# Variational Eigenvalues of the fractional $g$-Laplacian

Sabri Bahrouni  ,  Hichem Ounaies  and  Ariel Salort  Mathematics Department,
Faculty of Sciences, University of Monastir, 5019 Monastir, Tunisia  IMAS -
CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n, Buenos
Aires, Argentina<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the present work we study existence of sequences of variational eigenvalues
to non-local non-standard growth problems ruled by the fractional
$g-$Laplacian operator with different boundary conditions (Dirichlet, Neumann
and Robin). Due to the non-homogeneous nature of the operator several
drawbacks must be overcome, leading to some results that contrast with the
case of power functions.

###### Key words and phrases:

Fractional Orlicz-Sobolev spaces, Ljusternik-Schnirelman eigenvalues, Minimax
eigenvalues
2010 Mathematics Subject Classifications: 46E30, 35R11, 45G05

###### Contents

  1. 1 Introduction
  2. 2 Definitions and preliminary results
    1. 2.1 Young functions
    2. 2.2 Fractional Orlicz-Sobolev spaces
  3. 3 Lagrange multipliers and the eigenvalue problem
  4. 4 Ljusternik-Schnirelman eigenvalues
    1. 4.1 The Dirichlet case
    2. 4.2 The Neumann/Robin case
  5. 5 Minimax eigenvalues

##  1\. Introduction

Given two functionals $\mathcal{A}$ and $\mathcal{B}$ defined on a suitable
space $\mathcal{X}$ and a prescribed number $c$, the task of analyzing the
existence of numbers $\lambda\in\mathbb{R}$ and elements $u\in\mathcal{X}$
satisfying (in some appropriated sense) equations of the type

$\mathcal{B}^{\prime}(u)=\lambda\mathcal{A}^{\prime}(u),\qquad\mathcal{A}(u)=c,$

has been a challenging labor whose beginning dates back to the mid-20th
century (here $\mathcal{A}^{\prime}$ and $\mathcal{B}^{\prime}$ denote the
Fréchet derivatives of the functionals). The study on Hilbert spaces was
addressed by Krasnoselskij in [37]; for Banach spaces, it can be found in the
literature the works of Citlanadze [20] and Browder [15, 18, 19], where the
notion of _category of sets_ in the sense of Ljusternik and Schnirelman is
used. See also [30, 31] for some applications to partial differential
equations. The amount of research on these topics is nowadays huge. For
practical reasons, for an introduction to this theory and a comprehensive list
of references we refer to the books [16, 17, 47, 56].

After the introduction of the so-called monotonicity methods by Browder [13,
14], Minty [44] and Vaǐnberg and Kačurovskiǐ [54], the study of quasilinear
operators experimented an explosive growth, and both variational and
nonvariational techniques were introduced by Browder, Fučík, Ladyzhenskaya,
Leray, Lions, Morrey, Nečas, Rabinowicz, Schauder, Serrin, and Trudinger,
among several other mathematicians.

The prototypical $p-$Laplace operator ($p>1$) then became a focus of study,
and in particular, to understand its spectral structure: given an open and
bounded set $\Omega\subset\mathbb{R}^{n}$, to determine the existence couples
$(\lambda,u)$ satisfying the equation

(1.1) $-\mathop{\text{div}}(|\nabla u|^{p-2}u)=\lambda|u|^{p-2}u\text{ in
}\Omega,\quad u=0\text{ on }\partial\Omega$

in a suitable sense. In the seminal work of García Azorero and Peral Alonso
[35], it was proved the existence of a variational sequence of eigenvalues
tending to $+\infty$, which however, it is not known to exhaust the spectrum
unless $p=2$ or $n=1$. Several properties on eigenvalues (and their
corresponding eigenfunctions) were addressed by Anane et al [3, 4] and
Lindqvist [41], among others, and also for more general boundary conditions
than Dirichlet. See also [28, 42].

At this point, two possible generalizations of the eigenvalue problem (1.1)
could be considered. First, its non-local counterpart governed by the well-
known _fractional $p-$Laplace_ operator takes the form

(1.2) $(-\Delta_{p})^{s}u=\lambda|u|^{p-2}u\text{ in }\Omega,\quad u=0\text{
in }\mathbb{R}^{n}\setminus\Omega$

where $s\in(0,1)$ is a fractional parameter, $p>1$, and

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

The main difference here arises in the fact that this operator takes into
account interactions coming from the whole space. The same occurs with the
_boundary_ condition. Problem (1.2) was introduced in [40]. Existence of a
sequence of (variational) eigenvalues to (1.2) and its behavior as $s\uparrow
1$ was dealt in [11]. Several properties on eigenvalues and eigenfunctions
were obtained in [12, 22, 33, 40].

A second possible generalization of (1.1) can be obtained keeping the local
structure of the operator but allowing a growth behavior more general than a
power. These considerations lead to the well-known _$g-$ Laplace_ operator and
the problem

$-\Delta_{g}u=\lambda g(|u|)\frac{u}{|u|}\text{ in }\Omega,\quad u=0\text{ on
}\partial\Omega,$

where $\Delta_{g}u:=\mathop{\text{div}}\left(g(|\nabla u|)\frac{\nabla
u}{|\nabla u|}\right)$. The function $g=G^{\prime}$ is given in terms of a so-
called Young function $G$. Here the structure of the spectrum radically
changes due to the non-homogeneity of the operator, fact that is key in the
arguments corresponding to results related to problems (1.1) and (1.2). Here
several important differences appear: the spectrum may not be discrete, then
it is not clear the notion of a first eigenvalue nor of a sequence of them.
However, when restricting the _energy level_ of functions, some properties of
operators with $p-$structure are recovered: in [34, 36, 45, 46] existence of
eigenvalues was studied, and in [25, 53] the existence of a discrete sequence
of eigenvalues was obtained. It is worth of mention that under these settings,
eigenvalues are not in general variational, so less information can be
retrieved when using standard techniques.

The object of study of this manuscript is the eigenvalue problem ruled by the
_fractional $g-$Laplacian_, which can be seen as the non-local non-standard
growth counterpart of (1.1). Given an open and bounded set
$\Omega\subset\mathbb{R}^{n}$ and a fractional parameter $s\in(0,1)$, we are
concerned with the following eigenvalue problems:

  * •

Dirichlet problem:

($D(\Omega)$) $\displaystyle\begin{cases}(-\Delta_{g})^{s}u=\lambda
g(|u|)\frac{u}{|u|}&\text{ in }\Omega\\\ u=0&\text{ in
}\mathbb{R}^{n}\setminus\Omega,\end{cases}$

  * •

Neumann problem:

($N(\Omega)$) $\displaystyle\begin{cases}(-\Delta_{g})^{s}u=\lambda
g(|u|)\frac{u}{|u|}&\text{ in }\Omega\\\ \mathcal{N}_{g}u=0&\text{ in
}\mathbb{R}^{n}\setminus\Omega.\end{cases}$

  * •

Robin problem: given $\beta\in L^{\infty}(\mathbb{R}^{n}\setminus\Omega)$

($R(\Omega)$) $\displaystyle\begin{cases}(-\Delta_{g})^{s}u=\lambda
g(|u|)\frac{u}{|u|}&\text{ in }\Omega\\\ \mathcal{N}_{g}u+\beta
g(|u|)\frac{u}{|u|}=0&\text{ in }\mathbb{R}^{n}\setminus\Omega.\end{cases}$

The precise notion of _eigenvalues_ and _eigenfunctions_ to these problems is
defined in Section 3.

Here, the fractional $g-$Laplacian is defined as

(1.3)
$(-\Delta_{g})^{s}u:=\,\text{p.v.}\int_{\mathbb{R}^{n}}g\left(|D_{s}u|\right)\frac{D_{s}u}{|D_{s}u|}\frac{dy}{|x-y|^{n+s}},$

where $G$ is a Young function (see Section 2 for the precise definition) such
that $g=G^{\prime}$ and $D_{s}u:=\frac{u(x)-u(y)}{|x-y|^{s}}$. Clearly,
$(-\Delta_{g})^{s}$ boils down to the fractional $p-$Laplacian when $G$ is a
power function and to the $p-$Laplacian when $s\uparrow 1$. See [26]. The
boundary conditions in the previous problems reflect the non-local nature of
the operator $(-\Delta_{g})^{s}$: the Dirichlet case corresponds to functions
vanishing outside $\Omega$ and not only on $\partial\Omega$, whereas the
Neumann and Robin equations make use of the nonlocal normal derivative
$\mathcal{N}_{g}$ introduced in [6].

Throughout this article we assume the Young function $G=\int_{0}^{t}g(t)\,dt$
to satisfy the following structural conditions:

($G_{1}$) $1<p^{-}\leq\frac{tg(t)}{G(t)}\leq p^{+}<\infty\quad\forall t>0,$
($G_{2}$) $t\mapsto G(\sqrt{t}),\ t\in[0,\infty)\text{ is convex},$ ($G_{3}$)
$\displaystyle\int_{0}^{1}\frac{G^{-1}(\tau)}{\tau^{\frac{n+s}{n}}}d\tau<\infty\
\quad\text{and}\quad\displaystyle\int_{1}^{+\infty}\frac{G^{-1}(\tau)}{\tau^{\frac{n+s}{n}}}d\tau=\infty.$

In the case of powers (i.e. $G(t)=t^{p}$) these conditions mean that $2\leq
p\leq\frac{np}{n-sp}$.

The following functionals have an essential role in the study of eigenvalue
problems for the fractional $g-$Laplacian

(1.4)
$\mathcal{I}(u):=\int_{\Omega}G(|u|)\,dx,\qquad\mathcal{J}(u):=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}G(|D_{s}u|)\frac{dxdy}{|x-y|^{n}}.$

and

(1.5)
$\mathcal{J}_{\beta}(u):=\iint_{\mathbb{R}^{2n}\setminus(\Omega^{c})^{2}}G(|D_{s}u|)\frac{dxdy}{|x-y|^{n}}+\int_{\mathbb{R}^{n}\setminus\Omega}\beta
G(|u|)\,dx$

(see Section 3 for details).

Problem $D(\Omega)$ was recently studied in [49, 50], where, once established
existence of a minimizer of the associated problem under energy constraint

$\Lambda_{\alpha}^{D}:=\min\left\\{\mathcal{J}(u)\colon\mathcal{I}(u)=\alpha\right\\}$

for a given value $\alpha>0$, then by means of a version of the Lagrange
multipliers rule it is deduced existence of an eigenvalue
$\lambda_{\alpha}^{D}$ of $D(\Omega)$ in a suitable sense, and in general
$\Lambda_{\alpha}^{D}\neq\lambda_{\alpha}^{D}$. A further extension of this
result for problems $N(\Omega)$ and $R(\Omega)$ is provided in [6]. Other very
recent results involving eigenvalues of the fractional $g-$Laplacian can be
found in [1, 2, 5, 7, 8, 9, 10, 23, 27, 29, 49, 51].

