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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 04, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/04\hfil Oscillation of solutions]
{Annulus oscillation criteria for  second order
 nonlinear elliptic  differential equations with damping}

\author[R. K. Zhuang\hfil EJDE-2009/04\hfilneg]
{Rong-Kun Zhuang}

\address{Rong-Kun Zhuang \newline
Department of Mathematics, Huizhou University, Huizhou 516015,
China \newline
Department of Mathematics, Sun Yat-sen
University, Guangzhou 510275,  China}
\email{rkzhuang@163.com}

\thanks{Submitted May 28, 2008. Published January 2, 2009.}
\thanks{Supported by grant 10571184 from the NNSF of China}
\subjclass[2000]{35J60, 34C10}
\keywords{Nonlinear elliptic differential equation; second order;
 \hfill\break\indent oscillation; annulus criteria}

\begin{abstract}
 We establish oscillation criteria for the second-order
 elliptic differential equation
 $$
 \nabla\cdot(A(x)\nabla y)+B^T(x)\nabla y+q(x)f(y)=e(x), \quad
 x\in\Omega,
 $$
 where $\Omega $ is an exterior domain in $\mathbb{R}^N$.  These
 criteria are different from most known ones in the sense that they
 are based on the information only on  a sequence  of annulus
 of $\Omega$, rather than on the whole exterior domain $\Omega$.
 Both the cases when $\frac{\partial b_i}{\partial x_i}$ exists
 for all $i$  and when it does not exist for some $i$ are considered.
\end{abstract}

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

\section{Introduction}

 In this paper, we consider the oscillation  of
solutions to the second-order elliptic differential equation
\begin{equation} \label{e1.1}
\nabla\cdot(A(x)\nabla y)+B^T(x)\nabla y+q(x)f(y)=e(x),
\end{equation}
where $x\in\Omega$, an exterior domain in $\mathbb{R}^N$, $\nabla
=(\frac{\partial}{\partial x_1}, \frac{\partial}{\partial x_2},
\dots,\frac{\partial}{\partial x_N})$. The following notation
will be adopted in this article: $\mathbb{R}$ and $\mathbb{R}^+$ denote
the intervals $(-\infty, +\infty), (0,+\infty)$, respectively. The
norm of $x$ is denoted by
$|x|=[\sum_{i=1}^Nx_i^2]^{1/2}$. For a positive
constant $a>0$, let
\begin{gather*}
S_a=\{x\in \mathbb{R}^N:|x|=a\},\quad
G(a,+\infty)=\{x\in \mathbb{R}^N:|x|>a\},\\
G[a,b]=\{x\in \mathbb{R}^N:a\leq|x|\leq b\}, \quad
G(a, b)=\{x\in\mathbb{R}^N:a<|x|<b\}.
\end{gather*}
For the exterior domain $\Omega$ in
$\mathbb{R}^N$, there exists a positive number $a_0$ such that
$G(a_0,+\infty)\subset\Omega$.

A function $y\in C_{\rm loc}^{2+\mu}(\Omega,\mathbb{R}), \mu\in(0,1)$
is said to be a solution of \eqref{e1.1} in $\Omega$, if $y(x)$
satisfies \eqref{e1.1} for all $x\in\Omega$. For the existence of
solutions of \eqref{e1.1}, we refer the reader to the monograph
\cite{gil}. We restrict our attention only to the nontrivial
solution $y(x)$ of \eqref{e1.1}; i.e.,  for any $a>a_0$, $\sup
\{|y(x)|:|x|>a\}>0$ . A nontrivial solution $y(x)$ of \eqref{e1.1}
is called oscillatory if the zero set $\{x:y(x)=0\}$ of $y(x)$ is
unbounded, otherwise  it is called nonoscillatory. \eqref{e1.1} is
called oscillatory if all its nontrivial solutions are
oscillatory.

In the qualitative theory of nonlinear partial differential
equations, one of the important problems is to determine whether
or not solutions of the equation under consideration are
oscillatory. For the similinear elliptic equation
\begin{equation} \label{e1.2}
\nabla\cdot(A(x)\nabla y)+q(x)f(y)=0,
\end{equation}
the oscillation theory is fully developed by many authors.
Noussair and Swanson \cite{nou} first extended the Wintner theorem
by using the following partial Riccati type transformation
equation
\begin{equation} \label{e1.3}
W(x)=-\frac{\alpha(|x|)}{f(y(x))}(A\nabla y)(x),
\end{equation}
where $\alpha\in C^2$ is an arbitrary positive function. Swanson
[3] summarized the oscillation results for \eqref{e1.2} up to
1979. For recent contributions, we refer the reader  to
\cite{xu2,xu3,xu1}. However, as far as we know that the
\eqref{e1.1}  has never been the subject of systematic
investigations.

When $N = 1$,  \eqref{e1.1} reduces to second-order
ordinary differential equations such as:
\begin{gather}
 y''(t)+q(t)f(y)=e(t),\label{e1.4}\\
 (r(t)y'(t))'+q(t)y(t)=e(t),\label{e1.5}\\
(r(t)y'(t))'+q(t)f(y)=e(t),\label{e1.6}
\end{gather}

There is a great number of papers  devoted to
\eqref{e1.4}-\eqref{e1.6}
(see, for example, \cite{phi,swa,won1} and the
references quoted therein). Some of the known oscillation criteria
are established by making use of a technique introduced by
Kartsatos \cite{kar} where it is assumed that there exists a
second derivative function ``$h(t)$'' such that $h''(t) = e(t)$ in
order to reduce \eqref{e1.4} or \eqref{e1.5} to a second order
homogeneous equation. However, these results require the
information of ``$q$'' on the entire half-line $[t_0,\infty)$.

In 1993,  El-Sayed \cite{ei} gave an interval oscillation
criterion for \eqref{e1.4} which depends only on the behavior of
``$q$'' in certain subintervals of $[t_0,\infty)$. In 1999, Wong
\cite{won2} and Kong \cite{kon} have, respectively, noted that
interval criteria which Ei-Sayed \cite{ei} established for
oscillation of (\ref{e1.5}) are not very sharp, because a
comparison with a equation of constant coefficients is used in
Ei-Sayed's proof. Therefore, some other interval criteria for
oscillation,that is, criteria given by the behavior of
(\ref{e1.5}) and (\ref{e1.5}) with $e(t)=0$ only a sequence of
subintervals of $[t_0, \infty)$ are obtained by Wong \cite{won2}
and Kong \cite{kon}, respectively.

