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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 21, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/21\hfil Oscillation for quasilinear elliptic equations]
{Oscillation for quasilinear elliptic equations with $\boldsymbol{p(x)}$-Laplacians
in general domains}

\author[N. Yoshida \hfil EJDE-2013/21\hfilneg]
{Norio Yoshida}  % in alphabetical order

\address{Norio Yoshida\newline
Department of Mathematics, University of Toyama, Toyama,
930-8555 Japan}
\email{nori@sci.u-toyama.ac.jp}


\dedicatory{Dedicated to Professor Kusano Taka\^{s}i on
his eightieth birthday}

\thanks{Submitted  October 5, 2012. Published January 23, 2013.}
\subjclass[2000]{35B05, 35J92}
\keywords{ Oscillation; $p(x)$-Laplacian; quasilinear elliptic equations}

\begin{abstract}
 Oscillation of quasilinear elliptic equations with $p(x)$-Laplacians
 in general domains are derived by the variational approach as applications
 of Picone identity. Three examples are given, and generalizations to
 quasilinear elliptic equations with $p(x)$-Laplacians are shown.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Recently there has been much interest in establishing Picone identity
which plays an important role in Sturmian comparison theorems and
oscillation theorems for various differential equations.
We refer the reader to Allegretto \cite{all01},
Allegretto and Huang \cite{allhua98, allhua99},
Bogn\'{a}r and Do\v{s}l\'{y} \cite{bogdos03},
Do\v{s}l\'{y} and \v{R}eh\'{a}k \cite{dosreh05},
Dunninger \cite{dun95}, 
Kusano, Jaro\v{s}, and Yoshida \cite{kusjaryos00}, 
Yoshida \cite{yos09-1} for $p$-Laplace equations, 
and to Allegretto \cite{all07},
Bogn\'{a}r and Do\v{s}l\'{y} \cite{bogdos04},
Sahiner and Zafer \cite{sahiner},
Yoshida \cite{yos11-1, yos11-2, yos11-3, yos12}
for $p(x)$-Laplace equations.

The operator $\nabla\cdot\bigl(| \nabla u|^{p(x)-2}\nabla u\bigr)$ is
said to be $p(x)$-Laplacian ($p(x)>1$), and becomes $p$-Laplacian
$\nabla\cdot\bigl(| \nabla u|^{p-2}\nabla u\bigr)$ if
$p(x) = p\ (\text{constant})$, where the dot $\cdot$ denotes the
scalar product,$\nabla =(\partial/\partial x_1,\dots,\partial/\partial x_n)$
and $| x|$ denotes the Euclidean length of
$x = (x_1,x_2,\dots,x_n) \in \mathbb{R}^n$.

Various mathematical problems with variable exponent
growth condition have been received considerable attention in recent
years (see \cite{dieharhasruz11, harhas10}).
These problems arise from nonlinear elasticity theory,
electrorheological fluids (cf. \cite{ruz00, zhi87}) and
image processing (cf. \cite{chelevrao06}).

Oscillation results for half-linear elliptic equations have
been extensively developed, but most of them pertain to
unbounded domains which are exterior or general exterior
domains which are not ``small" at $\infty$.

In 1973 Swanson \cite{swa73} studied the linear elliptic equation
$$
   \nabla\cdot\bigl(A(x)\nabla v\bigr) + C(x)v = 0,
   \quad \Omega \subset \mathbb{R}^n,
$$
where $\Omega$ is an unbounded domain in $\mathbb{R}^n$ and
it is not required that $\Omega$ be quasiconical, quasicylindrical,
exterior, general exterior, or even connected.

