# Existence Results for double phase problem in Sobolev-Orlicz spaces with
variable exponents in Complete Manifold

Ahmed Aberqi1 Jaouad Bennouna2 Omar Benslimane2
Maria Alessandra Ragusa3
1Laboratory LAMA, Sidi Mohamed Ben Abdellah University, National School of
Applied Sciences, Fez, Morocco,
2Laboratory LAMA, Department of Mathematics, Sidi Mohamed Ben Abdellah
University, Faculty of Sciences Dhar El Mahraz, B.P 1796 Atlas Fez, Morocco,
3Dipartimento di Matematica e Informatica, Università di Catania, Catania,
Italy
RUDN University, 6 Miklukho-Maklay St, 117198, Moscow, Russia

###### Abstract

In this paper, we study the existence of non-negative non-trivial solutions
for a class of double-phase problems where the source term is a Caratheodory
function that satisfies the Ambrosetti-Rabinowitz type condition in the
framework of Sobolev-Orlicz spaces with variable exponents in complete compact
Riemannian n-manifolds. Our approach is based on the Nehari manifold and some
variational techniques. Further-more, the Hölder inequality, continuous and
compact embedding results are proved.

Mathematics Subject Classification(2000): 35J20, 35J47, 35J60.
Key words: Existence solutions; Double phase problem; Sobolev-Orlicz
Riemannian manifold; Nehari manifold.

##  1 Introduction

Let $(M,\,g)$ be a smooth, complete compact Riemannian n-manifold. In this
paper, we focused on the existence of non-trivial solutions of the following
double phase problem

$(\mathcal{P})\begin{cases}\,-\mbox{div}(\,|\,\nabla u(x)\,|^{p(x)-2}\nabla
u+\mu(x)\,|\,\nabla u(x)\,|^{q(x)-2}\,\nabla u)\\\ \hskip
8.5359pt=\lambda|\,u(x)\,|^{q(x)-2}\,u(x)-|\,u(x)\,|^{p(x)-2}\,u(x)+f(x,u(x))&\text{
in $M$ },\\\\[8.5359pt] \,u\,=\,0&\text{ on $\partial M$ },\end{cases}$

where $-\Delta_{p(x)}u(x)=-\,div\,(\,|\,\nabla u(x)\,|^{p(x)-2}\,.\,\nabla
u(x)),$ $-\Delta_{q(x)}u(x)=\\\ -\,div\,(\,|\,\nabla
u(x)\,|^{q(x)-2}\,.\,\nabla u(x))$ are the $p(x)$-laplacian and
$q(x)$-laplacian in $(M,g)$ respectively, $\lambda>0$ is a parameter specified
later, the function $\mu:\overline{M}\rightarrow\mathbb{R}^{+}_{*}$ is
supposed to be Lipschitz continuous, and the variables exponents $p,\,q\in
C(\overline{M})$ satisfy the assumption (3.1) in section 3.
The perturbation $f(x,u)$ is a Caratheodory function which satisfies the
Ambrosetti-Rabinowitz type condition:
$(f_{1}):$ There exists $\beta>p^{+}$ and some $A>0$ such as for each
$|\,\alpha\,|>A$ we have

$0<\int_{M}F(x,\alpha)\,\,dv_{g}(x)\leq\int_{M}f(x,\alpha)\,.\,\frac{\alpha}{\beta}\,\,dv_{g}(x)\,\,\mbox{a.e}\,\,x\in
M,$

where $\displaystyle F(x,\alpha)=\int_{0}^{\alpha}f(x,t)\,\,dt$ being the
primitive of $f(x,\alpha)$ and $dv_{g}=\sqrt{\mbox{det}(g_{ij})}\,\,dx$ is the
Riemannian volume element on $(M,g),$ where the $g_{ij}$ are the components of
the Riemannian metric $g$ in the chart and $dx$ is the Lebesgue volume element
of $\mathbb{R}^{N}.$
$(f_{2}):\,f(x,0)=0.$
And
$(f_{3}):\,\displaystyle\lim_{|\,\alpha\,|\rightarrow
0}\frac{f(x,\alpha)}{|\,\alpha\,|^{q(x)-1}}=0$ uniformly a.e $x\in M$.

Up to this day, several contributions have been devoted to study double phase
problems. This kind of operator was introduced, first, by Zhikov in his
relevant paper [25] in order to describe models with strongly anisotropic
materials by studying the functional

$u\longmapsto\int_{\Omega}(|\,\nabla u\,|^{p}+\mu(x)\,|\,\nabla
u\,|^{q})\,\,dx,$

where $1<p<q<N$ and with a nonnegative weight function $\mu\in
L^{\infty}(\Omega),$ see also the works of Zhikov [26, 27] and the monograph
of Zhikov-Kozlov-Oleinik [28]. Indeed, we can easily see that the previous
function reduces to p-laplacian if $\mu(x)=0$ or to the weighted laplacian
$(p(x),\,q(x))$ if $\displaystyle\inf_{x\in\overline{M}}\mu(x)>0,$
respectively.
In the case of single valued equations, Liu-Dai in [17] discussed double phase
problems and proved the existence and the multiplicity of the results, with
the sign-changing solutions by variational method. A similar treatment has
been recently done by Gasiński and Papageorgiou in [14] via the Nehari
manifold method. Following this direction, Papageorgiou, N. S. and Repovš, D.
D. and Vetro, C. in [19] studied the existence of positive solutions for a
class of double phase Dirichlet equations which has the combined effects of a
singular term and of a parametric super-linear term. In particular, in [24]
the author provides the Hölder continuity up to the boundary of minimizers of
so-called double phase functional with variable exponents, under suitable
Dirichlet boundary conditions. For more details, we refer the reader to [3,
10, 13, 16, 15, 21, 18, 23] and the references therein.

Also, there are many articles on nonstandard growth problems, especially on
$p(x)$-growth and double phase problems. About $p(x)$-growth problems, see [1,
2, 4, 5, 6, 7, 8, 9, 11, 12, 22] and the references given there.

Studying this type of problems is both significant and relevant. In the one
hand, we have the physical motivation; since the double phase operator has
been used to model the steady-state solutions of reaction-diffusion problems,
that arise in biophysic, plasma-physic and in the study of chemical reactions.
In the other hand, these operators provide a useful paradigm for describing
the behaviour of strongly anisotropic materials, whose hardening properties
are linked to the exponent governing the growth of the gradient change
radically with the point, where the coefficient $\mu(.)$ determines the
geometry of a composite made of two different materials.

Motivated by the aforementioned works, the aim of this paper, is to prove the
existence of non-negative non-trivial solutions of the problem $(\mathcal{P})$
where the perturbation $f(x,u)$ is a Caratheodory function, that satisfies the
Ambrosetti-Rabinowitz type condition. To the best of our knowledge, the
existence result for double-phase problems $(\mathcal{P})$ in the framework of
Sobolev-Orlicz spaces with variable exponents in complete manifold has not
been considered in the literature. The present paper is the first study
devoted to this type of problem in the setting of Sobolev-Orlicz spaces with
variable exponents in a complete manifold.

We would like to draw attention to the fact that the $p(x)-$laplacian operator
has more complicated non-linearity than the p-laplacian operator. For example,
they are non-homogeneous. Thus, we cannot use the Lagrange Multiplier Theorem
in many problems involving this operators, which prove that our problem is
more difficult than the operators p-Laplacian type.

The paper is organized as follows. In section 2, we recall the most important
and relevant properties and notations of Lebesgue spaces with variable
exponents and Sobolev-Orlicz spaces with variable exponents in complete
manifold. Moreover, we show two new results: the first one, the Hölder
inequality and the second one, the embedding result of these spaces into
Lebesgue space with variable exponent. In section 3, we introduce the Nehari
manifold associated with $(\mathcal{P})$ and we study three parts,
corresponding to local minima, local maxima and the points of inflection.
Finally, in section 4, we demonstrate the existence of two non-negative non-
trivial solutions of the problem $(\mathcal{P}).$

##  2 Mathematical background and auxiliary results

In this section, we recall the most important and relevant properties and
notations about Sobolev spaces with variable exponents and Sobolev spaces with
variable exponents on manifolds, and we prove some properties, that we will
need in our analysis of the problem $(\mathcal{P})$, by that, referring to [3,
16, 12, 20, 7, 13] for more details.

###  2.1 Sobolev spaces with variable exponents

Let $\Omega$ be a bounded open subset of $\mathbb{R}^{N}\,(N\geq 2)$, we
define the Lebesgue space with variable exponent $L^{q(\cdot)}(\Omega)$ as the
set of all measurable function $u:\Omega\longmapsto\mathbb{R}$ for which the
convex modular

$\rho_{q(\cdot)}(u)=\int_{\Omega}|\,u(x)\,|^{q(x)}\,\,dx,$

is finite. If the exponent is bounded, i.e if
$q^{+}=ess\,sup\\{\,q(x)/x\in\Omega\,\\}<+\infty,$ then the expression

$||\,u\,||_{q(\cdot)}=\inf\\{\,\lambda>0:\,\rho_{q(\cdot)}\bigg{(}\frac{u}{\lambda}\bigg{)}\leq
1\,\\},$

defines a norm in $L^{q(\cdot)}(\Omega)$, called the Luxemburg norm. The space
$(L^{q(\cdot)}(\Omega),\,||\,.\,||_{q(\cdot)})$ is a separable Banach space.
Moreover, if $1<q^{-}\leq q^{+}<+\infty,$ then $L^{q(\cdot)}(\Omega)$ is
uniformly convex, where $q^{-}=ess\,inf\\{\,q(x)/x\in\Omega\,\\},$ hence
reflexive, and its dual space is isomorphic to
$L^{q^{{}^{\prime}}(\cdot)}(\Omega)$ where
$\frac{1}{q(x)}+\frac{1}{q^{{}^{\prime}}(x)}=1.$
Finally, we have the Hölder type inequality:

$\bigg{|}\,\int_{\Omega}u\,v\,\,dx\,\bigg{|}\leq\bigg{(}\,\frac{1}{q^{-}}+\frac{1}{(q^{{}^{\prime}})^{-}}\bigg{)}\,||\,u\,||_{q(\cdot)}||\,v\,||_{q^{{}^{\prime}}(\cdot)},$

for all $u\in L^{q(\cdot)}(\Omega)$ and $v\in
L^{q^{{}^{\prime}}(\cdot)}(\Omega)$.
Now, we define the variable exponent Sobolev space by

$W^{1,q(\cdot)}(\Omega)=\\{u\in L^{q(\cdot)}(\Omega)\,\,\mbox{and}\,\,|\nabla
u|\in L^{q(\cdot)}(\Omega)\\}.$

which is a Banach space equipped with the following norm

$||u||_{1,q(\cdot)}=||u||_{q(\cdot)}+||\nabla u||_{q(\cdot)}\hskip
28.45274pt\forall u\in W^{1,q(\cdot)}(\Omega).$

The space $(W^{1,q(\cdot)}(\Omega),||\cdot||_{1,q(\cdot)})$ is separable and
reflexive Banach space.
We denote by $W_{0}^{1,q(\cdot)}(\Omega)$ the closure $C_{0}^{\infty}(\Omega)$
in $W^{1,q(\cdot)}(\Omega).$

######  Proposition 2.1.

[11] ( Poincaré inequality ) If $q\in C_{+}(\overline{\Omega}),$ then there is
a constant $c>0$ such that

$||\,u\,||_{q(x)}\leq c\,||\,|\,\nabla u\,|\,||_{q(x)},\hskip
14.22636pt\forall u\in W_{0}^{1,\,q(x)}(\Omega).$

Where, $\displaystyle C_{+}(\overline{\Omega})=\\{q/q\in
C(\overline{\Omega}),\,q(x)>1\,\,\mbox{for}\,\,x\in\overline{\Omega}\\}.$
Consequently, $||\,u\,||=||\,|\,\nabla u\,|\,||_{q(x)}$ and
$||\,u\,||_{1,\,q(x)}$ are equivalent norms on $W_{0}^{1,q(x)}(\Omega).$

###  2.2 Sobolev spaces with variable exponents on manifolds

In the following, all the manifolds we consider are smooth, and we will use
the following conditions on $(M,g)$, depending on the context:

######  Definition 2.1.

Let $(M,g)$ be a smooth Riemannain n-manifolds and let $\nabla$ be the Levi-
Civita connection. If $u$ is a smooth function on $M$, then $\nabla^{k}u$
denotes the $k-$th covariant derivative of $u$, and $|\,\nabla^{k}u\,|$ the
norm of $\nabla^{k}u$ defined in local coordinates by

$|\,\nabla^{k}u\,|^{2}=g^{i_{1}j_{1}}\cdots
g^{i_{k}j_{k}}\,(\nabla^{k}u)_{i_{1}\cdots i_{k}}\,(\nabla^{k}u)_{j_{1}\cdots
j_{k}}$

where Einstein’s convention is used.

