# Ljusternik-Schnirelmann eigenvalues for the fractional $m-$Laplacian without
the $\Delta_{2}$ condition

Julián Fernández Bonder  Instituto de Cálculo – CONICET and Departamento de
Matemática, FCEN – Universidad de Buenos Aires. Ciudad Universitaria, Pabellón
I, C1428EGA, Av. Cantilo s/n Buenos Aires, Argentina<EMAIL_ADDRESS>and
Juan F. Spedaletti  Departamento de Matemática, FCFMyN, Universidad Nacional
de San Luis Instituto DE Matemática Aplicada San Luis, IMASL, CONICET. Italia
avenue 1556,office 5, San Luis (5700), San Luis, Argentina.
<EMAIL_ADDRESS>
###### Abstract.

In this work we analyze the eigenvalue problem associated to the fractional
$m-$Laplacian, defined as

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

This operator serves as a model for nonlocal, nonstandard growth diffusion
problems. In contrast to previous analyses, we explore the eigenvalue problem
without presuming the $\Delta_{2}$ condition on $M$ – the primitive function
of $m$. Our results show the existence of a sequence of eigenvalues
$\lambda_{k}\to\infty$. This research contributes to advancing our
understanding of nonlocal diffusion models, specifically those characterized
by the fractional $m-$Laplacian, by relaxing the constraints imposed by the
$\Delta_{2}$ condition.

###### Key words and phrases:

Orlicz spaces, nonlinear eigenvalues, asymptotic behavior

###### 2020 Mathematics Subject Classification:

35J62; 35P30; 46E30

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries
  3. 3 The fractional $m-$laplacian $(-\Delta_{m})^{s}$
  4. 4 The eigenvalue problem
  5. A Density results
  6. B A Rellich-Kondrachov type result

##  1\. Introduction and main results

Eigenvalue problems stand as some of the most extensively investigated
challenges within Partial Differential Equations. This interest arises both
from their innate relevance to a wide array of natural phenomena, spanning
vibrating membranes, quantum physics, and signal processing, among others, and
from their intrinsic significance. Consider an open and bounded domain
$\Omega\subset\mathbb{R}^{n}$. The classical Courant minimax principle ensures
the existence of an infinite sequence of eigenvalues
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ for the classical Dirichlet eigenvalue
problem:

$\begin{cases}-\Delta u=\lambda u&\text{in }\Omega\\\ u=0&\text{in
}\partial\Omega,\end{cases}$

where $\lambda_{k}\to\infty$ (refer to [34] and [36]). The applications of
this problem extend across various branches of mathematics and natural
sciences.

In recent times, attention has shifted to nonlinear extensions and variations
of the eigenvalue problem associated with the Laplacian. One of the most
extensively studied is the eigenvalue problem for the $p-$Laplacian:

$\begin{cases}-\Delta_{p}u=\lambda|u|^{p-2}u&\text{in }\Omega\\\ u=0&\text{in
}\partial\Omega,\end{cases}$

introduced by [27] (also discussed in [28, 29]). For this problem, the
classical Ljusternik–Schnirelmann theory applied to the functionals $F$ and
$G$ defined on $W_{0}^{1,p}(\Omega)$:

$F,\,G\colon W_{0}^{1,p}(\Omega)\to\mathbb{R}$ $F(u)=\int_{\Omega}|\nabla
u|^{p}\,dx\qquad\text{ and }\qquad G(u)=\int_{\Omega}|u|^{p}\,dx$

yields a sequence of eigenvalues $\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ with
$\lambda_{k}\to\infty$. Crucially, in this case, the space
$W_{0}^{1,p}(\Omega)$ must be reflexive and separable, and the corresponding
functionals $F$ and $G$ must be differentiable (see [21, 22, 36]).

Another interesting nonlinear eigenvalue problem arises with the $m-$Laplacian
operator, defined as

$\Delta_{m}u=\mathop{\text{div}}\left(\frac{m(|\nabla u|)}{|\nabla u|}\nabla
u\right),$

where $m\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a nondecreasing function.
This operator generalizes the $p-$Laplacian operator when $m(t)=t^{p-1}$.

Consequently, the eigenvalue problem is given by

$\begin{cases}-\Delta_{m}u=\lambda g(u)&\text{in }\Omega\\\ u=0&\text{in
}\partial\Omega.\end{cases}$ (1.1)

Here, the function $g\colon\mathbb{R}\to\mathbb{R}$ satisfies certain growth
conditions. What makes these operators particularly appealing for applications
is the potential for distinct behaviors in diffusivity when $|\nabla u|\ll 1$
and $|\nabla u|\gg 1$, a phenomenon known in the literature as nonstandard
growth elliptic operators. See [26].

A key factor in addressing such problems is the primitive function of $m$,
denoted as $M(t)=\int_{0}^{t}m(s)\,ds$. When $M$ satisfies the so–called
$\Delta_{2}-$condition, that is

$M(2t)\leq CM(t),$

for some constant $C>1$ and all $t\geq T_{0}$, then the eigenvalue problem
(1.1) inherits many properties from the $p-$Laplacian case. In such instances,
the Ljusternik–Schnirelmann theory can be applied seamlessly to establish the
existence of a sequence of eigenvalues for (1.1).

However, when $M$ does not satisfy the $\Delta_{2}-$condition, the situation
becomes significantly more intrincate. In [35], the author analyzed problem
(1.1) and using ideas from [24] and employing a Galerkin–based approximation
method, the author successfully overcome the absence of the
$\Delta_{2}-$condition. Subsequently, the Ljusternik–Schinrelmann theory was
applied, resulting in the identification of an infinite sequence of
eigenvalues for (1.1).

In recent years, nonlocal diffusion models have garnered considerable
attention due to their diverse and novel applications in the natural sciences.
These operators naturally arise in the context of stochastic Lévy processes
with jumps and have been extensively investigated from both probabilistic and
analytical perspectives. Its applications range from physics, where it
describes nonlocal interactions in materials, to finance, where it captures
the memory effect in stochastic processes, and to image processing and
ecology, where it accounts for spatial interactions over long distances, as
documented in works such as [4, 10, 33] and the references therein. For the
mathematical background from the partial differential equation (PDE)
perspective adopted in this paper, readers can refer to [9, 23].

Arguably, one of the most significant nonlocal operators is the fractional
Laplacian, defined as

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

where $s\in(0,1)$ is the fractional parameter. The associated eigenvalue
problem takes the form

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

which can be analyzed using standard methods of functional analysis. This
fractional Laplacian has proven to be a powerful tool in capturing nonlocal
interactions and long-range dependencies, making it an invaluable tool for
modeling phenomena characterized by anomalous diffusion.

In the realm of nonlocal diffusion models, numerous nonlinear generalizations
of the fractional Laplacian eigenvalue problem have been explored in the
literature. One particularly well-studied extension is encapsulated by the
fractional $p-$Laplacian operator, defined as

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

It is noteworthy that when $p=2$, the fractional $p-$Laplacian reduces to the
standard fractional Laplacian.

The eigenvalue problem associated with the fractional $p-$Laplacian has been a
subject of investigation by various authors in recent years. Notable
contributions include works by [7, 19, 20], among others. This line of
research delves into understanding the spectral properties and the behavior of
solutions for this nonlinear nonlocal eigenvalue problem. The fractional
$p-$Laplacian offers a versatile framework that extends the capabilities of
the standard fractional Laplacian by incorporating additional nonlinearity
through the power $p$. These developments hold promise for applications in
modeling complex phenomena where both nonlocal interactions and nonlinear
effects play crucial roles. The exploration of such nonlocal operators
enriches the mathematical tools available for describing a wide range of
phenomena in different scientific disciplines.

In [18], the authors introduced a fractional counterpart of the $m-$Laplacian,
offering a model for nonlocal, nonstandard growth diffusion problems.
Specifically, for an increasing and continuous function $m(t)$, the fractional
$m-$Laplacian operator is defined as

$(-\Delta_{m})^{s}u(x)=\text{p.v.}\int_{\mathbb{R}^{n}}m\left(\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\frac{(u(x)-u(y))}{|u(x)-u(y)|}\frac{dy}{|x-y|^{n+s}}.$

It is worth noting that when $m(t)=t^{p-1}$, this fractional $m-$Laplacian
reduces to the fractional $p-$Laplacian. Subsequent to the pioneering work of
[18], numerous studies exploring this operator have emerged, as evidenced by
works such as [3, 16, 30], and references therein.

The associated eigenvalue problem for the fractional $m-$Laplacian is given by

$\begin{cases}(-\Delta_{m})^{s}u=\lambda g(u)&\text{in }\Omega\\\ u=0&\text{in
}\mathbb{R}^{n}\setminus\Omega.\end{cases}$ (1.2)

For an alternative eigenvalue problem associated with this operator, readers
are directed to [17].

Previous studies of Problem (1.2), such as those found in [5, 31], assumed the
$\Delta_{2}-$condition on $M(t)$. Notably, [32] stands as the sole work known
to address (1.2) without imposing the $\Delta_{2}-$condition, demonstrating
the existence of a first eigenvalue for this problem.

Thus, the primary focus of this article lies in the investigation of Problem
(1.2) without relying on the $\Delta_{2}-$condition for the function $M(t)$.
Our main result can be succinctly summarized as follows:

######  Theorem 1.1.

Under suitable assumptions on $\Omega$, $m(t)$, and $g(t)$ without
necessitating the $\Delta_{2}-$condition on $M(t)$ there exists a sequence
$\\{\lambda_{k}\\}_{k\in\mathbb{N}}$ of eigenvalues for (1.2). Moreover,
$\lambda_{k}\to\infty$ as $k\to\infty$.

For a precise and detailed statement of this result, please refer to Theorem
4.1 in Section 4. This contribution marks a significant advancement in our
understanding of nonlocal eigenvalue problems, specifically those associated
with the fractional $m-$Laplacian, by extending the analysis beyond the
constraints of the $\Delta_{2}-$condition.

##  2\. Preliminaries

In this section we present some preliminary definitions needed for the rest of
the paper. The first subsection is well know and does not contain any new
result being the book [25] the standard reference for the subject. The second
subsection contains the definitions and basic results regarding fractional
Orlicz-Sobolev spaces. See for instance [6] where these spaces were introduced
and [2, 1] where several properties of these spaces were analyzed whitout
requiring the $\Delta_{2}-$condition. The third subsection recall the
definition of complemantary pairs intruduced in [24] and construct a
complementary pair in the context of fractional Orlicz-Sobolev spaces. Finally
in the last subsection we recall an abstract result due to [35] that will be
helpful in the sequel.

###  2.1. Young functions and Orlicz spaces

Let $M\colon\mathbb{R}\to\mathbb{R}$ be a function, such that $M$ is even,
convex and continuous, $M(t)>0$ for $t>0$, $M(t)/t\to 0$ as $t\to 0$ and
$M(t)/t\to\infty$ as $t\to\infty$. Such a function $M$ is called a Young
function if it can be written as

$M(t)=\int_{0}^{|t|}m(s)\,ds,$

for $m\colon[0,\infty)\to[0,\infty)$ increasing, right continuous, $m(t)=0$ if
and only if $t=0$ and $m(t)\to\infty$ as $t\to\infty$.

It will be helpful to extend the function $m$ to the entire real line by
oddness, that is

$m(t)=\frac{m(|t|)}{|t|}t.$

We recall now some basic definitions on Orlicz spaces that can be found, for
instance, in [25].

Let $U\subset\mathbb{R}^{N}$ be a bounded domain and let $\mu$ be a Borel
measure in $U$. The Orlicz class $\mathcal{L}_{M}(U,d\mu)$ is defined as

$\mathcal{L}_{M}(U,d\mu):=\left\\{u\colon U\to\mathbb{R},\
\text{measurable}\colon\int_{U}M(u)\,d\mu<\infty\right\\}.$

The Orlicz space $L_{M}(U,d\mu)$ is then define as the linear hull of
$\mathcal{L}_{M}(U,d\mu)$. It follows that $L_{M}(U,d\mu)$ can be
characterized as

$L_{M}(U,d\mu)=\left\\{u\colon U\to\mathbb{R},\
\text{$\mu-$measurable}\colon\int_{U}M\left(\frac{u}{k}\right)\,d\mu<\infty,\text{
for some }k>0\right\\}.$

This space is a Banach space when it is equipped, for instance, with the
Luxemburg norm, i.e.

$\|u\|_{L_{M}(U,d\mu)}=\|u\|_{M,U,d\mu}=\|u\|_{M,d\mu}:=\inf\left\\{k>0\colon\int_{U}M\left(\frac{u}{k}\right)\,d\mu\leq
1\right\\}.$

A well known and interesting fact is that
$\mathcal{L}_{M}(U,d\mu)=L_{M}(U,d\mu)$ if and only if $M$ satisfies the so-
called $\Delta_{2}-$condition, i.e.

$M(2t)\leq CM(t),\quad\text{for }t\geq T.$ (2.1)

Also, the Orlicz space $L_{M}(U,d\mu)$ is separable, if and only if $M$
satisfies (2.1).

Next, we define the space $E_{M}(U,d\mu)$ as the closure of bounded
$\mu-$measurable functions in $L_{M}(U,d\mu)$, in the case $\mu(U)=\infty$ the
space $E_{M}(U,d\mu)$ is the closure in $L_{M}(U,d\mu)$ of bounded
$\mu-$measurable functions with bounded support. Again,
$E_{M}(U,d\mu)=L_{M}(U,d\mu)$ if and only if $M$ satisfies (2.1).

