∎

11institutetext: Ahmed Aberqi 22institutetext: Laboratory LAMA, Sidi Mohamed
Ben Abdellah University, National School of Applied Sciences Fez, Morocco.
22email<EMAIL_ADDRESS>33institutetext: Jaouad
Bennouna44institutetext: Laboratory LAMA, Department of Mathematics, Sidi
Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz, B.P 1796
Atlas Fez, Morocco.
44email<EMAIL_ADDRESS>55institutetext: Omar Benslimane
66institutetext: Laboratory LAMA, Department of Mathematics, Sidi Mohamed Ben
Abdellah University, Faculty of Sciences Dhar El Mahraz, B.P 1796 Atlas Fez,
Morocco.
66email<EMAIL_ADDRESS>77institutetext: Maria Alessandra Ragusa
88institutetext: Dipartimento di Matematica e Informatica, Universitá di
Catania, Viale A.Doria 6, 95125 Catania, Italy.
RUDN University, 6 Miklukho, Maklay St, Moscow, Russia 117198.
88email<EMAIL_ADDRESS>
# Existence of solutions for a singular double phase in Sobolev-Orlicz spaces
with variable exponents in a complete manifold

Ahmed Aberqi  Jaouad Bennouna  Omar Benslimane  Maria Alessandra Ragusa∗

(Received: date / Accepted: date)

###### Abstract

The purpose of this paper is to study a class of double phase problems, with a
singular term and a superlinear parametric term on the right-hand side. Using
the method of Nehari manifold combined with the fibering maps, we prove that
for all small values of the parameter $\lambda>0,$ there exist at least two
non-trivial positive solutions. Our results extend the previous works
Papageorgiou, Repovš, and Vetro papageorgiou2020positive  and Liu, Dai,
Papageorgiou, and Winkert liu2021existence , from the case of Musielak-Orlicz
Sobolev space, when exponents $p$ and $q$ are constant, to the case of
Sobolev-Orlicz spaces with variable exponents in a complete manifold.

###### Keywords:

Existence of solutions Double phase operator Nehari manifold singular problems
Sobolev-Orlicz Riemannian manifold with variable exponants

###### MSC:

MSC 35J60 MSC 58J05

##  1 Introduction

Let $(M,g)$ be a smooth complete compact Riemannian N-manifold. In this
article, we are addressed in proving the existence of at least two weak
solutions to the following singular double phase problem.

$(\mathcal{P})\,\begin{cases}-\,\mbox{div}(|D\mathrm{u}(z)|^{p(z)-2}D\mathrm{u}(z)+\mu(z)|D\mathrm{u}(z)|^{q(z)-2}D\mathrm{u}(z)\,)\\\
\hskip
17.07182pt=\frac{g(z)}{\mathrm{u}(z)^{\gamma(z)}}+\lambda\,|\mathrm{u}(z)|^{r(z)-1}&\text{in
\,\,M },\\\\[8.5359pt] \,\mathrm{u}\,>\,0&\text{in \,\,M },\\\
\,\mathrm{u}\,=\,0&\text{on \,$\partial$M }.\end{cases}$

We suppose the subsequent assumptions:

  * (i)

The variables exponents $p,q,r\in C(\overline{M})$ and satisfy the following
assumptions:

$1<q^{-}\leq q^{+}<p^{-}\leq p^{+}<r^{-}\leq r^{+}<+\infty,$ (1)

where
$q^{+}=\displaystyle\sup_{z\in\overline{M}}q(z),\,\,q^{-}=\inf_{z\in\overline{M}}q(z),\,\,p^{+}=\sup_{z\in\overline{M}}p(z),\,\,p^{-}=\inf_{z\in\overline{M}}p(z),\\\
r^{+}=\sup_{z\in\overline{M}}r(z),\,\,\mbox{and}\,\,r^{-}=\inf_{z\in\overline{M}}r(z).$

$\frac{p^{-}}{q^{+}}<1+\frac{1}{N}.$ (2)

  * (ii)

$\mu:\overline{M}\rightarrow[1,+\infty)$ is Lipschitz continuous.

  * (iii)

$g(z)\in L^{\infty}(M)$ and $g(z)\geq 0$ for a.a $z\in M$ with $g\not\equiv
0$.

  * (iv)

$0<\gamma(z)\in C(\overline{M}),\,\,0<\gamma^{-}\leq\gamma^{+}<1.$

  * (v)

$\lambda$ is a positive parameter.

Off late, nonlinear elliptic equations and variational problems involving
variable exponents growth conditions have become hugely popular area of
investigation owing to its manifold applications including the study of fluid
filtration in porous media, constrained heating, elasto-plasticity, optimal
control, financial mathematics and others. Interested readers may refer to
chen2006variable ; gwiazda2008non ; ruuvzivcka2004modeling ; zhikov2004density
and the references therein for more background of applications. Originally,
while studying the behaviour of strongly anisotropic materials, Zhikov
discovered that their hardening properties changed radically with the point,
which is called the Lavrentiev phenomenon, see for example zhikov1987averaging
; zhikov1995lavrentiev ; zhikov1997some ; jikov2012homogenization . In order
to describe this phenomenon, he introduced the functional

$\mathrm{u}\longmapsto\int_{\Omega}\big{(}|D\mathrm{u}|^{p}+\mu(z)\,|D\mathrm{u}|^{q}\big{)}\,\,dx,$
(3)

where the integrand switches two different elliptic behaviours.

Several interesting works have been carried out on the double phase problem
with a Dirichlet boundary condition. For example, Liu and Dai liu2018existence
studied the following problem

$\begin{cases}-\,\mbox{div}(|D\mathrm{u}(z)|^{p-2}D\mathrm{u}(z)+a(z)|D\mathrm{u}(z)|^{q-2}D\mathrm{u}(z)\,)=f(z,\mathrm{u}(z))&\text{in
\,\,$\Omega$ },\\\\[8.5359pt] \,\mathrm{u}\,=\,0&\text{on \,$\partial\Omega$
},\end{cases}$

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 gasinski2019constant  via the
Nehari manifold method. Following this direction, the authors in
papageorgiou2020ground  have shown that the double phase problem driven by the
sum of the $p$-Laplace operator and a weighted $q$-Laplacian $(q<p)$, with a
weight function which is not bounded away from zero and the reaction term, is
$(p-1)-$superlinear, has a ground state solution of constant sign and a nodal
(sign-changing) solution, employing the Nehari method. In particular, using
the theory of pseudomonotone operators, Gasiński, and Winkert in
gasinski2020existence  proved the existence and uniqueness of a weak solution
to quasilinear elliptic equations with double phase phenomena and a reaction
term depending on the gradient, under quite general assumptions on the
convection term and towering some linear conditions on the gradient variable.
Finally, for a deeper comprehension, we refer the reader to
aberqi2019existence ; acerbimingione ; benslimane2020existence1 ;
benslimane2020existence ; benslimane2020existence2 ; benslimane2020existence3
; benslimane2021existence ; duzaarmingione  and the references therein.

Next, we would like to mention some papers of double phase problems with
singularity. Papageorgiou, Repovš, and Vetro in papageorgiou2020positive
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. Readers may refer to papageorgiou2020positive ;
liu2021existence  for ideas and techniques developed in order to guarantee the
existence of at least two non-trivial positive solutions to double phase
problems driven by a singularity.

Considering another novel aspect, we go back to the double phase problem with
variable exponents which few authors consider. Ragusa and Tachikawa in
ragusa2019regularity  are the first ones who have achieved the regularity
theory for minimizers of (3) with variable exponents. Moreover, in
tachikawa2020boundary  Tachikawa, provides the Hölder continuity up to the
boundary of minimizers of so-called double phase functional with variable
exponents, under suitable Dirichlet boundary conditions. Recently, Aberqi,
Bennouna, Benslimane, and Ragusa in benslimane2020existence4  studied 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 manifold, using the Nehari manifold and
some variational techniques. Furthermore, they proved the Hölder inequality,
continuous and compact embedding. The readers may refer to the work due to
aubin1982nonlinear ; benslimane2020existence1 ; gaczkowski2016sobolev ;
hebey2000nonlinear ; guo2015dirichlet ; papageorgiou2019nonlinear  and the
references given there.