######  Remark 2.1.

A smooth manifold $M$ of dimension $n$ is a connected topological manifold $M$
of dimension $n$ together with a $C^{\infty}-$complete atlas.

######  Example 2.2.

The following examples are classical examples of smooth manifolds:
$\bullet\hskip 11.38092pt$ The Euclidean space $\mathbb{R}^{n}$ itself.
$\bullet\hskip 11.38092pt$ The torus $T^{n}.$
$\bullet\hskip 11.38092pt$ The unit sphere $S^{n}$ of $\mathbb{R}^{n+1}.$
$\bullet\hskip 11.38092pt$ The real projective space
$\mathbb{P}^{n}(\mathbb{R}).$

######  Definition 2.2.

To define variable Sobolev spaces, given a variable exponent $q$ in
$\mathcal{P}(M)$ ( the set of all measurable functions
$p(\cdot):M\rightarrow(1,\infty)$ ) and a natural number $k$, introduce

$C^{q(\cdot)}_{k}(M)=\\{\,u\in C^{\infty}(M)\,\,\mbox{such that }\,\,\forall
j\,\,0\leq j\leq k\,\,|\,\nabla^{j}u\,|\in L^{q(\cdot)}(M)\,\\}.$

On $C^{q(\cdot)}_{k}(M)$ define the norm

$||\,u\,||_{L^{q(\cdot)}_{k}(M)}=\sum_{j=0}^{k}||\,\nabla^{j}u\,||_{L^{q(\cdot)}(M)}.$

######  Definition 2.3.

The Sobolev spaces $L_{k}^{q(\cdot)}(M)$ is the completion of
$C^{q(\cdot)}_{k}(M)$ with respect to the norm $||\,u\,||_{L^{q(\cdot)}_{k}}$.
If $\Omega$ is a subset of $M,$ then $L_{k,0}^{q(\cdot)}(\Omega)$ is the
completion of $C_{k}^{q(\cdot)}(M)\cap C_{0}(\Omega)$ with respect to
$||.||_{L^{q(\cdot)}_{k}},$ where $C_{0}(\Omega)$ denotes the vector space of
continuous functions whose support is a compact subset of $\Omega.$

######  Definition 2.4.

Given $(M,g)$ a smooth Riemannian manifold, and
$\gamma:\,[\,a,\,b\,]\longrightarrow M$ is a curve of class $C^{1}$. The
length of $\gamma$ is

$l(\gamma)=\int_{a}^{b}\sqrt{g\,(\,\frac{d\gamma}{dt},\,\frac{d\gamma}{dt}\,)}\,\,dt,$

and for a pair of points $x,\,y\in M$, we define the distance $d_{g}(x,y)$
between $x$ and $y$ by

$d_{g}(x,y)=\inf\,\\{\,l(\gamma):\,\gamma:\,[\,a,\,b\,]\rightarrow
M\,\,\mbox{such that}\,\,\gamma(a)=x\,\,\mbox{and}\,\,\gamma(b)=y\,\\}.$

######  Definition 2.5.

A function $s:\,M\longrightarrow\mathbb{R}$ is log-Hölder continuous if there
exists a constant $c$ such that for every pair of points $\\{x,\,y\\}$ in $M$
we have

$|\,s(x)-s(y)\,|\leq\frac{c}{log(e+\frac{1}{d_{g}(x,y)}\,)}.$

We note by $\mathcal{P}^{log}(M)$ the set of log-Hölder continuous variable
exponents. The relation between $\mathcal{P}^{log}(M)$ and
$\mathcal{P}^{log}(\mathbb{R}^{N})$ is the following:

######  Proposition 2.2.

[3, 13] Let $q\in\mathcal{P}^{log}(M)$, and let $(\Omega,\phi)$ be a chart
such that

$\frac{1}{2}\delta_{ij}\leq g_{ij}\leq 2\,\delta_{ij}$

as bilinear forms, where $\delta_{ij}$ is the delta Kronecker symbol. Then
$qo\phi^{-1}\in\mathcal{P}^{log}(\phi(\Omega)).$

######  Proposition 2.3.

( Hölder’s inequality ) For all $u\in L^{q(\cdot)}(M)$ and $v\in
L^{q^{\prime}(\cdot)}(M)$ we have

$\int_{M}|\,u(x)\,v(x)\,|\,\,dv_{g}(x)\leq
r_{q}\,||\,u\,||_{L^{q(\cdot)}(M)}\,.\,||\,v\,||_{L^{q^{\prime}(\cdot)}(M)}.$

Where $r_{q}$ be a positive constant depend to $q^{-}$ and $q^{+}.$

###### Proof.

Obviously, we can suppose that $||u||_{L^{q(x)}(M)}\neq 0$ and
$||v||_{L^{q^{\prime}(x)}(M)}\neq 0,$ we have

$1<q(x)<\infty,\,\,|u(x)|<\infty,\,\,|v(x)|<\infty\,\,\mbox{a.e}\,\,x\in M.$

By young inequality, we have

$\displaystyle\frac{u(x)\,.\,v(x)}{||u(x)||_{L^{q(x)}(M)}\,.\,||v(x)||_{L^{q^{\prime}(x)}(M)}}\leq$
$\displaystyle\,\frac{1}{q(x)}\,\bigg{(}\frac{|u(x)|}{||u(x)||_{L^{q(x)}(M)}}\bigg{)}^{q(x)}$
$\displaystyle+\frac{1}{q^{\prime}(x)}\,\bigg{(}\frac{|v(x)|}{||v(x)||_{L^{q^{\prime}(x)}(M)}}\bigg{)}^{q^{\prime}(x)}$

Integrating over $M$, we obtain

$\displaystyle\int_{M}\frac{|\,u(x)\,.\,v(x)\,|}{||u(x)||_{L^{q(x)}(M)}\,.\,||v(x)||_{L^{q^{\prime}(x)}(M)}}\,\,dv_{g}(x)$
$\displaystyle\leq\frac{1}{q^{-}}\,\int_{M}\bigg{(}\frac{|u(x)|}{||u(x)||_{L^{q(x)}(M)}}\bigg{)}^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+\big{(}1-\frac{1}{q^{+}}\big{)}\,\int_{M}\bigg{(}\frac{|v(x)|}{||v(x)||_{L^{q^{\prime}(x)}(M)}}\bigg{)}^{q^{\prime}(x)}\,\,dv_{g}(x)$
$\displaystyle\leq 1+\frac{1}{q^{-}}-\frac{1}{q^{+}},$

then, using the same technique as in the proof of Theorem 1.15 in [12], we get
that

$\displaystyle\int_{M}|\,u(x)\,.\,v(x)\,|\,\,dv_{g}(x)$
$\displaystyle\leq\big{(}1+\frac{1}{q^{-}}+\frac{1}{q^{+}}\big{)}\,||u(x)||_{L^{q(x)}(M)}\,.\,||v(x)||_{L^{q^{\prime}(x)}(M)}$
$\displaystyle\leq
r_{q}||u(x)||_{L^{q(x)}(M)}\,.\,||v(x)||_{L^{q^{\prime}(x)}(M)},$

Which complete the proof. ∎

######  Remark 2.3.

If $a$ and $b$ are two positive functions on $M$, then by Hölder’s inequality
and [11, 13] we have

$\int_{q^{-}<2}a^{\frac{q^{-}}{2}}\,b^{\frac{2q^{-}-{q^{-}}^{2}}{2}}\,\leq
2\,||\,\mathbb{1}_{q^{-}<2}\,\,a^{\frac{q^{-}}{2}}\,||_{L^{\frac{2}{q^{-}}}}\,.\,||\,\mathbb{1}_{q^{-}<2}\,\,b^{\frac{2q^{-}-{q^{-}}^{2}}{2}}\,||_{L^{\frac{2}{2-q^{-}}}}.$
(2.1)

where $\mathbb{1}$ is the indicator function of $M$, moreover, since

$||\,\mathbb{1}_{q^{-}<2}\,\,a^{\frac{2}{q^{-}}}\,||_{L^{\frac{2}{q^{-}}}}\leq\max\\{\,\rho_{1}(a),\,\rho_{1}(a)^{\frac{q^{-}}{2}}\,\\}$

and

$||\,\mathbb{1}_{q^{-}<2}\,\,b^{\frac{2q^{-}-{q^{-}}^{2}}{2}}\,||_{L^{\frac{2}{2-q^{-}}}}\leq\max\\{\,\rho_{q(\cdot)}(b)^{\frac{2-q^{-}}{2}},\,1\,\\},$

we get,

$\int_{q^{-}<2}a^{\frac{q^{-}}{2}}\,.\,b^{\frac{2q^{-}-{q^{-}}^{2}}{2}}\leq
2\,\max\,\\{\,\rho_{1}(a),\rho_{1}(a)^{\frac{q^{-}}{2}}\,\\}\,\max\,\\{\,\rho_{q(\cdot)}(b)^{\frac{2-q^{-}}{2}},\,1\,\\}.$
(2.2)

######  Definition 2.6.

We say that the Riemannian n-manifold $(M,g)$ has property
$B_{vol}(\lambda,v)$ where $\lambda$ is a constant, if its geometry is bounded
in the following sense:
$\hskip 28.45274pt\bullet\,\,\mbox{The Ricci tensor of g noted by Rc ( g )
verifies,}\,\,Rc(g)\geq\lambda(n-1)\,g$ for some $\lambda,$ where $n$ is the
dimension of $M.$
$\hskip 28.45274pt\bullet$ There exists some $v>0$ such that
$|\,B_{1}(x)\,|_{g}\geq v\,\,\forall x\in M,$ where $B_{1}(x)$ are the balls
of radius 1 centered at some point $x$ in terms of the volume of smaller
concentric balls.

######  Remark 2.4.

If $M=\Omega\subseteq\mathbb{R}^{N}$ is a bounded open set, then the following
inequality is related to the two exponents $p,\,q$ ( isotropic case )

$\frac{q}{p}<1+\frac{1}{N}.$

This condition is essential, among others, for the embeddings of spaces to be
satisfied.

######  Proposition 2.4.

[3, 16] Let $(M,g)$ be a complete compact Riemannian n-manifold. Then, if the
embedding $L^{1}_{1}(M)\hookrightarrow L^{\frac{n}{n-1}}(M)$ holds, then
whenever the real numbers $q$ and $p$ satisfy

$1\leq q<n,$

and

$q\leq p\leq q*=\frac{nq}{n-q},$

the embedding $L^{q}_{1}(M)\hookrightarrow L^{p}(M)$ also holds.

######  Proposition 2.5.

[3, 16] Assume that the complete compact Riemannian n-manifold $(M,g)$ has
property $B_{vol}(\lambda,v)$ for some $(\lambda,v).$ Then there exist
positive constants $\delta_{0}=\delta_{0}(n,\,\lambda,\,v)$ and
$A=A(n,\,\lambda,\,v)$, we have, if $R\leq\delta_{0}$, if $x\in M$ if $1\leq
q\leq n$, and if $u\in L^{q}_{1,0}(\,B_{R}(x)\,)$ the estimate

$||\,u\,||_{L^{p}}\leq A\,p\,||\,\nabla u\,||_{L^{q}},$

where $\frac{1}{p}=\frac{1}{q}-\frac{1}{n}.$

We can extend the above proposition from the case when exponents $p$ and $q$
are constant, when $p(\cdot)$ and $q(\cdot)$ are functions.

