\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 156, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/156\hfil Fractional $p$-Laplacian equations]
{Fractional $p$-Laplacian equations on \\ Riemannian manifolds}

\author[L. Guo, B. Zhang, Y. Zhang \hfil EJDE-2018/156\hfilneg]
{Lifeng Guo, Binlin Zhang, Yadong Zhang}

\address{Lifeng Guo \newline
School of Mathematics and Statistics,
Northeast Petroleum University,
 Daqing 163318,  China}
\email{lfguo1981@126.com}

\address{Binlin Zhang (corresponding author)\newline
Department of Mathematics,
Heilongjiang Institute of Technology,
Harbin 150050,  China}
\email{zhangbinlin2012@outlook.com}

\address{Yadong Zhang \newline
School of Mathematics and Statistics,
Northeast Petroleum University,
Daqing 163318,  China}
\email{chichidong@foxmail.com}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted August 18, 2017. Published August 22, 2018.}
\subjclass[2010]{35R01, 35R11, 35A15}
\keywords{Fractional $p$-Laplacian; Riemannian manifolds;
variational methods}

\begin{abstract}
 In this article we establish the theory of fractional Sobolev spaces
 on Riemannian manifolds. As a consequence we investigate some important
 properties, such as the reflexivity, separability, the embedding theorem
 and so on. As an application, we consider fractional $p$-Laplacian
 equations with homogeneous Dirichlet boundary conditions
 \begin{gather*}
 (-\Delta_g)^s_p u(x)= f(x,u) \quad \text{in } \Omega,\\
 u=0 \quad \text{in } M\setminus\Omega,
 \end{gather*}
 where $N> ps$ with $s\in(0,1)$, $p\in (1, \infty)$, $(-\Delta_g)^s_p$ is
 the fractional $p$-Laplacian on Riemannian manifolds, $(M,g)$ is a compact
 Riemannian $N-$manifold, $\Omega$ is an open bounded subset of $M$ with
 smooth boundary $\partial\Omega$, and $f$ is a Carath\'{e}odory function
 satisfying the Ambrosetti-Rabinowitz type condition.
 By using variational methods, we obtain the existence of nontrivial weak
 solutions when the nonlinearity $f$ satisfies sub-linear or super-linear
 growth conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Recently, great attention has been paid on the study of problem involving 
fractional and non-local operators. This type of problem arises in many
applications, such as, continuum mechanics, phase transition phenomena, 
population dynamics and game theory, as they are the typical outcome of 
stochastically stabilization of L\'{e}vy processes, see \cite{r2, r-1, r-2} 
and the references therein. Here we would like to point out some 
interesting models involving the fractional Laplacian, such as, the 
fractional Lane-Emden equation (see \cite{FW}),
the fractional Schr\"{o}dinger equation (see \cite{ZZR, ZZX}), 
the fractional Kirchhoff equation (see \cite{r19, PXZ1, PXZ2, XZR2}), 
the fractional Cahn-Hilliard, Allen-Cahn and porous medium equations 
(see \cite{ASS, JLV}),
the fractional Yamabe problem (see \cite{CG}) and so on, have attracted recently 
considerable attention.
Indeed, the literature on non-local operators and their applications is 
very interesting and quite large, we refer the interested reader 
to \cite{AP, r15, r-4, MR} and the references therein. 
For the basic properties of fractional Sobolev spaces, we refer the 
interested reader to \cite{r8, MRS}.

In this article we deal with the  fractional $p$--Laplace problem
\begin{eqnarray}\label{k1}
\begin{gathered}
(-\Delta_g)^s_p u(x)= f(x,u) \quad \text{in } \Omega,\\
u=0 \quad \text{in } M\setminus\Omega,
\end{gathered}
\end{eqnarray}
where $N> ps$ with $s\in(0,1)$, $p\in (1, \infty)$, $(M,g)$ is a compact 
Riemannian $N-$manifold, $\Omega\subset M$ is an open bounded set with 
smooth boundary $\partial\Omega$, $f:\Omega\times\mathbb{R}\to\mathbb{R}$
is a Carath\'{e}odory function and $(-\Delta_g)^s_p u(x)$ is the fractional 
$p$-Laplace operator
which (up to normalization factors) may be defined as
\begin{equation*}
(-\Delta_g)^s_p u(x)=2\lim_{\varepsilon\to 0^{+}}
\int_{M\backslash B_\varepsilon(x)}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}
{(d_g(x,y))^{N+ps}}\,d\mu_g(y)
\end{equation*}
for $x\in M$, where $B_x(\varepsilon)$ denotes the geodesic ball of $M$ 
of center $x$ and radius $\varepsilon$ and $d_g(x,y)$ defines a distance 
on $M$ whose topology coincides with the original one of $M$, see Section 2 
for more details.

In the Euclidean case, problem \eqref{k1} reduces to the fractional Laplacian 
problem as $p=2$:
\begin{eqnarray}\label{k4}
\begin{gathered}
(-\Delta)^{s} u(x)= f(x,u) \quad \text{in } \Omega,\\
u=0 \quad \text{in } \mathbb{R}^{N}\setminus \Omega.
\end{gathered}
\end{eqnarray}
One typical feature of problem \eqref{k4} is the nonlocality, in the sense that
the value of $(-\Delta)^su(x)$ at any point $x\in \Omega$
depends not only on $\Omega$, but actually on the entire space $\mathbb{R}^N$.
The functional framework that takes into account problem \eqref{k4} with 
Dirichlet boundary condition was introduced in \cite{r17, r14}.
It is well known that problem \eqref{k1} has been used to model some physical 
phenomena occurring in nonlocal reaction-diffusion problems, non-Newtonian fluid, 
non-Newtonian filtration and turbulent flows of a gas in a porous medium, 
and so on. In the non-Newtonian fluid theory, the quantity $p$ is characteristic 
of the medium. Media with $p>2$ are called dilatant fluid and those with $p<2$ 
are called pseudoplastics. If $p=2$, they are Newtonian fluids.
Concerning the fractional Sobolev spaces in $\mathbb{R}^N$ and its applications 
to the  qualitative analysis of solutions for problem \eqref{k4}, 
we refer to \cite{ r12, r10, M, r-6, r-5, r16, XZR1} and the references therein 
for further details.


In recent years, the conformal fractional Laplacian  has received a lot
of attention. More precisely, the conformal fractional Laplacian is defined on 
the boundary of a Poincar\'{e}-Einstein
manifold in view of scattering theory, see \cite{MG} for all the necessary 
background.
Caffarelli and Silvestre \cite{r15} presented a construction for the standard
fractional Laplacian $(-{\Delta}_{\mathbb{R}^N})^s$ as a Dirichlet-to-Neumann 
operator of a uniformly degenerate
elliptic boundary value problem. In the manifold case, Chang and Gonz\'{a}lez 
\cite{CG} linked the
original definition of the conformal fractional Laplacian coming from scattering 
theory to a Dirichlet-to-Neumann operator for a related elliptic extension problem,
thus allowing for an analytic treatment of Yamabe-type problems in the non-local 
setting, see \cite{GQ}.
As for several definitions of fractional Laplace operator and their interrelation, 
we refer to \cite{r8} for more details.

Inspired by the above works, we are interested in considering the integral definition 
of fractional Laplacian from $\mathbb{R}^N$ to Riemannian manifolds. To our best 
knowledge, there is no result along this line.
It is worth to point out that our definition seems easier to be understood than  
the conformal fractional Laplacian, just from  the analytic points of view. 
In the mean time,  our definition would be convenient to generalize some related 
existence results on fractional Laplace equations exploited
by variational methods to those of Riemannian manifolds.

