\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 $$\label{e1.1} \nabla\cdot(A(x)\nabla y)+B^T(x)\nabla y+q(x)f(y)=e(x),$$ 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|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 $$\label{e1.2} \nabla\cdot(A(x)\nabla y)+q(x)f(y)=0,$$ 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 $$\label{e1.3} W(x)=-\frac{\alpha(|x|)}{f(y(x))}(A\nabla y)(x),$$ 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_10$; \item[(C2)]$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_10$for$|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{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} whereW^T$denotes the transpose of$W. Using Green's formula in (\ref{12}), we obtain \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} 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{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} By the assumption, we can choosea_1,b_1\geq T_0(a_10 $for all$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$c_10$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$, $$\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}H(b_i,s)Q_1(s)ds\leq V(c_i)+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}g_1(s)h_2^2(b_i,s)ds. \label{22}$$ \end{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} Lettingr\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_10$for$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$, $$\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}H(s,a_i)Q_1(s)ds\leq -V(c_i)+\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}g_1(s)h_1^2(s,a_i)ds. \label{24}$$ \end{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*} Lettingr\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_10 \end{aligned} \label{26} 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 $a_1, b_1\geq T_0(a_10, x\in G[a_1,b_1]$, 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{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} 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 $T\leq a_1a_1\geq T_j,j\in N$, 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 $T\leq a_10, \label{28} \\ \int_{c_i}^{b_i}\left[H(b_i,s)Q_1(s)-g_1(s)h_2^2(b_i,s)\right]ds>0. \label{29} \end{gather} \end{enumerate} Then \eqref{e1.1} is oscillatory. \end{corollary} Moreover, let$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_i0. \label{30} \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 $$R(r)=\int_{a_0}^r\frac{1}{g_1(s)}ds,\quad r\geq a_0, \label{31}$$ and let $$H(r,s)=[R(r)-R(s)]^\alpha, \quad r\geq s\geq a_0,\label{32}$$ 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 0; \end{aligned}\label{36} 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*} wherer=\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$\alpha\geq 3/2, then \begin{align*} M_1(H) &= \int_{a_1}^{b_1}[g_1(s)h^2(s)-Q_1(s)H(s)]ds\\ &= \pi\int_{n^2\pi^2}^{(n+1)^2\pi^2}\frac{\cos ^2\sqrt{s}}{2\sqrt{s}}ds -(2\alpha-1)\pi\int_{n^2\pi^2}^{(n+1)^2\pi^2} \frac{\sin ^2\sqrt{s}}{\sqrt{s}}ds \\ &= \pi\int_{n\pi}^{(n+1)\pi}\cos ^2sds -\frac{2\alpha-1}{2}\int_{n\pi}^{(n+1)\pi}\sin ^2sds \\ &= \frac{\pi^2}{2}-\frac{(2\alpha-1)\pi^2}{4}\leq 0. \end{align*} Similarly, fora_2,b_2$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 $$c u u^T + u v^T \geq \frac{c}{2} u u^T-\frac{1}{2c}vv^T.\label{s6}$$ \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$T\leq a_10$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$S_r. From \eqref{e1.1} and (\ref{s11}), it follows that \begin{aligned} \nabla\cdot W(x) &= -\frac{f'(y)}{f^2(y)}(\nabla y)^TA\nabla y-\frac{1}{f(y)}[q(x)f(y)+B^T\nabla y-e(x)] \\ &\leq -kW^TA^{-1}W-q(x)-B^TA^{-1} W +\frac{e(x)}{f(y)}\\ &\leq -\frac{k}{\lambda(x)}W^TW-q(x)-B^TA^{-1} W +\frac{e(x)}{f(y)}\quad(\mbox{By Lemma \ref{3.1}})\\ &\leq -\frac{k}{2\lambda(x)}|W|^2+\frac{1}{2k}\lambda(x) |B^TA^{-1}|^2-q(x)+\frac{e(x)}{f(y)}. \end{aligned}\label{s13} whereW^T$denotes the transpose of$W. Using Green's formula in (\ref{s12}), we get \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} 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}), $$V'(r)\leq-\int_{S_r}\Big[q(x)-\frac{1}{2k}\lambda(x)|B^TA^{-1}|^2\Big] d\sigma-\frac{1}{g_2(r)}V^2(r)+\int_{S_r}\frac{e(x)}{y(x)}d\sigma\label{s15}$$ 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 existc_10$for$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$, $$\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}H(b_i,s)Q_2(s)ds\leq V(c_i)+\frac{1}{H(b_i,c_i)}\int_{c_i}^{b_i}g_2(s)h_2^2(b_i,s)ds. \label{s22}$$ \end{lemma} \begin{lemma}\label{lem3.4} Let {\rm (C1), (C2)', (C3), (C4)} hold. Assume that there exist$a_10$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$, $$\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}H(s,a_i)Q_2(s)ds\leq -V(c_i)+\frac{1}{H(c_i,a_i)}\int_{a_i}^{c_i}g_2(s)h_1^2(s,a_i)ds. \label{s24}$$ \end{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_10,\quad i=1,2. \end{aligned}\label{s26} 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 $T\leq a_10, \label{s28} \\ \int_{c_i}^{b_i}\left[H(b_i,s)Q_2(s)-g_2(s)h_2^2(b_i,s)\right]ds>0. \label{s29} \end{gather} \end{enumerate} Then \eqref{e1.1} is oscillatory. \end{corollary} \begin{corollary}\label{coro 3.8} Let {\rm (C1), (C2)', (C3), (C4)} hold. Suppose that for any$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_i0. \label{s30} \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 $T\leq a_11$ 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. 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