)
$g_{\delta}(x,t)=\frac{f(x,t)}{1-\mu_{\delta}}\qquad(x\in\Omega,\,t\in\mathbb{R})$

satisfies _the same growth condition ( 1.2)_, with the same constant
$C_{f}>0$; we are then entitled to apply [38, Theorem 1.1], thus giving

(4.9)
$\|w_{\delta}\|_{L^{\infty}(\Omega)}\leq\kappa_{0}\Big{(}1+\int_{\Omega}|w_{\delta}|^{2^{*}\beta}\,dx\Big{)}^{\frac{1}{2^{*}(\beta-1)}},$

where $\beta=(2^{*}+1)/2$ and $\kappa_{0}>0$ is a constant independent of
$\delta$. With (4.9) at hand, to complete the proof of (4.5) we need to show
that

(4.10) $\int_{\Omega}|w_{\delta}|^{2^{*}\beta}\,dx\leq c$

for some constant $c>0$ independent of $\delta$, provided that $\delta$ is
sufficiently small. To prove (4.10), we closely follow the approach in [38,
Lemma 3.2]: defining

$\varphi(t):=\begin{cases}-\beta T^{\beta-1}(t+T)+T^{\beta},&t\leq-T\\\
|t|^{\beta},&-T<t<T\\\ \beta T^{\beta-1}(t-T)+T^{\beta},&t\geq T.\end{cases}$

(with $T>0$), and arguing exactly as in [38, Lemma 3.2], we get

(4.11)
$\displaystyle\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}\leq
c\beta\Big{(}\int_{\Omega}|w_{\delta}|^{2^{*}}\,dx+\int_{\Omega}(\varphi(w_{\delta}))^{2}|w_{\delta}|^{2^{\ast}-2}\,dx\Big{)},$

which is exactly to [38, Equation (3.5)] (here, $c>0$ is a constant only
depending on $n,\Omega$ and on the constant $C_{f}$). On the other hand, we
have

$\displaystyle\int_{\Omega}(\varphi(w_{\delta}))^{2}|w_{\delta}|^{2^{\ast}-2}\,dx$
$\displaystyle\qquad=\int_{\\{|w_{\delta}|\leq
1\\}}(\varphi(w_{\delta}))^{2}|w_{\delta}|^{2^{\ast}-2}\,dx+\int_{\\{|w_{\delta}|>1\\}}(\varphi(w_{\delta}))^{2}|w_{\delta}|^{2^{\ast}-2}\,dx$
$\displaystyle\qquad\leq\int_{\\{|w_{\delta}|\leq
1\\}}\frac{\varphi(w_{\delta})^{2}}{|w_{\delta}|}\,dx+\int_{\Omega}(\varphi(w_{\delta}))^{2}|w_{\delta}|^{2^{\ast}-2}\,dx=(\bigstar);$

then, using Hölder’s and Sobolev’s inequality, we obtain

$\displaystyle(\bigstar)$ $\displaystyle\leq\int_{\\{|w_{\delta}|\leq
1\\}}\frac{|\varphi(w_{\delta})|^{2}}{|w_{\delta}|}\,dx+\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}\Big{(}\int_{\Omega}|w_{\delta}|^{2^{\ast}}\,dx\Big{)}^{\frac{2^{*}-2}{2^{*}}}$
$\displaystyle\leq\int_{\\{|w_{\delta}|\leq
1\\}}\frac{|\varphi(w_{\delta})|^{2}}{|w_{\delta}|}\,dx+S_{n}^{\frac{2-2^{*}}{2}}\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}\|w_{\delta}\|_{H^{1}_{0}(\Omega)}^{2^{*}-2}$
$\displaystyle(\text{since
$\rho_{\varepsilon}(\cdot)\geq\|\cdot\|_{H^{1}_{0}}$, and since $w_{\delta}\in
K_{\delta}$})$ $\displaystyle\leq\int_{\\{|w_{\delta}|\leq
1\\}}\frac{|\varphi(w_{\delta})|^{2}}{|w_{\delta}|}\,dx+\delta^{2^{*}-2}S_{n}^{\frac{2-2^{*}}{2}}\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}.$

where $S_{n}>0$ is the best constant in the (local) Sobolev inequality. Using
this last estimate in the above (4.11), we then derive

(4.12)
$\begin{split}\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}&\leq
c\beta\Big{(}\int_{\Omega}|w_{\delta}|^{2^{*}}\,dx+\int_{\\{|w_{\delta}|\leq
1\\}}\frac{|\varphi(w_{\delta})|^{2}}{|w_{\delta}|}\,dx\Big{)}\\\
&\quad+c\beta\delta^{2^{*}-2}S_{n}^{\frac{2-2^{*}}{2}}\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}\Big{)}.\end{split}$

With (4.12) at hand, we can proceed toward the end of the proof of (4.10):
indeed, if we choose $\delta_{0}\in(0,1)$ so small that

$c\beta\delta^{2^{*}-2}S_{n}^{\frac{2-2^{*}}{2}}<\frac{1}{2}$

and if we let $0<\delta<\delta_{0}$, we can reabsorb the last integral in the
right-hand side of the cited (4.12), thus giving

$\begin{split}\Big{(}\int_{\Omega}|\varphi(w_{\delta})|^{2^{*}}\,dx\Big{)}^{2/2^{*}}&\leq
2c\beta\Big{(}\int_{\Omega}|w_{\delta}|^{2^{*}}\,dx+\int_{\\{|w_{\delta}|\leq
1\\}}\frac{|\varphi(w_{\delta})|^{2}}{|w_{\delta}|}\,dx\Big{)};\end{split}$

from this, by letting $T\to+\infty$ and by arguing as in [38, Lemma 3.2], we
get

$\displaystyle\Big{(}\int_{\Omega}|w_{\delta}|^{2^{*}\beta}\,dx\Big{)}^{2/2^{*}}\leq
4c\beta\int_{\Omega}|w_{\delta}|^{2^{*}}\,dx\quad\text{for every
$0<\delta<\delta_{0}$}.$

Finally, by using once again the Sobolev inequality, we conclude that

$\displaystyle\Big{(}\int_{\Omega}|w_{\delta}|^{2^{*}\beta}\,dx\Big{)}^{2/2^{*}}\leq
4c\beta\,S_{n}^{-2^{*}/2}\rho_{\varepsilon}(w_{\delta})\leq
4c\beta\,S_{n}^{-2^{*}/2},$

and this estimate holds for every $0<\delta<\delta_{0}$ (since
$\delta_{0}<1$). Recalling that $c>0$ only depends on $n,\Omega$ and on the
constant $C_{f}$, this completes the proof of (4.10).

Step II). In this second step we complete the proof of (4.4) - i). To this end
it suffices to observe that, since the function $g_{\delta}$ in (4.8)
satisfies the growth condition (1.2) _with the same constant $C_{f}$_
(independent of $\delta$), we are entitled to apply [38, Theorem 1.3]; this,
together with (4.5), ensures that

  * 1)

$w_{\delta}\in C^{1,\alpha}(\overline{\Omega})$ for every
$0<\alpha<\min\\{1,2-2s\\}$;

  * 2)

there exists $c>0$, independent of $\delta$, such that

(4.13) $\|w_{\delta}\|_{C^{1,\alpha}(\overline{\Omega})}\leq
c(1+\|w_{\delta}\|^{2^{*}}_{L^{\infty}(\Omega)})\leq c\quad\text{for every
$0<\delta<\delta_{0}$}.$

In particular, since $w_{\delta}\in\mathcal{X}^{1,2}(\Omega)$, we get
$w_{\delta}\equiv 0$ pointwise on $\partial\Omega$.

Step III). In this last step we give the proof of (4.4) - ii). To this end we
observe that, on account of (4.13), by the Arzelà-Ascoli Theorem we can find a
sequence $\\{\delta_{j}\\}\subseteq(0,\delta_{0})$ converging to $0$ as
$j\to+\infty$ and $w_{0}\in C_{0}^{1}(\overline{\Omega})$ such that

$\|w_{\delta_{j}}-w_{0}\|_{C^{1}(\overline{\Omega})}\to 0\,\,\text{as
$j\to+\infty$};$

thus, since $w_{\delta_{j}}\to 0$ in $\mathcal{X}^{1,2}(\Omega)$ as
$j\to+\infty$ (recall that $w_{\delta}\in K_{\delta}$ and that the two norms
$\rho_{\varepsilon}(\cdot),\,\rho(\cdot)$ are globally equivalent in
$\mathcal{X}^{1,2}(\Omega)$), we conclude that

$w_{0}\equiv 0.$

This ends the proof. ∎

##  5\. Proof of Theorem 1.4

Following the approach in [1], in this section we show how Theorem 1.3 can be
used in order to establish Theorem 1.4, that is, the existence of (at least)
_two distinct solutions_ of ($\mathrm{P}_{\varepsilon}$)λ provided that
$\varepsilon\in(0,1)$ is small enough.

We now turn to prove that, if $\varepsilon\in(0,1]$ is fixed and if
$\lambda\in(0,\mu_{\sharp})$ (with $\mu_{\sharp}>0$ as in Lemma 2.6), there
exists a solution $u_{\lambda,\varepsilon}$ of problem
($\mathrm{P}_{\varepsilon}$)λ which is a _local minimizer_ for the functional
associated with the problem, that is,

(5.1)
$I_{\lambda,\varepsilon}(u)=\dfrac{1}{2}\mathcal{\rho}_{\varepsilon}(u)^{2}-\dfrac{\lambda}{p+1}\int_{\Omega}|u|^{p+1}\,dx-\dfrac{1}{2^{\ast}}\int_{\Omega}|u|^{2^{\ast}}\,dx,$

where
$\mathcal{\rho}_{\varepsilon}(u)^{2}:=\|u\|_{H^{1}_{0}(\Omega)}+\varepsilon\,[u]^{2}_{s,\mathbb{R}^{n}}$.