Inspired by the above-mentioned papers, we study the problem $(\mathcal{P})$
which contains a singular term and a parametric superlinear term in the
framework of Sobolev spaces with variable exponents in Complete manifolds. To
the best of our knowledge, the results presented here are new. However, we
address the challenges presented by the fact that the $p(z)-$ laplacian and
$q(z)-$ laplacian operators possess more complicated nonlinearities than the
$p$-laplacian and $q$-laplacian operator, since $\Delta_{p(z)}$ and
$\Delta_{q(z)}$ are not homogeneous. Moreover, we can not use Lagrange
Multiplier Theorem in many problems involving this operator, which shows that
our problem has more difficulty than the operators $p$-Laplace type.

Now, we define the solution of the problem $(\mathcal{P})$ in a weaker sense.

###### Definition 1

A function $\mathrm{u}\in W_{0}^{1,q(z)}(M)$ is said to be a weak solution if
$\mathrm{u}>0$ for a.a $z\in M$ and

$\displaystyle\int_{M}\bigg{(}|D\mathrm{u}(z)|^{p(z)-2}D\mathrm{u}(z)+\mu(z)\,|D\mathrm{u}(z)|^{q(z)-2}D\mathrm{u}(z)\,\bigg{)}\,D\varphi(z)\,\,dv_{g}(z)$
$\displaystyle=\int_{M}g(z)\,|\mathrm{u}(z)|^{-\gamma(z)}\,\varphi(z)\,\,dv_{g}(z)+\lambda\int_{M}|\mathrm{u}(z)|^{r(z)-1}\varphi(z)\,\,dv_{g}(z),$

is satisfied for all $\varphi\in\mathcal{D}(M).$ Where $\mathcal{D}(M)$ denote
the space of $C^{\infty}$ functions with compact support in $M.$

###### Remark 1

For the singular term, the weak solution is by definition a function
$\mathrm{u}\in W_{0}^{1,q(z)}(M)$ such that
$\mathrm{u}(z)^{-\gamma(z)}\,\varphi(z)\in L^{1}(M)$ for every
$\varphi\in\mathcal{D}(M)$. Hence, the above definition of a weak solution is
well-defined.

###### Remark 2

In the case when $q(z)<r(z),$ we have the validity of the coercivity
properties of the functional energy associated with the problem
$(\mathcal{P})$ ( see Lemma 1 below ) which guaranteed the compactness for
sequences with uniformly bounded energy. Unfortunately, in the case when
$q(z)=r(z)$ would cause our coercivity properties to fail.

The main result of our paper can be stated as follows.

###### Theorem 1.1

Under assumptions (i)-(v), we assume that the complete N-manifold $(M,g)$ has
property $B_{vol}(\lambda,v)$. Then, there exists $\tilde{\lambda}^{*}_{0}>0$
such as for all $\lambda\in(0,\tilde{\lambda}^{*}_{0}]$, the problem
$(\mathcal{P})$ has at least two positive solutions
$\tilde{\mathrm{u}}^{*},\,\tilde{v}^{*}\in W_{0}^{1,q(z)}(M)$ such as
$J_{\lambda}(\tilde{\mathrm{u}}^{*})<0\leq J_{\lambda}(\tilde{v}^{*}).$

The context of the paper is organized as follows. In section 2 we will recall
the definitions and some properties of Sobolev spaces with variable exponents
and Sobolev spaces with variable exponents in complete manifolds. The readers
can consult the following papers aubin1982nonlinear ; benslimane2020existence1
; benslimane2020existence ; benslimane2020existence2 ;
benslimane2020existence3 ; hebey2000nonlinear ; trudinger1968remarks  for
details. 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. Moreover, we show the existence of two
non-trivial positive solutions when the parameter $\lambda>0$ is sufficiently
small.

##  2 Notations and Basic Properties

In order to discuss the problem $(\mathcal{P})$, we need some facts on spaces
$W_{0}^{1,q(x)}(\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^{N}$
and $W_{0}^{1,q(x)}(M)$ which are called the Sobolev spaces with variable
exponents and the Sobolev spaces with variable exponents in complete manifolds
setting. For this reason, we will recall some properties involving the above
spaces, which can be found in aubin1982nonlinear ; benslimane2020existence1 ;
fan2001spaces ; gaczkowski2016sobolev ; hebey2000nonlinear ; guo2015dirichlet
and references therein.

###  2.1 Sobolev spaces with variable exponents

Given $\Omega\subset\mathbb{R}^{N}$ a bounded open domain, with $N\geq 2.$
Let $q(\cdot):\Omega\rightarrow(1,\infty)$ be a measurable function, we define
the real numbers $q^{+}$ and $q^{-}$ as follows

$q^{+}=ess\,sup\\{\,q(z);\,z\in\Omega\,\\}\,\,\mbox{and}\,\,q^{-}=ess\,inf\\{\,q(z);\,z\in\Omega\,\\}.$

###### Definition 2

fan2001spaces  We define the Lebesgue space with variable exponent
$L^{q(\cdot)}(\Omega)$ as follows

$L^{q(\cdot)}(\Omega)=\bigg{\\{}\mathrm{u}:\Omega\rightarrow\mathbb{R};\,\rho_{q(\cdot)}(\mathrm{u})=\int_{\Omega}|\,\mathrm{u}(z)\,|^{q(z)}\,\,dz<+\infty\,\bigg{\\}},$

endowed with the Luxemburg norm

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

if $q^{+}<+\infty.$

###### Proposition 2.1

fan2001spaces  The space $(L^{q(\cdot)}(\Omega),\,||\,.\,||_{q(\cdot)})$ is a
separable Banach space, and uniformly convex for $1<q^{-}\leq q^{+}<+\infty,$
hence reflexive.

###### Proposition 2.2

fan2001spaces  (Hölder inequality)

$\bigg{|}\,\int_{\Omega}\mathrm{u}\,v\,\,dx\,\bigg{|}\leq\bigg{(}\,\frac{1}{q^{-}}+\frac{1}{(q^{{}^{\prime}})^{-}}\bigg{)}\,||\,\mathrm{u}\,||_{q(\cdot)}||\,v\,||_{q^{{}^{\prime}}(\cdot)},\hskip
14.22636pt\forall\mathrm{u},\,v\in L^{q(\cdot)}(\Omega)\times
L^{q^{{}^{\prime}}(\cdot)}(\mathcal{Q}),$

with $\frac{1}{q(z)}+\frac{1}{q^{{}^{\prime}}(z)}=1.$

###### Definition 3

fan2001spaces  We define the variable exponents Sobolev space by

$W^{1,q(z)}(\Omega)=\big{\\{}\,\mathrm{u}\in
L^{q(z)}(\Omega)\,\,\mbox{and}\,\,|\,D\mathrm{u}\,|\in
L^{q(z)}(\Omega)\,\big{\\}},$

with the norm

$||\,\mathrm{u}\,||_{W^{1,q(z)}(\Omega)}=||\,\mathrm{u}\,||_{L^{q(z)}(\Omega)}+||\,D\mathrm{u}\||_{L^{q(z)}(\Omega)},\,\,\forall\mathrm{u}\in
W^{1,q(z)}(\Omega).$

###### Remark 2.3

fan2001spaces

$W^{1,q(z)}_{0}(\Omega)=\overline{C^{\infty}_{0}(\Omega)}^{W^{1,q(z)}(\Omega)}.$

###  2.2 Sobolev spaces with variable exponents in complete manifolds

###### Definition 4

hebey2000nonlinear  Let $(M,g)$ be a smooth Riemannain N-manifolds and let $D$
be the Levi-Civita connection. If $\mathrm{u}$ is a smooth function on $M$,
then $D^{k}\mathrm{u}$ denotes the $k-$th covariant derivative of
$\mathrm{u}$, and $|\,D^{k}\mathrm{u}\,|$ the norm of $D^{k}\mathrm{u}$
defined in local coordinates by

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

where Einstein’s convention is used.