In the wake of the non-homogeneity of the operator, many of the properties
that eigenvalues of the fractional $p-$Laplacian fulfill are possibly not
inherited in the non-homogeneous case: for instance, isolation, simplicity and
a variational characterization of the first eigenvalue, or a variational
formula for the second one (see [22, 33, 40]). Moreover, the spectrum of
$D(\Omega)$ could be continuous when $G$ is a general Young function, and in
principle, it is not clear the meaning of a first or second eigenvalue. Due to
these drawbacks, the main aim of this manuscript is to understand under which
conditions it is possible to build a sequence of eigenvalues to problems
$D(\Omega)$, $N(\Omega)$ and $R(\Omega)$.

Our arguments are based in the fact that existence of sequences of variational
eigenvalues can be established when prescribing some energy level. First, when
the quantities $\mathcal{J}(u)$ or $\mathcal{J}_{\beta}(u)$ involving the
$s-$Hölder quotient are prescribed, by means of the Ljusternik-Schnirelman
theory we can infer existence of a discrete sequence of critical points of
$\mathcal{I}(u)$ which allow us to build a sequence of non-negative
eigenvalues. With the notation introduced in Section 2, our first result reads
as follows.

######  Theorem 1.1.

For any $\alpha>0$ there exist sequences of non-negative numbers
$\\{\lambda^{D}_{k,\alpha}\\}_{k\in\mathbb{N}},\\{\lambda^{N}_{k,\alpha}\\}_{k\in\mathbb{N}}$
and $\\{\lambda^{R}_{k,\alpha}\\}_{k\in\mathbb{N}}$ which are eigenvalues of
problems $D(\Omega)$, $N(\Omega)$ and $R(\Omega)$, respectively. Moreover,
these sequences diverge as $k\to\infty$.

The corresponding eigenfunctions

$\\{u_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}\subset
W^{s,G}_{0}(\Omega),\quad\\{u_{k,\alpha}^{N}\\}_{k\in\mathbb{N}}\subset
W^{s,G}_{*}(\Omega)\quad\text{ and
}\quad\\{u_{k,\alpha}^{R}\\}_{k\in\mathbb{N}}\subset\mathcal{X}_{\beta}(\Omega)$

satisfy the constraints

$\mathcal{J}(u_{k,\alpha}^{D})=\mathcal{J}_{0}(u_{k,\alpha}^{N})=\mathcal{J}_{\beta}(u_{k,\alpha}^{R})=\alpha$

and

$\mathcal{I}(u_{k,\alpha}^{D})=c_{k,\alpha}^{D},\qquad\mathcal{I}(u_{k,\alpha}^{N})=c_{k,\alpha}^{N},\qquad\mathcal{I}(u_{k,\alpha}^{R})=c_{k,\alpha}^{R}$

where the critical values are obtained as

$c_{k,\alpha}^{D}=\sup_{K\in\mathcal{C}_{k}^{D}}\inf_{u\in
K}\mathcal{I}(u),\qquad
c_{k,\alpha}^{N}=\sup_{K\in\mathcal{C}_{k}^{N}}\inf_{u\in
K}\mathcal{I}(u),\qquad
c_{k,\alpha}^{R}=\sup_{K\in\mathcal{C}_{k}^{R}}\inf_{u\in K}\mathcal{I}(u)$

where, denoting by $\gamma(K)$ the Krasnoselskii genus of $K$, we define

$\mathcal{C}_{k}^{D}:=\\{K\subset M_{\alpha}^{D}\text{ compact, symmetric with
}\mathcal{I}(u)>0\text{ on }K\text{ and }\gamma(K)\geq k\\},$

and

$\displaystyle M_{\alpha}^{D}$ $\displaystyle:=\\{u\in
W^{s,G}_{0}(\Omega)\colon\mathcal{J}(u)=\alpha\\},$ $\displaystyle
M_{\alpha}^{N}$ $\displaystyle:=\\{u\in
W^{s,G}_{*}(\Omega)\colon\mathcal{J}_{0}(u)=\alpha\\},$ $\displaystyle
M_{\alpha}^{R}$
$\displaystyle:=\\{u\in\mathcal{X}_{\beta}(\Omega)\colon\mathcal{J}_{\beta}(u)=\alpha\\}.$

The sets $\mathcal{C}_{k}^{N}$ and $\mathcal{C}_{k}^{R}$ are defined
analogously by changing the superscript $D$ by $N$ and $R$, respectively.

Secondly, when the quantity $\mathcal{I}(u)$ is prescribed, by using the so-
called minimax theory we obtain a sequence of critical points of
$\mathcal{J}(u)$ which provides for a (different) sequence of eigenvalues:

######  Theorem 1.2.

For any $\alpha>0$ there exist sequences of non-negative eigenvalues
$\\{\Lambda^{D}_{k,\alpha}\\}_{k\in\mathbb{N}}$,
$\\{\Lambda^{N}_{k,\alpha}\\}_{k\in\mathbb{N}}$ and
$\\{\Lambda^{R}_{k,\alpha}\\}_{k\in\mathbb{N}}$ of $D(\Omega)$, $N(\Omega)$
and $R(\Omega)$, respectively.

The corresponding eigenfunctions

$\\{u_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}\subset
W^{s,G}_{0}(\Omega),\quad\\{u_{k,\alpha}^{N}\\}_{k\in\mathbb{N}}\subset
W^{s,G}_{*}(\Omega)\quad\text{ and
}\quad\\{u_{k,\alpha}^{R}\\}_{k\in\mathbb{N}}\subset\mathcal{X}_{\beta}(\Omega)$

satisfy the constraints

$\mathcal{I}(u_{k,\alpha}^{D})=\mathcal{I}(u_{k,\alpha}^{N})=\mathcal{I}(u_{k,\alpha}^{R})=\alpha$

and

$\mathcal{J}(u_{k,\alpha}^{D})=C_{k,\alpha}^{D},\qquad\mathcal{J}_{0}(u_{k,\alpha}^{N})=C_{k,\alpha}^{N},\qquad\mathcal{J}_{\beta}(u_{k,\alpha}^{R})=C_{k,\alpha}^{R}$

where the critical values are obtained as

$C_{k,\alpha}^{D}=\inf_{h\in\Gamma(S^{k-1},M_{\alpha}^{D})}\sup_{w\in
S^{k-1}}\mathcal{J}(h(w))$

and $\Gamma(S^{k},M_{\alpha}^{D})=\\{h\in C(S^{k},M_{\alpha}^{D})\colon
h\text{ is odd}\\}$, being $S^{k}$ the unit sphere in $\mathbb{R}^{k+1}$.

The numbers $C_{k,\alpha}^{N}$ and $C_{k,\alpha}^{R}$ are defined analogously
by changing the superscript $D$ by $N$ and $R$, respectively, and the
functional $\mathcal{J}$ by $\mathcal{J}_{0}$ and $\mathcal{J}_{\beta}$,
respectively. $M_{\alpha}^{D}$, $M_{\alpha}^{N}$ and $M_{\alpha}^{R}$ denote
the sets

$\displaystyle M_{\alpha}^{D}$ $\displaystyle:=\\{u\in
W^{s,G}_{0}(\Omega)\colon\mathcal{I}(u)=\alpha\\},$ $\displaystyle
M_{\alpha}^{N}$ $\displaystyle:=\\{u\in
W^{s,G}_{*}(\Omega)\colon\mathcal{I}(u)=\alpha\\},$ $\displaystyle
M_{\alpha}^{R}$
$\displaystyle:=\\{u\in\mathcal{X}_{\beta}(\Omega)\colon\mathcal{I}(u)=\alpha\\}.$

In contrast to what happens in the case of powers, the Ljusternik-Schnirelman
and minimax eigenvalues obtained in Theorems 1.1 and 1.2, in general, are
neither the same nor easily comparable each other. As a consequence, in
principle we cannot ensure that the minimax sequences built in Theorems 1.2
diverge as $k\to\infty$. For a discussion of this situation in the case
$G(t)=t^{p}$ see [24, 43]. However, in the Dirichlet case we are able to
compare miminax eigenvalues of the $g-$Laplacian with eigenvalues of the
$p^{-}-$Laplacian (where $p^{-}$ is given in ($G_{1}$)), giving as a
consequence the following:

######  Theorem 1.3.

With the notation of Theorem 1.2,

$\Lambda_{k,\alpha}^{D}\to\infty\text{ as }k\to\infty,\qquad
C_{k,\alpha}^{D}\to\infty\text{ as }k\to\infty.$

In the case of powers, eigenvalues and critical points obtained in Theorems
1.1 and 1.2 coincide, i.e., $\lambda_{k}=c_{k}$ and $\Lambda_{k}=C_{k}$ for
all $k\in\mathbb{N}$. It does not occur for a general Young function, although
the following comparison result holds.

######  Theorem 1.4.

Let $\\{\lambda_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}$ and
$\\{c_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}$ be as in Theorem 1.1, and let
$\\{\Lambda_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}$ and
$\\{C_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}$ be as in Theorem 1.2. Then

$\frac{\alpha p^{-}}{p^{+}}\leq
c_{k,\alpha}^{D}\cdot\lambda_{k,\alpha}^{D}\leq\frac{\alpha
p^{+}}{p^{-}},\qquad\frac{p^{-}}{\alpha
p^{+}}\leq\frac{\Lambda_{k,\alpha}^{D}}{C_{k,\alpha}^{D}}\leq\frac{p^{+}}{\alpha
p^{-}},$

where $p^{+}$ and $p^{-}$ are the numbers defined in ($G_{1}$).

Finally, our last result establishes the closedness of the spectrum of the
fractional $g-$Laplacian in the following sense:

######  Theorem 1.5.

Fixed $\alpha_{0}>0$, let
$\\{\alpha_{k}\\}_{k\in\mathbb{N}}\subset(0,\alpha_{0})$. Let
$\\{\lambda_{\alpha_{k}}\\}_{k\in\mathbb{N}}$ be a sequence of eigenvalues of
$D(\Omega)$ with eigenfunctions $\\{u_{\alpha_{k}}\\}_{k\in\mathbb{N}}\subset
W^{s,G}_{0}(\Omega)$ such that $\Phi_{G,\Omega}(u_{\alpha_{k}})=\alpha_{k}$.