######  Theorem 5.1.

Let $\varepsilon\in(0,1]$ be fixed, and let $\lambda\in(0,\mu_{\sharp})$ _(_
with $\mu_{\sharp}>0$ as in Lemma 2.6 _)_. Then, there exists a solution
$u_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ of problem
($\mathrm{P}_{\varepsilon}$)λ which is a local minimizer for
$I_{\lambda,\varepsilon}$, that is, there exists $\varrho_{0}>0$ such that

$I_{\lambda,\varepsilon}(u)\geq
I_{\lambda,\varepsilon}(u_{\lambda,\varepsilon})\quad\text{for all
$u\in\mathcal{X}^{1,2}(\Omega)$ with
$\rho(u-u_{\lambda,\varepsilon})<\varrho_{0}$}.$

Moreover, the following assertions hold:

  * 1)

there exists a constant $C>0$, independent of $\varepsilon$ but possibly
depending on the fixed $\lambda\in(0,\mu_{\sharp})$, such that

(5.2) $\|u_{\lambda,\varepsilon}\|_{L^{\infty}(\Omega)}\leq C;$

  * 2)

if $w_{\lambda,\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ is the unique
solution of (3.1), we have

(5.3) $u_{\lambda,\varepsilon}\geq w_{\lambda,\varepsilon}\quad\text{a.e.\,in
$\Omega$}.$

###### Proof.

The proof is analogous to that of [1, Lemma 4.1], but we present it here for
the sake of completeness. To begin with, we fix
$\lambda_{1},\lambda_{2}\in(0,\mu_{\sharp})$ such that

(5.4) $\lambda_{1}<\lambda<\lambda_{2};$

accordingly, we let
$u_{i}=\bar{u}_{\lambda_{i},\varepsilon}\in\mathcal{X}^{1,2}(\Omega)$ be the
minimal solution of problem ($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{i}}$
(with $i=1,2$), whose existence is guaranteed by Lemma 2.6. We then observe
that, on account of Theorem 2.3, we have that $u_{1},u_{2}\in
C^{1}(\overline{\Omega})$; moreover, since the map
$t\mapsto\bar{u}_{t,\varepsilon}$ is non-decreasing on
$(0,\Lambda_{\varepsilon})$, by (5.4) we also have

$\text{$u_{1}\leq u_{2}$ pointwise in $\Omega$}.$

In view of these facts, and since $u_{i}$ solves
($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{i}}$ (for $i=1,2$), we infer that

  * a)

$w=u_{2}-u_{1}\in\mathcal{X}^{1,2}(\Omega)\cap C^{1}(\overline{\Omega})$;

  * b)

for every test function $\varphi\in
C_{0}^{\infty}(\Omega),\,\text{$\varphi\geq 0$ on $\Omega$}$, we have

$\displaystyle\mathcal{B}_{\varepsilon}(w,\varphi)$
$\displaystyle=\int_{\Omega}(\lambda_{2}u_{2}^{p}+u_{2}^{2^{*}-1})\varphi\,dx-\int_{\Omega}(\lambda_{1}u_{1}^{p}+u_{1}^{2^{*}-1})\varphi\,dx$
$\displaystyle(\text{by \eqref{eq:choicelambda12}, and since $u_{2}\geq
u_{1}\geq 0$})$
$\displaystyle\geq\lambda_{1}\int_{\Omega}(u_{2}^{p}-u_{1}^{p})\varphi\,dx+\int_{\Omega}(u_{2}^{2^{*}-1}-u_{1}^{2^{*}-1})\varphi\,dx\geq
0$

and this proves that $w$ is a (non-negative) _weak supersolution_ (in the
sense of Definition 2.2) of the following problem

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

Since $w\not\equiv 0$ in $\Omega$ (as $\lambda_{1}<\lambda_{2}$), by Theorem
2.4 we conclude that

(5.5) $\text{$w=u_{2}-u_{1}>0$ pointwise in
$\Omega$}\quad\text{and}\quad\text{$\partial_{\nu}(u_{2}-u_{1})<0$ on
$\partial\Omega$}.$

With (5.5) at hand, we now define the function

$f(x,t)=\begin{cases}\lambda u_{1}(x)^{p}+u_{1}(x)^{2^{*}-1}&\text{if $t\leq
u_{1}(x)$}\\\ \lambda t^{p}+t^{2^{*}-1}&\text{if $u_{1}(x)<t<u_{2}(x)$}\\\
\lambda u_{2}(x)^{p}+u_{2}(x)^{2^{*}-1}&\text{if $t\geq u_{2}(x)$}\end{cases}$

(for $x\in\Omega$ and $t\in\mathbb{R}$), and we consider the functional

$J(u)=\frac{1}{2}\rho_{\varepsilon}(u)^{2}-\int_{\Omega}F(x,u)\,dx,\qquad\text{with
$F(x,t)=\int_{0}^{t}f(x,s)\,ds$}.$

Since $u_{1},u_{2}\in C^{1}(\overline{\Omega})$, the function $f$ is _globally
bounded_ on $\Omega\times\mathbb{R}$; as a consequence of this fact, it is
easy to check that the functional $J$ achieves its (global) minimum at some
$u\in\mathcal{X}^{1,2}(\Omega)$, which is a weak solution of

(5.6) $\begin{cases}\mathcal{L}_{\varepsilon}u=f&\text{in $\Omega$},\\\
u=0&\text{on $\partial\Omega$}\end{cases}$

Moreover, from Theorem 2.3 we get $u\in C^{1}(\overline{\Omega})$. We then
claim that

(5.7) $\begin{split}\mathrm{i)}\,\,&\text{$u_{1}<u<u_{2}$ pointwise in
$\Omega$};\\\ \mathrm{ii)}\,\,&\text{$\partial_{\nu}(u-u_{1})<0$ and
$\partial_{\nu}(u-u_{2})>0$ pointwise on $\partial\Omega$}.\end{split}$

In fact, let $w_{1}=u-u_{1}$. On the one hand, we have
$w_{1}\in\mathcal{X}^{1,2}(\Omega)\cap C^{1}(\overline{\Omega})$ (as the same
is true of both $u$ and $u_{1}$); on the other hand, since $u_{1}$ solves
$\mathrm{(P)_{\lambda_{1},\varepsilon}}$ and since $u$ solves (5.6), by
definition of $f$ (and by recalling (5.5)) we also have

$\displaystyle\mathcal{B}_{\varepsilon}(w_{1},\varphi)$
$\displaystyle=\int_{\Omega}\big{(}f(x,u)-\lambda
u_{1}^{p}-u_{1}^{2^{*}-1}\big{)}\varphi\,dx\geq 0$

for every test function $\varphi\in
C_{0}^{\infty}(\Omega),\,\text{$\varphi\geq 0$ on $\Omega$}$. Hence, we see
that $w_{1}$ is a weak supersolution (in the sense of Definition 2.2) of the
problem

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

Since $w_{1}\not\equiv 0$ in $\Omega$ (as $\lambda_{1}<\lambda)$, again by
Theorem 2.4 we then get

$\text{$w_{1}=u-u_{1}>0$ pointwise in
$\Omega$}\quad\text{and}\quad\text{$\partial_{\nu}(u-u_{1})<0$ on
$\partial\Omega$}.$

In a totally analogous way one can prove that $w_{2}=u_{2}-u>0$ pointwise in
$\Omega$ and that $\partial_{\nu}w_{2}<0$ on $\partial\Omega$, and this
completes the proof of (5.7).

Now we have established (5.7), we are finally ready to complete the
demonstration of the theorem. Indeed, owing to (5.7), there exists
$\varrho_{0}>0$ so small that

$\|v-u\|_{C^{1}(\overline{\Omega})}<\varrho_{0}\,\,\Longrightarrow\,\,\text{$u_{1}\leq
v\leq u_{2}$ pointwise in $\Omega$};$

from this, taking into account that $J(v)=I_{\lambda,\varepsilon}(v)$ for
every $v\in\mathcal{X}^{1,2}(\Omega)$ satisfying $u_{1}\leq v\leq u_{2}$ in
$\Omega$ (see the explicit definition of $f$), we derive that

$I_{\lambda,\varepsilon}(v)=J(v)\geq
J(u)=I_{\lambda,\varepsilon}(u)\quad\text{for all $v\in
C^{1}(\overline{\Omega})$ with
$\|v-u\|_{C^{1}(\overline{\Omega})}<\varrho_{0}$},$

and this proves that $u$ is a local minimizer of $I_{\lambda,\varepsilon}$ in
the $C^{1}$-topology. This, together with Theorem 1.3, ensures that
$u\in\mathcal{X}^{1,2}(\Omega)\cap C^{1}(\overline{\Omega})$ is a local
minimizer of the functional $I_{\lambda,\varepsilon}$ also in the
$\mathcal{X}^{1,2}$-topology, as desired.

We now turn to prove the two assertions 1) - 2).

\-  _Proof of assertion_ 1). It is a direct consequence of (5.7) - i) and of
estimate (2.17) in Lemma 2.6, taking into account that
$u_{1}=\bar{u}_{\lambda_{1},\varepsilon}$.

\-  _Proof of assertion_ 2). First of all we observe that, since
$u_{\lambda,\varepsilon}$ is a weak solution of problem
($\mathrm{P}_{\varepsilon}$)λ (hence, in particular,
$u_{\lambda,\varepsilon}>0$ a.e. in $\Omega$), this function is a _weak
supersolution_ of problem (3.1) (in the sense of Definition 2.2 with
$f(t)=\lambda t^{p}$); thus, since $w_{\lambda,\varepsilon}$ solves (3.1), we
can use [2, Lemma 3.11] (which can certainly be applied to the operator
$\mathcal{L}_{\varepsilon}$, the role of $\varepsilon$ being immaterial),
obtaining

$w_{\lambda,\varepsilon}\leq u_{\lambda,\varepsilon}\quad\text{a.e.\,in
$\Omega$}.$

We explicitly mention that [2, Lemma 3.11] can be applied, since
$f(t)/t=\lambda t^{p-1}$ is decreasing, and since
$w_{\lambda,\varepsilon}\,\,u_{\lambda,\varepsilon}\in L^{2^{*}}(\Omega)$, see
Remark 2.1 \- 4).