###### Definition 5

hebey2000nonlinear  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

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

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

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

###### Definition 6

gaczkowski2016sobolev  The Sobolev spaces $L_{k}^{q(\cdot)}(M)$ is the
completion of $C^{q(\cdot)}_{k}(M)$ with respect to the norm
$||\,\mathrm{u}\,||_{L^{q(\cdot)}_{k}}$. If $\Omega$ is a subset of $M$, then
$L^{q(\cdot)}_{k,0}(\Omega)$ is the completion of $C^{q(\cdot)}_{k}(M)\cap
C_{0}(M)$ with respect to $||\,\cdot\,||_{L^{q(\cdot)}_{k}},$ where
$C_{0}(\Omega)$ denotes the vector space of continuous functions whose support
is a compact subset of $\Omega.$

###### Definition 7

hebey2000nonlinear  Given $(M,g)$ a smooth Riemannian manifold, and
$\gamma:\,[\,a,\,b\,]\longrightarrow M$ a curve of class $C^{1}$. The length
of $\gamma$ is

$\ell(\gamma)=\int_{a}^{b}\bigg{(}\,g\,\bigg{(}\,\frac{d\gamma}{dt},\,\frac{d\gamma}{dt}\,\bigg{)}\,\bigg{)}^{\frac{1}{2}}\,\,dt,$

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

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

###### Definition 8

hebey2000nonlinear  A function $s:\,M\longrightarrow\mathbb{R}$ is log-Hölder
continuous if there exists a constant $c$ such that for every pair of points
$\\{z,\,y\\}$ in $M$ we have

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

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

###### Proposition 1

aubin1982nonlinear ; gaczkowski2016sobolev  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
$qo\phi^{-1}\in\mathcal{P}^{log}(\phi(\Omega)).$

###### Definition 9

hebey2000nonlinear  We say that the N-manifold $(M,g)$ has property
$B_{vol}(\lambda,v),$ where $\lambda$ is a constant, if its geometry is
bounded in the following sense:
$\hskip 28.45274pt\bullet\,\,\mbox{The Ricci tensor of g noted by Rc ( g )
verify,}\,\,Rc(g)\geq\lambda(N-1)\,g$ for some $\lambda,$ where $N$ is the
dimension of $M.$
$\hskip 28.45274pt\bullet$ There exists some $v>0$ such that
$|\,B_{1}(z)\,|_{g}\geq v\,\,\forall z\in M,$ where $B_{1}(z)$ are the balls
of radius 1 centered at some point $z$ in terms of the volume of smaller
concentric balls.

###### Proposition 2

aubin1982nonlinear ; hebey2000nonlinear  Let $(M,g)$ be a complete 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 3

aubin1982nonlinear ; hebey2000nonlinear  Assume that the smooth 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 $z\in M$ if $1\leq q\leq N$, and if $\mathrm{u}\in
L^{q}_{1,0}(\,B_{R}(z)\,)$ the estimate

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

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

###### Proposition 4

aubin1982nonlinear ; hebey2000nonlinear ; gaczkowski2016sobolev  Assume that
for some $(\lambda,v)$ the smooth 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 $||\,\mathrm{u}\,||_{L^{q(\cdot)}_{1}}$
sufficiently small we have the estimate

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

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

###### Proposition 5

guo2015dirichlet  Let $\mathrm{u}\in
L^{q(z)}(M),\,\\{\,\mathrm{u}_{k}\,\\}\subset L^{q(z)}(M),\,k\in\mathbb{N},$
then we have

  1. (i)

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

  2. (ii)

For $\mathrm{u}\in
L^{q(z)}(M)\backslash\\{0\\},\,\,||\mathrm{u}||_{q(z)}=\lambda\Longleftrightarrow\rho_{q(z)}\big{(}\frac{\mathrm{u}}{\lambda}\big{)}=1.$

  3. (iii)

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

  4. (iv)

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

  5. (v)

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

###### Definition 10

guo2015dirichlet  The Sobolev space $W^{1,q(z)}(M)$ consists of such functions
$\mathrm{u}\in L^{q(z)}(M)$ for which $D^{k}\mathrm{u}\in L^{q(z)}(M)$
$k=1,2,\cdots,n.$ The norm is defined by

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

The space $W_{0}^{1,q(z)}(M)$ is defined as the closure of $C^{\infty}(M)$ in
$W^{1,q(z)}(M).$

###### Theorem 2.1

benslimane2020existence4  Let $M$ be a compact Riemannian manifold with a
smooth boundary or without boundary and $q(z),\,p(z)\in C(\overline{M})\cap
L^{\infty}(M).$ Assume that

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

Then,

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

is a continuous and compact embedding.

###### Theorem 2.2

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

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

Then,

$W^{1,q(z)}(M)\hookrightarrow L^{r(z)}(M),$

is a continuous and compact embedding.

###### Proof

The demonstration of this theorem is the same as the previous one.

###### Proposition 6

aubin1982nonlinear  If $(M,g)$ is complete, then
$W^{1,q(z)}(M)=W^{1,q(z)}_{0}(M).$

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

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

with the norm

$||\mathrm{u}||_{q(z),\mu(z)}=\inf\\{\alpha>0;\int_{M}\mu(z)\,\bigg{|}\frac{\mathrm{u}(z)}{\alpha}\bigg{|}^{q(z)}\,\,dv_{g}(z)\leq
1\,\\}.$

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

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

###### Example 1

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

where $C_{+}(\overline{M})=\\{q/q\in
C(\overline{M})\,\,\mbox{with}\,\,q(z)>1\,\,\mbox{for}\,\,z\in\overline{M}\\}.$

##  3 Existence results:

In this section, we are going to prove our main result stated as Theorem 1.1
in section 1. For this, we recall that
$\mathrm{J}_{\lambda}:W_{0}^{1,q(z)}(M)\rightarrow\mathbb{R}$ is the
corresponding energy function for the problem $(\mathcal{P})$ given by:

$\displaystyle\mathrm{J}_{\lambda}(\mathrm{u})=$
$\displaystyle\int_{M}\frac{1}{p(z)}\,|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\int_{M}\frac{\mu(z)}{q(z)}\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-\int_{M}\frac{1}{1-\gamma(z)}\,g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\int_{M}\frac{\lambda}{r(z)}\,|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z).$

From the presence of the singular term $g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)},$
we known that $\mathrm{J}_{\lambda}$ is not $C^{1}.$ In order to overcome
this, we will make use the Nehari manifold corresponding to the functional
$\mathrm{J}_{\lambda}$ defined as follows

$\displaystyle\mathcal{N}_{\lambda}=\\{\mathrm{u}\in
W_{0}^{1,q(z)}(M)\backslash\\{0\\};\,$
$\displaystyle||D\mathrm{u}||_{p(z)}^{p(z)}+\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(x)$
$\displaystyle=\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)+\lambda||\mathrm{u}||_{r(z)}^{r(z)}\,\\}.$

It is easy to see that $\mathcal{N}_{\lambda}$ is smaller than
$W_{0}^{1,q(z)}(M)$ and it contains the weak solutions of problem
$(\mathcal{P})$. The functional
$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}}$ can have nice
properties which fail to be true globally.
For further considerations we decompose the set $\mathcal{N}_{\lambda}$ into
three disjoint parts:

$\displaystyle\mathcal{N}_{\lambda}^{+}=\\{\mathrm{u}\in\mathcal{N}_{\lambda}:\int_{M}(p(z)-\gamma(z)-1)\,|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle+\int_{M}(q(z)+\gamma(z)-1)\,\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)-\lambda\int_{M}(r(z)+\gamma(z)-1)|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)>0\,\\},$
$\displaystyle\mathcal{N}_{\lambda}^{0}=\\{\mathrm{u}\in\mathcal{N}_{\lambda}:\int_{M}(p(z)-\gamma(z)-1)\,|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle+\int_{M}(q(z)+\gamma(z)-1)\,\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)=\lambda\int_{M}(r(z)+\gamma(z)-1)|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)\,\\},$
$\displaystyle\mathcal{N}_{\lambda}^{-}=\\{\mathrm{u}\in\mathcal{N}_{\lambda}:\int_{M}(p(z)-\gamma(z)-1)\,|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle+\int_{M}(q(z)+\gamma(z)-1)\,\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)-\lambda\int_{M}(r(z)+\gamma(z)-1)|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)<0\,\\}.$

Note that, since $\mu(\cdot):\overline{M}\longrightarrow[1,\,+\infty),$ then,
there exists $\mu_{0}>0,$ and for all $z\in M,$ we have that $\mu(z)>\mu_{0}.$
We start with the following Lemma about the coercivity of the energy
functional $\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}}$.

###### Lemma 1

If hypotheses (i)-(v) hold. Then,
$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}}$ is coercive.