So, in general, we have

$E_{M}(U,d\mu)\subset\mathcal{L}_{M}(U,d\mu)\subset L_{M}(U,d\mu),$

with equalities if and only if $M$ satisfies (2.1).

Observe that $E_{M}(U,d\mu)$ and $L_{M}(U,d\mu)$ are Banach spaces and
$\mathcal{L}_{M}(U,d\mu)$ is a convex set.

Given a Young function $M$, we define its complementary function $\bar{M}$ as

$\bar{M}(t):=\sup\\{\tau|t|-M(\tau)\colon\tau\geq 0\\}.$

Observe that $\bar{M}$ is also a Young function and is the optimal function in
the Young inequality

$\tau t\leq M(t)+\bar{M}(\tau),$ (2.2)

for all $\tau,t$. Observe that equality in (2.2) is achieved if and only if
$\tau=m(t)$ sign $t$ or $t=\bar{m}(\tau)$ sign $\tau$ where $\bar{m}(t)$ is
the derivative of $\bar{M}(t)$.

It follows directly from (2.2) that if $u\in L_{M}(U,d\mu)$ and $v\in
L_{\bar{M}}(U,d\mu)$, then $uv\in L^{1}(U)$ and

$\int_{U}|uv|\,d\mu\leq 2\|u\|_{M}\|v\|_{\bar{M}}.$

This fact allows one to define in $L_{M}(U,d\mu)$ the topology
$\sigma(L_{M},L_{\bar{M}})$ and it follows that $E_{M}(U,d\mu)$ is dense in
$L_{M}(U,d\mu)$ in this topology.

It is easy to check that $\bar{\bar{M}}=M$. The Orlicz space
$L_{\bar{M}}(U,d\mu)$ is the dual space of $E_{M}(U,d\mu)$ and so
$L_{M}(U,d\mu)$ is reflexive if and only if $M$ and $\bar{M}$ satisfy (2.1).

Finally, given $M$ a Young function, we define

$\mathop{\text{Dom}}(m):=\\{u\in L_{M}(U,d\mu)\colon m(|u|)\in
L_{\bar{M}}(U,d\mu)\\}.$ (2.3)

It can be checked that
$E_{M}(U,d\mu)\subset\mathop{\text{Dom}}(m)\subset\mathcal{L}_{M}(U,d\mu)$ and
hence, $\mathop{\text{Dom}}(m)=L_{M}(U,d\mu)$ if and only if $M$ satisfies
(2.1). Moreover, the map $u\mapsto m(u)$ from $E_{M}(U,d\mu)$ to
$L_{\bar{M}}(U,d\mu)$ is continuous if and only if $\bar{M}$ satisfies (2.1).

###  2.2. Fractional Orlicz-Sobolev spaces

In the product space $\mathbb{R}^{n}\times\mathbb{R}^{n}=\mathbb{R}^{2n}$ we
define de measure

$d\nu_{n}:=\frac{dxdy}{|x-y|^{n}}.$

Observe that this is a Borel measure and that if
$K\subset\mathbb{R}^{2n}\setminus\Delta$ is compact, then $\nu_{n}(K)<\infty$,
where $\Delta\subset\mathbb{R}^{n}\times\mathbb{R}^{n}$ is the diagonal
$\Delta:=\\{(x,x)\colon x\in\mathbb{R}^{n}\\}$.

We will consider two Orlicz spaces $L_{M}(U,d\mu)$. One with
$U=\mathbb{R}^{n}$ and $d\mu=dx$ (the Lebesgue measure) and other with
$U=\mathbb{R}^{2n}$ and $d\mu=d\nu_{n}$.

We will use the notations

$\mathcal{L}_{M}=\mathcal{L}_{M}(\mathbb{R}^{n},dx),\qquad
L_{M}=L_{M}(\mathbb{R}^{n},dx),\qquad E_{M}=E_{M}(\mathbb{R}^{n},dx);$
$\mathcal{L}_{M}(\nu_{n})=\mathcal{L}_{M}(\mathbb{R}^{2n},d\nu_{n}),\quad
L_{M}(\nu_{n})=L_{M}(\mathbb{R}^{2n},d\nu_{n}),\quad
E_{M}(\nu_{n})=E_{M}(\mathbb{R}^{2n},d\nu_{n}).$

Now, given a fractional parameter $s\in(0,1)$, we introduce the notation for
the Hölder quotient of a function $u\colon\mathbb{R}^{n}\to\mathbb{R}$.

$D^{s}u(x,y):=\frac{u(x)-u(y)}{|x-y|^{s}}.$

Then $D^{s}u\colon\mathbb{R}^{2n}\setminus\Delta\to\mathbb{R}$.

Now, with all the notation introduced, the fractional Orlicz-Sobolev spaces
are defined as

$W^{s}L_{M}:=\\{u\in L_{M}\colon D^{s}u\in L_{M}(\nu_{n})\\}$

and

$W^{s}E_{M}:=\\{u\in E_{M}\colon D^{s}u\in E_{M}(\nu_{n})\\}.$

These spaces are naturally equipped with the norms

$\|u\|_{s,M}=\|u\|_{M}+\|D^{s}u\|_{M,\nu_{n}}.$

Also, these spaces can be isometrically identified as closed subspaces of
$L_{M}\times L_{M}(\nu_{n})$ and $E_{M}\times E_{M}(\nu_{n})$ respectively
using the map

$u\mapsto(u,D^{s}u).$

Now, given $\Omega\subset\mathbb{R}^{n}$ a bounded open set, the space
$W^{s}_{0}L_{M}(\Omega)$ is then defined as the closure of
${\mathcal{D}}(\Omega)$ in $W^{s}L_{M}$ with respect to the topology
$\sigma(L_{M}\times L_{M}(\nu_{n}),E_{\bar{M}}\times E_{\bar{M}}(\nu_{n}))$.
The space $W_{0}^{s}L_{M}(\Omega)$ is equipped with the norm
$\|u\|_{W_{0}^{s}L_{M}(\Omega)}=\|u\|_{M,\Omega,dx}+\|D^{s}u\|_{M,\nu_{n}}$
and by Poincaré’s inequality (see [18, Corollary 6.2]) we can consider the
space $W_{0}^{s}L_{M}(\Omega)$ with the equivalent norm
$\|D^{s}u\|_{M,\nu_{n}}$.

The space $W^{s}_{0}E_{M}$ is defined as the closure of
${\mathcal{D}}(\Omega)$ in $W^{s}E_{M}$ in norm topology.

In order to define the dual spaces, we need to introduce the notion of
fractional divergence. See [15].

Given $F\in L_{\bar{M}}(\nu_{n})$, the fractional divergence of $F$ is defined
as

$\displaystyle{\mathrm{div}}^{s}F(x)$
$\displaystyle:=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{F(y,x)-F(x,y)}{|x-y|^{n+s}}\,dy$
$\displaystyle=\lim_{\varepsilon\to 0}\int_{\mathbb{R}^{n}\setminus
B_{\varepsilon}(x)}\frac{F(y,x)-F(x,y)}{|x-y|^{n+s}}\,dy.$

In [14] it is shown that for $F\in L_{\bar{M}}(\nu_{n})$, then
${\mathrm{div}}^{s}F\in(W^{s}_{0}L_{M}(\Omega))^{*}$ and the following
fractional integration by parts formula holds

$\langle{\mathrm{div}}^{s}F,u\rangle=\iint_{\mathbb{R}^{2n}}FD^{s}u\,d\nu_{n}.$

So, we define the following spaces of distributions

$W^{-s}L_{\bar{M}}(\Omega):=\\{\phi\in{\mathcal{D}}^{\prime}(\Omega)\colon\phi=f+\mathrm{div}^{s}F\text{
with }f\in L_{\bar{M}},\ F\in L_{\bar{M}}(\nu_{n})\\}$
$W^{-s}E_{\bar{M}}(\Omega):=\\{\phi\in{\mathcal{D}}^{\prime}(\Omega)\colon\phi=f+\mathrm{div}^{s}F\text{
with }f\in E_{\bar{M}},\ F\in E_{\bar{M}}(\nu_{n})\\}.$

Recall that since $E_{\bar{M}}$ and $E_{\bar{M}}(\nu_{n})$ are separable then
$W^{-s}E_{\bar{M}}(\Omega)$ is also separable.

These spaces are endowed with the usual quotient norms,

$\|\phi\|_{-s,\bar{M}}:=\inf\\{\|f\|_{\bar{M}}+\|F\|_{\bar{M},\nu_{v}}\colon\phi=f+\mathrm{div}^{s}F\\}.$

###  2.3. Complementary systems

In [12, 13] the authors introduce the notion of complementary systems in order
to work in spaces without the usual reflexivity assumption.

Let $Y$ and $Z$ be real Banach spaces with a duality pairing
$\langle\cdot,\cdot\rangle$. Let $Y_{0}\subset Y$ and $Z_{0}\subset Z$ be
closed and separable subspaces. We say $(Y,Y_{0};Z,Z_{0})$ is a complementary
system if $Y_{0}^{*}=Z$ and $Z_{0}^{*}=Y$ (where equality is understood in the
sense of a natural isometry via the duality pairing).

The first natural example of a complementary system is $Y=L_{M}$,
$Y_{0}=E_{M}$, $Z=L_{\bar{M}}$ and $Z_{0}=E_{\bar{M}}$.

We use the notation $(Y,Y_{0};Z,Z_{0})$ for a complementary system. Observe
that it is immediate to see that

$\left(L_{M}\times L_{M}(\nu_{n}),E_{M}\times E_{M}(\nu_{n});L_{\bar{M}}\times
L_{\bar{M}}(\nu_{n}),E_{\bar{M}}\times E_{\bar{M}}(\nu_{n})\right)$ (2.4)

is also a complementary system.

In [24] the author provides with a general method to generate complementary
systems from a previous one. More precisely

######  Lemma 2.1 ([24], Lemma 1.2).

Given a complementary system $(Y,Y_{0};Z,Z_{0})$ and a closed subspace $E$ of
$Y$, define $E_{0}=E\cap Y_{0}$, $F=Z/E_{0}^{\perp}$ and
$F_{0}=Z_{0}/E_{0}^{\perp}$.

Then, the pairing $\langle\cdot,\cdot\rangle$ between $Y$ and $Z$ induces a
pairing between $E$ and $F$ if and only if $E_{0}$ is $\sigma(Y,Z)$ dense in
$E$. In this case, $(E,E_{0};F,F_{0})$ is a complementary system if $E$ is
$\sigma(Y,Z_{0})$ closed, and conversely, when $Z_{0}$ is complete, $E$ is
$\sigma(Y,Z_{0})$ closed if $(E,E_{0};F,F_{0})$ is a complementary system.

Using this Lemma, in [24] it is shown that

$(W^{1}_{0}L_{M}(\Omega),W^{1}_{0}E_{M}(\Omega);W^{-1}L_{\bar{M}}(\Omega),W^{-1}E_{\bar{M}}(\Omega))$

is a complementary system when the domain $\Omega$ satisfies the segment
property (See Definition A.1 for a precise statement).

Let us now see that

$(W^{s}_{0}L_{M}(\Omega),W^{s}_{0}E_{M}(\Omega);W^{-s}L_{\bar{M}}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$
(2.5)

is also a complementary system under the same assumptions on $\Omega$.

In fact, since (2.4) is a complementary system, we use Lemma 2.1 to generate
(2.5) from (2.4).

So we take $E=W^{s}_{0}L_{M}(\Omega)$ and $E_{0}=E\cap
Y_{0}=W^{s}_{0}E_{M}(\Omega)$. It is also easy to see that
$F=Z/E_{0}^{\perp}=W^{-s}L_{\bar{M}}(\Omega)$ and
$F_{0}=W^{-s}E_{\bar{M}}(\Omega)$. So in order to see that (2.5) is a
complementary system it remains to check that $W^{s}_{0}E_{M}(\Omega)$ is
$\sigma(L_{M}\times L_{M}(\nu_{n}),L_{\bar{M}}\times L_{\bar{M}}(\nu_{n}))$
dense in $W^{s}_{0}L_{M}(\Omega)$ and that $W^{s}_{0}L_{M}(\Omega)$ is
$\sigma(W^{s}_{0}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ closed.

Now, $W^{s}_{0}L_{M}(\Omega)$ is
$\sigma(W^{s}_{0}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ closed by
definition. The proof of the density of $W^{s}_{0}E_{M}(\Omega)$ in
$W^{s}_{0}L_{M}(\Omega)$ with respect to the $\sigma(L_{M}\times
L_{M}(\nu_{n}),L_{\bar{M}}\times L_{\bar{M}}(\nu_{n}))$ topology follows
similarly as in [24, Theorem 1.3]. All these details are collected in Appendix
A for the reader convenience. See Theorem A.2.

###  2.4. An abstract result

In this subsection, we recall an abstract result from [35] where the author
construct in a complementary system a sequence of projector operators
converging to the identity. This abstract result will be of critical
importance in the application of the Ljusternik-Schnirelmann method.

