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\begin{document}
\title{\sc Existence of stationary states for $A$-Dirac equations with variable growth}
%\title{Solutions for $p(x)$-Laplacian equations without AR-condition via Morse theory}
\author{Giovanni Molica Bisci$^{\mathrm{a}}$,\ Vicen\c{t}iu D. R\u{a}dulescu$^{\mathrm{b},}$\footnote{Corresponding author.
E-mail address:\,{\tt vicentiu.radulescu@math.cnrs.fr}} \ and \ Binlin Zhang$^{\mathrm{c}}$\\
\small{$^{\mathrm{a}}$ Dipartimento P.A.U., Universit\`a  degli
Studi Mediterranea di Reggio Calabria}\\ \small{Salita Melissari - Feo di
Vito, 89124 Reggio Calabria, Italy}\\
\small $^{\mathrm{b}}$ Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia\\
\small $^{\mathrm{c}}$ Department of Mathematics, Heilongjiang Institute of Technology, 150050 Harbin, China} \date{ }
 \maketitle


\begin{abstract}
{\noindent In this paper, using
a Hodge-type decomposition of
variable exponent Lebesgue  spaces of Clifford-valued functions and variational methods,
we study the properties of weak solutions to the homogeneous and nonhomogeneous $A$-Dirac equations with variable growth in the setting
of variable exponent Sobolev spaces of Clifford-valued functions.\par}
\smallskip
\noindent \emph{\bf Keywords:} Clifford analysis; $A$-Dirac equation; variable exponent; Caccioppoli estimates; Hodge-type decomposition.\\
\emph{\bf 2010 Mathematics Subject Classification:} 30G35; 35J60; 35D30; 46E30.
\end{abstract}

%{\section*{{\bf 1. Introduction }}} \setcounter{section}{1}
 %\setcounter{equation}{0}
\section{Introduction}

%Variable exponent Lebesgue spaces $L^{p(x)}$ and the corresponding Sobolev spaces $W^{k, p(x)}$ have been studied extensively
% in the last twenty years since Kov\'{a}\v{c}ik and R\'{a}kosn\'{i}k specially investigated these spaces in \cite{1}.
% The study of the equations with variable growth is motivated by the problems in elastic mechanics, fluid dynamics, calculus of variations and differential %equations with variable growth conditions, for example, see \cite{1}, \cite{2}, \cite{3}, \cite{4}, \cite{5}, \cite{6} and references therein.

The Dirac equation arises in the study of nonlinear spinor fields in the unified theory of
elementary particles, see Heisenberg \cite{heise} and Weyl \cite{weyl}. The stationary states of the nonlinear Dirac field have been proposed as a model for elementary extended fermions and nucleons, see Thaller \cite{thaller}.

The paper is primarily concerned with the existence of weak solutions to the homogeneous $A$-Dirac
equation
$$DA(x,Du)=0,\eqno(1.1)$$
and the non-homogenous  $A$-Dirac equation
$$DA(x,Du)=Df,\eqno(1.2)$$
where $u$ is a function valued in the universal Clifford algebra $\mathrm{C}\ell _{n}$ over a bounded domain $\Omega$
with a sufficiently smooth boundary $\partial\Omega$ in $\mathbb{R}^{n}(n\geq 2)$,
$D$ is the usual Euclidean Dirac operator,
$f\in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ and the operator $A: \Omega\times\mathrm{C}\ell _{n} \rightarrow \mathrm{C}\ell _{n}$
satisfies the following conditions with variable growth:\\
\\
(A1) $A(x,\xi)$ is measurable with respect to $x$ for $\xi \in
\mathrm{C}\ell_{n}$
and continuous with respect to $\xi$ for a.e. $x\in \Omega$;\\
\\
(A2) $|A(x,\xi)|\leq C_{1}|\xi|^{p(x)-1}$+$g(x)$ for a.e. $x\in \Omega$ and $\xi \in \mathrm{C}\ell_{n}$;\\
\\
(A3) $\Big[\overline{A(x,\xi)}\xi\Big]_{0}\geq
C_{2}|\xi|^{p(x)}$+$h(x)$ for a.e. $x\in \Omega$ and $\xi \in \mathrm{C}\ell_{n}$;\\
\\
(A4) $\Big[\overline{(A(x,\xi_{1})-A(x,\xi_{2}))}(\xi_{1}-\xi_{2})\Big]_{0}\geq
0$ for a.e. $x\in \Omega$ and $\xi_{1}, \xi_{2}\in \mathrm{C}\ell_{n} $, \\
\\
where $C_{i}\  (i=1,2)$ are positive constants, $g(x)$ is bounded in $L^{p'(x)}(\Omega)$ and
$h(x)\in L^{1}(\Omega)$.
Of course, $DA(x,Du)=0$ and $DA(x,Du)=Df$ are meant in the distributional sense.

Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford, see \cite{clif}.

A study of the conformal $p$-Dirac equations, a special case of $A$-Dirac equations, appears in \cite{33}. These
equations are nonlinear generalizations of the Dirac Laplace equation as well
as generalizations of elliptic equations of $A$-harmonic type ${\rm div}\, A(x,\nabla u)= 0$.
The study of these equations is partially motivated by the fact that some
arise as the Euler-Lagrange equations to variational integrals.

In \cite{7, 8}, Nolder first introduced $A$-Dirac equations
(1.1) and developed some tools for the study of weak solutions
to nonlinear $A$-Dirac equations in the space
$W^{1,p}_{0}(\Omega,\mathrm{C}\ell _{n})$. In \cite{9}, Wang and Chen considered the nonhomogeneous $A$-Dirac equations $DA(x, Du)=f(x,Du)$ in space $W_{0}^{1,p}(\Omega, \mathrm{C}\ell_{n})$.
The authors proved that under certain conditions, the solutions to the inhomogeneous $A$-harmonic
equations if $f$ satisfies the controllable growth condition is in fact the scalar part of weak solutions to the corresponding inhomogeneous $A$-Dirac equations. In \cite{10},  Lu and  Bao were concerned with the regularity properties of weak solutions to the obstacle problem for homogeneous $A$-Dirac equations, such as a global reverse H\"{o}lder inequality and stability. However, the existence
of weak solutions to the $A$-Dirac equations has not been showed under the conditions (A1)--(A3) as $p(x)\equiv p$.
Inspired by their works, Fu and Zhang \cite{11, 12, 13} were interested in the
the existence of weak solutions for $A$-Dirac equations with variable growth.
Until now they have proved the existence of weak solutions to the scalar part of homogeneous and non-homogeneous $A$-Dirac equations under the assumptions (A1)--(A4).
Recently, Fu, Zhang and R\u{a}dulescu \cite{14} established a Hodge-type decomposition of variable exponent Lebesgue spaces of Clifford-valued functions. By using this decomposition, together with the Minty-Browder Theorem, existence and uniqueness of a weak solution to
the $A$-Dirac equations $DA(Du)=0$ were obtained under the following assumptions:\\

\noindent(H1) $|A(\xi) - A(\eta)| \leq C_{1}^{'}(|\xi| + |\eta|)^{p(x) - 2}|\xi -\eta|;$\\

\noindent(H2) $\Big[\overline{\big(A(\xi) - A(\eta)\big)}(\xi -\eta)\Big]_{0} \geq C_{2}^{'}(|\xi| + |\eta|)^{p(x) - 2}|\xi -\eta|^{2};$\\

\noindent(H3) $A(0) \in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, \\
\\
where $\xi$ and $\eta$ are arbitrary elements from $\mathrm{C}\ell _{n}$, both
$C_{1}^{'}$ and $C_{2}^{'}$ are positive constants independent of $\xi$ and
$\eta$. Obviously, conditions (A1)--(A4) are weaker than conditions (H1)--(H3).