###### Proof

Let $\mathrm{u}\in\mathcal{N}_{\lambda}$ with $||\mathrm{u}||>1,$ where
$||\,\cdot\,||$ is the induced norm of $W^{1,q(z)}_{0}(M)\backslash\\{0\\}.$
From the definition of the Nehari manifold $\mathcal{N}_{\lambda}$ we have

$\displaystyle\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)=$
$\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-\int_{M}g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z).$

Hence, according to (1), propositions 2, 5, Theorems 2.1, 2.2 and based on
inequality 2.3 in benslimane2020existence4  we obtain

$\displaystyle\mathrm{J}_{\lambda}(\mathrm{u})$
$\displaystyle=\int_{M}\frac{1}{p(z)}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\int_{M}\frac{\mu(z)}{q(z)}|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\int_{M}\frac{1}{1-\gamma(z)}\,g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\lambda\,\int_{M}\frac{1}{r(z)}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle\geq\frac{1}{p^{+}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\frac{1}{q^{+}}\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\frac{1}{1-\gamma^{+}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\frac{\lambda}{r^{-}}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle=\frac{1}{p^{+}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\frac{1}{q^{+}}\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\frac{1}{1-\gamma^{+}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\frac{1}{r^{-}}\bigg{[}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)-\int_{M}g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\,\bigg{]}$
$\displaystyle\geq\bigg{(}\frac{1}{p^{+}}-\frac{1}{r^{-}}\bigg{)}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\mu_{0}\,\bigg{(}\frac{1}{q^{+}}-\frac{1}{r^{-}}\bigg{)}\int_{M}|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\bigg{(}\frac{1}{r^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}\int_{M}g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle\geq\bigg{[}\frac{1}{c}\,\bigg{(}\frac{1}{p^{+}}-\frac{1}{r^{-}}\bigg{)}+\frac{\mu_{0}}{D^{p^{+}}(c+1)p^{+}}\,\bigg{(}\frac{1}{q^{+}}-\frac{1}{r^{-}}\bigg{)}\bigg{]}\,\rho_{p(.)}(\mathrm{u})$
$\displaystyle\hskip
8.5359pt+\bigg{(}\frac{1}{r^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}\int_{M}g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle\geq
c_{1}\,||\mathrm{u}||^{p^{+}}+c_{2}\,||\mathrm{u}||^{1-\gamma^{+}},$

for some $c_{1},c_{2}>0$ ( since $q^{+}<p^{+}<r^{-}$ ), and $c$ is the
Poincaré constant.
From that, since $0<\gamma^{+}<1$ and $1-\gamma^{+}<1<p^{+},$ we conclude that
$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}}$ is coercive.

Let $\sigma_{\lambda}^{+}=\inf_{\mathcal{N}_{\lambda}}\mathrm{J}_{\lambda}.$

###### Lemma 2

If the assumptions (i)-(v) are satisfied and $\mathcal{N}_{\lambda}^{+}\neq
0.$ Then $\sigma_{\lambda}^{+}<0.$

###### Proof

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

$\lambda||\mathrm{u}||_{r(z)}^{r(z)}<\frac{p^{+}+\gamma^{+}-1}{r^{-}+\gamma^{-}-1}||D\mathrm{u}||_{p(z)}^{p(z)}+\frac{q^{+}+\gamma^{+}-1}{r^{+}-\gamma^{-}-1}\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z).$

And since $\mathcal{N}_{\lambda}^{+}\subset\mathcal{N}_{\lambda},$ we get

$\displaystyle\mathrm{J}_{\lambda}(\mathrm{u})$
$\displaystyle=\int_{M}\frac{1}{p(z)}\,|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\int_{M}\frac{\mu(z)}{q(z)}\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\int_{M}\frac{g(z)}{1-\gamma(z)}\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\lambda\int_{M}\frac{1}{r(z)}\,|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle\leq\frac{1}{p^{-}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\frac{1}{q^{-}}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\frac{1}{1-\gamma^{+}}\int_{M}g(z)\,|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\frac{\lambda}{r^{+}}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle=\frac{1}{p^{-}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\frac{1}{q^{-}}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\frac{1}{1-\gamma^{+}}\bigg{[}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)-\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)\bigg{]}$
$\displaystyle\hskip
8.5359pt-\frac{\lambda}{r^{+}}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle=\bigg{(}\frac{1}{p^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}\,\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\bigg{(}\frac{1}{q^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}\,\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\lambda\,\bigg{(}\frac{1}{1-\gamma^{+}}-\frac{1}{r^{+}}\bigg{)}\,\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle\leq\bigg{[}\bigg{(}\frac{1}{p^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}+\bigg{(}\frac{1}{1-\gamma^{+}}-\frac{1}{r^{+}}\bigg{)}\frac{p^{+}+\gamma^{+}-1}{r^{-}+\gamma^{-}-1}\bigg{]}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\bigg{[}\bigg{(}\frac{1}{q^{-}}-\frac{1}{1-\gamma^{+}}\bigg{)}+\bigg{(}\frac{1}{1-\gamma^{+}}-\frac{1}{r^{+}}\bigg{)}\frac{q^{+}+\gamma^{+}-1}{r^{-}+\gamma^{-}-1}\bigg{]}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\leq\frac{p^{+}+\gamma^{+}-1}{1-\gamma^{-}}\bigg{[}\frac{1}{r^{+}}-\frac{1}{p^{-}}\bigg{]}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\mu_{0}\,\frac{q^{+}+\gamma^{+}-1}{1-\gamma^{-}}\bigg{[}\frac{1}{r^{+}}-\frac{1}{q^{-}}\bigg{]}\int_{M}|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle<0\hskip 28.45274pt\mbox{Since $q^{-}<p^{-}<r^{+}.$}$

Which implies that

$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}}<0.$

Hence,

$\sigma_{\lambda}^{+}<0.$

###### Lemma 3

Under assumptions (i)-(v), there exists $\lambda^{*}>0$ such as
$\mathcal{N}_{\lambda}^{0}=\emptyset$ for all $\lambda\in(0,\lambda^{*}).$

###### Proof

Suppose otherwise, that is $\mathcal{N}_{\lambda}^{0}\neq\emptyset$ for all
$\lambda>0,$ we can find $\mathrm{u}\in\mathcal{N}_{\lambda}^{0}$ such as

$\displaystyle(p^{+}+\gamma^{+}-1)\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+(q^{+}+\gamma^{+}-1)\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle=\lambda(r^{-}+\gamma^{-}-1)\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z),$
(4)

since $\mathrm{u}\in\mathcal{N}_{\lambda},$ we also have

$\displaystyle(r^{+}+\gamma^{+}-1)\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+(r^{+}+\gamma^{+}-1)\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle=(r^{-}+\gamma^{-}-1)\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)+\lambda(r^{-}+\gamma^{-}-1)\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z).$
(5)

Subtracting (3) from (3) yields

$\displaystyle(p^{+}-r^{+})\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+(q^{+}-r^{+})\int_{M}\mu(z)|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle=(r^{-}+\gamma^{-}-1)\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z).$
(6)

According to Theorems 13.17 of Hewitt-Stromberg [hewitt2013real , p. 196],
Theorems 2.1, 2.2 and propositions 2, 5, we deduce from (3) that

$\min\\{||\mathrm{u}||^{p^{+}},\,||\mathrm{u}||^{q^{+}}\,\\}\leq
c_{3}\,||\mathrm{u}||^{1-\gamma^{-}}\,\,\mbox{for
some}\,\,c_{3}>0.\,\,\mbox{Since}\,\,1-\gamma^{-}<q^{+}<p^{+}<r^{-}.$

Thus,

$||\mathrm{u}||\leq c_{4}\,\,\mbox{for some}\,\,c_{4}>0.$ (7)

From (3) and Theorems 2.1, we have

$||\mathrm{u}||^{r^{-}}\geq\frac{1}{\lambda\,c_{5}}\,\min\\{||\mathrm{u}||^{p^{+}},\,||\mathrm{u}||^{q^{+}}\,\\}\,\,\mbox{for
some}\,\,c_{5}>0,$

which implies that

$||\mathrm{u}||\geq\bigg{(}\frac{1}{\lambda\,c_{5}}\bigg{)}^{\frac{1}{r^{-}-p^{+}}}\hskip
28.45274pt\mbox{or}\hskip
28.45274pt||\mathrm{u}||\geq\bigg{(}\frac{1}{\lambda\,c_{5}}\bigg{)}^{\frac{1}{r^{-}-q^{+}}}.$

If $\lambda\rightarrow 0^{+}$ due to $q^{+}<p^{+}<r^{-},$ then
$||\mathrm{u}||\longrightarrow+\infty.$ Which contradicts (7).

