# Existence of Eigenvalues for Anisotropic and Fractional Anisotropic Problems
via Ljusternik-Schnirelmann Theory

Ignacio Ceresa Dussel and Julián Fernández Bonder  Instituto de Cálculo,
CONICET
Departamento de Matemática, FCEN - Universidad de Buenos Aires
Ciudad Universitaria, 0+$\infty$ building, C1428EGA, Av. Cantilo s/n
Buenos Aires, Argentina

###### Abstract.

In this work, our interest lies in proving the existence of critical values of
the following Rayleigh-type quotients

$\mathcal{Q}_{\mathbf{p}}(u)=\frac{\|\nabla
u\|_{\mathbf{p}}}{\|u\|_{\mathbf{p}}},\quad\text{and}\quad\mathcal{Q}_{\mathbf{s},\mathbf{p}}(u)=\frac{[u]_{\mathbf{s},\mathbf{p}}}{\|u\|_{\mathbf{p}}},$

where $\mathbf{p}=(p_{1},\dots,p_{n})$, $\mathbf{s}=(s_{1},\dots,s_{n})$ and

$\|\nabla u\|_{\mathbf{p}}=\sum_{i=1}^{n}\|u_{x_{i}}\|_{p_{i}}$

is an anisotropic Sobolev norm, $[u]_{\mathbf{s},\mathbf{p}}$ is a fractional
version of the same anisotropic norm, and

$\|u\|_{\mathbf{p}}=\left(\int_{\mathbb{R}}\left(\dots\left(\int_{\mathbb{R}}|u|^{p_{1}}dx_{1}\right)^{\frac{p_{2}}{p_{1}}}\,dx_{2}\dots\right)^{p_{n}/p_{n-1}}dx_{n}\right)^{1/p_{n}}$

is an anisotropic Lebesgue norm.

Using the Ljusternik-Schnirelmann theory, we prove the existence of a sequence
of critical values and we also find an associated Euler-Lagrange equation for
critical points. Additionally, we analyze the connection between the
fractional critical values and its local counterparts.

###### Key words and phrases:

Eigenvalues, mixed Lebesgue, anisotropic Sobolev spaces, Ljusternik-
Schnirelmann

###### 2020 Mathematics Subject Classification:

35J60, 35R11,35P,49R05

##  1\. Introduction

Eigenvalue problems are a well-established and widely studied subject that
spans across various fields, including analysis and partial differential
equations (PDEs). In the context of PDEs, the Laplacian eigenvalue problem
involves finding the eigenvalues and corresponding eigenfunctions that satisfy
the equation

$\Delta u+\lambda u=0.$

Solving this problem provides valuable insights into the behavior of functions
within a given domain. The eigenvalues offer information about the Laplacian’s
spectrum, while the eigenfunctions reveal spatial patterns associated with
different frequencies or modes.

Another well-known eigenvalues problem arises with the $p$-Laplacian, a
nonlinear generalization of the Laplacian defined by

$\Delta_{p}u=\mathop{\text{div}}(|\nabla u|^{p-2}\nabla u).$

The eigenvalue problem of the $p$-Laplacian, characterized by the nonlinearity
introduced through power exponentiation, is both intriguing and demanding, as
it entails solving the equation

$-\Delta_{p}u=\lambda|u|^{p-2}u$

Many authors have extensively explored this problem, as seen in [12, 17, 14,
2].

Even more, in recent decades, there has been a growing interest in fractional
operators due to their applications in various natural sciences models [9, 16,
21]. One prominent exponent of this family of fractional operators is the
fractional $p$-Laplacian, defined as

$(-\Delta_{p})^{s}u(x)=\text{p.v.}(1-s)K_{n,p}\int_{\mathbb{R}^{n}}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dy,$

where $K_{n,p}$ is a constant that depends only on $n$ and $p$.

The eigenvalue problem associated with this fractional operator,

$(-\Delta_{p})^{s}u(x)=\lambda|u|^{p-2}u,$

has also been studied extensively by several authors [5, 10, 22, 15].

The aim of this paper is to introduce an anisotropic feature to these
eigenvalue problems. This choice is motivated by the substantial attention
dedicated to investigating this phenomenon in signal processing and diffusion
studies [8, 24]. For those not familiar with the term, anisotropy can be
described as the characteristic of displaying directional dependence, where
various attributes or qualities manifest differently in distinct directions.
This stands in opposition to the isotropic nature of the Laplacian,
$p-$Laplacian, and fractional $p-$Laplacian, where these properties remain
uniform regardless of the direction.

On one hand, a strategy to address anisotropy involves emphasizing the
integrability of individual partial derivatives of a function $u$ by employing
the sum of standard $L^{p}$ norms,

$\|\nabla u\|_{\mathbf{p}}=\sum_{i=1}^{n}\left\|u_{x_{i}}\right\|_{p_{i}},$

see [19, 20, 23]. Hence, we naturally arrive at the following anisotropic
pseudo-laplace operator

$-\widetilde{\Delta}_{\mathbf{p}}u:=-\mathop{\text{div}}\left(\sum_{i=1}^{n}|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}\right)$

On the other hand, Benedek $\&$ Panzone [3] present the anisotropic
$L^{\mathbf{p}}$ ($\mathbf{p}=(p_{1},\dots,p_{n})$) space with a special norm
to address the anisotropy of a function $u$. The mixed Lebesgue space is
constructed by considering different exponents for each coordinate in the norm

$\|u\|_{\mathbf{p}}=\left(\int_{\mathbb{R}}\left(\dots\left(\int_{\mathbb{R}}|u|^{p_{1}}dx_{1}\right)^{\frac{p_{2}}{p_{1}}}\,dx_{2}\dots\right)^{p_{n}/p_{n-1}}dx_{n}\right)^{1/p_{n}}.$

By considering different exponents for each coordinate, the mixed Lebesgue
norm accounts for the anisotropy of the function $u$. It allows for a more
flexible and nuanced characterization of the integrability and decay
properties across different coordinates.

By combining this two perspective we can state the following eigenvalue
problem

$-\widetilde{\Delta}_{\mathbf{p}}u=\lambda\mathcal{F}_{\mathbf{p}}(u),$

where $\mathcal{F}_{\mathbf{p}}$ is a suitable functional related to
$\|u\|_{\mathbf{p}}$. See (3.3).

Unfortunately, this problem is hindered by its lack of homogeneity. It’s
important to observe that if $v$ is an eigenfunction associated with
$\lambda$, there’s a possibility that $tv$ may not qualify as an eigenfunction
of $\lambda$.

Note that a crucial approach to solving the Laplacian, $p-$Laplacian and
fractional $p-$Laplacian eigenvalue problems involves finding the critical
points of the Rayleigh quotient associated with each one, namely

$\mathcal{Q}_{2}(u)=\frac{\|\nabla
u\|^{2}_{2}}{\|u\|^{2}_{2}},\quad\mathcal{Q}_{p}(u)=\frac{\|\nabla
u\|_{p}^{p}}{\|u\|_{p}^{p}}\quad\text{and
}\mathcal{Q}_{sp}(u)=\frac{[u]_{sp}^{p}}{\|u\|_{p}^{p}}.$

Therefore, it is recommended to explore the following homogeneous Rayleigh
quotient

(1.1) $\mathcal{Q}_{\mathbf{p}}(u)=\frac{\|\nabla
u\|_{\mathbf{p}}}{\|u\|_{\mathbf{p}}}.$

As we will observe in Section 3, the associated Euler-Lagrange equation of
$\mathcal{Q}_{\mathbf{p}}(u)$ is the following homogeneous eigenvalue problem,

(1.2)
$-\mathcal{L}_{\mathbf{p}}u=-\mathop{\text{div}}\left(\sum_{i=1}^{n}\left|\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}\right|^{p_{i}-2}\frac{u_{x_{1}}}{\|u\|_{p_{i}}}\right)=\lambda\mathcal{F}_{\mathbf{p}}(u).$

In [7] fractional anisotropy is introduced through the utilization of
integrability parameters $\mathbf{p}=(p_{1},\ldots,p_{n})$, $1<p_{i}<\infty$,
and fractional parameters $\mathbf{s}=(s_{1},\ldots,s_{n})$, $0<s_{i}<1$, and
the subsequent norm,

$[u]_{\mathbf{s},\mathbf{p}}=\sum_{i=1}^{n}\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}(1-s_{i})\frac{|u(x+he_{i})-u(x)|^{p_{i}}}{|h|^{1+s_{i}p_{i}}}\,dh\,dx\right)^{1/p_{i}}.$

As in the non-fractional case, combining this perspective with the Benedek
$\&$ Panzone’s norm we arrive to the following eigenvalue problem

$(-\widetilde{\Delta}_{\mathbf{p}})^{\mathbf{s}}u(x)=\lambda\mathcal{F}_{\mathbf{p}}(u),$

where $(-\widetilde{\Delta}_{\mathbf{p}})^{\mathbf{s}}$ it the fractional
pseudo p-Laplacian operator defined as

$(-\widetilde{\Delta}_{\mathbf{p}})^{\mathbf{s}}u(x)=\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\frac{(1-s_{i})}{p_{i}}\frac{|u(x+he_{i})-u(x)|^{p_{i}-2}(u(x+he_{i})-u(x))}{|h|^{1+sp}}\,dh\,dx.$

Again, this is not an homogeneous problem, therefore we study the homogeneous
Rayleight quotient

(1.3)
$\mathcal{Q}_{\mathbf{s},\mathbf{p}}(u)=\frac{[u]_{\mathbf{s},\mathbf{p}}}{\|u\|_{\mathbf{p}}}.$

As we will see in Section 3 the Euler-Lagrange equation is

(1.4)
$-\mathcal{L}_{\mathbf{s}.\mathbf{p}}u=\lambda\mathcal{F}_{\mathbf{p}}(u),$

where $\mathcal{L}_{\mathbf{s},\mathbf{p}}$ is the fractional version of
$\mathcal{L}_{\mathbf{p}}$.