It is worth pointing out that an $A$-harmonic equation $\mathrm{div}A(x, \nabla u)=0$ is the scalar part of
the equation (1.1) under appropriate identifications, see \cite{7, 8}. When $u$ is a real function, $Du$ can be
identified with $\nabla u$. Hence the equation (1.2) corresponds to the nonhomogeneous $A$-harmonic equation
$$\mathrm{div}\,A(x, \nabla u)=\mathrm{div} f, \eqno(1.3)$$
Iwaniec and Sbordone \cite{15} introduced the definition of  very weak solutions for the equation (1.3)
in the  Sobolev space $W^{1, p}_{0}(\Omega)$ and
studied the existence and uniqueness of such solutions under the conditions (H1), (H2) and the homogeneity condition as $p(x)\equiv p$.
Many results have been obtained concerning existence, uniqueness and regularity results for this kinds of equations (1.3),
for example, see \cite{19, 20, 17, 16, 18, 15} and the references therein.

In \cite{21}, Diening, Kaplicky and Schwarzacher showed BMO  estimates of $p$--Laplace system given by
$-\mathrm{div}(|\nabla u|^{p-2}\nabla u)= -\mathrm{div}f$. In \cite{20},  Diening and Kaplicky studied the interior regularity of the local weak solutions
$u \in W^{1, \varphi}(\Omega)$ and $\pi \in L^{\varphi^{\ast}}(\Omega)$ of the following stationary generalized Stoked system:
\begin{equation*}
\begin{cases}
-\mathrm{div}A(Du) + \nabla \pi = -\mathrm{div}G\ \ \ \mathrm{in}\ \Omega\\
\mathrm{div}\  u = 0\ \ \ \mathrm{in}\ \Omega,
\end{cases}
\end{equation*}
where $D$ is the symmetric part of the gradient, the extra stress tensor $A$ determines properties of the fluid. The system originates in fluid mechanics. The authors showed optimal BMO and Campanato estimates for $A(Du)$. In \cite{23}, Diening and Kaplicky proceeded to study regularity theory of solution to the stationary generalized Stoked system, then apply these estimates
to the stationary generalized Navier-Stoke system. Therefore, it is reasonable to consider the nonhomogeneous $A$-Dirac equations $DA(x,Du)=Df$ as a national extension.


This paper is organized as follows. In section 2, we begin with a
brief summary of basic knowledge of Clifford algebras and variable
exponent spaces of Clifford-valued functions, which will be needed later.
In section 3, appealing to a Hodge-type decomposition as well as variational methods,
we prove the existence and uniqueness of solutions to the homogeneous $A$-Dirac equations with
variable growth in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$,
and a Caccioppoli-type estimate for weak solutions is obtained in the variable exponent context. In section 4,
we study the existence, uniqueness and stability of solutions to the nonhomogeneous $A$-Dirac equations with
variable growth in $W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell _{n})$.

%{\section*{{\bf 2. Preliminaries}}}
\section{Preliminaries}

\subsection{Clifford algebra}

 We first recall some related notions and results from Clifford
algebra. The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. For a detailed account we refer to \cite{28, 24, 30, 32, 27, 25, 26, 31, 33, 29}. Let $\mathrm{C}\ell _{n}$ be
the real universal Clifford algebra over $\mathbb{R}^{n}$. Denote
$$\mathrm{C}\ell _{n} =\mathrm{ span} \{\mathrm{e}_{0},\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n}, \mathrm{e}_{1}\mathrm{e}_{2},\ldots ,\mathrm{e}_{n-1}\mathrm{e}_{n},\ldots ,\mathrm{e}_{1}\mathrm{e}_{2}\ldots \mathrm{e}_{n}\}$$
where $\mathrm{e}_{0}=1$ (the identity element in $\mathbb{R}^{n}$),
$\{\mathrm{e}_{1},\mathrm{e}_{2},\ldots ,\mathrm{e}_{n}\}$ is an
orthonormal basis of $\mathbb{R}^{n}$ with the relation
$\mathrm{e}_{i}\mathrm{e}_{j}+\mathrm{e}_{j}\mathrm{e}_{i}=-2\delta_{ij}\mathrm{e}_{0}$.
Thus, the dimension of $\mathrm{C}\ell _{n}$ is $2^{n}$.
In particular, by $\mathbb{H}:=\mathrm{C}\ell _{2}$ we denote the algebra of real quaternions, see \cite{21} for further details about the algebra of real quaternions.
For
$I:=\{i_{1},\ldots,i_{r}\}\subset \{1,\ldots ,n\}$
    with $1\leq i_{1}<i_{2}<\ldots <i_{n}\leq n$, put $\mathrm{e}_{I}=\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}$, while for $I=\emptyset$, $\mathrm{e}_{\emptyset }=\mathrm{e}_{0}$.
     For $0\leq r\leq n$ fixed, the space $\mathrm{C}\ell _{n}^{r}$ is defined by
$$\mathrm{C}\ell _{n}^{r}=\mathrm{span}\{\mathrm{e}_{I}:|I|=r\},$$
where $|I|$ denotes cardinal number of the set $I$.
The Clifford algebra $\mathrm{C}\ell _{n}$ is a graded algebra as
$$\mathrm{C}\ell _{n}=\bigoplus_{0\leq r\leq n}\mathrm{C}\ell _{n}^{r}.$$
    Any element $a\in \mathrm{C}\ell _{n}$ may thus be written in a unique way as
$$a=[a]_{0}+[a]_{1}+\ldots +[a]_{n},$$
    where $[\ ]_{r}:\mathrm{C}\ell _{n}\rightarrow \mathrm{C}\ell _{n}^{r}$ denotes the projection of $\mathrm{C}\ell _{n}$ onto $\mathrm{C}\ell _{n}^{r}$. It is customary to identify $\mathbb{R}$ with $\mathrm{C}\ell _{n}^{0}$ and identify $\mathbb{R}^{n}$ with $\mathrm{C}\ell _{n}^{1}$ respectively.
     This means that each element $x$ of $\mathbb{R}^{n}$ may be represented by $x=\sum_{i=1}^{n}x_{i}\mathrm{e}_{i}.$
       From an analysis viewpoint, an  important property of the universal Clifford algebra is that every non-zero vector
 $x\in \mathbb{R}^{n}$ has a multiplicative inverse given by $-x/|x|^2$. Up to a sign, this inverse corresponds to the Kelvin inverse
 of a vector in Euclidean space.

 We point out that
 {\it Hamilton's quaternions} are constructed as the even sub algebra of the Clifford algebra, while {\it dual quaternions} are constructed as the even Clifford algebra of real four dimensional space with a degenerate quadratic form.