###### Lemma 4

Under assumptions (i)-(v), there exists
$\tilde{\lambda}^{*}\in(0,\lambda^{*}]$ such that
$\mathcal{N}_{\lambda}^{\pm}\neq\emptyset$ for all
$\lambda\in(0,\tilde{\lambda}^{*}).$ In addition, for all
$\lambda\in(0,\tilde{\lambda}^{*}),$ there exists
$\tilde{\mathrm{u}}^{*}\in\mathcal{N}_{\lambda}^{+}$ such that
$\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})=\sigma_{\lambda}^{+}<0$ and
$\tilde{\mathrm{u}}^{*}(z)\geq 0$ for a.a $z\in M.$

###### Proof

Let $\mathrm{u}\in W_{0}^{1,q(z)}(M)\backslash\\{0\\}$ and consider the
function $\zeta_{\mathrm{u}}:(0,+\infty)\rightarrow\mathbb{R}$ defined by

$\zeta_{\mathrm{u}}(t)=t^{p^{+}-r^{-}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)-t^{-r^{-}-\gamma^{-}+1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z).$

Since $r^{-}-p^{+}<r^{-}+\gamma^{-}-1$ we can find $\tilde{t}_{0}>0$ such as

$\zeta_{\mathrm{u}}(\tilde{t}_{0})=\max_{t>0}\zeta_{\mathrm{u}}(t).$

Then, we have

$\displaystyle\zeta^{\prime}_{\mathrm{u}}(\tilde{t}_{0})=0$
$\displaystyle\Rightarrow(p^{+}-r^{-})\,\tilde{t}_{0}^{p^{+}-r^{-}-1}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(x)$
$\displaystyle\hskip
8.5359pt+(r^{-}+\gamma^{-}-1)\,\tilde{t}_{0}^{-r^{-}-\gamma^{-}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)=0$
$\displaystyle\Rightarrow\tilde{t}_{0}=\bigg{(}\frac{(r^{-}+\gamma^{-}-1)\,\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)}{(r^{-}-p^{+})\,\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)}\bigg{)}^{\frac{1}{p^{+}+\gamma^{-}-1}}.$

Furthermore, we have

$\displaystyle\zeta_{\mathrm{u}}(\tilde{t}_{0})$
$\displaystyle=\frac{\bigg{[}(r^{-}-p^{+})\,\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{]}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}{\bigg{[}(r^{-}+\gamma^{-}-1)\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{]}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
2.84544pt-\frac{\bigg{[}(r^{-}-p^{+})\,\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{]}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{\bigg{[}(r^{-}+\gamma^{-}-1)\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{]}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle=\frac{(r^{-}-p^{+})^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}\,\bigg{(}\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{(r^{-}+\gamma^{-}-1)^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}\,\bigg{(}\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}$
$\displaystyle-\frac{(r^{-}-p^{+})^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}\,\bigg{(}\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{(r^{-}+\gamma^{-}-1)^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}\,\bigg{(}\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}$
$\displaystyle=\bigg{[}\frac{r^{-}+\gamma^{-}-1}{r^{-}-p^{+}}\,.\,\frac{(r^{-}-p^{+})^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{(r^{-}+\gamma^{-}-1)^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}-\frac{(r^{-}-p^{+})^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{(r^{-}+\gamma^{-}-1)^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}\,\bigg{]}$
$\displaystyle\hskip
8.5359pt\times\frac{\bigg{(}\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{\bigg{(}\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}$
$\displaystyle=\frac{p^{+}+\gamma^{-}-1}{r^{-}-p^{+}}\,\bigg{(}\frac{r^{-}-p^{+}}{r^{-}+\gamma^{-}-1}\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}\,\frac{\bigg{(}\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{\bigg{(}\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}.$
(8)

According to Proposition 5 and if $G$ denotes the best Sobolev constant, we
have

$G\,||\mathrm{u}||^{p^{+}}\leq\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z).$
(9)

Moreover, we have

$\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\leq
c_{6}\,||\mathrm{u}||^{1-\gamma^{-}}\,\,\mbox{for some}\,\,c_{6}>0.$ (10)

According to (3), (9) and (10) we have

$\displaystyle\zeta_{\mathrm{u}}(\tilde{t}_{0})-\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle=\frac{p^{+}+\gamma^{-}-1}{r^{-}-p^{+}}\,\bigg{(}\frac{r^{-}-p^{+}}{r^{-}+\gamma^{-}-1}\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}\,\frac{\bigg{(}\displaystyle\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}}{\bigg{(}\displaystyle\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}$
$\displaystyle\hskip
8.5359pt-\lambda\,\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)$
$\displaystyle\geq\frac{p^{+}+\gamma^{-}-1}{r^{-}-p^{+}}\,\bigg{(}\frac{r^{-}-p^{+}}{r^{-}+\gamma^{-}-1}\bigg{)}^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}\,\frac{G^{\frac{r^{-}+\gamma^{-}-1}{p^{+}+\gamma^{-}-1}}||u||^{\frac{p^{+}(r^{-}+\gamma^{-}-1)}{p^{+}+\gamma^{-}-1}}}{\bigg{(}c_{6}||\mathrm{u}||^{1-\gamma^{-}}\bigg{)}^{\frac{r^{-}-p^{+}}{p^{+}+\gamma^{-}-1}}}$
$\displaystyle\hskip 8.5359pt+\lambda\,c_{7}\,||\mathrm{u}||^{r^{+}}$
$\displaystyle\geq(c_{6}-\lambda c_{7})\,||\mathrm{u}||^{r^{+}}\,\,\,\mbox{for
some}\,\,\,c_{6},\,c_{7}>0.$

Hence, there exists $\tilde{\lambda}^{*}\in(0,\lambda^{*}]$ independent of
$\mathrm{u}$ such as

$\zeta_{\mathrm{u}}(\tilde{t}_{0})-\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)>0\,\,\,\mbox{for
all}\,\,\,\,\lambda\in(0,\tilde{\lambda}^{*}].$ (11)

Now, for $\mathrm{u}\in W_{0}^{1,q(z)}(M)$ we consider the function
$\tilde{\zeta}_{z}:(0,+\infty)\rightarrow\mathbb{R}$ defined by

$\displaystyle\tilde{\zeta}_{\mathrm{u}}(t)=$
$\displaystyle\,t^{p^{+}-r^{-}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+t^{q^{+}-r^{-}}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-t^{-r^{-}-\gamma^{-}+1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)\,\,\,\mbox{for
all}\,\,\,t>0.$

According to (1), we get that

$r^{-}-p^{+}<r^{-}-q^{+}<r^{-}+\gamma^{-}-1.$

Thus, we can find $t_{0}>0$ such as

$\tilde{\zeta}_{\mathrm{u}}(t_{0})=\max_{t>0}\tilde{\zeta}_{\mathrm{u}}(t).$

Obviously, we have $\tilde{\zeta}_{\mathrm{u}}\geq\zeta_{\mathrm{u}}$ and by
(11) we find the existence of $\tilde{\lambda}^{*}\in(0,\lambda^{*}]$
independent of $u$ such as

$\tilde{\zeta}_{\mathrm{u}}(t_{0})-\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)>0\,\,\,\mbox{for
all}\,\,\,\lambda\in(0,\tilde{\lambda}^{*}].$

Therefore, there exists $t_{1}<t_{0}<t_{2}$ such that

$\tilde{\zeta}_{\mathrm{u}}(t_{1})=\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)=\tilde{\zeta}_{\mathrm{u}}(t_{2}),$
(12)

and

$\tilde{\zeta^{\prime}}_{\mathrm{u}}(t_{2})<0<\tilde{\zeta^{\prime}}_{\mathrm{u}}(t_{1}),$
(13)

where

$\displaystyle\tilde{\zeta^{\prime}}_{\mathrm{u}}(t)$
$\displaystyle=(p^{+}-r^{-})\,t^{p^{+}-r^{-}-1}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+(q^{+}-r^{-})\,t^{q^{+}-r^{-}-1}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-(-r^{-}-\gamma^{-}+1)t^{-r^{-}-\gamma^{-}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z).$
(14)

Now, we consider the fibering function
$\psi_{\mathrm{u}}:[0,+\infty)\rightarrow\mathbb{R}$ defined by
$\psi_{\mathrm{u}}(t)=\mathrm{J}_{\lambda}(t\mathrm{u})$ for all $t\geq 0.$
Since $\psi_{\mathrm{u}}(t)\in C^{2}((0,\infty)).$ We deduce that

$\displaystyle\psi^{\prime}_{\mathrm{u}}(t_{1})=$
$\displaystyle\,t^{p^{+}-1}_{1}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+t_{1}^{q^{+}-1}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-
t_{1}^{-\gamma^{-}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\lambda\,t_{1}^{r^{-}-1}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z),$

and

$\displaystyle\psi^{\prime\prime}_{\mathrm{u}}(t_{1})=$
$\displaystyle(p^{+}-1)\,t_{1}^{p^{+}-2}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+(q^{+}-1)\,t_{1}^{q^{+}-2}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle+\gamma^{-}\,t^{-\gamma^{-}-1}_{1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\lambda(r^{-}-1)\,t_{1}^{r^{-}-2}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z).$
(15)