To address the problem of find criticals points of (1.1), and (1.3) and solve
the eigenvalues problems (1.2) and (1.4), the Ljusternik-Schnirelman theory
serves as a powerful framework for exploring critical point theory and the
existence of critical points for functionals as we will see in Section 5. See
[18].

The rest of the paper is organized as follows: In Section 2, we dive into
anisotropic Sobolev spaces and fractional anisotropic Sobolev spaces,
explaining them in more detail and discussing some interesting properties like
the Poincaré inequality and a Rellich-Kondrashov type theorem. Then, in
Section 3, we figure out the Euler-Lagrange equations associated with the
corresponding Rayleight-type quotients. In Section 4 we study the asymptotic
behavior of the sequence of eigenvalues as $\mathbf{s}\to(1,\dots,1)$ and
finally in Section 5 we use Ljusternik-Schnirelman theory to prove the
existence of eigenvalues.

##  2\. Mixed, anisotropic and fractional spaces

In this section, our objective is to establish the definition of the mixed
Lebesgue space, as introduced by [3]. This space will serve as a fundamental
building block for our analysis. Furthermore, we will define a suitable
anisotropic Sobolev space, $W_{0}^{1,\mathbf{p}}(\Omega)$, and a fractional
anistropic Sobolev space $W_{0}^{\mathbf{s},\mathbf{p}}(\Omega)$.

###  2.1. Mixed space

Let $\mathbf{p}=(p_{1},p_{2},\dots,p_{n})$ with $1<p_{i}<\infty$ for
$i=1,\dots,n$ be integral parameters. Without loss of generality, we can
assume that

(2.1) $1<p_{1}\leq p_{2}\leq\dots\leq p_{n}<\infty.$

We define the mixed Lebesgue space as

$L^{\mathbf{p}}(\mathbb{R}^{n})=\\{u\text{ measurable such that
}\|u\|_{L^{\mathbf{p}}(\mathbb{R}^{n})}<\infty\\}.$

Where

$\|u\|_{\mathbf{p}}=\left(\int_{\mathbb{R}}\left(\dots\left(\int_{\mathbb{R}}|u|^{p_{1}}dx_{1}\right)^{\frac{p_{2}}{p_{1}}}\,dx_{2}\dots\right)^{p_{n}/p_{n-1}}dx_{n}\right)^{1/p_{n}}.$

Furthermore, given $\Omega$ an open bounded subset of $\mathbb{R}^{n}$, we
define

$L^{\mathbf{p}}(\Omega)=\\{u\in L^{\mathbf{p}}(\mathbb{R}^{n})\text{ such that
}u=0\text{ in }\mathbb{R}^{n}\setminus\Omega\\}.$

Observe that $L^{\mathbf{p}}(\Omega)$ is a closed subspace of
$L^{\mathbf{p}}(\mathbb{R}^{n})$. This space $L^{\mathbf{p}}(\Omega)$ turns
out to be a reflexive Banach space and its properties were studied in [3, 1].

######  Remark 2.1.

The $\|.\|_{\mathbf{p}}$ norm can be defined by recurrence as

$\displaystyle I_{1}(u)$
$\displaystyle=\left(\int_{\mathbb{R}}|u|^{p_{1}}\,dx_{1}\right)^{1/p_{1}}$
$\displaystyle I_{2}(u)$
$\displaystyle=\left(\int_{\mathbb{R}}I_{1}(u)^{p_{2}}\,dx_{2}\right)^{1/p_{2}}$
$\displaystyle\vdots$ $\displaystyle I_{j}(u)$
$\displaystyle=\left(\int_{\mathbb{R}}I_{j-1}(u)^{p_{j}}\,dx_{j}\right)^{1/p_{j}}$
$\displaystyle\vdots$ $\displaystyle I_{n}(u)$
$\displaystyle=\left(\int_{\mathbb{R}}I_{n-1}(u)^{p_{n}}\,dx_{n}\right)^{1/p_{n}}$
$\displaystyle I(u)$ $\displaystyle=I_{n}(u)=\|u\|_{\mathbf{p}}$

######  Remark 2.2.

Observe that, given $u\in L^{\mathbf{p}}(\mathbb{R}^{n})$, $I_{j}(u)$ is a
function of $(x_{j+1},\dots,x_{n})$.

Moreover, for almost every $(y_{j+2},\dots,y_{n})\in\mathbb{R}^{n-j-2}$, the
function $I_{j}(u)$ (as a function of $x_{j+1}$), belongs to
$L^{p_{j+1}}(\mathbb{R})$.

Also, observe that if $\\{u_{k}\\}_{k\in\mathbb{N}}\subset
L^{\mathbf{p}}(\mathbb{R}^{n})$ is such that $u_{k}\to u\in
L^{\mathbf{p}}(\mathbb{R}^{n})$ as $k\to\infty$ then
$I_{j}(u_{k})(\cdot,y_{j+2},\dots,y_{n})\to
I_{j}(u)(\cdot,y_{j+2},\dots,y_{n})$ in $L^{p_{j+1}}(\mathbb{R})$ for a.e.
$(y_{j+2},\dots,y_{n})\in\mathbb{R}^{n-j-2}$.

###  2.2. Anisotropic Sobolev spaces

Our interest lies in functions whose partial derivatives have different
integrability. With this fact in mind, given $\mathbf{p}=(p_{1},\dots,p_{n})$
with $1<p_{i}<\infty$, the anisotropic Sobolev space is defined as follows:

$W^{1,\mathbf{p}}(\mathbb{R}^{n}):=\left\\{u\in
L^{\mathbf{p}}(\mathbb{R}^{n})\colon u_{x_{i}}\in L^{p_{i}}(\mathbb{R}^{n}),\
i=1,\dots,n\right\\},$

equipped with the following norm

$\|u\|_{1,\mathbf{p}}=\|u\|_{\mathbf{p}}+\sum_{i=1}^{n}\left\|u_{x_{i}}\right\|_{p_{i}}=\|u\|_{\mathbf{p}}+\|\nabla
u\|_{\mathbf{p}}.$

It is easy to prove that $W^{1,\mathbf{p}}(\mathbb{R}^{n})$ is a separable,
reflexive Banach space.

Now, given a bounded domain $\Omega\subset\mathbb{R}^{n}$ we define
$W^{1,\mathbf{p}}_{0}(\Omega)$ as the closure of $C_{c}^{\infty}(\Omega)$ in
$W^{1,\mathbf{p}}(\mathbb{R}^{n})$.

###  2.3. Fractional space

Next we present the fractional anisotropic Sobolev space.

First, given $i=1,\dots,n$, $s\in(0,1]$ and $p\in(1,\infty)$, for any
$u\colon\mathbb{R}^{n}\to\mathbb{R}$ measurable we define the quantity

$[u]_{s,p,i}=\left(\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\frac{|u(x+he_{i})-u(x)|^{p}}{|h|^{1+sp}}\,dhdx\right)^{\frac{1}{p}},$

where $e_{i}$ is the $i^{\text{th}}-$canonical vector base in
$\mathbb{R}^{n}$.

Now, given $\mathbf{p}=(p_{1},\dots,p_{n})$ and
$\mathbf{s}=(s_{1},\dots,s_{n})$ with $1<p_{i}<\infty$ and $0<s_{i}<1$, for
$i=1,\dots,n$, we define the anisotropic fractional order Sobolev space as

$W^{\mathbf{s},\mathbf{p}}(\mathbb{R}^{n}):=\left\\{u\in
L^{\mathbf{p}}(\mathbb{R}^{n})\colon[u]_{s_{i},p_{i},i}<\infty,\
i=1,\dots,n\right\\}.$

This space has a natural norm defined as

$\|u\|_{\mathbf{s},\mathbf{p}}:=\|u\|_{\mathbf{p}}+\sum_{i=1}^{n}[u]_{s_{i},p_{i},i}=\|u\|_{\mathbf{p}}+[u]_{\mathbf{s},\mathbf{p}}.$

It is easy to see that $W^{\mathbf{s},\mathbf{p}}(\mathbb{R}^{n})$ is a
separable and reflexive Banach space. See [6, 7].

