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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 227, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/227\hfil Weak solutions for A-Dirac equations]
{Weak solutions for A-Dirac equations with variable growth
 in Clifford analysis}

\author[B. Zhang \hfil EJDE-2012/227\hfilneg]
{Binlin Zhang, Yongqiang Fu}  

\address{Binlin Zhang\newline
 Department of Mathematics, Harbin Institute of Technology,
Harbin 150001, China}
\email{zhangbinlin2012@yahoo.cn}

\address{Yongqiang Fu \newline
 Department of Mathematics, Harbin Institute of Technology,
Harbin 150001, China}
\email{fuyqhagd@yahoo.cn}

\thanks{Submitted November 18, 2012. Published December 17, 2012.}
\subjclass[2000]{30G35, 35J60, 35D30, 46E35}
\keywords{Clifford analysis; variable exponent; 
$A$-Dirac equation; \hfill\break\indent obstacle problem}

\begin{abstract}
 In this article we show the existence of weak solutions for
 obstacle problems for $A$-Dirac equations with variable growth in
 the setting of variable exponent spaces of Clifford-valued
 functions. We also obtain the existence of weak solutions to the
 scalar part of $A$-Dirac equations in space
 $W_0^{1,p(x)}(\Omega,\mathrm{C}\ell_n)$.
\end{abstract}

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

\section{Introduction}

After  Kov\'{a}\v{c}ik and  R\'{a}kosn\'{i}k first discussed the
$L^{p(x)}$ space and $W^{k,p(x)}$ space in \cite{k2}, a lot of results
have been obtained concerning these kinds of variable exponent
spaces and their applications, for example, see \cite{e2,e3,f1,f2} and
references therein. Recently the theory of nonlinear partial
differential equations with nonstandard growth conditions has
important applications in elasticity (see \cite{z1}), eletrorheological
fluids (see \cite{r2}) and so on. For an overview of variable exponent
spaces with various applications to differential equations we refer
to \cite{h1} and the references quoted there.


Clifford algebras were introduced by Clifford as geometric
algebras in 1878, which are a generalization of the complex numbers,
the quaternions, and the exterior algebras, see \cite{g1}. As an active
branch of mathematics over the past 40 years, Clifford analysis
usually studies the solutions of the Dirac equations for functions
defined on domains in Euclidean space and taking value in Clifford
algebras, see \cite{h4}.  G\"{u}rlebeck and 
Spr\"{o}{\ss}ig  \cite{g2,g4}  developed the theory of Clifford analysis to
investigate elliptic boundary value problems of fluid dynamics, in
particular the Navier-Stokes equations and related equations. 
 Doran and Lasenby \cite{d2} gave in detail an overview of the intrinsic
value and usefulness of Clifford algebras and Clifford analysis for
mathematical physics.

Nolder \cite{n1,n2} introduced $A$-Dirac equations $DA(x,Du)=0$ and
investigated some properties of weak solutions to the scalar parts
of above-mentioned equations, for example, the Caccioppoli estimate
and the removability theorem.  Fu and Zhang \cite{f3} first
introduced the weighted variable exponent spaces in the context of
Clifford algebras, and then discussed the properties of these
spaces. As an application, they obtained the existence of weak
solutions in space $W^{D,p(x)}(\Omega,\mathrm{C}\ell_n)$ to the
scalar part of the nondegenerate $A$-Dirac equations
$DA(x,Du)+B(x,u)=0$. Unfortunately, the proof of 
\cite[Corollary 4.1]{f3} is invalid for the case in which $B(x,u)\equiv 0$.
Motivated by such problems, the aim of this paper is to investigate
the existence of solutions to the scalar part of A-Dirac equations.
Note that when $u$ is a real-valued function and $A: \Omega \times
\mathrm{C}\ell_n^{1}(\Omega) \to \mathrm{C}\ell_n^{1}(\Omega)$,
the scalar part of A-Dirac equations becomes 
$\operatorname{div} A(x, \nabla u)=0$; i.e., $A$-harmonic
equations. These equations have been extensively studied with many
applications, see \cite{h3}.