######  Proposition 2.6.

Assume that for some $(\lambda,v)$ the complete compact Riemannian n-manifold
$(M,g)$ has property $B_{vol}(\lambda,v)$. Then there exist positive constants
$\delta_{0}=\delta_{0}(n,\,\lambda,\,v)$ and $A=A(n,\,\lambda,\,v)$, we have,
if $R\leq\delta_{0}$, if $x\in M$ if $1\leq q(\cdot)\leq n$, and if $u\in
L^{q(\cdot)}_{1,0}(\,B_{R}(x)\,)$ the estimate

$||\,u\,||_{L^{p(\cdot)}}\leq A\,p^{-}\,||\,\nabla u\,||_{L^{q(\cdot)}},$

where $\frac{p(\cdot)}{q(\cdot)}<1+\frac{1}{n}.$

###### Proof.

To demonstrate this Proposition, we use the same technique as proposition 2.5,
for more detail see [3, 16]. ∎

In the following, we denote for all $u\in W_{0}^{1,q(x)}(M)$ that

$\rho_{p(\cdot)}(u)=\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)\,\,\,\mbox{and}\,\,\,\rho_{q(\cdot)}(u)=\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x).$

######  Proposition 2.7.

[3, 16, 13] Assume that for some $(\lambda,v)$ the complete compact Riemannian
n-manifold $(M,g)$ has property $B_{vol}(\lambda,v)$. Let $p\in\mathcal{P}(M)$
be uniformly continuous with $q^{+}<n.$ Then
$L^{q(\cdot)}_{1}(M)\hookrightarrow L^{p(\cdot)}(M),\,\forall
q\in\mathcal{P}(M)$ such that $q\ll p\ll q*=\frac{nq}{n-q}.$ In fact, for
$||\,u\,||_{L^{q(\cdot)}_{1}}$ sufficiently small we have the estimate

$\rho_{p(\cdot)}(u)\leq G\,(\,\rho_{q(\cdot)}(u)+\rho_{q(\cdot)}(|\,\nabla
u\,|)\,),$

where the positive constant $G$ depend on $n,\,\lambda,\,v,\,q$ and $p$.

######  Proposition 2.8.

[15] Let $u\in L^{q(x)}(M),\,\\{\,u_{k}\,\\}\subset
L^{q(x)}(M),\,k\in\mathbb{N},$ then we have

  1. (i)

$||u||_{q(x)}<1\,\,\mbox{( resp. = 1, $>$ 1 )}\iff\rho_{q(x)}(u)<1\,\,\mbox{(
resp. = 1, $>$ 1 )},$

  2. (ii)

$||u||_{q(x)}<1\Rightarrow||u||_{q(x)}^{q^{+}}\leq\rho_{q(x)}(u)\leq||u||_{q(x)}^{q^{-}},$

  3. (iii)

$||u||_{q(x)}>1\Rightarrow||u||_{q(x)}^{q^{-}}\leq\rho_{q(x)}(u)\leq||u||_{q(x)}^{q^{+}},$

  4. (iv)

$\lim_{k\rightarrow+\infty}||u_{k}-u||_{q(x)}=0\iff\lim_{k\rightarrow+\infty}\rho_{q(x)}(u_{k}-u)=0.$

To compare the functionals $||\,\,\,\,||_{q(\cdot)}$ and
$\rho_{q(\cdot)}(\,)$, one has the relation

$\min\\{\rho_{q(\cdot)}(u)^{\frac{1}{q^{-}}},\,\rho_{q(\cdot)}(u)^{\frac{1}{q^{+}}}\\}\leq||u||_{L^{q(\cdot)}}\leq\max\\{\rho_{q(\cdot)}(u)^{\frac{1}{q^{-}}},\,\rho_{q(\cdot)}(u)^{\frac{1}{q^{+}}}\\}.$

So, if $q^{+}<n,$ we have the embedding

$L^{q(\cdot)}_{1,0}(B_{R}(x))\hookrightarrow L^{p(\cdot)}(M),$

where, $p(x)=\frac{nq(x)}{n-q(x)}.$ In fact, there exists a positive constant
$D=D(n,\lambda,v,q^{+},q^{-})$ such as for every $u$ in
$L^{q(\cdot)}_{1,0}(B_{R}(x)),$ we have by Poincaré inequality and Proposition
2.8 that

$\displaystyle||u||_{L^{p(\cdot)}(M)}$ $\displaystyle\leq
D\,||u||_{L^{q(\cdot)}_{1}(M)}=D(||u||_{L^{q(\cdot)}(M)}+||\nabla
u||_{L^{q(\cdot)}(M)})$ $\displaystyle\leq D(c+1)\,||\nabla
u||_{L^{q(\cdot)}(M)},$

where $c$ is the Poincaré constant. Hence,

$\displaystyle\rho_{p(\cdot)}(u)$
$\displaystyle\leq||u||^{p^{+}}_{L^{p(\cdot)}(M)}\leq
D^{p^{+}}(c+1)^{p^{+}}\,||\nabla u||^{p^{+}}_{L^{q(\cdot)}(M)}$
$\displaystyle\leq D^{p^{+}}(c+1)^{p^{+}}\,\max\\{\rho_{q(\cdot)}(|\nabla
u|)^{\frac{p^{+}}{q^{-}}},\,\rho_{q(\cdot)}(|\nabla
u|)^{\frac{p^{+}}{q^{+}}}\\}$ $\displaystyle\leq
D^{p^{+}}(c+1)^{p^{+}}\rho_{q(\cdot)}(|\nabla u|)^{\frac{p^{+}}{q^{-}}}.$
(2.3)

######  Definition 2.7.

The Sobolev space $W^{1,q(x)}(M)$ consists of such functions $u\in
L^{q(x)}(M)$ for which $\nabla^{k}u\in L^{q(x)}(M)$ $k=1,2,\cdots,n.$ The norm
is defined by

$||\,u\,||_{W^{1,q(x)}(M)}=||\,u\,||_{L^{q(x)}(M)}+\sum_{k=1}^{n}||\,\nabla^{k}u\,||_{L^{q(x)}(M)}.$

The space $W_{0}^{1,q(x)}(M)$ is defined as the closure of $C^{\infty}_{c}(M)$
in $W^{1,q(x)}(M),$ with $C_{c}^{\infty}(M)$ be the vector space of smooth
functions with compact support on $M.$

######  Theorem 2.5.

Let $M$ be a compact Riemannian manifold with a smooth boundary or without
boundary and $q(x),\,p(x)\in C(\overline{M})\cap L^{\infty}(M).$ Assume that

$q(x)<N,\hskip
14.22636ptp(x)<\frac{N\,q(x)}{N-q(x)}\,\,\mbox{for}\,\,x\in\overline{M}.$

Then,

$W^{1,q(x)}(M)\hookrightarrow L^{p(x)}(M)$

is a continuous and compact embedding.

###### Proof.

This proof is based to an idea introduced in [12, 15]. Let $f:U(\subset
M)\longrightarrow\mathbb{R}^{N}$ be an arbitrary local chart on $M$, and $V$
be any open set in $M$, whose closure is compact and is contained in $U$.
Choosing a finite subcovering $\\{\,V_{\alpha}\,\\}_{\alpha=1,\cdots,k}$ of
$M$ such that $V_{\alpha}$ is homeomorphic to the open unit ball $B_{0}(1)$ of
$\mathbb{R}^{N}$ and for any $\alpha$ the components $g_{ij}^{\alpha}$ of $g$
in $(V_{\alpha},\,f_{\alpha})$ satisfy

$\frac{1}{\epsilon\,\delta_{ij}}\leq g_{ij}^{\alpha}<\epsilon\,\delta_{ij}$

as bilinear forms, where the constant $\epsilon>1$ is given. Let
$\\{\,\pi_{\alpha}\\}_{\alpha=1,\cdots,k}$ be a smooth partition of unity
subordinate to the finite covering $\\{\,V_{\alpha}\,\\}_{\alpha=1,\cdots,k}$.
It is obvious that if $u\in W^{1,q(x)}(M),$ then $\pi_{\alpha}\,u\in
W^{1,q(x)}(V_{\alpha})$ and $(f_{\alpha}^{-1})^{*}(\pi_{\alpha}u)\in
W^{1,q(f_{\alpha}^{-1}(x))}(B_{0}(1)).$ According to propsition 2.4 and the
Sobolev embeddings Theorem in [12, 13], we obtain the continuous and compact
embedding

$W^{1,q(x)}(V_{\alpha})\hookrightarrow L^{p(x)}(V_{\alpha})\hskip
14.22636pt\mbox{for each}\,\,\alpha=1,\cdots,k.$

Since $u=\displaystyle\sum_{\alpha=1}^{k}\pi_{\alpha}u,$ we can conclude that

$W^{1,q(x)}(M)\subset L^{p(x)}(M),$

and the embedding is continuous and compact. ∎

######  Proposition 2.9.

[3] If $(M,g)$ is complete, then $W^{1,q(x)}(M)=W^{1,q(x)}_{0}(M).$

The weighted variable exponent Lebesgue space $L_{\mu(x)}^{q(x)}(M)$ is
defined as follows:

$L_{\mu(x)}^{q(x)}(M)=\\{u:M\rightarrow\mathbb{R}\,\,\mbox{is measurable such
that,}\int_{M}\mu(x)|u(x)|^{q(x)}\,\,dv_{g}(x)<+\infty\,\\},$

with the norm

$||u||_{q(x),\mu(x)}=\inf\\{\gamma>0:\int_{M}\mu(x)\,\bigg{|}\frac{u(x)}{\gamma}\bigg{|}^{q(x)}\,\,dv_{g}(x)\leq
1\,\\}.$

Moreover, the weighted modular on $L_{\mu(x)}^{q(x)}(M)$ is the mapping
$\rho_{q(\cdot),\mu(\cdot)}:L_{\mu(x)}^{q(x)}(M)\rightarrow\mathbb{R}$ defined
like

$\rho_{q(\cdot),\mu(\cdot)}(u)=\int_{M}\mu(x)|u(x)|^{q(x)}\,\,dv_{g}(x).$

######  Example 2.6.

As a simple example of $\mu(x),$ we can take $\mu(x)=(1+|x|)^{\varepsilon(x)}$
with $\varepsilon(\cdot)\in C_{+}(\overline{M}).$

######  Proposition 2.10.

Let $u$ and $\\{u_{n}\\}\subset L^{q(x)}_{\mu(x)}(M),$ then we have the
following results:

  1. $(1)$

$||u||_{q(\cdot),\mu(\cdot)}<1$ (resp. $=1,>1$)
$\Longleftrightarrow\rho_{q(\cdot),\mu(\cdot)}(u)<1$ (resp. $=1,>1$).

  2. $(2)$

$||u||_{q(\cdot),\mu(\cdot)}<1\Rightarrow||u||^{q^{+}}_{q(\cdot),\mu(\cdot)}\leq\rho_{q(\cdot),\mu(\cdot)}(u)\leq||u||^{q^{-}}_{q(\cdot),\mu(\cdot)}.$

  3. $(3)$

$||u||_{q(\cdot),\mu(\cdot)}>1\Rightarrow||u||^{q^{-}}_{q(\cdot),\mu(\cdot)}\leq\rho_{q(\cdot),\mu(\cdot)}(u)\leq||u||^{q^{+}}_{q(\cdot),\mu(\cdot)}.$

  4. $(4)$

$\lim_{n\rightarrow+\infty}||u_{n}||_{q(\cdot),\mu(x)}=0\Longleftrightarrow\lim_{n\rightarrow+\infty}\rho_{q(\cdot),\mu(\cdot)}(u_{n})=0.$

  5. $(5)$

$\lim_{n\rightarrow+\infty}||u_{n}||_{q(\cdot),\mu(x)}=\infty\Longleftrightarrow\lim_{n\rightarrow+\infty}\rho_{q(\cdot),\mu(\cdot)}(u_{n})=\infty.$

Note that, the non-negative weighted function $\mu\in C(\overline{M})$ satisfy
the following hypothesis:
$\mu(\cdot):\overline{M}\longrightarrow\mathbb{R}^{+}_{*}$ such that
$\mu(\cdot)\in L^{\varepsilon(x)}(M)$ with

$\frac{Np(x)}{Np(x)-q(x)(N-p(x))}<\varepsilon(x)<\frac{p(x)}{p(x)-q(x)}\,\,\,\mbox{for
all}\,\,\,x\in\overline{M}.$ (2.4)

Indeed, since $\mu(\cdot):\overline{M}\longrightarrow\mathbb{R}^{+}_{*},$
then, there exists $\mu_{0}>0,$ and for all $x\in M,$ we have that
$\mu(x)>\mu_{0}.$

######  Theorem 2.7.

Let $M$ be a compact Riemannian manifold with a smooth boundary or without
boundary and $p(x),\,q(x)\in C(\overline{M})\cap L^{\infty}(M).$ Assume that
the assumption (2.4) is true. Then, the embedding