Then, if $\lim_{k\to\infty}\lambda_{\alpha_{k}}=\lambda$, we have that
$\lambda$ is an eigenvalue of $D(\Omega)$ with eigenfunction $u\in
W^{s,G}_{0}(\Omega)$ such that $\Phi_{G,\Omega}(u)=\beta\in(0,\alpha_{0})$.

A similar assertion holds for sequences of eigenvalues of $N(\Omega)$ and
$R(\Omega)$.

As a consequence we get the following.

######  Corollary 1.6.

Fixed $\alpha_{0}>0$, the numbers

$\hat{\lambda}_{k}^{D}:=\inf_{\alpha\in(0,\alpha_{0})}\lambda_{k,\alpha}^{D},\quad\hat{\lambda}_{k}^{N}:=\inf_{\alpha\in(0,\alpha_{0})}\lambda_{k,\alpha}^{N},\quad\hat{\lambda}_{k}^{R}:=\inf_{\alpha\in(0,\alpha_{0})}\lambda_{k,\alpha}^{R}$

are eigenvalues of $D(\Omega)$, $N(\Omega)$ and $R(\Omega)$, respectively.

The same claim is true for the numbers $\hat{\Lambda}_{k}^{D}$,
$\hat{\Lambda}_{k}^{N}$ and $\hat{\Lambda}_{k}^{R}$ defined in an analogous
way.

##  2\. Definitions and preliminary results

In this section we introduce the classes of Young function and fractional
Orlicz-Sobolev functions, the suitable class where the fractional
$g$-Laplacian is well defined.

###  2.1. Young functions

An application $G\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ is said to be a _Young
function_ if it admits the integral formulation
$G(t)=\int_{0}^{t}g(\tau)\,d\tau$, where the right continuous function $g$
defined on $[0,\infty)$ has the following properties:

($g_{1}$) $\displaystyle g(0)=0,\quad g(t)>0\text{ for }t>0,$ ($g_{2}$)
$\displaystyle g\text{ is non-decreasing on }(0,\infty),$ ($g_{3}$)
$\displaystyle\lim_{t\to\infty}g(t)=\infty.$

From these properties it is easy to see that a Young function $G$ is
continuous, nonnegative, strictly increasing and convex on $[0,\infty)$.

The following properties on Young functions are well-known. See for instance
[38] for the proof of these results.

######  Lemma 2.1.

Let $G$ be a Young function satisfying ($G_{1}$) and $a,b\geq 0$. Then

($L_{1}$) $\displaystyle\min\\{a^{p^{-}},a^{p^{+}}\\}G(b)\leq
G(ab)\leq\max\\{a^{p^{-}},a^{p^{+}}\\}G(b),$ ($L_{2}$) $\displaystyle
G(a+b)\leq\mathbf{C}(G(a)+G(b))\quad\text{with }\mathbf{C}:=2^{p^{+}},$
($L_{3}$) $\displaystyle G\text{ is Lipschitz continuous}.$

Condition ($G_{1}$) is known as the _$\Delta_{2}$ condition_ or _doubling
condition_ and, as it is showed in [38, Theorem 3.4.4], it is equivalent to
the right hand side inequality in ($G_{1}$).

The _complementary Young function_ $\tilde{G}$ of a Young function $G$ is
defined as

$\tilde{G}(t):=\sup\\{tw-G(w):w>0\\}.$

From this definition the following Young-type inequality holds

(2.1) $ab\leq G(a)+\tilde{G}(b)\qquad\text{for all }a,b\geq 0,$

and the following Hölder’s type inequality

$\int_{\Omega}|uv|\,dx\leq\|u\|_{L^{G}(\Omega)}\|v\|_{L^{\tilde{G}}(\Omega)}$

for all $u\in L^{G}(\Omega)$ and $v\in L^{\tilde{G}}(\Omega)$. Moreover, it is
not hard to see that $\tilde{G}$ can be written in terms of the inverse of $g$
as

(2.2) $\tilde{G}(t)=\int_{0}^{t}g^{-1}(\tau)\,d\tau,$

see [48, Theorem 2.6.8].

Since $g^{-1}$ is increasing, from (2.2) and ($G_{1}$) it is immediate the
following relation.

######  Lemma 2.2.

Let $G$ be an Young function satisfying ($G_{1}$) such that $g=G^{\prime}$ and
denote by $\tilde{G}$ its complementary function. Then

$\tilde{G}(g(t))\leq p^{+}G(t)$

holds for any $t\geq 0$.

The following convexity property proved in [39][Lemma 2.1] will be useful.

######  Lemma 2.3.

Let $G$ be a Young function satisfying ($G_{1}$) and ($G_{2}$). Then for every
$a,b\in\mathbb{R}$,

$\frac{G(|a|)+G(|b|)}{2}\geq
G\left(\left|\frac{a+b}{2}\right|\right)+G\left(\left|\frac{a-b}{2}\right|\right).$

###  2.2. Fractional Orlicz-Sobolev spaces

Given a Young function $G$ such that $G^{\prime}=g$, a parameter $s\in(0,1)$
and an open and bounded set $\Omega\subseteq\mathbb{R}^{n}$ we consider the
spaces

$\displaystyle L^{G}(\Omega):=\left\\{u\colon\Omega\to\mathbb{R}\text{
measurable such that }\Phi_{G,\Omega}(u)<\infty\right\\},$ $\displaystyle
W^{s,G}(\mathbb{R}^{n}):=\left\\{u\in L^{G}(\mathbb{R}^{n})\text{ such that
}\Phi_{s,G,\mathbb{R}^{n}}(u)<\infty\right\\},$ $\displaystyle
W^{s,G}_{0}(\Omega):=\left\\{u\in W^{s,G}(\mathbb{R}^{n}):\ u=0\ a.e.\
\text{in}\ \mathbb{R}^{n}\setminus\Omega\right\\}$

where the modulars $\Phi_{G,\Omega}$ and $\Phi_{s,G,\mathbb{R}^{n}}$ are
defined as

$\displaystyle\Phi_{G,\Omega}(u):=\int_{\Omega}G(|u(x)|)\,dx$
$\displaystyle\Phi_{s,G,\mathbb{R}^{n}}(u):=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}G(|D_{s}u(x,y)|)\,d\mu,$

where the _$s-$ Hölder quotient_ is defined as

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

being $d\mu(x,y):=\frac{dx\,dy}{|x-y|^{n}}$. These spaces are endowed with the
so-called _Luxemburg norms_

$\displaystyle\|u\|_{L^{G}(\Omega)}:=\inf\left\\{\lambda>0\colon\Phi_{G,\Omega}\left(\frac{u}{\lambda}\right)\leq
1\right\\},$
$\displaystyle\|u\|_{W^{s,G}(\Omega)}:=\|u\|_{L^{G}(\Omega)}+[u]_{W^{s,G}(\mathbb{R}^{n})},$

where the $(s,G)$-Gagliardo semi-norm is defined as

$\displaystyle[u]_{W^{s,G}(\mathbb{R}^{n})}:=\inf\left\\{\lambda>0\colon\Phi_{s,G,\mathbb{R}^{n}}\left(\frac{u}{\lambda}\right)\leq
1\right\\}.$

Since we assume ($G_{1}$), the space $W^{s,G}(\Omega)$ is a reflexive Banach
space. Moreover $C_{c}^{\infty}$ is dense in $W^{s,G}(\mathbb{R}^{n})$. See
[26, Proposition 2.11] and [23, Proposition 2.9] for details.

Observe that, in light of the modular Poincaré inequality for
$W^{s,G}(\Omega)$ (see [50]), it follows that
$[\,.\,]_{W^{s,G}(\mathbb{R}^{n})}$ is an equivalent norm in
$W_{0}^{s,G}(\Omega)$. We observe that $W^{s,G}_{0}(\Omega)$ is the natural
space to deal with Dirichlet problems.

The space of fractional Orlicz-Sobolev functions is the appropriated one where
to define the _fractional $g-$Laplacian operator_

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

where p.v. stands for in principal value. This operator is well defined
between $W^{s,G}(\mathbb{R}^{n})$ and its dual space
$W^{-s,\tilde{G}}(\mathbb{R}^{n})$ (see [26] for details). In fact, it follows
that

$\langle(-\Delta_{g})^{s}u,v\rangle=\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}D_{s}v\,d\mu,$

for any $v\in W^{s,G}(\mathbb{R}^{n})$.

As proved in [6, Proposition 2.6], the following integration by parts formula
arise naturally for $u\in C^{2}$ functions

$\langle(-\Delta_{g})^{s}u,v\rangle_{*}=\int_{\Omega}v(-\Delta_{g})^{s}u\
dx+\int_{\mathbb{R}^{n}\setminus\Omega}v\mathcal{N}_{g}u\ dx\quad\forall v\in
C^{2}$

where the _normal derivative_ $\mathcal{N}_{g}$ is defined as

$\mathcal{N}_{g}u(x)=\int_{\Omega}g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}\frac{dy}{|x-y|^{n+s}}.$

and the product $\langle\cdot,\cdot\rangle_{*}$ is defined as

$\langle(-\Delta_{g})^{s}u,v\rangle_{*}=\iint_{\mathbb{R}^{2n}\setminus(\mathbb{R}^{n}\setminus\Omega)^{2}}g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}D_{s}v\,d\mu.$

The previous definitions induce the following notation

$\displaystyle\Phi_{s,G,*}(u)$
$\displaystyle=\iint_{\mathbb{R}^{2n}\setminus(\Omega)^{2}}G(|D_{s}u(x,y)|)\,d\mu$
$\displaystyle[u]_{W^{s,G}_{*}(\mathbb{R}^{n})}$
$\displaystyle=\inf\left\\{\lambda>0\colon\Phi_{s,G,*}\left(\frac{u}{\lambda}\right)\leq
1\right\\}.$

Hence, it is natural to defined the space

$W^{s,G}_{*}(\Omega):=\\{u\in L^{G}(\Omega)\text{ such that
}\Phi_{s,G,*}(u)<\infty\\},$

which will be the appropriated one when dealing with the Neumann boundary
condition.