In 2003,  Yang \cite{yan} employed the technique in the work of
Philos \cite{phi} and Kong \cite{kon} for \eqref{e1.4}, and
presented several Interval oscillation criteria for \eqref{e1.6}.
One of the oscillation criteria of Kamenev's type in \cite{yan} is
as follows.

\begin{theorem} \label{thm1.1}
Suppose $f(y)/y\geq K|y|^{\nu-1}$ for $y\neq 0$,
$K>0$ and $\nu>1$. Then \eqref{e1.4} with $r(t)\equiv 1$ is
oscillatory provided that for each $t\geq t_0 $ and for some
$\lambda>1$, the following conditions hold
\begin{enumerate}

\item For any $T\geq t_0$, there exist $T\leq a_1<b_1\leq
a_2<b_2$ such that
\[
e(t)\begin{cases}
\leq 0, &t\in [a_1,b_1],\\
\geq 0, &t\in [a_2,b_2]\end{cases}
\]
and $q(t)\geq 0$ ($\not\equiv 0$), $t\in [a_1,b_1]\cup [a_2,b_2]$

\item there exist $c_i\in(a_i,b_i)$ for $i=1,2$, such
that $T\leq a_1<b_1\leq a_2<b_2$ and the following  inequalities
hold for $i=1,2$,
\begin{gather}
\frac{1}{(c_i-a_i)^{\lambda-1}}\int_{a_i}^{c_i}(s-a_i)^\lambda
|e(s)|^{1-(1/\nu)}[Kq(s)]^{1/\nu}ds\geq
\frac{\lambda^2}{4(\lambda-1)} \label{7}
\\
\frac{1}{(b_i-c_i)^{\lambda-1}}\int_{c_i}^{b_i}(b_i-s)^\lambda
|e(s)|^{1-(1/\nu)}[Kq(s)]^{1/\nu}ds\geq
\frac{\lambda^2}{4(\lambda-1)}.  \label{8}
\end{gather}
\end{enumerate}
\end{theorem}

Motivate by the ideas of Philos \cite{phi}, Kong \cite{kon}, and
Yang \cite{yan}. In this paper, by using generalized Riccati
techniques which are introduced by Noussair \cite{nou}, we obtain
several annulus criteria for oscillation, that is, criteria given
by the behavior of \eqref{e1.1} (or of $A,q,f$ and $e$) only on a
sequence of annulus of $\Omega$ in $ \mathbb{R}^N$. Our results
improve and extend the results of Ei-Sayed \cite{ei}, Kong
\cite{kon} and Yang \cite{yan}. Also information about the
distribution of the zero of solutions for\eqref{e1.1} is obtained.

\section{Oscillation results  when
$\frac{\partial b_i}{\partial x_i}$ exists for all $i$}

To establish oscillation theorems when $\frac{\partial
b_i}{\partial x_i}$ exists for all $i$ we shall impose the
following conditions:

\begin{itemize}

\item[(C1)] $A(x)=(A_{ij}(x))_{N\times N}$ is a real symmetric
positive definite matrix function (ellipticity condition) with
$A_{ij}\in C_{\rm loc}^{1+\mu} (\Omega(a_0), \mathbb{R}), \mu\in(0,1),
i, j=1, \dots, N$,  $\lambda_{\rm max}(x)$ denotes the largest
(necessarily positive) eigenvalue of the matrix $A(x)$;  there
exists a function $\lambda\in C^1(\mathbb{R}^+, \mathbb{R}^+)$
such that $\lambda(r)\geq\max_{|x|=r}\lambda_{\rm max}(x)$ for $
r>0$;

\item[(C2)] $B^T=(b_i(x))_{1\times N}, b_i\in
C_{\rm loc}^{1+\mu}(\Omega(a_0), \mathbb{R}),i=1, \dots, N$;

\item[(C3)] $ q\in C_{\rm loc}^{\mu}(\Omega(a_0),
\mathbb{R}),\mu\in(0,1)$ and $q(x)\not \equiv 0$ for $ |x|\geq
a_0$;

\item[(C4)] $f\in C^1(\mathbb{R},\mathbb{R}),yf(y)>0$ and
$f'(y)\geq k>0$ for all $y\not=0$ and some constant $k$.
\end{itemize}
For convenience, we let
\begin{gather*}
Q_1(r)=\int_{S_r}\big[q(x)-\frac{1}{4k}B^TA^{-1}B-\frac{1}{2k}\nabla\cdot
B\big]d\sigma,
\\
g_1(r)=\frac{\omega}{k}\lambda(r)r^{N-1},
\end{gather*}
where $S_r=\{x\in \mathbb{R}^N:|x|=r\},r>0, d\sigma$ denotes the
spherical integral element in $\mathbb{R}^N$, $\omega$ is the area
of unit sphere in $\mathbb{R}^N$ and $k$ is defined in (C4).