As before, given $\Omega\subset\mathbb{R}^{n}$ a bounded domain, we define
$W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$ as the closure of
$C^{\infty}_{c}(\Omega)$ in $W^{\mathbf{s},\mathbf{p}}(\mathbb{R}^{n})$.

The following two theorems represent analogs to the classical Poincaré
inequality and the Rellich-Kondrashov type theorem within the context of
$L^{\mathbf{p}}(\Omega)$ and anisotropic fractional Sobolev space.

######  Proposition 2.3 (Poincaré).

Given $\Omega$ an open bounded subset on $\mathbb{R}^{n}$, there exists
constants $C_{1}(\Omega,\mathbf{p},n)>0$ and
$C_{2}(\Omega,\mathbf{p},\mathbf{s},n)>0$ such that for every $u$ in
$W^{1,\mathbf{p}}_{0}(\Omega)$, the following inequality holds

(2.2) $\|u\|_{\mathbf{p}}\leq C_{1}\|\nabla u\|_{\mathbf{p}},$

and for every $u$ in $W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$, the following
inequality holds:

(2.3) $\|u\|_{\mathbf{p}}\leq C_{2}[u]_{\mathbf{s},\mathbf{p}}.$

###### Proof.

Let $u$ be a function in $W^{1,\mathbf{p}}_{0}(\Omega)$. On one hand, observe
that since $p_{i}\leq p_{n}$ for every $i=1,\dots,n-1$ and $|\Omega|<\infty$,
it follows by Hölder’s inequality that $L^{p_{n}}(\Omega)$ is continuously
embedded in $L^{\mathbf{p}}(\Omega)$, that is, there exists a positive
constant $C>0$ such that

$\|u\|_{\mathbf{p}}\leq C\|u\|_{L^{p_{n}}}.$

On the other hand, the Poincaré inequality for functions in
$W^{1,p_{n}}_{0}(\Omega)$

$\|u\|_{L^{p_{n}}(\Omega)}\leq C\|u_{x_{n}}\|_{L^{p_{n}}(\Omega)}\leq
C\sum_{i=1}^{n}\|u_{x_{i}}\|_{L^{p_{i}}(\Omega)}$

Therefore, by combining these results, we obtain (2.2).

For the second inequality, let $u$ be a function in
$W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$, we can assume that there exist $R>0$
such that $\mathop{\text{supp}}{u}\subset Q_{R}=[-R,R]^{n}$. Hence,

$\displaystyle[u]_{s_{1},p_{1},1}^{p_{1}}$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\frac{|u(x+he_{1})-u(x)|^{p_{1}}}{|h|^{1+s_{1}p_{1}}}\,dh\,dx$
$\displaystyle\geq\int_{Q_{R}}\int_{\mathbb{R}}\frac{|u(x+he_{1})-u(x)|^{p_{1}}}{|h|^{1+s_{1}p_{1}}}\,dh\,dx$
$\displaystyle\geq\int_{Q_{R}^{\prime}}\int_{|x_{1}|\leq
R}\int_{|x_{1}+he_{1}|\geq
R}\frac{|u(x)|^{p_{1}}}{|h|^{1+s_{1}p_{1}}}\,dh\,dx_{1}\,dx^{\prime}$
$\displaystyle\geq\int_{Q_{R}^{\prime}}\int_{|x_{1}|\leq
R}|u(x)|^{p_{1}}\int_{|h|\geq
2R}\frac{1}{|h|^{1+s_{1}p_{1}}}\,dh\,dx_{1}\,dx^{\prime}$ $\displaystyle\geq
C\|u\|_{p_{1}}^{p_{1}},$

where $Q_{R}^{\prime}=[-R,R]^{n-1}$ and $dx^{\prime}=dx_{2}\cdots dx_{n}$.

Arguing in a similar fashion we conclude that there exists
$C_{i}(\Omega,s_{i},p_{i})$ such that

$C_{i}[u]_{s_{i},p_{i},i}\geq\|u\|_{p_{i}}.$

Therefore taking $K=\max_{i}\\{C_{i}\\}$ we have that

$K\sum_{i=1}^{n}[u]_{s,p,i}\geq\sum_{i=1}^{n}\|u\|_{p_{i}}\geq\|u\|_{p_{n}}\geq
C\|u\|_{\mathbf{p}}.$

This fact concludes the proof of (2.3). ∎

The following notation will be used. Given a vector
$\mathbf{q}=(q_{1},\dots,q_{n})$ with $q_{i}>0$ for $i=1,\dots,n$, we denote
by $\bar{\mathbf{q}}$ the harmonic mean of the vector $\mathbf{q}$, i.e.

$\bar{\mathbf{q}}:=\left(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{q_{i}}\right)^{-1}.$

Next, given two vectors $\mathbf{q}=(q_{1},\dots,q_{n})$ and
$\mathbf{r}=(r_{1},\dots,r_{n})$ with $q_{i},r_{i}>0$ for $i=1,\dots,n$ we
define the product $\mathbf{q}\mathbf{r}$ as

$\mathbf{q}\mathbf{r}=(q_{1}r_{1},\dots,q_{n}r_{n}),$

the coordinate by coordinate multiplication.

######  Proposition 2.4 (Rellich-Kondrashov).

Let $\mathbf{p}=(p_{1},\dots,p_{n})$ with $1<p_{i}<\infty$, $i=1,\dots,n$ and
be such that

(2.4) $\bar{\mathbf{p}}<n.$

Define the critical exponent $\mathbf{p}^{*}$ as

(2.5) $\mathbf{p}^{*}:=\frac{n\bar{\mathbf{p}}}{n-\bar{\mathbf{p}}}.$

Then $W^{1,\mathbf{p}}_{0}(\Omega)\subset L^{q}(\Omega)$, for all $1\leq
q\leq\mathbf{p}^{*}$. Even more $W^{1,\mathbf{p}}_{0}(\Omega)\subset\subset
L^{q}(\Omega)$ if $1\leq q<\mathbf{p}^{*}$. In particular
$W^{1,\mathbf{p}}_{0}(\Omega)\subset\subset L^{\mathbf{p}}(\Omega)$.

Now, let $\mathbf{s}=(s_{1},\dots,s_{n})$ with $0<s_{i}<1$, for $i=1,\dots,n$
and $\mathbf{p}$ be as before. Assume that

(2.6) $\overline{\mathbf{s}\mathbf{p}}<n$

and define the fractional critical exponent

(2.7)
$\mathbf{p}^{*}_{\mathbf{s}}=\frac{n\frac{\overline{\mathbf{s}\mathbf{p}}}{\bar{\mathbf{s}}}}{n-\overline{\mathbf{s}\mathbf{p}}}.$

Moreover, assume that

(2.8) $p_{n}<\mathbf{p}^{*}_{\mathbf{s}}.$

Then $W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\subset L^{q}(\Omega)$, for all
$1\leq q\leq\mathbf{p}^{*}_{\mathbf{s}}$. Even more,
$W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\subset\subset L^{q}(\Omega)$ for $1\leq
q<\mathbf{p}^{*}_{\mathbf{s}}$. In particular
$W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\subset\subset L^{\mathbf{p}}(\Omega)$.

###### Proof.

The proof of $W^{1,\mathbf{p}}_{0}(\Omega)\subset\subset L^{q}$ for all
$1<q<\mathbf{p}^{*}$ is studied in the previous references [23, 11]. To prove
that $W^{1,\mathbf{p}}_{0}(\Omega)\subset\subset L^{\mathbf{p}}(\Omega)$
observe that as $p_{n}<\mathbf{p}^{*}$ then $L^{p_{n}}(\Omega)\subset
L^{\mathbf{p}}(\Omega)$ continuously.

The proof of fractional case is immediate of [7, Theorem 2.1] and the previous
idea. ∎

Without loss of generality, we can always assume that (2.1) is satisfied.

In the rest of the paper, it will always be assumed that conditions (2.4),
(2.6) and (2.8) hold.