The suitable space in which to define a Robin boundary condition of the type
$\mathcal{N}_{g}u+\beta g(|u|)u/|u|$ in $\mathbb{R}^{n}\setminus\Omega$ (where
$\beta$ is a fixed function in $L^{\infty}(\mathbb{R}^{n}\setminus\Omega)$)
is:

$\mathcal{X}_{\beta}(\Omega)=\left\\{u\text{
measurable}\colon\Phi_{s,G,*}(u)+\Phi_{G,\Omega}(u)+\Phi_{G,\beta,\mathbb{R}^{n}\setminus\Omega}(u)<\infty\right\\}$

where

$\Phi_{G,\beta,\mathbb{R}^{n}\setminus\Omega}(u)=\int_{\mathbb{R}^{n}\setminus\Omega}\beta
G(|u|)\,dx.$

This space can be proved to be a reflexive Banach space endowed with the norm

$\|u\|_{\mathcal{X_{\beta}}}:=[u]_{W^{s,G}_{*}(\mathbb{R}^{n})}+\|u\|_{L^{G}(\Omega)}+\|u\|_{L^{G,\beta}(\mathbb{R}^{n}\setminus\Omega)},$

being

$\|u\|_{L^{G,\beta}(\mathbb{R}^{n}\setminus\Omega)}=\inf\left\\{\lambda>0\colon\int_{\mathbb{R}^{n}\setminus\Omega}\beta\
G\left(\frac{|u|}{\lambda}\right)\,d\mu\leq 1\right\\}.$

See [6] for more details.

######  Remark 2.4.

Observe that, with the notation introduced in this section, the functionals
introduced in (1.4) and (1.5) can be identified as

(2.3) $\displaystyle\begin{split}&\mathcal{I}\colon
W^{s,G}(\Omega)\to\mathbb{R},\quad\mathcal{I}(u)=\Phi_{G,\Omega}(u),\\\
&\mathcal{J}\colon
W^{s,G}_{0}(\Omega)\to\mathbb{R},\quad\mathcal{J}(u)=\Phi_{s,G,\mathbb{R}^{n}}(u),\\\
&\mathcal{J}_{\beta}\colon(\Omega)\to\mathbb{R},\quad\mathcal{J}_{\beta}(u)=\Phi_{s,G,*}(u)+\Phi_{G,\beta,\mathbb{R}^{n}\setminus\Omega}.\end{split}$

In order to state some embedding results for fractional Orlicz-Sobolev spaces
we recall that given two Young functions $A$ and $B$, we say that _$B$ is
essentially stronger than $A$_ or equivalently that _$A$ decreases essentially
more rapidly than $B$_, and denoted by $A\prec\prec B$, if for each $a>0$
there exists $x_{a}\geq 0$ such that $A(x)\leq B(ax)$ for $x\geq x_{a}$.

When the Young function $G$ fulfills condition ($G_{3}$), the critical
function for the fractional Orlicz-Sobolev embedding is given by

$G_{*}^{-1}(t)=\int_{0}^{t}\frac{G^{-1}(\tau)}{\tau^{\frac{n+s}{n}}}d\tau.$

With these preliminaries the following compact embeddings hold.

######  Proposition 2.5.

Let $G$ be a Young function satisfying ($G_{1}$) and ($G_{3}$) and let
$s\in(0,1)$. Let $\Omega\subset\mathbb{R}^{n}$ be a $C^{0,1}$ bounded open
subset. Then for any Young function $B$ such that $B\prec\prec G_{*}$ it holds
that

  * (i)

the embedding $W^{s,G}(\Omega)\hookrightarrow L^{B}(\Omega)$ is compact;

  * (ii)

the embedding $\mathcal{X}_{\beta}(\Omega)\hookrightarrow L^{B}(\Omega)$ is
compact.

The proof of (i) can be found in [1][Theorem 9.1], [8][Theorem 1.2]; for the
proof of (ii) see [6][Lemma 3.2].

The following relation between modulars and norms holds. See [32, Lemma 2.1].

######  Lemma 2.6.

Let $G$ be a Young function satisfying ($G_{1}$) and let
$\xi^{-}(t)=\min\\{t^{p^{-}},t^{p^{+}}\\}$,
$\xi^{+}(t)=\max\\{t^{p^{-}},t^{p^{+}}\\}$, for all $t\geq 0$. Then

  * (i)

$\xi^{-}(\|u\|_{L^{G}(\mathbb{R}^{n})})\leq\Phi_{G,\mathbb{R}^{n}}(u)\leq\xi^{+}(\|u\|_{L^{G}(\mathbb{R}^{n})})\
\text{for}\ u\in L^{G}(\mathbb{R}^{n})$,

  * (ii)

$\xi^{-}([u]_{W^{s,G}(\mathbb{R}^{n})})\leq\Phi_{s,G,\mathbb{R}^{n}}(u)\leq\xi^{+}([u]_{W^{s,G}(\mathbb{R}^{n})})\
\text{for}\ u\in W^{s,G}(\mathbb{R}^{n})$.

##  3\. Lagrange multipliers and the eigenvalue problem

In this section we define the notion of Dirichlet, Neumann and Robin
eigenvalue problems in the context of fractional Orlicz-Sobolev spaces. We
recall some existence results already proved in [49] and [50] for the
Dirichlet case, and state further extension to more general boundary
conditions.

We say that

  * $\lambda$ is an _eigenvalue_ of $D(\Omega)$ with _eigenfunction_ $u\in W_{0}^{s,G}(\Omega)$ if

(3.1)
$\langle(-\Delta_{g})^{s}u,v\rangle=\lambda\int_{\Omega}g(|u|)\frac{u}{|u|}v\,dx\qquad\forall
v\in W_{0}^{s,G}(\Omega).$

  * $\lambda$ is an _eigenvalue_ of $N(\Omega)$ with _eigenfunction_ $u\in W^{s,G}_{*}(\Omega)$ if

$\langle(-\Delta_{g})^{s}u,v\rangle_{*}=\lambda\int_{\Omega}g(|u|)\frac{u}{|u|}v\,dx\qquad\forall
v\in W^{s,G}_{*}(\Omega).$

  * $\lambda$ is an _eigenvalue_ of $R(\Omega)$ with with _eigenfunction_ $u\in\mathcal{X}_{\beta}(\Omega)$ if

$\langle(-\Delta_{g})^{s}u,v\rangle_{*}=\lambda\int_{\Omega}g(|u|)\frac{u}{|u|}v\,dx-\int_{\mathbb{R}^{n}\setminus\Omega}\beta
g(|u|)\frac{u}{|u|}v\,dx\qquad\forall v\in\mathcal{X}_{\beta}(\Omega).$

As mentioned, eigenvalues for non-homogeneous eigenproblems strongly depend on
the energy level $\alpha>0$. It is possible to prove existence of an
eigenvalue by a multiplier argument once it is proved existence of the
following associated minimization problems

$\displaystyle\Lambda^{D}_{\alpha}$
$\displaystyle:=\min\\{\mathcal{J}(u)\colon u\in M^{D}_{\alpha}\\},\quad\text{
where }M^{D}_{\alpha}=\\{u\in
W^{s,G}_{0}(\Omega)\colon\mathcal{I}(u)=\alpha\\}$
$\displaystyle\Lambda^{N}_{\alpha}$
$\displaystyle=\min\\{\mathcal{J}_{0}(u)\colon u\in
M^{N}_{\alpha}\\},\quad\text{ where }M^{N}_{\alpha}=\\{u\in
W^{s,G}_{*}(\Omega)\colon\mathcal{I}(u)=\alpha\\}$
$\displaystyle\Lambda^{R}_{\alpha}$
$\displaystyle:=\min\\{\mathcal{J}_{\beta}(u)\colon u\in
M^{R}_{\alpha}\\},\quad\text{ where
}M^{R}_{\alpha}=\\{u\in\mathcal{X}_{\beta}(\Omega)\colon\mathcal{I}(u)=\alpha\\}.$

As pointed out in [6], by standard computations of the calculus of variations,
for each $\alpha>0$ the quantities $\Lambda_{\alpha}^{D}$,
$\Lambda_{\alpha}^{N}$ and $\Lambda_{\alpha}^{R}$ are attained by suitable
functions $u_{\alpha}^{D}$, $u_{\alpha}^{N}$ and $u_{\alpha}^{R}$,
respectively. Since condition ($G_{1}$) is assumed on $G$, the functionals
$\mathcal{J}$, $\mathcal{J_{\beta}}$ and $\mathcal{I}$ are class $C^{1}$ with
Frechét derivatives given by

(3.2)
$\displaystyle\begin{split}\langle(\mathcal{J}^{\prime}(u),v\rangle&=\langle(-\Delta_{g})^{s}u,v\rangle\qquad\forall
v\in W^{s,G}_{0}(\Omega),\\\
\langle(\mathcal{J}_{0}^{\prime}(u),v\rangle&=\langle(-\Delta_{g})^{s}u,v\rangle_{*}\qquad\forall
v\in W^{s,G}_{*}(\Omega),\\\
\langle(\mathcal{I}^{\prime}(u),v\rangle&=\int_{\Omega}g(|u|)\frac{u}{|u|}v\,dx\qquad\forall
v\in L^{G}(\Omega),\\\
\langle(\mathcal{J}^{\prime}_{\beta}(u),v\rangle&=\langle(-\Delta_{g})^{s}u,v\rangle_{*}+\int_{\mathbb{R}^{n}\setminus\Omega}\beta
g(|u|)\frac{u}{|u|}v\,dx\qquad\forall v\in W^{s,G}_{*}(\Omega),\end{split}$

and therefore, by an application of the Lagrange multipliers rule, there exist
numbers $\lambda^{D}_{\alpha}$, $\lambda^{N}_{\alpha}$ and
$\lambda^{R}_{\alpha}$ which are eigenvalues of $D(\Omega)$, $N(\Omega)$ and
$R(\Omega)$ with eigenfunctions $u_{\alpha}^{D}$, $u_{\alpha}^{N}$ and
$u_{\alpha}^{R}$, respectively. Furthermore, as proved in [6, Proposition 5.6
and 5.8], minimizers are comparable each other; more precisely,
$\Lambda_{\alpha}^{N}\leq\Lambda_{\alpha}^{R}\leq\Lambda_{\alpha}^{D}$, and a
similar relation for the eigenvalues holds, up to a multiplicative constant.

Proceeding as in [49, Theorem 1.3] (see Theorem 1.5) it can be proved that for
each fixed $\alpha_{0}>0$ it holds that

$\displaystyle\hat{\lambda}^{D}$
$\displaystyle:=\inf\\{\lambda^{D}_{\alpha}\colon\alpha\in(0,\alpha_{0})\\},$
$\displaystyle\hat{\lambda}^{N}$
$\displaystyle:=\inf\\{\lambda^{N}_{\alpha}\colon\alpha\in(0,\alpha_{0})\\},$
$\displaystyle\hat{\lambda}^{R}$
$\displaystyle:=\inf\\{\lambda^{R}_{\alpha}\colon\alpha\in(0,\alpha_{0})\\}$

are eigenvalues of $D(\Omega)$, $N(\Omega)$ and $R(\Omega)$, respectively.