$W^{1,q(x)}(M)\hookrightarrow L^{q(x)}_{\mu(x)}(M),$

is compact

###### Proof.

Let
$\theta(x)=\frac{\varepsilon(x)}{\varepsilon(x)-1}q(x)=\hat{\varepsilon}(x)q(x),$
where $\frac{1}{\varepsilon(x)}+\frac{1}{\hat{\varepsilon}(x)}=1.$ From,
(2.4), we deduce that $\theta(x)<p^{*}(x)$ for all $x\in\overline{M},$ which
implies by Theorem 2.5, that $W^{1,q(x)}(M)\hookrightarrow L^{\theta(x)}(M).$
Hence, we have that $|u|^{q(x)}\in L^{\theta(x)}(M)$ for any $u\in
W^{1,q(x)}(M)$. Now, using the Hölder inequality, we get

$\rho_{q(\cdot),\mu(\cdot)}(u)\leq
r_{q}\,.\,||\mu(x)||_{\varepsilon(x)}||\,|u|^{q(x)}\,||_{\hat{\varepsilon}(x)}<+\infty.$
(2.5)

It follows that $u\in L^{q(x)}_{\mu(x)}(M),$ that is

$W^{1,q(x)}(M)\hookrightarrow L^{q(x)}_{\mu(x)}(M).$

Next, we prove that this embedding is compact. For that, we consider
$\\{u_{n}\\}\subset W^{1,q(x)}(M)$ such that $u_{n}\rightharpoonup 0$ weakly
in $W^{1,q(x)}(M)$ and since $W^{1,q(x)}(M)\hookrightarrow\hookrightarrow
L^{\theta(x)}(M),$ we obtain that

$u_{n}\longrightarrow 0\,\,\,\mbox{in}\,\,\,L^{\theta(x)}(M).$

Then, it follows that $||\,|u|^{q(x)}\,||_{\hat{\varepsilon}(x)}\rightarrow 0$
as $n\rightarrow+\infty.$ By Hölder inequality and (2.5), we have

$\rho_{q(\cdot),\mu(\cdot)}(u_{n})\longrightarrow 0.$

From proposition 2.10, result ( 4 ), we deduce that

$||u_{n}||_{q(\cdot),\mu(\cdot)}\longrightarrow
0\,\,\,\mbox{as}\,\,\,n\rightarrow+\infty.$ (2.6)

Hence, the embedding $W^{1,q(x)}(M)\hookrightarrow L^{q(x)}_{\mu(x)}(M)$ is
compact. ∎

##  3 Nehari Manifold Analysis for $(\mathcal{P})$

In this section, we note by $D(M)$ the space of $C_{c}^{\infty}$ functions
with compact support in $M$.

######  Definition 3.1.

$u\in W^{1,q(x)}_{0}(M)$ is said to be a weak solution of the problem
$(\mathcal{P})$ if for every $\phi\in D(M)$ we have

$\displaystyle\int_{M}\big{(}\,|\,\nabla u(x)\,|^{p(x)-2}+\mu(x)|\,\nabla
u(x)\,|^{q(x)-2}\big{)}\,.\,g(\nabla u(x),\,\nabla\phi(x))\,\,dv_{g}(x)$
$\displaystyle=\lambda\int_{M}|\,u(x)|^{q(x)-2}\,.\,u(x)\,\phi(x)\,\,dv_{g}(x)-\int_{M}|\,u(x)\,|^{p(x)-2}\,.\,u(x)\,.\,\phi(x)\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt+\int_{M}f(x,u(x))\,.\,\phi(x)\,\,dv_{g}(x).$

The variable exponents $p,q\in C(\overline{M})$, are assumed to satisfy the
following assumption:

$1<q^{-}\leq q^{+}<p^{-}\leq p^{+}<N.$ (3.1)

Then, we have

$\frac{p^{+}}{q^{+}-q^{-}}<\frac{p^{+}-q^{+}}{p^{+}-q^{-}}-\frac{(q^{+}-q^{-})\,.\,(p^{+}-q^{+})}{(p^{+}-q^{-})\,.\,(p^{-}-q^{-})}.$
(3.2)

We suppose $\frac{p^{-}}{q^{+}}\leq 1+\frac{1}{N},$ and the function
$\mu:\overline{M}\rightarrow\mathbb{R}_{*}^{+}$ is Lipschitz continuous.

Let us consider the energy functional
$J_{\lambda}:W^{1,q(x)}_{0}(M)\longrightarrow\mathbb{R}$ associated to problem
$(\mathcal{P})$ which is defined by

$\displaystyle J_{\lambda}(u)$
$\displaystyle=\int_{M}\frac{1}{p(x)}\,|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{\mu(x)}{q(x)}\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\int_{M}\frac{\lambda}{q(x)}\,|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{p(x)}\,|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{M}F(x,u(x))\,\,dv_{g}(x).$

And, for any $u\in W^{1,q(x)}_{0}(M)$ with $||\,u\,||_{W^{1,q(x)}_{0}(M)}>1,$
we have by $(f_{1}),(f_{2}),$ (2.2), Proposition 2.6 and Poincaré inequality
that

$\displaystyle J_{\lambda}(u)$
$\displaystyle=\int_{M}\frac{1}{p(x)}\,|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{\mu(x)}{q(x)}\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\int_{M}\frac{\lambda}{q(x)}\,|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\int_{M}\frac{1}{p(x)}\,|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq\frac{1}{p^{+}}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}q^{+}}\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt-\frac{\lambda}{q^{-}}\int_{M}|\,u(x)|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{+}}\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt-\frac{1}{\beta}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)$
$\displaystyle\geq\frac{1}{cp^{+}}\rho_{p(\cdot)}(u)+\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}q^{+}}\rho_{p(\cdot)}(u)-\frac{\lambda}{q^{-}}\rho_{q(\cdot)}(u)+\frac{1}{p^{+}}\rho_{p(\cdot)}(u)-\frac{1}{p^{+}}\rho_{q(\cdot)}(u)$
$\displaystyle\hskip
8.5359pt(\mbox{since}\,\,\beta>p^{+}\,\,\mbox{from}\,\,(f_{1}),\,\
\mbox{and}\,\,c\,\,\mbox{is the Poincaré constant}).$

According to the proposition 2.8, we have that

$\displaystyle J_{\lambda}(u)$
$\displaystyle\geq\bigg{(}\frac{1}{cp^{+}}+\frac{1}{p^{+}}+\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}q^{+}}\bigg{)}\,\rho_{p(\cdot)}(u)-\frac{\lambda}{q^{-}}\rho_{q(\cdot)}(u)-\frac{1}{p^{+}}\rho_{q(\cdot)}(u)$
$\displaystyle\geq\bigg{(}\frac{1}{cp^{+}}+\frac{1}{p^{+}}+\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}q^{+}}\bigg{)}\,||u||^{p^{-}}_{W^{1,q(x)}_{0}(M)}-\bigg{(}\frac{\lambda}{q^{-}}+\frac{1}{p^{+}}\bigg{)}\,||u||^{q^{+}}_{W^{1,q(x)}_{0}(M)}.$

From (3.1), we have that $J_{\lambda}$ is not bounded below on the whole space
$W^{1,q(x)}_{0}(M),$ but it is bounded above on an appropriate subset of
$W^{1,q(x)}_{0}(M)$ which is the Nehari manifold associated to $J_{\lambda}$
defined by

$\mathcal{N}_{\lambda}=\\{\,u\in
W_{0}^{1,q(x)}(M)\backslash\\{0\\}:\langle\,J_{\lambda}^{{}^{\prime}}(u),\,u\,\rangle=0\,\\}.$

Indeed, if we take for example $X$ a Banach space, and $J$ the Euler (energy)
functional associated with a variational problem on $X.$ If $J$ is bounded
above and has a minimizer, then, this minimizer is a critical point of $J.$
Therefore, it is a weak solution of the variational problem. However, in many
problems, $J$ is not bounded on the whole space $X,$ but is bounded on an
appropriate subset of $X,$ which is the case of our problem.
So, it is clear that the critical points of the functional $J_{\lambda}$ must
lie on $\mathcal{N}_{\lambda}$ and local minimizers on $\mathcal{N}_{\lambda}$
are usually critical points of $J_{\lambda}$. Thus,
$u\in\mathcal{N}_{\lambda}$ if and only if

$\displaystyle\langle J_{\lambda}^{{}^{\prime}}(u),\,u\rangle$
$\displaystyle=\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle-\lambda\,\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle-\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)=0.$ (3.3)

Hence, $\mathcal{N}_{\lambda}$ contains every nontrivial weak solution of
problem $(\mathcal{P})$ (see definition 3.1). Moreover, we have the following
result

######  Lemma 3.1.

Under assumptions $(f_{1})-(f_{3}).$ The energy functional $J_{\lambda}$ is
coercive and bounded below on $W^{1,q(x)}_{0}(M)$.

###### Proof.

Let $u\in\mathcal{N}_{\lambda}$ and $||\,u\,||>1,$ where $||\,.\,||$ is the
induced norm of $W_{0}^{1,q(x)}(\Omega)\backslash\\{0\\}.$ Then, by (3),
(3.1), (2.2), $(f_{1}),\,(f_{3}),$ propositions 2.6 and 2.8, we have

$\displaystyle J_{\lambda}(u)\geq\frac{1}{p^{+}}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{+}}\,\int_{M}\mu(x)|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\frac{\lambda}{q^{-}}\,\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{+}}\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle=\frac{1}{p^{+}}\int_{M}\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{+}}\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\frac{\lambda}{q^{-}}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{+}}\bigg{[}-\int_{M}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip 8.5359pt-\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)+\lambda\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)\bigg{]}-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq\mu_{0}\bigg{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\bigg{)}\int_{M}|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)+\lambda\bigg{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\bigg{)}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+\frac{1}{\beta}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)+\int_{M}F(x,u(x))\,\,dv_{g}(x)$
(since $\beta>p^{+},$ then $\frac{1}{p^{+}}>\frac{1}{\beta}$ and by $(f_{1})$
we get the following inequality)
$\displaystyle\geq\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\bigg{(}\frac{p^{+}-q^{+}}{p^{+}q^{+}}\bigg{)}\rho_{p(\cdot)}(u)+\lambda\bigg{(}\frac{q^{-}-p^{+}}{p^{+}q^{-}}\bigg{)}\rho_{q(\cdot)}(u)\,\,\mbox{(
from Proposition \ref{prop61} )}$
$\displaystyle\geq\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\bigg{(}\frac{p^{+}-q^{+}}{p^{+}q^{+}}\bigg{)}||u||^{p^{-}}+\lambda\bigg{(}\frac{q^{-}-p^{+}}{p^{+}q^{-}}\bigg{)}||u||^{q^{+}}\,\,\mbox{(
from Proposition \ref{prop13} )}$

As $p^{-}>q^{+},$ then $J_{\lambda}(u)\longrightarrow+\infty$ as
$||\,u\,||\rightarrow\infty.$ It follows that $J_{\lambda}$ is coercive and
bounded below on $\mathcal{N}_{\lambda}.$ ∎

Next, we consider the functional
$\psi:\mathcal{N}_{\lambda}\longrightarrow\mathbb{R}$ defined by

$\psi_{\lambda}(u)=\,\langle
J^{{}^{\prime}}_{\lambda}(u),\,u\rangle\,\,\mbox{for
all}\,\,u\in\mathcal{N}_{\lambda}.$

Hence, it is natural to split $\mathcal{N}_{\lambda}$ into three part: the
first set corresponding to local minima, the second set corresponding to local
maxima, and the third one corresponding to points of inflection which defined
respectively as follows

$\mathcal{N}_{\lambda}^{+}=\\{\,u\in\mathcal{N}_{\lambda}:\,\langle\psi^{{}^{\prime}}_{\lambda}(u),\,u\rangle>0\,\\},$
$\mathcal{N}_{\lambda}^{-}=\\{\,u\in\mathcal{N}_{\lambda}:\,\langle\psi^{{}^{\prime}}_{\lambda}(u),\,u\rangle<0\,\\},$
$\mathcal{N}_{\lambda}^{0}=\\{\,u\in\mathcal{N}_{\lambda}:\,\langle\psi^{{}^{\prime}}_{\lambda}(u),\,u\rangle=0\,\\}.$

######  Lemma 3.2.

Under assumptions $(f_{1})-(f_{3})$. There exists $\lambda^{*}>0$ such that
for any $\lambda\in(0,\,\lambda^{*})$ we have
$\mathcal{N}_{\lambda}^{0}=\emptyset.$