##  3\. The Euler-Lagrange equation

###  3.1. Non-fractional case

In this subsection we will establish the Euler-Lagrange equation associated to
the Rayleigh-type quotient $\mathcal{Q}_{\mathbf{p}}$ defined in (1.1). In
fact, following ideas from [14] (see also [13]), we show that the EL equation
turns out to be the following

(3.1)
$\begin{cases}-\mathcal{L}_{\mathbf{p}}u:=\lambda\mathcal{F}_{\mathbf{p}}(u)\quad&\text{
in }\Omega\\\ u=0&\text{ in }\mathbb{R}^{n}\setminus\Omega,\end{cases}$

where

(3.2)
$-\mathcal{L}_{\mathbf{p}}u:=-\mathop{\text{div}}\left(\sum_{i=1}^{n}\left|\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}\right|^{p_{i}-2}\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}\right)$

and

(3.3)
$\mathcal{F}_{\mathbf{p}}(u)=\prod_{i=1}^{n}I_{i}(u)^{p_{i+1}-p_{i}}|u|^{p_{1}-2}u,$

where $p_{n+1}=1$.

######  Definition 3.1.

Let $u$ be a function in $W^{1,\mathbf{p}}_{0}(\Omega)$, then $u$ is a weak
solution of (3.1) if and only if $u$ verifies

$\int_{\Omega}\sum_{i=1}^{n}\left|\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}\right|^{p_{i}-2}\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}v_{x_{i}}\,dx=\lambda\int_{\Omega}\mathcal{F}_{\mathbf{p}}(u)v\,dx,$

for all $v\in W^{1,\mathbf{p}}_{0}(\Omega).$

We will need the following lemma regarding the behavior of the functional
$\mathcal{F}_{\mathbf{p}}$.

######  Lemma 3.2.

Let $\mathbf{p}=(p_{1},\dots,p_{n})$ be such that $1<p_{i}<\infty$ and let
$\mathbf{p}^{\prime}=(p_{1}^{\prime},\dots,p_{n}^{\prime})$. Let
$\mathcal{F}_{\mathbf{p}}$ be the functional defined in (3.3).

Then $\mathcal{F}_{\mathbf{p}}\colon L^{\mathbf{p}}(\mathbb{R}^{n})\to
L^{\mathbf{p}^{\prime}}(\mathbb{R}^{n})$ is continuous.

###### Proof.

To see that it is well defined, just observe that if $u\in
L^{\mathbf{p}}(\mathbb{R}^{n})$, then

$\displaystyle\left(\int_{\mathbb{R}}|\mathcal{F}_{\mathbf{p}}(u)|^{p_{1}^{\prime}}\,dx_{1}\right)^{1/p_{1}^{\prime}}$
$\displaystyle=\prod_{i=1}^{n}I_{i}(u)^{p_{i+1}-p_{1}}\left(\int_{\mathbb{R}}|u|^{(p_{1}-1)p_{1}^{\prime}}\,dx_{1}\right)^{1/p_{1}^{\prime}}$
$\displaystyle=\prod_{i=1}^{n}I_{i}(u)^{p_{i+1}-p_{1}}I_{1}(u)^{p_{1}/p_{1}^{\prime}}$
$\displaystyle=\prod_{i=2}^{n}I_{i}(u)^{p_{i+1}-p_{1}}I_{1}(u)^{p_{2}-1}.$

Iterating this procedure, one easily conclude that

$\|\mathcal{F}_{\mathbf{p}}(u)\|_{\mathbf{p}^{\prime}}=\|u\|_{\mathbf{p}}.$

In order to see the continuity of $\mathcal{F}_{\mathbf{p}}$, let
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset L^{\mathbf{p}}(\mathbb{R}^{n})$ be such
that $u_{k}\to u$ in $L^{\mathbf{p}}(\mathbb{R}^{n})$. Then, define

$\displaystyle\tilde{I}_{1}(k):=\left(\int_{\mathbb{R}}|\mathcal{F}_{\mathbf{p}}(u_{k})-\mathcal{F}_{\mathbf{p}}(u)|^{p_{1}^{\prime}}\,dx_{1}\right)^{1/p_{1}^{\prime}}$
$\displaystyle\tilde{I}_{i+1}(k):=\left(\int_{\mathbb{R}}\tilde{I}_{i}(k)^{p_{i+1}^{\prime}}\,dx_{i+1}\right)^{1/p_{i+1}^{\prime}},\qquad
i=1,\dots,n-1.$

Observe that
$\|\mathcal{F}_{\mathbf{p}}(u_{k})-\mathcal{F}_{\mathbf{p}}(u)\|_{\mathbf{p}^{\prime}}=\tilde{I}_{n}(k)$,
so it is enough to show that, up to a subsequence,

(3.4) $\begin{split}&\tilde{I}_{i}(k)\to 0\text{ as }k\to\infty\quad\text{a.e.
}(x_{i+1},\dots,x_{n}),\qquad i=1,\dots,n\\\ &\text{and}\\\ &\text{a.e.
}(x_{i+2},\dots,x_{n}),\ \tilde{I}_{i}(k)(x_{i+1})\leq
h_{i}(x_{i+1}),\quad\text{with }h\in
L^{p_{i+1}^{\prime}}(\mathbb{R}).\end{split}$

In fact, let us see (3.4) for $i=1$ and the rest will follow by induction.

By Remark 2.2, it is easy see that
$\mathcal{F}_{\mathbf{p}}(u_{k})\to\mathcal{F}_{\mathbf{p}}(u)$ a.e. So in
order to see that $\tilde{I}_{1}(k)\to 0$ for a.e.
$x^{\prime}=(x_{2},\dots,x_{n})$ we need to find an integrable majorant for
$|\mathcal{F}_{\mathbf{p}}(u_{k})-\mathcal{F}_{\mathbf{p}}(u)|^{p_{1}^{\prime}}$
for a.e. $x^{\prime}\in\mathbb{R}^{n-1}$.

Hence,

$\displaystyle|\mathcal{F}_{\mathbf{p}}(u_{k})-\mathcal{F}_{\mathbf{p}}(u)|^{p_{1}^{\prime}}\leq
C\left(\prod_{i=1}^{n}I_{i}(u_{k})^{(p_{i+1}-p_{i})p_{1}^{\prime}}|u_{k}|^{p_{1}}+\prod_{i=1}^{n}I_{i}(u)^{(p_{i+1}-p_{i})p_{1}^{\prime}}|u|^{p_{1}}\right)$

As, by Remark 2.2 $I_{i}(u_{k})(\cdot,x_{i+2},\dots,x_{n})\to
I_{i}(u)(\cdot,x_{i+2},\dots,x_{n})$ in $L^{p_{i+i}}(\mathbb{R})$ for a.e.
$(x_{i+2},\dots,x_{n})$, using [4, Theorem 4.9], there exists
$h_{i}=h_{i}(\cdot,x_{i+2},\dots,x_{n})\in L^{p_{i+1}}(\mathbb{R})$ such that

$I_{i}(u_{k})(x_{1},x_{i+2},\dots,x_{n})\leq
h_{i}(x_{1},x_{i+2},\dots,x_{n}).$

Moreover, since $u_{k}(\cdot,x^{\prime})\to u(\cdot,x^{\prime})$ in
$L^{p_{1}}(\mathbb{R})$ we obtain the existence of $h_{0}(x)$,
$h_{0}(\cdot,x^{\prime})\in L^{p_{1}}(\mathbb{R})$ such that

$|u_{k}(x)|\leq h_{0}(x).$

Hence

$|\mathcal{F}_{\mathbf{p}}(u_{k})-\mathcal{F}_{\mathbf{p}}(u)|^{p_{1}^{\prime}}\leq
C\left(\prod_{i=1}^{n}h_{i}^{(p_{i+1}-p_{i})p_{1}^{\prime}}h_{0}^{p_{1}}+\prod_{i=1}^{n}I_{i}(u)^{(p_{i+1}-p_{i})p_{1}^{\prime}}|u|^{p_{1}}\right)=:\Phi(x_{1},x^{\prime}).$

Since $\Phi(\cdot,x^{\prime})\in L^{1}(\mathbb{R})$ for a.e.
$x^{\prime}\in\mathbb{R}^{n-1}$ we obtain that $\tilde{I}_{1}(k)\to 0$.

The proof of (3.4) now follows by induction and the details are left to the
reader. ∎

Knowing the definition of weak solution we can state our main result of this
section.

######  Theorem 3.3.

Let $u$ be a function in $W^{1,\mathbf{p}}_{0}(\Omega).$ Then $u$ is a
critical point of (1.1) if and only if $u$ is a weak solution of (3.1).

To prove this theorem we will use the following notation

(3.5) $H(u)=\|\nabla u\|_{\mathbf{p}}\quad\text{and}\quad
I(u)=\|u\|_{\mathbf{p}}$

and we need to establish some lemmas that will facilitate the proof.

First, we have to show that the functionals $H$ and $I$ are Fréchet
differentiable.