\begin{theorem} \label{thm2.1}
 Let {\rm (C1)--(C4)} hold. Suppose that for
any $T\geq a_0$, there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$, $x\in G[a_1,b_1]\cup G[a_2,b_2]$.
Denote by $\Psi(a_i,b_i)$ the set
$$
\big\{H\in C^1[a_i,b_i],H(r)\geq 0(\not\equiv
0),H(a_i)=H(b_i)=0, H_r'=2h(r)\sqrt{H(r)}\big\},
$$
$i=1,2$. If there exist $H\in \Psi(a_i,b_i)$ such that
$$
M_i(H)=\int_{a_i}^{b_i}\Big\{g_1(s)h^2(s)-Q_1(s)H(s)\Big\}ds<0,
$$
for $i=1,2$, then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that there exists a solution $y(x)$ of
\eqref{e1.1} such that $y(x)>0$ for $|x|\geq a_1\geq a_0$. Define
\begin{gather}
W(x)=\frac{1}{f(y)}(A\nabla y)(x)+\frac{1}{2k}B,\quad x\in
G[a_1, +\infty), \label{11} \\
V(r)=\int_{S_r}W(x)\cdot\gamma(x)d\sigma,\quad x\in G[a_1,
+\infty) ,\label{12}
\end{gather}
where $\nabla y$ denotes the gradient of $y(x)$,
$\gamma(x)=\frac{x}{|x|},|x|\not=0$ is the outward unit normal to
$S_r$.
From \eqref{e1.1} and (\ref{11}), it follows that
\begin{equation}
\begin{aligned}
\nabla\cdot W(x)&= -\frac{f'(y)}{f^2(y)}(\nabla
y)^TA\nabla y-\frac{1}{f(y)}[q(x)f(y)+B^T\nabla
y-e(x)]+\frac{1}{2k}\nabla\cdot B  \\
&\leq -k[W-\frac{1}{2k}B]^TA^{-1}[W-\frac{1}{2k}B]-q(x)-B^TA^{-1}
[W-\frac{1}{2k}B]\\
&\quad +\frac{1}{2k}\nabla\cdot B +\frac{e(x)}{f(y)}\\
&= -kW^TA^{-1}W-q(x)+\frac{1}{4k}B^TA^{-1}B+\frac{1}{2k}\nabla\cdot
B+\frac{e(x)}{f(y)}.
\end{aligned} \label{13}
\end{equation}
where $W^T$ denotes the transpose of $W$.
Using Green's formula in
(\ref{12}), we obtain
\begin{equation}
\begin{aligned}
V'(r)&=\int_{S_r}\nabla\cdot W(x)d\sigma  \\
&\leq-\int_{S_r}q(x)
d\sigma+\int_{S_r}\big[\frac{1}{4k}B^TA^{-1}B
 +\frac{1}{2k}\nabla\cdot B\big]d\sigma\\
&\quad-k\int_{S_r}(W^TA^{-1}W)(x)d\sigma+\int_{S_r}\frac{e(x)}{f(y)}d\sigma.
\end{aligned}\label{14}
\end{equation}
In view of (C1), we have
$(W^TA^{-1}W)(x)\geq \lambda_{\rm max}^{-1}(x)|W(x)|^2$. Then, by Cauchy-Schwartz
inequality, we obtain
$$
\int_{S_r}|W(x)|^2d\sigma\geq\frac{r^{1-N}}{\omega}\Big[\int_{S_r}W(x)\cdot
\gamma(x)d\sigma\Big]^2.
$$
Moreover, by  (\ref{14}) and (\ref{12}), we get
\begin{equation}
\begin{aligned}
V'(r)&\leq -\int_{S_r}\Big[q(x)
-\frac{1}{4k}B^TA^{-1}B-\frac{1}{2k}\nabla\cdot
B\Big]d\sigma-\frac{1}{g_1(r)}V^2(r)+\int_{S_r}\frac{e(x)}{f(y)}d\sigma\\
&= -Q_1(r)-\frac{1}{g_1(r)}V^2(r)+\int_{S_r}\frac{e(x)}{f(y)}d\sigma.
\end{aligned}\label{15}
\end{equation}
 By the assumption, we can choose $a_1,b_1\geq
T_0(a_1<b_1)$ such that $e(x)\leq 0, x\in G[a_1,b_1]$, then we
have for $x\in G[a_1,b_1]$,
\begin{equation}
V'(r)\leq -Q_1(r)-\frac{1}{g_1(r)}V^2(r).\label{17}
\end{equation}
Let $H(r)\in \Psi(a_1,b_1)$ be given as in the hypothesis,
Multiplying $H(r)$ throughout (\ref{17}) and integrating from
$a_1$ to $b_1$, we obtain
\begin{equation}
\int_{a_1}^{b_1}H(s)V'(s)ds\leq-\int_{a_1}^{b_1}Q_1(s)H(s)ds
-\int_{a_1}^{b_1}H(s)\frac{1}{g_1(s)}V^2(s)ds.\label{18}
\end{equation}
Integrating  by parts and using the fact
$H(a_1)=H(b_1)=0$, we find
\begin{equation}
-\int_{a_1}^{b_1}2h(s)\sqrt{H(s)}V(s)ds
\leq-\int_{a_1}^{b_1}Q_1(s)H(s)ds-\int_{a_1}^{b_1}
H(s)\frac{1}{g_1(s)}V^2(s)ds.\label{19}
\end{equation}
which is equivalent to
\begin{equation}
\begin{aligned}
0&\leq -\int_{a_1}^{b_1}Q_1(s)H(s)ds+\int_{a_1}^{b_1}
 \Big[2h(s)\sqrt{H(s)}V(s)-\frac{H(s)}{g_1(s)}V^2(s)\Big]ds\\
 &= \int_{a_1}^{b_1}[g_1(s)h^2(s)-Q_1(s)H(s)]ds-\int_{a_1}^{b_1}
 \Big[\sqrt{\frac{H(s)}{g_1(s)}}V(s) -\sqrt{g_1(s)}h(s)\Big]^2ds\\
 &= M_1(H)-\int_{a_1}^{b_1}\Big[\sqrt{\frac{H(s)}{g_1(s)}}V(s)
 -\sqrt{g_1(s)}h(s)\Big]^2ds
\end{aligned}\label{20}
\end{equation}
Because $M_1(H)<0$, (\ref{20}) is incompatible. This contradiction
proves that $y(x)$ must be oscillatory.

When $y(x)$ is eventually negative, we use $H(r)\in \Psi(a_2,b_2)$
and $e(x)\geq 0, x\in G[a_2,b_2]$ to reach a similar
contradiction. the proof is complete.
\end{proof}

Following Philos \cite{phi} and Kong \cite{kon}, we introduce the
class of function $\Re$ which will be extensively and use in the
sequel.

Let $D=\{(r,s):-\infty<s\leq r<\infty\}$, a function $H=H(r,s)$ is
said to belong to $\Re$, if $H\in C(D,\mathbb{R})$ and satisfies

\begin{itemize}
\item[(H1)] $H(r,r)=0, r\geq a_0; H(r,s)>0 $ for all $r>s\geq
a_0$;

\item[(H2)] $H$ has partial derivatives $\partial H/\partial r$
and $\partial H/\partial s$ on $D$ such that:
$$
\frac{\partial H}{\partial r}=2h_1(r,s)\sqrt{H(r,s)}
\quad\mbox{บอ}\quad
\frac{\partial H}{\partial s}=-2h_2(r,s)\sqrt{H(r,s)},
$$
where $h_1,h_2\in L_{\rm loc}(D,\mathbb{R})$.
\end{itemize}

\begin{lemma} \label{lem2.2}
Let {\rm (C1)--(C4)} hold. Assume that there exist
$c_1<b_1<c_2<b_2$ such that $q(x)\geq 0$ for $x\in G[c_1, b_1]\cup
G[c_2,b_2]$ and
\[
e(x) \begin{cases}
\leq 0, &x\in G[c_1,b_1],\\
\geq 0, &x\in G[c_2,b_2],
\end{cases}
\]
 $y(x)$ is a solution of \eqref{e1.1} such that $y(x)>0$ for
$x\in G[c_1,b_1]$ and $ y(x)<0$ for $ x\in G[c_2,b_2]$.
Then for any $H\in \Re$ and $i=1,2$,
\begin{equation}
\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}H(b_i,s)Q_1(s)ds\leq
V(c_i)+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}g_1(s)h_2^2(b_i,s)ds.
\label{22}
\end{equation}
\end{lemma}