      For $u\in \mathrm{C}\ell _{n}$, we denote by $[u]_{0}$ the scalar part of $u$, that is, the coefficient of the element $\mathrm{e}_{0}$. We define the Clifford conjugation as follows:
$$\overline{\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}}=(-1)^{\frac{r(r+1)}{2}}\mathrm{e}_{i_{1}}\mathrm{e}_{i_{2}}\ldots \mathrm{e}_{i_{r}}.$$
For $A\in \mathrm{C}\ell _{n}$, $B\in \mathrm{C}\ell _{n}$, we have
$$\overline{AB}=\overline{B}\ \overline{A},\ \
\overline{\overline{A}}=A.$$
 We denote
$$(A,B)=[\overline{A}B]_{0}.$$
Then an inner product is thus obtained, give rising to the norm $|\cdot|$ on $\mathrm{C}\ell _{n}$ given by
$$|A|^{2}=[\overline{A}A]_{0}.$$
By \cite{26}, we know that this norm is submultiplicative:
$$|AB|\leq C_{n}|A||B|, \eqno(2.1)$$ where $C_{n}$ is a positive constant depending only on ${n}$ and smaller than $2^{n/2}$.




A Clifford-valued function $u:\Omega\rightarrow \mathrm{C}\ell _{n}$ can be
written as $u=\displaystyle\sum_{I}u_{I}\mathrm{e}_{I}$, where the coefficients
$u_{I}:\Omega\rightarrow \mathbb{R}$ are real-valued functions.


    The Dirac operator on Euclidean space used here is introduced by
$$D=\sum_{j=1}^{n}\mathrm{e}_{j}\frac{\partial}{\partial x_{j}}.$$
This is a special case of the Atiyah-Singer-Dirac operator acting on sections of a spinor bundle.
We also point out that the most famous Dirac operator describes the propagation of a free fermion in three dimensions.

    If $u$ is a real-valued function defined on a domain $\Omega$ in $\mathbb{R}^{n}$, then $Du=\nabla u$.
    Moreover, $D^{2}=-\Delta$, where $\Delta$ is the
Laplace operator which operates only on coefficients. A function is
left monogenic if it satisfies the equation $Du(x)=0$ for each
$x\in\Omega$. A similar definition can be given for right monogenic
function. An important example of a left monogenic function is the
generalized Cauchy kernel
$$G(x)=\frac{1}{\omega_{n}}\frac{\overline{x}}{|x|^{n}},$$
where $\omega_{n}$ denotes the surface area of the unit ball in $\mathbb{R}^{n}$. This function is a fundamental solution of the Dirac operator. Basic properties of left monogenic functions one can refer to \cite{24, 27, 25, 26}.



%{\section*{{\bf 3. Verification of the $(C)$ condition and $(PS)$
%condition}}}
\subsection{Variable exponent spaces of Clifford-valued functions}
Next we investigate some basic properties of variable exponent
spaces of Clifford-valued functions. Note that in what follows, we
use the short notation $L^{p(x)}(\Omega)$, $W^{1,p(x)}(\Omega)$,
 instead of $L^{p(x)}(\Omega, \mathbb{R})$,
$W^{1,p(x)}(\Omega,\mathbb{R})$. Throughout this paper we
always assume (unless declared specially)
$$p \in \mathrm{\mathcal{P}^{log}}(\Omega) \ \mathrm{and} \ 1 < p_{-}:= \inf_{x \in \overline{\Omega}}p(x) \leq p(x) \leq \sup_{x \in \overline{\Omega}}p(x) =:p_{+} < \infty.\eqno(2.2)$$
where  $\mathrm{\mathcal{P}^{log}}(\Omega)$ is the set of exponent $p$
satisfying the so-called log-$\mathrm{H\ddot{o}lder}$ continuity,
that is,
$$|p(x) - p(y)| \leq \frac{C}{\mathrm{log}(\mathrm{e} + |x-y|^{-1})}$$
holds for all $x,y \in \Omega$, see \cite{1, 2}. Let $\mathcal{P}(\Omega)$  be the
set of all Lebesgue measurable functions $p: \Omega \rightarrow
(1,\infty)$. Given $p \in \mathcal{P}(\Omega)$ we define the conjugate
function $p'(x)\in \mathcal{P}(\Omega)$ by
$$p'(x)=\frac{p(x)}{p(x)-1},\ \ \ \ \   \mathrm{for}\ \mathrm{all} \
x\in\Omega.$$

The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is defined
by
$$L^{p(x)}(\Omega)=\Big\{ u\in \mathcal{P}(\Omega):\int_{\Omega}|u(x)|^{p(x)}dx <\infty \Big\},$$
with the norm
$$\|u\|_{L^{p(x)}(\Omega)}=\mathrm{inf} \Big\{ t>0:\int_{\Omega}\big|\frac{u(x)}{t}\big|^{p(x)}dx \leq 1 \Big\},$$
and the variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined
by
$$W^{1,p(x)}(\Omega)=\Big\{u\in L^{p(x)}(\Omega): |\nabla  u| \in L^{p(x)}(\Omega)\Big\},$$
with the norm
$$\|u\|_{W^{1,p(x)}(\Omega)}=\|\nabla  u\|_{L^{p(x)}(\Omega)} + \|u\|_{L^{p(x)}(\Omega)}.\eqno(2.3)$$
Let $W_{0}^{1,p(x)}(\Omega)$ be the completion of
$C_{0}^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$ with respect to the
norm (2.3). The space $W^{-1,p(x)}(\Omega)$ is defined as the dual
of the space $W_{0}^{1,p'(x)}(\Omega)$.
For more details we refer to \cite{ 2, 4, 3, 1} and the references therein.

In the sequel, we say that $u\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$  can be
understood coordinatewisely.
 For example, $u\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ means
 that $\{u_{I}\} \subset L^{p(x)}(\Omega)$ for $u=\Sigma_{I}u_{I}e_{I}\in \mathrm{C}\ell _{n}$
 with the norm $\|u\|_{L^{p(x)}(\Omega, \mathrm{C}\ell _{n})}= \sum_{I}\|u_{I}\|_{L^{p(x)}(\Omega)}$.
 In this way, spaces $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$,
 $W_{0}^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$, $C_{0}^{\infty}(\Omega,\mathrm{C}\ell _{n})$, etc., can be understood similarly.
 In particular, the space $L^{2}(\Omega,\mathrm{C}\ell _{n})$ can be converted into a right Hilbert $\mathrm{C}\ell _{n}$-module by defining
 the following Clifford-valued inner product (see \cite[Definition 3. 74]{25})
 $$\big(f,g\big)_{Cl_{n}}=\int_{\Omega}{\overline{f(x)}g(x)}dx.  \eqno(2.4)$$