This ends the proof. ∎

Now we have established Theorem 5.1, we are ready to embark on the
demonstration of Theorem 1.4. To keep the notation as simple as possible,
throughout what follows we avoid to keep explicit track of the dependence on
$\varepsilon$ of the functional $I_{\lambda,\varepsilon}$ in (5.1) and of the
solution $u_{\lambda,\varepsilon}$ obtained in Theorem 5.1. Thus, as in the
previous sections we simply write $I_{\lambda}$ and $u_{\lambda}$, the
dependence on $\varepsilon$ being understood.

We begin with the following key lemma.

######  Lemma 5.2.

Let $\lambda\in(0,\mu_{\sharp})$ be fixed _(_ with $\mu_{\sharp}>0$ as in
(2.18)_)_. Then, there exist $\varepsilon_{0}\in(0,1)$, $R_{0}>0$ and a
positive function $\Psi\in\mathcal{X}^{1,2}(\Omega)$ such that

(5.8)
$\left\\{\begin{array}[]{lr}I_{\lambda}(u_{\lambda}+R\Psi)<I_{\lambda}(u_{\lambda})&\textrm{for
all $\varepsilon\in(0,\varepsilon_{0})$ and $R\geq R_{0}$},\\\
I_{\lambda}(u_{\lambda}+tR_{0}\Psi)<I_{\lambda}(u_{\lambda})+\tfrac{1}{n}S_{n}^{n/2}&\textrm{for
all $\varepsilon\in(0,\varepsilon_{0})$ and $t\in[0,1]$}.\end{array}\right.$

###### Proof.

First of all, we choose a _Lebesgue point_ of $u_{\lambda}$ in $\Omega$, say
$y$, and we let $r>0$ be such that $B_{r}(y)\Subset\Omega$; we then choose a
cut-off function $\varphi\in C_{0}^{\infty}(\Omega)$ such that

  * $(\ast)$

$0\leq\varphi\leq 1$ in $\Omega$;

  * $(\ast)$

$\varphi\equiv 1$ on $B_{r}(y)$;

and we consider the one-parameter family of functions

(5.9)
$U_{\varepsilon}=V_{\varepsilon}\,\varphi,\qquad\text{where}\,\,V_{\varepsilon}(x)=\frac{\varepsilon^{\frac{\alpha(n-2)}{2}}}{(\varepsilon^{2\alpha}+|x-y|^{2})^{\frac{n-2}{2}}},$

where $\alpha>0$ will be appropriately chosen later on. Notice that this
family $\\{V_{\varepsilon}\\}_{\varepsilon}$ is the well-known family of the
_Aubin-Talenti functions_ , which are the unique (up to translation) extremals
in the (local) Sobolev inequality; this means that

(5.10) $\frac{\||\nabla
V_{\varepsilon}|\|^{2}_{L^{2}(\mathbb{R}^{n})}}{\|V_{\varepsilon}\|^{2}_{L^{2^{*}}(\mathbb{R}^{n})}}=S_{n},$

where $S_{n}>0$ is the _best constant_ in the Sobolev inequality.

We now recall that, owing to [16, Lemma 1.1], we have (as $\varepsilon\to
0^{+}$)

(5.11)
$\begin{split}\mathrm{i)}&\,\,\|U_{\varepsilon}\|^{2}_{H^{1}_{0}(\Omega)}=\varepsilon^{\alpha(n-2)}\Big{(}\frac{K_{1}}{\varepsilon^{\alpha(n-2)}}+O(1)\Big{)}=K_{1}+o(\varepsilon^{\frac{\alpha(n-2)}{2}});\\\\[2.84544pt]
\mathrm{ii)}&\,\,\|U_{\varepsilon}\|^{2^{*}}_{L^{2^{*}}(\Omega)}=\varepsilon^{\alpha
n}\Big{(}\frac{K_{2}}{\varepsilon^{\alpha
n}}+O(1)\Big{)}=K_{2}+o(\varepsilon^{\alpha n/2});\end{split}$

where the constants $K_{1},K_{2}>0$ are given, respectively, by

$K_{1}=\||\nabla V_{1}|\|^{2}_{L^{2}(\mathbb{R}^{n})},\qquad
K_{2}=\int_{\mathbb{R}^{n}}\frac{1}{(1+|x|^{2})^{n}}\,dx$

and they satisfy $K_{1}/K_{2}^{1-2/n}=S_{n}$, see the above (5.10). On the
other hand, by combining (5.11) - ii) with [10, Lemma 4.11], we also have

(5.12)
$\begin{split}\varepsilon\,[U_{\varepsilon}]^{2}_{s}&\leq\varepsilon\,\|U_{\varepsilon}\|^{2}_{L^{2^{*}}(\Omega)}\cdot
O(\varepsilon^{\alpha(2-2s)})=\|U_{\varepsilon}\|^{2}_{L^{2^{*}}(\Omega)}\cdot
O(\varepsilon^{1+\alpha(2-2s)})\\\ &=\big{(}K_{2}+o(\varepsilon^{\alpha
n/2})\big{)}^{2/2^{*}}\cdot
O(\varepsilon^{1+\alpha(2-2s)})=o(\varepsilon^{1+\alpha(1-s)}).\end{split}$

Defining $w=u_{\lambda}+tRU_{\varepsilon}$ (for $R\geq 1$ and $t\in[0,1]$), we
then obtain the following estimate (as $\varepsilon\to 0^{+}$):

$\begin{split}I_{\lambda,\varepsilon}(w)&=I_{\lambda,\varepsilon}(u_{\lambda})+\frac{t^{2}R^{2}}{2}\rho_{\varepsilon}(U_{\varepsilon})^{2}+tR\mathcal{B}_{\rho}(u_{\lambda},U_{\varepsilon})\\\
&\qquad-\frac{1}{2^{*}}\big{(}\|u_{\lambda}+tRU_{\varepsilon}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\|u_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}\big{)}\\\
&\qquad-\frac{\lambda}{p+1}\Big{(}\int_{\Omega}|u_{\lambda}+tRU_{\varepsilon}|^{p+1}\,dx-\int_{\Omega}|u_{\lambda}|^{p+1}\,dx\Big{)},\end{split}$

and recalling that $u_{\lambda}$ solves ($\mathrm{P}_{\varepsilon}$)λ, we get

$\begin{split}I_{\lambda,\varepsilon}(w)&=I_{\lambda,\varepsilon}(u_{\lambda})+\frac{t^{2}R^{2}}{2}\rho_{\varepsilon}(U_{\varepsilon})^{2}+tR\int_{\Omega}(\lambda
u_{\lambda}^{p}+u_{\lambda}^{2^{*}-1})U_{\varepsilon}\,dx\\\
&\qquad-\frac{1}{2^{*}}\big{(}\|u_{\lambda}+tRU_{\varepsilon}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\|u_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}\big{)}\\\
&\qquad-\frac{\lambda}{p+1}\Big{(}\int_{\Omega}|u_{\lambda}+tRU_{\varepsilon}|^{p+1}\,dx-\int_{\Omega}|u_{\lambda}|^{p+1}\,dx\Big{)}\\\
&=I_{\lambda,\varepsilon}(u_{\lambda})+\frac{t^{2}R^{2}}{2}\rho_{\varepsilon}(U_{\varepsilon})^{2}-\frac{t^{2^{*}}R^{2^{*}}}{2^{*}}\|U_{\varepsilon}\|^{2^{*}}_{L^{2^{*}}(\Omega)}\\\
&\qquad-t^{2^{*}-1}R^{2^{*}-1}\int_{\Omega}U_{\varepsilon}^{2^{*}-1}u_{\lambda}\,dx-\mathcal{R}_{\varepsilon}-\mathcal{D}_{\varepsilon}\end{split}$

where we have introduced the notation

$\displaystyle(\ast)$
$\displaystyle\,\,\mathcal{R}_{\varepsilon}=\frac{1}{2^{*}}\int_{\Omega}\Big{\\{}|u_{\lambda}+tRU_{\varepsilon}|^{2^{*}}-u_{\lambda}^{2^{*}}-(tRU_{\varepsilon})^{2^{*}}$
$\displaystyle\qquad\qquad\qquad-2^{*}u_{\lambda}(tRU_{\varepsilon})\big{(}u_{\lambda}^{2^{*}-2}+(tRU_{\varepsilon})^{2^{*}-2}\big{)}\Big{\\}}dx;$
$\displaystyle(\ast)$
$\displaystyle\,\,\mathcal{D}_{\varepsilon}=\frac{\lambda}{p+1}\int_{\Omega}\Big{\\{}|u_{\lambda}+tRU_{\varepsilon}|^{p+1}-|u_{\lambda}|^{p+1}-tR(p+1)u_{\lambda}^{p}U_{\varepsilon}\Big{\\}}dx.$

We start estimating $\mathcal{R}_{\varepsilon}$. To this aim, we follow [17,
Proof of Theorem 1] (from equation (17) on) where the main difference is due
to the fact that $u_{\lambda}$ is actually dependent on $\varepsilon$ as well.
However, using the uniform upper bound of the $L^{\infty}$-norm of
$u_{\lambda}$ in Theorem 5.1, one can get

(5.13) $|-\mathcal{R}_{\varepsilon}|\leq
R^{\beta}o(\varepsilon^{\frac{\alpha(n-2)}{2}})\quad\text{as $\varepsilon\to
0^{+}$},$

for some $\beta\in(0,2^{*})$. Regarding $\mathcal{D}_{\varepsilon}$, since the
map