\begin{proof}
Suppose that $y(x)$ is a solution of \eqref{e1.1} such that
$y(x)>0$ for $x\in G[c_1,b_1]$ and $ y(x)<0$ for $ x\in
G[c_2,b_2]$. Then, similar to the proof of Theorem 2.1, we
multiply
 (\ref{17}) by $H(r,s)$, integrate it with respect to s from $r$ to
 $c_i$, we get for $s\in [c_i, r)$
\begin{align*}
&\int_{c_i}^{r}H(r,s)Q_1(s)ds\\
&\leq -\int_{c_i}^{r}H(r,s)V'(s)ds-\int_{c_i}^{r}H(r,s)
\frac{1}{g_1(s)}V^2(s)ds
\\
&=H(r,c_i)V(c_i)-\int_{c_i}^{r}2h_2(r,s)\sqrt{H(r,s)}V(s)ds
 -\int_{c_i}^{r}H(r,s)\frac{1}{g_1(s)}V^2(s)ds\\
&=H(r,c_i)V(c_i)+\int_{c_i}^{r}g_1(s)h_2^2(r,s)ds-\int_{c_i}^{r}
\Big[\sqrt{\frac{H(r,s)}{g_1(s)}}V(s)
+\sqrt{g_1(s)h_2^2(r,s)}\Big]^2ds\\
&\leq H(r,c_i)V(c_i)+\int_{c_i}^{r}g_1(s)h_2^2(r,s)ds
\end{align*} %\label{23}
Letting $r\to b_i^-$ and dividing both sides by
$H(b_i,c_i)$ we obtain (\ref{22}).
\end{proof}

\begin{lemma} \label{lem2.3}
Let {\rm (C1)--(C4)} hold. Assume that there exist
$a_1<c_1<a_2<c_2$ such that $q(x)\geq 0$ for $x\in G[a_1, c_1]\cup
G[a_2,c_2]$ and
\[
e(x)\begin{cases}
\leq 0, &x\in G[a_1,c_1],\\
\geq 0, &x\in G[a_2,c_2],
\end{cases}
\]
$y(x)$ is a solution of \eqref{e1.1} such that $y(x)>0$ for
$x\in G[a_1,c_1]$ and $ y(x)<0$ for $ x\in G[a_2,c_2]$.  Then for any
$H\in \Re$ and $i=1,2$,
\begin{equation}
\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}H(s,a_i)Q_1(s)ds\leq
-V(c_i)+\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}g_1(s)h_1^2(s,a_i)ds.
\label{24}
\end{equation}
\end{lemma}

\begin{proof}
As in the proof of Lemma \ref{lem2.2}, we multiply
(\ref{17}) by $H(s,r) $ and integrate it with respect to $s$ from
$r$ to $c_i$. We have
\begin{align*}
&\int_{r}^{c_i}H(s,r)Q_1(s)ds\\
&\leq -\int_{r}^{c_i}H(s,r)V'(s)ds-\int_{r}^{c_i}H(r,s)
\frac{1}{g_1(s)}V^2(s)ds
\\
&=-H(c_i,r)V(c_i)+\int_{r}^{c_i}2h_1(s,r)\sqrt{H(s,r)}V(s)ds
 -\int_{r}^{c_i}H(s,r)\frac{1}{g_1(s)}V^2(s)ds\\
&=-H(c_i,r)V(c_i)+\int_{r}^{c_i}g_1(s)h_1^2(s,r)ds\\
&\quad -\int_{c_i}^{r}\Big[\sqrt{\frac{H(s,r)}{g_1(s)}}V(s)
-\sqrt{g_1(s)h_2^2(r,s)}\Big]^2ds\\
&\leq -H(c_i,r)V(c_i)+\int_{r}^{c_i}g_1(s)h_1^2(s,r)ds
%\label{25}
\end{align*}
Letting $r\to a_i^+$ and dividing both sides by
$H(c_i,a_i)$ we obtain (\ref{24}).
\end{proof}

The following theorem is an immediate result from Lemmas
\ref{lem2.2} and \ref{lem2.3}.

\begin{theorem} \label{thm2.4} Let {\rm (C1)--(C4)} hold. Suppose that there
exist $a_1<b_1\leq a_2<b_2$ such that $q(x)\geq 0$ for $ x\in
G[a_1,b_1]\cup G[a_2,b_2]$ and
\[
e(x)\begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
further, there exist some $c_i\in (a_i, b_i)$ and some $H\in \Re$
such that
\begin{equation}
\begin{aligned}
&\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}
 [H(s,a_i)Q_1(s)-g_1(s)h_1(s,a_i)]ds
 \\
&+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}
 [H(b_i,s)Q_1(s)-g_1(s)h_2(b_i,s)]ds>0
\end{aligned} \label{26}
\end{equation}
holds for $i=1,2$, then every nontrivial solution of \eqref{e1.1}
has at least one zero either in $G(a_1,b_1)$ or in $G(a_2,b_2)$.
\end{theorem}

\begin{proof}
Suppose to the contrary that there exists a solution $y(x)$ of
\eqref{e1.1} such that $y(x)>0$ for $ x\in G[T_0, +\infty)(T_0\geq
a_0)$, by the assumption, we can choose $a_1, b_1\geq
T_0(a_1<b_1)$ such that $e(x)>0, x\in  G[a_1,b_1]$, then from
Lemma 2.2 and Lemma 2.3 we see that
 (\ref{22}) and (\ref{24}) with $i=1$ hold. Adding (\ref{22}) and
 (\ref{24}), we have that
\begin{equation}
\begin{aligned}
&\frac{1}{H(c_1,a_1)}\int_{a_1}^{c_1}
 [H(s,a_1)Q_1(s)-g_1(s)h_1(s,a_1)]ds
 \\
&+\frac{1}{H(b_1,c_1)}\int_{c_1}^{b_1}
 [H(b_1,s)Q_1(s)-g_1(s)h_2(b_1,s)]ds\leq 0. \label{27}
\end{aligned}
\end{equation}
which contradicts the assumption (\ref{26}) with $i=1$.