In recent years, obstacle problems in the variable exponent setting
have attracted a lot of interest, we refer to \cite{e3,f3,f4,h2,r1} and
references therein. Inspired by their works, we are interested in
the following obstacle problems:
\begin{equation}
\int_{\Omega}\big[\overline{A(x,Du)}D(v-u)\big]_0\geq 0 \label{e1.1}
\end{equation}
for $v$ belonging to
\begin{equation}
K_{\psi}=\{v\in W_0^{1,p(x)}(\Omega,\mathrm{C}\ell_n): v \geq \psi
\text{ a.e. }   \mathrm{in}\ \Omega \}\label{e1.2}
\end{equation}
where $\psi (x)=\Sigma \psi_{I}e_{I}\in\mathrm{C}\ell_n(\Omega)$,
$\psi_{I}:\Omega \to [-\infty,\ +\infty]$,
 $v\geq \psi$, a.e. in $\Omega$ means that for any $I$, we have $v_{I}\geq
\psi_{I}$ a.e. in $\Omega$.

We will study the solution $u\in K_{\psi}$ for \eqref{e1.1}-\eqref{e1.2} as
$A(x,\xi): \Omega \times \mathrm{C}\ell_n\to
\mathrm{C}\ell_n$ satisfies
the following growth conditions:
\begin{itemize}

\item[(A1)] $A(x,\xi)$ is measurable with respect $x$ for $\xi \in
\mathrm{C}\ell_n$
and continuous with respect to $\xi$ for a.e. $x\in \Omega$,

\item[(A2)] $|A(x,\xi)|\leq C_1|\xi|^{p(x)-1}+g(x)$ for a.e. $x\in \Omega$ and
$\xi \in \mathrm{C}\ell_n$,

\item[(A3)] $\big[\overline{A(x,\xi)}\xi\big]_0\geq
C_2|\xi|^{p(x)}+h(x)$ for a.e. $x\in \Omega$ and $\xi \in \mathrm{C}\ell_n$,

\item[(A4)]
$\big[\overline{(A(x,\xi_1)-A(x,\xi_2))}(\xi_1-\xi_2)\big]_0>
0$ for a.e. $x\in \Omega$ and $\xi_1\neq \xi_2\in \mathrm{C}\ell_n $,
\end{itemize}
where $g \in L^{p'(x)}(\Omega)$, $h \in L^{1}(\Omega)$, 
$C_{i}$ $(i=1,2)$ are positive constants. Throughout the paper we always assume
that $\Omega$ is a bounded domain in $\mathbb{R}^{n}(n\geq 2)$. And
that (unless declare specially)
\begin{equation}
p\in P^{\log }(\Omega) \quad\text{and}\quad
 1<p_{-}=:\inf_{x\in\bar{\Omega}}p(x)\leq p(x)
 \leq \sup_{x\in\bar{\Omega}}p(x):=p_{+}< \infty  \label{e1.3}
\end{equation}


This article is divided into four sections. In Section 2, we will
recall some basic knowledge of Clifford algebras and variable
exponent spaces of Clifford-valued functions, which will be needed
later. In Section 3, we will prove the existence of solutions for
the above-mentioned obstacle problems for $A$-Dirac equations with
nonstandard growth. Furthermore, we also obtain the existence of
solutions to the scalar part of  $A$-Dirac equations in
$W_0^{1,p(x)}(\Omega ,\mathrm{C}\ell_n)$.


\section{Preliminaries}

\subsection{Clifford algebra}

In this section we first recall some related notions and results
from Clifford algebras. For a detailed account we refer to
\cite{g1,g2,g3,g4,h1,h2,h3,h4}.