As an application of the fractional Sobelev spaces on Riemannian manifolds, 
we will consider the existence of weak solutions for problem \eqref{k1}.
 For this purpose,  we assume that $\Omega\subset M$ is a open bounded set
 and $f: \Omega \times \mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory 
function satisfying the following:
\begin{itemize}
 \item[(A1)]
 There exist $a>0$ and $1<q<p_s^*=Np/(N-ps)$ such that
$$
|f(x,\eta)|\leq a(1+|\eta|^{q-1}),
$$
for a.e.\ $x\in\Omega$,\ $\eta\in\mathbb{R}$;

 \item[(A2)]
There exist $\gamma>\displaystyle p$ and $r>0$ such that for a.e. $x\in\Omega$ 
and $r\in\mathbb{R}$, $|\xi|\geq r,$
$$
0<\gamma F(x,\xi)\leq \xi f(x,\xi),
$$
where $F(x,\xi)=\int_0^\xi f(x,\tau)d\tau$;

\item[(A3)] It holds
$$\lim_{\zeta\to0}\frac{f(x,\zeta)}{|\zeta|^{p-1}}=0\text{ uniformly for a. e. } 
 x\in \Omega;
$$
 
\item[(A4)]
There exist $a_1>0$ and an open bounded set $\Omega_0\subset\Omega$ such that
$$
|f(x,\rho)|\geq a_1|\rho|^{q-1}\quad\text{for a.e. } x\in\Omega_0\text{ and all }
 \rho\in \mathbb{R}.
$$
 \end{itemize}

Now, we give the definition of weak solutions for problem \eqref{k1}.

\begin{definition} \rm
We say that $u\in W_0^{s,p}(\Omega)$ is a weak solution of problem \eqref{k1}, if
\begin{align*}
&\iint_{M \times M}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{(d_g(x,y))^{N+ps}}
 \,d\mu  _g(x)d\mu_g(y) \\
&=\int_{\Omega}f(x,u(x))\varphi(x)d\mu  _g(x),
\end{align*}
for any $\varphi\in W_0^{s,p}(M)$, where space $W_0^{s,p}(M)$ will be introduced 
in Section 2.
\end{definition}

Then, by variational methods, we can get the following existence results for 
problem \eqref{k1}.

\begin{theorem}\label{k5}
Let {\rm (A1)} and {\rm (A4)} hold. If $1<q<p$, then the problem \eqref{k1}
 has a nontrivial weak solution in $W_0^{s,p}(M)$.
\end{theorem}

\begin{theorem}\label{k6}
Let  {\rm (A1)--(A3)} hold. If $p<q<p_s^*$, then problem \eqref{k1} 
has a nontrivial weak solution in $W_0^{s,p}(M)$.
\end{theorem}

\begin{remark} \rm 
Theorems \ref{k5} and \ref{k6} can be viewed as the counterpart of 
\cite[Theorems 1.1 and 1.2]{XZF} on compact Riemannian $N$-manifold
 in the non-Kirchhoff case.
\end{remark}


This article is organized as follows. In Section 2, we will present 
some necessary definitions and properties of space $W_0^{s,p}(M)$. 
In Section 3, using variational methods, we obtain the existence of 
weak solutions for problem \eqref{k1}  in two cases: $1<q < p$ and $p<q<p_s^*$.

\section{Fractional Sobolev space on Riemannian manifolds}

Let we first recall some basic material on Riemannian geometry 
(see \cite{guo1, guo2}). Let $(M,g)$ be a smooth Riemannian $N-$manifold, 
and let $\nabla$ be the Levi-Civita connection.
For $u\in C^\infty(M)$, then $\nabla^k u$ denotes the $k-$th covariant 
derivative of $u$. In local coordinates, the pointwise norm of $\nabla^k u$ 
is given by
$$
|\nabla^k u|=g^{i_1j_1}\cdot\cdot\cdot g^{i_kj_k}(\nabla^k u)_{i_1i_2\dots i_k}
(\nabla^k u)_{j_1j_2\dots j_k}
$$
When $k=1$, the components of $\nabla u$  in local coordinates are given by 
$(\nabla u)_i=\nabla^iu$. By definition one has that
$$
   |\nabla u|=\sum_{i,j=1}^\infty g^{ij}\nabla^iu\nabla^ju
$$

Given $(M,g)$ a smooth Riemannian $N-$manifold, and $\gamma:[a, b]\to M$
a curve of class $C^1$, the length of $\gamma$ is
$$
L(\gamma)=\int_a^b\sqrt{g \big(\gamma(t)\big)\Big(\big(\frac{d\gamma}{dt}\big)(t),
\big(\frac{d\gamma}{dt}\big)(t)\Big)}d\mu.
$$
For $x,y\in M$, let $C^{1}_{x,y}$ be the space of piecewise
$C^1$ curves $\gamma:[a, b]\to M$ such that $\gamma(a)=x$
and $\gamma(b)=y$. Then $d_g(x,y)=\inf_{C^{1}_{x,y}}L(\gamma)$ defines a 
distance on $M$ whose topology coincides with the original one of $M$.
In particular, by Stine's theorem, a smooth Riemannian manifold is paracompact.
By definition, $d_g$ is the distance associated to $g$.

Given $(M,g)$ a smooth Riemannian $N$-manifold, one can define a natural 
positive Radon measure on $M$. In particular, the theory of the Lebesgue 
integral can be applied. For $(\Omega_i,\varphi_i)_{i\in I}$ some atlas 
of $M$, we shall say that a family  $(\Omega_j,\varphi_j,\eta_j)_{j\in J}$ 
is a partition of unity subordinate to $(\Omega_i,\varphi_i)_{i\in I}$. 
As one can easily check, for any atlas $(\Omega_i,\varphi_i)_{i\in I}$ of  $M$, 
there exists a partition of unity $(\Omega_j,\varphi_j,\eta_j)_{j\in J}$ 
subordinate $(\Omega_i,\varphi_i)_{i\in I}$. Then we can define the Riemannian 
measure as follows: given $u:M\to\mathbb{R}$ is continuous with compact support, 
and given $(\Omega_i,\varphi_i)_{i\in I}$ is an atlas of $M$,
$$
\int_Mu(x)d\mu_g(x)=\sum_{k\in J}\int_{\varphi_k(\Omega_k)}
\Big(\sqrt{\det(g_{ij})}\eta_k u\Big)\circ \varphi_k^{-1}(x)dx,
$$
where  $(\Omega_j,\varphi_j,\eta_j)_{j\in J}$ is a partition of unity subordinate 
to $(\Omega_i,\varphi_i)_{i\in I}$,
$d\mu_g(x)=\sqrt{\text{det}(g_{ij})}dx$ 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
$\mathbb{R}^N$.

In what follows, we give some basic results that will be used in the next section.
In the Euclidean case, we refer to \cite{FSV, r17, r14, XZF} for related results.
Let $0<s<1<p<\infty$ be real numbers and the fractional critical exponent $p^*_s$ 
be defined as
\begin{align*}
p_s^*=
\begin{cases}
\frac{Np}{N-sp} &\text{if } sp<N\\
\infty &\text{if } sp\geq N.
\end{cases}
\end{align*}

This section is devoted to the definition of the fractional Sobolev spaces 
on Riemannian manifolds.
We start by fixing the fractional exponent s in $(0,1)$. For any
$p\in[1,+\infty)$, we define $W^{s,p}(M)$ as follows:
\begin{equation*}%\label{equation1}
W^{s,p}(M)=\big\{u\in L^p{(M)}: \frac{|u(x)-u(y)|}{(d_g(x,y)
)^{\frac{n}{p}+s}}\in L^p(M\times M)\big\}
\end{equation*}
i.e, an intermediary Banach space between $L^p{(M)}$ and $W^{s,p}(M)$, 
endowed with the natural norm
\begin{equation*}%\label{equation2}
\|u\|_{W^{s,p}(M)}=\Big(\int_M|u(x)|^pd\mu_g(x)+[u]_{W^{s,p}(M)}^p\Big)^{1/p},
\end{equation*}
where the term
\begin{equation*}%\label{equation3}
[u]_{W^{s,p}(M)}=\Big(\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}d\mu 
 _g(x)d\mu  _g(y)\Big)^{1/p}.
\end{equation*}
is the so-called Gagliardo (semi)norm of $u$.

