p-1>0$. \begin{thm}[Liouville Type Property] Suppose $u \in C^{2}(\mathbb{R}^{n})$ is a solution of (1.2). Furthermore $u$ doesn't change sign. Then $u$ is a constant (hence,$u\equiv1$,or $u\equiv0$,or $u\equiv-1$). \end{thm} \begin{thm}[Global Boundedness] Suppose $u\in C^{2}(\mathbb{R}^{n})$ is a changing-sign solution of (1.2). Then $u\in L^{\infty}(\mathbb{R}^{n})$, moreover $|u|<1$ on $\mathbb{R}^{n}$. \end{thm} \begin{thm}[One-dimensional Property] Suppose that $u\in C^{2}(\mathbb{R}^{n})$ solves (1.2) on $\mathbb{R}^{n}$ and $|Du|>0$. If $u^{-1}(0)$ lies on one side of a hyperplane and touches that hyperplane, i.e., there exists $\nu \in S^{n-1}$ and $x_{0}\in u^{-1}(0)$ such that $\nu \cdot (x-x_{0})\geq 0$ for all $x\in u^{-1}(0)$,then $u$ depends on one variable only (in the direction of $\nu$). \end{thm} Similar results to our Theorems 1.1 and 1.2 are obtained by Dancer and Du \cite{DD}, Du and Gu \cite{DGu} by different methods. Now we compare our results with the very interesting works of P.Pucci, J.Serrin, and H.Zou \cite{PS, PSZ, SZ}. In their papers \cite{PS, PSZ}, the aim is to find conditions which make the Maximum Principle to be true. So they have to assume the behavior at infinity or at some point of solutions. Since one of our aim is to get Liouville type result, we need only to use the Comparison Principles (see Theorem 2.1-2.4 below). In the paper \cite{SZ}, the authors consider the radial symmetry of the solutions with the assumption about the behavior at infinity of the solutions. But in our Theorem 1.3, we study the one dimensional property of solutions under different conditions of the solutions. Throughout this paper, for simplicity, we assume that $u\in C^{2}(\mathbb{R}^{N})$. The rest of this paper is organized as follows. Some preliminary results are given in section2. In section 3, we prove Theorem 1.1. Theorem 1.2 is proved in section 4. In section 5, we prove two lemmas which are needed in the proof of Theorem 1.3. Theorem 1.3 is proved in section 6. \section{Preliminary Results} \label{ev2} In this section, we collect the related weak and strong comparison principles. Let $\Omega$ be a domain in $\mathbb{R}^{N},N\geq 2$, and let $u,v\in C^{2}(\Omega)$ be solutions of \begin{equation} \begin{gathered} -\Delta_{p}u \leq f(u) \quad\text{in $\Omega$} \\ -\Delta_{p}v \geq f(v) \quad\text{in $\Omega$} \end{gathered} \end{equation} For a set $A \subseteq\Omega$ we define \begin{equation} \begin{aligned} M_{A}&=M_{A}(u,v)&=\sup_{A}(|Du|+|Dv|) \\ m_{A}&=m_{A}(u,v)&=\inf_{A}(|Du|+|Dv|) \end{aligned} \end{equation} Firstly we state the weak maximum principles. \begin{thm}[Weak Comparison Principle] Let $u,v$ be solutions of (2.1) in a bounded domain $\Omega$ and $f \in C[0,\infty),f(0)=0$ and $f$ is non-decreasing on some interval $[0,\delta]$. Suppose also that $u$ and $v$ are continuous in $D$, with $v<\delta$ in $\Omega$ and $u\geq v$ on $\partial \Omega$. Then $u\geq v$ in $\Omega$. \end{thm} \begin{thm}[Weak Comparison Principle] Let $u,v$ be respective solutions of (2.1) in D(maybe unbounded). Suppose that $u$ and $v$ are continuous in $D$,that $m_{D}>0$,and that $u\geq v$ on $\partial D$. Then $u\geq v$ in $D$ \end{thm} \begin{thm}[Weak Comparison Principle] Suppose that $1

0$, depending on
$p$, $|\Omega|$, $M_{\Omega}$ and the $L^{\infty}$ norms of $u$ and $v$
such that: if an open set
$\Omega ' \subseteq \Omega$ satisfies
$\Omega '=A_{1}\cup A_{2},|A_{1}\cap A_{2}|=0,|A_{1}<\alpha|,M_{A_{2}} 1$ is a constant and $m>p-1$. Then
$u$ must be a constant.