 Let $\mathrm{C}\ell_n$ for the real universal Clifford algebras over
$\mathbb{R}^{n}$, then
$$
\mathrm{C}\ell_n =\operatorname{span} \{e_0,e_1,e_2,\dots ,e_n,e_1e_2,\dots 
,e_{n-1}e_n,\dots ,e_1e_2\dots e_n\}
$$
where $e_0=1$ (the identity element in $\mathbb{R}^{n}$),
$\{e_1,e_2,\dots ,e_n\}$ is an orthonormal basis of
$\mathbb{R}^{n}$ with the relation
$e_{i}e_{j}+e_{j}e_{i}=-2\delta_{ij}$. Thus the dimension of
$\mathrm{C}\ell_n$ is $2^{n}$. For
$I=\{i_1,\dots,i_{r}\}\subset \{1,\dots ,n\}$
    with $1\leq i_1<i_2<\dots <i_n\leq n$, put $e_{I}=e_{i_1}e_{i_2}\dots e_{i_{r}}$, while for $I=\emptyset$, $e_{\emptyset}=e_0$.
     For $0\leq r\leq n$ fixed, the space $C l_n^{r}$ is defined by
$$
\mathrm{C}\ell_n^{r}=\operatorname{span}\{e_{I}:|I|:=\operatorname{card}(I)=r\}.
$$
The Clifford algebras $\mathrm{C}\ell_n$ is a graded algebra as
$$
\mathrm{C}\ell_n=\oplus_{r}\mathrm{C}\ell_n^{r}.
$$
Any element $a\in \mathrm{C}\ell_n$ may thus be written in a unique way as
$$
a=[a]_0+[a]_1+\dots +[a]_n
$$
    where $[\ ]_{r}:\mathrm{C}\ell_n\to \mathrm{C}\ell_n^{r}$ 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.
    For $u\in \mathrm{C}\ell_n$, we know that $[u]_0$ denotes the scalar 
part of $u$, that is the coefficient of the element $e_0$. 
We define the Clifford conjugation as follows:
$$
\overline{(e_{i_1}e_{i_2}\dots e_{i_{r}})}
=(-1)^{\frac{r(r+1)}{2}}e_{i_1}e_{i_2}\dots e_{i_{r}}
$$
For $A\in \mathrm{C}\ell_n$, $B\in\mathrm{C}\ell_n$, we have
$$
\overline{AB}=\overline{B}\ \overline{A},\quad
\overline{\overline{A}}=A.
$$
 We denote
$$
(A,B)=[\overline{A}B]_0.
$$
Then an inner product is thus obtained, leading to the norm
$|\cdot|$ on $\mathrm{C}\ell_n$ given by
$$
|A|^{2}=[\overline{A}A]_0.
$$
From \cite{g3} we know that this norm is submultiplicative:
\begin{equation}
|AB|\leq C_{3}|A\|B|.\label{e2.1}
\end{equation}
where $C_{3}\in[1,2^{n/2}]$ is a constant.

A Clifford-valued function $u:\Omega\to\mathrm{C}\ell_n$
can be written as $u=\Sigma_{I}u_{I}e_{I}$, where the coefficients
$u_{I}:\Omega\to \mathbb{R}$ are real valued functions.

 The Dirac operator on Euclidean space used here is as follows:
$$
D=\sum_{j=1}^{n}e_{j}\frac{\partial}{\partial x_{j}}
=\sum_{j=1}^{n}e_{j}\partial_{j}.
$$


If $u$ is $C^{1}$ real-valued function defined on a domain $\Omega$
in $\mathbb{R}^{n}$, then $Du=\nabla u=(\partial_1 u,\partial_2
u,\dots,\partial_n u)$, where $\nabla$ is the distributional
gradient. Further $D^{2}=-\Delta$, 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{g1,g2,g3,g4}.

Next we recall some basic properties of variable exponent spaces.
Let $P(\Omega)$ be the set of all Lebesgue measurable functions
$p:\Omega\to(1,\infty)$. Given $p\in P(\Omega)$ we define
the conjugate function $p'(x)\in P(\Omega)$ by
$$
p'(x)=\frac{p(x)}{p(x)-1},\quad  x\in\Omega.
$$

  We define the variable exponent Lebesgue spaces $L^{p(x)}(\Omega)$ by
$$
L^{p(x)}(\Omega)=\{u\in P(\Omega):\int_{\Omega}|u|^{p(x)}dx<\infty\}.
$$
with the norm
\begin{equation}
\|u\|_{L^{p(x)}(\Omega)}=\inf \{t>0:\int_{\Omega}\Big|
\frac{u}{t}\Big|^{p(x)}dx\leq1\}.\label{e2.2}
\end{equation}

\begin{definition}[\cite{d2}] \label{def2.1}\rm
A function $a:\Omega\to \mathbb{R}$ is globally log-H\"{o}lder continuous 
in $\Omega$ if there exist  $L_{i}>0$ $(i=1,2)$ and 
$a_{\infty}\in \mathbb{R}^{n}$
such that
$$
|a(x)-a(y)|\leq \frac{L_1}{\log (e+1/|x-y|)},\quad
|a(x)-a_\infty|\leq \frac{L_2}{\log (e+|x|)}
$$
hold for all $x,y\in \Omega$. We define the following class of
variable exponents
$$
P^{\rm log} (\Omega)=\big\{p\in P(\Omega): \frac{1}{p} 
\text{ is globally log-H\"older continuous }\}.
$$
\end{definition}