######  Theorem 2.2 ([35], Theorem 3.1).

Assume $(E,E_{0};F,F_{0})$ is a complementary system, the norm $\|\cdot\|_{F}$
is dual to $\|\cdot\|_{E_{0}}$, the norm $\|\cdot\|_{E}$ is dual to
$\|\cdot\|_{F_{0}}$ and $V\subset E_{0}$ is a norm-dense linear subspace. Then
there exists a sequence of mappings $P_{k}\colon E_{0}\to E_{0},k=1,2,\dots$
satisfying

  * •

$P_{k}$ is odd and norm-continuous for all $k=1,2,\dots$

  * •

$P_{k}(E_{0})$ is contained in a finite-dimensional subespace of $V$ for all
$k=1,2,\dots$

  * •

If $\\{u_{k}\\}\in E_{0}$ and $u_{k}\to u\in E$ for $\sigma(E,F_{0})$, then
$P_{k}(u_{k})\to u$ for $\sigma(E,F_{0})$.

  * •

If $\\{u_{k}\\}\in E_{0}$ and $u_{k}\to u\in E$ strongly, then
$\|P_{k}(u_{k})\|_{E}\to\|u\|_{E}$.

##  3\. The fractional $m-$laplacian $(-\Delta_{m})^{s}$

In this section we introduce the integro–differential operator appearing in
our eigenvalue problem (1.2). This operator was first introduced in [6] and
was analyze in the case where the Young function $M$ satisfies the
$\Delta_{2}-$condition.

Let $M$ be a Young function and $\Omega\subset\mathbb{R}^{n}$ be a bounded,
open set with the segment property. Recall that for $0<s<1$ the fractional
$m-$Laplacian of a function $u$ is defined as

$\displaystyle(-\Delta_{m})^{s}u(x)$
$\displaystyle:=2\,\text{p.v}.\int_{\mathbb{R}^{n}}m(D^{s}u)\,\frac{dy}{|x-y|^{n+s}}$
$\displaystyle=2\lim_{\varepsilon\downarrow
0}\int_{|x-y|\geq\varepsilon}m(D^{s}u)\,\frac{dy}{|x-y|^{n+s}}.$

Let us see now that this operator is well defined between the spaces
$\mathop{\text{Dom}}((-\Delta_{m})^{s})$ and $W^{-s}L_{\bar{M}}(\Omega)$
respectively where

$\mathop{\text{Dom}}((-\Delta_{m})^{s}):=\\{u\in W^{s}_{0}L_{M}(\Omega)\colon
m(D^{s}u)\in L_{\bar{M}}(\nu_{n})\\}.$

To do this we consider for $\varepsilon>0$

$(-\Delta_{m})_{\varepsilon}^{s}u(x):=2\int_{|x-y|\geq\varepsilon}m(D^{s}u)\,\frac{dy}{|x-y|^{n+s}}.$

######  Theorem 3.1.

Let $0<s<1$ be fixed. For $u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$ the
limit $(-\Delta_{m})^{s}u:=\lim_{\varepsilon\downarrow
0}(-\Delta_{m})_{\varepsilon}^{s}u$ exists in $W^{-s}L_{\bar{M}}(\Omega)$,
that is

$\langle(-\Delta_{m})^{s}u,v\rangle:=\lim_{\varepsilon\downarrow
0}\langle(-\Delta_{m})_{\varepsilon}^{s}u,v\rangle<\infty,$

forall $v\in W_{0}^{s}E_{M}(\Omega)$.

Moreover the following representation formula holds

$\langle(-\Delta_{m})^{s}u,v\rangle=\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}v\,d\nu_{n},$

for all $v\in W_{0}^{s}E_{M}(\Omega)$.

###### Proof.

Let $0<\varepsilon<1$. We begin by proving that
$(-\Delta_{m})_{\varepsilon}^{s}u\in L_{\bar{M}}$ for
$u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$. If
$u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$ then there exists a constant
$k>0$ such that

$\iint_{\mathbb{R}^{2n}}\bar{M}\left(\frac{m(D^{s}u)}{k}\right)d\nu_{n}<\infty,$

therefore by Jensen’s inequality

$\displaystyle\int_{\mathbb{R}^{n}}\bar{M}\left(\frac{(-\Delta_{m})_{\varepsilon}^{s}u}{\left({\frac{2k\varepsilon^{-s}\omega_{n-1}}{s}}\right)}\right)\,dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\bar{M}\left(\frac{\int_{|x-y|\geq\varepsilon}\frac{m\left(\frac{u(x)-u(y)}{|x-y|^{s}}\right)}{k}\,\frac{dy}{|x-y|^{n+s}}}{\frac{\varepsilon^{-s}\omega_{n-1}}{s}}\right)\,dx$
$\displaystyle\leq\frac{\varepsilon^{s}s}{\omega_{n-1}}\int_{\mathbb{R}^{n}}\int_{|x-y|\geq\varepsilon}\bar{M}\left(\frac{m\left(\frac{u(x)-u(y)}{|x-y|^{s}}\right)}{k}\right)\,\frac{dy}{|x-y|^{n+s}}\,dx$
$\displaystyle\leq\frac{\varepsilon^{s}s}{\omega_{n-1}}\iint_{\mathbb{R}^{2n}}\bar{M}\left(\frac{m(D^{s}u)}{k}\right)\,d\nu_{n}$
$\displaystyle<\infty,$

where $\omega_{n-1}$ denotes the measure of the $(n-1)-$dimensional sphere
$S^{n-1}$. The above inequality implies $(-\Delta_{m})_{\varepsilon}^{s}u\in
L_{\bar{M}}$ .

Let $v\in W_{0}^{s}E_{M}(\Omega)$, using Fubini’s theorem and change variables
we obtain

$\displaystyle\langle(-\Delta_{m})_{\varepsilon}^{s}u,v\rangle$
$\displaystyle=2\int_{\mathbb{R}^{n}}\int_{|x-y|\geq\varepsilon}m\left(\frac{u(x)-u(y)}{|x-y|^{s}}\right)v(x)\,\frac{dy}{|x-y|^{n+s}}\,dx$
$\displaystyle=2\int_{\mathbb{R}^{n}}\int_{|x-y|\geq\varepsilon}m\left(\frac{u(y)-u(x)}{|x-y|^{s}}\right)v(y)\,\frac{dy}{|x-y|^{n+s}}\,dx,$

and so

$\displaystyle\langle(-\Delta_{m})_{\varepsilon}^{s}u,v\rangle$
$\displaystyle=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}m\left(\frac{u(x)-u(y)}{|x-y|^{s}}\right)\left(\frac{v(x)-v(y)}{|x-y|^{s}}\right)\,\chi_{\\{|x-y|\geq\varepsilon\\}}(x,y)\,\frac{dxdy}{|x-y|^{n}}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}v\chi_{\\{|x-y|\geq\varepsilon\\}}\,d\nu_{n},$

since $u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$ and $v\in
W_{0}^{s}E_{M}(\Omega)$ we have $m(D^{s}u)\in L_{\bar{M}}(\nu_{n})$ and
$D^{s}v\in L_{M}(\nu_{n})$ respectively, therefore by the dominated
convergence theorem we conclude the proof. ∎

By our remarks after the definition of $\mathop{\text{Dom}}(m)$, (2.3), it
follows that

$W^{s}_{0}E_{M}(\Omega)\subset\mathop{\text{Dom}}((-\Delta_{m})^{s})\subset
W^{s}_{0}L_{M}(\Omega).$

Recall now that the monotonicity of $m$ implies that, for any
$a,b\in\mathbb{R}$,

$(m(a)-m(b))(a-b)\geq(m(|a|)-m(|b|))(|a|-|b|)\geq 0,$

from where it follows that the operator $(-\Delta_{m})^{s}$ is monotone. That
is

$\langle(-\Delta_{m})^{s}u-(-\Delta_{m})^{s}v,u-v\rangle\geq 0,\quad\text{for
}u,v\in\mathop{\text{Dom}}((-\Delta_{m})^{s}).$

Another key property of the fractional $m-$laplacian is that it is
pseudomonotone, this is the content of the following theorem.

######  Theorem 3.2.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain that satisfies the
segment property. If
$\\{u_{i}\\}_{i\in\mathbb{N}}\subset\mathop{\text{Dom}}((-\Delta_{m})^{s})$ is
a sequence such that fulfills the conditions

$\begin{cases}&u_{i}\to u\text{ for
}\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))\\\
&(-\Delta_{m})^{s}u_{i}\to f\in W^{-s}L_{\bar{M}}(\Omega)\text{ for
}\sigma(W^{-s}L_{\bar{M}}(\Omega),W_{0}^{s}E_{M}(\Omega))\\\
&\limsup_{i\to\infty}\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle\leq\langle
f,u\rangle\end{cases}$

then

$\begin{cases}&u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})\\\
&(-\Delta_{m})^{s}u=f\\\ &\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle\to\langle
f,u\rangle\text{ if }i\to\infty.\end{cases}$

It will be convenient to introduce the following notation. Given a function
$u$, we denote the sets $\\{R_{j}(u)\\}_{j\in\mathbb{N}}$ as

$R_{j}(u):=\\{(x,y)\in\mathbb{R}^{2n}\colon|(x,y)|\leq j\text{ and
}|D^{s}u(x,y)|\leq j\\}.$

To prove Theorem 3.2 we need first the following two lemmas.

######  Lemma 3.3.

Let $u$ be a function in $W_{0}^{s}L_{M}(\Omega)$ and $v\in
W_{0}^{s}E_{M}(\Omega)$ respectively. If $0<|\lambda|<1$ then for each
$j\in\mathbb{N}$

$m(D^{s}u+\lambda D^{s}v)D^{s}v,\quad m(D^{s}u)D^{s}v\in
L^{1}(R_{j}(u),\nu_{n}).$

Moreover

$\lim_{\lambda\to 0}\iint_{R_{j}(u)}m(D^{s}u+\lambda
D^{s}v)D^{s}v\,d\nu_{n}=\iint_{R_{j}(u)}m(D^{s}u)D^{s}v\,d\nu_{n}.$

###### Proof.

Let $v\in W_{0}^{s}E_{M}(\Omega)$ and $u\in W_{0}^{s}L_{M}(\Omega)$, then
$m(2|D^{s}v|)\in L_{\bar{M}}(\nu_{n})$ and $D^{s}v\in L_{M}(\nu_{n})$ then
there exists constants $k,\tilde{k}>0$ such that

$\iint_{\mathbb{R}^{2n}}\bar{M}\left(\frac{m(2|D^{s}v|)}{k}\right)\,d\nu_{n}<\infty$
(3.1)

and

$\iint_{\mathbb{R}^{2n}}M\left(\frac{|D^{s}v|}{\tilde{k}}\right)\,d\nu_{n}<\infty.$
(3.2)

We observe by using Young’s inequality that

$\begin{split}|m(D^{s}u+\lambda D^{s}v)D^{s}v|&\leq
k\tilde{k}\left\\{\bar{M}\left(\frac{m(|D^{s}u|+|D^{s}v|)}{k}\right)+M\left(\frac{|D^{s}v|}{\tilde{k}}\right)\right\\}\\\
&\leq
k\tilde{k}\left\\{\bar{M}\left(\frac{m(j+|D^{s}v|)}{k}\right)+M\left(\frac{|D^{s}v|}{\tilde{k}}\right)\right\\}\end{split}$
(3.3)

in $R_{j}(u)$. Therefore for each $j$

$\displaystyle\iint_{R_{j}(u)}\left|\bar{M}\left(\frac{m(j+|D^{s}v|)}{k}\right)\right|\,d\nu_{n}=$
$\displaystyle\left\\{\iint_{\\{|D^{s}v|\leq j\\}\cap
R_{j}(u)}+\iint_{\\{|D^{s}v|>j\\}\cap
R_{j}(u)}\right\\}\left|\bar{M}\left(\frac{m(j+|D^{s}v|)}{k}\right)\right|\,d\nu_{n}\leq$
$\displaystyle\bar{M}(m(2j/k))|R_{j}(u)|+\iint_{\\{|D^{s}v|>j\\}\cap
R_{j}(u)}\bar{M}\left(\frac{m(2|D^{s}v|)}{k}\right)\,d\nu_{n}\leq$
$\displaystyle\bar{M}(m(2j/k))|R_{j}(u)|+\iint_{\mathbb{R}^{2n}}\bar{M}\left(\frac{m(2|D^{s}v|)}{k}\right)\,d\nu_{n}<\infty,$

from this inequality together with (3.1)-(3.3) it follows that
$m(D^{s}u+\lambda D^{s}v)D^{s}v\in L^{1}(R_{j}(u),d\nu_{n})$.

Moreover observe that in $R_{j}(u)$,

$|m(D^{s}u)D^{s}v|\leq
m(j)|D^{s}v|\leq\bar{M}(\tilde{k}m(j))+M\left(\frac{|D^{s}v|}{\tilde{k}}\right).$

So using (3.2) it follows that $m(D^{s}u)D^{s}v\in L^{1}(R_{j}(u),\nu_{n})$.