The objective of this paper is to provide oscillation criteria for
the half-linear elliptic inequality with $p(x)$-Laplacian
\begin{equation}
   vQ[v] \leq 0  \label{eq1}
\end{equation}
in a general domain $\Omega \subset \mathbb{R}^n$, where
\begin{equation}
\begin{aligned}
   Q[v] & :=  \nabla\cdot\bigl(A(x)|\nabla v|^
               {\alpha(x)-1}\nabla v\bigr)
               - A(x)(\log | v|)|\nabla v|^
               {\alpha(x)-1}\nabla\alpha(x)\cdot\nabla v   \\
        & \quad + |\nabla v|^{\alpha(x)-1}
               B(x)\cdot \nabla v
               + C(x)| v|^{\alpha(x)-1}v.
\end{aligned}  \label{eq2}
\end{equation}
We note that $\log | v|$ in \eqref{eq2} has
singularities at zeros of $v$, but $v \log | v|$
becomes continuous at the zeros of $v$ if we define
$v \log | v| = 0$ at the zeros, in light of the fact
that $\lim_{\varepsilon \to +0} \varepsilon \log \varepsilon = 0$.
Therefore, we conclude that $vQ[v]$ has no singularities and
is continuous in $\Omega$. From the relation
$$
   (kv)Q[kv] = | k|^{\alpha(x)+1}vQ[v]\ (k \in \mathbb{R})
$$
we see that \eqref{eq1} is half-linear in the sense that
a constant multiple of a solution
$v$ of \eqref{eq1} is also a solution of \eqref{eq1}
(cf. Yoshida \cite[Proposition 2.1]{yos11-1}).

Our approach is an adaptation of variational method which is
based on Picone identity.
We refer to Ma\v{r}\'{\i}k \cite{mar05} which investigates
oscillation of half-linear elliptic equations with $p$-Laplacian
in general exterior domains by the variational approach.

In Section 2 we present oscillation results based on Picone
identity.  Section 3 is devoted to examples which illustrate
a main oscillation theorem in Section 2.
Generalizations to more general elliptic inequalities
are given in Section 4.



\section{Oscillation results}

In this section we derive a main oscillation theorem by
using Lemmas \ref{lem3} and \ref{lem4} which are deduced from
Picone identity.

Let $\Omega$ be an unbounded domain in $\mathbb{R}^n$.
It is assumed that
$A(x) \in C(\Omega; (0,\infty))$,
$B(x) \in C(\Omega; \mathbb{R}^n)$,
$C(x) \in C(\Omega; \mathbb{R})$,
$\alpha(x) \in C^{1}(\Omega; (0,\infty))$,
and $\alpha(x) > 0$.

The domain $\mathcal{D}_{Q}(\Omega)$ of $Q$ is defined to be the
set of all functions $v$ of class $C^{1}(\Omega; \mathbb{R})$ such that
$A(x)|\nabla v|^{\alpha(x)-1}\nabla v
\in C^{1}(\Omega; \mathbb{R}^n)$.

A solution $v \in \mathcal{D}_{Q}(\Omega)$ of \eqref{eq1} is said to be
{\it oscillatory} in $\Omega$ if it has a zero in $\Omega_{r}$ for any $r > 0$, where
$$
   \Omega_{r} = \Omega \cap \{ x \in \mathbb{R}^n;\ | x| > r \}.
$$

Let $G$ be a bounded domain with piecewise smooth boundary
$\partial G$ such that $G \subset \Omega$.
We need four lemmas.
The following lemma was established by Yoshida
\cite[Theorem 2.1]{yos11-2}.

\begin{lemma}[Picone identity for $Q$]\label{lem1}
If $v \in \mathcal{D}_{Q}(G)$ and $v$ has no zero in $G$,
then we obtain the following Picone identity for any
$u \in C^{1}(G; \mathbb{R})${\rm :}
\begin{equation}
\begin{aligned}
   & - \nabla \cdot \Big(u\varphi(u)
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}\Big) \\
   &= - A(x)\Big| \nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x)\Big|^{\alpha(x)+1}
  + C(x)| u |^{\alpha(x)+1}
         \\
   &\quad  + A(x)\Bigl[\Big|\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x)\Big|^{\alpha(x)+1} \\
   &\quad   + \alpha(x) \big|\frac{u}{v}\nabla v
         \big|^{\alpha(x)+1}
 - (\alpha(x)+1)\big|\frac{u}{v}\nabla v
         \big|^{\alpha(x)-1}\Big(\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)   \\
   &\quad   - \frac{u}{(\alpha(x)+1)A(x)}B(x)
         \Big)\cdot\big(\frac{u}{v}\nabla v   \big)\Bigr]
  - \frac{| u|^{\alpha(x)+1}}{| v|^{\alpha(x)+1}}
         \bigl(vQ[v]\bigr) \quad \text{in }G,
\end{aligned} \label{eq3}
\end{equation}
where $\varphi(u) = | u |^{\alpha(x)-1}u$.
\end{lemma}