 \begin{theorem}[\cite{d1}] \label{thm2.1}
 If $p(x)\in P(\Omega)$, then the inequality
$$
\int_{\Omega}|uv|dx\leq 2\|u\|_{L^{p(x)}(\Omega)}\|v\|_{L^{p'(x)}(\Omega)}
$$
holds for every $u\in L^{p(x)}(\Omega)$, 
$v\in  L^{p'(x)}(\Omega)$.
\end{theorem}

\begin{theorem}[\cite{d1}] \label{thm2.2}
If $p(x)\in P(\Omega)$, then space $L^{p(x)}(\Omega)$ is complete and
reflexive.
\end{theorem}

\begin{remark} \label{rmk2.1} \rm
We shall say that $f_n\in L^{p(x)}(\Omega)$
converge modularly to $f\in L^{p(x)}(\Omega)$ if
$\lim_{n\to\infty}\int_{\Omega}|f_n-f|^{p(x)}dx=0$.
 In \cite{k2} it is shown that the topology of $L^{p(x)}(\Omega)$ given
 by the norm \eqref{e2.3} coincides with topology of modular convergence.
\end{remark}


\subsection*{Variable exponent spaces of
Clifford-valued functions}

In this section, we first recall some notation of variable exponent
spaces of Clifford-valued functions, for a detailed treat we refer
to \cite{f3,f4}.

We define the space
$$
L^{p(x)}(\Omega, \mathrm{C}\ell_n)=\{u \in \mathrm{C}\ell_n: 
u = \sum_{I}u_{I}e_{I}, u_{I}\in L^{p(x)}(\Omega)\}
$$
with the norm
$$
\|u\|_{L^{p(x)}(\Omega, \mathrm{C}\ell_n)}
=\big\|\sum_{I}u_{I}e_{I}\big\|_{L^{p(x)}(\Omega, \mathrm{C}\ell_n)}
=\sum_{I}\|u_{I}\|_{L^{p(x)}(\Omega)}
$$
and the Sobolev space 
$$
W^{1,p(x)}(\Omega, \mathrm{C}\ell_n)=\{u \in L^{p(x)}(\Omega, 
\mathrm{C}\ell_n): \nabla u \in (L^{p(x)}(\Omega, \mathrm{C}\ell_n))^{n}\}
$$
with the norm
\begin{equation}
\|u\|_{W^{1,p(x)}(\Omega, \mathrm{C}\ell_n)}
=\|u\|_{L^{p(x)}(\Omega, \mathrm{C}\ell_n)}
+\|\nabla u\|_{({L^{p(x)}(\Omega, \mathrm{C}\ell_n)})^{n}}\label{e2.3}
\end{equation}

By $C^{\infty}(\Omega, \mathrm{C}\ell_n)$ denote the space of
Clifford-valued functions in $\Omega$ whose coefficients are
infinitely differentiable in $\Omega$ and by 
$C^{\infty}_0(\Omega,\mathrm{C}\ell_n)$
denote the subspace of $C^{\infty}(\Omega, \mathrm{C}\ell_n)$ with 
compact support in $\Omega$. Denote $W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$ 
by the closure of $C^{\infty}_0(\Omega, \mathrm{C}\ell_n)$ in 
$W^{1, p(x)}(\Omega, \mathrm{C}\ell_n)$ with respect to the norm \eqref{e2.3}.

\begin{remark}[\cite{f3}] \label{rmk2.2} \rm
A simple computation shows that
$$
2^{-\frac{n(1+p_{+})}{p_{-}}}\||u|\|_{{L^{p(x)}(\Omega)}} 
\leq \|u\|_{L^{p(x)}(\Omega,Cl_n)} \leq 2^{n}\||u|\|_{{L^{p(x)}(\Omega)}},
$$
from which we can obtain that
$\|u\|_{L^{p(x)}(\Omega,\mathrm{C}\ell_n)}$ and 
$\||u\||_{L^{p(x)}(\Omega)}$ are
equivalent norms on $L^{p(x)}(\Omega, \mathrm{C}\ell_n)$.
\end{remark}

\begin{theorem}[\cite{f3}] \label{thm2.3}
 If $p(x)\in P(\Omega)$, then the inequality 
$$
\int_{\Omega}|uv|dx \leq C(p,n)\|u\|_{L^{p(x)}(\Omega,
\mathrm{C}\ell_n)}\|v\|_{L^{p'(x)}(\Omega, \mathrm{C}\ell_n)}
$$
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{theorem}