$[0,+\infty)\ni r\mapsto\gamma(r)=r^{p+1}$

is _convex on $[0,+\infty)$_, for every $\varepsilon\in(0,1)$ we have

(5.14)
$\begin{split}-\mathcal{D}_{\varepsilon}&=\frac{\lambda}{p+1}\int_{\Omega}\Big{\\{}tR(p+1)u_{\lambda}^{p}U_{\varepsilon}+|u_{\lambda}|^{p+1}-|u_{\lambda}+tRU_{\varepsilon}|^{p+1}\\\
&=-\frac{\lambda}{p+1}\int_{\Omega}\Big{\\{}\gamma(u_{\lambda}+tRU_{\varepsilon})-\gamma(u_{\lambda})-\gamma^{\prime}(u_{\lambda})(tRU_{\varepsilon})\Big{\\}}dx\leq
0.\end{split}$

Finally, combining once again (5.3) and Proposition 3.1 \- iii), and arguing
as in the proof of [10, Lemma 4.11], we get

(5.15) $\displaystyle\int_{\Omega}U_{\varepsilon}^{2^{*}-1}u_{\lambda}\,dx$
$\displaystyle\geq\int_{B_{r}(y)}V_{\varepsilon}^{2^{*}-1}w_{\lambda,\varepsilon}\,dx\geq
c_{2}\,\int_{B_{r}(y)}V_{\varepsilon}^{2^{*}-1}\,dx$
$\displaystyle\geq\varepsilon^{\alpha(n-2)/2}D_{0}+o(\varepsilon^{\alpha(n-2)/2})\quad\text{as
$\varepsilon\to 0^{+}$},$

where $D_{0}>0$ is a constant only depending on $n$.

Gathering (5.11)-(5.12), we obtain

(5.16) $\displaystyle I_{\lambda,\varepsilon}\leq
I_{\lambda,\varepsilon}(u_{\lambda})+\frac{t^{2}R^{2}}{2}K_{1}-\frac{t^{2^{*}}R^{2^{*}}}{2^{*}}K_{2}$
$\displaystyle\qquad-t^{2^{*}-1}R^{2^{*}-1}D_{0}\varepsilon^{\alpha(n-2)/2}-R^{\beta}o(\varepsilon^{\alpha(n-2)/2})+C(tR+t^{2}R^{2})\,o(\varepsilon^{\frac{\alpha(n-2)}{2}})$
$\displaystyle\qquad\qquad+\big{(}t^{2}R^{2}+t^{2^{*}}R^{2^{*}}\big{)}\big{(}o(\varepsilon^{1+\alpha(1-s)})+o(\varepsilon^{\alpha(n-2)/2})\big{)}.$

Thus, if we choose $\alpha\in(0,1]$ so small that
$1+\alpha(1-s)>\alpha(n-2)/2$ (notice that this is always possible, since
$n\geq 3$ and $s\in(0,1)$), we get

(5.17)
$\begin{split}I_{\lambda,\varepsilon}(u_{\lambda}+tRU_{\varepsilon})&\leq
I_{\lambda,\varepsilon}(u_{\lambda})+\frac{t^{2}R^{2}}{2}K_{1}-\frac{t^{2^{*}}R^{2^{*}}}{2^{*}}K_{2}\\\
&\qquad-t^{2^{*}-1}R^{2^{*}-1}D_{0}\,\varepsilon^{\alpha(n-2)/2}\\\
&\qquad\qquad+C\big{(}t^{2}R^{2}+t^{2^{*}}R^{2^{*}}\big{)}o(\varepsilon^{\alpha(n-2)/2}).\end{split}$

Now, thanks to estimate (5.17) we are finally ready to complete the proof of
the lemma: in fact, starting from this estimate and repeating word by word the
argument in [39, Lemma 3.1], we find $\varepsilon_{0}>0$ and $R_{0}>0$ such
that

$\begin{cases}I_{\lambda}(u_{\lambda}+RU_{\varepsilon})<I_{\lambda}(u_{\lambda})&\text{for
all $\varepsilon\in(0,\varepsilon_{0})$ and $R\geq R_{0}$},\\\
I_{\lambda}(u_{\lambda}+tR_{0}U_{\varepsilon})<I_{\lambda}(u_{\lambda})+\frac{1}{n}S_{n}^{n/2}&\text{for
all $\varepsilon\in(0,\varepsilon_{0})$ and $t\in[0,1]$}.\end{cases}$

Thus, the lemma is proved by choosing $\Psi=U_{\varepsilon}$ (with
$\varepsilon<\varepsilon_{0}$ and $y,a$ as above). ∎

Thanks to Lemma 5.2, we can now proceed toward the proof of Theorem 1.4: in
fact, we turn to show that problem ($\mathrm{P}_{\varepsilon}$)λ possesses a
second solution $v_{\lambda}\neq u_{\lambda}$, provided that
$0<\lambda<\lambda_{*}$ and $0<\varepsilon<\varepsilon_{0}$, _where
$\varepsilon_{0}>0$ is as in Lemma 5.2_.

Let then $0<\varepsilon<\varepsilon_{0}$ be arbitrarily but fixed (with
$\varepsilon_{0}>0$ as in Lemma 5.2), and let $0<\lambda<\lambda_{*}$. Since
we know that $u_{\lambda}$ is a local minimizer of the functional
$I_{\lambda}$, there exists some
$0<\varrho_{0}=\varrho_{0}(\varepsilon)\leq\rho_{\varepsilon}(u_{\lambda})$
such that

(5.18) $I_{\lambda}(u)\geq I_{\lambda}(u_{\lambda})\quad\text{for every
$u\in\mathcal{X}^{1,2}(\Omega)$ with
$\rho_{\varepsilon}(u-u_{\lambda})<\varrho_{0}$}.$

As a consequence of (5.18), if we consider the cone

$T=\\{u\in\mathcal{X}^{1,2}(\Omega):\,\text{$u\geq u_{\lambda}>0$ a.e.\,in
$\Omega$}\\},$

only one of the following two cases hold:

  * A)

$\inf\\{I_{\lambda}(u):\,\text{$u\in T$ and
$\rho_{\varepsilon}(u-u_{\lambda})=\varrho$}\\}=I_{\lambda}(u_{\lambda})$ for
every $0<\varrho<\varrho_{0}$;

  * B)

there exists $\varrho_{1}\in(0,\varrho_{0})$ such that

(5.19) $\inf\\{I_{\lambda}(u):\,\text{$u\in T$ and
$\rho_{\varepsilon}(u-u_{\lambda})=\varrho_{1}$}\\}>I_{\lambda}(u_{\lambda}).$

We explicitly notice that use of $\rho_{\varepsilon}(\cdot)$ to define an open
neighborhood of $u_{\lambda}$ in (5.18) is motivated by the fact that the norm
$\rho_{\varepsilon}(\cdot)$ is globally equivalent to $\rho(\cdot)$ (and to
the $H_{0}^{1}$-norm), uniformly w.r.t. $\varepsilon$, see (2.6).

We then turn to consider the two cases A) - B) separately. In doing this, we
will repeatedly use the following elementary result of Real Analysis.

######  Lemma 5.3.

Let $A\subseteq\mathbb{R}^{n}$ be an arbitrary measurable set, and let $1\leq
m<\infty$. Moreover, let $f\in L^{m}(A)$ and let $\\{f_{j}\\}_{j}\subseteq
L^{m}(A)$ be such that

$\text{$f_{j}\to f$ in $L^{m}(A)$ as $j\to+\infty$}.$

Then, for every $0<\vartheta\leq m$ we have

(5.20)
$\lim_{j\to+\infty}\int_{A}|f_{j}|^{\vartheta}\varphi\,dx=\int_{A}|f|^{\vartheta}\varphi\,dx\quad\text{for
every $\varphi\in L^{m^{\prime}_{\vartheta}}(A)$},$

where $m^{\prime}_{\vartheta}=m/(m-\vartheta)$ _(_ with the convention
$m^{\prime}_{\vartheta}=\infty$ if $\vartheta=m$_)_.

###### Proof.

We preliminary observe that, taking into account the very definition of
$m^{\prime}_{\vartheta}$, a direct application of Hölder’s inequality shows
that

(5.21) $|u|^{\vartheta}|v|\in L^{1}(A)\quad\text{for every $u\in L^{m}(A)$ and
every $v\in L^{m^{\prime}_{\vartheta}}(A)$};$

hence, all the integrals in (5.20) are well-defined and finite.

With (5.21) at hand, we now turn to establish (5.20). To this end, we
arbitrarily choose a sub-sequence $\\{f_{j_{k}}\\}_{k}$ of $\\{f_{j}\\}_{j}$
and we prove that, by possibly choosing a further sub-sequence, identity
(5.20) holds for $\\{f_{j_{k}}\\}_{k}$.

Let then $\varphi\in L^{m^{\prime}_{\vartheta}}(A)$ be fixed. Since, by
assumption, $f_{j}\to f$ strongly in $L^{m}(A)$ as $j\to+\infty$, we can find
a function $h\in L^{m}(A)$ such that (up to a sub-sequence)

  * i)

$f_{j_{k}}\to f$ a.e. in $A$ as $k\to+\infty$;

  * ii)

$0\leq|f_{j_{k}}|\leq h$ a.e. in $A$, for every $k\in\mathbb{N}$;

thus, since $\vartheta>0$, by assertion ii) we have the estimate

(5.22) $\text{$0\leq|f_{j_{k}}|^{\vartheta}|\varphi|\leq
h^{\vartheta}|\varphi|$ a.e.\,in $A$ and for every $k\in\mathbb{N}$}.$

Now, by (5.21) we know that $g=h^{\vartheta}|\varphi|\in L^{1}(A)$; this,
together with assertion i) and estimate (5.22), allows us to apply the
Lebesgue Theorem, giving

$\lim_{k\to+\infty}\int_{A}|f_{j_{k}}|^{\vartheta}\varphi\,dx=\int_{A}|f|^{\vartheta}\varphi\,dx.$

This ends the proof. ∎

We are now ready to prove the following propositions.