\end{thm}
The basic ingredients in the proof consist of the following three
lemmas. For use in later sections and possible future
applications,these lemmas are given in much more general form than
what is required in the proof of Theorem 3.1.
We consider the problem
\begin{equation}
\Delta_{p}u+\alpha(x) u^{p-1}-\beta(x)u^{m}=0 \quad \text{on
$\mathbb{R}^{n}$}
\end{equation}
Here $p>1$ is a constant and $m>p-1$.
\begin{lem}[Comparison Principle]
Suppose that $\Omega$ is a
bounded domain in $\mathbb{R}^{N}$, $\alpha(x)$ and $\beta(x)$ are
continuous functions on $\Omega$ with $\|\alpha\|_{\infty}<\infty$
and $\beta(x)$ positive, $p>2$. Let $u_{1},u_{2}\in
C^{2}(\Omega)$ be positive in $\Omega$ and satisfy
\begin{equation}
\Delta_{p}u_{1}+\alpha(x)u_{1}^{p-1}-\beta(x)u_{1}^{m}\leq 0 \leq
\Delta_{p}u_{2}+\alpha(x)u_{2}^{p-1}-\beta(x)u_{2}^{m},x\in \Omega
\end{equation}
and $\limsup_{x\rightarrow \partial \Omega}(u_{2}-u_{1})\leq 0$,
and $\alpha(x)\leq \beta(x)$. Then $u_{2}\leq u_{1}$ in $\Omega$
\end{lem}
\begin{proof}
Let $\varepsilon_{1}>\varepsilon_{2}>0$ and denote
$w_{i}=(u_{i}+\varepsilon_{i})^{-1}((u_{2}+\varepsilon_{2})^{2}-(u_{1}+\varepsilon_{1})^{2})_{+}(i=1,2)$.
Observe $w_{i}$ be $C^{2}$ nonnegative functions on $\Omega$ and
vanishing near $\partial \Omega$. Using (3.2), applying
integration by parts and subtracting, we obtain
\begin{equation}
\begin{aligned}
&-\int_{\Omega}[|\nabla u_{2}|^{p-2}\nabla u_{2}\nabla
w_{2}-|\nabla u_{1}|^{p-2}\nabla u_{1}\nabla w_{1}]dx \\
&\geq \int_{\Omega}\beta(x)[u_{2}^{m}w_{2}-u_{1}^{m}w_{2}]+\int_{\Omega}\alpha(x)(u_{1}^{p-1}w_{1}-u_{2}^{p-1}w_{2})
\end{aligned}
\end{equation}
Denote $\Omega_{+}(\varepsilon_{1},\varepsilon_{2})=\{x\in
\Omega:u_{2}(x)+\varepsilon_{2}>u_{1}(x)+\varepsilon_{1}\}$ and
note that the integrands in (3.3) vanishing outside this set. The
left side of (3.3) equals
\begin{equation}
\begin{aligned}
&-\int_{\Omega_{+}(\varepsilon_{1},\varepsilon_{2})}[|\nabla u_{1}
|^{p-2}|\nabla
u_{2}-\frac{u_{2}+\epsilon_{2}}{u_{1}+\varepsilon_{1}}\nabla
u_{1}|^{2}+|\nabla u_{2}|^{p-2}|\nabla
u_{1}-\frac{u_{2}+\epsilon_{2}}{u_{1}+\varepsilon_{1}}\nabla
u_{2}|^{2}]\\
&-\int_{\Omega_{+}(\varepsilon_{1},\varepsilon_{2})}(|\nabla
u_{2}|^{p-2}-|\nabla u_{1}|^{p-2})(\nabla u_{2}\nabla u_{2}-\nabla
u_{1}\nabla u_{1})dx
\end{aligned}
\end{equation}
Noting that $w_{1}>w_{2}$ in
$\Omega_{+}(\varepsilon_{1},\varepsilon_{2})$. We conclude that
the left side of (3.4) is not positive. On the other hand as
$\varepsilon_{1}\rightarrow 0$ the right side of (3.3) converges
to
$$
\int_{\Omega_{+}(0,0)}[\beta(x)(u_{2}^{m-1}-u_{1}^{m-1})-\alpha(x)(u_{2}^{p-1}-u_{1}^{p-1})](u^{2}_{2}-u^{2}_{1})
$$
while last term in (3.3) converge to $0$. Unless $\Omega_{+}(0,0)$
is empty, the limiting value of the right side of (3.3) is
positive.