Finally, by using

$|m(D^{s}u+\lambda D^{s}v)D^{s}v|\leq
k\tilde{k}\left\\{\bar{M}\left(\frac{m(j+|D^{s}v|)}{k}\right)+M\left(\frac{|D^{s}v|}{\tilde{k}}\right)\right\\}\in
L^{1}(R_{j}(u),\nu_{n}),$

the fact

$m(D^{s}u+\lambda D^{s}v)D^{s}v\to m(D^{s}u)D^{s}v$

if $\lambda\to 0$ a.e. in $R_{j}(u)$ and the dominated convergence theorem we
conclude the proof. ∎

######  Lemma 3.4.

If there exists $u\in W_{0}^{s}L_{M}(\Omega)$ and $\phi\in
L_{\bar{M}}(\nu_{n})$ such that

$\iint_{R^{2n}}\left(m(W)-\phi\right)(W-D^{s}u)\,d\nu_{n}\geq 0,$ (3.4)

for all $W\in L^{\infty}(\mathbb{R}^{2n},d\nu_{n})$ with compact support then
$m(D^{s}u)=\phi$ in $(W_{0}^{s}L_{M}(\Omega))^{*}$ that is

$\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}v\,d\nu_{n}=\iint_{\mathbb{R}^{2n}}\phi
D^{s}v\,d\nu_{n}\qquad\forall v\in W_{0}^{s}L_{M}(\Omega).$

###### Proof.

Let $w\in W_{0}^{s}L_{M}(\Omega)$ be such that $D^{s}w\in
L^{\infty}(R_{j}(u),\nu_{n})$. For $l\geq j$ we take

$W\equiv
D^{s}w\chi_{R_{j}(u)}-D^{s}u\chi_{R_{j}(u)}+D^{s}u\chi_{R_{l}(u)}=D^{s,j}w-D^{s,j}u+D^{s,l}u,$

now using $W$ in (3.4) we obtain

$\iint_{\mathbb{R}^{2n}}(m(D^{s,j}w-D^{s,j}u+D^{s,l}u)-\phi)((D^{s,j}w-D^{s,j}u+D^{s,l}u)-D^{s}u)d\nu_{n}\geq
0$ (3.5)

The left hand side in the above inequality can be written as

$\displaystyle\iint_{\mathbb{R}^{2n}}(m(D^{s,j}w-D^{s,j}u+D^{s,l}u)-\phi)(D^{s,j}w-D^{s,j}u)\,d\nu_{n}+$
$\displaystyle\iint_{\mathbb{R}^{2n}}m(D^{s,j}w-D^{s,j}u+D^{s,l}u)(D^{s,l}u-D^{s}u)\,d\nu_{n}-$
$\displaystyle\iint_{\mathbb{R}^{2n}}\phi(D^{s,l}u-D^{s}u)\,d\nu_{n}=$
$\displaystyle I+II+III.$

The first integral $I$ is zero outside $R_{j}(u)$, therefore

$\displaystyle I=$
$\displaystyle\iint_{R_{j}(u)}(m(D^{s,j}w-D^{s,j}u+D^{s,l}u)-\phi)(D^{s,j}w-D^{s,j}u)\,d\nu_{n}$
$\displaystyle=$
$\displaystyle\iint_{R_{j}(u)}(m(D^{s}w)-\phi)(D^{s}w-D^{s}u)\,d\nu_{n}.$

For the second integral $II$ we observe that $D^{s,l}u-D^{s}u$ is zero inside
$R_{l}(u)$, and outside $R_{l}(u)$ we have
$m(D^{s,j}w-D^{s,j}u+D^{s,l}u)=m(0)=0$ and so $II=0$. The third integral $III$
goes to zero as $l\to+\infty$. Hence, letting $l\to\infty$ in (3.5), we obtain

$\iint_{R_{j}(u)}(m(D^{s}w)-\phi)(D^{s}w-D^{s}u)\,d\nu_{n}\geq 0,$ (3.6)

$\forall w\in W_{0}^{s}L_{M}(\Omega)$ with $D^{s}w\in
L^{\infty}(R_{j}(u),\nu_{n})$.

Now let $v\in\mathcal{D}(\Omega)$, using (3.6) and Lemma 3.3 with $\lambda>0$
first with $w=u+\lambda v$ and then $w=u-\lambda v$ we have

$\iint_{R_{j}(u)}\left(m(D^{s}u)-\phi\right)D^{s}v\,d\nu_{n}=0\qquad\forall
v\in\mathcal{D}(\Omega),$

taking limit $j\to\infty$ we have

$\iint_{\mathbb{R}^{2n}}\left(m(D^{s}u)-\phi\right)D^{s}v\,d\nu_{n}=0\qquad\forall
v\in\mathcal{D}(\Omega),$

and by density $\forall v\in W_{0}^{s}E_{M}(\Omega)$. Using the density of
$W^{s}_{0}E_{M}(\Omega)$ in $W^{s}_{0}L_{M}(\Omega)$ with respect to the
$\sigma(L_{M}\times L_{M}(\nu_{n}),L_{\bar{M}}\times L_{\bar{M}}(\nu_{n}))$
topology we conclude the proof. ∎

With this preliminaries we are ready to prove the pseudomonotonicity.

###### Proof of Theorem 3.2.

Given a sequence
$\\{u_{i}\\}_{i\in\mathbb{N}}\subset\mathop{\text{Dom}}((-\Delta_{m})^{s})$
such that $u_{i}\to u\in W_{0}^{s}L_{M}(\Omega)$ for
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$,
$(-\Delta_{m})^{s}u_{i}\to f\in W^{-s}L_{\bar{M}}(\Omega)$ for
$\sigma(W^{-s}L_{\bar{M}}(\Omega),W_{0}^{s}E_{M}(\Omega))$ and

$\limsup_{i\to\infty}\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle\leq\langle
f,u\rangle.$ (3.7)

We must prove that
$u\in\mathop{\text{Dom}}((-\Delta_{m})^{s}),(-\Delta_{m})^{s}u=f$ and
$\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle\to\langle f,u\rangle$ if
$i\to\infty$.

We prove first that the sequence $\\{m(D^{s}u_{i})\\}_{i\in\mathbb{N}}$
remains bounded in $L_{\bar{M}}(\nu_{n})$.

Using Young’s inequality we have

$m(D^{s}u_{i})D^{s}u_{i}=\bar{M}(m(D^{s}u_{i}))+M(D^{s}u_{i}),$

then

$\displaystyle\iint_{\mathbb{R}^{2n}}\bar{M}(m(D^{s}u_{i}))\,d\nu_{n}$
$\displaystyle\leq\iint_{\mathbb{R}^{2n}}m(D^{s}u_{i})D^{s}u_{i}\,d\nu_{n}$
$\displaystyle=\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle$
$\displaystyle\leq|\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle|\leq C,$

by (3.7). Therefore the sequence $\\{m(D^{s}u_{i})\\}_{i\in\mathbb{N}}$ is
bounded in $L_{\bar{M}}(\nu_{n})$ hence there exists a subsequence that we
still denote $\\{m(D^{s}u_{i})\\}_{i\in\mathbb{N}}$ and $\phi\in
L_{\bar{M}}(\nu_{n})$ such that $m(D^{s}u_{i})\to\phi$ in
$\sigma(L_{\bar{M}}(\nu_{n}),E_{M}(\nu_{n}))$ combining this fact with the
convergence $(-\Delta_{m})^{s}u_{i}\to f\in W^{-s}L_{\bar{M}}(\Omega)$ for
$\sigma(W^{-s}L_{\bar{M}}(\Omega),W_{0}^{s}E_{M}(\Omega))$ implies that for
any $v\in W_{0}^{s}E_{M}(\Omega)$,

$\displaystyle\langle f,v\rangle$
$\displaystyle=\lim_{i\to\infty}\iint_{\mathbb{R}^{2n}}m(D^{s}u_{i})D^{s}v\,d\nu_{n}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}\phi D^{s}v\,d\nu_{n}.$

This formula togheter with the density of $W^{s}_{0}E_{M}(\Omega)$ in
$W^{s}_{0}L_{M}(\Omega)$ with respect to the $\sigma(L_{M}\times
L_{M}(\nu_{n}),L_{\bar{M}}\times L_{\bar{M}}(\nu_{n}))$ topology allow us to
extend $f$ to the space $W_{0}^{s}L_{M}(\Omega)$.

Let $W\in L^{\infty}(\mathbb{R}^{2n},d\nu_{n})$ with compact support, using
the monotonicity property of $(-\Delta_{m})^{s}$

$\iint_{\mathbb{R}^{2n}}(m(D^{s}u_{i})-m(W))(D^{s}u_{i}-W)\,d\nu_{n}\geq 0.$
(3.8)

All the above discussion allow us to pass to the limit in (3.8) and we obtain

$\iint_{\mathbb{R}^{2n}}(\phi-m(W))(D^{s}u-W)\,d\nu_{n}\geq 0.$

It then follows from Lemma 3.4 that $m(D^{s}u)=\phi$ in
$(W_{0}^{s}L_{M}(\Omega))^{*}$ and so by Young’s inequality

$\displaystyle\iint_{\mathbb{R}^{2n}}\phi D^{s}u\,d\nu_{n}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}u\,d\nu_{n}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}\bar{M}(m(D^{s}u))\,d\nu_{n}+\iint_{\mathbb{R}^{2n}}M(D^{s}u)\,d\nu_{n}$
$\displaystyle\geq\iint_{\mathbb{R}^{2n}}\bar{M}(m(D^{s}u))\,d\nu_{n}$

the above says that $m(D^{s}u)\in L_{\bar{M}}(\nu_{n})$ and
$u\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$. Also

$\langle f,v\rangle=\iint_{\mathbb{R}^{2n}}\phi
D^{s}v\,d\nu_{n}=\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}v\,d\nu_{n}=\langle(-\Delta_{m})^{s}u,v\rangle,$

for all $v\in W_{0}^{s}L_{M}(\Omega)$, then $(-\Delta_{m})^{s}u=f$ in
$(W_{0}^{s}L_{M}(\Omega))^{*}$.

Finally, we prove that $\langle(-\Delta_{m})^{s}u_{i},u_{i}\rangle\to\langle
f,u\rangle$ for $i\to\infty$. Let

$L=\liminf_{i\to\infty}\iint_{\mathbb{R}^{2n}}m(D^{s}u_{i})D^{s}u_{i}\,d\nu_{n},$

we only need to prove that $L\geq\langle f,u\rangle$. Using the monotonicity
property again we have

$\iint_{R^{2n}}(m(D^{s}u_{i})-m(D^{s,j}u))(D^{s}u_{i}-D^{s,j}u)\,d\nu_{n}\geq
0,$

where $D^{s,j}u=D^{s}u\chi_{R_{j}(u)}$ taking limit in $i$ and rewriting we
obtain

$L\geq\iint_{\mathbb{R}^{2n}}m(D^{s,j}u)(D^{s}u-D^{s,j}u)\,d\nu_{n}+\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s,j}u\,d\nu_{n}.$

In $R_{j}(u)$ the factor $D^{s}u-D^{s,j}u=0$, and in $(R_{j}(u))^{c}$ the
factor $m(D^{s,j}u)=m(0)=0$ so the first integral in the above inequality is
zero, then

$L\geq\iint_{R_{j}(u)}m(D^{s}u)D^{s}u\,d\nu_{n}$

for arbitrary $j$ therefore

$L\geq\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}u\,d\nu_{n}=\langle f,u\rangle.$

This concludes the proof of the theorem. ∎

##  4\. The eigenvalue problem

In this section we study the main result of the paper, namely the existence of
a sequence $\\{(\lambda_{k},u_{k})\\}_{k\in\mathbb{N}}$ of eigenpairs of the
equation (1.2) and, moreover, $\lambda_{k}\to\infty$ as $k\to\infty$.

We say that $(\lambda,u)$ is an eigenpair of (1.2) if

$\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}\phi\,d\nu_{n}=\lambda\int_{\Omega}g(u)\phi\,dx,$
(4.1)

for every $\phi\in\mathcal{D}(\Omega)$, provided that both integrals are
defined.

The strategy of the proof is to apply the Ljusternik-Schnirelmann method that
have been proved to be succesful in previous works. However, the lack of
reflexivity of the spaces involved prevent us to apply directly the
Ljusternik-Schnirelmann method. This fact was already observe by [35] where
the author is able to reduce the problem to a finite dimensional one and then
pass to the limit. Here we apply the same idea to the context of fractional
order spaces.

In fact after the work performed in the previous sections the ideas of [35]
can be applied almost straightforward. Howewer we include the details for the
reader convenience and to make the paper selfcontained.

Therefore we look for the existence of a sequence
$(\lambda_{k},u_{k})\subset\mathbb{R}\times W^{s}_{0}E_{M}(\Omega)$ of
eigenpairs for the problem (1.2). Observe that, since $u_{k}\in
W^{s}_{0}E_{M}(\Omega)\subset\mathop{\text{Dom}}((-\Delta_{m})^{s})$,
$m(D^{s}u)\in L_{\bar{M}}(\nu_{n})$. Hence the left hand side of (4.1) is well
defined forall $\phi\in W_{0}^{s}E_{M}(\Omega)$.

On the function $g\colon\mathbb{R}\to\mathbb{R}$ we assume that is an odd and
continuous function satisfying $g(t)t>0$ for all $t\neq 0$ and

$|g(t)|\leq a_{1}+a_{2}m(a_{3}t)\qquad\text{ for all }t\geq 0,$ (4.2)

where $a_{1},a_{2}$ and $a_{3}$ are positive constants. Observe that if $u\in
W_{0}^{s}E_{M}(\Omega)$ the right hand side of (4.1) is well defined.