######  Proposition 5.4.

Assume that Case A) holds. Then, for every $\varrho\in(0,\varrho_{0})$ there
exists a solution $v_{\lambda}$ of problem ($\mathrm{P}_{\varepsilon}$)λ such
that

$\text{$\rho_{\varepsilon}(u_{\lambda}-v_{\lambda})=\varrho$}.$

In particular, $v_{\lambda}\not\equiv u_{\lambda}$.

###### Proof.

Let $0<\varrho<\varrho_{0}$ be arbitrarily fixed. Since we are assuming that
Case A) holds, we can find a sequence $\\{u_{k}\\}_{k}\subseteq T$ satisfying
the following properties:

  * a)

$\rho_{\varepsilon}(u_{k}-u_{\lambda})=\varrho$ for every $k\geq 1$;

  * b)

$I_{\lambda}(u_{k})\to I_{\lambda}(u_{\lambda})=:\mathbf{c}_{\lambda}$ as
$k\to+\infty$.

We then choose $\delta>0$ so small that $\varrho-\delta>0$ and
$\varrho+\delta<\varrho_{0}$ and, accordingly, we consider the subset of $T$
defined as follows:

$X=\\{u\in
T:\,\varrho-\delta\leq\rho_{\varepsilon}(u-u_{\lambda})\leq\varrho+\delta\\}\subseteq
T$

(note that $u_{k}\in X$ for every $k\geq 1$, see a)). Since it is _closed_ ,
this set $X$ is a _complete metric space_ when endowed with the distance
induced by $\rho$; moreover, since $I_{\lambda}$ is a _real-valued and
continuous functional_ on $X$, and since

$\textstyle\inf_{X}I_{\lambda}=I_{\lambda}(u_{\lambda})$

we are entitled to apply the Ekeland Variational Principle (see [5]) to the
functional $I_{\lambda}$ on $X$, providing us with a sequence
$\\{v_{k}\\}_{k}\subseteq X$ such that

(5.23) $\begin{split}\mathrm{i)}&\,\,I_{\lambda}(v_{k})\leq
I_{\lambda}(u_{k})\leq I_{\lambda}(u_{\lambda})+1/k^{2},\\\
\mathrm{ii)}&\,\,\rho_{\varepsilon}(v_{k}-u_{k})\leq 1/k,\\\
\mathrm{iii)}&\,\,I_{\lambda}(v_{k})\leq
I_{\lambda}(u)+1/k\cdot\rho(v_{k}-u)\quad\text{for every $u\in
X$}.\end{split}$

We now observe that, since $\\{v_{k}\\}_{k}\subseteq X$ and since the set $X$
is _bounded_ in $\mathcal{X}^{1,2}(\Omega)$, there exists
$v_{\lambda}\in\mathcal{X}^{1,2}(\Omega)$ such that (as $k\to+\infty$ and up
to a sub-sequence)

(5.24) $\begin{split}\mathrm{i)}&\,\,\text{$v_{k}\to v_{\lambda}$ weakly in
$\mathcal{X}^{1,2}(\Omega)$};\\\ \mathrm{ii)}&\,\,\text{$v_{k}\to v_{\lambda}$
strongly in $L^{m}(\Omega)$ for every $1\leq m<2^{*}$};\\\
\mathrm{iii)}&\,\,\text{$v_{k}\to v_{\lambda}$ pointwise a.e.\,in
$\Omega$}.\end{split}$

where we have also used the _compact embedding_
$\mathcal{X}^{1,2}(\Omega)\hookrightarrow L^{2}(\Omega)$. To complete the
proof, we then turn to prove the following two facts:

  * 1)

$v_{\lambda}$ is a weak solution of ($\mathrm{P}_{\varepsilon}$)λ;

  * 2)

$\rho_{\varepsilon}(v_{\lambda}-u_{\lambda})=\varrho>0$.

_Proof of_ 1). To begin with, we fix $w\in T$ and we choose
$\nu_{0}=\nu_{0}(w,\lambda)>0$ so small that $u_{\nu}=v_{k}+\nu(w-v_{k})\in X$
for every $0<\nu<\nu_{0}$. We explicitly stress that the existence of such an
$\nu_{0}$ easily follows from (5.23)-ii) and property a) above.

On account of (5.23)-iii) (with $u=u_{\nu}$), we have

$\frac{I_{\lambda}(v_{k}+\nu(w-v_{k}))-I_{\lambda}(v_{k})}{\nu}\geq-\frac{1}{k}\rho_{\varepsilon}(w-v_{k});$

From this, by letting $\nu\to 0^{+}$ we obtain

(5.25)
$\begin{split}-\frac{1}{k}\rho_{\varepsilon}(w-v_{k})&\leq{\mathcal{B}_{\varepsilon}}(v_{k},w-v_{k})-\int_{\Omega}v_{k}^{2^{*}-1}(w-v_{k})\,dx\\\
&\qquad-\lambda\int_{\Omega}v_{k}^{p}(w-v_{k})\,dx,\end{split}$

Now, given any $\varphi\in C_{0}^{\infty}(\Omega)$ and any $\nu>0$, we define

  * $(\ast)$

$\psi_{k,\nu}=v_{k}+\nu\varphi-u_{\lambda}$ and
$\phi_{k,\nu}=(\psi_{k,\nu})_{-}$;

  * $(\ast)$

$\psi_{\nu}=v_{\lambda}+\nu\varphi-u_{\lambda}$ and
$\phi_{\nu}=(\psi_{\nu})_{-}$.

Since, obviously, $w=v_{k}+\nu\varphi+\phi_{k,\nu}\in T$, by exploiting (5.25)
we get

(5.26)
$\begin{split}-\frac{1}{k}\rho_{\varepsilon}(\nu\varphi+\phi_{k,\nu})&\leq{\mathcal{B}_{\varepsilon}}(v_{k},\nu\varphi+\phi_{k,\nu})-\int_{\Omega}v_{k}^{2^{*}-1}(\nu\varphi+\phi_{k,\nu})\,dx\\\
&\qquad-\lambda\int_{\Omega}v_{k}^{p}(\nu\varphi+\phi_{k,\nu})\,dx.\end{split}$

Then, we aim to pass to the limit as $k\to+\infty$ and $\nu\to 0^{+}$ in the
above (5.26). To this end we first observe that, on account of (5.24)-iii), we
have

(5.27) $\text{$\phi_{k,\nu}\to\phi_{\nu}$ pointwise a.e.\,in $\Omega$ as
$k\to+\infty$};$

moreover, by the very definition of $\phi_{k,\nu}$ we also have the following
estimate

$|\phi_{k,\nu}|=(u_{\lambda}-\nu\varphi-
v_{k})\cdot\mathbf{1}_{\\{u_{\lambda}-\nu\varphi-v_{k}\geq 0\\}}\leq
u_{\lambda}+\nu|\varphi|;$

as a consequence, we get

(5.28)
$\begin{split}\mathrm{i)}&\,\,v_{k}^{2^{*}-1}|\phi_{k,\nu}|=v_{k}^{2^{*}-1}(u_{\lambda}-\nu\varphi-
v_{k})\cdot\mathbf{1}_{\\{u_{\lambda}-\nu\varphi-v_{k}\geq
0\\}}\leq(u_{\lambda}+\nu|\varphi|)^{2^{*}};\\\
\mathrm{ii)}&\,\,v_{k}^{p}|\phi_{k,\nu}|=v_{k}^{p}(u_{\lambda}-\nu\varphi-
v_{k})\cdot\mathbf{1}_{\\{u_{\lambda}-\nu\varphi-v_{k}\geq
0\\}}\leq(u_{\lambda}+\nu|\varphi|)^{p+1}.\end{split}$

Using a standard dominated-convergence argument based on (5.27) - (5.28),
jointly with Lemma 5.3 (see (5.24) - ii) and remind that $\varphi\in
C_{0}^{\infty}(\Omega)$), we then obtain

(5.29)
$\begin{split}&\lim_{k\to+\infty}\Big{(}\int_{\Omega}v_{k}^{2^{*}-1}(\nu\varphi+\phi_{k,\nu})\,dx+\lambda\int_{\Omega}v_{k}^{p}(\nu\varphi+\phi_{k,\nu})\,dx\Big{)}\\\
&\qquad\qquad=\int_{\Omega}v_{\lambda}^{2^{*}-1}(\nu\varphi+\phi_{\nu})\,dx+\lambda\int_{\Omega}v_{\lambda}^{p}(\nu\varphi+\phi_{\nu})\,dx.\end{split}$

As regards the _operator term_
${\mathcal{B}_{\varepsilon}}(v_{k},\nu\varphi+\phi_{k,\nu})$ by proceeding
_exactly_ as in [13, Proposition 5.2] (where the same operator
$\mathcal{L}_{\varepsilon}$ is considered), we get

(5.30)
${\mathcal{B}_{\varepsilon}}(v_{k},\nu\varphi+\phi_{k,\nu})\leq{\mathcal{B}_{\varepsilon}}(v_{\lambda},\nu\varphi+\phi_{\nu})+o(1)\qquad\text{as
$k\to+\infty$},$

where we have used the fact that $v_{k}\to v_{\lambda}$ weakly in
$\mathcal{X}^{1,2}(\Omega)$.