It is easy to prove that $\|\cdot\|_{W^{s,p}(M)}$ is a norm on $W^{s,p}(M)$. 
We will work in the closed linear subspace
\[
W_0^{s,p}(M)=\big\{u\in W^{s,p}(M) :\operatorname{supp} (u)\text{ is a compact subset
 of }  M\big\},
\]
where $\operatorname{supp}(u)=\overline{\{x\in M: u(x)\neq 0\}}$.

\begin{lemma}\label{lemma1}
Let $(M,d_g)$ be a complete Riemannian $N-$manifold with finite volume, then 
$C_0^\infty(M)\subset W_0^{s,p}(M)$.
\end{lemma}

\begin{proof}
For $v\in C_0^\infty(M)$, we only need to check that
 $$
\iint_{M\times M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)
d\mu  _g(y)<\infty. 
$$
Notice that
\begin{gather*}
|v(x)-v(y)|\leq\|\nabla v\|_{L^\infty(M)}d_g(x,y), \\
|v(x)-v(y)|\leq2 \|v\|_{L^\infty(M)}
\end{gather*}
for all $x,y\in M$. Thus,
\begin{align*}
|v(x))-v(y))|^p\leq (2\|v\|_{C^1(M)})^p\min\{(d_g(x,y))^p,1\}.
\end{align*}
Therefore, 
\begin{align*}
& \iint_{M\times M}\frac{|(\eta_sv)(x)-(\eta_sv)(y)|^p}
{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y)\\
&\leq   \operatorname{Vol}(M)(2\|v\|_{C^1(M)})^p\iint_{M\times M}
\frac{\min\{(d_g(x,y))^p,1\}}{(d_g(x,y))^{N+sp}}d\mu  _g(x)d\mu  _g(y)
<\infty.
\end{align*}
Consequently, for $v\in C_0^\infty(M)$ we have
\[
\iint_{M\times M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y)
<\infty.
\]
This implies $v\in W_0^{s,p}(M)$.
\end{proof}

\begin{remark}\label{r3} \rm
 Lemma \ref{lemma1} and the fact that $C^\infty_0(M)$ is dense 
in $L^p(M)$ (see for example \cite{guo2}), imply that $C^\infty_0(M)$ is 
dense also in $W^{s,p}(M)$.
\end{remark}

\begin{remark} \rm
The space $W_0^{s,p}(M)$ is  the closure of
$C_0^\infty( M)$ in $W^{s,p}(M)$.
\end{remark}

\begin{lemma}\label{lemma3}
Let $(M,d_g)$ be a compact Riemannian $N-$manifold. Then
\begin{itemize}
\item[(1)] there exists a positive constant $C_1=C_1(N,p,q,s)$ such that for any
 $v\in W_0^{s,p}(M)$ and $1\leq q \leq p^*_s$,
\[
\|v\|^{p}_{L^{q}(M)}\leq C_1\iint_{M\times M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}
d\mu  _g(x)d\mu  _g(y).
\]

\item[(2)] there exists a constant $\widetilde{C}=\widetilde{C}(N,p,q,s)$ such that
 for any $v\in W_0^{s,p}(M)$,
\begin{align*}
&\iint_{M\times M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y) \\
&\leq \|v\|^p_{W^{s,p}(M)} \\
&\leq \widetilde{C}\iint_{M\times M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y).
\end{align*}
\end{itemize}
\end{lemma}

\begin{proof}
Let $v\in W_0^{s,p}(M)$.  Since $M$ is compact, $M$ can be covered by a 
finite number of charts
\[
(B_{x_k}(r),\varphi_k)_{k=1,2,\dots ,m}
\]
satisfying
\begin{equation} \label{l1.00}
B_0(r/2)\subset\varphi_k(B_{x_k}(r))\subset B_0(2r)\quad\text{and}\quad
\frac{1}{Q}\delta_{ij}\leq g_{ij}^s\leq Q\delta_{ij},
\end{equation}
where $g_{ij}^s$ are bilinear forms, $Q>1$ is given, $B_{x_k}(r)$ denotes the
 ball of $M$ of center $x_k$ and radius $r$, $B_0(2r)$ denotes the Euclidean ball
of $\mathbb{R}^N$ of center 0 and radius $2r$. Moreover, we have
\begin{equation} \label{l1.000}
\frac{1}{C}|\varphi_k^{-1}(y_1)-\varphi_k^{-1}(y_2)|\leq d_g(y_1,y_2)
\leq C|\varphi_k^{-1}(y_1)-\varphi_k^{-1}(y_2)|,
\end{equation}
for $y_1,y_2\in B_{x_k}(r)$ where $C>1$ is given.

Let $(\eta_k)$ be a smooth partition of unity subordinate to the covering 
$B_{x_k}(r)$. For any $k$, using \cite[Theorem 6.5]{r8}, we obtain
\begin{equation} \label{l1.0}
\begin{aligned}
&\|v\|^{p}_{L^{q}(M)}\leq 2^p\sum_{k=1}^m\|\eta_kv\|^{p}_{L^{q}(M)} \\
&\leq 2^\frac{pq+pN}{q}\sum_{k=1}^m\|(\eta_kv)\circ \varphi^{-1}\|^{p}_{L^{q}
 (\mathbb{R}^N)} \\
&\leq  C_02^\frac{pq+pN}{q}\sum_{k=1}^m\iint_{\mathbb{R}^N\times \mathbb{R}^N}
 \frac{|(\eta_kv)(\varphi_k^{-1}(x))-(\eta_kv)(\varphi_k^{-1}(y))|^{p}}{|x-y|^{N+ps}}
\,dx\,dy \\
&\leq  CC_02^{\frac{pq+pN}{q}+2N}\sum_{k=1}^m\iint_{M\times M}\frac{|(\eta_kv)
 (\bar{x})-(\eta_kv)(\bar{y})|^{p}}{(d_g(\bar{x},\bar{y}))^{N+ps}}d\mu  _g
 (\bar{x})d\mu  _g(\bar{y}) \\
&\leq  mCC_02^{\frac{pq+pN}{q}+2N}\iint_{M\times M}\frac{|v(\bar{x})-v(\bar{y})|^{p}}
 {(d_g(\bar{x},\bar{y}))^{N+ps}}d\mu  _g(\bar{x})d\mu  _g(\bar{y}) \\
&=C_1\iint_{M\times M}\frac{|v(\bar{x})-v(\bar{y})|^{p}}
 {(d_g(\bar{x},\bar{y}))^{N+ps}}d\mu  _g(\bar{x})d\mu  _g(\bar{y}),
\end{aligned}
\end{equation}
where $C_1=mCC_02^{\frac{pq+pN}{q}+2N}$ is a positive constant depending only on
$N,s,p,q$. Thus, we obtain the assertion (1). The assertion (2) easily follows
 by combining the definition of norm of $W^{s,p}(M)$ with \eqref{l1.0}.
\end{proof}


\begin{remark} \rm
 By Lemma \ref{lemma3}, we obtain an equivalent norm on $W_0^{s,p}(M)$ defined as
\[
\|v\|_{W_0^{s,p}(M)}=\Big(\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}
d\mu  _g(x)d\mu  _g(y)\Big)^{1/p},
\]
 for all $ v\in W_0^{s,p}(M)$.
\end{remark}


\begin{lemma}\label{lemma1.1}
Let $(M,d_g)$ be a compact Riemannian $N$-manifold, $p\in [1,\infty]$ and 
$s\in(0,1)$. Then
\[
 \|u\|_{W^{s,p}(M)}\leq  \|u\|_{W^{1,p}(M)}
\]
for some suitable positive constant $C=C(N,s,p)\geq1$. In particular,
\begin{equation*}
  W^{1,p}(M)\subseteq W^{s,p}(M).
\end{equation*}
\end{lemma}