 \noindent\textbf{Remark 2.1.} A simple calculation shows that
$$2^{-\frac{n(1+p_{+})}{p_{-}}}\| |u| \|_{{L^{p(x)}(\Omega)}} \leq \|u\|_{L^{p(x)}(\Omega,\mathrm{C}\ell_{n})} \leq 2^{n}\| |u| \|_{{L^{p(x)}(\Omega)}}.$$
holds for each $u\in L^{p(x)}(\Omega,\mathrm{C}\ell_{n})$, see \cite[Remark 1]{9}. Thus, $\|u\|_{L^{p(x)}(\Omega,\mathrm{C}\ell
_{n})}$ and $\||u|\|_{L^{p(x)}(\Omega)}$ are
equivalent norms on $L^{p(x)}(\Omega, \mathrm{C}\ell _{n}).$
Furthermore, we have that $\|Du\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}$ and
$\|u\|_{W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})}$ are equivalent norms
on $ W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$,
see \cite[Definition 2.9]{13}.\\

The following definitions and lemmas will be crucial in the sequel.\\



\noindent\textbf{Lemma 2.1.} (See \cite{2}) \begin{itshape}Let $\displaystyle\rho (u) =
\int_{\Omega} |u(x)|^{p(x)}dx$. For $u\in L^{p(x)}(\Omega),$ we
have

$(1)$ If  $\|u\|_{L^{p(x)}(\Omega)} \geq 1$, then
$\|u\|^{p_{-}}_{L^{p(x)}(\Omega)} \leq \rho (u) \leq
\|u\|^{p_{+}}_{L^{p(x)}(\Omega)}$.

$(2)$ If  $\|u\|_{L^{p(x)}(\Omega)} \leq 1$, then $\|u\|^{p_{+}}_{L^{p(x)}(\Omega)} \leq \rho (u) \leq \|u\|^{p_{-}}_{L^{p(x)}(\Omega)}$.\end{itshape}\\

\noindent\textbf{Lemma 2.2.} (See \cite{11}) \begin{itshape} Assume that $p(x) \in \mathcal{P}(\Omega)$.
Then the inequality
$$\int_{\Omega}|uv|dx \leq C(n,p)\|u\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}\|v\|_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}$$
holds for every $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ and $ v \in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$.\end{itshape}\\

\noindent\textbf{Lemma 2.3.} (See \cite{11, 12}) \begin{itshape}  Assume that $p(x) \in
\mathcal{P}(\Omega)$. Then

$(1)$ The dual of the space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is
$L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$. That is,
$\big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$ $=
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$. Thus, the space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$
is a reflexive and separable Banach space.

$(2)$ The space $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is a reflexive and separable Banach
space.\end{itshape}\\


\noindent\textbf{Definition 2.1.} (See \cite{25}) (i) Let $u \in C(\Omega, \mathrm{C}\ell_{n})$. The {\it Teodorescu operator} is defined by
$$Tu(x) = \int_{\Omega}G(x-y)u(y)dy,$$
where $G(x)$ is the generalized Cauchy kernel above mentioned.

(ii) Let $u \in C^{1}(\Omega, \mathrm{C}\ell_{n}) \bigcap C(\overline{\Omega},
\mathrm{C}\ell_{n})$. The {\it boundary operator} is defined by
$$Fu(x) = \int_{\partial \Omega}G(y-x)\alpha(y)u(y)dS_{y},$$
where $\alpha(y)$ denotes the outward normal unit vector at $y$.\\

\noindent\textbf{Lemma 2.4.} (See \cite{11}) \begin{itshape}The operator $D: W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is bounded.\end{itshape}\\

\noindent\textbf{Lemma 2.5.} (See \cite{12}) \begin{itshape}The operator $T: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is bounded.\end{itshape}\\

\noindent\textbf{Lemma 2.6.}\begin{itshape} Let $p(x) \in \mathcal{P}(\Omega)$. If $u\in
W^{1,p(x)}(\Omega, \mathrm{C}\ell_{n}),$ then the Borel-Pompeiu formula
$Fu(x) + TDu(x) = u(x)$
holds for all $x \in\Omega$. Moreover, if  $u \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n}),$ then the equation $DTu(x) = u(x)$ holds for all
$x \in \Omega$.\end{itshape}

\begin{proof} By Remark 4.21 in \cite{25}, the conclusions are
implied by $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ $\hookrightarrow
W^{1,p_{-}}(\Omega,\mathrm{C}\ell _{n})$ and
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\hookrightarrow
L^{p_{-}}(\Omega,\mathrm{C}\ell _{n})$.
\qed

\noindent\textbf{Lemma 2.7.}\begin{itshape} The operator $F:
\mathrm{tr}\big(W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})\big) \rightarrow
W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \cap \mathrm{ker}D$ is bounded. Here,
the trace space $\mathrm{tr}\big(W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})\big)$ is
defined by
$$\mathrm{tr}\big(W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})\big) = \big\{f \in L^{1}(\partial \Omega,\mathrm{C}\ell _{n}): \exists u \in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}), s. t. \ u|_{\partial \Omega} =f \big\}.$$\end{itshape}
\begin{proof} Let $u \in \mathrm{tr}\big(W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})\big)$. Then there exists
$v \in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$  such that $v|_{\partial \Omega} =
u$. Using Borel-Pompeiu formula, we know that $Fu = v - TDv$.
 By Lemma 2.4 and Lemma 2.5,  $Fu \in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ and $I - TD$ is continuous in $W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
Since $DFu =Du - DTDu = 0$, we have $Fu \in \mathrm{ker}D$. \qed

\noindent\textbf{Lemma 2.8.} (See \cite{14})\begin{itshape} There exists a unique linear
extension $\widetilde{T}$ of the operator $T$ such that the operator
$\widetilde{T}: W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is bounded. \end{itshape}\\

Diening,  Lengeler and  Ru\v{z}i\v{c}ka \cite{34} showed that
the Dirichlet problem of the Poisson equation with homogeneous
boundary data
$$\left\{
  \begin{array}{ll}
    -\Delta u = f &\mathrm{in} \ \Omega \\
    u = 0 &\mathrm{on} \ \partial \Omega,
  \end{array}
\right.\eqno(2.5)$$
possesses a unique weak solution $u\in
W^{1,p(x)}(\Omega)$ for each $f \in W^{-1,p(x)}(\Omega)$. Moreover, there is the
estimate
 $$\|u\|_{W^{1,p(x)}(\Omega)} \leq C(n,p,\Omega)\|f\|_{W^{-1,p(x)}(\Omega)}$$
where we call $u$ a weak solution of (2.5) provided that
$$\langle f,\varphi\rangle=\int_{\Omega}\nabla u\cdot \nabla \varphi dx ,\ \ \forall\  \varphi \in W^{1,p'(x)}_{0}(\Omega).$$ Then it is easy to see
that for all $f \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ the problem (2.5)
still has a unique weak solution $u\in W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n})$.
We denote by $\Delta^{-1}_{0}$ the solution operator. On the other
hand, it is clear that the operator
$$\Delta : W^{1,p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$$
is continuous, so we obtain that the operator $\widetilde{D}=-\Delta
T:L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$ is
continuous from Lemma 2.5, where the operator $\widetilde{D}$ can
be considered as a unique continuous linear extension of the Dirac
operator.\\

\noindent\textbf{Lemma 2.9.} (See \cite{14})\begin{itshape} Assume that $p(x)$ satisfies relation $(2.2)$.

$(i)$ If $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n}),$ then the equation
$\widetilde{T}\widetilde{D}u(x) = u(x)$ holds for all $x \in
\Omega$.