###### Proof.

Suppose otherwise, that is $\mathcal{N}_{\lambda}^{0}\neq\emptyset$ for all
$\lambda\in\mathbb{R}\backslash\\{0\\}.$ Let $u\in\mathcal{N}_{\lambda}^{0}$
such that $||\,u\,||>1.$ Then by (3), (3.1), $(f_{1})$ and the definition of
$\mathcal{N}_{\lambda}^{0},$ we have

$\displaystyle 0=\langle\psi^{{}^{\prime}}_{\lambda}(u),\,u\rangle$
$\displaystyle\geq\,\,p^{-}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+q^{-}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-q^{+}\bigg{[}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)\,\bigg{]}$
$\displaystyle\hskip
5.69046pt+p^{-}\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq(p^{-}-q^{+})\int_{M}|\nabla u(x)|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt+(q^{-}-q^{+})\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+(p^{-}-q^{+})\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
5.69046pt+q^{+}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
(since $\beta>p^{+}>q^{+}$, and by (3.1) we have
$q^{+}>\frac{1}{q^{+}}>\frac{1}{\beta}.$ Then, by $(f_{1})$ we get the
following inequality) $\displaystyle\geq(q^{-}-q^{+})\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+(p^{-}-q^{+})\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x).$

Then,

$0\geq(p^{-}-q^{+})\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)+(q^{-}-q^{+})\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x).$

Therefore, by Propositions 2.8, 2.10 and Theorem 2.7, we have

$0\geq(p^{-}-q^{+})||u||^{p^{-}}+c_{1}(q^{-}-q^{+})||u||^{q^{+}},$

where $c_{1}$ being the constant of the embedding Theorem 2.7.
Hence,

$||\,u\,||\leq\bigg{(}\frac{c_{1}(q^{+}-q^{-})}{p^{-}-q^{+}}\,\bigg{)}^{\frac{1}{p^{-}-q^{+}}}.$
(3.4)

Analogously:

$\displaystyle 0=\langle\psi^{{}^{\prime}}_{\lambda}(u),u\rangle$
$\displaystyle\leq\,p^{+}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+q^{+}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\lambda\,q^{-}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+p^{+}\,\bigg{[}\,-\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)+\lambda\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
5.69046pt+\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)\,\bigg{]}-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\leq(q^{+}-p^{+})\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\lambda(p^{+}-q^{-})\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+p^{+}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)$
$\displaystyle\leq(q^{+}-p^{+})\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\lambda(p^{+}-q^{-})\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+p^{+}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x).$

Then,

$\displaystyle\mu_{0}(p^{+}-q^{+})\int_{M}|\nabla u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\leq(p^{+}-q^{+})\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\leq\big{[}\lambda(p^{+}-q^{-})+p^{+}\big{]}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x).$

By (2.2) and proposition 2.6 we deduce that

$\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,(p^{+}-q^{+})\,||\,u\,||^{p^{-}}\leq\big{[}\lambda\,(p^{+}-q^{-})+p^{+}\big{]}\,||\,u\,||^{q^{+}}.$

Thus,

$||\,u\,||\leq\bigg{(}\frac{D^{p^{+}}(c+1)^{p^{+}}\big{[}\lambda\,(p^{+}-q^{-})+p^{+}\big{]}}{\mu_{0}\,(p^{+}-q^{+})}\,\bigg{)}^{\frac{1}{p^{-}-q^{+}}}.$
(3.5)

For $\lambda$ sufficiently small
$\big{(}\,\lambda<\frac{2\,\mu_{0}\,(p^{+}-q^{+})}{D^{p^{+}}(c+1)^{p^{+}}(p^{+}-q^{-})}\,-\,\frac{\mu_{0}\,c_{1}\,(q^{+}-q^{-})(p^{+}-q^{+})}{D^{p^{+}}(c+1)^{p^{+}}(p^{+}-q^{-})(p^{-}-q^{-})}-\frac{p^{+}}{p^{+}-q^{-}}\,\big{)}$,
if we combining (3.2), (3.4) and (3.5) we find $||\,u\,||<1$ for $\mu_{0}$
sufficiently large, which contradicts our assumption. Consequently, we can
conclude that there exists $\lambda^{*}>0$ such that
$\mathcal{N}_{\lambda}^{0}=\emptyset$ for any $\lambda\in(0,\,\lambda^{*}).$ ∎

######  Remark 3.3.

As a consequence of Lemma 3.2, for $0<\lambda<\lambda^{*},$ we can write
$\mathcal{N}_{\lambda}=\mathcal{N}_{\lambda}^{+}\cup\mathcal{N}_{\lambda}^{-},$
and we define

$\theta_{\lambda}^{+}=\inf_{u\in\mathcal{N}_{\lambda}^{+}}J_{\lambda}(u),\hskip
28.45274pt\theta_{\lambda}^{-}=\inf_{u\in\mathcal{N}_{\lambda}^{-}}J_{\lambda}(u).$

######  Lemma 3.4.

Suppose that $(f_{1})-(f_{3})$ are true. If $0<\lambda<\lambda^{*}$ with
$\lambda^{*}>0,$ then for all $u\in\mathcal{N}_{\lambda}^{+}$ we have
$J_{\lambda}(u)<0.$

###### Proof.

Suppose $u\in\mathcal{N}_{\lambda}^{+},$ from the definition of $J_{\lambda},$
we have

$\displaystyle J_{\lambda}(u)$
$\displaystyle\leq\frac{1}{p^{-}}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{-}}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\frac{\lambda}{q^{+}}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{-}}\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt-\int_{M}F(x,u(x))\,\,dv_{g}(x),$ (3.6)

from (3) and (3) we have

$\displaystyle J_{\lambda}(u)\leq$
$\displaystyle\frac{1}{p^{-}}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{-}}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle-\frac{1}{q^{+}}\,\bigg{[}\,\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle+\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)\,\bigg{]}$
$\displaystyle+\frac{1}{p^{-}}\int_{M}|\,u(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\leq\big{(}\frac{1}{p^{-}}-\frac{1}{q^{+}}\big{)}\int_{M}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+\big{(}\frac{1}{q^{-}}-\frac{1}{q^{+}}\big{)}\int_{M}\mu(x)\,|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\big{(}\frac{1}{p^{-}}-\frac{1}{q^{+}}\big{)}\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{+}}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt-\int_{M}F(x,u(x))\,\,dv_{g}(x),$

using Poincaré inequality and Theorem 2.7, we find that:

$J_{\lambda}(u)\leq-\bigg{(}\frac{p^{-}-q^{+}}{p^{-}q^{+}}\bigg{)}\bigg{(}\frac{1}{c}+1\bigg{)}||u||^{p^{-}}+\bigg{[}c_{1}\bigg{(}\frac{q^{+}-q^{-}}{q^{-}q^{+}}\bigg{)}+\frac{1}{q^{+}}\bigg{]}\,||u||^{q^{+}},$
(3.7)

where $c_{1}$ being the constant of the embedding Theorem 2.7.
As $u\in\mathcal{N}_{\lambda}^{+}$ we have

$\displaystyle p^{+}\int_{M}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)+q^{+}\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle-\lambda
q^{-}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)+p^{+}\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)-\int_{M}F(x,u(x))\,\,dv_{g}(x)>0,$
(3.8)

we multiply (3) by $(-p^{+}),$ we obtain

$\displaystyle-p^{+}\int_{M}|\nabla
u(x)|^{p(x)}\,\,dv_{g}(x)-p^{+}\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle+\lambda
p^{+}\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)-p^{+}\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle+p^{+}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)=0,$ (3.9)

we add (3) to (3), then according to the fact that $\beta>p^{+}$ and by (3.1)
we have $p^{+}>\frac{1}{p^{+}}>\frac{1}{\beta}$. Therefore, we obtain that

$(q^{+}-p^{+})\int_{M}\mu(x)|\nabla
u(x)|^{q(x)}\,\,dv_{g}(x)+\lambda(p^{+}-q^{-})\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)>0.$

Then, by (2.2) and Proposition 2.6 we get

$\displaystyle\lambda(p^{+}-q^{-})\int_{M}|u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle>(p^{+}-q^{+})\int_{M}\mu(x)\,|\nabla u(x)|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle>\frac{\mu_{0}(p^{+}-q^{+})}{D^{p^{+}}(c+1)^{p^{+}}}\int_{M}|u(x)|^{p(x)}\,\,dv_{g}(x).$

Hence,

$||u||^{p^{-}}<\frac{\lambda\,D^{p^{+}}(c+1)^{p^{+}}(p^{+}-q^{-})}{\mu_{0}(p^{+}-q^{+})}\,||u||^{q^{+}}.$

According to (3.7), we get

$\displaystyle J_{\lambda}(u)\leq\bigg{[}$
$\displaystyle-\bigg{(}\frac{p^{-}-q^{+}}{p^{-}q^{+}}\bigg{)}\bigg{(}\frac{1}{c}+1\bigg{)}\,.\,\frac{\lambda\,D^{p^{+}}(c+1)^{p^{+}}(p^{+}-q^{-})}{\mu_{0}(p^{+}-q^{+})}$
$\displaystyle+c_{1}\bigg{(}\frac{q^{+}-q^{-}}{q^{-}q^{+}}\bigg{)}+\frac{1}{q^{+}}\bigg{]}||u||^{q^{+}}.$

Finally, for $\lambda$ sufficiently large, we deduce that
$\theta_{\lambda}^{+}=\inf_{u\in\mathcal{N}_{\lambda}^{+}}J_{\lambda}(u)<0.$ ∎

######  Lemma 3.5.

Under assumptions $(f_{1})-(f_{3})$. If $0<\lambda<\lambda^{**},$ then for all
$u\in\mathcal{N}_{\lambda}^{-}$ we have $J_{\lambda}(u)>0.$

###### Proof.

Let $u\in\mathcal{N}_{\lambda}^{-}.$ By (3.1), $(f_{1}),$ (3) and the
definition of $J_{\lambda},$ we find that

$\displaystyle J_{\lambda}(u)$
$\displaystyle\geq\frac{1}{p^{+}}\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{1}{q^{+}}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\frac{\lambda}{q^{-}}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{+}}\bigg{[}\,-\int_{M}|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)+\lambda\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
5.69046pt+\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)\,\bigg{]}-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{1}{p^{+}}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt-\int_{M}F(x,u(x))\,\,dv_{g}(x)$
$\displaystyle\geq\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\int_{M}\mu(x)\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{1}{\beta}\int_{M}f(x,u(x))\,.\,u(x)\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt-\int_{M}F(x,u(x))\,\,dv_{g}(x)$ (since
$\beta>p^{+},$ then $\frac{1}{p^{+}}>\frac{1}{\beta}$ and by $(f_{1})$ we get
the following inequality)
$\displaystyle\geq\mu_{0}\,\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\int_{M}|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\int_{M}|\,u(x)\,|^{q(x)}\,\,dv_{g}(x),$

according to (2.2) we deduce that

$J_{\lambda}(u)\geq\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\,||\,u\,||^{p^{-}}+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\,||\,u\,||^{q^{+}}.$

Since, $p^{-}>q^{+}$ we have

$J_{\lambda}(u)\geq\bigg{(}\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,.\,\frac{p^{+}-q^{+}}{q^{+}\,p^{+}}+\lambda\,\frac{q^{-}-p^{+}}{p^{+}\,q^{-}}\,\bigg{)}\,||\,u\,||^{p^{-}}.$

Thus, if we choose
$\lambda<\frac{\mu_{0}\,q^{-}\,(p^{+}-q^{+})}{D^{p^{+}}(c+1)^{p^{+}}q^{+}\,(p^{+}-q^{-})}=\lambda^{**},$
we deduce that $J_{\lambda}(u)>0.$
It follows that
$\theta_{\lambda}^{-}=\inf_{u\in\mathcal{N}_{\lambda}^{-}}J_{\lambda}(u)>0.$ ∎

Hence,
$\mathcal{N}_{\lambda}=\mathcal{N}_{\lambda}^{+}\cup\mathcal{N}_{\lambda}^{-}$
and $\mathcal{N}_{\lambda}^{+}\cap\mathcal{N}_{\lambda}^{-}=\emptyset,$ by
above Lemma, we must have $u\in\mathcal{N}_{\lambda}^{-}.$

##  4 Existence of non-negative solutions:

In this section, we prove the existence of two non-negative solutions of
problem $(\mathcal{P}).$ For this, we first show the existence of minimizers
in $\mathcal{N}_{\lambda}^{+}$ and $\mathcal{N}_{\lambda}^{-}$ for all
$\lambda\in(0,\,\bar{\lambda}),$ where
$\bar{\lambda}=\min\,\\{\,\lambda^{*},\,\lambda^{**}\,\\}.$