######  Lemma 3.4.

$I\colon L^{\mathbf{p}}(\Omega)\to\mathbb{R}$ and $H\colon
W^{1,\mathbf{p}}_{0}(\Omega)\to\mathbb{R}$ are Gateaux differentiable away
from zero and its derivatives are given by

(3.6) $\frac{d}{dt}I(u+tv)|_{t=0}=\langle
I^{\prime}(u),v\rangle=\int_{\mathbb{R}^{n}}\prod_{i=1}^{n}I_{i}(u)^{p_{i+1}-p_{i}}|u|^{p_{1}-2}uv\,dx,$

where $p_{n+1}=1$ and

(3.7) $\frac{d}{dt}H(u+tv)|_{t=0}=\langle
H^{\prime}(u),v\rangle=\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\left|\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}\right|^{p_{i}-2}\frac{u_{x_{i}}}{\|u_{x_{i}}\|_{p_{i}}}v_{x_{i}}\,dx.$

That is, $H^{\prime}=\mathcal{L}_{\mathbf{p}}$ and
$I^{\prime}=\mathcal{F}_{\mathbf{p}}$.

###### Proof.

To prove (3.6), let $u,v\in L^{\mathbf{p}}(\Omega)$ and $t\in\mathbb{R}$.
Then, recalling Remark 2.1, we compute

$\displaystyle\left.\frac{d}{dt}I_{1}(u+tv)\right|_{t=0}=I_{1}(u)^{1-p_{1}}\int_{\mathbb{R}}|u|^{p_{1}-2}uv\,dx_{1}.$

Next,

$\displaystyle\left.\frac{d}{dt}I_{2}(u+tv)\right|_{t=0}$
$\displaystyle=I_{2}(u)^{1-p_{2}}\int_{\mathbb{R}}I_{1}(u+tv)^{p_{2}-1}\left.\frac{d}{dt}I_{1}(u+tv)\right|_{t=0}\,dx_{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}I_{2}(u)^{1-p_{2}}I_{1}(u)^{p_{2}-p_{1}}|u|^{p_{1}-2}uv\,dx_{1}dx_{2}$

Therefore, by induction, we arrive at

$\left.\frac{d}{dt}I_{n}(u+tv)\right|_{t=0}=\int_{\mathbb{R}^{n}}\prod_{i=1}^{n}I_{i}(u)^{p_{i+1}-p_{i}}|u|^{p_{1}-2}uv\,dx,$

where $p_{n+1}=1$ and the proof of (3.6) follows observing that $I_{n}=I$.

The proof of (3.7) is standard and the details are left to the reader. ∎

######  Theorem 3.5.

The functionals $I$ and $H$ given in (3.5) are Fréchet differentiable.

###### Proof.

The proof follows easily from Lemma 3.4 just observing that
$I^{\prime}=\mathcal{F}_{\mathbf{p}}$ and
$H^{\prime}=\mathcal{L}_{\mathbf{p}}$ are continuous. In fact, the continuity
of $\mathcal{F}_{\mathbf{p}}$ is proved in Lemma 3.2 and the continuity of
$\mathcal{L}_{\mathbf{p}}$ is an easy excercise. ∎

At this point we can give a rigorous proof of Theorem 3.3.

###### Proof of Theorem 3.3 .

Recall that, since

$\mathcal{Q}_{\mathbf{p}}(u)=\frac{H(u)}{I(u)},$

using Lemma 3.4, one obtain that, if $u\neq 0$,

$\langle\mathcal{Q}_{\mathbf{p}}^{\prime}(u),v\rangle=\frac{1}{I(u)}\left(\langle
H^{\prime}(u),v\rangle-\mathcal{Q}_{\mathbf{p}}(u)\langle
I^{\prime}(u),v\rangle\right).$

Hence, $u\in W^{1,\mathbf{p}}_{0}(\Omega)$ is a critical point of
$\mathcal{Q}_{\mathbf{p}}$ if and only if

$\langle H^{\prime}(u),v\rangle=\mathcal{Q}_{\mathbf{p}}(u)\langle
I^{\prime}(u),v\rangle.$

But this is the same as saying that $u$ is a weak solution to (3.1) with
$\lambda=\mathcal{Q}_{\mathbf{p}}(u)$. ∎

###  3.2. The fractional case

Now, we will analyze the fractional case. So, we consider the Rayleigh-type
quotient $\mathcal{Q_{\mathbf{s},\mathbf{p}}}$ defined in (1.3), and look for
the Euler-Lagrange equation associated to it.

The main result of this section is to show that the E-L equation is given by

(3.8)
$\begin{cases}-\mathcal{L}_{\mathbf{s},\mathbf{p}}u=\lambda\mathcal{F}_{\mathbf{p}}(u)\quad&\text{
in }\Omega\\\ u=0&\text{ in }\mathbb{R}^{n}\setminus\Omega,\end{cases}$

where $\mathcal{F}_{\mathbf{p}}$ is given by (3.3) and

$\displaystyle\mathcal{L}_{\mathbf{s},\mathbf{p}}u=\text{p.v.}\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\left|\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\right|^{p_{i}-2}\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\frac{dh}{|h|^{1+s_{i}}}\,dx,$
$\displaystyle=\lim_{\varepsilon\to
0}\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{|h|>\varepsilon}\left|\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\right|^{p_{i}-2}\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\frac{dh}{|h|^{1+s_{i}}}\,dx$

and

$D^{s,i}_{h}u(x)=\frac{u(x+he_{i})-u(x)}{|h|^{s}}.$

It is shown in [6] that the operator $\mathcal{L}_{\mathbf{s},\mathbf{p}}$ is
the fractional version of $\mathcal{L}_{\mathbf{p}}$. This operator
$\mathcal{L}_{\mathbf{s},\mathbf{p}}$ has to be understood in the weak sense,
i.e. given $u,v\in W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$,

$\langle-\mathcal{L}_{\mathbf{s},\mathbf{p}}u,v\rangle=\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\left|\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\right|^{p_{i}-2}\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}D^{s_{i},i}_{h}v(x)\frac{dh}{|h|}\,dx.$

Again, we have to give a definition of weak solution.

######  Definition 3.6.

Let $u$ be a function in $W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$, then $u$ is
a weak solution of (3.8) if $u$ verifies

$\displaystyle\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\left|\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\right|^{p_{i}-2}\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}D^{s_{i},i}_{h}v(x)\frac{dh}{|h|}\,dx=\lambda\int_{\mathbb{R}^{n}}\mathcal{F}_{\mathbf{p}}(u)v\,dx,$

for all $v\in W^{\mathbf{s},\mathbf{p}}_{0}(\Omega).$

Again, we introduce the notation

(3.9) $H_{\mathbf{s}}(u)=[u]_{\mathbf{s},\mathbf{p}}$

and in an analogous form as in Lemma 3.4 we have the following lemma, whose
proof is left to the reader.

######  Lemma 3.7.

The functional $H_{\mathbf{s}}\colon
W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\to\mathbb{R}$ is Gateaux differentiable
away from zero and its derivative is given by

$\displaystyle\langle
H_{\mathbf{s}}^{\prime}(u),v\rangle=\sum_{i=1}^{n}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}}\left|\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}\right|^{p_{i}-2}\frac{D^{s_{i},i}_{h}u(x)}{[u]_{s_{i},p_{i},i}}D^{s_{i},i}_{h}v(x)\frac{dh}{|h|}\,dx.$

That is $H_{\mathbf{s}}^{\prime}=\mathcal{L}_{\mathbf{s},\mathbf{p}}$.

Finally, we can state the Euler-Lagrange Theorem for the fractional case, and
its proof is analogous to the non-fractional case and therefore is ommitted.

######  Theorem 3.8.

Let $u$ be a function in $W^{\mathbf{s},\mathbf{p}}_{0}(\Omega).$ Then $u$ is
a critical point of (1.3) if and only if $u$ is a weak solution of the Euler
Lagrange equation (3.8)

##  4\. General properties of eigenvalues

After having derived the Euler-Lagrange equation for each case, it becomes
evident that these are eigenvalue problems, for which we can explore some
properties.

We say that $\lambda\in\mathbb{R}$ is an eigenvalue of
$\mathcal{L}_{\mathbf{p}}$ under Dirichlet boundary conditions in the domain
$\Omega$, if problem (3.1) admits a nontrivial weak solution $u\in
W^{1,\mathbf{p}}_{0}(\Omega)$. Then $u$ is called an eigenfunction of
$\mathcal{L}_{\mathbf{p}}$ corresponding to $\lambda$. We will denote
$\Sigma^{\mathbf{p}}$ the collection of these eigenvalues.

Similarly, we say that $\lambda\in\mathbb{R}$ is an eigenvalue of
$\mathcal{L}_{\mathbf{s},\mathbf{p}}$ under Dirichlet boundary conditions in
the domain $\Omega$, if problem (3.8) admits a nontrivial weak solution $u\in
W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$. Then $u$ is called an eigenfunction of
$\mathcal{L}_{\mathbf{s},\mathbf{p}}$ corresponding to $\lambda$. We will
denote $\Sigma^{\mathbf{s},\mathbf{p}}$ the collection of these eigenvalues.