According to (12) and (13), we get

$\displaystyle
t_{1}^{p^{+}-r^{-}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+t_{1}^{q^{+}-r^{-}}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-
t_{1}^{-r^{-}-\gamma^{-}+1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)=\lambda\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z),$

which implies by multiplying with $\gamma^{-}t_{1}^{r^{-}-2}$ and
$-(r^{-}-1)\,t_{1}^{r^{-}-2},$ respectively, that

$\displaystyle\gamma^{-}t_{1}^{p^{+}-2}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)+\gamma^{-}t_{1}^{q^{+}-2}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-\gamma^{-}\lambda
t_{1}^{r^{-}-2}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z)=\gamma^{-}t_{1}^{-\gamma^{-}-1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z),$
(16)

and

$\displaystyle-(r^{-}-1)t_{1}^{p^{+}-2}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)-(r^{-}-1)t_{1}^{q^{+}-2}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle+(r^{-}-1)t_{1}^{-\gamma^{-}-1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)=-\lambda(r^{-}-1)t_{1}^{r^{-}-2}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z).$
(17)

Applying (3) in (3) we obtain

$\displaystyle\psi^{\prime\prime}_{\mathrm{u}}(t_{1})$
$\displaystyle=(p^{+}+\gamma^{-}-1)t_{1}^{p^{+}-2}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+(q^{+}+\gamma^{-}-1)t_{1}^{q^{+}-2}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\lambda\,(r^{-}+\gamma^{-}-1)t_{1}^{-\gamma^{-}-1}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z).$
(18)

On the other hand, by using the same technique, we apply (3) in (3), we deduce
that

$\psi^{\prime\prime}_{\mathrm{u}}(t_{1})=t_{1}^{1-r^{-}}\zeta^{\prime}_{\mathrm{u}}(t_{1})>0.$
(19)

From (3) and (19) we conclude that

$\displaystyle\psi^{\prime\prime}_{\mathrm{u}}(t_{1})=$
$\displaystyle(p^{+}+\gamma^{-}-1)t_{1}^{p^{+}}\int_{M}|D\mathrm{u}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle+(q^{+}+\gamma^{-}-1)t_{1}^{q^{+}}\int_{M}\mu(z)\,|D\mathrm{u}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-\lambda(r^{-}+\gamma^{-}-1)t_{1}^{r^{-}}\int_{M}|\mathrm{u}(z)|^{r(z)}\,\,dv_{g}(z).$

Thus,

$t_{1}\mathrm{u}\in\mathcal{N}_{\lambda}^{+}\,\,\,\mbox{for
all}\,\,\,\lambda\in(0,\tilde{\lambda}^{*}].$

Hence,

$\mathcal{N}_{\lambda}^{+}\neq\emptyset.$

Using the same method for the point $t_{2}$ and according to (12) and (13), we
can see that $\mathcal{N}_{\lambda}^{-}\neq\emptyset.$
This shows the first assertion of the proposition. Now consider a minimizer
sequence
$\\{\mathrm{u}_{m}\\}_{m\in\mathbb{N}}\subset\mathcal{N}_{\lambda}^{+}$ such
as

$J_{\lambda}(\mathrm{u}_{m})\searrow\sigma_{\lambda}^{+}<0\,\,\,\,\mbox{as}\,\,\,\,m\rightarrow\infty.$
(20)

According to the fact that
$\mathcal{N}_{\lambda}^{+}\subseteq\mathcal{N}_{\lambda}$ and Lemma 1, we have
that

$\\{\mathrm{u}_{m}\\}_{m\in\mathbb{N}}\subseteq
W_{0}^{1,q(z)}(M)\,\,\,\,\mbox{is bounded}.$

Therefore, we may assume that

$\mathrm{u}_{m}\rightharpoonup\tilde{\mathrm{u}}^{*}\,\,\,\mbox{in}\,\,\,W_{0}^{1,q(z)}(M)\,\,\,\,\mbox{and}\,\,\,\,\mathrm{u}_{m}\rightarrow\tilde{\mathrm{u}}^{*}\,\,\,\,\mbox{in}\,\,\,\,L^{r(z)}(M).$
(21)

From (20) and (21) we know that

$\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})\leq\lim_{m\rightarrow+\infty}\inf\,\mathrm{J}_{\lambda}(\mathrm{u}_{m})<0=\mathrm{J}_{\lambda}(0).$

Hence,

$\tilde{\mathrm{u}}^{*}\neq 0.$

Arguing by contradiction, suppose that
$\mathrm{u}_{m}\not\rightarrow\tilde{\mathrm{u}}^{*}$ in $W_{0}^{1,q(z)}(M).$
Then we will have

$\lim_{m\rightarrow+\infty}\inf\int_{M}|D\mathrm{u}_{m}(z)|^{p(z)}\,\,dv_{g}(z)>\int_{M}|D\tilde{\mathrm{u}}^{*}(z)|^{p(z)}\,\,dv_{g}(z).$
(22)

Thus, using (22), we have

$\displaystyle\lim_{m\rightarrow+\infty}\inf\psi^{\prime}_{\mathrm{u}_{m}}(t_{1})$
$\displaystyle=\lim_{m\rightarrow+\infty}\inf\bigg{[}t_{1}^{p^{+}-1}\int_{M}|D\mathrm{u}_{m}(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip
5.69046pt+t_{1}^{q^{+}-1}\int_{M}\mu(z)\,|D\mathrm{u}_{m}(z)|^{q(z)}\,\,dv_{g}(z)-t_{1}^{-\gamma^{-}}\int_{M}g(z)|\mathrm{u}_{m}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip 5.69046pt-\lambda
t_{1}^{r^{-}-1}\int_{M}|\mathrm{u}_{m}(z)|^{r(z)}\,\,dv_{g}(z)\bigg{]}$
$\displaystyle>t_{1}^{p^{+}-1}\int_{M}|D\tilde{\mathrm{u}}^{*}(z)|^{p(z)}\,\,dv_{g}(z)+t_{1}^{q^{+}-1}\int_{M}\mu(z)\,|D\tilde{\mathrm{u}}^{*}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle\hskip 2.84544pt-
t_{1}^{-\gamma^{-}}\int_{M}g(z)|\tilde{\mathrm{u}}^{*}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)-\lambda
t_{1}^{r^{-}-1}\int_{M}|\tilde{\mathrm{u}}^{*}(z)|^{r(z)}\,\,dv_{g}(z).$ (23)

According to (12) and (13) we obtain that

$\lim_{m\rightarrow+\infty}\inf\psi^{\prime}_{\mathrm{u}_{m}}(t_{1})>\psi^{\prime}_{\tilde{\mathrm{u}}^{*}}(t_{1})=0.$
(24)

Which implies that, there exists $m_{0}\in\mathbb{N}$ such as

$\psi^{\prime}_{\mathrm{u}_{m}}(t_{1})>0\,\,\,\mbox{for all}\,\,\,m>m_{0}.$

Since
$\mathrm{u}_{m}\in\mathcal{N}_{\lambda}^{+}\subseteq\mathcal{N}_{\lambda}$ and
$\psi^{\prime}_{\mathrm{u}_{m}}=t^{r^{-}-1}\bigg{[}\zeta_{\mathrm{u}_{m}}-\lambda\displaystyle\int_{M}|\mathrm{u}_{m}(z)|^{r(z)}\,\,dv_{g}(z)\bigg{]},$
we have

$\psi^{\prime}_{\mathrm{u}_{m}}(t)<0\,\,\mbox{for
all}\,\,t\in(0,1)\,\,\mbox{and}\,\,\psi^{\prime}_{\mathrm{u}_{m}}(1)=0.$

Then , by (24) we have $t_{1}>0.$
Then function $\psi^{\prime}_{\tilde{\mathrm{u}}^{*}}(\cdot)$ is decreasing on
$(0,1).$ Hence, from (24) we have

$\mathrm{J}_{\lambda}(t_{1}\tilde{\mathrm{u}}^{*})\leq\mathrm{J}_{\lambda}(t\tilde{\mathrm{u}}^{*})<\sigma_{\lambda}^{+}.$
(25)

However, $t_{1}\tilde{\mathrm{u}}^{*}\in\mathcal{N}_{\lambda}^{+}.$ Hence by
(25) we get