Gathering (5.29) and (5.30), and taking into account that
${\rho_{\varepsilon}}(\phi_{k,\nu})$ is _uniformly bounded_ with respect to
$k$ (as the same is true of $v_{k}$), we can finally pass to the limit as
$k\to+\infty$ in (5.26), obtaining

(5.31)
$\begin{split}{\mathcal{B}_{\varepsilon}}(v_{\lambda},\nu\varphi+\phi_{\nu})\geq\int_{\Omega}v_{\lambda}^{2^{*}-1}(\nu\varphi+\phi_{\nu})\,dx+\lambda\int_{\Omega}v_{\lambda}^{p}(\nu\varphi+\phi_{\nu})\,dx.\end{split}$

With (5.31) at hand, we can now exploit once again the computations carried
out in [13, Proposition 5.2], getting

(5.32)
$\begin{split}&{\mathcal{B}_{\varepsilon}}(v_{\lambda},\varphi)-\lambda\int_{\Omega}v_{\lambda}^{p}\varphi\,dx-\int_{\Omega}v_{\lambda}^{2^{*}-1}\varphi\,dx\\\
&\qquad\geq-\frac{1}{\nu}\Big{(}{\mathcal{B}_{\varepsilon}}(v_{\lambda},\phi_{\nu})-\lambda\int_{\Omega}v_{\lambda}^{p}\phi_{\nu}\,dx-\int_{\Omega}v_{\lambda}^{2^{*}-1}\phi_{\nu}\,dx\Big{)}\\\
&\qquad(\text{since $u_{\lambda}$ is a solution of
\eqref{eq:Problem}${}_{\lambda}$})\\\
&\qquad=\frac{1}{\nu}\Big{(}-{\mathcal{B}_{\varepsilon}}(v_{\lambda}-u_{\lambda},\phi_{\nu})+\lambda\int_{\Omega}(v_{\lambda}^{p}-u_{\lambda}^{p})\phi_{\nu}\,dx\\\
&\qquad\qquad+\int_{\Omega}(v_{\lambda}^{2^{*}-1}-u_{\lambda}^{2^{*}-1})\phi_{\nu}\,dx\Big{)}\\\
&\qquad(\text{since $v_{\lambda}=\textstyle\lim_{k\to+\infty}v_{k}\geq
u_{\lambda}$})\\\
&\qquad\geq\frac{1}{\nu}\Big{(}-\int_{\\{v_{\lambda}+\nu\varphi\leq
u_{\lambda}\\}}\nabla(v_{\lambda}-u_{\lambda})\cdot\nabla(v_{\lambda}-u_{\lambda}+\nu\varphi)\,dx\\\
&\qquad\qquad-\varepsilon\iint_{\mathbb{R}^{2n}}\frac{((v_{\lambda}-u_{\lambda})(x)-(v_{\lambda}-u_{\lambda})(y))(\phi_{\nu}(x)-\phi_{\nu}(y))}{|x-y|^{n+2s}}\,dx\,dy\Big{)}\\\
&\qquad\geq\text{$o(1)$ as $\nu\to
0^{+}$};\phantom{\iint_{\mathbb{R}^{2n}}}\end{split}$

as a consequence, by letting $\nu\to 0^{+}$ in (5.32), we obtain

${\mathcal{B}_{\varepsilon}}(v_{\lambda},\varphi)-\lambda\int_{\Omega}v_{\lambda}^{p}\varphi\,dx-\int_{\Omega}v_{\lambda}^{2^{*}-1}\varphi\,dx\geq
0.$

This, together with the _arbitrariness_ of the fixed $\varphi\in
C_{0}^{\infty}(\Omega)$, finally proves that the function $v_{\lambda}$ is a
weak solution of problem ($\mathrm{P}_{\varepsilon}$)λ as claimed.

In particular, from Theorem 2.3 we derive that

(5.33) $v_{\lambda}\in L^{\infty}(\Omega).$

_Proof of_ 2). To prove assertion 2) it suffices to show that

(5.34) $\text{$v_{k}\to v_{\lambda}$ strongly in $\mathcal{X}^{1,2}(\Omega)$
as $k\to+\infty$}.$

In fact, owing to property a) of $\\{u_{k}\\}_{k}$ we have

$\varrho-{\rho_{\varepsilon}}(u_{k}-v_{k})\leq{\rho_{\varepsilon}}(v_{k}-u_{\lambda})\leq{\rho_{\varepsilon}}(v_{k}-u_{k})+\varrho;$

this, together with (5.34) and (5.23)-ii), ensures that
${\rho_{\varepsilon}}(u_{\lambda}-v_{\lambda})=\varrho.$ Hence, we turn to to
prove (5.34), namely the _strong convergence_ of $\\{v_{k}\\}_{k}$ to
$v_{\lambda}$.

First of all, by (5.24) and the Brezis - Lieb Lemma [15], we have

(5.35)
$\begin{split}\mathrm{i)}&\,\,\text{$\|v_{k}-v_{\lambda}\|_{L^{p+1}(\Omega)}\to
0$ as $k\to+\infty$},\\\
\mathrm{ii)}&\,\,\|v_{k}\|^{2^{*}}_{L^{2^{*}}(\Omega)}=\|v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+o(1);\\\
\mathrm{iii)}&\,\,{\rho_{\varepsilon}}(v_{k})^{2}={\rho_{\varepsilon}}(v_{\lambda})^{2}+{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}+o(1).\end{split}$

In particular, from (5.35) - i) we get

(5.36)
$\int_{\Omega}v_{k}^{p+1}\,dx=\int_{\Omega}v_{\lambda}^{p+1}\,dx+o(1)\quad\text{as
$k\to+\infty$}.$

Owing to (5.35), and choosing $w=v_{\lambda}\in T$ in (5.25), we then get

$\begin{split}&{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}=-{\mathcal{B}_{\varepsilon}}(v_{k},v_{\lambda}-v_{k})+{\mathcal{B}_{\varepsilon}}(v_{\lambda},v_{\lambda}-v_{k})\\\
&\qquad\leq\frac{1}{k}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})+\lambda\int_{\Omega}v_{k}^{p}(v_{k}-v_{\lambda})\,dx\\\
&\qquad\qquad+\int_{\Omega}v_{k}^{2^{*}-1}(v_{k}-v_{\lambda})\,dx+{\mathcal{B}_{\varepsilon}}(v_{\lambda},v_{\lambda}-v_{k})\\\
&\qquad(\text{since $\\{v_{k}\\}_{k}$ is bounded and $v_{k}\to v_{\lambda}$
weakly in $\mathcal{X}^{1,2}(\Omega)$})\\\
&\qquad=\lambda\int_{\Omega}v_{k}^{p+1}\,dx+\int_{\Omega}v_{k}^{2^{*}-1}(v_{k}-v_{\lambda})\,dx-\lambda\int_{\Omega}v_{k}^{p}v_{\lambda}\,dx+o(1)\\\
&\qquad=\lambda\int_{\Omega}v_{\lambda}^{p+1}\,dx+\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+\|v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\int_{\Omega}v_{k}^{2^{*}-1}v_{\lambda}\,dx\\\
&\qquad\qquad-\lambda\int_{\Omega}v_{k}^{p}v_{\lambda}\,dx+o(1)\quad\text{as
$k\to+\infty$}.\end{split}$

On the other hand, owing to (5.24) - (5.33) (and reminding that $0<p<1$), we
can exploit once again Lemma 5.3, thus deriving

$\displaystyle\int_{\Omega}v_{k}^{p}v_{\lambda}\,dx\to\|v_{\lambda}\|^{p+1}_{L^{p+1}(\Omega)}\qquad\text{and}\qquad\int_{\Omega}v_{k}^{2^{*}-1}v_{\lambda}\,dx\to\|v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}$

as $k\to+\infty$; as a consequence, we obtain

(5.37)
${\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}\leq\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+o(1)\quad\text{as
$k\to+\infty$}.$

To proceed further, we now choose $w=2v_{k}\in T$ in (5.25): this yields

${\rho_{\varepsilon}}(v_{k})^{2}-\|v_{k}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\lambda\int_{\Omega}v_{k}^{p+1}\,dx\geq-\frac{1}{k}{\rho_{\varepsilon}}(v_{k})^{2}=o(1);$

thus, recalling that $v_{\lambda}$ is a weak solution of problem
($\mathrm{P}_{\varepsilon}$)λ, we get

(5.38)
$\begin{split}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}&={\rho_{\varepsilon}}(v_{k})^{2}-{\rho_{\varepsilon}}(v_{\lambda})^{2}+o(1)\\\
&\geq\Big{(}\|v_{k}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+\lambda\int_{\Omega}v_{k}^{p+1}\,dx\Big{)}-{\mathcal{B}_{\varepsilon}}(v_{\lambda},v_{\lambda})\\\
&=\|v_{k}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+\lambda\int_{\Omega}v_{k}^{p+1}\,dx-\|v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\lambda\int_{\Omega}v_{\lambda}^{p+1}\,dx\\\
&=\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+o(1)\quad\text{as
$k\to+\infty$},\end{split}$

where we have also used (5.36). Gathering (5.37)-(5.38), we then obtain

(5.39)
${\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}=\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}+o(1)\quad\text{as
$k\to+\infty$}.$

With (5.39) at hand, we can finally end the proof of (5.34). In fact, assuming
(to fix the ideas) that $I_{\lambda}(u_{\lambda})\leq
I_{\lambda}(v_{\lambda})$, from (5.23) and (5.35) - i) we get

$\displaystyle I_{\lambda}(v_{k}-v_{\lambda})$
$\displaystyle=\frac{1}{2}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}+\frac{\lambda}{p+1}\int_{\Omega}|v_{k}-v_{\lambda}|^{p+1}\,dx+\frac{1}{2^{*}}\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}$
$\displaystyle=I(v_{k})-I(v_{\lambda})+o(1)\leq
I(u_{\lambda})-I(v_{\lambda})+\frac{1}{k^{2}}+o(1)$ $\displaystyle\leq
o(1)\qquad\text{as $k\to+\infty$};$

this, together with (5.35) - i), gives

(5.40)
$\begin{split}&\frac{1}{2}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}-\frac{1}{2^{*}}\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}\\\
&\qquad=I_{\lambda}(v_{k}-v_{\lambda})+\frac{\lambda}{p+1}\int_{\Omega}|v_{k}-v_{\lambda}|^{p+1}\,dx\leq
o(1).\end{split}$

Thus, by combining (5.39)-(5.40), we easily obtain

$\lim_{k\to+\infty}\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}=\lim_{k\to+\infty}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}=0,$

and this proves (5.34). ∎

######  Proposition 5.5.

Assume that Case B) holds. Then, there exists a second solution $v_{\lambda}$
of problem ($\mathrm{P}_{\varepsilon}$)λ such that $v_{\lambda}\not\equiv
u_{\lambda}$.