Therefore, the main result of this paper reads as follows:

######  Theorem 4.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open and bounded domain with the
segment property. Then there exists a sequence of eigenpairs
$\\{(\lambda_{k},u_{k})\\}_{k\in\mathbb{N}}\subset\mathbb{R}_{+}\times
W_{0}^{s}L_{M}(\Omega)$ of (1.2). Moreover, $\lambda_{k}\to+\infty$ and
$u_{k}\to 0$ in the topology
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ as $k\to\infty$.

We consider the even functionals $\mathcal{M}_{s}\colon
D_{\mathcal{M}_{s}}\to\mathbb{R}$ and $\mathcal{G}\colon
D_{\mathcal{G}}\to\mathbb{R}$ defined as

$\displaystyle\mathcal{M}_{s}(u)$
$\displaystyle:=\iint_{\mathbb{R}^{2n}}M(D^{s}u)\,d\nu_{n}$ (4.3)
$\displaystyle\mathcal{G}(u)$ $\displaystyle:=\int_{\Omega}G(u)\,dx,$ (4.4)

where $G(t)=\int_{0}^{|t|}g(\tau)\,d\tau$ and

$D_{\mathcal{M}_{s}}:=\\{u\in
W^{s}_{0}L_{M}(\Omega)\colon\mathcal{M}(u)<\infty\\}\quad\text{and}\quad
D_{\mathcal{G}}:=\\{u\in
W^{s}_{0}L_{M}(\Omega)\colon\mathcal{G}(u)<\infty\\}.$

It is clear that
$W_{0}^{s}E_{M}(\Omega)\subset\mathop{\text{Dom}}((-\Delta_{m})^{s})\subset
D_{\mathcal{M}_{s}}\subset W_{0}^{s}L_{M}(\Omega)$ and both functionals
$\mathcal{M}_{s}$ and $\mathcal{G}$ vanish only at zero.

Let $B=\\{\phi_{1},\phi_{2},\dots\\}\subset{\mathcal{D}}(\Omega)$ be a
countable linearly independent subset in $W_{0}^{s}E_{M}(\Omega)$ such that
the linear hull is norm dense, and we define the sets $V$ and $V_{k}$ as the
linear hull of the sets $B$ and
$B_{k}:=\\{\phi_{1},\phi_{2},\dots,\phi_{k}\\}$ respectively. We denote the
continuous pairing between $W^{-s}L_{\bar{M}}(\Omega)$ and
$W^{s}_{0}E_{M}(\Omega)$ by $\langle\cdot,\cdot\rangle$, and the one between
$V_{k}$ and $(V_{k})^{*}$ by $\langle\cdot,\cdot\rangle_{k}$.

A straightforward calculation gives $\mathcal{M}_{s},\mathcal{G}\in
C^{1}(V_{k})$ and

$\displaystyle\langle\mathcal{M}_{s}^{\prime}(u),v\rangle_{k}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}m(D^{s}u)D^{s}v\,d\nu_{n}\qquad$
$\displaystyle\text{for all }u,v\in V_{k}$
$\displaystyle\langle\mathcal{G}^{\prime}(u),v\rangle_{k}$
$\displaystyle=\int_{\Omega}g(u)v\,dx\qquad$ $\displaystyle\text{for all
}u,v\in V_{k},$

for each $k=1,2,\dots$

By the growth condition (4.2) and the compact immersion
$W^{s}_{0}L_{M}(\Omega)\subset\subset E_{M}(\Omega)$ (see Theorem B.3), it
follows that $D_{\mathcal{G}}=W_{0}^{s}L_{M}(\Omega)$ and

$\displaystyle\mathcal{G}(u_{k})$ $\displaystyle\to\mathcal{G}(u)$
$\displaystyle\langle\mathcal{G}^{\prime}(u_{k}),u_{k}\rangle$
$\displaystyle\to\int_{\Omega}g(u)u\,dx$
$\displaystyle\langle\mathcal{G}^{\prime}(u_{k}),v\rangle$
$\displaystyle\to\int_{\Omega}g(u)v\,dx,$

whenever $u_{k}\in V_{k},u_{k}\to u\in W_{0}^{s}L_{M}(\Omega)$ in
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega)),$ and $v\in V$.

A first step to attack the problem (1.2) is to consider the following problem
on the finite dimensional space $V_{k}$

$\begin{split}&\mathcal{M}_{s}^{{}^{\prime}}(u)=\lambda\mathcal{G}^{\prime}(u)\text{
in }V_{k}^{*}\\\ &u\in\mathcal{N}_{k}^{s},\lambda\in\mathbb{R},\end{split}$
(4.5)

where $\mathcal{N}_{k}^{s}=\\{u\in V_{k}\colon\mathcal{M}_{s}(u)=1\\}$.

Now, in order to study the problem (4.5), we use the following notation

$\displaystyle\mathcal{N}^{s}$ $\displaystyle=\\{u\in
W_{0}^{s}E_{M}(\Omega)\colon\mathcal{M}_{s}(u)=1\\}$
$\displaystyle\mathcal{K}_{i}^{s}$
$\displaystyle=\\{K\subset\mathcal{N}^{s}\text{ compact and symmetric }\colon
gen(K)\geq i\\}$ $\displaystyle\mathcal{K}_{i,k}^{s}$
$\displaystyle=\\{K\subset\mathcal{N}_{k}^{s}\text{ compact and symmetric
}\colon gen(K)\geq i\\}$ $\displaystyle c_{i}$
$\displaystyle=\sup_{K\in\mathcal{K}_{i}^{s}}\inf_{u\in K}\mathcal{G}(u)$
$\displaystyle c_{i,k}$
$\displaystyle=\sup_{K\in\mathcal{K}_{i,k}^{s}}\inf_{u\in K}\mathcal{G}(u),$

observe that the critical levels $c_{i,k}$ are increasing in $k$ and choosing
$S_{i}$ to be the unit sphere of $V_{i}$, we have that $gen(S_{i})=i$ and
$\inf_{u\in S_{i}}\mathcal{G}(u)>0$ therefore $c_{i,k}>0$ when $i\leq k$. If
$i>k$ then $\mathcal{K}_{i,k}^{s}$ are empty and so $c_{i,k}=0$. The next
lemma provides solution for the problem (4.5).

######  Lemma 4.2.

Let $i\in\mathbb{N}$ be given. Then there exist sequences
$\\{u_{k}\\}_{k=i}^{\infty}\subset W_{0}^{s}E_{M}(\Omega)$, and
$\\{\lambda_{k}\\}_{k=i}^{\infty}\subset(0,+\infty)$ such that

$\displaystyle u_{k}$ $\displaystyle\in\mathcal{N}_{k}^{s}\subset V_{k}$
$\displaystyle\mathcal{M}_{s}^{{}^{\prime}}(u_{k})$
$\displaystyle=\lambda_{k}\mathcal{G}^{\prime}(u_{k})\qquad\text{in
}V_{k}^{*}$ $\displaystyle\mathcal{G}(u_{k})$ $\displaystyle=c_{i,k},$

for all $k=i,i+1,\dots$

###### Proof.

As we observe above for each $k$ we have $\mathcal{M}_{s},\mathcal{G}\in
C^{1}(V_{k})$ and $\mathcal{G}(0)=0$.

On the other hand let $u\in V_{k}$ with $u\neq 0$ putting
$r(u)=1/\|D^{s}u\|_{M,\nu_{n}}$ we have

$\iint_{\mathbb{R}^{2n}}M(r(u)D^{s}u)\,d\nu_{n}=1.$

Now from the finite dimensional Ljusternik-Schnirelmann theory ([36, Theorem 2
and Corollary 7.1]) the lemma follows. ∎

We are interested in the study of the asymptotic behavior of the sequences
$\\{u_{k}\\}_{k}$ and $\\{\lambda_{k}\\}_{k}$ in $k$ with fixed $i$. The
following tools are needed first.

######  Lemma 4.3.

Let $G_{k}\colon\mathbb{R}^{2n}\to\mathbb{R}$ be a sequence of functions such
that $G_{k}\to G$ $\nu_{n}-$a.e. for some function
$G\colon\mathbb{R}^{2n}\to\mathbb{R}$. Suppose that there exists a sequence of
functions $\\{\Phi_{k}\\}_{k}$ in $L^{1}(\mathbb{R}^{2n},\nu_{n})$ such that

$|G_{k}|\leq\Phi_{k}\quad\nu_{n}-\text{a.e. in }\mathbb{R}^{2n}$ (4.6)

$\Phi_{k}\to\Phi$ $\nu_{n}-$a.e. in $\mathbb{R}^{2n}$ and

$\iint_{\mathbb{R}^{2n}}\Phi_{k}\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n}$

for some function $\Phi\in L^{1}(\mathbb{R}^{2n},\nu_{n})$. Then

$\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}.$

###### Proof.

The condition (4.6) implies $G_{k}+\Phi_{k},\Phi_{k}-G_{k}\geq 0$ a.e. in
$\mathbb{R}^{2n}$, and by Fatou’s lemma we obtain

$\displaystyle\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}+\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n}$
$\displaystyle=\iint_{\mathbb{R}^{2n}}(G+\Phi)\,d\nu_{n}$
$\displaystyle\leq\liminf_{k\to\infty}\iint_{\mathbb{R}^{2n}}(G_{k}+\Phi_{k})\,d\nu_{n}$
$\displaystyle\leq\liminf_{k\to\infty}\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}+\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n},$

from where we obtain

$\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}\leq\liminf_{k\to\infty}\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}.$

In similar way from $\Phi_{k}-G_{k}\geq 0$ a.e. in $\mathbb{R}^{2n}$ we obtain

$\displaystyle\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n}-\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}$
$\displaystyle\leq\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n}+\liminf_{k\to\infty}\left(-\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}\right)$
$\displaystyle=\iint_{\mathbb{R}^{2n}}\Phi\,d\nu_{n}-\limsup_{k\to\infty}\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n},$

and so

$\limsup_{k\to\infty}\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}\leq\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}.$

The proof is completed. ∎

######  Lemma 4.4.

Let $G_{k}\colon\mathbb{R}^{2n}\to\mathbb{R}$ be a sequence of a nonnegative
functions in $L^{1}(\mathbb{R}^{2n},\nu_{n})$ and $G\in
L^{1}(\mathbb{R}^{2n},\nu_{n})$ such that $G_{k}\to G$ $\nu_{n}-$a.e. in
$\mathbb{R}^{2n}$ and

$\iint_{\mathbb{R}^{2n}}G_{k}\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}G\,d\nu_{n}.$

Then $G_{k}\to G$ in $L^{1}(\mathbb{R}^{2n},\nu_{n})$.

###### Proof.

If we define $\Phi_{k}:=G_{k}+G$ then $\Phi_{k}\in
L^{1}(\mathbb{R}^{2n},\nu_{n}),\Phi_{k}\to 2G$ a.e. in $\mathbb{R}^{2n}$ and

$\iint_{\mathbb{R}^{2n}}\Phi_{k}\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}2G\,d\nu_{n}.$

We observe that $|G_{k}-G|\leq\Phi_{k}$ a.e. in $\mathbb{R}^{2n}$, therefore
by Lemma 4.3

$\iint_{\mathbb{R}^{2n}}|G_{k}-G|\,d\nu_{n}\to 0,$

and so $G_{k}\to G$ in $L^{1}(\mathbb{R}^{2n},\nu_{n})$. ∎

We are in position to give the asymptotic behavior of the sequence
$\\{(\lambda_{k},u_{k})\\}_{k}$ .

######  Theorem 4.5.

Let $i\in\mathbb{N}$ be fixed and the sequences $\\{u_{k}\\}_{k=i}^{\infty}\in
W_{0}^{s}E_{M}(\Omega)$ and $\\{\lambda_{k}\\}_{k=i}^{\infty}\in(0,+\infty)$
as given by Lemma 4.2. Then there exists
$\bar{u}_{i}\in\mathop{\text{Dom}}((-\Delta_{m})^{s})$ and
$\bar{\lambda}_{i}\in(0,+\infty)$ such that up to a subsequence,
$\lambda_{k}\to\bar{\lambda}_{i}$ and $u_{k}\to\bar{u}_{i}$ for
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$. Moreover,
$\mathcal{M}_{s}(\bar{u}_{i})=1,\mathcal{G}(\bar{u}_{i})=\lim_{k\to\infty}c_{i,k}$,
and $(\bar{\lambda}_{i},\bar{u}_{i})$ is an eigenpair for (1.2).