The next lemma is stated in Yoshida \cite[Lemma 3.1]{yos11-2}.

\begin{lemma}\label{lem2} The inequality
$$
   | \xi |^{\alpha(x)+1} + \alpha(x)\,| \eta
   |^{\alpha(x)+1}
   - (\alpha(x)+1)| \eta |^{\alpha(x)-1}\xi\cdot\eta \geq 0
$$
is valid for $x \in G$, $\xi, \eta \in \mathbb{R}^n$, where the
equality holds if and only if $\xi = \eta$.
\end{lemma}

For a nonempty piecewise smooth bounded domain $G \subset \Omega$,
we define the functional
\begin{align*}
   M[u;G] & :=  \int_{G}\Bigl[
                 A(x)\Big| \nabla u + \frac{u\log | u|}
                 {\alpha(x)+1}\nabla\alpha(x)
                 - \frac{u}{(\alpha(x)+1)A(x)}B(x)
                  \Big|^{\alpha(x)+1} \\
          &\quad  - C(x)| u |^{\alpha(x)+1} \Bigr]dx
\end{align*}
with domain consisting of all real-valued piecewise $C^{1}$-functions
$u$ on $\overline{G}$.

\begin{lemma}\label{lem3}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$ and
$B(x)/A(x) \in C^{1}(G; \mathbb{R}^n)$.
Assume that there exists a function
$u \in C^{1}(\overline{G}; \mathbb{R})$ such that
$u = 0$ on $\partial G$ and $u$ has no zero in $G$.
If $M[u;G] \leq 0$ and
\begin{itemize}
   \item[(H1)] there is a function $F \in C(\overline{G}; \mathbb{R})$ such that
    $F \in C^{1}(G; \mathbb{R})$ and
   $$
   \nabla F = \frac{\log | u|} {\alpha(x)+1}\nabla\alpha(x)
   - \frac{B(x)}{(\alpha(x)+1)A(x)} \quad \text{in }G,
   $$
\end{itemize}
then every solution $v \in \mathcal{D}_{Q}(G)$ of \eqref{eq1} must
vanish at some point of $\overline{G}$.
\end{lemma}