\begin{proof}
Let $\gamma:[0,1]\to M$ be the minimizing geodesic from $x$ and $y$, where 
$x,y\in M$. Then for $v\in W^{s,p}(M)$ we have
\begin{equation} \label{l01}
\begin{aligned}
&\int_{M}\int_{M\cap\{d_g(x,y)<1\}}\frac{|v(x)-v(y)|^{p}}{(d_g(x,y))^{N+ps}}
 d\mu_g(x)d\mu_g(y) \\
&\leq\int_{M}\int_{M\cap\{d_g(x,y)<1\}}\int_0^1\frac{|\nabla v(\gamma(t))|^{p}}
 {(d_g(x,y))^{N+ps-p}}dtd\mu_g(y)d\mu_g(x) \\
&\leq \int_{M}\frac{\|\nabla v\|_{L^{p}(M)}}{(d_g(x,y))^{N+ps-p}}d\mu  _g(y) \\
&\leq C{(N,s,p)}\|\nabla v\|_{L^{p}(M)}
\end{aligned}
\end{equation}
and
\begin{equation} \label{l02}
\begin{aligned}
&\int_{M}\int_{M\cap\{d_g(x,y)\geq1\}}\frac{|v(x)-v(y)|^{p}}
 {(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y) \\
&\leq2^{p-1}\int_{M}\int_{M\cap\{d_g(x,y)\geq1\}}
 \frac{|v(x)|^{p}-|v(y)|^{p}}{(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y) \\
&\leq  C{(N,p)}\|v\|_{L^{p}(M)}.
\end{aligned}
\end{equation}
From \eqref{l01} and \eqref{l02} it follows that
\begin{align*}
\|v\|_{W^{s,p}(M)}\leq C{(N,s,p)}\|v\|_{W^{1,p}(M)}.
\end{align*}
Thus the proof is complete.
\end{proof}

\begin{remark} \rm
Remark \ref{r3} and Lemma \ref{lemma1.1}, imply that $W^{1,p}(M)$ is dense 
also in $W^{s,p}(M)$.
\end{remark}

\begin{lemma}\label{lemma2}
Let $(M,d_g)$ be a compact Riemannian $N-$manifold. Then $W^{s,p}(M)$ is separable.
\end{lemma}

\begin{proof}
Since $W^{1,p}(M)$ is a separable Banach space (see  \cite{guo2}), 
there exists a countable dense subset $\mathfrak{A}$ of
$W^{1,p}(M)$. We claim that $\mathfrak{A}$ is also dense in $W^{s,p}(M)$. 
For each $u\in W^{s,p}(M)$, there exists a sequence $\{u_n\}_n$
in $W^{1,p}(M)$ such that $u_n\to u$ strongly in $W^{s,p}(M)$, by the density 
of $W^{1,p}(M)$ in $W^{s,p}(M)$. Hence, for
each $n\geq1$, there exists a sequence $\{u_{m,n}\}_m$ in $\mathfrak{A}$ such that
$$
\lim_{m\to\infty}\|u_{m,n}-u_{n}\|_{W^{1,p}(M)}=0.
$$
By the standard diagonal process, there exists a sequence 
$\{u_{m_n,n}\}_n\subseteq \{u_{m,n}\}_m$ such that
$$
\lim_{n\to\infty}\|u_{m_n,n}-u_{n}\|_{W^{1,p}(M)}=0.
$$
Therefore, Lemma \ref{lemma1.1} yields 
\begin{align*}
\|u_{m_n,n}-u\|_{W^{s,p}(M)}
&\leq\|u_{m_n,n}-u_n\|_{W^{s,p}(M)}+\|u_n-u\|_{W^{s,p}(M)}\\
&\leq C\|u_{m_n,n}-u_n\|_{W^{1,p}(M)}+\|u_n-u\|_{W^{s,p}(M)}.
\end{align*}
This implies that $u_{m_n,n}\to u$ strongly in $W^{s,p}(M)$ as $n\to \infty$. 
Hence $\mathfrak{A}$ is dense in $W^{s,p}(M)$. This, together
with the countability of $\mathfrak{A}$, completes the proof.
\end{proof}


\begin{lemma}\label{lemma2b}
If $(M,d_g)$ is a complete Riemannian $N$-manifold, then $W_0^{s,p}( M)$ 
is a Banach space.
\end{lemma}

\begin{proof}
 We only need to check that $W_0^{s,p}( M)$ is complete with respect to the 
norm $\|\cdot\|_{W_0^{s,p}( M)}$. Let $\{u_t\}$ be a cauchy sequence in 
$W_0^{s,p}( M)$. Thus, for any $\varepsilon>0$ there exists $N_\varepsilon$ 
such that if $n,m\geq N_\varepsilon$, then
\begin{equation} \label{l}
\|u_n-u_m\|_{L^p(M)}^p\leq\|u_n-u_m\|^p_{W_0^{s,p}(M)}<\varepsilon.
\end{equation}
 Let $\{G_l\}$ be a sequence of compact sets such that $G_l\subset G_{l+1}\subset M$
for $l\in\mathbb{N}$ and $M=\cup_{l=1}^\infty G_l$.
Then the sequence $\{u_t\}$ is Cauchy in each $L^p(G_l)$ for $l\in\mathbb{N}$.
By induction we may find subsequences
$\{u_t^{(l)}\}_t$ and  $u^{(l)}\in L^p(G_l)$ such that
$u_t^{(l)}\to u^{(l)}$ a.e. on $G_l$ for $l\in\mathbb{N}$, and
$u^{(l+1)}\chi_{G_l}=u^{(l)}$. Thus,
$\lim_{\tau\to\infty}u_\tau^{(\tau)}=\lim_{\tau\to\infty}u^{(\tau)}\chi_{G_\tau}=u$
a.e. on  $M$. Therefore, by the Fatou Lemma and the second inequality
in \eqref{l} with $\varepsilon=1$, we have
\begin{align*}
&\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y) \\
&\leq \liminf_{\tau\to\infty}\iint_{M\times M}\frac{|u_\tau^{(\tau)}(x)
 -u_\tau^{(\tau)}(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y)\\
&\leq \liminf_{\tau\to\infty}\left(\|u_\tau^{(\tau)}-u_{\mu_1}\|_{W_0^{s,p}(M)}
 +\|u_{\mu_1}\|_{W_0^{s,p}(M)}\right)^p\\
&\leq \left(1+\|u_{\mu_1}\|_{W_0^{s,p}(M)}\right)^p<\infty.
\end{align*}
Thus, $u\in W_0^{s,p}(M)$. Let $t\geq \mu_\varepsilon$, by the second inequality
in \eqref{l} and Fatou's lemma, we obtain
\[
\|u_t-u\|_{W_0^{s,p}(M)}^p\leq\liminf_{\tau\to\infty}
\|u_t-u_\tau^{(\tau)}\|_{W_0^{s,p}(M)}^p\leq \varepsilon,
\]
that is, $u_n\to u$ strongly in $W_0^{s,p}(M)$ as $n\to\infty$.
\end{proof}

\begin{lemma}\label{lemma2c}
Let $(M,d_g)$ be a complete Riemannian $N$-manifold. Then $W_0^{s,p}( M)$  
is uniformly convex.
\end{lemma}

\begin{proof}
Let $u,v\in W_0^{s,p}(M)$ satisfy $\|u\|_{W_0^{s,p}(M)}=\|v\|_{W_0^{s,p}(M)}=1$ 
and $\|u-v\|_{W_0^{s,p}(M)}\geq\varepsilon$, where $\varepsilon\in(0,2)$.
\smallskip