######  Theorem 4.1.

Suppose that $(f_{1})-(f_{3})$ are true, then for all
$\lambda\in(0,\,\lambda^{*}),$ there exists a minimizer $u_{0}^{+}$ of
$J_{\lambda}(u)$ on $\mathcal{N}_{\lambda}^{+}$ such that
$J_{\lambda}(u_{0}^{+})=\theta_{\lambda}^{+}.$

###### Proof.

From Lemma 3.1, $J_{\lambda}$ is bounded below on $\mathcal{N}_{\lambda},$ in
particular is bounded below on $\mathcal{N}_{\lambda}^{+}.$ Then there exists
a minimizing sequence $\\{\,u_{n}^{+}\,\\}\subset\mathcal{N}_{\lambda}^{+}$
such that

$\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{+})=\inf_{u\in\mathcal{N}^{+}}J_{\lambda}(u)=\theta_{\lambda}^{+}<0.$

Since, $J_{\lambda}$ is coercive, $\\{\,u_{n}^{+}\,\\}$ is bounded in
$W^{1,q(x)}_{0}(M).$ Hence we assume that, without loss generality,
$u_{n}^{+}\rightharpoonup u_{0}^{+}$ in $W^{1,q(x)}_{0}(M)$ and by the compact
embedding ( Theorem 2.5 ) we have

$u_{n}^{+}\longrightarrow u_{0}^{+}\,\,\mbox{in}\,\,L^{p(x)}(M).$ (4.1)

Now, we shall prove $u_{n}^{+}\longrightarrow u_{0}^{+}$ in
$W^{1,q(x)}_{0}(M).$ Otherwise, let $u_{n}^{+}\not\to u_{0}^{+}$ in
$W^{1,q(x)}_{0}(M).$ Then, we have

$\rho_{q(.)}(u_{0}^{+})<\lim_{n\rightarrow+\infty}\inf\,\rho_{q(.)}(u_{n}^{+}),$
(4.2)

using (4.1) we obtain

$\int_{M}|\,u_{0}^{+}\,|^{p(x)}\,\,dv_{g}(x)=\lim_{n\rightarrow+\infty}\inf\,\int_{M}|\,u_{n}^{+}\,|^{p(x)}\,\,dv_{g}(x),$

since $\langle J^{{}^{\prime}}_{\lambda}(u_{n}^{+}),\,u_{n}^{+}\rangle=0,$ and
using the same technique as in Lemma 3.5, we get by (2.2) that

$J_{\lambda}(u_{n}^{+})\geq\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\,\rho_{p(.)}(u_{n}^{+})+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\,\rho_{q(.)}(u_{n}^{+}).$

That is

$\displaystyle\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{+})\geq$
$\displaystyle\,\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\,\lim_{n\rightarrow+\infty}\,\rho_{p(.)}(u_{n}^{+})$
$\displaystyle+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\,\lim_{n\rightarrow+\infty}\rho_{q(.)}(u_{n}^{+}).$

By (4.1) and (4.2), we have

$\theta_{\lambda}^{+}>\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,\big{(}\frac{1}{q^{+}}-\frac{1}{p^{+}}\big{)}\,||\,u_{0}^{+}\,||^{p^{-}}+\lambda\,\big{(}\frac{1}{p^{+}}-\frac{1}{q^{-}}\big{)}\,||\,u_{0}^{+}\,||^{q^{+}},$

since $p^{-}>q^{+},$ for $||\,u_{0}^{+}\,||>1,$ we deduce

$\theta_{\lambda}^{+}=\inf_{u\,\in\,\mathcal{N}_{\lambda}^{+}}J_{\lambda}(u)>0,$

which is a contradiction with Lemma 3.4. Hence

$u_{n}^{+}\longrightarrow u_{0}^{+}\,\,\mbox{in}\,\,W_{0}^{1,q(x)}(M),$

and

$\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{+})=J_{\lambda}(u_{0}^{+})=\theta_{\lambda}^{+}.$

Consequently, $u_{0}^{+}$ is a minimizer of $J_{\lambda}$ on
$\mathcal{N}_{\lambda}^{+}.$ ∎

######  Theorem 4.2.

Suppose that conditions $(f_{1})-(f_{3})$ are true, and for all
$\lambda\in(0,\,\lambda^{**}),$ there exists a minimizer $u_{0}^{-}$ of
$J_{\lambda}$ on $\mathcal{N}_{\lambda}^{-}$ such that
$J_{\lambda}(u_{0}^{-})=\theta_{\lambda}^{-}.$

###### Proof.

Since $J_{\lambda}$ is bounded below on $\mathcal{N}_{\lambda}$ and so on
$\mathcal{N}_{\lambda}^{-}.$ Then, there exists a minimizing sequence
$\\{\,u_{n}^{-}\,\\}\subseteq\mathcal{N}_{\lambda}^{-}$ such that

$\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{-})=\inf_{u\in\mathcal{N}_{\lambda}^{-}}J_{\lambda}(u)=\theta_{\lambda}^{-}>0.$

As $J_{\lambda}$ is coercive, $\\{\,u_{n}^{-}\,\\}$ is bounded in
$W_{0}^{1,q(x)}(M).$ Thus without loss of generality, we may assume that,
$u_{n}^{-}\rightharpoonup u_{0}^{-}$ in $W_{0}^{1,q(x)}(M)$ and by Theorem 2.5
we have

$u_{n}^{-}\longrightarrow u_{0}^{-}\,\,\mbox{in}\,\,L^{p(x)}(M).$ (4.3)

On the other hand, if $u_{0}^{-}\in\mathcal{N}_{\lambda}^{-},$ then there
exists a constant $t>0$ such that $t\,u_{0}^{-}\in\mathcal{N}_{\lambda}^{-}$
and $J_{\lambda}(u_{0}^{-})\geq J_{\lambda}(t\,u_{0}^{-}).$ According to
$(f_{1})$ and the definition of $\psi_{\lambda}^{{}^{\prime}},$ we have

$\displaystyle\langle\psi_{\lambda}^{{}^{\prime}}(t\,u_{0}^{-}),\,t\,u_{0}^{-}\rangle$
$\displaystyle=\,\int_{M}p(x)\,|\,\nabla tu_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 5.69046pt+q(x)\int_{M}\mu(x)\,|\,\nabla
tu_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\lambda\,q(x)\int_{M}|\,tu_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
5.69046pt+p(x)\int_{M}|\,tu_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}F(x,\,tu_{0}^{-}(x))\,\,dv_{g}(x)$
$\displaystyle\leq p^{+}\,t^{p^{+}}\int_{M}|\,\nabla
u_{0}^{-}\,|^{p(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt+\,q^{+}\,t^{q^{+}}\int_{M}\mu(x)|\,\nabla
u_{0}^{-}\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
5.69046pt-\lambda\,q^{-}\,t^{q^{-}}\int_{M}|\,u_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
5.69046pt+p^{+}\,t^{p^{+}}\int_{M}|\,u_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x).$

Since $q^{-}\leq q^{+}<p^{+},$ and by (2.2), propositions 2.6 and 2.8, it
follows that
$\langle\psi_{\lambda}^{{}^{\prime}}(t\,u_{0}^{-},\,t\,u_{0}^{-}\rangle<0.$
Hence by the definition of
$\mathcal{N}_{\lambda}^{-},\,\,t\,u_{0}^{-}\in\mathcal{N}_{\lambda}^{-}.$
Next, we show that $u_{n}^{-}\longrightarrow u_{0}^{-}$ in
$W_{0}^{1,q(x)}(M).$ Otherwise, suppose $u_{n}^{-}\not\to u_{0}^{-}$ in
$W_{0}^{1,q(x)}(M).$ Then by Fatou’s Lemma we have

$\int_{M}\mu(x)\,|\,\nabla
u_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)\leq\lim_{n\rightarrow+\infty}\int_{M}\mu(x)\,|\,\nabla
u_{n}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x).$

By (4.3) we get

$\int_{M}|\,u_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)\leq\lim_{n\rightarrow+\infty}\int_{M}|\,u_{n}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x),$

and

$\int_{M}|\,\nabla
u_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)\leq\lim_{n\rightarrow+\infty}\int_{M}|\,\nabla
u_{n}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x).$

Then, according the above inequalities and $(f_{1})$, we obtain

$\displaystyle J_{\lambda}(t\,u_{0}^{-})$
$\displaystyle\leq\frac{t^{p^{+}}}{p^{-}}\int_{M}|\,\nabla
u_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)+\frac{t^{q^{+}}}{q^{+}}\int_{M}\mu(x)\,|\,\nabla
u_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt-\frac{\lambda\,t^{q^{-}}}{q^{+}}\int_{M}|\,u_{0}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)+\frac{t^{p^{+}}}{p^{+}}\int_{M}|\,u_{0}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip 8.5359pt-\int_{M}F(x,t\,u_{0}^{-}(x))\,\,dv_{g}(x)$
$\displaystyle\leq\lim_{n\rightarrow+\infty}\bigg{[}\,\frac{t^{p^{+}}}{q^{-}}\int_{M}|\,\nabla
u_{n}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)$ $\displaystyle\hskip
8.5359pt+\frac{t^{q^{+}}}{q^{+}}\int_{M}\mu(x)\,|\,\nabla
u_{n}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)-\frac{\lambda\,t^{q^{-}}}{q^{+}}\int_{M}|\,u_{n}^{-}(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+\frac{t^{p^{+}}}{p^{+}}\int_{M}|\,u_{n}^{-}(x)\,|^{p(x)}\,\,dv_{g}(x)-\int_{M}F(x,t\,u_{0}^{-}(x))\,\,dv_{g}(x)\,\bigg{]}$
$\displaystyle\leq\lim_{n\rightarrow+\infty}J_{\lambda}(t\,u_{n}^{-})<\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{-})=\inf_{u\,\in\,\mathcal{N}_{\lambda}^{-}}J(u)=\theta_{\lambda}^{-}.$

Hence,
$J_{\lambda}(t\,u_{0}^{-})<\inf_{u\,\in\,\mathcal{N}_{\lambda}^{-}}J_{\lambda}(u)=\theta_{\lambda}^{-},$
which is a contradiction. Consequently

$u_{n}^{-}\longrightarrow
u_{0}^{-}\,\,\mbox{in}\,\,W^{1,q(x)}_{0}(M)\,\,\mbox{and}\,\,\lim_{n\rightarrow+\infty}J_{\lambda}(u_{n}^{-})=J_{\lambda}(u_{0}^{-})=\theta_{\lambda}^{-}.$

Then, we conclude that $u_{0}^{-}$ is a minimizer of $J_{\lambda}$ on
$\mathcal{N}_{\lambda}^{-}.$ ∎

######  Theorem 4.3.

Under assumptions $(f_{1})-(f_{3})$ we assume that the smooth complete compact
Riemannian n-manifold $(M,\,g)$ has property $B_{vol}(\lambda,\,v)$. Then,
there exists $\bar{\lambda}$ such that for all
$\lambda\in(0,\,\bar{\lambda}),$ the problem $(\mathcal{P})$ has at least two
non-negative weak solutions.