\begin{proof}
Suppose to the contrary that there exists a solution
$v \in \mathcal{D}_{Q}(G)$ of \eqref{eq1} such that
$v$ has no zero on $\overline{G}$.
Integrating the Picone identity \eqref{eq3} over $G$ and using
the divergence theorem,
we arrive at
\begin{equation}
   0  \geq  -M[u;G] + \int_{G}W(u,v)\,dx,  \label{eq4}
\end{equation}
where
\begin{align*}
   &  W(u,v) \\
   & :=  A(x)\Bigl[\Big|\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x)\Big|^{\alpha(x)+1} \\
   &\quad  + \alpha(x) \big|\frac{u}{v}\nabla v
         \big|^{\alpha(x)+1}
  - (\alpha(x)+1)\big|\frac{u}{v}\nabla v
         \big|^{\alpha(x)-1}\Big(\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)   \\
   &\quad  - \frac{u}{(\alpha(x)+1)A(x)}B(x)
         \Big)\cdot\big(\frac{u}{v}\nabla v\big)\Bigr].
\end{align*}
Lemma \ref{lem2} implies that $W(u,v) \geq 0$ in $G$, and hence
$\int_{G}W(u,v)\,dx \geq 0$.  Since $-M[u;G] \geq 0$,
we see that the right-hand side of \eqref{eq4} is
non-negative.  Therefore we conclude that
$\int_{G}W(u,v)\,dx = 0$, which yields
$$
   \nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x) \equiv \frac{u}{v}\nabla v
   \quad \text{in } G
$$
from Lemma \ref{lem2}.  Using the hypothesis (H1), we obtain
$$
   \nabla u + u\nabla F \equiv \frac{u}{v}\nabla v
   \quad \text{in }G,
$$
from which we get
$$
   e^{-F}v\nabla\left(e^{F}\frac{u}{v}\right) \equiv 0 \quad \text{in }G.
$$
Hence, there is a constant $c_{0}$ such that $e^{F}u/v = c_{0}$ in $G$,
and therefore on $\overline{G}$ by continuity.
Since $u = 0$ on $\partial G$, we find that $c_{0} = 0$,
that is, $u \equiv 0$ in $G$,
which contradicts the hypothesis that $u$ has no zero in $G$.
This completes the proof.
\end{proof}

\begin{lemma}\label{lem4}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$ and
$B(x)/A(x) \in C^{1}(G; \mathbb{R}^n)$.
Assume that there exists a nontrivial function
$u \in C^{1}(\overline{G}; \mathbb{R})$ such that
$u = 0$ on $\partial G$.
If $M[u;G] < 0$,
then every solution $v \in \mathcal{D}_{Q}(G)$ of {\rm \eqref{eq1}} must
vanish at some point of $\overline{G}$.
\end{lemma}

\begin{proof}
Suppose that there is a solution
$v \in \mathcal{D}_{Q}(G)$ of \eqref{eq1} such that
$v$ has no zero on $\overline{G}$.
As in the proof of Lemma \ref{lem3}, we observe that
\eqref{eq4} holds.  Since $- M[u;G] > 0$ and
$\int_{G}W(u,v)\,dx \geq 0$, it can be shown that
the right-hand side of \eqref{eq4} is positive.
The contradiction proves the lemma.
\end{proof}

\begin{remark}\label{rem1}\rm
In the hypothesis (H1) of Lemma \ref{lem3}, the vector-valued
function
$$
   \frac{\log | u|}{\alpha(x)+1}\nabla\alpha(x)
   - \frac{B(x)}{(\alpha(x)+1)A(x)}
$$
must be a $C^{1}$-function (cf. \cite[Proposition 2.2]{yos11-2}),
and therefore we suppose that
$\alpha(x) \in C^{2}$, $B(x)/A(x) \in C^{1}$ in
Lemmas \ref{lem3} and \ref{lem4}.
\end{remark}

\begin{theorem}\label{thm1}
Assume that $\Omega$ contains a sequence of domains
$G_k$  $(k=1,2,\dots)$ such that $G_k$ are nonempty
bounded domains with piecewise smooth boundaries $\partial G_k$.
If for any $r > 0$
there exist a natural number $m=m(r) \in \mathbb{N}$
such that $\overline{G_m} \subset \Omega_{r}$ and
a nontrivial piecewise $C^{1}$-function
$u_m$ on $\overline{G_m}$ with the
following properties:
\begin{itemize}
   \item[(i)] $\alpha(x) = \alpha_m(x) \in C^{2}(G_m; (0,\infty))$
      and 
\[B(x)/A(x) = B_m(x)/A_m(x)
      \in C^{1}(G_m; \mathbb{R}^n) \quad \text{on } G_m;
\] 