$(ii)$ If $u \in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n}),$ then the equation
$\widetilde{D}\widetilde{T}u(x) = u(x)$ holds
for all $x \in \Omega$.\end{itshape}\\

\noindent\textbf{Lemma 2.10.} (See \cite{14})\begin{itshape} The space
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ allows the Hodge-type decomposition
$$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) =(\mathrm{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})) \oplus DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$$
with respect to the Clifford-valued product $(2.4)$.\end{itshape}\par

\begin{proof} The proof is given in \cite[Theorem 3.1]{14}. We sketch it here for the reader's convenience.
First, it is easy to prove that
$$\big(\mathrm{ker}\widetilde{D} \cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})\big) \cap DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) = \{0\}.$$
Now let $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. Then we have $u_{2} =
D\Delta^{-1}_{0}\widetilde{D}u \in DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Let $u_{1} = u - u_{2}$. Then $u_{1} \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.
Furthermore, we have
$$Du_{1}=Du -\widetilde{D}D\Delta^{-1}_{0}Du=Du +\Delta\Delta^{-1}_{0}Du=Du-Du=0.$$
Thus, $u_{1} \in \mathrm{ker}\widetilde{D}$. Since $u \in
L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ is arbitrary, the desired result
follows immediately. \qed

Beginning with this decomposition we can get the following
projections
$$P: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \mathrm{ker}\widetilde{D}\cap L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$$
$$Q: L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \rightarrow DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$$
For $p(x) \equiv 2$, these are ortho-projections. Notice that directly from the proof of Theorem 3.1 we obtain
$$Q = D\Delta^{-1}_{0}\widetilde{D}, P = I - Q.\eqno(2.6)$$
It follows from (2.6) that the operator $Q$ as well as $P$ maps the
space $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$ into itself.\\


\noindent\textbf{Lemma 2.11.} (See \cite{14}). \begin{itshape}The space
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}) \cap \mathrm{im}Q$ is a closed subspace of
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n}),$ that is, the space
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
is closed in $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.\end{itshape}\\

\noindent\textbf{Lemma 2.12.} (See \cite{14}).  \begin{itshape}$\big(L^{p(x)}(\Omega,\mathrm{C}\ell _{n})
\cap \mathrm{im}Q\big)^{\ast} = L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}) \cap
\mathrm{im}Q,$ that is, the linear operator
$$\Phi : DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}) \rightarrow \big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast}$$
given by
$$\Phi(Du)(D\varphi) = (D\varphi , Du)_{Sc}:= \int_{\Omega}[\overline{D\varphi}Du]_{0}dx$$
is a Banach space isomorphism.\end{itshape}

%{\section*{{\bf 4. Computation of critical groups}}}
\section{The homogeneous $A$-Dirac equations}
In this section, we are interested in the existence of weak solutions for the homogeneous $A$-Dirac equations (1.1).
In order to get the existence of weak solutions to (1.1),
we need the following result, see \cite[Theorem 2.1]{35} for the proof.\\


\noindent\textbf{Proposition 3.1.} \begin{itshape}Let
$X$ be a reflexive, separable Banach space, and assume that $G: X\rightarrow X^{\ast}$ is \\
$(i)\ (monotone)\ (Gv-Gw, v-w)\geq0\ \ \forall\  v, w\in X;$\\
$(ii)$\ (bounded)\ $G$ maps bounded sets to bounded sets; \\
$(iii)\ (demicontinuous)\ (G(u+\lambda v),w)\rightarrow(G(u+v),w),\ as\ \lambda \in \mathbb{R},\ \lambda\rightarrow 0,\ u,v,w\in X;$\\
$(iv)\ (coercive)\  \lim_{\|v\| \rightarrow \infty}
{{\|v\|}^{-1}}{(Gv,v)}=\infty.$\\
Then $G$ is surjective.\end{itshape}\\

 Now we are ready to prove our result as follows.\\

\noindent\textbf{Theorem 3.1.} \begin{itshape}Under the conditions $(A1)$--$(A4)$, there exists a weak solution
 $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ to the A-Dirac equations $(1.1)$, that is to say, there exists a
 Clifford-valued function $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that $$\int_{\Omega}\overline{A(x,Du)}Dvdx=0 \eqno(3.1)$$
 for any $v \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Furthermore, the solution to the scalar part of
$(1.1)$ is unique up to a monogenic function.\end{itshape}\\

\begin{proof} We divide the proof into four steps:

\begin{itshape}Step 1.\end{itshape} We first claim that $A(x,u) \in
L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$ for every $u \in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$.
Indeed, from (A2) we obtain
$$\int_{\Omega}|A(x,u)|^{p'(x)}dx \leq 2^{n-1}C_{1}\int_{\Omega}|u|^{p(x)}dx+2^{n-1}\int_{\Omega}|g|^{p'(x)}dx.$$
This estimate together with Remark 2.1 and Lemma 2.1 yields the
previous assertion.


\begin{itshape}Step 2.\end{itshape} By Lemma 2.11, we know that
$DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ is a reflexive and separable Banach space.
By Lemma 2.12, we get that
$\big(DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big)^{\ast} =
DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Obviously, it follows from (2.6)
that $QA\big(\Omega \times DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})\big) \subset
DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Now, we define the nonlinear mapping $\mathcal{F}:DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})
\rightarrow DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
as follows:
$$\mathcal{F}(Du)=QA(x,Du),\ \  \mathrm{for\  each}\  u\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$$
 In the following, to get surjectivity of the operator $\mathcal{F}$, we need to
verify the conditions of Proposition 3.1 respectively.

(1) The operator $\mathcal{F}$ is demicontinuous. Obviously, it suffice to show that the operator $\mathcal{F}$
is strongly-weakly continuous. Suppose that $Du_{k}, Du \in
DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ and $||Du_{k} -
Du||_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow 0$ as $k \rightarrow
\infty$. Then  $\{Du_{k}\}$ is uniformly
bounded in $L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. By (A1) we can deduce that for
each $v\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
$$\Big[\overline{QA(x,Du_{k})}Dv\Big]_{0}\rightarrow
\Big[\overline{QA(x,Du)}Dv\Big]_{0}\ \ \ \  \mathrm{a.e.\ on}\
\Omega,\ \  \ \ \mathrm{as}\ k\rightarrow\infty.$$
On the other
hand, to see equi-continuous integrability of the sequence
$\big\{\big[\overline{A(x,Du_{k})}Dv\big]_{0}\big\}$, we take a measurable
subset $\Omega^{'}\subset\Omega$, by (2.1), (A2) and the H\"{o}lder inequality we have for
each $v\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
\begin{align*}
&\Big|\int _{\Omega^{'}}\Big[\overline{QA(x,Du_{k})}Dv\Big]_{0}dx\Big|\\
&\leq 2C_{n}\|Q\|_{L^{p'(x)}(\Omega^{'})}
\big\|\ |Dv|\ \big\|_{L^{p(x)}(\Omega^{'})}
\big(C_{1}\big\||Du_{k}|^{p(x)-1}\big \|_{L^{p'(x)}(\Omega^{'})}
+\|g\|_{L^{p'(x)}(\Omega^{'})}\big).\tag{3.2}
\end{align*}
In terms of Remark 2.1, Lemma 2.1 and boundedness of the operator $Q$, we obtain that
the third part of (3.2) is uniformly bounded in $k$. The second norm
of (3.2) is arbitrarily small if the measure of $\Omega^{'}$ is
chosen small enough. By the Vitali Convergence Theorem, we have for
each $v\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
$$\aligned
(\mathcal{F}(Du_{k}), Dv):=&\int_{\Omega}\Big[ \overline{QA(x, Du_{k})}Dv\Big]_{0}dx\\
&\rightarrow \int_{\Omega}\Big[ \overline{QA(x, Du)}Dv\Big]_{0}dx
=(\mathcal{F}(Du), Dv)
\endaligned$$
as $k\rightarrow\infty.$ That is to say, $\mathcal{F}$ is strongly-weakly continuous.