\begin{theorem}[\cite{f4}] \label{thm2.4}
 If $p(x)\in P(\Omega)$, then $W^{1,p(x)}(\Omega, \mathrm{C}\ell_n)$ is a
reflexive Banach space. 
\end{theorem}

\begin{theorem}[\cite{f4}] \label{thm2.5}
 If $u \in W_0^{1,p(x)}(\Omega,Cl_n)$, then 
$$
\| u\|_{L^{p(x)}(\Omega,
Cl_n)}\leq C(n,\Omega) \|\partial u\|_{L^{p(x)}(\Omega,Cl_n)}.
$$
\end{theorem}

\begin{theorem}[\cite{f4}] \label{thm2.6}
 If $p(x)$ satisfies \eqref{e1.1} and 
$u \in W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$, then the norms
$\big|\big|u\big|\big|_{W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)}$
and $\big|\big|Du\big|\big|_{L^{p(x)}(\Omega, \mathrm{C}\ell_n)}$
are equivalent
 on $W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$.
\end{theorem}
 
\begin{proof}  By  \cite[Remark 2.3]{f4}, we know that
$\big|\big|\nabla u\big|\big|_{({L^{p(x)}(\Omega, Cl_n)})^{n}}$ is
equivalent to $\big|\big|Du\big|\big|_{L^{p(x)}(\Omega, Cl_n)}$
for $u \in W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$. According to
Theorem \ref{thm2.5}, the norms
$\big|\big|u\big|\big|_{W^{1,p(x)}_0(\Omega,
\mathrm{C}\ell_n)}$ and $\big|\big|\nabla
u\big|\big|_{({L^{p(x)}(\Omega, Cl_n)})^{n}}$ are equivalent
on $W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$. Thus we obtain the
desired conclusion.
\end{proof}

\section{Weak solutions for obstacle problems for A-Dirac
equations}

In this section we will establish the existence of weak solutions
for obstacle problems for $A$-Dirac equations with variable growth.
As a corollary, the existence of weak solutions to the scalar part
of A-Dirac equations is obtained. We first introduce a theorem of
Kinderlehrer and Stampacchia.

Let $X$ be a reflexive Banach space with dual $X^{\ast}$ and let
$\langle \cdot,\cdot\rangle$ denote a pairing between $X$ and
$X^{\ast}$. If $K \subset X$ is a closed convex set, then a mapping
$A:K \to X^{\ast}$ is called monotone if
$$
\langle Au-Av, u-v\rangle \geq 0
$$
for all $u,v \in K$. Further $A$ is called coercive on $K$ if there
exists $\varphi \in K$ such that
$$
\frac{\langle Au_n-A\varphi, u_n-\varphi\rangle}{\|u_n-\varphi\|_{X}} \to \infty
$$
whenever $\{u_n\} \subset K$ with $\|u_n-\varphi\|_{X} \to \infty$ as 
$n \to \infty$.
Moreover $A$ is called strongly-weakly continuous on $K$ if 
$u_n \to \varphi$ in $K$, then $Au_n \to A\varphi$  weakly in $X^{\ast}$.

\begin{proposition}[\cite{k1}] \label{prop3.1}
Let $K$ be a nonempty closed convex subset of $X$ and
let $A:K \to X^{\ast}$ be monotone, coercive and
strongly-weakly continuous on $K$. Then there exists an element $u
\in K$ such that
$$\langle Au, v-u\rangle \geq 0$$
for all $v \in K$. 
\end{proposition}

 In the following discuss we set 
$X = W_0^{1,p(x)}(\Omega, \mathrm{C}\ell_n), K=K_{\psi}$ and let
 $\langle \cdot,\cdot\rangle$ be
the usual pairing between $X$ and $X^{\ast}$; i.e.,
$$
\langle u,v\rangle = \int _{\Omega}\big[\overline{u} v\big]_0dx,
$$
where $u \in X, v \in X^{\ast}$. By the definition of $K_{\psi}$, 
it is immediate to obtain the following lemma.

\begin{lemma} \label{lem3.1}
 $K$ is a closed convex set in $X$. 
\end{lemma}

Next we define a mapping $T:K \to X^{\ast}$ by
$$
\langle Tu,v\rangle = \int _{\Omega}\big[\overline{A(x,Du)}Dv\big]_0
$$
for $v \in X$.