Since this leads to a contradiction we conclude that $u_{2}\leq
u_{1}$in $\Omega$
\end{proof}
\begin{lem}[Locally uniformly Boundedness]
$u\in C^{2}$ is a positive solution of (3.1). Then we have the bound
$$
\max_{G}u(x)\leq c_{0}
$$
For every compact subset $G\subset \mathbb{R}^{n}$ and $c_{0}$ is a
constant.
\end{lem}
\begin{proof}
Suppose that $max_{G}u(x)=u_{x_{0}}$ for some $x_{0}\in
G$. If $|Du(x_{0})|=0$, Then $u\leq \max_{G}(\alpha(x)/\beta(x))^{m-p+1}$.
Otherwise, we may assume
that there is a ball $B_{2r}:=B_{2r}(x_{0})\subset \mathbb{R}^{n}$with
center $x_{0}\subset G$ such that
$$
\max_{\bar{G}}=\max_{B_{r}}u(x):=M(r) \quad
\mbox{and}\quad
\min_{B_{2r}}|Du|>0
$$
Since,on $B_{2r}$, we have
$$
\Delta_{p}u \geq -\alpha(x)u
$$
Then as pointed out in \cite{DM1}, $u$ locally uniformly bounded.
\end{proof}
\begin{lem}
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with
smooth boundary. Suppose $\alpha$ and $\beta$ are smooth positive
functions on $\bar{\Omega}$, and let $\mu_{1}$ denote the first
eigenvalue of $-\Delta_{p}u=\mu \alpha(x)u^{p-1}$ on $\Omega$
under Dirichlet boundary conditions on $\partial \Omega$. Then the
problem
$$
-\Delta_{p}u=\mu u[\alpha (x)u^{p-2}-\beta(x)u^{m-1}],u|_{\partial
\Omega}=0
$$
has a unique positive solution for every $\mu>\mu_{1}$,and the
unique positive solution $u_{\mu}$ satisfies $u_{\mu}\rightarrow
[\alpha(x)/\beta(x)]^{1/(m-p+1)}$
\end{lem}
\begin{proof}
The existence from a simple upper and lower
solution argument. clearly any constant greater that or equal to
$M=\max_{\bar{\Omega}}[\alpha(x)/\beta(x)]^{1/(m-p+1)}$ is an upper
solution. Let $\phi$ be a positive eigenfunction corresponding to
$\mu_{1}$, then for each fixed $\mu>\mu_{1}$ and all small
positive $\epsilon,\epsilon \phi__0\}$. If $f$
is a function defined in $\Theta$, set $f_{\sigma}(x)=f(x+\sigma
\xi)$ for $x\in \Theta_{\sigma}$. Assume that $\sigma$ is a
sufficiently small positive number so that
$\Theta_{\sigma}\subset\subset\Omega$. Then
$v_{1,\sigma}+w_{\sigma}$ is a supersolution in $\Theta_{\sigma}$
and hence $v_{1,\sigma}+w_{\sigma}>u$ there. On the other hand,
by(3.11), $v_{2,-\sigma} 0$,
$$
\lim_{x\rightarrow \partial
\Omega}[(1+\epsilon)u_{1}-u_{2}]=\infty
$$
As $(1+\epsilon)u_{1}$ is an upper solution to (3.10), we can
apply comparison principle to conclude that $(1+\epsilon)u_{1}\geq
u_{2}$ on $\Omega$. As $\epsilon >0$ is arbitrary, we deduce
$u_{2}\geq u_{1}$. Thus $u_{1}=u_{2}$ on $\Omega$. This proves the
uniqueness.
\noindent 3. Asymptotic behavior. Now we know that
the positive solution $u_{\mu}$ constructed above is the unique
positive solution. Let $K$ be an arbitrary compact subset of
$\Omega,v_{0}=(\alpha/\beta)^{1/(m-p+1)}$ and $\epsilon>0$ any
small positive number satisfying $\epsilon__