We begin this section by collecting some simple properties for the eigensets
$\Sigma^{\mathbf{p}}$ and $\Sigma^{\mathbf{s},\mathbf{p}}$.

######  Proposition 4.1.

$\Sigma^{\mathbf{p}},\Sigma^{\mathbf{s},\mathbf{p}}\subset(0,\infty)$ are
closed sets.

###### Proof.

As we have done throughout the article, we will only provide the proof for the
non-fractional case, leaving the fractional case for the reader.

First, let $\lambda\in\Sigma^{\mathbf{p}}$ and $u\in
W^{1,\mathbf{p}}_{0}(\Omega)$ be an associated eigenfunction. So, if we take
the same $u$ as a test function in the weak formulation of (3.1), we obtain
that

$\|\nabla u\|_{\mathbf{p}}=\lambda\|u\|_{\mathbf{p}}.$

Therefore, $\lambda>0$

Next, let us see that $\Sigma^{\mathbf{p}}$ is closed. To this end, let
$\\{\lambda_{k}\\}$ be a sequence of eigenvalues such that
$\lambda_{k}\to\lambda\in\mathbb{R}$ as $k\to\infty$ and
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset W^{1,\mathbf{p}}_{0}(\Omega)$ be a
corresponding sequence of $L^{\mathbf{p}}$-normalized eigenfunctions. Observe
that

$\|\nabla u_{k}\|_{\mathbf{p}}=\lambda_{k}\|u_{k}\|_{\mathbf{p}}=\lambda_{k},$

from where it follows that $\\{u_{k}\\}_{k\in\mathbb{N}}$ is a bounded
sequence in $W^{1,\mathbf{p}}_{0}(\Omega)$. Hence, passing to a subsequence,
we get that

$u_{k}\rightharpoonup u\text{ in }W^{1,\mathbf{p}}_{0}(\Omega)\text{ and
}u_{k}\to u\text{ in }L^{\mathbf{p}}(\Omega).$

From Lemma 3.2, $\mathcal{F}_{\mathbf{p}}$ is continuous and we get that

$\langle\mathcal{L}_{\mathbf{p}}u_{k},v\rangle=\lambda_{k}\langle\mathcal{F}_{\mathbf{p}}(u_{k}),v\rangle\to\lambda\langle\mathcal{F}_{p}(u),v\rangle.$

As each $u_{k}$ is an $L^{\mathbf{p}}$-normalized eigenfunction, then

$\langle\mathcal{L}_{\mathbf{p}}u_{k},u_{k}\rangle=\lambda_{k}\to\lambda.$

Now, we make use of Lemma 5.3, that is proved in the next section, to obtain
that $u_{k}\to u$ in $W^{1,\mathbf{p}}_{0}(\Omega)$.

Hence, we can pass to the limit $k\to\infty$ in the weak formulation

$\langle\mathcal{L}_{\mathbf{p}}u_{k},v\rangle=\lambda_{k}\langle\mathcal{F}_{p}(u_{k}),v\rangle$

and get that

$\langle\mathcal{L}_{\mathbf{p}}u,v\rangle=\lambda\langle\mathcal{F}_{\mathbf{p}}(u),v\rangle$

for any $v\in W^{1,\mathbf{p}}_{0}(\Omega)$. That is, $u$ is eigenfunction
associated to $\lambda$ and the proof is complete. ∎

Now, we arrive at the main point of this section, that is the asymptotic
behavior of the eigenset $\Sigma^{\mathbf{s},\mathbf{p}}$ as the fractional
parameters $\mathbf{s}=(s_{1},\dots,s_{n})$ verify that $s_{i}\to 1$,
$i=1,\dots,n$.

To this end, we will make use of the following result which is a particular
case of [6, Theorem 3.3].

######  Proposition 4.2.

[6, Theorem 3.3] Let $\\{\mathbf{s}_{k}\\}_{k\in\mathbb{N}}$ be a sequence of
fractional parameters $\mathbf{s}_{k}\to(1,\dots,1)$ as $k\to\infty$. Let
$\mathbf{p}=(p_{1},\dots,p_{n})$ be such that $1<p_{i}<\infty$ for each
$i=1,\dots,n$ and for each $k\in\mathbb{N}$ let $u_{k}\in
W^{\mathbf{s}_{k},\mathbf{p}}_{0}(\Omega)$ be such that

(4.1) $\sup_{k\in\mathbb{N}}\|u_{k}\|_{\mathbf{s}_{k},\mathbf{p}}<\infty.$

Then, there exists a function $u\in W^{1,\mathbf{p}}_{0}(\Omega)$ and a
subsequence
$\\{u_{k_{j}}\\}_{j\in\mathbb{N}}\subset\\{u_{k}\\}_{k\in\mathbb{N}}$ such
that

$u_{k_{j}}\to u\quad\text{in
}L^{\mathbf{p}}(\Omega)\quad\text{and}\quad\|\nabla
u\|_{\mathbf{p}}\leq\liminf_{k\to\infty}\,[u_{k}]_{\mathbf{s}_{k},\mathbf{p}}.$

Moreover, we also need to borrow a Lemma from [6].

######  Lemma 4.3.

[6, Lemma 5.7] Let $\\{\mathbf{s}_{k}\\}_{k\in\mathbb{N}}$ be a sequence of
fractional parameters satisfying that $\mathbf{s}_{k}\to(1,\dots,1)$ as
$k\to\infty$ and let $v_{k}\in W^{\mathbf{s}_{k},\mathbf{p}}_{0}(\Omega)$.
Assume that $\\{v_{k}\\}_{k\in\mathbb{N}}$ satisfy (4.1), and let $u\in
W^{1,\mathbf{p}}_{0}(\Omega)$ be fixed.

Without loss of generality, we can assume that there existes $v\in
W^{1,\mathbf{p}}_{0}(\Omega)$ such that $v_{k}\to v$ in
$L^{\mathbf{p}}(\Omega)$ as $k\to\infty$.

Then

$\langle\mathcal{L}_{\mathbf{p}}^{\mathbf{s}_{k}}u,v_{k}\rangle\to\langle\mathcal{L}_{\mathbf{p}}u,v\rangle\quad\text{as
}k\to\infty.$

Hence we can state and prove the following theorem.

######  Theorem 4.4.

Let $\\{\mathbf{s}_{k}\\}_{k\in\mathbb{N}}$ be a sequence of fractional
parameters $\mathbf{s}_{k}\to(1,\dots,1)$ as $k\to\infty$. Let
$\mathbf{p}=(p_{1},\dots,p_{n})$ be such that $1<p_{i}<\infty$ for each
$i=1,\dots,n$. Let $\lambda_{k}\in\Sigma^{\mathbf{s}_{k},\mathbf{p}}$ be an
eigenvalue of (3.8) such that $\lambda_{k}\to\lambda$ as $k\to\infty$. Then
$\lambda\in\Sigma^{\mathbf{p}}$. Moreover, if $u_{k}\in
W^{\mathbf{s}_{k},\mathbf{p}}_{0}(\Omega)$ is a normalized eigenfunction
associated to $\lambda_{k}$ then any $L^{\mathbf{p}}(\Omega)-$accumulation
point $u$ of the sequence $\\{u_{k}\\}_{k\in\mathbb{N}}$, satisfy that $u\in
W^{1,\mathbf{p}}_{0}(\Omega)$ and is an eigenfunction associated to $\lambda$.

###### Proof.

Let $\\{\mathbf{s}_{k}\\}_{k\in\mathbb{N}}$ be sequence of fractional
parameters such that $\mathbf{s}_{k}\to(1,\dots,1)$ as $k\to\infty$ and et
$\\{\lambda_{\mathbf{s}_{k}}\\}_{k\in\mathbb{N}}$ be a sequence of eigenvalues
that converge to $\lambda$ as $k\to\infty$. For each
$\lambda_{\mathbf{s}_{k}}$ there is a eigenfunction $u_{k}$, that we can
assume to be $L^{\mathbf{p}}$-normalized.

Note that since $\|u_{k}\|_{\mathbf{p}}=1$ and $\lambda_{\mathbf{s}_{k}}$ is
convergent, and therefore bounded, the sequence $\\{u_{k}\\}_{k\in\mathbb{N}}$
satisfy (4.1). Therefore we can apply Proposition 4.2 and obtain a
subsequence, that we still denote $\\{u_{k}\\}_{k\in\mathbb{N}}$ and a
function $u\in W^{1,\mathbf{p}}_{0}(\Omega)$ such that $u_{k}\to u$ in
$L^{\mathbf{p}}(\Omega)$.

Now, using Lemma 4.3, the proof follows by using a classical monotonicity
argument. ∎

##  5\. Existence of eigenvalues

In this section, we establish the existence of eigenvalues using the
Ljusternik-Schnirelman theory. This is the main part of the article.

First we recall some abstract results from critical point theory that will be
essential in our proof of the existence of eigenvalues.