$\sigma_{\lambda}^{+}\leq\mathrm{J}_{\lambda}(t_{1}\tilde{\mathrm{u}}^{*})<\sigma_{\lambda}^{+},$

which is a contradiction. Hence
$\mathrm{u}_{m}\rightarrow\tilde{\mathrm{u}}^{*}$ in $W_{0}^{1,q(z)}(M)$
holds, and we have

$\mathrm{J}_{\lambda}(\mathrm{u}_{m})\longrightarrow\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*}),$

which implies that

$\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})=\sigma_{\lambda}^{+}.$

Since $\mathrm{u}_{m}\in\mathcal{N}_{\lambda}^{+}$ for all $m\in\mathbb{N},$
we have

$\displaystyle(p^{+}+\gamma^{-}-1)\int_{M}|D\mathrm{u}_{m}(z)|^{p(z)}\,\,dv_{g}(z)+(q^{+}+\gamma^{-}-1)\int_{M}\mu(z)\,|D\mathrm{u}_{m}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle>\lambda(r^{-}+\gamma^{-}-1)\int_{M}|\mathrm{u}_{m}(z)|^{r(z)}\,\,dv_{g}(z).$

Letting $m\rightarrow+\infty,$ gives

$\displaystyle(p^{+}+\gamma^{-}-1)\int_{M}|D\tilde{\mathrm{u}}^{*}(z)|^{p(z)}\,\,dv_{g}(z)+(q^{+}+\gamma^{-}-1)\int_{M}\mu(z)\,|D\tilde{\mathrm{u}}^{*}(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle>\lambda(r^{-}+\gamma^{-}-1)\int_{M}|\tilde{\mathrm{u}}^{*}(z)|^{r(z)}\,\,dv_{g}(z).$
(26)

Remind that $\lambda\in(0,\tilde{\lambda}^{*})$ and
$\tilde{\lambda}^{*}\leq\lambda^{*}.$ Then, by Lemma 2, we find that equality
in (3) cannot hold. Therefore, we conclude that $\tilde{\mathrm{u}}^{*}(z)>0$
for a.a $z\in M$ with $\tilde{\mathrm{u}}^{*}\neq 0.$ This completes the proof
of Lemma 4.

###### Lemma 5

Suppose that hypotheses (i)-(v) hold, suppose $w\in W_{0}^{1,q(z)}(M)$ and let
$\lambda\in(0,\tilde{\lambda}^{*}].$ Then, there exists $\alpha>0$ such as for
all $t\in[0,\alpha],$ we have

$\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})\leq\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*}+tw).$

###### Proof

We introduce the function $\eta_{w}:[0,+\infty]\longrightarrow\mathbb{R}$
defined by

$\displaystyle\eta_{w}(t)=$
$\displaystyle\,(p^{+}+\gamma^{-}-1)\int_{M}|D\tilde{\mathrm{u}}^{*}(z)+tw(z)|^{p(z)}\,\,dv_{g}(z)$
$\displaystyle+(q^{+}+\gamma^{-}-1)\int_{M}\mu(z)\,|D\tilde{\mathrm{u}}^{*}(z)+tw(z)|^{q(z)}\,\,dv_{g}(z)$
$\displaystyle-\lambda(r^{-}+\gamma^{-}-1)\int_{M}|\tilde{\mathrm{u}}^{*}(z)+tw(z)|^{r(z)}\,\,dv_{g}(z).$
(27)

Since $\tilde{\mathrm{u}}^{*}\in\mathcal{N}_{\lambda}^{+},$ we have
$\eta_{w}(0)>0.$ According to the continuity of $\eta_{w}(\cdot)$ we find
$\alpha>0$ such as

$\eta_{w}(t)>0\,\,\,\mbox{for all}\,\,\,t\in[0,\alpha].$

Thus, $\tilde{\mathrm{u}}^{*}+tw\in\mathcal{N}_{\lambda}^{+}$ for all
$t\in[0,\alpha].$ Hence, by Lemma 4, we deduce that

$\sigma_{\lambda}^{+}=\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})\leq\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*}+tw)\,\,\,\mbox{for
all}\,\,\,\,t\in[0,\alpha].$

This ends the demonstration.

The next Lemma proves that $\mathcal{N}_{\lambda}^{+}$ is a natural constraint
for the energy functional $\mathrm{J}_{\lambda},$ see Papageorgiou-Rădulescu-
Repovš (papageorgiou2019nonlinear  p.426 ).

###### Lemma 6

Under assumptions (i)-(v), let $\lambda\in(0,\tilde{\lambda}^{*}].$ Then
$\tilde{\mathrm{u}}^{*}$ is a weak solution of $(\mathcal{P}).$

###### Proof

Let $w\in W_{0}^{1,q(z)}(M),$ by Lemma 5 we have for all $t\in[0,\alpha]$ that

$0<\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*}+tw)-\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*}).$

Then,

$\displaystyle\frac{1}{1-\gamma^{-}}\int_{M}g(z)\big{(}|\tilde{\mathrm{u}}^{*}(z)+tw(z)|^{1-\gamma(z)}-|\tilde{\mathrm{u}}^{*}(z)|^{1-\gamma(z)}\,\big{)}\,\,dv_{g}(z)$
$\displaystyle\leq\frac{1}{p^{-}}\int_{M}\big{(}|D\tilde{\mathrm{u}}^{*}(z)+tDw(z)|^{p(z)}-|D\tilde{\mathrm{u}}^{*}(z)|^{p(z)}\big{)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt+\frac{1}{q^{-}}\int_{M}\mu(z)\,\big{(}|D\tilde{\mathrm{u}}^{*}(z)+tDw(z)|^{q(z)}-|D\tilde{\mathrm{u}}^{*}(z)|^{q(z)}\big{)}\,\,dv_{g}(z)$
$\displaystyle\hskip
8.5359pt-\frac{\lambda}{q^{-}}\int_{M}\big{(}|\tilde{\mathrm{u}}^{*}(z)+tw(z)|^{r(z)}-|\tilde{\mathrm{u}}^{*}(z)|^{r(z)}\big{)}\,\,dv_{g}(z).$
(28)

Dividing the above inequality by $t,$ then $t\rightarrow 0^{+}.$ We deduce
that

$\displaystyle\int_{M}g(z)(\tilde{\mathrm{u}}^{*})^{-\gamma(z)}w(z)\,\,dv_{g}(z)\leq$
$\displaystyle\int_{M}|D\tilde{\mathrm{u}}^{*}(z)|^{p(z)-2}D\tilde{\mathrm{u}}^{*}(z)\,.\,Dw(z)\,\,dv_{g}(z)$
$\displaystyle+\int_{M}\mu(z)\,|D\tilde{\mathrm{u}}^{*}(z)|^{q(z)-2}D\tilde{\mathrm{u}}^{*}(z)\,.\,Dw(z)\,\,dv_{g}(z)$
$\displaystyle-\lambda\int_{M}(\tilde{\mathrm{u}}^{*}(z))^{r(z)-1}\,.\,w\,\,dv_{g}(z).$

Since, $w\in W_{0}^{1,q(z)}(M)$ is arbitrary. Then that equality must hold,
and so $\tilde{\mathrm{u}}^{*}$ is a weak solutions of $(\mathcal{P})$ for all
$\lambda\in(0,\tilde{\lambda}^{*}].$

Now, using the manifold $\mathcal{N}_{\lambda}^{-},$ we will achieve a second
weak solution when the parameter $\lambda>0$ is sufficiently small.