###### Proof.

Using that

$\iint_{\mathbb{R}^{2n}}M(D^{s}u_{k})\,d\nu_{n}=1,$

for all $k\geq i$ we have that the sequence $\\{u_{k}\\}_{k}$ is bounded in
$W_{0}^{s}L_{M}(\Omega)$ therefore there exists $\bar{u}_{i}\in
W_{0}^{s}L_{M}(\Omega)$ such that $u_{k}\to\bar{u}_{i}$ in
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ for a subsequence
then $\mathcal{G}(u_{k})\to\mathcal{G}(\bar{u}_{i})$ and so by Lemma 4.2

$\mathcal{G}(\bar{u}_{i})=\lim_{k\to\infty}\mathcal{G}(u_{k})=\lim_{k\to\infty}c_{i,k},$

since $\\{c_{i,k}\\}_{k}$ are increasing and positive we have
$\mathcal{G}(\bar{u}_{i})>0$ implying $g(\bar{u}_{i})\not\equiv 0$ (and
therefore $\bar{u}_{i}\not\equiv 0$). Then there exists $k_{0}>i$ and a
function $\phi\in V_{k_{0}}$ such that

$\int_{\Omega}g(\bar{u}_{i})(\bar{u}_{i}-\phi)\,dx<0.$

Now assume that $\lambda_{k}\to\infty$, when $k\to\infty$. Using the
monotonicity of the operator $(-\Delta_{m})^{s}$ for $k\geq k_{0}$

$\iint_{\mathbb{R}^{2n}}(m(D^{s}u_{k})-m(D^{s}\phi))(D^{s}u_{k}-D^{s}\phi)\,d\nu_{n}\geq
0,$

the above inequality and Lemma 4.2 produces

$\displaystyle\iint_{\mathbb{R}^{2n}}m(D^{s}\phi)(D^{s}u_{k}-D^{s}\phi)\,d\nu_{n}$
$\displaystyle\leq\iint_{\mathbb{R}^{2n}}m(D^{s}u_{k})(D^{s}u_{k}-D^{s}\phi)\,d\nu_{n}$
$\displaystyle=\lambda_{k}\int_{\Omega}g(u_{k})(u_{k}-\phi)\,dx,$

since

$\int_{\Omega}g(u_{k})(u_{k}-\phi)\,dx\to\int_{\Omega}g(\bar{u}_{i})(\bar{u}_{i}-\phi)\,dx<0$

and

$\iint_{\mathbb{R}^{2n}}m(D^{s}\phi)(D^{s}u_{k}-D^{s}\phi)\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}m(D^{s}\phi)(D^{s}\bar{u}_{i}-D^{s}\phi)\,d\nu_{n},$

we have a contradiction and so the sequence $\\{\lambda_{k}\\}$ is bounded.
Therefore we may assume for a common subsequence that
$\lambda_{k}\to\bar{\lambda}_{i}$ in $\mathbb{R}$ and $u_{k}\to\bar{u}_{i}$
for $\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$.

Recall that Young’s inequality implies that $\bar{M}(m(t))=m(t)t-M(t)\leq
m(t)t$ forall $t\in\mathbb{R}$. Therefore

$\displaystyle\iint_{\mathbb{R}^{2n}}\bar{M}(m(D^{s}u_{k}))\,d\nu_{n}$
$\displaystyle\leq\iint_{\mathbb{R}^{2n}}m(D^{s}u_{k})D^{s}u_{k}\,\nu_{n}$
$\displaystyle=\lambda_{k}\int_{\Omega}g(u_{k})u_{k}\,dx$
$\displaystyle\to\bar{\lambda}_{i}\int_{\Omega}g(\bar{u}_{i})\bar{u}_{i}\,dx<\infty,$

where we used Lemma 4.2. Then the sequence $\\{m(D^{s}u_{k})\\}$ is bounded in
$L_{\bar{M}}(\nu_{n})$ and there exists $F\in L_{\bar{M}}(\nu_{n})$ such that
$m(D^{s}u_{k})\to F$ in $\sigma(L_{\bar{M}}(\nu_{n}),E_{M}(\nu_{n}))$
therefore $(-\Delta_{m})^{s}u_{k}\to f=\mathop{\text{div}}^{s}F\in
W^{-s}L_{\bar{M}}(\Omega)$ in
$\sigma(W^{-s}L_{\bar{M}}(\Omega),W_{0}^{s}E_{M}(\Omega))$. For any $\phi\in
V$ we have

$\displaystyle\langle f,\phi\rangle$
$\displaystyle=\lim_{k\to\infty}\langle(-\Delta_{m})^{s}u_{k},\phi\rangle$
$\displaystyle=\lim_{k\to\infty}\iint_{\mathbb{R}^{2n}}m(D^{s}u_{k})D^{s}\phi\,d\nu_{n}$
$\displaystyle=\lim_{k\to\infty}\lambda_{k}\int_{\Omega}g(u_{k})\phi\,dx$
$\displaystyle=\bar{\lambda}_{i}\int_{\Omega}g(\bar{u}_{i})\phi\,dx.$

Since $V$ is norm dense in $W_{0}^{s}E_{M}(\Omega)$ and
$W_{0}^{s}E_{M}(\Omega)$ is dense in $W_{0}^{s}L_{M}(\Omega)$ in
$\sigma(L_{M}\times L_{M}(\nu_{n}),L_{\bar{M}}\times L_{\bar{M}}(\nu_{n}))$ it
follows that

$\langle
f,\phi\rangle=\bar{\lambda}_{i}\int_{\Omega}g(\bar{u}_{i})\phi\,dx\qquad\forall\phi\in
W_{0}^{s}L_{M}(\Omega).$ (4.7)

Also we have

$\displaystyle\lim_{k\to\infty}\langle(-\Delta_{m})^{s}u_{k},u_{k}\rangle$
$\displaystyle=\lim_{k\to\infty}\iint_{\mathbb{R}^{2n}}m(D^{s}u_{k})D^{s}u_{k}\,d\nu_{n}$
$\displaystyle=\lim_{k\to\infty}\lambda_{k}\int_{\Omega}g(u_{k})u_{k}\,dx$
$\displaystyle=\bar{\lambda}_{i}\int_{\Omega}g(\bar{u}_{i})\bar{u}_{i}\,dx$
$\displaystyle=\langle f,\bar{u}_{i}\rangle.$

Now we are in position to apply the pseudomonotonicity property of
$(-\Delta_{m})^{s}$ (Theorem 3.2) and conclude that
$\bar{u}_{i}\in\mathop{\text{Dom}}((-\Delta_{m})^{s}),(-\Delta_{m})^{s}\bar{u}_{i}=f$
and

$\iint_{\mathbb{R}^{2n}}m(D^{s}u_{k})D^{s}u_{k}\,d\nu_{n}\to\iint_{\mathbb{R}^{2n}}m(D^{s}\bar{u}_{i})D^{s}\bar{u}_{i}\,d\nu_{n}.$
(4.8)

Therefore by (4.7)

$\iint_{\mathbb{R}^{2n}}m(D^{s}\bar{u}_{i})D^{s}\phi\,d\nu_{n}=\bar{\lambda}_{i}\int_{\Omega}g(\bar{u}_{i})\phi\,dx\qquad\forall\phi\in
W_{0}^{s}L_{M}(\Omega),$

that is $(\bar{\lambda}_{i},\bar{u}_{i})$ is an eigenpair for the problem
(1.2). Observe that the above implies $\bar{\lambda}_{i}>0$.

On the other hand using Lemma 4.4 and (4.8) we have that
$m(D^{s}u_{k})D^{s}u_{k}\to m(D^{s}\bar{u}_{i})D^{s}\bar{u}_{i}$ in
$L^{1}(\mathbb{R}^{2n},\nu_{n})$ and so there exists a majorant integrable
$h\in L^{1}(\mathbb{R}^{2n},\nu_{n})$ such that $m(D^{s}u_{k})|D^{s}u_{k}|\leq
h$ $\nu_{n}-$a.e. in $\mathbb{R}^{2n}$ ([8, Theorem 4.9]), then by Young’s
inequality we have

$\displaystyle M(D^{s}u_{k})$
$\displaystyle\leq\bar{M}(m(D^{s}u_{k}))+M(D^{s}u_{k})$
$\displaystyle=m(D^{s}u_{k})|D^{s}u_{k}|$ $\displaystyle\leq h,$

$\nu_{n}-$a.e. in $\mathbb{R}^{2n}$. By the compact immersion
$W^{s}_{0}L_{M}(\Omega)\subset\subset E_{M}(\Omega)$ we can assume for a
subsequence that $u_{k}\to\bar{u}_{i}$ a.e. in $\Omega$, and extending
$\bar{u}_{i}=0$ in $\Omega^{c}$ we have $D^{s}u_{k}\to D^{s}\bar{u}_{i}$
$\nu_{n}-$a.e. in $\mathbb{R}^{2n}$, and consequently $M(D^{s}u_{k})\to
M(D^{s}\bar{u}_{i})$ $\nu_{n}-$a.e. in $\mathbb{R}^{2n}$. Therefore by the
dominated convergence theorem

$\mathcal{M}_{s}(\bar{u}_{i})=\lim_{k\to\infty}\iint_{\mathbb{R}^{2n}}M(D^{s}u_{k})\,d\nu_{n}=1.$

This finishes the proof of the theorem. ∎

In order to complete the proof of the main theorem we need to analize the
asymptotic behavior of the eigenvalues $\bar{\lambda}_{i}$ as $i\to\infty$. In
order to complet this fact we first study the behavior of the constants
$c_{i,k}$ and $c_{i}$. This is the content of the next two lemmas. The proofs
of this lemmas follow the same ideas of the lemmas 4.3 and 4.4 in [35].

######  Lemma 4.6.

Let $i\in\mathbb{N}$ be fixed. Then $c_{i,k}\to c_{i}$ as $k\to\infty$.

###### Proof.

Using the definition of the constants $c_{i,k}$ we have $c_{i,k}\leq
c_{i,k+1}\leq\cdots\leq c_{i}$ for all $k\in\mathbb{N}$. We argue by
contradiction, suppose that there exists $\varepsilon>0$ such that
$c_{i,k}<c_{i}-\varepsilon$ for all $k\in\mathbb{N}$. By the definition of
$c_{i}$ and the supremum property there exists
$K_{\varepsilon}\in\mathcal{K}_{i,k}^{s}$ such that

$c_{i}-\varepsilon/2<\inf_{w\in K_{\varepsilon}}\mathcal{G}(w).$ (4.9)

We consider now the mappings $P_{k}\colon W_{0}^{s}E_{M}(\Omega)\to
W_{0}^{s}E_{M}(\Omega)$ given by Theorem 2.2. It is easy to check that
$0\notin P_{k}(K_{\varepsilon})$ for every $k$ large enough. Indeed, suppose
$P_{k_{j}}(w_{k_{j}})=0$ with $w_{k_{j}}\in K_{\varepsilon}$ and
$k_{j}\to\infty$. By compactness of the set $K_{\varepsilon}$, $w_{k_{j}}\to
w\in K_{\varepsilon}$ for a subsequence. Theorem 2.2 implies $w=0$ wich
contradicts $0\notin K_{\varepsilon}$. Therefore the map $\Psi_{k}\colon
K_{\varepsilon}\to\mathcal{N}^{s}\cap V_{m_{k}}$ defined as

$\Psi_{k}(w)=\frac{P_{k}(w)}{\|P_{k}(w)\|_{s,M}}$

is odd and continuous for $k$ large enough. Hence
$\Psi_{k}(K_{\varepsilon})\subset\mathcal{K}_{i,m_{k}}^{s}$ implying

$\inf_{w\in\Psi_{k}(K_{\varepsilon})}\mathcal{G}(w)\leq c_{i,m_{k}},$

for every $k$ large enough. Thus, for every $k$ large enough there exists
$w_{k}\in K_{\varepsilon}$ such that

$\mathcal{G}(\Psi_{k}(w_{k}))<c_{i}-\varepsilon.$ (4.10)

by the compactness we have $w_{k}\to w\in K_{\varepsilon}$ in
$W_{0}^{s}E_{M}(\Omega)$ for a subsequence implying, again by Theorem 2.2,
that $\|P_{k}(w_{k})\|_{s,M}\to\|w\|_{s,M}=1$ therefore

$\Psi_{k}(w_{k})\to w\qquad\text{for
}\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega)),$

and so $\mathcal{G}(\Psi_{k}(w_{k}))\to\mathcal{G}(w)$, which contradicts
(4.9) and (4.10). ∎

######  Lemma 4.7.

$c_{i}\to 0$ as $i\to\infty$.

###### Proof.

Let $\varepsilon>0$ be arbitraty and $\\{P_{k}\\}_{k\in\mathbb{N}}$ the
mappings given by Theorem 2.2. The continuity properties of the mappings
$P_{k}$ and $\mathcal{G}$ imply the existence of $k_{0}\in\mathbb{N}$ and
$\delta>0$ such that

$|\mathcal{G}(P_{k_{0}}(w))-\mathcal{G}(w)|<\varepsilon/2\qquad\text{ for all
}w\in\mathcal{N}^{s}$

and

$\mathcal{G}(w)<\varepsilon/2\qquad\text{ for all }\|w\|_{M,s}\leq\delta.$

Therefore if $K\subset\mathcal{N}^{s}$ is compact and symmetric with
$\inf_{w\in K}\mathcal{G}(w)>\varepsilon$ then

$\|P_{k_{0}}(w)\|_{M,s}\geq\delta\qquad\text{ for all }w\in K.$

Hence $gen(K)\leq gen(P_{k_{0}}(w))\leq m_{k_{0}}$ for some
$m_{k_{0}}\in\mathbb{N}$. Thus, if $i>m_{k_{0}}$ and $K\in\mathcal{K}_{i}^{s}$
then

$\inf_{w\in K}\mathcal{G}(w)\leq\varepsilon,$

and recalling that

$c_{i}=\sup_{K\in\mathcal{K}_{i}^{s}}\inf_{u\in K}\mathcal{G}(u)$

we have $c_{i}\leq\varepsilon$ for $i>m_{k_{0}}$. ∎

Now we are ready to complete the proof of the main result, namely Theorem 4.1.

######  Theorem 4.8.

With the previous assumptions and notations, $\bar{\lambda}_{i}\to+\infty$ and
$\bar{u}_{i}\to 0$ for
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ as $i\to\infty$.