###### Proof.

Form Theorems 4.1 and 4.2, we deduce that for any
$\lambda\in(0,\bar{\lambda}),$ there exist
$u_{0}^{+}\in\mathcal{N}_{\lambda}^{+}$ and
$u_{0}^{-}\in\mathcal{N}_{\lambda}^{-}$ such as

$J_{\lambda}(u_{0}^{+})=\inf_{u\,\in\,\mathcal{N}_{\lambda}^{+}}J_{\lambda}(u)\,\,\mbox{and}\,\,J_{\lambda}(u_{0}^{-})=\inf_{u\,\in\,\mathcal{N}_{\lambda}^{-}}J_{\lambda}(u).$

Then, the problem $(\mathcal{P})$ has two solutions
$u_{0}^{+}\in\mathcal{N}_{\lambda}^{+}$ and
$u_{0}^{-}\in\mathcal{N}_{\lambda}^{-}$ in $W_{0}^{1,q(x)}(M).$ By Lemma 3.2,
it follows that
$\mathcal{N}_{\lambda}^{-}\cap\mathcal{N}_{\lambda}^{+}=\emptyset.$ Then,
$u_{0}^{-}\neq u_{0}^{+}.$ Thus these two solutions are distinct.
Next, we prove that $u_{0}^{-}$ and $u_{0}^{+}$ are non-negative in $M$. For
this, we introduce the truncation function
$h_{+}:M\times\mathbb{R}\longrightarrow\mathbb{R}$ defined by

$h_{+}(x,s)=\begin{cases}0&\text{ if $s<0$ },\\\\[8.5359pt] h(x,s)&\text{if
$s\geq 0$ }.\end{cases}$

We set $H_{+}(x,s)=\displaystyle\int_{0}^{s}f(x,t)\,\,dt$ and consider the
$C^{1}-$functional
$J_{\lambda}^{+}:W_{0}^{1,q(x)}(M)\longrightarrow\mathbb{R}$ given by

$\displaystyle J_{\lambda}^{+}(u)=$
$\displaystyle\int_{M}\frac{1}{p(x)}\,|\,\nabla
u(x)\,|^{p(x)}\,\,dv_{g}(x)+\int_{M}\frac{\mu(x)}{q(x)}\,|\,\nabla
u(x)\,|^{q(x)}\,\,dv_{g}(x)$
$\displaystyle-\int_{M}H_{+}(x,u(x))\,\,dv_{g}(x).$

Then, by (2.2) and proposition 2.8 we have for all
$u_{-}=\min\,\\{\,0,\,u(x)\,\\}$ that

$\displaystyle
0=\langle(J_{\lambda}^{+})^{{}^{\prime}}(u_{-}),\,u_{-}\rangle\,$
$\displaystyle\geq p^{-}\,\rho_{p(.)}(|\,\nabla
u_{-}\,|)+\frac{\mu_{0}}{D^{p^{+}}(c+1)^{p^{+}}}\,q^{-}\,\rho_{p(.)}(u_{-})$
$\displaystyle\geq\rho_{p(.)}(u_{-})\geq||\,u_{-}\,||^{p^{-}}.$

Hence, $||\,u_{-}\,||=0,$ and thus $u=u_{+}$. Then, by taking $u=u_{0}^{-}$
and $u=u_{0}^{+}$ respectively, we deduce that $u_{0}^{-}$ and $u_{0}^{+}$ are
non-negative solutions of problem $(\mathcal{P})$. ∎

Conclusion: According to the above results, we can then say that $u^{\pm}$ are
critical points of $J_{\lambda}$ and hence are non-negative weak solutions of
problem $(\mathcal{P}).$

## Acknowledgements

This paper has been supported by the RUDN University Strategic Academic
Leadership Program and P.R.I.N. 2019.

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NANOGrav signal from first-order confinement/deconfinement phase transition in
different QCD matters

Shou-Long Li$\,{}^{1}$, Lijing Shao$\,{}^{2,3}$, Puxun Wu$\,{}^{1}$ and
Hongwei Yu$\,{}^{1}$

$\,{}^{1}$Department of Physics and Synergetic Innovation Center for Quantum
Effect and Applications, Hunan Normal University, Changsha 410081, China

$\,{}^{2}$Kavli Institute for Astronomy and Astrophysics, Peking University,
Beijing 100871, China

$\,{}^{3}$National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100012, China

ABSTRACT

Recently, an indicative evidence of a stochastic process, reported by the
NANOGrav Collaboration based on the analysis of 12.5-year pulsar timing array
data which might be interpreted as a potential stochastic gravitational wave
signal, has aroused keen interest of theorists. The first-order color charge
confinement phase transition at the QCD scale could be one of the cosmological
sources for the NANOGrav signal. If the phase transition is flavor dependent
and happens sequentially, it is important to find that what kind of QCD matter
in which the first-order confinement/deconfinement phase transition happens is
more likely to be the potential source of the NANOGrav signal during the
evolution of the universe. In this paper, we would like to illustrate that the
NANOGrav signal could be generated from confinement/deconfinement transition
in either heavy static quarks with a zero baryon chemical potential, or quarks
with a finite baryon chemical potential. In contrast, the gluon confinement
could not possibly be the source for the NANOGrav signal according to the
current observation. Future observation will help to distinguish between
different scenarios.

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##  1 Introduction

Recently, the North American Nanohertz Observatory for Gravitational Wave
(NANOGrav) Collaboration [1] has reported an analysis of 12.5-year pulsar
timing array (PTA) data. According to the analysis [1], a possible evidence is
found for a stochastic common-spectrum process which may be interpreted as a
gravitational wave (GW) signal with its frequency in $1$–$10$ nHz, and its
average GW energy density
$\langle\Omega_{\textup{GW}}h^{2}\rangle_{\textup{NANOGrav}}\sim 10^{-10}$
with an almost flat GW spectrum $\Omega_{\rm GW}h^{2}\sim f^{-1.5\pm 0.5}$ at
1-$\sigma$ level. Although the observational results need further analyses,
such as a joint analysis with data from the other PTA collaborations (such as
EPTA and PPTA) [2, 3, 4], the potentiality of its being a stochastic GW
background (SGWB) signal has aroused keen interest of theorists [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30, 31, 32]. On the one hand, one may explain it as a potential SGWB
signal by considering different astrophysical and cosmological sources. On the
other hand, the possible SGWB signal could serve as a potential new probe to
studying new physics.

During the evolution of the universe, as the temperature decreases, the
universe may undergo several phase transitions from the metastable vacuums to
stable vacuums. At the quantum chromodynamics (QCD) energy scale, there are
two important transitions, i.e. the spontaneous chiral symmetry breaking and
the color charge confinement. For the confinement transition, the universe
will go from a quark-gluon plasma (QGP) phase to a hadron phase as the
temperature decreases. The numerical lattice simulation shows [33, 34, 35]
that the QCD transition is likely a crossover for three dynamical quark
flavors when the baryon and charge chemical potential is negligible, i.e., in
the absence of the baryon and lepton asymmetries. If the transition is first-
order, GWs could be produced due to the violent process of vacuum bubble
nucleation and subsequent bubble collisions [36, 42, 37, 38, 39, 40, 41],
sound waves [43, 44, 45, 46] and magnetohydrodynamic (MHD) turbulence [47, 48,
49, 50, 51, 52, 53]. The GWs produced are within the frequency range of the
PTA observation [54, 55, 56]. The three processes in the first-order phase
transition could be potential sources of the NANOGrav signal [19, 17, 22].
Therefore, the next issue is what are the QCD matters in which the first-order
phase transition can occur. In this regard, let us note that for the case of a
zero baryon chemical potential, the transition is first-order in a non-
dynamical (static) heavy quark system [57, 58]. For the case of a finite
baryon chemical potential, a large lepton asymmetry might affect the dynamics
of the QCD phase transition in a way to render it first-order in the early
universe [59]. Besides, for a pure gluon system, the first-order phase
transition might occur as well.

Different QCD matter has a different temperature of phase transition which is
a crucial parameter determining the features of the GWs produced. Then a
question arises naturally as to what kind of QCD matter in which the first-
order confinement/deconfinement phase transition happens is more likely to be
the potential source of the NANOGrav signal during the evolution of the
universe. In this paper, we will try to answer this question. We consider
three types of QCD matter systems: (i) heavy static quarks with a zero baryon
chemical potential, (ii) quarks with a finite baryon chemical potential, and
(iii) a pure gluon system. We match the NANOGrav signal with the GW spectra
from the first-order confinement/deconfinement phase transitions in three QCD
matter systems with holographic models [56, 60, 61]. We show that the GW
spectra from the phase transitions in pure quark systems, regardless of
whether the chemical potential is finite or zero, could explain the NANOGrav
signal according to the current observation. In contrast, the signal could not
possibly come from the gluon confinement.

##  2 GWs from first-order phase transitions

When first-order phase transition occurs, the universe transfers from a
metastable vacuum to a stable vacuum. This process can be described as bubble
nucleation. Generally, the GW signal from cosmological first-order phase
transitions mainly comes from three processes: collisions of vacuum bubble
walls, sound waves, and the MHD turbulence in the plasma [36, 42, 37, 38, 39,
40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. So, the total energy
density of the GW, which is a sum of the three, is given by

$h^{2}\Omega(f)\simeq
h^{2}\Omega_{\textup{en}}(f)+h^{2}\Omega_{\textup{sw}}(f)+h^{2}\Omega_{\textup{tu}}(f)\,,$
(2.1)

where $h=H_{0}/100\ \textup{km}^{-1}\ \textup{s}\ \textup{Mpc}$ in which
$H_{0}$ is the Hubble constant today. $h^{2}\Omega_{\textup{en}}$,
$h^{2}\Omega_{\textup{sw}}$ and $h^{2}\Omega_{\textup{tu}}$ are the
contributions from bubble collision, sound waves, and the turbulence,
respectively, which are given by [41, 42, 46, 52]

$\displaystyle h^{2}\Omega_{\textup{en}}$ $\displaystyle=$ $\displaystyle
3.6\times
10^{-5}\left(\frac{H_{\ast}}{\beta}\right)^{2}\left(\frac{\kappa_{1}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}\left(\frac{0.11v_{w}^{3}}{0.42+v_{w}^{2}}\right)S_{\textup{en}}\,,$
$\displaystyle h^{2}\Omega_{\textup{sw}}$ $\displaystyle=$ $\displaystyle
5.7\times
10^{-6}\left(\frac{H_{\ast}}{\beta}\right)\left(\frac{\kappa_{2}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}v_{w}S_{\textup{sw}}\,,$
$\displaystyle h^{2}\Omega_{\textup{tu}}$ $\displaystyle=$ $\displaystyle
7.2\times
10^{-4}\left(\frac{H_{\ast}}{\beta}\right)\left(\frac{\kappa_{3}\alpha}{1+\alpha}\right)^{2}\left(\frac{10}{g_{\ast}}\right)^{\frac{1}{3}}v_{w}S_{\textup{tu}}\,.$
(2.2)

In above equations, $\beta$ is the inverse time duration of the phase
transition, $v_{w}$ is the velocity of bubble wall, $g_{\ast}$ and $H_{\ast}$
represent the number of active degrees of freedom and Hubble parameter at the
time of production of GWs respectively,
$\alpha=30\epsilon_{\ast}/(\pi^{2}g_{\ast}T_{\ast}^{4})$ is the ratio of the
vacuum energy density $\epsilon_{\ast}=\big{[}\Delta F(T)-Td\Delta
F(T)/dT\big{]}\Big{|}_{T=T_{\ast}}$ over radiation energy density where
$T_{\ast}$ and $\Delta F(T)$ are the temperature of the thermal bath at time
of production of GWs and the difference of the free energy between two phases
respectively, and $S_{\textup{en}},S_{\textup{sw}}$, and $S_{\textup{tu}}$ are
spectral shapes of GWs which are characterized from numerical fits as [41, 42,
46, 52]

$\displaystyle S_{\textup{en}}$ $\displaystyle=$
$\displaystyle\frac{3.8\left(\frac{f}{f_{\textup{en}}}\right)^{2.8}}{1+2.8\left(\frac{f}{f_{\textup{en}}}\right)^{3.8}}\,,$
$\displaystyle S_{\textup{sw}}$ $\displaystyle=$
$\displaystyle\left(\frac{f}{f_{\textup{sw}}}\right)^{3}\left(\frac{7}{4+3\left(\frac{f}{f_{\textup{sw}}}\right)^{2}}\right)^{\frac{7}{2}}\,,$
$\displaystyle S_{\textup{tu}}$ $\displaystyle=$
$\displaystyle\frac{\left(\frac{f}{f_{\textup{tu}}}\right)^{3}}{\left(1+\frac{f}{f_{\textup{tu}}}\right)^{\frac{11}{3}}\left(1+\frac{8\pi
f}{h_{\ast}}\right)}\,,$ (2.3)

where $h_{\ast}$ is Hubble rate at $T_{\ast}$,
$f_{\textup{en}},f_{\textup{sw}}$ and $f_{\textup{tu}}$ are peak frequencies
in three cases, which are given by [39, 42, 52]

$\displaystyle h_{\ast}$ $\displaystyle=$ $\displaystyle 11.2\ \textup{nHz}\
\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{en}}$ $\displaystyle=$ $\displaystyle 11.2\
\textup{nHz}\
\left(\frac{0.62}{1.8-0.1v_{w}+v_{w}^{2}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{sw}}$ $\displaystyle=$ $\displaystyle 12.9\
\textup{nHz}\
\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,,$
$\displaystyle f_{\textup{tu}}$ $\displaystyle=$ $\displaystyle 18.4\
\textup{nHz}\
\left(\frac{1}{v_{w}}\right)\left(\frac{\beta}{H_{\ast}}\right)\left(\frac{T_{\ast}}{100\textup{MeV}}\right)\left(\frac{g_{\ast}}{10}\right)^{\frac{1}{6}}\,.$
(2.4)