   \item[(ii)] $u_m = 0$ on $\partial G_m$;
   \item[(iii)] $u_m$ has no zero in $G_m$;
   \item[(iv)] $M[u_m;G_m] \leq 0$;
   \item[(v)] there exists a function
               $F_m \in C(\overline{G_m}; \mathbb{R})$ such that
                    $F_m \in C^{1}(G_m; \mathbb{R})$ and
                    \begin{equation}
                       \nabla F_m = \frac{\log | u_m|}
                       {\alpha_m(x)+1}\nabla\alpha_m(x)
                        - \frac{B_m(x)}{(\alpha_m(x)+1)A_m(x)} \quad
                       \text{in }G_m,
                       \label{eq5}
                    \end{equation}
\end{itemize}
then every solution $v \in \mathcal{D}_{Q}(\Omega)$ of \eqref{eq1}
is oscillatory in $\Omega$.  Furthermore, if $M[u_m;G_m] < 0$ in the
hypothesis {\rm (iv)},
then the conclusion is valid without the hypotheses {\rm (iii), (v)}.
\end{theorem}

\begin{proof}
For any $r > 0$ there exist a natural number $m=m(r) \in \mathbb{N}$
such that $\overline{G_m} \subset \Omega_{r}$ and
a nontrivial piecewise $C^{1}$-function
$u_m$ on $\overline{G_m}$ satisfying (i)--(v).
It follows from Lemma \ref{lem3} that
every solution $v \in \mathcal{D}_{Q}(G_m)$ of {\rm \eqref{eq1}}
has a zero on $\overline{G_m} \subset \Omega_{r}$, that is,
$v$ has a zero in $\Omega_{r}$ for any $r > 0$, which implies that
$v$ is oscillatory in $\Omega$.
This completes the proof of the first statement of the theorem.
If $M[u_m;G_m] < 0$, the second statement follows from
Lemma \ref{lem4} by the same arguments as were used in the first
statement.
\end{proof}

\begin{remark}\label{rem2}\rm
Let $p \geq 1$. It follows from Jensen's inequality that
$$
   \Big(\frac{a+b+c}{3}\Big)^{p} \leq \frac{a^{p}+b^{p}+c^{p}}{3}
$$
for any $a, b, c \geq 0$.  We let $\alpha(x) > 0$, and obtain
\begin{align*}
 | a+b+c |^{\alpha(x)+1}
   & \leq
      \left(| a| + | b| + | c|\right)^{\alpha(x)+1} \\
   & \leq  3^{\alpha(x)}
      \left(| a|^{\alpha(x)+1} + | b|^{\alpha(x)+1}
      + | c|^{\alpha(x)+1}\right)
\end{align*}
for any $a, b, c \in \mathbb{R}$.  Hence, we obtain
\begin{align*}
   & \Big| \nabla u + \frac{u\log | u|}
            {\alpha(x)+1}\nabla\alpha(x)
            - \frac{u}{(\alpha(x)+1)A(x)}B(x)
            \Big|^{\alpha(x)+1} \\
   & \leq  3^{\alpha(x)}
            \Bigl(| \nabla u|^{\alpha(x)+1}
            + \left| \frac{u\log | u|}
            {\alpha(x)+1}\nabla\alpha(x)\right|^{\alpha(x)+1}
   + \big| \frac{u}{(\alpha(x)+1)A(x)}B(x)\big|^{\alpha(x)+1}\Bigr).
\end{align*}
Defining
\begin{align*}
   \tilde{M}[u;G]
   & :=  \int_{G}\Bigl[
          A(x)3^{\alpha(x)}
          \Bigl(| \nabla u|^{\alpha(x)+1}
          + \big| \frac{u\log | u|}
          {\alpha(x)+1}\nabla\alpha(x)\big|^{\alpha(x)+1} \\
   &\quad   + \big| \frac{u}{(\alpha(x)+1)A(x)}B(x)
          \big|^{\alpha(x)+1}\Bigr)
          - C(x)| u |^{\alpha(x)+1} \Bigr]dx,
\end{align*}
we observe that $M[u;G] \leq \tilde{M}[u;G]$.
Consequently, Lemmas \ref{lem3}, \ref{lem4} and Theorem \ref{thm1}
remain true if $M[u;G], M[u_m;G_m]$ are replaced by
$\tilde{M}[u;G], \tilde{M}[u_m;G_m]$, respectively.
\end{remark}