\noindent\textbf{Case $p\geq2$.} 
By  \cite[inequality (28)]{r22}, we have
\begin{equation} \label{l1.1}
\begin{aligned}
&\|\frac{u+v}{2}\|^p_{W_0^{s,p}(M)}+\|\frac{u-v}{2}\|^p_{W_0^{s,p}(M)} \\
&\leq \frac{1}{2}\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y) \\
&\quad + \frac{1}{2}\iint_{M\times  M}\frac{|v(x)-v(y)|^p}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y) \\
&=\frac{1}{2}\|u\|^p_{W_0^{s,p}(M)}+\frac{1}{2}\|v\|^p_{W_0^{s,p}(M)}=1.
\end{aligned}
\end{equation}
From \eqref{l1.1} it follows that
$\|\frac{u+v}{2}\|^p_{W_0^{s,p}(M)}\leq 1-(\varepsilon/{2})^p$.
Taking $\delta = \delta(\varepsilon)$ such that
$1-(\varepsilon/2)^p = (1-\delta)^p$, we obtain
$\|\frac{u+v}{2}\|_{W_0^{s,p}(M)}\leq (1-\delta)$.
\smallskip

\noindent\textbf{Case $1<p<2$.} 
Note that
\[
\|u\|^{p'}_{W_0^{s,p}(M)}
=\Big[\iint_{M\times  M}\Big(\Big(\frac{|u(x)-u(y)|}{(d_g(x,y))^{\frac{N}{p}+s}}
\Big)^{p'}\Big)^{p-1}d\mu  _g(x)d\mu  _g(y)\Big]^{\frac{1}{p-1}},
\]
where $p'=p/(p-1)$. With the help of the reverse Minkowski inequality 
(see \cite[Theorem 2.13]{r22}) and the inequality (27) in \cite{r22}, we obtain
\begin{equation} \label{l1.2}
\begin{aligned}
&\|\frac{u+v}{2}\|_{W_0^{s,p}(M)}^{p'}+\|\frac{u-v}{2}\|_{W_0^{s,p}(M)}^{p'} \\
&\leq \Big\{\iint_{M\times  M}
\Big[\Big(|\frac{(u(x)-u(y))+(v(x)-v(y))}{2(d_g(x,y))^{\frac{N}{p}+s}}|\Big)^{p'} \\
&\quad +\Big(|\frac{(u(x)-u(y))-(v(x)-v(y))}{2(d_g(x,y))^{\frac{N}{p}+s}}|
 \Big)^{p'}\Big]^{p-1}d\mu  _g(x)d\mu  _g(y)\Big\}^{\frac{1}{p-1}} \\
&\leq\Big(\frac{1}{2}\|u\|^p_{W_0^{s,p}(M)}+\frac{1}{2}\|v\|^p_{W_0^{s,p}(M)}
 \Big)^{p'-1}=1.
\end{aligned}
\end{equation}
By \eqref{l1.2}, we have
\begin{align*}
\|\frac{u+v}{2}\|^{p'}_{W_0^{s,p}(M)}\leq 1-\frac{\varepsilon^{p'}}{2^{p'}}.
\end{align*}
Taking $\delta=\delta(\varepsilon)$ such that
$1-(\varepsilon/2)^{p'}=(1-\delta)^{p'}$, we obtain the desired conclusion.
\end{proof}

\begin{remark} \rm
According to \cite[Theorem 1.21]{r22}, $W_0^{s,p}(M)$ is a reflexive Banach space.
\end{remark}

\begin{lemma}\label{lemma5}
Let $(M,d_g)$ be a compact Riemannian $N$-manifold and $\{v_j\}$ be a bounded 
sequence in $W_0^{s,p}(M)$. Then, there exists $v\in L^{q}(M)$  such that 
up to a subsequence,
\[
v_j\to v\quad\text{strongly in $L^q(M)$, as } j\to\infty,
\]
for any $q\in[1,p_s^*)$.
\end{lemma}

\begin{proof}
For any $\{v_j\}$, which is a bounded sequence in $W_0^{s,p}(M)$. 
Since $M$ is compact, $M$ can be covered by a finite number of charts
$(\Omega_k,\varphi_k)_{k=1,2,\dots ,m}$
such that for any $k$ the components $g_{ij}^k$ of $g$ in $(\Omega_k,\varphi_k)$ 
satisfying
\[
  \frac{1}{2}\delta_{ij}\leq g_{ij}^k\leq 2\delta_{ij}
\]
are bilinear forms. Let $(\eta_k)$ be a smooth partition of unity subordinate 
to the covering $(\Omega_k)$. By means of Corollary 7.2 in \cite{r8}, for any $k$, 
there exists $\omega_k\in L^{q}(\mathbb{R}^N)$ such that
$$
(\eta_kv_j)\circ \varphi^{-1}\to \omega_k\quad\text{strongly in 
  $L^q(\varphi_k(\Omega_k))$,  as } j\to\infty.
$$
 Then
 $$
\eta_kv_j\to \omega_k\circ\varphi=u_k\quad\text{strongly in $L^q(\Omega_k)$,
 as } j\to\infty.
$$
 Furthermore, we can define $v=\sum_{k=1}^mu_s\in  L^q(M)$ satisfying
\[
v_j\to v\quad\text{strongly in $L^q(M)$, as } j\to\infty.
\]
Thus, the proof is complete.
 \end{proof}


\section{Proofs of main restults}

Following the approach of \cite{XZF},  we will give the proofs of 
Theorems \ref{k5} and \ref{k6}.
For the reader's convenience, here we  give a detailed treatment.
For $u\in W_0^{s,p}(M)$, we define
\begin{gather*}
J(u)=\frac{1}{p}\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}d\mu  _g(x)
d\mu  _g(y),\quad
 H(u)=\int_\Omega F(x,u)d\mu  _g(x), \\
I(u)=J(u)-H(u).
\end{gather*}
Obviously, the energy functional $I:W_0^{s,p}(M)\to\mathbb{R}$ associated 
with problem \eqref{k1} is well defined.

\begin{lemma}\label{lemma6}
If $f$ satisfies {\rm (A1)}, then the functional $H\in C^1(W_0^{s,p}(M),\mathbb{R})$
 and
\[
\langle H'(u),v\rangle=\int_\Omega f(x,u)vd\mu  _g(x)\quad
\text{for all } u,\ v\in W_0^{s,p}(\Omega).
\]
\end{lemma}

\begin{proof}
 (i) $H$ is G\^{a}teaux-differentiable in $W_0^{s,p}(M)$.
Let $u, v\in W_0^{s,p}(M)$. For each $x\in\Omega$ and $0<|t|<1$, 
by the mean value theorem, there exits $0<\delta<1$,
\begin{align*}
\frac{1}{t}(F(x,u+tv)-F(x,u))
&=\frac{1}{t}\int_0^{u+tv}f(x,s)ds -\frac{1}{t}\int_0^{u}f(x,s)ds\\
&=\frac{1}{t}\int_u^{u+tv}f(x,s)ds\\
&=f(x,u+\delta tv)v.
\end{align*}
Combining (A1) with Young's inequality, we obtain
\begin{align*}
|f(x,u+\delta tv)v|
&\leq a(|v|+|u+\delta tv|^{q-1}|v|)\\
&\leq a(2|v|^q+|u+\delta tv|^q+1)\leq a 2^{q}(|v|^q+|u|^q+1).
\end{align*}
Since $1<q<p_s^*$, by Lemma \ref{lemma3} we have $u,v\in L^q(M)$.
 Moreover, the Lebesgue's dominated convergence theorem implies
\begin{align*}
\lim_{t\to0}\frac{1}{t}(H(u+tv)-H(u))
&=\lim_{t\to0}\int_\Omega f(x,u+\delta tv)vd\mu  _g(x)\\
&=\int_\Omega\lim_{t\to0}f(x,u+\delta tv)vd\mu  _g(x)=\int_\Omega f(x,u)vd\mu  _g(x).
\end{align*}