When $y(x)$ is eventually negative, we choose $a_2,b_2\geq T_0$
such that $e(x)\leq 0, x\in G[a_2,b_2]$ to reach a similar
contradiction and hence completes the proof.
\end{proof}

\begin{theorem}\label{thm2.5}
 Let {\rm (C1)--(C4)} hold. Suppose that for any $T\geq a_0$, the
following conditions hold:
\begin{enumerate}
\item there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for $ x\in G[a_1,b_1]\cup G[a_2,b_2]$

\item there exist some $c_i\in(a_i,b_i),i=1,2$, and some
$H\in \Re$ such that $T\leq a_1<b_1\leq a_2<b_2$ and (\ref{26})
holds.
\end{enumerate}
 Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Pick up a sequence $\{T_j\}\subset[a_0,+\infty)$, such that
$j\to\infty$, $T_j\to\infty$. By the assumption, for each $j\in
N$, there exist $a_1,b_1,c_1,a_2,b_2,c_2\in \mathbb{R}$ such that
$T_j\leq a_1<c_1<b_1\leq a_2<c_2<b_2$ and (\ref{26}) holds. From
Theorem \ref{thm2.4}, every solution $y(x)$ has at least one zero
on $G(a_1,b_1)$ or $G(a_2,b_2)$. Noting that $ |x|>a_1\geq
T_j,j\in N$, we see that the zero set $\{x\in \Omega:y(x)=0\}$ of
$y(x)$ is is unbounded. Thus, every nontrivial solution of
\eqref{e1.1} is oscillatory. The proof is complete.
\end{proof}

\begin{remark} \label{rmk1} \rm
With an appropriate choice of function $H$ one can
derive a number of oscillation criteria for \eqref{e1.1}.
\end{remark}

As an immediate consequence of Theorem \ref{thm2.5} we get the
following oscillation criteria for \eqref{e1.1}.

\begin{corollary} \label{coro2.7}
Let {\rm (C1)--(C4)} hold. Suppose that for any
$T\geq a_0$, the following conditions hold:
\begin{enumerate}
\item there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x)\begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2],
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for $ x\in G[a_1,b_1]\cup
G[a_2,b_2]$.

\item there exist some $c_i\in(a_i,b_i),i=1,2$, and some
$H\in \Re$ such that $T\leq a_1<b_1\leq a_2<b_2$ and the following
two inequalities hold for $i=1,2$,
\begin{gather}
\int_{a_i}^{c_i}[H(s,a_i)Q_1(s)-g_1(s)h_1^2(s,a_i)]ds>0, \label{28}
\\
\int_{c_i}^{b_i}\left[H(b_i,s)Q_1(s)-g_1(s)h_2^2(b_i,s)\right]ds>0.
\label{29}
\end{gather}
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

Moreover, let $H=H(r-s)\in \Re $, we have tha $\frac{\partial
H(r-s)}{\partial r}=-\frac{\partial H(r-s)}{\partial s}$, and
denote them by $h(r-s)$. The subclass of $\Re$ containing such
$H(r-s)$ is denoted by $\Re_0$. Applying Theorem \ref{thm2.5} to
$\Re_0$, we obtain the following result.

\begin{corollary}\label{coro2.8}
Let {\rm (C1)--(C4)} hold. Suppose that for any
$T\geq a_0$, the following conditions hold:
\begin{enumerate}
\item there exist $T\leq a_1<2c_1-a_1\leq a_2<2c_2-a_2$ such
that
\[
e(x)\begin{cases}
\leq 0, &x\in G[a_1,2c_1-a_1],\\
\geq 0, &x\in G[a_2,2c_2-a_2],
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for
$ x\in G[a_1,2c_1-a_1]\cup G[a_2,2c_2-a_2]$.

\item there exist some $H\in \Re_0$ such that
$T\leq a_i<c_i$ for $ i=1,2$ and the following inequality holds
\begin{equation}
\int_{a_i}^{c_i}\left\{H(s-a_i)[Q_1(s)+Q_1(2c_i-s)]
-[g_1(s)+g_1(2c_i-s)]h^2(s-a_i)\right\}ds>0. \label{30}
\end{equation}
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{proof}
Let $b_i=2c_i-a_i$, then $H(b_i-c_i)=H(c_i-a_i)=H((b_i-a_i)/2)$,
and for any $f \in L[a,b]$, we have
\[
\int_{c_i}^{b_i}H(b_i-s)f(s)ds=\int_{a_i}^{c_i}H(s-a_i)f(2c_i-s)ds.
\]
Thus that (\ref{30}) holds implies that (\ref{26}) holds for $H\in
\Phi_0$ and therefor \eqref{e1.1} is oscillatory by Theorem
\ref{thm2.4}.
\end{proof}

Define
\begin{equation}
R(r)=\int_{a_0}^r\frac{1}{g_1(s)}ds,\quad r\geq a_0, \label{31}
\end{equation}
and let
\begin{equation}
H(r,s)=[R(r)-R(s)]^\alpha, \quad r\geq s\geq a_0,\label{32}
\end{equation}
where  $\alpha>1$ is a constant. Based on the above results, we
obtain the following oscillation criteria of Kamenev's type.

\begin{theorem}\label{thm2.9}
Let {\rm (C1)--(C4)} hold. Assume that
$\lim_{r\to\infty}R(r)=\infty$. If for each $T\geq a_0$, the
following conditions hold:
\begin{enumerate}
\item there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2],
\end{cases}
\]
and $q(x)\geq 0$ ($\not\equiv 0$) for
$ x\in G[a_1,b_1]\cup G[a_2,b_2]$