###### Lemma 7

If hypotheses (i)-(v) are satisfied. Then there exists
$\tilde{\lambda}^{*}_{0}\in(0,\tilde{\lambda}^{*}]$ such as for all
$\lambda\in(0,\tilde{\lambda}^{*}_{0}]$ we have
$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}^{-}}\geq 0.$

###### Proof

Let $\mathrm{u}\in\mathcal{N}_{\lambda}^{-}.$ According to Lemma 4, Theorem
2.2 and the definition of $\mathcal{N}_{\lambda}^{-}$ we obtain that

$\displaystyle(p^{+}+\gamma^{-}-1)||D\mathrm{u}||_{p(z)}^{p^{+}}+(q^{+}+\gamma^{-}-1)\,||D\mathrm{u}||_{q(z),\mu(z)}^{q^{+}}<\lambda\,(r^{-}+\gamma^{-}-1)||\mathrm{u}||_{r(z)}^{r^{-}}.$

Then,

$\lambda(r^{-}+\gamma^{-}-1)||\mathrm{u}||_{r(z)}^{r^{-}}>(q^{+}+\gamma^{-}-1)\,\mu_{0}\,||D\mathrm{u}||_{q(z)}^{q^{+}}>\mu_{0}\,(q^{+}+\gamma^{-}-1)\,.\,c\,||\mathrm{u}||^{q^{+}}_{q(z)},$

where, $c$ being the Poincaré canstant. Thus, by Theorem 2.2 we have that

$||\mathrm{u}||^{r^{-}-q^{+}}_{r(x)}>\frac{(q^{+}+\gamma^{-}-1)\,\mu_{0}\,.\,c\,.\,c_{8}}{\lambda\,(r^{-}+\gamma^{-}-1)},$

where, $c_{8}$ being the constant of the embedding Theorem 2.2. Hence,

$||\mathrm{u}||_{r(z)}>\bigg{[}\frac{(q^{+}+\gamma^{-}-1)\,\mu_{0}\,.\,c\,.\,c_{8}}{\lambda\,(r^{-}+\gamma^{-}-1)}\,\bigg{]}^{\frac{1}{r^{-}-q^{+}}}.$
(29)

Arguing by contradiction, suppose that the Lemma is not true. Then, we can
find $\mathrm{u}\in\mathcal{N}_{\lambda}^{-}$ such as
$\mathrm{J}_{\lambda}(\mathrm{u})<0,$ that is

$\displaystyle\frac{1}{p^{-}}||D\mathrm{u}||_{p(z)}^{p^{+}}+\frac{1}{q^{-}}||D\mathrm{u}||_{q(z),\mu(z)}^{q^{+}}-\frac{1}{1-\gamma^{-}}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle-\frac{\lambda}{r^{+}}||\mathrm{u}||_{r(z)}^{r^{-}}<0.$ (30)

Since $\mathrm{u}\in\mathcal{N}_{\lambda}^{-}\subseteq\mathcal{N}_{\lambda},$
we know that

$||D\mathrm{u}||_{q(z),\mu(z)}^{q^{+}}=-||D\mathrm{u}||_{p(z)}^{p^{+}}+\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)+\lambda||\mathrm{u}||_{r(z)}^{r^{-}},$
(31)

from (3) and (31) we have

$\displaystyle\big{(}\frac{1}{p^{-}}-\frac{1}{q^{-}}\big{)}||D\mathrm{u}||_{p(z)}^{p^{+}}+\big{(}\frac{1}{q^{-}}-\frac{1}{1-\gamma^{-}}\big{)}\int_{M}g(z)|\mathrm{u}(z)|^{1-\gamma(z)}\,\,dv_{g}(z)$
$\displaystyle+\lambda\,\big{(}\frac{1}{q^{-}}-\frac{1}{r^{+}}\big{)}||\mathrm{u}||_{r(z)}^{r^{-}}<0,$

which implies that

$\lambda\,\big{(}\frac{1}{q^{-}}-\frac{1}{r^{+}}\big{)}||\mathrm{u}||_{r(z)}^{r^{-}}<\big{(}\frac{1}{1-\gamma^{-}}-\frac{1}{q^{-}}\big{)}\,c_{9}\,||\mathrm{u}||_{r(z)}^{1-\gamma^{-}}\,\,\,\,\mbox{for
some}\,\,\,\,c_{9}>0.$

Thus, since $q^{-}<p^{-}<r^{+}$ we have

$||\mathrm{u}||_{r(z)}^{r^{-}+\gamma^{-}-1}\leq\frac{c_{9}\,(q^{-}+\gamma^{-}-1)\,r^{+}}{\lambda\,(1-\gamma^{-})(r^{+}-q^{-})}.$

Hence,

$||\mathrm{u}||_{r(z)}\leq
c_{10}\,\bigg{(}\frac{1}{\lambda}\bigg{)}^{\frac{1}{r^{-}+\gamma^{-}-1}}\,\,\,\mbox{for
some}\,\,\,\,c_{10}>0.$ (32)

Applying (32) in (29) we get

$c_{11}\,\bigg{(}\frac{1}{\lambda}\bigg{)}^{\frac{1}{r^{-}-q^{+}}}\leq
c_{10}\bigg{(}\frac{1}{\lambda}\bigg{)}^{\frac{1}{r^{-}+\gamma^{-}-1}}\,\,\,\,\mbox{with}\,\,\,\,c_{11}=\bigg{(}\frac{\mu_{0}\,.\,c\,.\,c_{8}\,(q^{+}+\gamma^{-}-1)}{r^{-}+\gamma^{-}-1}\bigg{)}^{\frac{1}{r^{-}-q^{+}}}.$

Hence,

$0<\frac{c_{11}}{c_{10}}<\lambda^{\frac{1}{r^{-}-q^{+}}-\frac{1}{r^{-}+\gamma^{-}-1}}=\lambda^{\frac{q^{+}+\gamma^{-}-1}{(r^{-}-q^{+})(r^{-}+\gamma^{-}-1)}}\longrightarrow
0\,\,\,\mbox{as}\,\,\lambda\rightarrow 0^{+}.$

Since $1<q^{+}<r^{-}$ and $\gamma(\cdot)\in(0,1),$ which is a contradiction.
Thus, we conclude that we can find
$\tilde{\lambda}^{*}_{0}\in(0,\tilde{\lambda}^{*}]$ such as for all
$\lambda\in(0,\tilde{\lambda}^{*}_{0}]$ we have

$\mathrm{J}_{\lambda}\bigg{|}_{\mathcal{N}_{\lambda}^{-}}\geq 0.$

###### Lemma 8

Under assumptions (i)-(v), let $\lambda\in(0,\tilde{\lambda}^{*}_{0}].$ Then,
there exists $\tilde{v}^{*}\in\mathcal{N}_{\lambda}^{-}$ with
$\tilde{v}^{*}\geq 0$ such as

$\sigma_{\lambda}^{-}=\inf_{\mathcal{N}_{\lambda}^{-}}=\mathrm{J}_{\lambda}(\tilde{v}^{*})>0.$

###### Proof

Using the same method as Lemma 4. If
$\\{v_{m}\\}_{m\in\mathbb{N}}\subseteq\mathcal{N}_{\lambda}^{-}$ is a
minimizing sequence, then, by Lemma 1, we have that
$\\{\mathrm{u}_{m}\\}_{m\in\mathbb{N}}\subseteq W_{0}^{1,q(z)}(M)$ is bounded.
Then, we may assume that

$v_{m}\rightharpoonup\tilde{v}^{*}\,\,\,\mbox{weakly
in}\,\,\,W_{0}^{1,q(z)}(M)\,\,\,\mbox{and}\,\,\,v_{m}\rightarrow\tilde{v}^{*}\,\,\,\mbox{in}\,\,\,L^{r(z)}(M).$

From (12) and (13) we can find $0<t_{2}$ such as

$\tilde{\zeta^{\prime}}_{\tilde{v}^{*}}(t_{2})<0\,\,\,\mbox{and}\,\,\,\tilde{\zeta}_{\tilde{v}^{*}}(t_{2})=\lambda||\tilde{v}^{*}||_{r(z)}^{r^{-}}.$
(33)

We contend as in the proof of Lemma 4 and using (33), we obtain that
$\tilde{v}^{*}\in\mathcal{N}_{\lambda}^{-},\,\ \tilde{v}^{*}\geq
0,\,\,\,\sigma_{\lambda}^{-}=\mathrm{J}_{\lambda}(\tilde{v}^{*}).$

###### Lemma 9

Under assumptions (i)-(v) and
$\lambda\in(0,\tilde{\lambda}^{*}],\,\,\tilde{v}^{*}$ is a weak solution of
the problem $(\mathcal{P})$.

###### Proof

To demonstrate this Lemma, we use the same reasoning as in the proofs of
Lemmas 5 and 6.

## Conclusion

According to the above Lemmas, our problem $(\mathcal{P})$ has at least two
positive solutions $\tilde{\mathrm{u}}^{*},\,\tilde{v}^{*}\in
W_{0}^{1,q(z)}(M),$ such as
$\mathrm{J}_{\lambda}(\tilde{\mathrm{u}}^{*})<0\leq\mathrm{J}_{\lambda}(\tilde{v}^{*})$
for all $\lambda\in(0,\,\tilde{\lambda}^{*}_{0}]$ where
$\tilde{\lambda}^{*}_{0}>0.$

## Authors’ contributions

The authors declare that their contributions are equal.

## Acknowledgments

This paper has been supported by the RUDN University Strategic Academic
Leadership Program and P.R.I.N. 2019.
The authors would like to thank the anonymous referees for the valuable
suggestions and comments which improved the quality of the presentation.

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