 (ii) The continuity of Gateaux-derivative.
Let $\{u_n\}\subset W_0^{s,p}(M), u\in W_0^{s,p}(M)$ such that $u_n\to u$ strongly 
in $W_0^{s,p}(M)$ as $n\to\infty$. Without loss of generality, we assume that 
$u_n\to u$ a.e. in $\Omega$.
In view of  (A1), for any measurable subset $U\subset \Omega$,
\[
\int_U |f(x,u_n)|^{q'}d\mu  _g(x)\leq 2^{\frac{q+1}{q-1}}a^{\frac{q}{q-1}}
\Big(\int_U|u_n|^{q}d\mu  _g(x)+\mu(U)\Big),
\]
where $\mu(U)$ denotes the $N$ dimensional Radon measure of set $U$. 
Since $1<q<p_s^*$, by Lemma \ref{lemma3} and H\"{o}lder's inequality, we have
\begin{equation} \label{m1.3}
\begin{aligned}
\int_U |f(x,u_n)|^{q'}d\mu  _g(x)
&\leq 2^{\frac{q+1}{q-1}}a^{\frac{q}{q-1}}
 \Big(\|u_n|^{q}\|_{L^{\frac{p_s^*}{q}}(U)}\|1\|_{L^{\frac{p_s^*}{p_s^*-q}}(U)}
 +\mu(U)\Big) \\
&\leq C(\mu(U))^{\frac{p_s^*-q}{p_s^*}}+C\mu(U).
\end{aligned}
\end{equation}
It follows from \eqref{m1.3} that the sequence $\{|f(x,u_n)-f(x,u)|^{q'}\}$ is
uniformly bounded and equi-integrable in $L^1(\Omega)$.
 The Vitali convergence theorem  implies
\[
\lim_{n\to\infty}\int_\Omega |f(x,u_n)-f(x,u)|^{q\prime}d\mu  _g(x)=0.
\]
Thus, by H\"{o}lder's inequality and Lemma \ref{lemma3}(1), we obtain
\begin{align*}
\|H'(u_n)-H'(u)\|
&\leq \|f(x,u_n)-f(x,u)\|_{L^{q'}(\Omega)}\|\varphi\|_{L^q(\Omega)}\\
&\leq C_1^{1/p}\|f(x,u_n)-f(x,u)\|_{L^{q'}(\Omega)}\to 0,
\end{align*}
as $n\to\infty$.
Hence, we complete the proof.
\end{proof}

Using the same strategy as in Lemma \ref{lemma6}, we have

\begin{lemma}\label{lemma7}
The functional $J\in C^1(W_0^{s,p}(M),\mathbb{R})$ and
\begin{align*}
\langle J'(u),v\rangle=\iint_{M\times M} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)
-v(y))}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y),
\end{align*}
for all $u,v\in W_0^{s,p}(M)$. Moreover, for each $u\in W_0^{s,p}(M)$,
 $J'(u)\in W_0^{s,p}(M)^*$, where $W_0^{s,p}(M)^*$ denotes the dual space of
 $W_0^{s,p}(M)$.
\end{lemma}

\begin{proof}
Firstly, it is easy to see that
\begin{equation} \label{m1.4}
\begin{aligned}
&\langle J'(u),v\rangle \\
&=\iint_{M\times M} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y),
\end{aligned}
\end{equation}
for all $\ u,\ v\in W_0^{s,p}(M)$. It follows from \eqref{m1.4} that for each
$u\in W_0^{s,p}(M)$, $J'(u)\in W_0^{s,p}(M)^*$.

Next, we prove that $J\in C^1(W_0^{s,p}(M),\mathbb{R})$.  
Let $\{u_n\}\subset W_0^{s,p}(M),\ u\in W_0^{s,p}(M) $ with $u_n\to u$ strongly 
in $W_0^{s,p}(M)$ as $n\to\infty$.
By Lemma \ref{lemma5} there exists a subsequence of $\{u_n\}$ still denoted 
by $\{u_n\}$ such that $u_n\to u$ a.e. in $\Omega$.
Then the sequence
\[
\Big\{\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{(d_g(x,y))^{\frac{N+ps}{p'}}}
\Big\}_{n}\quad\text{is bounded in }L^{p'}(\Omega\times \Omega),
\]
and
\begin{align*}
\mathcal{M}_n(x,y)&:=\frac{|u_n(x)-u_n(y)|^{p-2}(u_n(x)-u_n(y))}{(d_g(x,y)
 )^{\frac{N+ps}{p'}}} \\
&\to \mathcal{M}(x,y):=\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}
 {(d_g(x,y))^{\frac{N+ps}{p'}}}
\end{align*}
a.e. in $M\times M$.
Thus, the Br\'{e}zis-Lieb Lemma (see \cite{r21}) implies
\begin{equation} \label{three1}
\begin{aligned}
&\lim_{n\to\infty}\iint _{M\times M}(\mathcal{M}_n(x,y)
 -\mathcal{M}(x,y))^{p'}d\mu  _g(x)d\mu  _g(y)  \\
&=\lim_{n\to\infty}\iint _{M\times M}([u_n]_{W^{s,p}(M)}
 -[u]_{W^{s,p}(M)})d\mu  _g(x)d\mu  _g(y).
\end{aligned}
\end{equation}
The fact that $u_n\to u$ strongly in $W_0^{s,p}(M)$ implies
\begin{equation} \label{three3}
\lim_{n\to\infty}\iint _{M\times M}(\mathcal{M}_n(x,y)
-\mathcal{M}(x,y))^{p'}d\mu  _g(x)d\mu  _g(y)=0.
\end{equation}
Combining  \eqref{three3} with the H\"{o}lder inequality, we have
\[
\|J'(u_n)-J'(u)\|=\sup_{v\in W_0^{s,p}(\Omega),\, \|v\|_{W_0^{s,p}(\Omega)}\leq1}
|\langle J'(u_n)-J'(u),v\rangle|
\to 0,
\]
as $n\to\infty$.
\end{proof}


Combining Lemmas \ref{lemma6} and \ref{lemma7}, we obtain that
 $I\in C^1(W_0^{s,p}(M),\mathbb{R})$ and
\begin{align*}
\langle I'(u),v\rangle
&= \iint_{M\times M} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}
 {(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y) \\
&\quad -\int_\Omega f(x,u)vd\mu  _g(x),
\end{align*}
for all $u, v\in W_0^{s,p}(\Omega)$.

\subsection*{Case 1: $1<q< p$}
In this subsection, we prove the existence of weak solutions of problem \eqref{k1},
 where the growth exponent $q$ of function $f$ satisfies $1<q<p$.

\begin{lemma}\label{lemma8}
Let {\rm (A1)} be satisfied. Then the functional $I\in C^1(W_0^{s,p}(M),\mathbb{R})$ 
is weakly lower semi-continuous.
\end{lemma}

\begin{proof}
Firstly, we notice that the map $v\mapsto \|v\|^p_{W_0^{s,p}(M)}$ is lower
semi-continuous in the weak topology of $W_0^{s,p}(M)$. Indeed, we define 
a functional $\psi: W_0^{s,p}(M)\to\mathbb{R}$ as
\[
\psi(v)=\iint_{M\times M} \frac{|v(x)-v(y)|^{p}}{(d_g(x,y))^{N+ps}}d\mu  _g(x)
d\mu  _g(y).
\]
Similar to Lemma \ref{lemma7}, we obtain $\psi\in C^1(W_0^{s,p}(M))$ and 
\begin{align*}
&\langle \psi'(w),v\rangle \\
&=p\iint_{M\times M}\frac{ |w(x)-w(y)|^{p-2}(w(x)-w(y))(v(x)-v(y))}
 {(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y) ,
\end{align*}
for all $w, v\in W_0^{s,p}(M)$. Note that
\begin{align*}
\psi\big(\frac{w+v}{2}\big)
&\leq\iint_{M\times M}\frac{2^{-1}|w(x)-w(y)|^p+2^{-1}|v(x)-v(y)|^{p}}
 {(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y)\\
&=\frac{1}{2}\psi(w)+\frac{1}{2}\psi(v).
\end{align*}
Thus, $\psi$ is a convex functional in $W_0^{s,p}(M)$. 
Furthermore, $\psi$ is subdifferentiable and the subdifferential denoted by
 $\partial\psi$ satisfies $\partial\psi(u)=\{\psi'(u)\}$ for each 
$u\in W_0^{s,p}(M)$ (see \cite[Proposition 1.1]{r25}).
Now, let $\{v_n\}\subset W_0^{s,p}(M), v\in W_0^{s,p}(M)$ with 
$v_n\rightharpoonup v$ weakly in $W_0^{s,p}(M)$ as $n\to\infty$. 
Then it follows from the definition of subdifferential that
\begin{align*}
\psi(v_n)-\psi(v)\geq \langle \psi'(v), v_n-v\rangle.
\end{align*}
Hence, we obtain $\psi(v)\leq \liminf_{n\to\infty} \psi(v_n)$, that is, 
the map $v\mapsto \|v\|^p_{W_0^{s,p}(M)}$ is weakly lower semi-continuous.