\begin{lemma} \label{lem3.2}
 For any $u \in K$, we have $Tu \in X^{\ast}$. 
\end{lemma}

\begin{proof}
 In view of (A2), \eqref{e2.1} and the H\"{o}lder inequality, we obtain
\begin{align*}
\big|\int _{\Omega}\big[\overline{A(x,Du)}Dv\big]_0\big| 
&\leq \int _{\Omega}\big|\overline{A(x,Du)}Dv\big|\\
&\leq C_{3}\int _{\Omega}\big(C_1|Du|^{p(x)-1} + g(x)\big)|Dv|\\
&\leq 2C_1C_{3} \big\|\,|Du|^{p(x)-1} \big\|_{L^{p'(x)}(\Omega)}\||Dv|\|_{L^{p(x)}(\Omega)}\\
&\quad +2C_{3}\|g\|_{L^{p'(x)}(\Omega)}\||Dv\||_{L^{p(x)}(\Omega)}.
\end{align*}
Moreover,
\begin{align*}
\|\,| Du|^{p(x)-1} \big\|_{L^{p'(x)}(\Omega)} 
&=\inf  \Big\{ t>0: \int _{\Omega}\frac{|Du|^{p(x)}}{t^{p'(x)}}dx \leq 1\Big\}\\
&= \inf  \Big\{ t>0: \int _{\Omega} 
 \Big( \frac{|Du|}{\lambda^{\frac{1}{p(x)-1}}}\Big)^{p(x)}dx \leq 1\Big\}\\
&\leq \mathrm{max} \Big\{ \||Du\||^{p_{+}-1}_{L^{p(x)}(\Omega)}, 
\||Du\||^{p_{-}-1}_{L^{p(x)}(\Omega)}\Big\}.
\end{align*}
Then the assertion immediately follows from Remark \ref{rmk2.2}
 and Theorem \ref{thm2.6}.
\end{proof}