In contrast with $p-$Laplacian type problems, when dealing with non-
homogeneous eigenproblems, eigenvalues are not variational in general. One
could consider the variational quantity

$\bar{\lambda}^{D}=\displaystyle\inf_{u\in
W^{s,G}_{0}(\Omega)\setminus\\{0\\}}\frac{\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}g(|D_{s}u|)|D_{s}u|\,d\mu}{\int_{\Omega}g(|u|)|u|\,dx},$

but in general this number cannot be probed to be an eigenvalue. In an
analogous way, we can define $\bar{\lambda}^{R}$ and $\bar{\lambda}^{N}$, and
the same assertion holds. However, the following result is true:

######  Theorem 3.1.

It holds that $\bar{\lambda}^{D}\leq\hat{\lambda}^{D}$ and there is no
eigenvalue $\lambda$ of $D(\Omega)$ such that $\lambda<\bar{\lambda}^{D}$.

The same holds by changing the superscript $D$ with $N$ or $R$.

###### Proof.

First observe that, since $\hat{\lambda}^{D}$ is an eigenvalue of $D(\Omega)$,
there exists a nontrivial eigenfunction $\hat{u}\in W^{s,G}_{0}(\Omega)$ such
that (3.1) holds, therefore

$\lambda_{0}=\inf\left\\{\frac{\langle(-\Delta_{g})^{s}u,u\rangle}{\int_{\Omega}g(|u|)|u|\,dx}\colon
u\in
W^{s,G}_{0}(\Omega)\right\\}\leq\frac{\langle(-\Delta_{g})^{s}\hat{u},\hat{u}\rangle}{\int_{\Omega}g(|u|)|u|\,dx}=\hat{\lambda}^{D}.$

If we suppose that there exists $\lambda<\lambda_{0}$ which is an eigenvalue
of problem $D(\Omega)$ with eigenfunction $u_{\lambda}\in
W^{s,G}_{0}(\Omega)$, we arrive to a contradiction since

$\lambda=\frac{\langle(-\Delta_{g})^{s}u_{\lambda},\hat{u}_{\lambda}\rangle}{\int_{\Omega}g(|u_{\lambda}|)|u_{\lambda}|\,dx}<\lambda_{0}=\inf\left\\{\frac{\langle(-\Delta_{g})^{s}u,u\rangle}{\int_{\Omega}g(|u|)|u|\,dx}\colon
u\in W^{s,G}_{0}(\Omega)\right\\}.$

The proofs of the Robin and Neumann case run analogously. ∎

We finish this section by proving Theorem 1.5.

###### Proof of Theorem 1.5.

Let $\\{\lambda_{\alpha_{k}}\\}_{k\in\mathbb{N}}$ be a sequence of eigenvalues
of $D(\Omega)$ such that $\lambda_{\alpha_{k}}\to\lambda$ and let
$\\{u_{\alpha_{k}}\\}_{k\in\mathbb{N}}\subset W^{s,G}_{0}(\Omega)$ be the
corresponding sequence of associated eigenfunctions such that
$\Phi_{G,\Omega}(u_{\alpha_{k}})=\alpha_{k}$ with
$\alpha_{k}\in(0,\alpha_{0})$ for all $k\in\mathbb{N}$. Then, it holds that

(3.3)
$\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}g(|D_{s}u_{\alpha_{k}}|)\frac{D_{s}u_{\alpha_{k}}}{|D_{s}u_{\alpha_{k}}|}D_{s}v\,d\mu=\lambda_{\alpha_{k}}\int_{\Omega}g(|u_{\alpha_{k}}|)\frac{u_{\alpha_{k}}}{|u_{\alpha_{k}}|}v\qquad\forall
v\in W^{s,G}_{0}(\Omega).$

Observe that $\\{u_{\alpha_{k}}\\}_{k\in\mathbb{N}}$ is bounded in
$W^{s,G}_{0}(\Omega)$ since by ($G_{1}$) and (3.3)

$\displaystyle\Phi_{s,G,\mathbb{R}^{n}}(u_{\alpha_{k}})$
$\displaystyle\leq\frac{1}{p^{-}}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}g(|D_{s}u_{\alpha_{k}}|)|D_{s}u_{\alpha_{k}}|\,d\mu$
$\displaystyle=\frac{\lambda_{\alpha_{k}}}{p^{-}}\int_{\Omega}g(|u_{\alpha_{k}}|)|u_{\alpha_{k}}|\leq\frac{p^{+}}{p^{-}}\lambda_{\alpha_{k}}\alpha_{k}$
$\displaystyle\leq\frac{p^{+}}{p^{-}}(1+\lambda)\alpha_{0}$

for $k$ big enough. Then, by using the compact embedding given in Proposition
2.5 (i), up to a subsequence, there exists $u\in W^{s,G}_{0}(\Omega)$ such
that

(3.4) $\begin{array}[]{ll}u_{\alpha_{k}}\to u&\text{ strongly in
}L^{G}(\Omega),\\\ u_{\alpha_{k}}\to u&\text{ a.e. in
}\mathbb{R}^{n}.\end{array}$

From the continuity of $t\mapsto g(t)\frac{t}{|t|}$ and (3.4) we deduce that

$g(|D_{s}u_{k}|)\frac{D_{s}u_{k}}{|D_{s}u_{k}|}\to
g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}\quad\text{ a.e. in }\Omega$

and also from (3.4)

$\limsup_{k\to\infty}\Phi_{G,\mathbb{R}^{n}}(u_{\alpha_{k}})\in(0,\alpha_{0}).$

Hence, taking limit as $k\to\infty$ in (3.3) we obtain that

$\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}D_{s}v\,d\mu=\lambda\int_{\Omega}g(|u|)\frac{u}{|u|}v\qquad\text{
for all }v\in W^{s,G}_{0}(\Omega)$

from where the proof in the Dirichlet case follows.

The proof for the Neumann and Robin case follows analogously by replacing
$W^{s,G}_{0}(\Omega)$ with $W^{s,G}_{*}(\Omega)$ or
$\mathcal{X}_{\beta}(\Omega)$, respectively, and using the compact embedding
given in Proposition 2.5 item (ii) to deduce (3.4). ∎

##  4\. Ljusternik-Schnirelman eigenvalues

In order to build a sequence of eigenvalues for problems $D(\Omega)$,
$N(\Omega)$ and $R(\Omega)$, the idea is to apply an abstract theorem of the
so-called Ljusternik-Schnirelman theory. See for instance [17, 47, 56]. We
will particularly use the result stated in [47, Theorem 9.27].

Given $\alpha>0$, assume that $\mathcal{A}$, $\mathcal{B}$ are two functionals
defined in a reflexive Banach space $\mathcal{X}$, such that

  * ($h_{1}$)

$\mathcal{A}$,$\mathcal{B}$ are $C^{1}(\mathcal{X},\mathbb{R})$ even
functionals with $\mathcal{I}(0)=\mathcal{B}(0)=0$ and the level set

$M_{\alpha}:=\\{u\in\mathcal{X}\colon\mathcal{B}(u)=\alpha\\}$

is bounded.

  * ($h_{2}$)

$\mathcal{A}^{\prime}$ is strongly continuous, i.e.,

$u_{k}\rightharpoonup u\text{ in
}\mathcal{X}\implies\mathcal{A}^{\prime}(u_{k})\to\mathcal{A}^{\prime}(u).$

Moreover, for any $u$ in the closure of the convex hull of $M_{\alpha}$,

$\langle\mathcal{A}^{\prime}(u),u\rangle=0\iff\mathcal{A}(u)=0\iff u=0.$

  * ($h_{3}$)

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

$u_{k}\rightharpoonup u,\quad\mathcal{B}^{\prime}(u_{k})\rightharpoonup
v,\quad\langle\mathcal{B}^{\prime}(u_{k}),u_{k}\rangle\to\langle
v,u\rangle\implies u_{k}\to u\text{ in }\mathcal{X}.$

  * ($h_{4}$)

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

$\langle\mathcal{B}^{\prime}(u),u\rangle>0,\qquad\displaystyle\lim_{t\to+\infty}\mathcal{B}(tu)=+\infty,\qquad\displaystyle\inf_{u\in
M_{\alpha}}\langle\mathcal{B}^{\prime}(u),u\rangle>0.$

Define max-min values

$c_{k,\alpha}=\begin{cases}\sup_{K\in\mathcal{C}_{k}}\inf_{u\in
K}\mathcal{A}(u),&\mathcal{C}_{k}\neq\emptyset,\\\
0,&\mathcal{C}_{k}=\emptyset,\end{cases}$

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

$\mathcal{C}_{k}:=\\{K\subset M_{\alpha}\text{ compact, symmetric with
}\mathcal{A}(u)>0\text{ on }K\text{ and }\gamma(K)\geq k\\},$

and the Krasnoselskii genus of $K$ is defined as

$\gamma(K):=\inf\\{p\in\mathbb{N}\colon\exists h\colon
K\to\mathbb{R}^{p}\setminus\\{0\\}\ \text{such that}\ h\text{ is continuous
and odd}\\}$

see [37] for details.

Thus, $\\{c_{k,\alpha}\\}_{k\geq 1}$ forms a nonincreasing sequence

$+\infty>c_{1,\alpha}\geq c_{2,\alpha}\geq\ldots\geq
c_{k,\alpha}\geq\ldots\geq 0.$

Under these considerations, the Ljusternik–Schnirelmann principle stated in
[47, Theorem 9.27] establishes that there exists a sequence
$\\{(\mu_{k,\alpha},u_{k,\alpha})\\}_{k\geq 1}$ such that

$\langle\mathcal{A}^{\prime}(u_{k,\alpha}),v\rangle=\mu_{k,\alpha}\langle\mathcal{B}^{\prime}(u),v\rangle\qquad\forall
v\in\mathcal{X}^{*}$

such that $u_{k,\alpha}\in M_{\alpha}$,
$\mathcal{A}(u_{k,\alpha})=c_{k,\alpha}$, $\mu_{k,\alpha}\neq 0$,
$\mu_{k,\alpha}\to 0$, and $u_{k,\alpha}\rightharpoonup 0$ in $\mathcal{X}$.

###  4.1. The Dirichlet case

With these preliminaries, we are in position to prove Theorem 1.1.

We consider the space $\mathcal{X}:=W^{s,G}_{0}(\Omega)$ and the functionals
$\mathcal{B}(u):=\mathcal{J}(u)$ and $\mathcal{A}(u):=\mathcal{I}(u)$ defined
in (2.3). As mentioned, these functionals are $C^{1}$ and their Frechét
derivatives are given in (3.2).