Let $u_n\rightharpoonup u$ weakly in $W_0^{s,p}(M)$. 
By assumption (H1) and Lemma \ref{lemma5}, up to a subsequence, $u_n\to u$ 
strongly in $L^q(\Omega)$. Without loss of generality, we assume that 
$u_n\to u$ a.e. in $\Omega$. Assumption (A1) implies
\[
F(x,t)\leq a\left(|t|+q^{-1}|t|^{q}\right)\leq 2a(|t|^q+1).
\]
Thus, for any measurable subset $U\subset \Omega$,
\[
\int_U |F(x,u_n)|d\mu  _g(x)\leq 2a\int_U|u_n|^{q}dx+2a\mu(U).
\]
From $1<q<p_s^*$, Lemma \ref{lemma3} and H\"{o}lder's inequality, we have
\begin{align*}
\int_U |F(x,u_n)|d\mu  _g(x)
&\leq 2a\|u_n|^{q}\|_{L^{\frac{p_s^*}{q}}(U)}\|1\|_{L^{\frac{p_s^*}
 {p_s^*-q}}(U)}+2a\mu(U)\\
&\leq 2aC\|u_n\|^q_{W_0^{s,p}(M)}(\mu(U))^{\frac{p_s^*-q}{p_s^*}}+2a\mu(U).
\end{align*}
Similar to the proof of Lemma \ref{lemma6}, we obtain
\[
\lim_{n\to\infty}\int_\Omega F(x,u_n)d\mu  _g(x)=\int_\Omega F(x,u)d\mu  _g(x).
\]
Thus, the functional $H$ is weakly continuous. Furthermore, we obtain that $I$ 
is weakly lower semi-continuous.
\end{proof}

\begin{proof}[Proof of Theorem \ref{k5}]
 By (A1), we have $|F(x,t)|\leq 2a(|t|^p+1)$. Thus, by Lemma \ref{lemma3}, 
we obtain
\begin{align*}
I(u)
&\geq \frac{1}{p}\iint_{M\times M}\frac{|u(x)-u(y)|^{p}}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y)-2a\int_{\Omega}|u|^{q}d\mu  _g(x)-2a\mu(\Omega)\\
&\geq \frac{1}{p}\|u\|^p_{W_0^{s,p}(M)}-2aC_1^{\frac{q}{p}}\|u\|^q_{W_0^{s,p}(M)}
 -2a\mu(\Omega).
\end{align*}
Since $q<p$, we have $I(u)\to\infty$ as $\|u\|_{W_0^{s,p}(M)}\to\infty$. 
By Lemma \ref{lemma8}, $I$ is weakly lower semi-continuous on $W_0^{s,p}(M)$. 
So the functional $I$ has a minimum point $u_0$ in $W_0^{s,p}(\Omega)$ 
(see \cite[Theorem 1.2]{r23}) and $u_0\in W_0^{s,p}(M)$ is a weak solution 
of problem \eqref{k1}.
\end{proof}


\subsection*{Case 2: $p<q<p_s^*$}

\begin{lemma}\label{lemma9}
Let  $f$ satisfy {\rm (A1)} and {\rm (A3)}. 
If $p<q<p_s^*$, then there exist $\rho>0$ and $\alpha>0$ such that  
$$
I(u)\geq\alpha>0,
$$
for any $u\in W_0^{s,p}(M)$ with $\|u\|_{W_0^{s,p}(M)}=\rho$.
\end{lemma}

\begin{proof}
In view of (A1) and (A3), for any $\varepsilon>0$, there exists 
$C(\varepsilon)>0$ such that for any $\xi\in\mathbb{R}$ and a.e. $x\in\Omega$, 
we have
\begin{equation} \label{l3}
|F(x,\xi)|\leq \varepsilon|\xi|^{p}+C(\varepsilon)|\xi|^{q}.
\end{equation}
Let $u\in W_0^{s,p}(M)$. By \eqref{l3} and Lemma \ref{lemma3}, we obtain
\begin{equation} \label{l4}
\begin{aligned}
I(u)&\geq \frac{1}{p}\iint_{M\times M}\frac{|u(x)-u(y)|^{p}}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y)
 -\varepsilon\int_\Omega |u(x)|^{p}d\mu  _g(x) \\
&\quad -C(\varepsilon)\int_\Omega|u(x)|^qd\mu  _g(x) \\
&\geq \frac{1}{p}\|u\|^p_{W_0^{s,p}(M)}-\varepsilon C_1\|u\|^p_{W_0^{s,p}(M)}
 -C(\varepsilon)C_1^{\frac{q}{p}}\|u\|^q_{W_0^{s,p}(M)}.
\end{aligned}
\end{equation}
Choosing $\varepsilon=1/(2pC_1)$, from \eqref{l4} we have
\[
I(u)\geq \frac{1}{2p}\|u\|^p_{W_0^{s,p}(M)}-C\|u\|^q_{W_0^{s,p}(M)}
\geq  \|u\|^p_{W_0^{s,p}(M)}\Big(\frac{1}{2p}-C\|u\|^{q-p}_{W_0^{s,p}(M)}\Big),
\]
where $C$ is a constant only depending on $N, s, p$.
Now, let $\|u\|_{W_0^{s,p}(M)}=\rho>0$. Since $q>p$, we can choose $\rho$
sufficiently small such that $1/(2p)-C\rho^{q-p}>0$, so that
\[
I(u)\geq \rho^p\Big(\frac{1}{2p}-C\rho^{q-p}\Big)=:\alpha>0.
\]
As desired.
\end{proof}

\begin{lemma}\label{lemma10}
Let $f$ satisfies {\rm (A1)--(A3)}. If $p<q<p_s^*$, then there exists 
$e\in C^\infty_0(\Omega)$ such that  $\|e\|_{W_0^{s,p}(M)}\geq\rho$ and 
$I(\rho)<\alpha$, where $\rho$ and $\alpha$ are given in Lemma \ref{lemma9}.
\end{lemma}

\begin{proof}
From (A2) it follows that
\begin{align}\label{l6}
F(x,\xi)\geq r^{-\gamma}\min\{F(x,r),F(x,-r)\}|\xi|^{\gamma},
\end{align}
for all $|\xi|>r$ and a.e. $x\in\Omega$. Thus, by \eqref{l6} and 
$F(x,\xi)\leq \max_{|\xi|\leq r}F(x,\xi)$ for all $|\xi|\leq r$, we obtain
\begin{equation} \label{l7}
\begin{aligned}
F(x,\xi)
&\geq r^{-\gamma}\min\{F(x,r),F(x,-r)\}|\xi|^{\gamma}
 -\max_{|\xi|\leq r}F(x,\xi) \\
&\quad -\min\{F(x,r),F(x,-r)\},
\end{aligned}
\end{equation}
for any $\xi\in\mathbb{R}$ and a.e. $x\in\Omega$.