\item there exist $c_i\in(a_i,b_i)$ for $i=1,2$, such that
$T\leq a_1<b_1\leq a_2<b_2$ and the following  inequalities hold
for $i=1,2$,
\begin{gather}
\frac{1}{[R(c_i)-R(a_i)]^{\alpha-1}}\int_{a_i}^{c_i}[R(s)-R(a_i)]^\alpha
Q_1(s)ds\geq \frac{\alpha^2}{4(\alpha-1)}, \label{33}
\\
\frac{1}{[R(b_i)-R(c_i)]^{\alpha-1}}\int_{c_i}^{b_i}[R(b_i)-R(s)]^\alpha
Q_1(s)ds\geq \frac{\alpha^2}{4(\alpha-1)}.  \label{34}
\end{gather}
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
It is easy to see that
\[
h_1(r,s)=\alpha[R(r)-R(s)]^{\frac{\alpha-2}{2}}\frac{1}{2g_1(r)},\quad
h_2(r,s)=\alpha[R(r)-R(s)]^{\frac{\alpha-2}{2}}\frac{1}{2g_1(s)},
\]
Hence we have
\begin{equation}
\begin{aligned}
\int_{a_i}^{c_i}g_1(s)h_1^2(s,a_i)ds
&= \int_{a_i}^{c_i}g_1(s)\alpha^2[R(s)-
R(a_i)]^{\alpha-2}\frac{1}{4g_1^2(s)}ds \\
 &= \int_{a_i}^{c_i}[R(s)-R(a_i)]^{\alpha-2}\frac{\alpha^2}{4g_1(s)}ds \\
 &= \frac{\alpha^2}{4(\alpha-1)}[R(c_i)-R(a_i)]^{\alpha-1}.\label{35}
 \end{aligned}
\end{equation}
 From (\ref{33}) and (\ref{35}) we have
\begin{equation}
 \begin{aligned}
&\frac{1}{[R(c_i)-R(a_i)]^{\alpha-1}}\int_{a_i}^{c_i}
\left[H(s,a_i)Q_1(s)-g_1(s)h_1^2(s,a_i)\right]ds\\
&= \frac{1}{[R(c_i)-R(a_i)]^{\alpha-1}}\int_{a_i}^{c_i}
[R(s)-R(a_i)]^\alpha Q_1(s)ds- \frac{\alpha^2}{4(\alpha-1)}> 0;
\end{aligned}\label{36}
\end{equation}
i.e., (\ref{28}) holds. Similarly, (\ref{34}) implies (\ref{29})
holds. From Corollary \ref{coro2.7}, \eqref{e1.1} is oscillatory.
\end{proof}

\subsection*{Example}
Consider \eqref{e1.1} with
\begin{gather*}
A=\mathop{\rm diag}\Big(\frac{1}{\sqrt{r}},\frac{1}{\sqrt{r}}\Big),\quad
B^T=\Big(-\frac{2x_1}{r^2},-\frac{2x_2}{r^2}\Big),
\\
q(x)=\frac{\alpha}{ r\sqrt{r}},\quad
f(y)=y+y^3,\quad e(x)= \frac{1}{r\sqrt{r}}\sin\sqrt{r},
\end{gather*}
where $r=\sqrt{x_1^2+x_2^2}$, $r\geq 1$, $N=2$. Let $k=1$, hence
\[
\lambda(r)=\frac{1}{\sqrt{r}},\quad
Q_1(r)=\frac{(2\alpha-1)\pi}{\sqrt{r}},\quad
g_1(r)=2\pi\sqrt{r}.
\]
Choose $a_1=n^2\pi^2$, $b_1=(n+1)^2\pi^2$, $a_2=(n+1)^2\pi^2$,
$b_2=(n+2)^2\pi^2$, and $H(r)=\sin^2\sqrt{r}$. It is easy to see
that if $\alpha\geq 3/2$, then
\begin{align*}
M_1(H)
&= \int_{a_1}^{b_1}[g_1(s)h^2(s)-Q_1(s)H(s)]ds\\
&= \pi\int_{n^2\pi^2}^{(n+1)^2\pi^2}\frac{\cos ^2\sqrt{s}}{2\sqrt{s}}ds
-(2\alpha-1)\pi\int_{n^2\pi^2}^{(n+1)^2\pi^2}
\frac{\sin ^2\sqrt{s}}{\sqrt{s}}ds
 \\
&= \pi\int_{n\pi}^{(n+1)\pi}\cos ^2sds
-\frac{2\alpha-1}{2}\int_{n\pi}^{(n+1)\pi}\sin ^2sds \\
&= \frac{\pi^2}{2}-\frac{(2\alpha-1)\pi^2}{4}\leq 0.
\end{align*}
Similarly, for $a_2,b_2$ we can show that $M_2(H)\leq 0$. It
follows from Theorem \ref{thm2.1} that \eqref{e1.1} is oscillatory
when $\alpha\geq 3/2$.

\section{Oscillation results  when
$\frac{\partial b_i}{\partial x_i}$ does not exist for some $i$}

In this section, we establish oscillation criteria for
\eqref{e1.1} in case when $\frac{\partial b_i}{\partial x_i}$ does
not exist for some $i$. For convenience, we let
\[
Q_2(r)=\int_{S_r}\big[q(x)-\frac{1}{2k}\lambda(x)|B^TA^{-1}|^2
\big]d\sigma,\quad g_2(r)
=\frac{2\lambda(r)}{k}\omega r^{N-1},
\]
We begin with the following lemma, the proof of this lemma is easy
and thus omitted.

\begin{lemma}\label{3.1}
For two $n$-dimensional vectors $u, v \in \mathbb{R}^N$, and a
positive constant $c$, then
\begin{equation} c u u^T +
u v^T \geq \frac{c}{2} u u^T-\frac{1}{2c}vv^T.\label{s6}
\end{equation}
\end{lemma}