###### Proof.

Using Theorem 4.5, Lemma 4.6 and Lemma 4.7 we obtain that
$\mathcal{G}(\bar{u}_{i})\to 0$ as $i\to\infty$. On the other by Theorem 4.5
we can assume that $\bar{u}_{i}\to\tilde{u}\in W_{0}^{s}L_{M}(\Omega)$ for
$\sigma(W_{0}^{s}L_{M}(\Omega),W^{-s}E_{\bar{M}}(\Omega))$ as $i\to\infty$.
From these facts we easily deduce that $\tilde{u}=0$.

Now taking $\bar{u}_{i}$ as test function in (4.1) we obtain

$\bar{\lambda}_{i}=\frac{\iint_{\mathbb{R}^{2n}}m(D^{s}\bar{u}_{i})D^{s}\bar{u}_{i}\,d\nu_{n}}{\int_{\Omega}g(\bar{u}_{i})\bar{u}_{i}\,dx}\geq\frac{1}{\int_{\Omega}g(\bar{u}_{i})\bar{u}_{i}\,dx}.$

Finally by the compactness of the embedding $\bar{u}_{i}\to 0$ in $E_{M}$ as
$i\to\infty$ from where it follows that $\bar{\lambda}_{i}\to\infty$ as
$i\to\infty$. ∎

##  Appendix A Density results

In this Section, we prove some density results regarding the fractional order
Orlicz-Sobolev spaces that are needed in this work. We mention that these
results follow the same approach as the analog ones for the classical Orlicz-
Sobolev spaces proved in [24]. So we only sketch the arguments, including some
detail where the differences arise.

One of the key properties that we assume on the domain $\Omega$ is that it
satisfies the so called segment property.

######  Definition A.1.

Let $\Omega\subset\mathbb{R}^{n}$ be an open and bounded domain. We say that
$\Omega$ has the segment property if there exists a locally finite open
covering of $\partial\Omega$ with balls $\\{B_{t}(x_{j})\\}$ centered in
$x_{j}\in\partial\Omega$ with radius $t$, a corresponding sequence of units
vectors $n_{j}$, and a number $t^{*}\in(0,1)$ such that

$x\in\bar{\Omega}\cap B_{t}(x_{j})\Longrightarrow x+tn_{j}\in\Omega\text{ for
all }t\in(0,t^{*}).$

This condition about the domain $\Omega$ says, in some sense, that the domain
lie locally on one side of its boundary. Observe that the segment property
does not impply any smoothness on the boundary $\partial\Omega$. Conversely,
if a domain is of class $C^{1}$ does not imply that the domain satisfy the
segment property. See Section 8.1 in [11] for more details.

The main result in this section is the following.

######  Theorem A.2.

Assume that $\Omega\subset\mathbb{R}^{n}$ has the segment property. Then
${\mathcal{D}}(\Omega)$ is dense in $W^{s}_{0}L_{M}(\Omega)$ with respect to
the $\sigma(W^{s}_{0}L_{M}(\Omega),W^{-s}L_{\bar{M}}(\Omega))$ topology.

In the rest of the paper we allways assume that the domain $\Omega$ satisfies
the segment property.

For the proof of Theorem A.2 we use a series of lemmas from [24].

######  Lemma A.3 ([24], Lemma 1.4).

Let $u_{k}\in\mathcal{L}_{M}(\mathbb{R}^{N},d\mu)$ be such that $u_{k}\to u$
$\mu-$a.e. in $\mathbb{R}^{N}$ and $M(u_{k})\leq w_{k}$ $\mu-$a.e. in
$\mathbb{R}^{N}$ where $w_{k}\to w$ in $L^{1}(\mathbb{R}^{N},d\mu)$. Then
$u\in\mathcal{L}_{M}(\mathbb{R}^{N},d\mu)$ and $u_{k}\to u$ in the
$\sigma(L_{M}(\mathbb{R}^{N},d\mu),L_{\bar{M}}(\mathbb{R}^{N},d\mu))$
topology.

######  Lemma A.4 ([24], Lemma 1.5).

Let $u\in L_{M}(\mathbb{R}^{N},d\mu)$ and denote by $u_{t}$ the translated
function: $u_{t}(x)=u(x-t)$. Then, $u_{t}\to u$ in the
$\sigma(L_{M}(\mathbb{R}^{N},d\mu),L_{\bar{M}}(\mathbb{R}^{N},d\mu))$ topology
as $t\to 0$.

######  Lemma A.5 ([24], Lemma 1.6).

Let $u\in L_{M}(\mathbb{R}^{n},dx)$ and denote by $u_{\varepsilon}$ the
regularized function: $u_{\varepsilon}=u*\rho_{\varepsilon}$, where
$\rho_{\varepsilon}(x)=\varepsilon^{-n}\rho(x/\varepsilon)$ is the standard
mollifier. Then $u_{\varepsilon}\to u$ in the
$\sigma(L_{M}(\mathbb{R}^{n},dx),L_{\bar{M}}(\mathbb{R}^{n},dx))$ topology as
$\varepsilon\to 0$.

Now, we need a modification of Lemma A.5 to deal with functions $F\in
L_{M}(\mathbb{R}^{2n},d\nu_{n})$.

######  Lemma A.6.

Let $F\in L_{M}(\mathbb{R}^{2n},d\nu_{n})$ and denote by $F_{\varepsilon}$

$F_{\varepsilon}(x,y)=\int_{\mathbb{R}^{n}}F(x-z,y-z)\rho_{\varepsilon}(z)\,dz,$

where $\rho_{\varepsilon}(z)=\varepsilon^{-n}\rho(z/\varepsilon)$ is the
standard mollifier. Then, if $F$ has compact support, $F_{\varepsilon}\to F$
as $\varepsilon\to 0$ in the
$\sigma(L_{M}(\mathbb{R}^{2n},d\nu_{n}),L_{\bar{M}}(\mathbb{R}^{2n},d\nu_{n}))$
topology.

###### Proof.

The proof of this lemma is a modification of Lemma A.5. In fact, first observe
that it is enough to prove the lemma in the case where
$F\in\mathcal{L}_{M}(\mathbb{R}^{2n},d\nu_{n})$. Then, using Jensen’s
inequality, one can easily verify that

$M(F_{\varepsilon})\leq
M(F)_{\varepsilon}=\int_{\mathbb{R}^{n}}M(F(x-z,y-z))\rho_{\varepsilon}(z)\,dz.$

So, to finish the proof it remains to see that $M(F)_{\varepsilon}\to M(F)$ in
$L^{1}(\mathbb{R}^{2n},d\nu_{n})$, that $F_{\varepsilon}\to F$ $\nu_{n}-$a.e.
and apply Lemma A.3.

First, observe that if $G\in L^{1}(\mathbb{R}^{2n},d\nu_{n})$ then
$G_{\varepsilon}\to G$ in $L^{1}(\mathbb{R}^{2n},d\nu_{n})$. In fact,

$\displaystyle\iint_{\mathbb{R}^{2n}}|G_{\varepsilon}-G|\,d\nu_{n}$
$\displaystyle\leq\iint_{\mathbb{R}^{2n}}|G(x-z,y-z)-G(x,y)|\rho_{\varepsilon}(z)\,dz\,d\nu_{n}(x,y)$
$\displaystyle=\int_{|z|\leq\varepsilon}\rho_{\varepsilon}(z)\left(\iint_{\mathbb{R}^{2n}}|G(x-z,y-z)-G(x,y)|\,d\nu_{n}(x,y)\right)\,dz.$

From this inequality, the result follows by the continuity of the
$L^{1}-$norm.

Applying this result to $G=M(F)_{\varepsilon}$ gives that
$M(F)_{\varepsilon}\to M(F)$ in $L^{1}(\mathbb{R}^{2n},d\nu_{n})$, Finally,
observe that since $F$ has compact support,
$F\in\mathcal{L}_{M}(\mathbb{R}^{2n},d\nu_{n})$ implies that $F\in
L^{1}(\mathbb{R}^{2n},d\nu_{n})$, then if we apply the same result to $G=F$ we
get that $F_{\varepsilon}\to F$ in $L^{1}(\mathbb{R}^{2n},d\nu_{n})$, so
passing to a subsequence $\varepsilon_{k}\to 0$ if necessary, we get the
desired result. ∎

Now we are ready to prove the main result of this appendix

###### Proof of Theorem A.2.

Let $u\in W^{s}_{0}L_{M}(\Omega)$. We can assume, without loss of generality,
that $u\in W^{s}_{0}L_{M}(\mathbb{R}^{n})$ and that $u=0$ in
$\mathbb{R}^{n}\setminus\Omega$.

Now, using Lemma A.3, the segment property of $\Omega$, and observing that
$D^{s}u_{t}=(D^{s}u)_{t}$, we can argue exactly as in the proof of [24,
Theorem 1.3] and assume, without loss of generality, that $u$ has compact
support in $\Omega$.

Now, we can regularize $u$ by convolution
$u_{\varepsilon}=u*\rho_{\varepsilon}$ and apply Lemmas A.5 and A.6 to
conclude the desired result. ∎

##  Appendix B A Rellich-Kondrachov type result

In this appendix we prove a Rellich-Kondrachov compactness result for the
inclusion $W_{0}^{s}L_{M}(\Omega)\subset L_{M}(\Omega)$. In the case where $M$
satisfies the $\Delta_{2}-$condition this result was proof in [18, Theorem
3.1].

It is worth mention that in [1] optimal embeddings of the form
$W^{s}L_{M}(\Omega)\subset L_{N}(\Omega)$ were obtained when $\Omega$ is
Lipschitz and $M$ satisfies some subcritical conditions. See Theorem 9.1 in
[1].

The purpose of this appendix is to obtain the compact embedding result of [18]
whitout requiring the $\Delta_{2}-$condition on $M$. We want to stress that
the main ideas in order to accomplish this task are taken from [1].

######  Lemma B.1.

For all $u\in W^{s}L_{M}$ and $|h|<1/2$ we have

$\int_{\mathbb{R}^{n}}M\left(|u(x+h)-u(x)|\right)\,dx\leq\frac{2^{n+1}}{\omega_{n}}\iint_{\mathbb{R}^{2n}}M(2^{s+1}|h|^{s}D^{s}u)\,d\nu_{n}.$

###### Proof.

Let $x,h\in\mathbb{R}^{n}$ with $|h|<1/2$ and $u\in W^{s}L_{M}$. We define the
following sets

$\displaystyle S_{1}$ $\displaystyle=\left\\{y\in
B_{|h|}(x)\colon|u(x+h)-u(y)|\geq\frac{1}{2}|u(x+h)-u(x)|\right\\},$
$\displaystyle S_{2}$ $\displaystyle=\left\\{y\in
B_{|h|}(x)\colon|u(x)-u(y)|\geq\frac{1}{2}|u(x+h)-u(x)|\right\\}.$

Then $B_{|h|}(x)\subset S_{1}\cup S_{2}$. Therefore it follows that

$|S_{1}|\geq\frac{1}{2}|B_{|h|}(x)|\quad\text{ or
}\quad|S_{2}|\geq\frac{1}{2}|B_{|h|}(x)|.$

Without loss of generality we may assume that

$\frac{1}{2}\omega_{n}|h|^{n}\leq|S_{1}|\leq\omega_{n}|h|^{n}.$

Hence we have

$\displaystyle\int_{\mathbb{R}^{n}}M\left(|u(x+h)-u(x)|\right)\,dx$
$\displaystyle=\int_{\mathbb{R}^{n}}\frac{1}{|S_{1}|}\int_{S_{1}}M\left(|u(x+h)-u(x)|\right)\,dy\,dx$
$\displaystyle\leq\frac{2}{\omega_{n}|h|^{n}}\iint_{\mathbb{R}^{n}\times
B_{|h|}(x)}M(2|u(x+h)-u(y)|)\,dy\,dx.$

The last integral is bounded by

$\displaystyle\frac{2}{\omega_{n}|h|^{n}}\iint_{\mathbb{R}^{n}\times
B_{|h|}(x)}M\left(2\frac{|u(x+h)-u(y)|}{|x+h-y|^{s}}|x+h-y|^{s}\right)|x+h-y|^{n}\frac{dydx}{|x+h-y|^{n}}$
$\displaystyle\leq\frac{2^{n+1}}{\omega_{n}}\iint_{\mathbb{R}^{2n}}M\left(2^{s+1}|h|^{s}\frac{|u(x)-u(y)|}{|x-y|^{s}}\right)\,d\nu_{n}.$

This finish the proof. ∎

######  Corollary B.2.

There exists a constant $C=C(n,s)>0$ such that

$\|\tau_{h}u-u\|_{M}\leq C|h|^{s}\|D^{s}u\|_{M,\nu_{n}},$

for every $u\in W^{s}L_{M}$ and $|h|<1/2$

###### Proof.

Take $\lambda=\|D^{s}u\|_{M,\nu_{n}}2^{s+1}|h|^{s}A$ where
$A=\max\\{1,2^{n+1}/\omega_{n}\\}$, and apply the previous lemma to the
function $u/\lambda$. And we get

$\displaystyle\int_{\mathbb{R}^{n}}M\left(\frac{|u(x+h)-u(x)|}{\lambda}\right)\,dx$
$\displaystyle\leq\frac{2^{n+1}}{\omega_{n}}\iint_{\mathbb{R}^{2n}}M\left(\frac{D^{s}u}{\|D^{s}u\|_{M,\nu_{n}}A}\right)$
$\displaystyle\leq 1.$

This finish the proof taking $C=2^{s+1}A$. ∎

Whit these preliminaries we are ready to prove the main result of this
appendix.