By Lemma \ref{lemma1}, we can fix $u_0\in C^\infty_0(\Omega)$ such that
 $\|u_0\|_{W_0^{s,p}(M)}=1$. Now, let $t\geq1$. By \eqref{l7}, we have
\begin{align*}
I(tu_0)
&=\frac{1}{p}\iint_{M\times M}\frac{|tu_0(x)-tu_0(y)|^{p}}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y)-\int_\Omega F(x,tu_0(x))d\mu  _g(x) \\
&\leq \frac{t^p}{p}-r^{-\gamma}t^\gamma\int_\Omega \min\{F(x,r),F(x,-r)\}
 |u_0(x)|^\gamma d\mu  _g(x) \\
&\quad +\int_\Omega\max_{|\xi|\leq r}F(x,\xi)+\min\{F(x,r),F(x,-r)\}d\mu  _g(x).
\end{align*}
Using (A1) and (A2), we obtain that $0<F(x,\xi)\leq a(|r|+|r|^q)$ for 
$|\xi|\leq r$ a. e. $x\in\Omega$. Thus, $0<\min\{F(x,r),F(x,-r)\}<a(|r|+|r|^q)$ 
a.e.\ $x\in\Omega$.
Since $\gamma>p$ by assumption (A2), passing to the limit as $t\to\infty$, 
we obtain that $I(tu_0)\to-\infty$. Thus, the assertion follows by taking 
$e=Tu_0$ with $T$ sufficiently large.
\end{proof}

\begin{definition} \rm
We say that $I$ satisfies (PS) condition in $W_0^{s,p}(M)$, if for any sequence 
$\{u_n\}\subset W_0^{s,p}(M)$ such that $I(u_n)$ is bounded and $I'(u_n)\to0$ 
as $n\to\infty$, there exists a convergent subsequence of $\{u_n\}$.
\end{definition}


\begin{lemma}\label{lemma11}
Let $f$ satisfy {\rm (A1)--(A3)}. If $p<q<p_s^*$, then the functional $I$ 
satisfies the {\rm (PS)} condition.
\end{lemma}

\begin{proof}
For any sequence $\{u_n\}\subset W_0^{s,p}(M)$ such that $I(u_n)$ is bounded 
and $I'(u_n)\to0$ as $n\to\infty$, there exits $C>0$ such that
$|\langle I'(u_n),u_n\rangle|\leq C\|u_n\|_{W_0^{s,p}(M)}$
and $|I(u_n)|\leq C$.
By (A1), we have
\begin{equation} \label{l8}
\begin{aligned}
&\big|\int_{\Omega\bigcap\{|u_n|\leq r\}}(F(x,u_n)-\gamma^{-1}f(x,u_n)u_n)
 d\mu  _g(x)| \\
&\leq (a+\gamma^{-1})(r+r^q)|\Omega|\leq C,
\end{aligned}
\end{equation}
where $\{|u_n|\leq r\}=\{x\in\Omega:|u_n(x)|\leq r\}$.
Thus, by  (H2) and \eqref{l8}, we obtain
\begin{align*}
C+C\|u_n\|_{W_0^{s,p}(M)}
&\geq I(u_n)-\frac{1}{\gamma}\langle I'(u_n),u_n\rangle\\
&\geq\Big(\frac{1}{p}-\frac{1}{\gamma}\Big)
 \iint_{M\times M}\frac{|u_n(x)-u_n(y)|^{p}}{(d_g(x,y))^{N+ps}}d\mu  _g(x)d\mu  _g(y)\\
&\quad -\int_{\Omega\bigcap\{|u_n|\leq r\}}
 \left(F(x,u_n)-\gamma^{-1}f(x,u_n)u_n\right)d\mu  _g(x)\\
&\geq \Big(\frac{1}{p}-\frac{1}{\gamma}\Big)
 \iint_{M\times M}\frac{|u_n(x)-u_n(y)|^{p}}{(d_g(x,y))^{N+ps}}
 d\mu  _g(x)d\mu  _g(y)-C,
\end{align*}
where $C$ denotes various positive constants.
Hence, $\{u_n\}$ is bounded in $W_0^{s,p}(M)$. Since $W_0^{s,p}(M)$
is a reflexive Banach space, up to a subsequence, still denoted by $\{u_n\}$
such that $u_n\rightharpoonup u$ weakly in $W_0^{s,p}(M)$.
Then $\langle I'(u_n),u_n-u\rangle\to0$.

For each $\varphi\in W_0^{s,p}(M)$ we define a functional 
$T:W_0^{s,p}(M)\to (W_0^{s,p}(M))'$ by
\[
\langle T(\varphi),v\rangle=\iint_{M\times M}
\frac{ |\varphi(x)-\varphi(y)|^{p-2}(\varphi(x)-\varphi(y))}{(d_g(x,y))^{N+ps}}
 (v(x)-v(y))d\mu  _g(x)d\mu  _g(y),
\]
for all $v\in W_0^{s,p}(M)$. Clearly, by the H\"{o}lder inequality, 
$T(\varphi)$ is also continuous, being
\[
|\langle T(\varphi),v\rangle|\leq \|\varphi\|^{p-1}_{W_0^{s,p}(M)}
\|v\|_{W_0^{s,p}(M)}\quad\text{for all }v\in W_0^{s,p}(M).
\]
Thus, we have
\begin{align}\label{l9}
\langle I'(u_n),u_n-u\rangle
=\langle T(u_n), u_n-u\rangle-\int_\Omega f(x,u_n)(u_n-u)d\mu_g(x)\to 0
\end{align}
as $n\to\infty$. Moreover, by Lemma \ref{lemma5}, up to a subsequence,
\[
u_n\to u\quad\text{strongly in $L^q(\Omega)$ and  a.e.\ in } \Omega.
\]
Thus, $f(x,u_n)(u_n-u)\to 0$ a.e. in $\Omega$ as $n\to\infty$.
 It is easy to check that sequence $\{f(x,u_n)(u_n-u)\}$ is uniformly 
bounded and equi-integrable in $L^1(\Omega)$. Hence, the Vitali convergence 
theorem implies
\[
\lim_{n\to\infty}\int_\Omega f(x,u_n)(u_n-u)d\mu  _g(x)=0.
\]
Therefore, from \eqref{l9} it follows that
\[
\lim_{n\to\infty}\langle T(u_n), u_n-u\rangle=0.
\]
Furthermore, by the weak convergence of $\{u_n\}$ in $W_0^{s,p}(M)$, we obtain
\[
\lim_{n\to\infty}\langle T(u_n)-T(u), u_n-u\rangle=0.
\]
Let us recall  the well--known vector inequalities:
\begin{gather*}
\left(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta\right)\cdot(\xi-\eta)\geq C_p|\xi-\eta|^{p},
\quad  p\geq2;\\
\left(|\xi|^{p-2}\xi-|\eta|^{p-2}\eta\right)\cdot(\xi-\eta)
\geq \widetilde{C}_p\frac{|\xi-\eta|^2}{(|\xi|+|\eta|)^{2-p}}, \quad 1<p<2,
\end{gather*}
for all $\xi,\eta\in\mathbb{R}^N$, where $C_p, \widetilde{C}_p$ are constants 
depending only on $p$. From which it is easy to verify that
for $p>2$ and $1<p<2$, we have
\begin{align}\label{l10}
\iint_{M\times M}\frac{|u_n(x)-u_n(y)-u(x)+u(y)|^p}{(d_g(x,y))^{N+ps}}
d\mu  _g(x)d\mu  _g(y)
\to0,
\end{align}
as $n\to\infty$.
Hence, from \eqref{l10} we obtain that $u_n\to u$ strongly in $W_0^{s,p}(M)$ 
as $n\to\infty$. Therefore, the proof is complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{k6}]
According to Lemmas \ref{lemma9}--\ref{lemma11}, the Mountain Pass Theorem 
\cite[Theorem 6.1]{r23} implies that there exists a critical point 
$u\in W_0^{s,p}(M)$ for problem \eqref{k1}.
\end{proof}



\subsection*{Acknowledgements}
 L. Guo was supported by Youth Science Foundation of Heilongjiang Province 
of China  (No. QC2016002) and Northeast Petroleum University Teaching 
Reform Foundation for Postgraduate (No. JYCX-JG10\_2018). 
B. Zhang was partially supported by the National Natural Science Foundation 
of China (No. 11601515, 11701178, 11871199).

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\end{document}