\begin{theorem}\label{thm3.2}
Assume  {\rm (C1),(C3),(C4)} and
\begin{itemize}
\item[(C2)'] $b_i\in C_{\rm loc}^{\mu}(\Omega,\mathbb{R}),\mu\in(0,1)$,
 $i=1, \dots, N$.
\end{itemize}
Suppose that for any $T\geq a_0$, there exist $T\leq
a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$, $x\in G[a_1,b_1]\cup G[a_2,b_2]$
If there
exist $H\in \Psi(a_i,b_i)$ such that
$$
M_i(H)=\int_{a_i}^{b_i}\big\{g_2(s)h^2(s)-Q_2(s)H(s)\big\}ds<0,
 \quad \mbox{for } i=1,2,
$$
where $\Psi(a_i,b_i)$ is defined in Theorem \ref{thm2.1}.
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{proof}
Suppose to the contrary that there exists a solution $y(x)$ of
\eqref{e1.1} such that $y(x)>0$ for $|x|\geq a_1\geq a_0$. Define
\begin{gather}
W(x)=\frac{1}{f(y)}(A\nabla y)(x),\quad x\in G[a_1, +\infty), \label{s11}
\\
V(r)=\int_{S_r}W(x)\cdot\gamma(x)d\sigma,\quad x\in G[a_1,
+\infty) ,\label{s12}
\end{gather}
where $\nabla y$ denotes the gradient of $y(x)$,
$\gamma(x)=\frac{x}{|x|},|x|\not=0$ is the outward unit normal to
$S_r$.
 From \eqref{e1.1} and (\ref{s11}), it follows that
\begin{equation}
\begin{aligned}
\nabla\cdot W(x)
&= -\frac{f'(y)}{f^2(y)}(\nabla
y)^TA\nabla y-\frac{1}{f(y)}[q(x)f(y)+B^T\nabla y-e(x)]  \\
&\leq -kW^TA^{-1}W-q(x)-B^TA^{-1} W +\frac{e(x)}{f(y)}\\
&\leq -\frac{k}{\lambda(x)}W^TW-q(x)-B^TA^{-1}
W +\frac{e(x)}{f(y)}\quad(\mbox{By Lemma \ref{3.1}})\\
&\leq -\frac{k}{2\lambda(x)}|W|^2+\frac{1}{2k}\lambda(x)
 |B^TA^{-1}|^2-q(x)+\frac{e(x)}{f(y)}.
\end{aligned}\label{s13}
\end{equation}
where $W^T$ denotes the transpose of $W$.
Using Green's formula in
(\ref{s12}), we get
\begin{equation}
\begin{aligned}
V'(r)&=\int_{S_r}\nabla\cdot W(x)d\sigma  \\
&\leq-\int_{S_r}q(x)d\sigma
+\frac{1}{2k}\int_{S_r}\lambda(x)|B^TA^{-1}|^2d\sigma\\
&\quad -\frac{k}{2\lambda(r)}\int_{S_r}|W|^2d\sigma
+\int_{S_r}\frac{e(x)}{y(x)}d\sigma.
\end{aligned}\label{s14}
\end{equation}
 By Cauchy-Schwartz inequality,
$$
\int_{S_r}|W(x)|^2d\sigma\geq\frac{r^{1-N}}{\omega}
\Big[\int_{S_r}W(x)\cdot \gamma(x)d\sigma\Big]^2.
$$
Moreover, by  (\ref{s14}) and (\ref{s12}),
\begin{equation}
V'(r)\leq-\int_{S_r}\Big[q(x)-\frac{1}{2k}\lambda(x)|B^TA^{-1}|^2\Big]
d\sigma-\frac{1}{g_2(r)}V^2(r)+\int_{S_r}\frac{e(x)}{y(x)}d\sigma\label{s15}
\end{equation}
The rest of proof is similar to that of Theorem \ref{thm2.1} and
hence omitted.
\end{proof}

Similar to the discussions in Section 2, we have the following
results.

\begin{lemma}\label{lem3.3}
Let {\rm (C1), (C2)', (C3), (C4)}
hold. Assume that there exist $c_1<b_1<c_2<b_2$ such that
$q(x)\geq 0$ for $x\in G[c_1, b_1]\cup G[c_2,b_2]$ and
\[
e(x)\begin{cases}
\leq 0,  &x\in G[c_1,b_1],\\
\geq 0,  &x\in G[c_2,b_2],
\end{cases}
\]
$y(x)$ is a solution of \eqref{e1.1} such that $y(x)>0$ for $x\in
G[c_1,b_1]$ and $ y(x)<0$ for $ x\in G[c_2,b_2]$. Then for any
$H\in \Re$, and $i=1,2$,
\begin{equation}
\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}H(b_i,s)Q_2(s)ds\leq
V(c_i)+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}g_2(s)h_2^2(b_i,s)ds.
\label{s22}
\end{equation}
\end{lemma}

\begin{lemma}\label{lem3.4}
 Let {\rm (C1), (C2)', (C3), (C4)} hold.
Assume that there exist $a_1<c_1<a_2<c_2$ such that $q(x)\geq 0$
for $x\in G[a_1, c_1]\cup G[a_2,c_2]$ and
\[
e(x) \begin{cases}
\leq 0,  &x\in G[a_1,c_1],\\
\geq 0,  &x\in G[a_2,c_2],
\end{cases}
\]
$y(x)$ is a solution of \eqref{e1.1} such that $y(x)>0$ for $x\in
G[a_1,c_1]$ and $ y(x)<0$ for $ x\in G[a_2,c_2]$.  Then for any
$H\in \Re$ and $i=1,2$,
\begin{equation}
\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}H(s,a_i)Q_2(s)ds\leq
-V(c_i)+\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}g_2(s)h_1^2(s,a_i)ds.
\label{s24}
\end{equation}
\end{lemma}
The following theorem is an immediate result from Lemmas
\ref{lem3.3} and  \ref{lem3.4}.

\begin{theorem}\label{thm3.5}
 Let {\rm (C1), (C2)', (C3), (C4)} hold. Suppose that there exist
$a_1<b_1\leq a_2<b_2$ such
that $q(x)\geq 0$ for $ x\in G[a_1,b_1]\cup G[a_2,b_2]$ and
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
further, there exist some $c_i\in (a_i, b_i)$ and some $H\in \Re$
such that
\begin{equation}
\begin{aligned}
&\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}
 [H(s,a_i)Q_2(s)-g_2(s)h_1(s,a_i)]ds  \\
&+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}
 [H(b_i,s)Q_2(s)-g_2(s)h_2(b_i,s)]ds>0,\quad
i=1,2.
\end{aligned}\label{s26}
\end{equation}
Then every nontrivial solution of \eqref{e1.1} has at least
one zero either in $G(a_1,b_1)$ or in $G(a_2,b_2)$.
\end{theorem}

\begin{theorem}\label{thm3.6}
 Let {\rm (C1), (C2)', (C3), (C4)} hold.
Suppose that for any $T\geq a_0$, the following conditions hold:
\begin{enumerate}
\item there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2]
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for $ x\in G[a_1,b_1]\cup G[a_2,b_2]$

\item there exist some $c_i\in(a_i,b_i),i=1,2$, and some
$H\in \Re$ such that $T\leq a_1<b_1\leq a_2<b_2$ and (\ref{s26})
holds.
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{corollary} \label{coro3.7}
Let {\rm (C1), (C2)', (C3), (C4)} hold. Suppose that for any
$T\geq a_0$, the following conditions
hold:
\begin{enumerate}
\item there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0,  &x\in G[a_1,b_1],\\
\geq 0,  &x\in G[a_2,b_2],
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for $ x\in G[a_1,b_1]\cup G[a_2,b_2]$.

\item there exist some $c_i\in(a_i,b_i),i=1,2$, and some
$H\in \Re$ such that $T\leq a_1<b_1\leq a_2<b_2$ and the following
two inequalities hold for $i=1,2$,
\begin{gather}
\int_{a_i}^{c_i}\left[H(s,a_i)Q_2(s)-g_2(s)h_1^2(s,a_i)\right]ds>0,
\label{s28}
\\
\int_{c_i}^{b_i}\left[H(b_i,s)Q_2(s)-g_2(s)h_2^2(b_i,s)\right]ds>0.
\label{s29}
\end{gather}
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{corollary}\label{coro 3.8}
Let {\rm (C1), (C2)', (C3), (C4)} hold. Suppose that
for any $T\geq a_0$, the following conditions hold:
\begin{enumerate}
\item  there exist $T\leq a_1<2c_1-a_1\leq a_2<2c_2-a_2$
such that
\[
e(x)\begin{cases}
\leq 0, & x\in G[a_1,2c_1-a_1],\\
\geq 0, & x\in G[a_2,2c_2-a_2],
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for
$ x\in G[a_1,2c_1-a_1]\cup G[a_2,2c_2-a_2]$.