\begin{lemma} \label{lem3.3}
 $T$ is monotone and coercive on $K$.
\end{lemma}

\begin{proof}
 In view of (A4), it is immediate that $T$ is monotone.
Next we show that $T$ is coercive. Given $\varphi \in K$. Then by
(A2), (A3) and \eqref{e2.1}, we obtain
\begin{align*}
&\langle Tu-T\varphi, u-\varphi \rangle\\
&\geq C_2\int _{\Omega}|Du|^{p(x)}dx + C_2\int _{\Omega}|D\varphi|^{p(x)}dx - 2\int _{\Omega}|h|dx-C_{3}\int _{\Omega}|Du\|g|dx \\
&\quad -C_{3}\int _{\Omega}|D\varphi\|g|dx - C_1C_{3}\int
_{\Omega}|Du|^{p(x)-1}|D\varphi|dx - C_1C_{3}\int
_{\Omega}|Du\|D\varphi|^{p(x)-1}dx
\\
&\geq C_2\int _{\Omega}|Du|^{p(x)}dx + C_2\int _{\Omega}|D\varphi|^{p(x)}dx - 2\int _{\Omega}|h|dx\\
&\quad -C_{3}\varepsilon \int _{\Omega}\frac{1}{p(x)}|Du|^{p(x)}dx - C_{3}\int _{\Omega}\frac{1}{p'(x)} \varepsilon^{\frac{1}{1-p'(x)}}|g|^{p'(x)}dx -C_{3}\int _{\Omega}|D\varphi\|g|dx\\
&\quad -\varepsilon C_1C_{3} \int _{\Omega}\frac{1}{p'(x)}|Du|^{p(x)}dx - C_1C_{3} \int _{\Omega}\frac{1}{p(x)} \varepsilon^{\frac{1}{1-p'(x)}}|D\varphi|^{p(x)}dx\\
&\quad -\varepsilon C_1C_{3} \int _{\Omega}\frac{1}{p(x)}|Du|^{p(x)}dx - C_1C_{3} \int _{\Omega}\frac{1}{p'(x)} \varepsilon^{\frac{1}{1-p'(x)}}|D\varphi|^{p(x)}dx\\
&\geq \Big( C_2 - C_{3}\Big( \frac{1}{p_{-}} +
C_1\Big)\varepsilon\Big)\int _{\Omega}|Du|^{p(x)}dx - C_{5}.
\end{align*}
Taking $\varepsilon = \frac{C_2p_{-}}{2C_{3}(1+C_1p_{-})}$, we
have
\begin{align*}
\langle Tu-T\varphi, u-\varphi \rangle
& \geq \frac{C_2}{2}\int _{\Omega}|Du|^{p(x)}dx - C_{5}\\
& \geq \frac{C_2}{2}\int _{\Omega} 2^{-p_{+}}\Big( |Du-D\varphi|^{p(x)} 
 - |D\varphi|^{p(x)}\Big)dx - C_{5}\\
& \geq \frac{C_2}{2^{1+p_{+}}}\int _{\Omega}|Du-D\varphi|^{p(x)}dx
- C_{6}.
\end{align*}
Since
\begin{align*}
&\frac{\int _{\Omega}\big| Du-D\varphi \big|^{p(x)}dx}
{\||Du-D\varphi|\|_{L^{p(x)}(\Omega)}}\\
&= \int _{\Omega}\big( \frac{\big| Du-D\varphi  \big|}{
2^{-1}\||Du-D\varphi |\|_{L^{p(x)}(\Omega)}}\big)^{p(x)}
\cdot \frac{\big(2^{-1}\||Du-D\varphi |\|_{L^{p(x)}(\Omega)}
\big)^{p(x)}}{\||Du-D\varphi |\|_{L^{p(x)}(\Omega)}}dx,
\end{align*}
 as $\||Du-D\varphi |\|_{L^{p(x)}(\Omega)} \geq 1$
we have
$$
\frac{\int _{\Omega}\big| Du-D\varphi  \big|^{p(x)}dx}
{{\||Du-D\varphi |\|}_{L^{p(x)}(\Omega)}} 
\geq 2^{-p_{+}}\||Du-D\varphi |\|^{p_{-}-1}_{L^{p(x)}(\Omega)}.
$$
Therefore, in view of Remark \ref{rmk2.2} and Theorem \ref{thm2.6}, we obtain 
$$
\frac{\langle Tu-Tv, u-v \rangle}{\|u-\varphi\|_{X}} \to \infty
$$
as $\|u-\varphi\|_{X} \to \infty$. That is to say, $T$ is coercive on $K$.  
\end{proof}

\begin{lemma} \label{lem3.4}
The operator  $T$ is strongly-weakly
continuous on $K$. 
\end{lemma}

\begin{proof}
 Let $\{ u_{k}(x)\} \subset K$ be a sequence that
converges to an element $u(x)\in K$ in $X$. Then $\{u_{k}\}$ is
uniformly bounded in $X$. Moreover, by (A1) we can deduce that for
each $v\in X$
$$
\big[\overline{A(x,Du_{k})}Dv\big]_0\to
\big[\overline{A(x,Du)}Dv\big]_0\quad  \text{a.e. on }
\Omega,\text{ as } k\to\infty.
$$ 
To see the equi-continuous integrability of the sequence
$\big\{[\overline{A(x,Du_{k})}Dv]_0\big\}$, we take a measurable
subset $\Omega'\subset\Omega$, by \eqref{e2.1} and (A2),  for
each $v\in X$, we have
\begin{equation}
\begin{aligned}
&\big|\int _{\Omega'}\big[\overline{A(x,Du_{k})}Dv\big]_0\big|\\
&\leq 2C_1\big(C_1 \big\| \,|Du_{k}|^{p(x)-1}\big\|_{L^{p'(x)}(\Omega')}
+\|g\|_{L^{p'(x)}(\Omega')}\big)\||Dv\||_{L^{p(x)}(\Omega')}.
\end{aligned} \label{e3.1}
\end{equation}
In view of Remark \ref{rmk2.1}, Remark \ref{rmk2.2} and 
Theorem \ref{thm2.6}, we obtain that
the first term of \eqref{e3.1} is uniformly bounded in $k$. The second term
of \eqref{e3.1} is arbitrarily small if the measure of $\Omega'$ is
chosen small enough. By the Vitali convergence theorem, we have
$$
\langle Tu_{k}, v\rangle
=\int_{\Omega}\big[ \overline{A(x, Du_{k})}Dv\big]_0
\;\to\; \int_{\Omega}\big[ \overline{A(x, Du)}Dv\big]_0
=\langle Tu, v\rangle
$$
as $k\to\infty$. That is to say, $T$ is strongly-weakly continuous.
\end{proof}