######  Theorem B.3.

Let $\Omega\subset\mathbb{R}^{n}$ be a bounded domain that satisfies the
segment property and let $M$ be a Young function. Then the inclusion
$W_{0}^{s}L_{M}(\Omega)\subset E_{M}(\Omega)$ is compact. That is if
$\\{u_{k}\\}_{k\in\mathbb{N}}\subset W_{0}^{s}L_{M}(\Omega)$ is bounded, there
exists $u\in E_{M}(\Omega)$ and a subsequence
$\\{u_{k_{j}}\\}_{j\in\mathbb{N}}\subset\\{u_{k}\\}_{k\in\mathbb{N}}$ such
that $u_{k_{j}}\to u$ in $E_{M}(\Omega)$ as $j\to\infty$.

###### Proof.

With the help of Corollary B.2 the proof of the theorem is an immediate
consequence of [25, Theorem 11.4]. ∎

### Acknowledgements.

This work was partially supported by CONICET under grant PIP 11220150100032CO
and PIP 11220210100238CO, by ANPCyT under grants PICT 2019-3837 and PICT
2019-3530 and by University of San Luis under PROICO 03-2023. Both authors are
members of CONICET and are grateful for the support.

During the creation of this work, the second author benefitted from the
collaborative environment at Instituto de Calculo – CONICET, where a portion
of the research was conducted. The gracious hospitality extended by the first
author, along with the conducive work atmosphere, is sincerely acknowledged
and greatly valued.

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# Some detection tests for low complexity data models
and unknown background distribution.

David Mary1, Sébastien Bourguignon2, Étienne Roquain3,
Sophia Sulis4 and Marie Perrot-Dockes3.
1 Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte
d’Azur.
2 Laboratoire des Sciences du Numérique de Nantes, École Centrale de Nantes.
3 Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne
Université.
4 Laboratoire d’Astrophysique de Marseille, Aix Marseille Université.

###### Abstract

We consider several detection situations where, under the alternative
hypothesis, the signal admits a low complexity model and, under both the null
and the alternative hypotheses, the distribution of the background noise is
unknown. We present several detection strategies for such cases, whose design
relies on exogenous or on endogenous data. These testing procedures have been
inspired by and are applied to two specific problems in Astrophysics, namely
the detection of exoplanets from radial velocity curves and of distant
galaxies in hyperspectral datacubes.

##  1 Introduction

Several fields like Genomics or Astrophysics often require to perform a large
number of statistical tests simultaneously. During the last two decades, these
applications have favored the emergence of theoretical works in multiple
testing. Low-complexity- or sparsity-adaptive tests have been designed,
including for instance the Benjamini-Hochberg [1] procedure, which controls
the  false discovery rate (FDR), or several variants of Donoho and Jin’s
higher criticism (HC) [2] and of the Berk-Jones (BJ) tests [3, 4, 5, 6, 7]. In
practice however, the distribution of the noise background may not be well
known or even unknown. This may explain why such multiple testing procedures,
which require the knowledge of the distribution under the null hypothesis, are
not used in practical applications where they would yet be useful. Users may
prefer in practice considering more ad hoc procedures with the hope, sometimes
illusory, that the test is performed at a controlled level. A recent example
in Astrophysics is the claimed exoplanet detection around $\alpha$ Centauri
Bb, with a P-value (i.e., the probability, under ${\cal H}_{0}$, of observing
a test statistic at least as extreme as the one observed) evaluated at
$0.02\%$ [8]. This detection was strongly discarded by further analyses [9,
10], stressing out the necessity of considering effects that are difficult to
control, such as the noise background created by the star hosting the planet.

This papers presents some testing procedures with correctly controlled error
risks, even when the distribution of the test statistics is not well known. We
consider two cases: exoplanet detection from radial velocity data [11] and the
detection of distant galaxies in the integral field spectrograph MUSE
installed at the Very Large Telescope in Chile [12, 13].

##  2 Exoplanet detection from time series

The presence of a planet orbiting a star induces a reflex motion of the star
around the star-planet mass centre. This motion induces a quasi-periodic
modulation of the radial velocity of the star with respect to a distant
observer and imprints its signature on the stellar spectrum by the Doppler
effect [14]. By measuring the Doppler shift in the lines of the stellar
spectrum, one obtains the radial velocity of the star at a given time. Data
thus consist of a time series containing such measurements.

One way to setup the detection problem considers the following composite
(i.e., with unknown parameters) hypotheses:

$\left\\{\begin{aligned} \text{ ${\cal{H}}_{0}$ :
}{X}(t_{j})&={\displaystyle{{{E}}}}(t_{j})\\\ \text{ ${\cal{H}}_{1}$ :
}X(t_{j})&=\sum_{q=1}^{N_{s}}\alpha_{q}\sin(2\pi
f_{q}t_{j}+\varphi_{q})+E(t_{j})\\\ \end{aligned}\right.$

where $X(t_{j}=j\Delta t),\;j=1,\ldots,N$, are the data points and $E(t_{j})$
is a zero-mean second-order stationary Gaussian noise. The signal under the
alternative hypothesis ${\cal H}_{1}$ has low complexity since it is modeled
by a few sine waves, in some unknown number and with unknown parameters. That
is, the planetary signature is sparse in the Fourier space. Under both
hypotheses, the noise covariance matrix $\Sigma$ (or equivalently the power
spectrum density $S$) is unknown, owing to the adversarial presence of
correlated stellar noise. As a consequence, the distribution of the
periodogram $P$ at each frequency $\nu$, which depends on $S$, is also
unknown. For instance, under $\mathcal{H}_{0}$, $P(\nu)$ is (asymptotically) a
$\chi^{2}$ random variable scaled by $S(\nu)$:
$P(\nu\;;\;{\mathcal{H}_{0}})\sim{S(\nu)}\chi^{2}_{2}/2$.

To circumvent this difficulty we propose an exogenous approach, i.e., based on
ancillary data under $H_{0}$. These data are simulated time series
corresponding to stellar noise, available via astrophysical simulations (see
our recent work [15]). The computation load of such simulations is however
very demanding, so the number (say, $L$) of training time series is small (up
to a few tens, typically).

The considered framework is also relevant in other contexts. For instance, in
asteroseismology, one wants to detect periodic signals in photometric time
series, where the noise is mainly due to atmospheric perturbations. In modern
photometers, auxiliary optical channels are often devoted to monitor “empty”
regions of the sky and stable calibration stars [16]. Note that radar systems
also use training noise samples for detection since ages. However, an
important difference is that adaptive test statistics in radar typically rely
on the estimation of the noise covariance matrix and therefore require $L>N$
[17]. This is a very different regime from that considered here, where $L\ll
N$.

The method simply consists in building an averaged periodogram (say,
$\overline{P}_{L}(\nu)$) from the $L$ training time series. Then, the
standardized periodogram is defined as $Z(\nu):=P(\nu)/\overline{P}_{L}(\nu)$.
Let $Z_{1},\dots,Z_{N}$ denote a subset of its components. For evenly sampled
data, the distributions of $Z$ admit analytical expressions under both
hypotheses [18]. Under standard assumptions [19], one can show that
$Z_{1},\dots,Z_{N}$ are asymptotically independent with
$Z_{j}\sim\mathcal{F}(2,2L)$ (a Fisher F-distribution with parameters $2$ and
$2L$) under the null hypothesis and $Z_{j}\sim\mathcal{F}_{\lambda_{j}}(2,2L)$
(non central F-distribution), whose non-centrality parameters $\lambda_{j}$
depend on the planetary signature.

Under $H_{0}$, the distribution of the standardized periodogram is independent
of $S$. This property, together with asymptotic independence, allows one to
build global hypothesis tests of level $\alpha$ from values of $Z_{j}$, e.g.,
with the test of the maximum: this test rejects the null hypothesis if
$\max(Z_{j},1\leq j\leq N)$ is larger than a threshold $c_{\alpha}$,
calibrated so that [18]

$1-\Big{(}1-\Big{(}\frac{L}{c_{\alpha}+L}\Big{)}^{L}\Big{)}^{N}=\alpha.$

Similar results can be obtained for several other tests like that of Fisher
[20] or its variants [21, 22, 23]. This also allows one to use approaches such
as HC et BJ in a context where the exact distribution under
${\mathcal{H}_{0}}$ is not known.

In simulations, we have shown that the asymptotic approximation is accurate
when the length of the time series is sufficiently large with respect to the
typical correlation time of the noise, and that the tests are conducted at the
target false alarm rates, with the predicted probabilities of detection.
Moreover, the analytical models allow to study the asymptotic powers of the
tests, whose expressions are also accurate for moderate values of $N$. Such
results allow for detectability studies combining specific planetary
parameters and instrumental configurations [11]. An example is shown in the
left panel of Fig. 1, for the test defined by the maximum of the periodogram
[11].

|
---|---
Figure 1: Left panel: exoplanet detection (Section 2).: Power of the detection
test for a fixed false alarm rate of $1\%$ as a function of the observing
duration, for a planet orbiting circularly a solar-type star in 17.5 h. The
considered exoplanet masses are $0.1$ (blue), 0.5 (black) and 1 (red) Earth
mass. The sampling step is $2$ hours and $L=20$. Solid lines correspond to
correlated noise (realistic case) and dashed lines to white noise. The noise
standard deviation is $49$ cm.s-1 in both cases. The planetary signal
amplitude typically lies in the range $[1\;10]$ cm.s-1, depending on the
planet mass. The detectability decreases for correlated noise because the
frequency of the searched signal lies in a region where the stellar noise is
stronger. In the considered configuration, the detection of a planet with half
an Earth mass would be possible with probability $80\%$ if observed during
$12.4$ days. With white noise, this duration would be of only $3$ days (blue
disk vs. circle). Right panel: galaxy detection (Section 3). True purity
versus purity estimated by the proposed method, for the emission line
detection algorithm. The purity is defined as the fraction of true discoveries
(true emission lines) over the total number of discoveries. The purity
estimation method is evaluated on simulated MUSE data.

##  3 Detection of galaxies in MUSE

The integral field spectrograph MUSE, installed on one of the 8-meter
telescopes of the Very Large Telescope (Chile) acquires very rich
hyperspectral images (typically $300\times 300$ pixels in 3 600 optical
channels). In such a data “cube”, we seek to detect very distant and faint
galaxies, whose signatures essentially consist of a weak increase of the light
flux corresponding to an emission line, convolved by the spatial point spread
function of the instrument. The shape of the lines is approximately known, but
not their amplitudes, their positions in the data cube, and nor their number—a
few hundreds typically, against several hundreds of millions of voxels. In
practice, the lines amplitudes can be very weak with respect to other
astrophysical sources and to the background level, caused by the detector
noise, residual sky emission, instrumental artifacts, etc.

In this framework, we have considered a detection approach in three steps :
the nuisance sources are first eliminated by linear subspace filtering
techniques; a specific matched filtering step is applied to the residual; the
detection is performed in the filtered data residuals. In order to focus on
the sparse and localized nature of the target signatures, the considered test
statistics are the local maxima of the resulting filtered data cube. Because
of the numerous and complex preprocessing steps applied to the data before the
detection step, the distribution of the test statistics under $H_{0}$ is
unknown, with little hope to ever establish a reliable theoretical
characterization.

In the considered problem there are several null hypotheses, each one linked
to the position of a local maximum [24]. If we denote by $x,y,z$ the position
of a local maximum ($x$ and $y$ are spatial coordinates, while $z$ is the
wavelength channel), we consider the two hypotheses :

$\left\\{\begin{aligned} \text{ $H_{0,x,y,z}$ : \text{there is no emission
line at position} $(x,y,z)$},\\\ \text{ $H_{1,x,y,z}$ : \text{ there is one
such line at this position.} \;\;\;\;\;\;\;\;\;}\\\ \end{aligned}\right.$

The error criterion considered here is that of the FDR, that is, the expected
value of the proportion of false discoveries over the total number of
positions found as possibly corresponding to a line [1, 13]. In contrast to
the previous application, the approach considered here is endogenous, in the
sense that it does not rely on auxiliary data. We have shown on simulated data
that the distribution of the local maxima under the joint null hypothesis
(i.e., there is no line in the whole data cube) can be estimated from the data
itself, and that the whole procedure (including nuisance cancellation and
matched filtering) controls the FDR (see Fig. 1, right panel). A detailed
presentation of astrophysical results obtained with this algorithm can be
found in [13].

##  4 Conclusions and perspectives

In the signal processing community, there is a large literature on constant
false alarm rate tests—see for instance [25, 26]. To our knowledge, however,
the regime, approach and results described in Section 2 are new. In the
statistics community, the validation of testing procedures learning
automatically the null distribution traces back to permutation and
randomization tests. In the framework of multiple testing and of the FDR
considered here, some results have been brought recently, for instance in [27,
28, 29]. In the studies considered in Section 3, the validation of the
approach is merely done by numerical simulations so far. One direction of
research is to obtain theoretical guarantees, possibly by modifying or
simplifying the procedures and models, inspired by such recent results.

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