\item  there exist some $H\in \Re_0$ such that $T\leq
a_i<c_i $ for $i=1,2$ and the following inequality holds
\begin{equation}
\int_{a_i}^{c_i}\left\{H(s-a_i)[Q_2(s)+Q_2(2c_i-s)]
-[g_2(s)+g_2(2c_i-s)]h^2(s-a_i)\right\}ds>0.
\label{s30}
\end{equation}
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{corollary}

\begin{theorem}\label{thm3.9}
Let {\rm (C1), (C2)', (C3),  (C4)} hold. Assume that
$\lim_{r\to\infty}R(r)=\infty$. If for each $T\geq a_0$, the
following conditions hold:
\begin{enumerate}
\item  there exist $T\leq a_1<b_1\leq a_2<b_2$ such that
\[
e(x) \begin{cases}
\leq 0, &x\in G[a_1,b_1],\\
\geq 0, &x\in G[a_2,b_2],
\end{cases}
\]
and $q(x)\geq 0(\not\equiv 0)$ for $ x\in G[a_1,b_1]\cup G[a_2,b_2]$

\item there exist $c_i\in(a_i,b_i)$ for $i=1,2$, such that
$T\leq a_1<b_1\leq a_2<b_2$ and the following  inequalities hold
for $i=1,2$,
\begin{gather}
\frac{1}{[R(c_i)-R(a_i)]^{\alpha-1}}\int_{a_i}^{c_i}[R(s)-R(a_i)]^\alpha
Q_2(s)ds\geq \frac{\alpha^2}{4(\alpha-1)}, \label{s33}
\\
\frac{1}{[R(b_i)-R(c_i)]^{\alpha-1}}\int_{c_i}^{b_i}[R(b_i)-R(s)]^\alpha
Q_2(s)ds\geq \frac{\alpha^2}{4(\alpha-1)}. \label{s34}
\end{gather}
Where $R(r)=\int_{a_0}^r\frac{1}{g_2(s)}ds$.
\end{enumerate}
Then \eqref{e1.1} is oscillatory.
\end{theorem}

\begin{remark} \label{rmk3.10} \rm
The results of the paper are presented in the
form of a high degree of generality and thus they give wide
possibilities of deriving the different oscillation criteria with
an appropriate choice of the functions $H$. For instance, if we
choose $H(r,s)=(r-s)^\alpha$, $[R(r)-R(s)]^\alpha$,
$[\log({G(r)}/{G(s)})]^\alpha$, or
$[\int_s^r{dz}/{\rho(z)}]^\alpha$, etc.,  for
$r\ge s\ge a_0$, where $\alpha>1$ is a constant,
$R(r)=\int_{a_0}^r{\,ds}/{g_1(s)}$, or
$R(r)=\int_{a_0}^r{\,ds}/{g_2(s)}$,
$G(r)=\int_{r}^{\infty}{\,ds}/{g_1(s)}<\infty$, or
$G(r)=\int_{r}^{\infty}{\,ds}/{g_2(s)}<\infty$, for $r\ge a_0$,
$~\rho\in C([a_0,\infty), \mathbb{R}^+)$ satisfying
$\int_{a_0}^{\infty}{dz}/{\rho(z)}=\infty$, then we can derive
various explicit oscillation criteria.
\end{remark}


\begin{thebibliography}{00}

\bibitem{ei} M. A. Ei-Sayed; \emph{An oscillation criterion for a forced second order linear
         differential equations}, Pro. Amer. Math. Soc. 118(1993),813-817.

\bibitem{el} A. Elbert; \emph{Oscillation/nonoscillation for linear second
order differential equations}, J. Math. Anal. Appl.
226(1998)207-219.

\bibitem{gil} D. Gi1bag, N.S.Trudinger; \emph{Elliptic partial differential equations
of second
         order}, Spinger-Verlag, New York, 1983


\bibitem{kam} I. V. Kamenev; \emph{An integral criteria for oscillation of linear
differential equations}, Mat. Zametki, 23(1978)249-278.

\bibitem{kar} A. G. Kartsatos; \emph{Maintenance of oscillation under the effect of a
periodic
         forcing term}, Proc. Amer. Math. Soc.33(1972),377-383.

\bibitem{kon}  Q. Kong; \emph{Interval criteria for oscillation of second
order linear differential equations},  J. Math. Anal. Appl. 229
(1999) 258--270.

\bibitem{nou} E. S.Noussair, C. A. Swanson; \emph{Oscillation of simlinear
elliptic inequalities by
         Riccati trandformation}, Canad. J. Math., 1980, 32(4):908-923.

\bibitem{phi}  Gh. G. Philos; \emph{Oscillation theorems for linear
differential equations of second order}, Arch. Math. 53(1989),
483-492.

\bibitem{swa} C. A Swanson; \emph{Semilinear second order elliptic oscillation}, Canada
Math. Bull. 22 (1979) 139-157.

\bibitem{won1}J. S. Wong; \emph{Second order nonlinear forced oscillations}, SIMA J.
Math. Anal. 19(1988),667-675.

\bibitem{won2} J. S. Wong; \emph{Oscillation criteria for a forced second
order linear differential
         equation},  J. Math. Anal. Appl., 231(1999), 235-240.

\bibitem{xu1} H. Y. Xing, Z. T. Xu; \emph{Kamenev-type oscillation criteria
for semilinear elliptic differential equation}, Northeast. Math.
J, 20(2)(2004), 153-160.

\bibitem{xu2}Z. T. Xu, Riccati techniques and oscillation of semilinear
elliptic equations, Chin. J. Contemp. Math, (4)24(2003), 329-340.

\bibitem{xu3}  Z. T. Xu; \emph{The Oscillatory Behavior of Second Order
Nonlinear Elliptic Equations}, Electronic Journal of Qualitative
Theory of Differential Equations, 8(2005), 1-11.

 \bibitem{yan} Q. Yang; \emph{Interval oscillation criteria for a forced
second order nonlinear ordinary differential equations with
oscillatory potential}, App. Math.Comput. 135(2003), 49-64.

\end{thebibliography}

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