(2) The operator $\mathcal{F}$ is bounded. In terms of the H\"{o}lder inequality, the boundedness of $Q$ and (A2),
together Remark 2.1 and Lemma 2.1, we obtain for
each $v\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
$$\aligned
\big|(\mathcal{F}(Du),Dv)\big|
&=\Big|\int _{\Omega}\Big[\overline{QA(x,Du)}Dv\Big]_{0}dx\Big|\\
&\leq C_{3}\Big(C_{4}\mathrm{max}\Big\{\|Du\|_{L^{p(x)}(\Omega, \mathrm{C}\ell_{n})}^{p_{+}-1}
, \|Du\|^{p_{-}-1}_{L^{p(x)}(\Omega, \mathrm{C}\ell_{n})}\Big\}+\|g(x)\|_{L^{p'(x)}(\Omega)}\Big)\|Dv\|_{L^{p(x)}(\Omega,\mathrm{C}\ell_{n})},
\endaligned$$
where $C_3$ and $C_{4}$ are two positive constants. This implies that $\mathcal{F}$ is bounded.

(3) The operator $\mathcal{F}$ is monotone. In view of Lemma 2.10,
we have
$$ QA(x,Du) = A(x,Du) - PA(x,Du)$$
for each $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
Thus, for any $u,v \in
W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$, (3.2) gives
$$\big(QA(Du), Dv\big)_{Sc} = \big(A(Du), Dv\big)_{Sc} - \big(PA(Du), Dv\big)_{Sc} = \big(A(Du), Dv\big)_{Sc}.\eqno(3.3)$$
Then the condition (A4) yields
$$\aligned
\big(\mathcal{F}(Du) - \mathcal{F}(Dv), Du - Dv\big)
&=\big(QA(Du) - QA(Dv), Du - Dv\big)_{Sc} \\
&= \big(A(Du) - A(Dv), Du - Dv\big)_{Sc}\\
&= \int_{\Omega} \Big[\overline{\big(A(Du) - A(Dv)\big)}(Du - Dv)\Big]_{0}dx \geq 0.
\endaligned$$


(4) The operator $\mathcal{F}$ is coercive. By means of (3.3) and (A3) we
have
$$\aligned
\frac{\big(\mathcal{F}(Du),
Du\big)}{\| |Du|\|_{L^{p(x)}(\Omega)}}= \frac{\big(QA(x,Du),
Du\big)_{Sc}}{\||Du|\|_{L^{p(x)}(\Omega)}}
&= \frac{\big(A(x,Du), Du\big)_{Sc}}{\||Du|\|_{L^{p(x)}(\Omega)}}\\
&=\frac{\displaystyle\int_{\Omega} \Big[\overline{A(x,Du)}Du\Big]_{0}dx}{\|||Du|\|_{L^{p(x)}(\Omega)}}\\
&\geq\frac{\displaystyle C_{2}\int_{\Omega}
|Du|^{p(x)}dx+\int_{\Omega}h(x)dx}{\||Du|\|_{L^{p(x)}(\Omega)}}.
\endaligned$$
Since
$$\frac{ \displaystyle\int_{\Omega} |Du|^{p(x)}dx}{\||Du|\|_{L^{p(x)}(\Omega)}} = \int_{\Omega}\bigg(\frac{|Du|}{2^{-1}\||Du|\|_{L^{p(x)}(\Omega)}}\bigg)^{p(x)} \frac{ \Big(2^{-1}\||Du|\|_{L^{p(x)}(\Omega)}\Big)^{p(x)}}{\||Du|\|_{L^{p(x)}(\Omega)}}dx.$$
When $\||Du|\|_{L^{p(x)}(\Omega)} \geq 1$, we have
$$\frac{\displaystyle \int_{\Omega} |Du|^{p(x)}dx}{\||Du|\|_{L^{p(x)}(\Omega)}} \geq 2^{-p_{+}}\||Du|\|_{L^{p(x)}(\Omega)}^{p_{-}-1}.$$
Hence, by Remark 2.1, it easily follows that
$$\frac{(\mathcal{F}(Du), Du)}{\|Du\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}} \rightarrow \infty$$
as $\|Du\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})} \rightarrow \infty$.

\begin{itshape}Step 3.\end{itshape} Let $X=DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. According to Proposition 3.1,
we get that the operator $\mathcal{F}$ is surjective. Consequently,
there exists $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that $\mathcal{F}(Du)=QA(x,Du)= 0$.
 Furthermore, Lemma 2.10 deduces
 $$\int_{\Omega}\overline{A(x,Du)}D\varphi dx=\int_{\Omega}\overline{\big(QA(x,Du) + PA(x,Du)\big)}D\varphi dx=\int_{\Omega}\overline{\widetilde{D}PA(x,Du)}\varphi dx=0,$$
for any $\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Therefore, $u$
is a weak solution of the $A$-Dirac equations (1.1).

\begin{itshape}Step 4.\end{itshape} If $u_{1},u_{2}$ are solutions to the $A$-Dirac equations (1.1), then
$\big(A(x,Du_{i}), D\varphi\big)_{Sc} = 0\ (i=1,2)$ hold for any
$\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
 Set $\varphi = u_{1} - u_{2}$, then the condition (A4) yields
$$\aligned
0 &= \big(A(x,Du_{1}) - A(x,Du_{2}), Du_{1} - Du_{2}\big)_{Sc}\\
&= \int_{\Omega} \Big[\overline{\big(A(x,Du_{1}) - A(x,Du_{2})\big)}(Du_{1} - Du_{2})\Big]_{0}dx \geq 0.
\endaligned$$
Thus, $[Du_{1}]_{0} = [Du_{2}]_{0}$. The proof is now complete. \qed

As $p(x)$ is a constant function, Nolder \cite[Theorem 3.1]{7} proved a Caccioppoli-type estimate for weak solutions for (1.1) under the assumptions
(A2) with $g(x)\equiv 0$ and (A3) with $h(x)\equiv 0$.
Harjulehto, H\"{a}sto and Latvala \cite[Lemma 5.3]{36} showed
a Caccioppoli-type estimate for weak solutions to the equation $-\mathrm{div}(p(x)|\nabla u|^{p(x)-2}\nabla u)=0$ as $u$ is a real function.
Thus it is natural to study a  Caccioppoli-type estimate for weak solutions to (1.1) in the variable exponent setting.
Taking the similar approach presented in \cite{36} we prove the following result:\\