Let $X$ be a reflexive Banach space and maps $\phi,\psi\in
C^{1}(X,\mathbb{R})$. We will assume that $\phi,\psi$ verify the assumptions
(H1)–(H4) below:

  1. (H1)

$\phi,\psi\in C^{1}(X,\mathbb{R})$, are even maps with $\phi(0)=\psi(0)=0$ and
the level set

${\mathcal{M}}=\\{u\in X:\psi(u)=1\\}$

is bounded.

  2. (H2)

$\phi^{\prime}$ is completely continuous. Moreover, for any $u\in X$ it holds
that

$\langle\phi^{\prime}(u),u\rangle=0\iff\phi(u)=0,$

where $\langle\cdot,\cdot\rangle$ denote the duality brackets for the pair
$(X,X^{*}).$

  3. (H3)

$\psi^{\prime}$ is continuous, bounded and, as $k\to\infty$, it holds that

$u_{k}\rightharpoonup u,\psi^{\prime}(u_{k})\rightharpoonup v\text{ and
}\langle\psi^{\prime}(u_{k}),u_{k}\rangle\to\langle v,u\rangle\Rightarrow
u_{k}\to u\text{ in }X.$

  4. (H4)

For every $u\in X\setminus\\{0\\}$ it holds that

$\langle\psi^{\prime}(u),u\rangle>0,\ \lim_{t\to\infty}\psi(tu)=\infty\text{
and }\inf_{u\in M}\langle\psi^{\prime}(u),u\rangle>0.$

Now, for any $n\in\mathbb{N}$, we define

${\mathcal{K}}_{n}=\\{K\subset\mathcal{M}\colon K\text{ is symmetric, compact,
with }\phi|_{K}>0\text{ and }\gamma(K)\geq n\\},$

where $\gamma(K)$ is the Krasnoselskii genus of the set $K$.

Finally, let

$c_{n}=\begin{cases}\sup_{K\in{\mathcal{K}}_{n}}\min_{u\in K}\phi(u)&\text{if
}{\mathcal{K}}_{n}\neq\emptyset\\\ 0&\text{if
}{\mathcal{K}}_{n}=\emptyset\end{cases}$

The following general abstract result is proved in [25] (see also [18, Theorem
9.27]).

######  Theorem 5.1.

Let $X$ be a reflexive Banach space and $\phi,\psi\in C^{1}(X,\mathbb{R})$.
Assume that $\phi,\psi$ satisfy (H1)–(H4). Then

  1. (1)

$c_{1}<\infty$ and $c_{n}\to 0$ as $n\to\infty$.

  2. (2)

If $c=c_{n}>0$, then we can find an element $u\in\mathcal{M}$ that is a
solution of

(5.1)
$\mu\psi^{\prime}(u)=\phi^{\prime}(u),\quad(\mu,u)\in\mathbb{R}\times{\mathcal{M}},$

for an eigenvalue $\mu\neq 0$ and such that $\phi(u)=c$.

  3. (3)

More generally, if $c=c_{n}=c_{n+k}>0$ for some $k\geq 0$, then the set of
solutions $u\in{\mathcal{M}}$ of (5.1) such that $\phi(u)=c$ has genus $\geq
k+1$.

  4. (4)

If $c_{n}>0$ for all $n\geq 1$, then there is a sequence
$\\{(\mu_{n},u_{n})\\}_{n\in\mathbb{N}}$ of solutions of (5.1) with
$\phi(u_{n})=c_{n}$, $\mu_{n}\neq 0$ for all $n\geq 1$, and $\mu_{n}\to 0$ as
$n\to\infty$.

  5. (5)

If we further require that

$\langle\phi^{\prime}(u),u\rangle=0\text{ if and only if }\phi(u)=0\text{ if
and only if }u=0,$

then, $c_{n}>0$ for all $n\geq 1$, and there is a sequence
$\\{(\mu_{n},u_{n})\\}_{n\in\mathbb{N}}$ of solutions of (5.1) such that
$\phi(u_{n})=c_{n}$, $\mu_{n}\neq 0$, $\mu_{n}\to 0$, and
$u_{n}\rightharpoonup 0$ in $X$ as $n\to\infty$.

Now we will apply Theorem 5.1 in the case where
$X=W^{1,\mathbf{p}}_{0}(\Omega)$, $(\phi,\psi)=(I,H)$ and in the case where
$X=W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$ and
$(\phi,\psi)=(I,H_{\mathbf{s}})$, where the operators $I,H$ and
$H_{\mathbf{s}}$ where introduced in (3.5) and (3.9).

For enhanced readability of the work, we will demonstrate the properties of
$H$ and $H_{\mathbf{s}}$ through a series of lemmas.

######  Lemma 5.2.

Let $H\colon W^{1,\mathbf{p}}_{0}(\Omega)\to\mathbb{R}$ be defined in (3.5)
and $H_{\mathbf{s}}\colon W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\to\mathbb{R}$
be defined in (3.9).

Then $H^{\prime}\colon W^{1,\mathbf{p}}_{0}(\Omega)\to
W^{-1,\mathbf{p}^{\prime}}(\Omega)$ and $H_{\mathbf{s}}^{\prime}\colon
W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\to
W^{-\mathbf{s},\mathbf{p}^{\prime}}(\Omega)$ are bounded and monotone.

###### Proof.

Let us first demonstrate that $H^{\prime}$ is bounded. In fact

$\displaystyle|\langle H^{\prime}(u),v\rangle|$
$\displaystyle=\left|\sum_{i=1}^{n}\frac{1}{\|u_{x_{i}}\|_{p_{i}}^{p_{i}-1}}\int_{\mathbb{R}}|u_{x_{i}}|^{p_{i}-2}u_{x_{i}}v_{x_{i}}\,dx\right|$
$\displaystyle\leq\sum_{i=1}^{n}\frac{1}{\|u_{x_{i}}\|_{p_{i}}^{p_{i}-1}}\int_{\mathbb{R}}|u_{x_{i}}|^{p_{i}-1}|v_{x_{i}}|\,dx$
$\displaystyle\leq\sum_{i=1}^{n}\frac{\|u_{x_{i}}\|_{p_{i}}^{p_{i}/p_{i}^{\prime}}\|v_{x_{i}}\|_{p_{i}}}{\|u_{x_{i}}\|_{p_{i}}^{p_{i}-1}}$
$\displaystyle\leq\sum_{i=1}^{n}\|v_{x_{i}}\|_{p_{i}}=\|\nabla
v\|_{\mathbf{p}}.$

Therefore, $H^{\prime}$ is a bounded operator. Moreover, from this result we
can obtain the monotonicity of it.

$\langle H^{\prime}(u),v\rangle+\langle
H^{\prime}(v),u\rangle\leq\left|\langle
H^{\prime}(u),v\rangle\right|+\left|\langle
H^{\prime}(v),u\rangle\right|\leq\|\nabla v\|_{\mathbf{p}}+\|\nabla
u\|_{\mathbf{p}}.$

Hence,

$\displaystyle\langle H^{\prime}(u)-H^{\prime}(v),u-v\rangle$
$\displaystyle=\langle H^{\prime}(u),u\rangle+\langle
H^{\prime}(v),v\rangle-\left(\langle H^{\prime}(u),v\rangle+\langle
H^{\prime}(v),u\rangle\right)$ $\displaystyle=\|\nabla
u\|_{\mathbf{p}}+\|\nabla v\|_{\mathbf{p}}-\left(\langle
H^{\prime}(u),v\rangle+\langle H^{\prime}(v),u\rangle\right)\geq 0.$

The proof for $H^{\prime}$ is complete.

The proof for $H_{\mathbf{s}}^{\prime}$ is analogous and the details are left
to the reader. ∎

######  Lemma 5.3.

The operators $H$ and $H_{\mathbf{s}}$ given in (3.5) and (3.9) respectively,
verify hypothesis (H3).

That is $H^{\prime}\colon W^{1,\mathbf{p}}_{0}(\Omega)\to
W^{-1,\mathbf{p}^{\prime}}(\Omega)$ and $H_{\mathbf{s}}^{\prime}\colon
W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)\to
W^{-\mathbf{s},\mathbf{p}^{\prime}}(\Omega)$ are continuous and bounded
operator, and moreover, as $k\to\infty$, it holds that

(5.2) $\displaystyle u_{k}\rightharpoonup u,\ H^{\prime}(u_{k})\rightharpoonup
v\text{ and }\langle H^{\prime}(u_{k}),u_{k}\rangle\to\langle
v,u\rangle\Rightarrow u_{k}\to u\text{ in }W^{1,\mathbf{p}}_{0}(\Omega)$ (5.3)
$\displaystyle u_{k}\rightharpoonup u,\
H_{\mathbf{s}}^{\prime}(u_{k})\rightharpoonup v\text{ and }\langle
H_{\mathbf{s}}^{\prime}(u_{k}),u_{k}\rangle\to\langle v,u\rangle\Rightarrow
u_{k}\to u\text{ in }W^{\mathbf{s},\mathbf{p}}_{0}(\Omega).$

###### Proof.

In view of Lemma 5.2 it remains to see that $H^{\prime}$ and
$H_{\mathbf{s}}^{\prime}$ are continuous and that verify (5.2) and (5.3)
respectively.