######  Lemma 4.1.

The functionals $\mathcal{J}$ and $\mathcal{I}$ defined above fulfill
hypotheses $(h_{1})$–$(h_{4})$.

###### Proof.

$(i)$ Clearly, the maps $\mathcal{I},\mathcal{J}$ are even and
$\mathcal{I}(0)=\mathcal{J}(0)=0$.

$(ii)$ We notice that

$p^{-}\xi^{-}(\|u\|_{L^{G}(\Omega)})\leq
p^{-}\mathcal{I}(u)\leq\langle\mathcal{I}^{\prime}(u),u\rangle\leq
p^{+}\mathcal{I}(u)\leq p^{+}\xi^{+}(\|u\|_{L^{G}(\Omega)}).$

Then immediately we obtain

$\langle\mathcal{I}^{\prime}(u),u\rangle=0\ \Leftrightarrow\ \mathcal{I}(u)=0\
\Leftrightarrow\ u=0.$

Thus, it remains to check that $\mathcal{I}^{\prime}$ is strongly continuous.
Let $u_{k}\rightharpoonup u$ in $W^{s,G}_{0}(\Omega)$, then
$\\{u_{k}\\}_{k\in\mathbb{N}}$ is bounded in $W^{s,G}_{0}(\Omega)$. We need to
show that $\mathcal{I}^{\prime}(u_{k})\to\mathcal{I}^{\prime}(u)$ in
$W^{-s,\tilde{G}}(\Omega)$.

$\displaystyle|\langle\mathcal{I}^{\prime}(u_{k})-\mathcal{I}^{\prime}(u),v\rangle|$
$\displaystyle=\left|\int_{\Omega}\left(g(|u_{k}|)\frac{u_{k}}{|u_{k}|}-g(|u|)\frac{u}{|u|}\right)v\,dx\right|$
$\displaystyle\leq\left|\int_{\Omega}g(|u_{k}|)\left(\frac{u_{k}}{|u_{k}|}-\frac{u}{|u|}\right)v\,dx\right|+\left|\int_{\Omega}(g(|u_{k}|)-g(|u|))\frac{u}{|u|}v\,dx\right|$
$\displaystyle:=I_{1,k}+I_{2,k}.$

By using Lemma 2.2 and Hölder’s inequality, the first term can be bounded as

$I_{1,k}\leq\|g(|u_{k}|)\|_{\tilde{G}}\|2v\|_{G}\rightarrow 0,\
k\rightarrow+\infty.$

Similarly, by using the $\Delta_{2}$ condition, Lemma 2.2 and Hölder’s
inequality we get

$I_{2,k}\leq\|g(|u_{k}|)-g(|u|)\|_{\tilde{G}}\|v\|_{G}.$

Since $u_{k}\rightharpoonup u$ in $W^{s,G}_{0}(\Omega)$, in light of
Proposition 2.5, $u_{k}\to u$ strongly in $L^{G}(\Omega)$ and a.e. in
$\mathbb{R}^{n}$, moreover, from Lemma 2.3, $g(|u_{k}|)-g(|u|)\in
L^{1}(\Omega)$, therefore, by dominated convergence theorem,
$\int_{\Omega}\tilde{G}(|g(|u_{k}|)-g(|u|)|)\,dx\to 0$ and hence
$\|g(|u_{k}|)-g(|u|)\|_{L^{\tilde{G}}(\Omega)}\to 0$ as $k\to\infty$.

From the last relations it follows that
$\|\mathcal{I}^{\prime}(u_{k})-\mathcal{I}^{\prime}(u)\|_{W^{-s,\tilde{G}}(\Omega)}\rightarrow
0$ as required.

$(iii)$ One can easily see that $\mathcal{J}^{\prime}$ is continuous (see for
instance [50, Proposition 4.1]). From Lemma 2.2 and Holder’s inequality

$|\langle\mathcal{J}^{\prime}(u),v|\rangle\leq\|g(|u|)\|_{L^{\tilde{G}}(\mathbb{R}^{n}\times\mathbb{R}^{n},d\mu)}\|v\|_{L^{G}(\mathbb{R}^{n}\times\mathbb{R}^{n},d\mu)}\leq
p^{+}[u]_{W^{s,G}(\mathbb{R}^{n})}[v]_{W^{s,G}(\mathbb{R}^{n})}$

from there $\mathcal{J}^{\prime}$ is bounded.

It remains to be showed that if $\\{u_{k}\\}_{k\in\mathbb{N}}$ is a sequence
in $W^{s,G}_{0}(\Omega)$ such that

(4.1) $u_{k}\rightharpoonup u,\quad\mathcal{J}^{\prime}(u_{k})\rightharpoonup
v,\quad\langle\mathcal{J}^{\prime}(u_{k}),u_{k}\rangle\to\langle v,u\rangle\ $

then $u_{k}\to u$ in $W^{s,G}_{0}(\Omega)$.

Since $G$ is convex, we have

$G(|D_{s}u|)\leq
G\bigg{(}\bigg{|}\frac{D_{s}u+D_{s}u_{k}}{2}\bigg{|}\bigg{)}+g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}.\frac{D_{s}u-D_{s}u_{k}}{2}$

and

$G(|D_{s}u_{k}|)\leq
G\bigg{(}\bigg{|}\frac{D_{s}u+D_{s}u_{k}}{2}\bigg{|}\bigg{)}+g(|D_{s}u_{k}|)\frac{D_{s}u_{k}}{|D_{s}u_{k}|}.\frac{D_{s}u_{k}-D_{s}u}{2}$

for every $u,v\in W^{s,G}_{0}(\Omega)$. Adding the above two relations and
integrating over $\mathbb{R}^{n}$ we find that

$\displaystyle\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\Big{(}g(|D_{s}u|)\frac{D_{s}u}{|D_{s}u|}$
$\displaystyle-g(|D_{s}u_{k}|)\frac{D_{s}u_{k}}{|D_{s}u_{k}|}\Big{)}(D_{s}u-D_{s}u_{k})\
d\mu$ (4.2)
$\displaystyle\geq\Phi_{s,G,\mathbb{R}^{n}}(u)+\Phi_{s,G,\mathbb{R}^{n}}(u_{k})-2\Phi_{s,G,\mathbb{R}^{n}}\left(\frac{u+u_{k}}{2}\right)$

for every $u,v\in W^{s,G}_{0}(\Omega)$. By applying Lemma 2.3, the right part
of the inequality above can be bounded by below by
$2\Phi_{s,G,\mathbb{R}^{n}}\left(\frac{u-u_{k}}{2}\right)$, and hence we get

$\langle(-\Delta_{g})(u)-(-\Delta_{g})(u_{k}),u-u_{k}\rangle\geq
4\Phi_{s,G,\mathbb{R}^{n}}(u-u_{k}).$

This, together with Lemma 2.6 yields

(4.3)
$\langle\mathcal{J}^{\prime}(u)-\mathcal{J}^{\prime}(u_{k}),u-u_{k}\rangle\geq\xi^{-}([u-u_{k}]_{W^{s,G}(\mathbb{R}^{n})}).$

On the other hand, Proposition 2.5 gives that $u_{k}\to u$ in $L^{G}(\Omega)$
and a.e. in $\mathbb{R}^{n}$, which, mixed up with the assumptions (4.1)
allows us to deduce that

$\displaystyle\lim_{k\to\infty}\langle\mathcal{J}^{\prime}(u_{k})-\mathcal{J}^{\prime}(u),u_{k}-u\rangle=\displaystyle\lim_{k\to\infty}\left(\langle\mathcal{J}^{\prime}(u_{k}),u_{k}\rangle-\langle\mathcal{J}^{\prime}(u_{k}),u\rangle-\langle\mathcal{J}^{\prime}(u),u_{k}-u\rangle\right)=0.$

Hence, from (4.3), $[u_{k}-u]_{W^{s,G}(\mathbb{R}^{n})}\to 0$ as $k\to\infty$
as required.

$(iv)$ It is clear that, for any $u\in W^{s,G}_{0}(\Omega)\setminus\\{0\\}$,

$\langle\mathcal{J}^{\prime}(u),u\rangle>0,\qquad\displaystyle\lim_{t\to+\infty}\mathcal{J}(tu)=+\infty,\qquad\displaystyle\inf_{u\in
M_{\alpha}}\langle\mathcal{J}^{\prime}(u),u\rangle>0.$

This concludes the proof. ∎

###### Proof of Theorem 1.1 (Dirichlet case).

In light of Lemma 4.1 and in virtue of [47, Theorem 9.27], there exist a
sequence of positive numbers $\\{\mu_{k}\\}_{k\in\mathbb{N}}$ tending to 0 and
a corresponding sequence of functions
$\\{u_{k,\alpha}^{D}\\}_{k\in\mathbb{N}}\in W^{s,G}_{0}(\Omega)$ such that

$\langle(-\Delta_{g})^{s}u_{k,\alpha}^{D},v\rangle=\frac{1}{\mu_{k,\alpha}^{D}}\int_{\Omega}g(|u_{k,\alpha}^{D}|)\frac{u_{k,\alpha}^{D}}{|u_{k,\alpha}^{D}|}v\,dx\quad\forall
v\in W^{s,G}_{0}(\Omega).$

Moreover, $\mathcal{J}(u_{k,\alpha}^{D})=\alpha$ and

$\mathcal{I}(u_{k,\alpha}^{D}):=c_{k,\alpha}^{D}=\sup_{K\in\mathcal{C}_{k}}\inf_{u\in
K}\mathcal{I}(u)>0.$

Consequently, $\lambda_{k,\alpha}^{D}=1/\mu_{k,\alpha}$ is an eigenvalue of
$D(\Omega)$ with eigenfunction $u_{k,\alpha}^{D}$. ∎

######  Remark 4.2.

We mention some observations regarding the Ljusternik-Schnirelman sequence of
eigenvalues obtained in Theorem 1.1.

  * (a)

Since in general these functionals are not homogeneous, we cannot claim that
$\lambda_{k,\alpha}^{D}=1/c_{k,\alpha}^{D}$.

  * (b)

The first Ljusternik-Schnirelman eigenvalue $\lambda_{1,\alpha}^{D}$ is in
fact different to the eigenvalue $\lambda^{D}_{\alpha}$ obtained in Section 3
by means of the Lagrange multipliers rule.

  * (c)

The sequence $\\{\lambda^{D}_{k,\alpha}\\}_{k\in\mathbb{N}}$ does not exhaust
the spectrum of $D(\Omega)$. In fact, the spectrum is not discrete since the
parameter $\alpha$ can be taken in $\mathbb{R}^{+}$.