###### Proof.

To begin with, we consider the set

$\Gamma=\big{\\{}\eta\in
C([0,1];T):\,\text{$\eta(0)=u_{\lambda},\,I_{\lambda}(\eta(1))<I_{\lambda}(u_{\lambda})$
and ${\rho_{\varepsilon}}(\eta(1)-u_{\lambda})>\varrho_{1}$}\big{\\}},$

(where $\varrho_{1}>0$ is as in (5.19)), and we claim that
$\Gamma\neq\varnothing$.

In fact, since the fixed $\varepsilon$ satisfies
$0<\varepsilon<\varepsilon_{0}$ (with $\varepsilon_{0}>0$ as in Lemma 5.2), by
the cited Lemma 5.2 we know that there exists $R_{0}>0$ such that

(5.41)
$\begin{cases}I_{\lambda}(u_{\lambda}+RU_{\varepsilon})<I_{\lambda}(u_{\lambda})&\text{for
all $R\geq R_{0}$},\\\
I_{\lambda}(u_{\lambda}+tR_{0}U_{\varepsilon})<I_{\lambda}(u_{\lambda})+\frac{1}{n}S_{n}^{n/2}&\text{for
all $t\in[0,1]$}.\end{cases}$

In particular, from (5.41) we easily see that

$\eta_{0}(t)=u_{\lambda}+tR_{0}U_{\varepsilon}\in\Gamma$

(by enlarging $R_{0}$ if needed), and thus $\Gamma\neq\varnothing$, as
claimed.

Now we have proved that $\Gamma\neq\varnothing$, we can proceed towards the
end of the proof. To this end we first observe that, since it is non-empty,
this set $\Gamma$ is a _complete metric space_ , when endowed with the
distance

$d_{\Gamma}(\eta_{1},\eta_{2}):=\max_{0\leq t\leq
1}{\rho_{\varepsilon}}\big{(}\eta_{1}(t)-\eta_{2}(t)\big{)};$

moreover, since $I_{\lambda}$ is _real-valued and continuous_ on
$\mathcal{X}^{1,2}(\Omega)$, it is easy to recognize that the functional
$\Phi:\Gamma\to\mathbb{R}$ defined as

$\Phi(\eta):=\max_{0\leq t\leq 1}I_{\lambda}(\eta(t)),$

is (well-defined and) continuous on $\Gamma$. In view of these facts, we are
then entitled to apply the Ekeland Variational Principle to this functional
$\Phi$ on $\Gamma$: setting

$\gamma_{0}:=\inf{\Gamma}\Phi(\eta),$

there exists a sequence $\\{\eta_{k}\\}_{k}\subseteq\Gamma$ such that

(5.42) $\begin{split}\mathrm{i)}&\,\,\Phi(\eta_{k})\leq\gamma_{0}+1/k,\\\
\mathrm{ii)}&\,\,\Phi(\eta_{k})\leq\Phi(\eta)+1/k\,d_{\Gamma}(\eta_{k},\eta)\quad\text{for
every $\eta\in\Gamma$}.\end{split}$

Now, starting from (5.42) and proceeding exactly as in the proof of [6, Lemma
3.5], we can find another sequence

$v_{k}=\eta_{k}(t_{k})\in T$

(for some $t_{k}\in[0,1]$) such that

a) $I_{\lambda}(v_{k})\to\gamma_{0}$ as $k\to+\infty$;

b) there exists some $C>0$ such that, for every $w\in T$, one has

(5.43)
$\begin{split}&{\mathcal{B}_{\varepsilon}}(v_{k},w-v_{k})-\lambda\int_{\Omega}v_{k}^{p}(w-v_{k})\,dx\\\
&\qquad\qquad-\int_{\Omega}v_{k}^{2^{*}-1}(w-v_{k})\,dx\geq-\frac{C}{k}(1+{\rho_{\varepsilon}}(w)).\end{split}$

In particular, choosing $w=2v_{k}$ in (5.43), we get

(5.44)
${\rho_{\varepsilon}}(v_{k})^{2}-\lambda\int_{\Omega}v^{p+1}\,dx-\int_{\Omega}v_{k}^{2^{*}}\,dx\geq-\frac{C}{k}(1+2{\rho_{\varepsilon}}(v_{k})).$

By combining (5.44) with assertion a), and exploiting Hölder’s and Sobolev’s
inequalities, we then obtain the following estimate

(5.45)
$\begin{split}\gamma_{0}+o(1)&=\frac{1}{2}{\rho_{\varepsilon}}(v_{k})^{2}-\frac{\lambda}{p+1}\int_{\Omega}v_{k}^{p+1}\,dx-\frac{1}{2^{*}}\int_{\Omega}v_{k}^{2^{*}}\,dx\\\
&\geq\Big{(}\frac{1}{2}-\frac{1}{2^{*}}\Big{)}{\rho_{\varepsilon}}(v_{k})^{2}-\lambda\Big{(}\frac{1}{p+1}-\frac{1}{2^{*}}\Big{)}\int_{\Omega}v_{k}^{p+1}\,dx\\\
&\qquad-\frac{C}{2^{*}\,k}(1+2{\rho_{\varepsilon}}(v_{k}))\\\
&\geq\Big{(}\frac{1}{2}-\frac{1}{2^{*}}\Big{)}{\rho_{\varepsilon}}(v_{k})^{2}-C\big{(}{\rho_{\varepsilon}}(v_{k})^{p+1}-2{\rho_{\varepsilon}}(v_{k})-1\big{)},\end{split}$

where $C>0$ is a constant depending on $n$ and on $|\Omega|$. Since,
obviously,

$c_{0}=\frac{1}{2}-\frac{1}{2^{*}}>0,$

it is readily seen from (5.45) that the sequence $\\{v_{k}\\}_{k}$ is _bounded
in $\mathcal{X}^{1,2}(\Omega)$_ (otherwise, by possibly choosing a sub-
sequence we would have ${\rho_{\varepsilon}}(v_{k})\to+\infty$, and hence the
right-hand side of (5.45) would diverges as $k\to+\infty$, which is not
possible).

In view of this fact, we can thus proceed as in the proof of Lemma 5.4 to show
that $\\{v_{k}\\}_{k}$ weakly converges (up to a sub-sequence) to a _weak
solution_ $v_{\lambda}\in\mathcal{X}^{1,2}(\Omega)$ of problem
($\mathrm{P}_{\varepsilon}$)λ, further satisfying the identity

(5.46)
${\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}-\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}=o(1)\quad\text{as
$k\to+\infty$}.$

In view of these facts, to complete the proof we are left to show that
$v_{\lambda}\not\equiv u_{\lambda}$. To this end we first observe that, given
any $\eta\in\Gamma$, we have

$\text{${\rho_{\varepsilon}}(\eta(0)-u_{\lambda})=0$\quad and
\quad${\rho_{\varepsilon}}(\eta(1)-u_{\lambda})>\varrho_{1}$},$

and hence there exists a point $t_{\eta}\in[0,1]$ such that
${\rho_{\varepsilon}}(\eta(t_{\eta})-u_{\lambda})=\varrho_{1}$; as a
consequence, since _we are assuming that_ Case B) holds, we obtain

$\displaystyle\gamma_{0}$
$\displaystyle=\inf_{\eta\in\Gamma}\Phi(\eta)\geq\inf\big{\\{}I_{\lambda}(\eta(t_{\eta})):\,\eta\in\Gamma\big{\\}}$
$\displaystyle\geq\inf\\{I_{\lambda}(u):\text{$u\in T$ and
${\rho_{\varepsilon}}(u-u_{\lambda})=\varrho_{1}$}\\}>I_{\lambda}(u_{\lambda}).$

On the other hand, since we already know that
$\eta_{0}(t)=u_{\lambda}+tR_{0}U_{\varepsilon}\in\Gamma$, from (5.41) (and the
very definition of $\gamma_{0}$) we derive the following estimate

$\displaystyle\gamma_{0}$ $\displaystyle\leq\Phi(\eta_{0})=\max_{0\leq t\leq
1}I_{\lambda}(\eta_{0}(t))<I_{\lambda}(u_{\lambda})+\frac{1}{n}S_{n}^{n/2}.$

Summing up, we have

(5.47)
$I_{\lambda}(u_{\lambda})<\gamma_{0}<I_{\lambda}(u_{\lambda})+\frac{1}{n}S_{n}^{n/2}.$

Now, since the sequence $\\{v_{k}\\}_{k}$ weakly converges in
$\mathcal{X}^{1,2}(\Omega)$ to $v_{\lambda}$ as $k\to+\infty$, _the same
assertions_ in (5.35) hold also in this context; this, together with (5.47)
and the above property a) of the sequence $\\{v_{k}\\}_{k}$, gives

(5.48)
$\begin{split}&\frac{1}{2}{\rho_{\varepsilon}}(v_{k}-v_{\lambda})^{2}-\frac{1}{2^{*}}\|v_{k}-v_{\lambda}\|^{2^{*}}_{L^{2^{*}}(\Omega)}\\\
&\qquad=\frac{1}{2}\big{(}{\rho_{\varepsilon}}(v_{k})^{2}-{\rho_{\varepsilon}}(v_{\lambda})^{2}\big{)}-\frac{1}{2^{*}}\big{(}\|v_{k}\|^{2^{*}}_{L^{2^{*}}(\Omega)}-\|v\|^{2^{*}}_{L^{2^{*}}(\Omega)}\big{)}+o(1)\\\\[4.26773pt]
&\qquad=I_{\lambda}(v_{k})-I_{\lambda}(u_{\lambda})+o(1)=\gamma_{0}-I_{\lambda}(u_{\lambda})+o(1)\\\
&\qquad<\frac{1}{n}S_{n}^{n/2}-\delta_{0},\end{split}$

for some $\delta_{0}>0$ such that $1/n\,S_{n}^{n/2}-\delta_{0}>0$ (provided
that $k$ is large enough).