First we claim that $H^{\prime}$ is a continuous operator. In fact just
observe that we can rewrite $H^{\prime}$ as

$H^{\prime}(u)=\sum_{i=1}^{n}\frac{J_{p_{i}}(u_{x_{i}})}{\|u_{x_{i}}\|_{p_{i}}^{p_{i}-1}},$

where $J_{p}(u):=|u|^{p-2}u$.

Since $J_{p}\colon L^{p}(\mathbb{R}^{n})\to L^{p^{\prime}}(\mathbb{R}^{n})$ is
continuous, the claim follows

To verify (5.2) let $\\{u_{k}\\}_{k\in\mathbb{N}}$ be a sequence in
$W^{1,\mathbf{p}}_{0}(\Omega)$ such that $u_{k}\rightharpoonup u$ in
$W^{1,\mathbf{p}}_{0}(\Omega)$, $H^{\prime}(u_{k})\rightharpoonup v$ in
$W^{-1,\mathbf{p}^{\prime}}(\Omega)$ and $\langle
H^{\prime}(u_{k}),u_{k}\rangle\to\langle v,u\rangle$ then we need to show that
$u_{k}\to u$ in $W^{1,\mathbf{p}}_{0}(\Omega).$

Given $w\in W^{1,\mathbf{p}}_{0}(\Omega)$ arbitrary, by the monotonicity of
$H^{\prime}$ (Lemma 5.2) we get that

$0\leq\langle H^{\prime}(w)-H^{\prime}(u_{k}),w-u_{k}\rangle.$

Taking the limit as $k\to\infty$, we arrive at

$0\leq\langle H^{\prime}(w),w-u\rangle-\langle v,w-u\rangle.$

Now we can take $w=u+tz$ with $t>0$ and we find that

$0\leq\langle H^{\prime}(u+tz)-v,tz\rangle$

Dividing by $t$ and taking limit $t\to 0+$ we get that for all $z\in
W^{1,\mathbf{p}}_{0}(\Omega)$,

$0\leq\langle H^{\prime}(u)-v,z\rangle.$

Therefore $H^{\prime}(u)=v$. Moreover

$\|\nabla u_{k}\|_{\mathbf{p}}=\langle
H^{\prime}(u_{k}),u_{k}\rangle\to\langle v,u\rangle=\langle
H^{\prime}(u),u\rangle=\|\nabla u\|_{\mathbf{p}}.$

As the space $W^{1,\mathbf{p}}_{0}(\Omega)$ is uniformly convex, weak
convergence and norm convergence implies that $u_{k}\to u$ strongly as
desired.

The proof for $H_{\mathbf{s}}^{\prime}$ is analogous. ∎

The upcoming theorem is the main important result of this section, as it
assures the existence of eigenvalues.

######  Theorem 5.4.

There exist a sequence $\\{u_{k}\\}_{k\in\mathbb{N}}\subset
W^{1,\mathbf{p}}_{0}(\Omega)$ of critical points of $\mathcal{Q}_{\mathbf{p}}$
with associated critical values
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ such that
$\lambda_{k}\to\infty$ as $k\to\infty$. Moreover, these critical values have
the following variational characterization

(5.4) $\lambda_{k}=\inf_{K\in{\mathcal{K}}_{k}}\sup_{u\in K}H(u)$

where, for any $k\in\mathbb{N}$

${\mathcal{K}}_{k}=\\{K\subset M\text{ compact, symmetric with
}H^{\prime}(u)>0\text{ on }K\text{ and }\gamma(K)\geq k\\}$ $M=\\{u\in
W_{0}^{1,\mathbf{p}}(\Omega):I(u)=1\\}$

and $\gamma$ is the Krasnoselskii genus of $K$.

In particular, $u_{k}$ is a weak solution to (3.1) with eigenvalue
$\lambda_{k}$.

###### Proof.

We must confirm that the functionals $I$ and $H$ satisfy the hypotheses of
Theorem 5.1.

Note that conditions (H1) and (H3) are direct consequences of Lemmas 3.4 and
5.3, respectively. Condition (H4) follows directly from the definition of
$H^{\prime}(u)$.

In order to show that (H2) holds, just observe that if $u_{k}\rightharpoonup
u\text{ in }W^{1,\mathbf{p}}_{0}(\Omega)$, by the compactness of the immersion
$W^{1,\mathbf{p}}_{0}(\Omega)\subset\subset L^{\mathbf{p}}(\Omega)$, it
follows that $u_{k}\to u$ in $L^{\mathbf{p}}(\Omega)$ and using Lemma 3.2, we
get that $I^{\prime}(u_{k})\to I^{\prime}(u)$ in
$L^{\mathbf{p}^{\prime}}(\Omega)\subset W^{-1,\mathbf{p}^{\prime}}(\Omega)$.

Finally observe that $\langle I^{\prime}(u),u\rangle=I(u)=\|u\|_{\mathbf{p}}$.
Therefore each one is zero if and only if $u=0$.

We then apply the Ljusternik-Schnirelman theory, Theorem 5.1, to the
functionals $I$ and $H$ on the level set ${\mathcal{M}}=\\{u\in
W^{1,\mathbf{p}}_{0}(\Omega)\colon H(u)=1\\}$.

By Theorem 5.1 there exist a sequence of numbers
$\\{\mu_{k}\\}_{k\in\mathbb{N}}\searrow 0$ and functions
$\\{u_{k}\\}_{k\in\mathbb{N}}\in W^{1,\mathbf{p}}_{0}(\Omega)$ normalized such
that $H(u_{k})=1$, and

(5.5) $\mu_{k}\langle H^{\prime}(u_{k}),v\rangle=\langle
I^{\prime}(u_{k}),v\rangle\quad\forall v\in W^{1,\mathbf{p}}_{0}(\Omega)$

and $I(u_{k})=c_{k}$ with

(5.6) $c_{k}=\sup_{K\subset{\mathcal{K}_{k}}}\min_{u\in K}I(u).$

Using that $\langle H^{\prime}(u),u\rangle=H(u)$ and $\langle
I^{\prime}(u),u\rangle=I(u)$ one immediately obtain that $c_{k}=\mu_{k}$.

So, if we denote $\lambda_{k}=\mu_{k}^{-1}$, using (5.5), we have that $u_{k}$
is a weak solution to (3.1) with eigenvalue $\lambda_{k}$ and from (5.6) one
also obtain the validity of (5.4). ∎

######  Remark 5.5.

The eigenvalues obtained in Theorem 5.4 are commonly called the Ljusternik-
Schnirelman eigenvalues or simply the LS-eigenvalues and are denoted by
$\Sigma^{\mathbf{p}}_{LS}$.

Similarly, we state a Theorem for the fractional counterpart. The proof is a
slight variation of Theorem 5.4 and is omitted.

######  Theorem 5.6.

There exist a sequence $\\{u_{k}^{\mathbf{s}}\\}_{k\in\mathbb{N}}\subset
W^{\mathbf{s},\mathbf{p}}_{0}(\Omega)$ of critical points of
$\mathcal{Q}_{\mathbf{s},\mathbf{p}}$ with critical values
$\\{\lambda_{k}^{\mathbf{s}}\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ such that
$\lambda_{k}^{\mathbf{s}}\to\infty$ as $k\to\infty$. Moreover, this critical
values have the following variational characterization

(5.7)
$\lambda_{k}^{\mathbf{s}}=\inf_{K\in{\mathcal{K}}_{k}^{\mathbf{s}}}\sup_{u\in
K}H_{\mathbf{s}}(u)$

where, for any $k\in\mathbb{N}$

${\mathcal{K}}_{k}^{\mathbf{s}}=\\{K\subset M_{\mathbf{s}}\text{ compact,
symmetric with }H_{\mathbf{s}}^{\prime}(u)>0\text{ on }K\text{ and
}\gamma(K)\geq k\\}$ $M_{\mathbf{s}}=\\{u\in
W_{0}^{\mathbf{s},\mathbf{p}}(\Omega):I(u)=1\\}$

and $\gamma$ is the Krasnoselskii genus of $K$.

This eigenvalues related to the fractional problem will be denoted by
$\Sigma_{LS}^{\mathbf{s},\mathbf{p}}$.

## Acknowledgements

This work was partially supported by UBACYT Prog. 2018 20020170100445BA, by
ANPCyT PICT 2016-1022 and by PIP No. 11220150100032CO.

J. Fernández Bonder is a members of CONICET and I. Ceresa Dussel is a doctoral
fellow of CONICET.

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