  * (d)

The eigenfunction associated to $\lambda_{k,\alpha}^{D}$ is one-signed since
the $\Phi_{s,G,\mathbb{R}^{n}}(\cdot)$ and $\Phi_{G,\Omega}(\cdot)$ are
invariant by replacing $u_{k,\alpha}^{D}$ with $|u_{k,\alpha}^{D}|$.

  * (e)

As mentioned in Section 3, due to the closedness of the spectrum $\Sigma_{D}$
we have that the quantity $\inf\\{\lambda_{k,\alpha}^{D}\colon
0<\alpha<\alpha_{0}\\}$ is also an eigenvalue of $D(\Omega)$, for any
$\alpha_{0}>0$ fixed.

###  4.2. The Neumann/Robin case

To deal with the Robin case we take
$\mathcal{X}:=\mathcal{X}_{\beta}(\Omega)$, $\mathcal{A}:=\mathcal{I}(u)$ and
$\mathcal{B}:=\mathcal{J}_{\beta}(u)$. The Neumann case it follows just by
setting $\beta=0$.

###### Proof of Theorem 1.1 (Neumann/Robin case).

The proof of this result is similar to the proof of the Dirichlet case, just
noticing that, the embedding $\mathcal{X}_{\beta}(\Omega)\hookrightarrow
L^{G}(\Omega)$ is compact (see Proposition 2.5 (ii)) and the quantities
$[u]_{W^{s,G}(\mathbb{R}^{n})}$ and $[u]_{W^{s,G}_{*}(\mathbb{R}^{n})}$ play a
symmetrical role. ∎

##  5\. Minimax eigenvalues

This section is devoted to prove Theorem 1.2. We start with the Dirichlet
case.

Given $\alpha>0$ we recall that

$M_{\alpha}^{D}:=\\{u\in W^{s,G}_{0}(\Omega)\colon\mathcal{I}(u)=\alpha\\}$

defines a $C^{1}$ manifold. We denote $\tau_{\alpha}$ the restriction of
$(-\Delta_{g})^{s}$ to $M_{\alpha}^{D}$, i.e., for each $u\in M_{\alpha}^{D}$,

$\langle\tau_{\alpha}(u),v\rangle:=\langle(-\Delta_{g})^{s}u,v\rangle\quad\forall
v\in T_{u}M_{\alpha}^{D}$

where the tangent space of $M_{\alpha}^{D}$ at $u$ is defined as

$T_{u}M_{\alpha}^{D}:=\left\\{v\in
W^{s,G}_{0}(\Omega)\colon\langle\mathcal{I}^{\prime}(u),v\rangle=0\right\\}.$

Thus, we define

$\|\tau_{\alpha}u\|:=\|\tau_{\alpha}u\|_{(T_{u}M_{\alpha}^{D})^{*}}$

where $(T_{u}M_{\alpha}^{D})^{*}$ is the dual space of $T_{u}M_{\alpha}^{D}$.

We recall that the _duality mapping_ $J\colon
W^{s,G}_{0}(\Omega)\to(W^{s,G}_{0}(\Omega))^{*}$ (where we have denoted the
dual space of the space $\mathcal{X}$ as $\mathcal{X}^{*}$) is defined as a
bijective isometry such that

(5.1)
$\|Ju\|_{(W^{s,G}_{0}(\Omega))^{*}}=[u]_{W^{s,G}_{0}(\Omega)},\qquad\langle
Ju,u\rangle=[u]_{W^{s,G}(\mathbb{R}^{n})}^{2}\qquad\forall u\in
W^{s,G}_{0}(\Omega).$

See [55, Chapter 18.11c] and [56, Proposition 47.18] for details.

Given $\alpha>0$, the idea is to apply the minimax theorem stated in [21] to
the functional $\mathcal{J}(u)$ under the constraint $\mathcal{I}(u)=\alpha$
to obtain a sequence of critical points of the form

$C_{k,\alpha}^{D}=\inf_{h\in\Gamma(S^{k-1},M_{\alpha}^{D})}\sup_{w\in
S^{k-1}}\Phi_{s,G,\mathbb{R}^{n}}(h(w)).$

being $\Gamma(S^{k-1}k,M_{\alpha}^{D})=\\{h\in C(S^{k-1},M_{\alpha}^{D})\colon
h\text{ is odd}\\}$.

Recall that the derivatives of $\mathcal{I}$ and $\mathcal{J}$ are given by
$\mathcal{J}^{\prime}(u)=(-\Delta_{g})^{s}u$ and
$\mathcal{I}^{\prime}(u)=\frac{g(|u|)u}{|u|}$ for any $u\in L^{G}(\Omega)$.

######  Definition 5.1.

We say that $\mathcal{J}$ satisfies the _Palais-Smale condition on
$M_{\alpha}^{D}$ at level $c$_ if any sequence
$\\{u_{n}\\}_{n\in\mathbb{N}}\subset M_{\alpha}^{D}$ such that
$\mathcal{J}(u_{n})\to c$ and $\|\tau_{\alpha}u_{n}\|\to 0$, possesses a
convergent subsequence.

The key ingredient is to analyze the validity of the Palais-Smale condition.

######  Lemma 5.2.

Given $\alpha>0$ the functional $\mathcal{J}$ satisfies the Palais-Smale
condition on $M_{\alpha}^{D}$ at level $C_{k,\alpha}^{D}$.

###### Proof.

We follow closely the construction of [52, Theorem 5.3]. Given
$k\in\mathbb{N}$, let $\\{u_{n}\\}_{n\in\mathbb{N}}$ be a sequence in
$M_{\alpha}^{D}$ such that

(5.2) $\mathcal{I}(u_{n})\to C_{k,\alpha}^{D}\quad\text{ and
}\quad\|\tau_{\alpha}(u_{n})\|\to 0.$

For each $u\in M_{\alpha}^{D}$ define the projection $P_{u}\colon
W^{s,G}_{0}(\Omega)\to T_{u}M_{\alpha}^{D}$ such that

$P_{u}v=v-\frac{\langle\mathcal{I}^{\prime}(u),v\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}J^{-1}\left(\mathcal{I}^{\prime}(u)\right).$

Observe first that $P_{u}$ is well defined: given $v\in W^{s,G}_{0}(\Omega)$
and $u\in M_{\alpha}^{D}$ we have

$\langle\mathcal{I}^{\prime}(u),P_{u}v\rangle=\langle\mathcal{I}^{\prime}(u),v\rangle-\frac{\langle\mathcal{I}^{\prime}(u),v\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}\langle\mathcal{I}^{\prime}(u),J^{-1}\left(\mathcal{I}^{\prime}(u)\right)\rangle=0$

since due to (5.2),
$\langle\mathcal{I}^{\prime}(u),J^{-1}\left(\mathcal{I}^{\prime}(u)\right)\rangle=\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}$.

Moreover, for every $v\in W^{s,G}_{0}(\Omega)$ we get the inequality

$\displaystyle\langle\mathcal{I}^{\prime}(u),v\rangle\leq\|\mathcal{I}^{\prime}(u)\|_{(W^{s,G}_{0}(\Omega))^{*}}[v]_{W^{s,G}(\mathbb{R}^{n})}.$

and by (5.1) it follows that

$\left[J^{-1}\left(\mathcal{I}^{\prime}(u)\right)\right]_{W^{s,G}(\mathbb{R}^{n})}=\|\mathcal{I}^{\prime}(u)\|_{(W^{s,G}_{0}(\Omega))^{*}}.$

Therefore we get that $[P_{u}v]_{W^{s,G}(\mathbb{R}^{n})}\leq
2[v]_{{}_{W^{s,G}(\mathbb{R}^{n})}}$, which implies that, for any $v\in
W^{s,G}_{0}(\Omega)$

$\langle(-\Delta_{g})^{s}u,P_{u}v\rangle=\langle\tau_{\alpha}(u),P_{u}v\rangle\leq
2\|\tau_{\alpha}(u)\|[v]_{W^{s,G}(\mathbb{R}^{n})}.$

Consequently, by (5.2)

$\sup_{[v]_{W^{s,G}(\mathbb{R}^{n})}\leq
1}|\langle(-\Delta_{g})^{s}u_{n},P_{u_{n}}v\rangle|\leq\sup_{[v]_{W^{s,G}(\mathbb{R}^{n})}\leq
1}2\|\tau_{\alpha}(u_{n})\|[v]_{W^{s,G}(\mathbb{R}^{n})}\to 0\quad\text{as
}n\to\infty.$

Hence we get that

$\displaystyle\langle(-\Delta_{g})^{s}u_{n},P_{u_{n}}v\rangle$
$\displaystyle=\langle(-\Delta_{g})^{s}u_{n},v\rangle-\left\langle(-\Delta_{g})^{s}u_{n},\frac{\langle\mathcal{I}^{\prime}u_{n},v\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}J^{-1}(\mathcal{I}^{\prime}(u_{n}))\right\rangle$
$\displaystyle=\langle(-\Delta_{g})^{s}u_{n},v\rangle-\left\langle(-\Delta_{g})^{s}u_{n},J^{-1}(\mathcal{I}^{\prime}(u_{n}))\right\rangle\frac{\langle\mathcal{I}^{\prime}(u_{n}),v\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}$
$\displaystyle=\langle(-\Delta_{g})^{s}u_{n},v\rangle-\left\langle\frac{\left\langle(-\Delta_{g})^{s}u_{n},J^{-1}(\mathcal{I}^{\prime}(u_{n}))\right\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}\mathcal{I}^{\prime}(u_{n}),v\right\rangle\to
0,$

that is

$(-\Delta_{g})^{s}u_{n}-\frac{\left\langle(-\Delta_{g})^{s}u_{n},J^{-1}(\mathcal{I}^{\prime}(u_{n}))\right\rangle}{\|\mathcal{I}^{\prime}(u)\|^{2}_{(W^{s,G}_{0}(\Omega))^{*}}}\mathcal{I}^{\prime}(u_{n})\rightharpoonup
0\quad\text{ weakly in }W^{s,G}_{0}(\Omega).$

From (5.2), up to a subsequence $u_{n}\rightharpoonup u$ weakly in
$W^{s,G}_{0}(\Omega)$ and strongly in $L^{G}(\Omega)$ to some $u\in
W^{s,G}_{0}(\Omega)$ due to Proposition 2.5. Observe that in Lemma 4.1 we have
proved that $\mathcal{I}^{\prime}$ satisfies property $(h_{2})$ of Section 4,
i.e., $\mathcal{I}$ is strongly continuous, which implies that
$\mathcal{I}^{\prime}(u_{n})\to\mathcal{I}^{\prime}(u)$.