Gathering (5.46)-(5.48), and arguing as in [39, Proposition 3.1], it is then
easy to recognize that $v_{k}\to v_{\lambda}$ _strongly in
$\mathcal{X}^{1,2}(\Omega)$_; as a consequence, by the continuity of the
functional $I_{\lambda}$ and by (5.43)-(5.47), we get

$I_{\lambda}(u_{\lambda})<\gamma_{0}=\lim_{k\to+\infty}I_{\lambda}(v_{k})=I_{\lambda}(v_{\lambda}),$

and this finally proves that $v_{\lambda}\not\equiv u_{\lambda}$, as desired.
∎

##  6\. Proof of Theorem 1.6

As anticipated in the Introduction, in this last section we exploit Theorem
1.2 to prove Theorem 1.6. The main reason why we postponed this proof at the
very end of the paper is _philosophical_ : since Theorem 1.6 concerns the
behavior of the $L^{\infty}$-norm of any solution of problem
($\mathrm{P}_{\varepsilon}$)λ _different from its minimal solution_
$\bar{u}_{\lambda,\varepsilon}$, we need to know that there exists (at least)
_one bounded solution of ( $\mathrm{P}_{\varepsilon}$)λ distinct from such
$\bar{u}_{\lambda,\varepsilon}$_; this is precisely the content of our Theorem
1.4 (at least for $\varepsilon\ll 1$), taking into account Theorem 2.3.

###### Proof (of Theorem 1.6).

We closely follow the approach in [1, Theorem 2.4].

Arguing by contradiction, we assume that there exist a sequence
$\\{\lambda_{j}\\}_{j}\subseteq(0,\Lambda_{\varepsilon})$ and a family
$\\{v_{j}=v_{\lambda_{j},\varepsilon}\\}_{j}$ of weak solutions of problem
($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{j}}$ such that

  * a)

$\lambda_{j}\to 0$ as $j\to+\infty$;

  * b)

$v_{j}\not\equiv\bar{u}_{\lambda_{j},\varepsilon}$ (where
$\bar{u}_{\lambda_{j},\varepsilon}$ is the unique minimal solution of
($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{j}}$), and

(6.1) $\|v_{j}\|_{L^{\infty}(\Omega)}\leq\mathbf{c}.$

for some constant $\mathbf{c}>0$ (possibly depending on $\varepsilon$).

In particular, we have

(6.2) $\begin{split}\bullet)\,\,&\text{$v_{j}>0$ pointwise in $\Omega$ and
$u=0$ pointwise in $\partial\Omega$};\\\
\bullet)\,\,&\text{$\partial_{\nu}v_{j}<0$ pointwise on
$\partial\Omega$}\end{split}$

We then observe that, since $v_{j}$ is a weak solution of problem
($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{j}}$ different from its minimal
solution $\bar{u}_{\lambda_{j},\varepsilon}$ (that is, problem by
($\mathrm{P}_{\varepsilon}$)${}_{\lambda_{j}}$ possesses at least two distinct
solutions), by Theorem 1.2 we necessarily have that

(6.3) $\|v_{j}\|_{L^{\infty}(\Omega)}\geq M_{\varepsilon},$

for some constant $M_{\varepsilon}>0$ (independent of $j$, but possibly
depending on $\varepsilon$).

On the other hand, by combining (6.1) with the global regularity result proved
in [38, Theorem 1.3] (see also the proof of Theorem 2.3), we derive that

(6.4) $\|v_{j}\|_{C^{1,\alpha}(\overline{\Omega})}\leq
c\,\big{(}1+\|v_{j}\|^{2^{*}}_{L^{\infty}(\Omega)}\big{)}\leq c_{1},$

and such an estimate holds for every fixed $\alpha\in(0,\min\\{1,2-2s\\})$,
with a suitable constant $c>0$ independent of $j$. Now, owing to (6.4) (and
bearing in mind (6.2)), we easily derive that the sequence $\\{v_{j}\\}_{j}$
_uniformly converges_ (up to a sub-sequence) as $j\to+\infty$ to some _non-
negative function_

$v_{0}\in\mathcal{X}^{1,2}(\Omega)\cap C^{1}(\overline{\Omega}),$

which is a weak solution of the _purely critical problem_

$(\star)\qquad\quad\begin{cases}\mathcal{L}_{\varepsilon}u=|u|^{2^{*}-1}&\text{in
$\Omega$}\\\ u=0&\text{in $\mathbb{R}^{n}\setminus\Omega$}\end{cases}$

(recall that $\lambda_{j}\to 0$ as $j\to+\infty$); moreover, by (6.4) we get

$\|v_{0}\|_{L^{\infty}(\Omega)}\geq
M_{\varepsilon}\,\,\Longrightarrow\,\,v_{0}\not\equiv 0.$

Summing up, the function $v_{0}$ is a _non-trivial and non-negative weak
solution of $(\star)$_ (in the sense of Definition 2.2), but this is in
contradiction with [10, Theorem 1.3] (since, by assumption, $\Omega$ is star-
shaped). This ends the proof. ∎

## References

  * [1] A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122(2), (1994), 519–543.
  * [2] D. Amundsen, A. Moameni, R.Y. Temgoua, Mixed local and nonlocal supercritical Dirichlet, Commun. Pure Appl. Anal. 22(10), (2023), 3139–3164.
  * [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_ , to appear in Proc. Roy. Soc. Edinburgh Sect. A, (2023). doi:10.1017/prm.2023.80
  * [5] J.-P. Aubin, I. Ekeland, Applied nonlinear analysis. Pure Appl. Math. (N.Y.) Wiley-Intersci. Publ. John Wiley & Sons, Inc., New York, 1984. xi+518 pp.
  * [6] M. Badiale, G. Tarantello, Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities, Nonlinear Anal. Theory Methods Appl. 29, (1997), 639–677.
  * [7] B. Barrios, E. Colorado, R. Servadei, F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H.Poincaré Anal. Non Linéaire 32(4), (2015), 875–900.
  * [8] 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.
  * [9] 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.
  * [10] 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
  * [11] S. Biagi, F. Esposito, L. Montoro, E. Vecchi, On mixed local-nonlocal problems with Hardy potential, forthcoming.
  * [12] S. Biagi, D. Mugnai, E. Vecchi, A Brezis-Oswald approach for mixed local and nonlocal operators, Commun. Contemp. Math. 26(2), (2024), 2250057, 28 pp.
  * [13] S. Biagi, E. Vecchi, Multiplicity of positive solutions for mixed local-nonlocal singular critical problems, preprint. https://arxiv.org/abs/2308.09794
  * [14] 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.
  * [15] H. Brezis, E. Lieb, A relations between pointwise convergence of functions and convergence of integrals, Proc. Amer. Math. Soc. 88, (1983), 486–490.
  * [16] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equation involving the critical Sobolev exponent, Comm. Pure Appl. Math. 36, (1983), 437–477.
  * [17] H. Brezis, L. Nirenberg, A minimization problem with critical exponent and nonzero data, in Symmetry in Nature (a volume in honor of L. Radicati), Scuola Normale Superiore Pisa, (1989), Volume I, 129–140.
  * [18] H. Brezis, L. Nirenberg, $H^{1}$ versus $C^{1}$ local minimizers, C. R. Acad. Sci. Paris 317, (1993), 465–472.
  * [19] H. Brezis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal. 10 (1986), 55–64.
  * [20] X. Cabré, S. Dipierro, E. Valdinoci, The Bernstein Technique for Integro-Differential Equations, Arch. Rational Mech. Anal. 243, (2022), 1597–1652.
  * [21] C. Cancelier, Problèmes aux limites pseudo-différentiels donnant lieu au principe du maximum, Comm. Partial Differential Equations 11(15), (1986), 1677–1726.
  * [22] F. Charro, E. Colorado, I. Peral, Multiplicity of solutions to uniformly elliptic fully nonlinear equations with concave-convex right-hand side, J. Differential Equations 246(11), (2009), 4221–4248.
  * [23] 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.
  * [24] E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. Funct. Anal. 199(2), (2003), 468–507.
  * [25] J.V. da Silva, A. Fiscella, V.A.B. Viloria, Mixed local-nonlocal quasilinear problems with critical nonlinearities, preprint. https://arxiv.org/abs/2308.07460
  * [26] J.V. da Silva, A.M. Salort, A limiting problem for local/non-local p-Laplacians with concave-convex nonlinearities, Z. Angew. Math. Phys.71(6), (2020), Paper No. 191, 27 pp.
  * [27] C. De Filippis, G. Mingione, Gradient regularity in mixed local and nonlocal problems, Math. Ann. (2022). https://doi.org/10.1007/s00208-022-02512-7.
  * [28] 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 .
  * [29] S. Dipierro, E. Valdinoci, Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes, Phys. A. 575, (2021), 126052.
  * [30] 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.
  * [31] P. Garain, E. Lindgren, Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations, Calc. Var. 62, 67, (2023).
  * [32] J. García Azorero, I. Peral, J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math. 2(3), (2000), 385–404.
  * [33] D. Kumar, V. Radulescu, K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity 33(7), (2020), 3336–3369.
  * [34] G. Leoni, _A First Course in Fractional Sobolev Spaces_ , Graduate Studies in Mathematics (2023).
  * [35] A. Moameni, Critical point theory on convex subsets with applications in differential equations and analysis, J. Math. Pures Appl. 141, (2020), 266–315.
  * [36] T. Mukherjee, L. Sharma On nonlocal problems with mixed operators and Dirichlet-Neumann mixed boundary conditions, preprint. https://arxiv.org/abs/2311.02567
  * [37] 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.
  * [38] 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
  * [39] G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9, (1992), 281–304.