\noindent\textbf{Theorem 3.2.} \begin{itshape} Let $p(x)\in \mathcal{P}(\Omega)$
and $A$ satisfies the hypotheses $(\mathrm{A}2)$  with $g(x)\equiv 0$ and $(\mathrm{A}3)$ with $h(x)\equiv 0$. If $u$ be a weak solution to $(1.1)$ and $\eta \in C^{\infty}_{0}(\Omega)$ with
$0< \eta \leq 1,$ then
$$\int_{\Omega}|Du|^{p(x)}\eta^{p_{+}}dx\leq \Big(1+\frac{2C_{1}p_{+}}{C_{2}}\Big)^{p_{+}} \int_{\Omega}|u|^{p(x)}|\nabla\eta|^{p(x)}dx.$$  \end{itshape}

\begin{proof} Choose $\varphi =-u\eta^{p_{+}}$. Then $D\varphi =-p_{+}\eta^{p_{+}-1}(D\eta)u-\eta^{p_{+}}Du$.
Hence, according to (3.1) and (A3) we obtain
\begin{align*}
0=\int_{\Omega}\Big[\overline{A(x,Du)}D\varphi\Big]_{0} dx
&=\int_{\Omega}\Big[\overline{A(x,Du)}(-p_{+}{\eta}^{p_{+}-1}(D\eta)u-{\eta}^{p_{+}}Du)\Big]_{0}dx\\
&\leq -C_{2}\int_{\Omega} |Du|^{p(x)}{\eta}^{p_{+}}dx+ p_{+}\int_{\Omega} \big|\overline{A(x,Du)}\big| |u| |D\eta| |\eta|^{p_{+}-1}dx.
\end{align*}
Then from (A2) we have
$$C_{2}\int_{\Omega} |Du|^{p(x)}{\eta}^{p_{+}}dx\leq p_{+}\int_{\Omega} \big|\overline{A(x,Du)}\big| |u| |D\eta| |\eta|^{p_{+}-1}dx
\leq C_{1}p_{+}\int_{\Omega} \big|Du\big|^{p(x)-1} |u| |D\eta| |\eta|^{p_{+}-1}dx.$$
By the Young's inequality, for any $\varepsilon \in (0,1]$ we have
$$ ab\leq \Big(\frac{1}{\varepsilon}\Big)^{p(x)-1}\frac{a^{p(x)}}{p(x)}+\varepsilon \frac{b^{p'(x)}}{p'(x)}
\leq \Big(\frac{1}{\varepsilon}\Big)^{p(x)-1}a^{p(x)}+\varepsilon b^{p'(x)}.$$
Take $a=|u| |D\eta| \eta^{p_{+}-\frac{p_{+}}{p(x)}-1}$ and $b=\big|Du\big|^{p(x)-1}\eta^{\frac{p_{+}}{p(x)}}$, we get
$$\int_{\Omega} \big|Du\big|^{p(x)-1} |u| |D\eta| \eta^{p_{+}-1}dx
\leq \Big(\frac{1}{\varepsilon}\Big)^{p(x)-1} \int_{\Omega}|u|^{p(x)} |D\eta|^{p(x)} \eta^{p_{+}-p(x)}dx
+\varepsilon \int_{\Omega} |Du|^{p(x)}{\eta}^{p_{+}}dx.$$
Let $ \varepsilon =\mathrm{min}\big\{1,\frac{C_{2}}{2C_{1}p_{+}}\big\}$. Then
$$\int_{\Omega} |Du|^{p(x)}{\eta}^{p_{+}}dx\leq
\frac{2C_{1}p_{+}}{C_{2}}\Big(1+\frac{2C_{1}p_{+}}{C_{2}}\Big)^{p_{+}-1}\int_{\Omega}|u|^{p(x)}|\nabla\eta|^{p(x)}dx.$$
Thus the proof is complete.
\qed
\noindent\textbf{Remark 3.1.} From the proof of Theorem 3.2, we know that the conclusion in Theorem 3.2 still holds
if $u$ is just a weak solution to the scalar part of the equations (1.1).


\section{The nonhomogeneous $A$-Dirac equations}
%{\section*{{\bf 5. Proof of main results}}}
In this section we are concerned with the existence of solutions for the following nonlinear $A$-Dirac equations
with right hand side in Dirac equations form:
$$DA(x,Du)=Df.  \eqno(4.1)$$
The natural space in which to consider the weak solutions of (4.1) is the Sobolev spaces
$W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ under the conditions (A1)--(A4). Thus,
we suppose that $f$ belongs to the dual space $L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$.
Note that for every $f\in \mathbb{R}^{m\times n}$,
the existence and uniqueness of the solution $u:\Omega\rightarrow \mathbb{R}^{m}$
were established by general principles of monotone operators in Iwaniec and Sbordone \cite[Proposition 4.1]{15}.
For further details we refer to \cite{20, 17, 18, 15} and the references therein. \\

\noindent\textbf{Theorem 4.1.} \begin{itshape}Under the assumptions $(A1)$--$(A4),$ for each $f\in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}),$ there exists a weak solution
 $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ to the A-Dirac equations $(4.2),$ that is to say, there exists a
 Clifford-valued function $u \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ such that $$\int_{\Omega}\overline{A(x,Du)}Dvdx=\int_{\Omega}\overline{f}Dvdx$$ for any $v \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.
 Furthermore, the solution to the scalar part of $(4.2)$ is unique up to a monogenic function.\end{itshape}

\begin{proof} We notice that \begin{itshape} Step 1\end{itshape},\begin{itshape} Step 2\end{itshape},
\begin{itshape} Step 4\end{itshape} of the proof are completely similar to those of the proof in Theorem 3.1. The only difference lies in \begin{itshape} Step 3\end{itshape}.
In fact, let $X=DW^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. In view of Proposition 3.1,
the operator $\mathcal{F}$ is surjective. By means of Lemma 2.10,
for each $f\in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})$, we have the following decomposition:
$$f=f_{1}+f_{2}, \ \ f_{1} \in
\mathrm{ker}\widetilde{D} \cap L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}), f_{2}
\in DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n}).$$
Then for $f_{2}\in DW^{1,p'(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$,
there exists $u \in W^{1,p(x)}_{0}(\Omega,Cl_{n})$ such that $\mathcal{F}(Du)=QA(x,Du)
= f_{2}$.
Furthermore, Lemma 2.10 gives
\begin{align*}
 &\int_{\Omega}\overline{A(x,Du)}D\varphi dx=\int_{\Omega}\overline{\big(QA(x,Du) + PA(x,Du)\big)}D\varphi dx\\
 =&\int_{\Omega}\overline{f-f_{1}}D\varphi dx+\int_{\Omega}\overline{PA(x,Du)}D\varphi dx\\
 =&\int_{\Omega}\overline{f}D\varphi dx-\int_{\Omega}\overline{\widetilde{D}f_{1}}\varphi dx+\int_{\Omega}\overline{\widetilde{D}PA(x,Du)}\varphi dx =\int_{\Omega}\overline{f}D\varphi dx,
 \end{align*}
for any $\varphi \in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$. Therefore, $u$
is a weak solution of the $A$-Dirac equations (4.1).
\qed

In what follows we first consider the solvability of the following Dirac equation with homogeneous
boundary data.
$$\left\{
  \begin{array}{ll}
    Du = f &\mathrm{in} \ \Omega \\
    u = 0 &\mathrm{on} \ \partial \Omega,
  \end{array}
\right.\eqno(4.2)$$
Let us show the following results.\\

\noindent\textbf{Theorem 4.2.}  \begin{itshape}Let $f\in L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. Then the equations $(4.2)$
is solvable in $ W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ if and only if $\mathrm{tr}\ Tf=0$. Furthermore,
if a solution $u$ exists, then it can be represented by $u=Tf$. \end{itshape}
\begin{proof} On the one hand, if the equations (4.2) is solvable, then from the Borel-Pompeiu formula we have
$$u=Fu+TDu=TDu=Tf.$$ Thus we get $\mathrm{tr}\ Tf=0$ due to $u\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$.