Moreover, (5.2) also gives that $(-\Delta_{g})^{s}u_{n}$ is bounded and
$u_{n}$ is bounded away from zero. Therefore, up to a subsequence

(5.3) $(-\Delta_{g})^{s}u_{n}\rightharpoonup v\quad\text{ weakly in
}(W^{s,G}_{0}(\Omega))^{*}$

In Lemma 4.1 we have proved that $\mathcal{J}^{\prime}=(-\Delta_{g})^{s}$
satisfies property $(h_{3})$ of Section 4, from where, in view of (5.3) we
have that $u_{n}\to u$ strongly in $W^{s,G}_{0}(\Omega)$. This proves that
$\mathcal{J}$ satisfies the Palais-Smale condition on $M_{\alpha}^{D}$ at
level $C_{k,\alpha}^{D}$ which concludes the proof. ∎

######  Remark 5.3.

Observe that in the proof of Lemma 5.2 we have not used any particular
property for functions in $W^{s,G}_{0}(\Omega)$. With the pertinent changes,
the same arguments can be applied to deduce that, given $\alpha>0$, the
functional $\mathcal{J}$ satisfies the Palais-Smale condition on
$M_{\alpha}^{N}$ (resp. $M_{\alpha}^{R}$) at level $C_{k,\alpha}^{N}$ (resp.
$C_{k,\alpha}^{R}$), where $C_{k,\alpha}^{N}$ (resp. $C_{k,\alpha}^{R}$) is
defined just by changing the superscript $D$ by $N$ (resp. $R$) in the
definition of $C_{k,\alpha}^{D}$. We leave to the reader the remaining
details.

###### Proof of Theorem 1.2 .

Due to Lemma 5.2, $\mathcal{J}$ (resp. $\mathcal{J}_{0}$
,$\mathcal{J}_{\beta}$) satisfies the Palais-Smale condition on
$M_{\alpha}^{D}$ (resp. $M_{\alpha}^{N}$, $M_{\alpha}^{R}$) at level
$C_{\alpha,k}^{D}$ (resp. $C_{\alpha,k}^{N}$, $C_{\alpha,k}^{R}$) for each
$k\in\mathbb{N}$, then by [21, Proposition 2.7] there exists
$u_{\alpha,k}^{D}\in M_{\alpha}^{D}$ (resp. $u_{\alpha,k}^{N}\in
M_{\alpha}^{N}$, $u_{\alpha,k}^{R}\in M_{\alpha}^{R}$) such that

$\mathcal{J}(u_{k,\alpha}^{D})=C_{k,\alpha}^{D},\qquad(\text{resp.}\
\mathcal{J}_{0}(u_{k,\alpha}^{N})=C_{k,\alpha}^{N},\,\,\mathcal{J}_{\beta}(u_{k,\alpha}^{R})=C_{k,\alpha}^{R})$

and

$\mathcal{J}^{\prime}(u_{k,\alpha}^{D})=\mathcal{J}_{0}^{\prime}(u_{k,\alpha}^{N})=\mathcal{J}_{\beta}^{\prime}(u_{k,\alpha}^{R})=0.$

Therefore, by the Lagrange multipliers rule, there exists
$\Lambda_{\alpha,k}^{D}\in\mathbb{R}$ (resp.
$\Lambda_{\alpha,k}^{N}\in\mathbb{R},\Lambda_{\alpha,k}^{R}\in\mathbb{R})$
such that

$\mathcal{J}^{\prime}(u_{\alpha,k}^{D})=\Lambda_{\alpha,k}^{D}\mathcal{I}^{\prime}(u_{\alpha,k}^{D})\quad\text{
weakly in }\Omega,$
$(\text{resp.}\quad\mathcal{J}_{0}^{\prime}(u_{\alpha,k}^{N})=\Lambda_{\alpha,k}^{N}\mathcal{I}^{\prime}(u_{\alpha,k}^{N}),\qquad\mathcal{J}_{\beta}^{\prime}(u_{\alpha,k}^{R})=\Lambda_{\alpha,k}^{R}\mathcal{I}^{\prime}(u_{\alpha,k}^{R})),$

that is, $\\{\Lambda^{D}_{k,\alpha}\\}_{k\in\mathbb{N}}$,
$\\{\Lambda^{N}_{k,\alpha}\\}_{k\in\mathbb{N}}$ and
$\\{\Lambda^{R}_{k,\alpha}\\}_{k\in\mathbb{N}}$ are eigenvalues of
$D(\Omega)$, $N(\Omega)$ and $R(\Omega)$, respectively, satisfying that

$\mathcal{J}(u_{k,\alpha}^{D})=C_{k,\alpha}^{D},\qquad\mathcal{J}_{0}(u_{k,\alpha}^{N})=C_{k,\alpha}^{N},\qquad\mathcal{J}_{\beta}(u_{k,\alpha}^{R})=C_{k,\alpha}^{R}.$

The proof is concluded. ∎

Finally we provide for a proof of the comparison result Theorem 1.4.

###### Proof of Theorem 1.4.

Let $u_{k,\alpha}^{D}$ be the eigenfunctions corresponding to
$\lambda_{k,\alpha}^{D}$. By definition of $\lambda_{k,\alpha}^{D}$ and
property ($G_{1}$) we get

$\lambda_{k,\alpha}^{D}=\frac{\langle(-\Delta_{g})^{s}u_{k,\alpha}^{D},u_{k,\alpha}^{D}\rangle}{\int_{\Omega}g(|u_{k,\alpha}^{D}|)|u_{k,\alpha}^{D}|\,dx}\leq\frac{p^{+}}{p^{-}}\frac{\Phi_{s,G,\mathbb{R}^{n}}(u_{k,\alpha}^{D})}{\Phi_{G,\Omega}(u_{k,\alpha}^{D})}=\frac{p^{+}}{p^{-}}\frac{\alpha}{c_{k,\alpha}^{D}}.$

In a similar way, if we consider the sequence $v_{k,\alpha}^{D}$ of
eigenfunctions corresponding to $\Lambda_{k,\alpha}^{D}$, by using ($G_{1}$)
we get

$\Lambda_{k,\alpha}=\frac{\langle(-\Delta_{g})^{s}v_{k,\alpha}^{D},v_{k,\alpha}^{D}\rangle}{\int_{\Omega}g(|v_{k,\alpha}^{D}|)|v_{k,\alpha}^{D}|\,dx}\leq\frac{p^{+}}{p^{-}}\frac{\Phi_{s,G,\mathbb{R}^{n}}(v_{k,\alpha}^{D})}{\Phi_{G,\Omega}(v_{k,\alpha}^{D})}=\frac{p^{+}}{p^{-}}\frac{C_{k,\alpha}^{D}}{\alpha}.$

The lower bounds follow analogously. ∎

Finally, we provide for a proof of Theorem 1.3

###### Proof of Theorem 1.3.

Step 1. In [27, Corollary 2.10] it is proved that $W^{s,G}_{0}(\Omega)\subset
W^{s,p^{-}}_{0}(\Omega)$, therefore

$[u]_{W^{s,p^{-}}(\mathbb{R}^{n})}\leq C[u]_{W^{s,G}(\mathbb{R}^{n})}.$

Given $u\in M_{\alpha}^{D}$ we get

$\mathcal{J}(u)=\Phi_{s,G,\mathbb{R}^{n}}(u)\geq\xi^{-}([u]_{W^{s,G}(\mathbb{R}^{n})})\geq
C\xi^{-}([u]_{W^{s,p^{-}}(\mathbb{R}^{n})})$

where we have used Lemma 2.6. Then

(5.4) $C[u]_{W^{s,p^{-}}(\mathbb{R}^{n})}^{\beta}\leq\mathcal{J}(u)$

for some $\beta=\beta(p^{\pm})$.

Step 2. Define

$\mathcal{M}_{\delta}^{D}:=\\{u\in
W^{s,p^{-}}_{0}(\Omega)\colon[u]^{\beta}_{L^{p^{-}}(\Omega)}\leq\delta\\}$

where $\beta$ is the same of step 1.

As in [27, Lemma 2.7] it can be seen that $L^{G}(\Omega)\subset
L^{p^{-}}(\Omega)$. Then $\|u\|_{L^{p^{-}}(\Omega)}\leq
C\|u\|_{L^{G}(\Omega)}$.

Given $u\in M_{\alpha}^{D}$,

$\alpha=\mathcal{I}(u)\geq\xi^{-}(\|u\|_{L^{G}(\Omega)})\geq
C\xi^{-}(\|u\|_{L^{p^{-}}(\Omega)}).$

Therefore, there exists some $\delta=\delta(\alpha,p^{\pm})$ such that
$\|u\|^{\beta}_{L^{p^{-}}(\Omega)}\leq\delta$ and then

(5.5) $M_{\alpha}^{D}\subset\mathcal{M}_{\delta}^{D}.$

Step 3. By (5.4) and (5.5)

(5.6)
$\displaystyle\begin{split}C_{k,\alpha}^{D}&=\inf_{h\in\Gamma(S^{k-1},M_{\alpha}^{D})}\sup_{w\in
S^{k-1}}\mathcal{J}(h(w))\\\ &\geq
C\inf_{h\in\Gamma(S^{k-1},\mathcal{M}_{\delta}^{D})}\sup_{w\in
S^{k-1}}[h(w)]_{W^{s,p^{-}}}^{\beta}:=C(\mu_{k}^{D})^{\beta}\end{split}$

where $\mu_{k}^{D}$ is the minimax eigenvalue of the $p^{-}-$Laplacian with
Dirichlet boundary conditions obtained in [43, Theorem 4.1] (observe that
since the $p^{-}-$Laplacian is a homogeneous operator, in fact the same
eigenvalue is obtained for any $\delta$).

Step 4. From Theorem 1.4 and (5.6) we get

$C\frac{p^{-}}{\alpha p^{+}}(\mu_{k}^{D})^{\beta}\leq\frac{p^{-}}{\alpha
p^{+}}C_{k,\alpha}^{D}\leq\Lambda_{k,\alpha}^{D}.$

Since $\mu_{k}^{D}\to\infty$ as $k\to\infty$ we obtain that
$C_{k,\alpha}^{D},\Lambda_{k,\alpha}^{D}\to\infty$ as $k\to\infty$. ∎

### Acknowledgements.

The third author was partially supported by Consejo Nacional de
Investigaciones Científicas y Técnicas (CONICET-Argentina).

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