In Eq. (2.2), $\kappa_{1},\kappa_{2}$, and $\kappa_{3}$ are the fractions of
the vacuum energy converted to the kinetic energy of the bubbles, bulk fluid
motion, and the MHD turbulence, respectively. These factors are model-
dependent. In this work, we consider two cases of bubble: Jouguet detonations
and non-runaway bubbles. For the case of Jouguet detonations [62, 42, 39, 63,
64, 46], we have

$\displaystyle\kappa_{1}$ $\displaystyle=$
$\displaystyle\frac{0.715\alpha+0.181\sqrt{\alpha}}{1+0.715\alpha}\,,$
$\displaystyle\kappa_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\alpha}}{0.135+\sqrt{\alpha+0.98}}\,,$
$\displaystyle\kappa_{3}$ $\displaystyle=$ $\displaystyle 0.05\kappa_{2}\,,$
$\displaystyle v_{w}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1/3}+\sqrt{\alpha^{2}+2\alpha/3}}{1+\alpha}\,,$
(2.5)

and for the case of non-runaway bubbles [64, 42, 46, 39],

$\displaystyle\kappa_{1}$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle\kappa_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{0.73+0.083\sqrt{\alpha}+\alpha}\,,$
$\displaystyle\kappa_{3}$ $\displaystyle=$ $\displaystyle 0.05\kappa_{2}\,,$
$\displaystyle v_{w}$ $\displaystyle=$ $\displaystyle 0.95\,.$ (2.6)

With all these expressions, there are still four unknown parameters in Eq.
(2.1), i.e., $g_{\ast},\beta/H_{\ast}$, $T_{\ast}$ and $\alpha$, which
characterize the first-order cosmological QCD phase transition. Generally, for
QCD phase transitions, the temperature $T_{\ast}$ is around several hundreds
MeV, of which the concrete value depends on the types of the QCD matter and
the phase transition. We will obtain, in the following section, $T_{\ast}$ for
the phase transition in different QCD matters by the method of the holographic
QCD. One can also calculate $\alpha$ from different holographic models. For
simplicity, we assume that the GW can be generated soon after the
confinement/deconfinement phase transition occurs, so $T_{\ast}$ is
approximated by the critical temperature of the phase transition. Besides, we
fix $g_{\ast}$ and $\beta/H_{\ast}$ at their typical values which could be
chosen as $g_{\ast}=10$ and $\beta/H_{\ast}=10$ [56, 60, 61] at the QCD scale.

##  3 GWs from holographic QCD models and the NANOGrav signal

In this section, we will obtain the GW spectra from the first-order
confinement/deconfinement transition in heavy static quarks with a zero baryon
chemical potential, quarks with a finite baryon chemical potential, and a pure
gluon system, and match them with the NANOGrav signal. First, we start with
finding the corresponding critical temperatures by use of holographic QCD
models.

Following the Anti-de Sitter/conformal field theory (AdS/CFT) correspondence
principle [65, 66, 67], AdS/QCD offers some new insights to the non-
perturbative hadron dynamics from the dual gravitational field [68].
Especially, the first-order confinement/deconfinement phase transitions could
be interpreted by Hawking-Page (HP) phase transitions [69] in five-dimensional
spacetime in the AdS/QCD models [70], where the high-temperature QGP
corresponds to the AdS black hole, while the low-temperature hadron
corresponds to the thermal AdS space. In the absence of the baryon chemical
potential, the transition temperature calculated by the soft-wall model is
consistent with numerical results [70]. In the case of the finite baryon
chemical potential, AdS/QCD models can also give a good explanation of the
phase transition [71, 72, 73, 74], while the standard lattice QCD simulation
suffers from the famous sign problem [75, 76, 77] and could not provide many
useful results. The GW produced by the first-order QCD phase transition was
estimated in the case of the heavy static quark system with a zero baryon
chemical potential via hard-wall and soft-wall models of AdS/QCD in Ref. [56]
for the first time, and then studied in the finite chemical potential system
[60, 78] and pure gluon system [61] via different models. It is also worth
mentioning that the initial idea of explaining GWs generated from cosmological
phase transitions by holographic method could be traced back to Randall and
Servant’s seminal work [79] in 2006 to the best of our knowledge.

We consider three types of QCD matter systems in five different holographic
models: (i) heavy static quarks with a zero baryon chemical potential in hard-
wall model $S_{1}$ and (ii) soft-wall model $S_{2}$, (iii) quarks with a
finite baryon chemical potential in hard-wall model $S_{3}$ and (iv) soft-wall
model $S_{4}$, and (v) pure gluons in the quenched dynamical holographic model
$S_{5}$. The corresponding five-dimensional gravitational actions are given by
[70, 60, 56, 61]

$\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}\left(R+\frac{12}{\ell^{2}}\right)\,,$ (3.1) $\displaystyle
S_{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-\phi}\left(R+\frac{12}{\ell^{2}}\right)\,,$ (3.2)
$\displaystyle S_{3}$ $\displaystyle=$ $\displaystyle\int
d^{5}x\sqrt{-g}\left[\frac{1}{2\kappa^{2}}\left(R+\frac{12}{\ell^{2}}\right)-\frac{1}{4g_{5}^{2}}F^{2}\right]-\frac{1}{\kappa^{2}}\int
d^{4}x\sqrt{h}\nabla_{\mu}n^{\mu}\,,$ (3.3) $\displaystyle S_{4}$
$\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-\phi}\left[\frac{1}{2\kappa^{2}}\left(R+\frac{12}{\ell^{2}}\right)-\frac{1}{4g_{5}^{2}}F^{2}\right]-\frac{1}{\kappa^{2}}\int
d^{4}x\sqrt{h}e^{-\phi}\nabla_{\mu}n^{\mu}\,,$ (3.4) $\displaystyle S_{5}$
$\displaystyle=$ $\displaystyle\frac{1}{2\kappa^{2}}\int
d^{5}x\sqrt{-g}e^{-2\varphi}\left(R-\frac{4}{3}\partial_{\mu}\varphi\partial^{\mu}\varphi-V(\varphi)\right)\,,$
(3.5)

where $\kappa^{2}=8\pi G$, $g_{5}$ is coupling constant, $\ell$ is the radius
of five-dimensional AdS space, $g$ and $h$ are determinants of bulk and
boundary metrics respectively, $n^{\mu}$ is the unit vector normal to the
hypersurface, $\phi$ is a non-dynamical dilaton, and $\varphi$ is a dynamical
dilaton. The confinement/deconfinement transition in the cases of heavy static
quarks with a zero baryon chemical potential and pure gluons are analogous to
the HP transition between the static AdS black hole and the thermal AdS
vacuum. For the case of quarks with a finite baryon chemical potential, the
confinement/deconfinement transition is analogous to the HP transition between
the charged AdS black hole and the thermal charged AdS vacuum, and the
chemical potential is related to the electric charge of the black hole. For a
specific model, one can calculate the free energies of the black hole and AdS
vacuum respectively, and obtain the difference of the free energy $\Delta F$
between two phases. Then one can obtain the value of the temperature
$T_{\ast}$ and $\alpha$ via some holographic techniques. Here, we refer
readers to Refs. [70, 60, 56, 61] for detailed calculations and list the
corresponding critical temperatures in these models in Table 1.

Table 1: Critical temperatures from five holographical QCD models. QCD matters | Holographic QCD models | Temperature
---|---|---
heavy static quarks with | hard wall | 122 MeV [70, 56]
a zero chemical potential | soft wall | 191 MeV [70, 56]
quarks with a finite | hard wall | 112 MeV [60]
chemical potential | soft wall | 192 MeV [60]
pure gluons | quenched dynamical holographic QCD | 255 MeV [61]

We assume that GW is generated quickly after the phase transition occurs and
temperature $T_{\ast}$ is approximated as the critical phase transition
temperature. Now we match the NANOGrav results with the GW produced from the
confinement/deconfinement phase transition in different holographic models. We
plot the GW spectra in two bubble models: Jouguet detonations and non-runaway
bubbles. The results are illustrated respectively in Fig. 1 and Fig. 2.

Figure 1: GW spectra from QCD matter confinement/deconfinement phase
transition in the Jouguet detonation bubble case from five holographical
models. The results are compared with the current sensitivity of NANOGrav
12.5-yr [1, 8], EPTA [2], and PPTA [3].

Figure 2: Same as Fig. 1, for the non-runaway bubble case.

From Fig. 1 and Fig. 2, we show that the power spectra of GWs from the quark
confinement phase transitions enter the 95% confidence interval from the
NANOGrav 12.5-yr observation [1, 8], which indicates that the
confinement/deconfinement phase transition in pure quark systems (cases of
heavy static quarks with a zero baryon chemical potential and quarks with a
finite baryon chemical potential) could possibly be the potential cosmological
sources of the NANOGrav signal for both the Jouguet detonation and non-runaway
bubble cases. The power spectra of GWs calculated from different holographic
models (hard wall and soft wall) in a specific QCD matter system are
different, but the difference does not change the conclusion. In contrast, the
confinement/deconfinement transition in the pure gluon system could not
possibly be the source of the NANOGrav signal. Since the critical phase
transition temperatures for the cases of a finite chemical potential and a
zero chemical potential are very close, the chemical potential has little
effect on the power spectrum of GWs, and the quark confinement dominates the
cosmological QCD transition according to the current NANOGrav observation. But
these conclusions need to be further supported by more accurate observations
of GWs in the future. We also calculate the spectrum indices from different
holographic models, and find that the values are similar in the same bubble
model. The values are about 2.78 and 2.91 for the Jouguet and non-runaway
cases, respectively. These values could be used to check our conclusions with
more accurate future observations.

##  4 Conclusions and Discussions

In this paper, we have showed that a possible stochastic common-spectrum
process reported by the NANOGrav Collaboration based on the 12.5-yr PTA data
can be explained potentially as a GW signal from the first-order cosmological
confinement/deconfinement phase transition in the cases of (i) heavy static
quarks with a zero baryon chemical potential and (ii) quarks with a finite
baryon chemical potential. We also find that the gluon confinement could not
possibly be the potential source of the NANOGrav signal based on the current
observation. We match the GW spectra from the first-order phase transition in
five different holographic QCD models with the NANOGrav signal. By considering
both the Jouguet detonation and non-runaway bubble growth models, we find that
the GW spectra from the confinement/deconfinement phase transition in pure
quark systems, irrespective of whether the baryon chemical potential is finite
or zero, enter the 95% confidence interval from the NANOGrav 12.5-yr
observation [1, 8]. The baryon chemical potential has little influence on the
power spectra of GWs produced by confinement/deconfinement phase transitions
since the phase transition temperatures in both the finite baryon chemical
potential case and zero chemical potential case are very close.

We must point out that we have assumed in our analysis that GWs are generated
soon after the phase transition happens, so the temperature at which the GWs
are produced is approximately the transition temperature. Although this is an
acceptable assumption, it remains interesting to find a more credible
holographic method to calculate the temperature $T_{\ast}$, analogous to what
was done in Ref. [79]. Moreover, it is also worth considering the
confinement/deconfinement phase transitions in possible QCD matter systems
other than those we have looked at in the present paper and chiral symmetry
breaking phase transitions. We would like to leave these to future studies.

## Acknowledgement

We are grateful to Jie-Wen Chen, Muyang Chen, Chengjie Fu, Yong Gao, Long-
Cheng Gui, Hongbo Li, Chang Liu, Jing Liu, H. Lü, Xueli Miao, Shi Pi, Shao-
Jiang Wang, Hao Wei, Rui Xu, and Junjie Zhao for useful discussions. SL thanks
Lijing for his warm hospitality during the visit to KIAA. SL, PW and HY were
supported in part by the NSFC under Grants No. 11947216, No. 11690034, No.
11805063, No. 11775077 and No. 12075084, and China Postdoctoral Science
Foundation 2019M662785. LS was supported by the National SKA Program of China
(2020SKA0120300), the National Natural Science Foundation of China (11975027,
11991053, 11721303), the Young Elite Scientists Sponsorship Program by the
China Association for Science and Technology (2018QNRC001), and the Max Planck
Partner Group Program funded by the Max Planck Society.

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