\noindent On the other hand, let $\mathrm{tr}\ Tf=0$. In view of Lemma 2.10, we know that $f=Pf+Qf$. Then we have
$$\mathrm{tr}\ Tf=\mathrm{tr}\ TPf+\mathrm{tr}\ TQf=0.$$ From (2.6), it follows that $\mathrm{tr}\ TQf=0$.
Thus, $\mathrm{tr}\ TPf=0$. Note that $-\Delta TPf= \widetilde{D}DTPf=0$ due to Lemma 2.6 and the definition of the operator $P$.
According to the uniqueness of solutions of the problem (2.5), we obtain $TPf=0$. And hence $DTPf=Pf=0$.
Further, $f=Qf$. Therefore, by the definition of the operator $Q$, there exists $u\in W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$
such that $Du=f$. The proof is now completed.
\qed

\noindent\textbf{Corollary 4.1.} \begin{itshape}Let $f\in W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$. Then the equations $(4.2)$
is solvable in $ W^{1,p(x)}_{0}(\Omega,\mathrm{C}\ell _{n})$ if and only if $\mathrm{tr}\ \widetilde{T}f=0$. Furthermore,
if a solution $u$ exists, then it can be represented by $u=\widetilde{T}f$. \end{itshape}

\begin{proof} By means of Proposition 12.3.4 in \cite{2}, it is easy to show that that the
space $C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ is dense in
$W^{-1,p(x)}(\Omega,\mathrm{C}\ell _{n})$. According to Theorem 2.1 in \cite{10}, we know that
$C^{\infty}_{0}(\Omega,\mathrm{C}\ell _{n})$ is dense in
$L^{p(x)}(\Omega,\mathrm{C}\ell _{n})$. Therefore, by the density arguments it is easy to see that
the desired conclusion follows from Lemma 2.8 and Theorem 4.2.
\qed

We would like to point out that the uniqueness of weak solutions for the equations (1.1) in Theorem 3.1 and (4.1) in Theorem 4.1 can be obtained
if the condition (A4) is replaced by the following strong monotonicity:\par
\begin{itemize}
\item[$(A4')$]\ $ \Big[\overline{(A(x,\xi_{1})-A(x,\xi_{2}))}(\xi_{1}-\xi_{2})\Big]_{0}\geq C_{3} |\xi_{1}-\xi_{2}|^{p(x)},$
\end{itemize}
where $C_{3}>0$ is a constant. Indeed, it is easy to deduce the assertion from Step 4 in the proof of Theorem 3.1 and Remark 2.1.
Furthermore, we can consider the stability of weak solutions to the non-homogeneous $A$-Dirac equations. In other words, we have
the following result.\\

\noindent\textbf{Theorem 4.3.}\begin{itshape} Under the assumptions $(A1)$--$(A3)$ and
$(A4'),$ for given $f, g \in L^{p'(x)}(\Omega,\mathrm{C}\ell _{n}),$ each of the two equations
$$\left\{
  \begin{array}{ll}
    DA(x,Du)=Df \\
    u\in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}),
  \end{array}
\right.\eqno(4.3)$$

$$\left\{
  \begin{array}{ll}
    DA(x,Dv)=Dg \\
    v\in W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n}),
  \end{array}
\right.\eqno(4.4)$$
has a unique weak solution and
$$\mathrm{min}\bigg\{\|u-v\|_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n})}^{p_{+}-1}
, \|u-v\|^{p_{-}-1}_{W^{1,p(x)}_{0}(\Omega, \mathrm{C}\ell_{n})}\bigg\}\leq C(n,p,\Omega)\|f-g\|_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
\end{itshape}
\begin{proof} In view of Theorem 4.1 and the above arguments, we know that the equations (4.3) and (4.4) has a unique weak solution
$u$ and $v$ respectively. Then we have
$$\big(A(x,Du), D(u-v)\big)_{Sc} =(f, D(u-v))_{Sc}, \ \big(A(x,Dv), D(u-v)\big)_{Sc} =(g, D(u-v))_{Sc}.$$
Therefore, we obtain
$$\big(A(x,Du)-A(x,Dv), Du-Dv\big)_{Sc} =(f-g, Du-Dv)_{Sc}.$$
From $(A4')$ and Lemma 2.2 it follows that
\begin{align*}
\int_{\Omega}|Du-Dv|^{p(x)}dx
&\leq \frac{1}{C_{3}}\int_{\Omega}\Big[\overline{\big(A(x,Du) - A(x,Dv)\big)}(Du - Dv)\Big]_{0}dx \\
&\leq \frac{1}{C_{3}}\int_{\Omega}\Big[\overline{(f-g)}(Du - Dv)\Big]_{0}dx \\
&\leq C(n,p,\Omega)\|f-g\|_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}\|Du-Dv\|_{L^{p(x)}(\Omega,\mathrm{C}\ell _{n})}.
\end{align*}
Using Remark 2.1 and Lemma 2.1 we have
$$\mathrm{min}\bigg\{\|Du-Dv\|_{L^{p(x)}(\Omega, \mathrm{C}\ell_{n})}^{p_{+}-1}
, \|Du-Dv\|^{p_{-}-1}_{L^{p(x)}(\Omega, \mathrm{C}\ell_{n})}\bigg\}\leq C(n,p,\Omega)\|f-g\|_{L^{p'(x)}(\Omega,\mathrm{C}\ell _{n})}.$$
Then the desired inequality immediately follows from Remark 2.1.
\qed

\noindent \textbf{Acknowlegements.} The authors are grateful to the reviewers for helpful comments and suggestions.
B. Zhang was supported by the project titled ``Variational, topological and algebraic methods for nonlinear elliptic problems"
when he visited University of Reggio Calabria and Medalics Research Centre and was supported by Natural Science Foundation of Heilongjiang Province of China (A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667).\par

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\end{document}
The Dirac Equation is a modern presentation of the relativistic quantum mechanics of electrons intended to make new mathematical results accessible to a wider audience. It treats in some depth the relativistic invariance of a quantum theory, self-adjointness and spectral theory, qualitative features of relativistic bound and scattering states, and the external field problem in quantum electrodynamics, without neglecting the interpretational difficulties and limitations of the theory.