\begin{theorem} \label{thm3.1}
Suppose $K \neq \emptyset$.
Under conditions {\rm (A1)-(A4)}, there exists a Clif\-ford-valued solution
$u \in K$ for the obstacle problems \eqref{e1.1}-\eqref{e1.2}. That is to say,
there exists a Clifford-valued function $u \in K$ such that
$$
\int_{\Omega}\big[ \overline{A(x, Du)}D(v-u)\big]_0 \geq 0
$$
for any $v \in K$. Moreover, the solution to the scalar part of
\eqref{e1.1} and \eqref{e1.2} is unique up to a monogenic function, namely, if
$u_1,u_2 \in K$ are solutions to the obstacle problem
\eqref{e1.1}--\eqref{e1.2}, then $[Du_1]_0=
[Du_2]_0$ on $\Omega$. 
\end{theorem}

\begin{proof}
Using Lemma \ref{lem3.1}, Lemma \ref{lem3.4} and 
Proposition \ref{prop3.1}, it is
immediate to obtain the existence of weak solutions for the obstacle
problems \eqref{e1.1}--\eqref{e1.2}.   If there are two solutions
 $u_1,u_2 \in K$ to the obstacle problem \eqref{e1.1}--\eqref{e1.2}, then
\begin{gather*}
\int_{\Omega}\Big[ \overline{A(x, Du_1)}D(u_2-u_1)\Big]_0dx \geq 0,
\\
\int_{\Omega}\Big[ \overline{A(x, Du_2)}D(u_1-u_2)\Big]_0dx \geq 0
\end{gather*}
So we have
$$
\int_{\Omega}\Big[ \overline{A(x, Du_1) - A(x, Du_2)}D(u_1-u_2)\Big]_0dx \leq 0
$$
According to (A4), we can infer that
$$
\int_{\Omega}\Big[ \overline{A(x, Du_1) - A(x, Du_2)}D(u_1-u_2)\Big]_0dx = 0
\quad\text{on }\Omega.
$$
That is to say, $[Du_1]_0 = [Du_2]_0$ on $\Omega$.
\end{proof}

\begin{corollary} \label{coro3.1}
 Under the conditions  in Theorem \ref{thm3.1}, 
there exists one weak solution $u \in X$ to the scalar
part of $DA(x,Du)=0$. Namely, there exists at least one $u \in X$
satisfying
\begin{equation}
\int_{\Omega}\big[ \overline{A(x, Du)}D\varphi\big]_0 = 0\label{e3.2}
\end{equation}
for any $\varphi \in W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$.
\end{corollary}

\begin{proof} 
Let $\psi = \sum_{I}\psi_{I}e_{I}$, where $\psi_{I}= -\infty$ for any $I$. 
Let $u$ be a solution for the obstacle
problem of \eqref{e1.1}--\eqref{e1.2} in $K$. Since the Clifford-valued functions
$u-\varphi \in K, u+\varphi \in K$ for any $\varphi \in
W^{1,p(x)}_0(\Omega, \mathrm{C}\ell_n)$, we have
\begin{gather*}
\int_{\Omega}\big[ \overline{A(x, Du)}D\varphi \big]_0 \geq 0,
\\
- \int_{\Omega}\big[ \overline{A(x, Du)}D\varphi \big]_0 \geq 0.
\end{gather*}
Thus
$$
\int_{\Omega}\big[ \overline{A(x, Du)}D\varphi \big]_0 = 0.
$$
The proof is complete.
\end{proof}

\subsection*{Example} 
If $A(x,\xi)=\xi$, then $A$-Dirac equations
$DA(x,Du)=0$ becomes $-\triangle u=0,$ that is, Clifford Laplacian
equation. If $A(x,\xi)=|\xi|^{p-2}\xi$, then $A$-Dirac equations
becomes $D(|Du|^{p-2}Du)=0$, that is, $p$-Dirac equation (see \cite{n3}).
Moreover, if $u$ is real-valued function, then the scalar part of
$A$-Dirac equations is A-harmonic equations
$-\operatorname{div}(A(x,\nabla u))=0$ (see \cite{n1,n2}). Therefore, by
Corollary \ref{coro3.1},
we obtain the existence of a weak solution of the A-harmonic equations
 under the required conditions as in Theorem \ref{thm3.1}.


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\end{document}