\section{Examples}

Three examples which illustrate Theorem \ref{thm1} are given in this section.
We consider the cases where $\Omega$ contains a sequence of
rectangular domains, annular domains, or balls.
Using the similar arguments as in \cite[Remark 2.4]{yos11-2},
we give the following example.

\begin{example}\label{ex1}\rm
We consider the case where $n=2$ and $\Omega$ contains a sequence of
rectangular domains
$$
   G_k = \bigl(a_k, a_k+\pi\bigr)\times \bigl(b_k, b_k+\pi\bigr)
   \quad (k=1,2,\dots),
$$
where $\lim_{k\to\infty} a_k = \infty$ or $\lim_{k\to\infty} b_k = \infty$. Assume that
$$
   A(x) \leq A_k, \quad C(x) \geq C_k \quad \text{on }  G_k
$$
for some constants $A_k > 0$, $C_k > 0$, and let
\begin{gather*}
  \alpha(x) = \alpha_k(x) = e^{\sin (x_1-a_k) \sin (x_2-b_k)+1}-1, \\
\begin{aligned}
 \frac{B(x)}{A(x)} &= \frac{B_k(x)}{A_k(x)}\\
   & = e^{\sin (x_1-a_k)\sin (x_2-b_k)+1}
 \Bigl(-\cos (x_1-a_k)\sin (x_2-b_k),\\
 &\quad  -\sin (x_1-a_k)\cos (x_2-b_k)\Bigr)
\end{aligned}
\end{gather*}
on each $G_k$.  For any $r > 0$ there exists an integer
$m = m(r) \in \mathbb{N}$ such that $\overline{G_m} \subset
\Omega_{r}$, because
$\lim_{k\to\infty} a_k = \infty$ or $\lim_{k\to\infty} b_k = \infty$.
Letting
$$
   u_m = \sin \bigl(x_1-a_m\bigr)\sin \bigl(x_2-b_m\bigr),
$$
we see that $u_m \in C^{1}(\overline{G_m};\mathbb{R})$,
$u_m = 0$ on $\partial G_m$, $u_m > 0$ in $G_m$,
and that there exists a function $F_m \in C(\overline{G_m}; \mathbb{R})
\cap C^{1}(G_m; \mathbb{R})$ satisfying \eqref{eq5}. Since
\[
  \frac{B_m(x)}{A_m(x)} = - (\alpha_m(x)+1)\nabla u_m, \quad
  \frac{\nabla\alpha_m(x)}{\alpha_m(x)+1} = \nabla u_m,
\]
we obtain
\begin{align*}
 \frac{\log | u_m|}
         {\alpha_m(x)+1}\nabla\alpha_m(x)
         - \frac{B_m(x)}{(\alpha_m(x)+1)A_m(x)}
   &= (\log u_m)\nabla u_m + \nabla u_m \\
   &= \nabla F_m \quad \text{in }G_m,
\end{align*}
where
$$
   F_m = u_m\log u_m.
$$
Moreover, we observe that $F_m \in C(\overline{G_m};\mathbb{R})$.
It is easy to see that
\begin{equation}
\begin{aligned}
  &  M[u_m;G_m]  \\
  & \leq  \int_{a_m}^{a_m+\pi}\int_{b_m}^{b_m+\pi}
      \Bigl[A_m\Bigl| \nabla u_m + \frac{u_m\log | u_m|}
                 {\alpha_m(x)+1}\nabla\alpha_m(x)  \\
  &\quad   - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
                  \Bigr|^{\alpha_m(x)+1}
   - C_m| u_m |^{\alpha_m(x)+1} \Bigr]dx_1dx_2.
 \end{aligned} \label{eq6}
\end{equation}
We easily obtain
\begin{align*}
   &   \nabla u_m + \frac{u_m\log | u_m|}
        {\alpha_m(x)+1}\nabla\alpha_m(x)
        - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)   \\
   &= (1 + u_m\log u_m + u_m)\nabla u_m
\end{align*}
and hence
\begin{equation}
\begin{aligned}
   &  \Bigl| \nabla u_m + \frac{u_m\log | u_m|}
        {\alpha_m(x)+1}\nabla\alpha_m(x)
        - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
                  \Bigr|  \\
   & \leq  \sqrt{2}| 1 + u_m\log u_m + u_m |
 \end{aligned}      \label{eq7}
\end{equation}
in view of the inequality $| \nabla u_m| \leq \sqrt{2}$.
Since
\begin{align*}
   u_m\log u_m &= \sin (x_2-b_m)\Bigl[\sin (x_1-a_m)
                         \log\sin (x_1-a_m)\Bigr] \\
        &\quad  +\sin (x_1-a_m)\Bigl[\sin (x_2-b_m)
               \log\sin (x_2-b_m)\Bigr],
\end{align*}
we see that
\begin{equation}
   | u_m\log u_m | \leq \frac{2}{e}   \label{eq8}
\end{equation}
in light of the inequality
$$
   | x\log\,x | \leq \frac{1}{e}    \quad (0 \leq x \leq 1).
$$
Since $\alpha_m(x)+1 \leq e^{2}$, it follows from \eqref{eq7} and
\eqref{eq8} that
\begin{equation}
\begin{aligned}
   &  \Bigl| \nabla u_m + \frac{u_m\log | u_m|}
        {\alpha_m(x)+1}\nabla\alpha_m(x)
        - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
                  \Bigr|^{\alpha_m(x)+1}  \\
   & \leq  \Big(\sqrt{2}\Big(1+\frac{2}{e} + 1\Big)\Big)^{e^{2}}
            = \Big(2\sqrt{2}\Big(1+\frac{1}{e}\Big)\Big)^{e^{2}}.
\end{aligned}   \label{eq9}
\end{equation}
Since $\alpha_m(x)+1 \leq e^{2} \le 8$, it can be shown that
\begin{align*}
   & \int_{a_m}^{a_m+\pi}\int_{b_m}^{b_m+\pi}
       | u_m|^{\alpha_m(x)+1} dx_1dx_2 \\
   &= \int_{a_m}^{a_m+\pi}\int_{b_m}^{b_m+\pi}
         \left| \sin (x_1-a_m)\sin (x_2-b_m)
         \right|^{\alpha_m(x)+1}dx_1dx_2 \\
   & \geq  \int_{a_m}^{a_m+\pi}\int_{b_m}^{b_m+\pi}
            \sin^{8}(x_1-a_m)\sin^{8}(x_2-b_m)dx_1dx_2 \\
   &= \Big(\int_{0}^{\pi} \sin^{8}x\,dx\Big)^{2}.
\end{align*}
It is known that
\begin{align*}
   \int_{0}^{\pi} \sin^{8}x\,dx
   &= 2 \int_{0}^{\pi/2} \sin^{8}x\,dx \\
   &= 2\cdot \frac{\pi}{2}\frac{7!!}{8!!}
         = \pi\cdot \frac{7\cdot 5 \cdot 3 \cdot 1}{8\cdot 6 \cdot 4 \cdot 2}
         = \frac{105}{384}\pi,
\end{align*}
and therefore we have
\begin{equation}
   \int_{a_m}^{a_m+\pi}\int_{b_m}^{b_m+\pi}
       | u_m|^{\alpha_m(x)+1} dx_1dx_2
   \geq \left(\frac{105}{384}\right)^{2}\pi^{2}.
   \label{eq10}
\end{equation}
Combining \eqref{eq6}, \eqref{eq9} and \eqref{eq10} yields
$$
   M[u_m;G_m] \leq A_m
   \Big(2\sqrt{2}\Big(1+\frac{1}{e}\Big)\Big)^{e^{2}}\pi^{2}
   - C_m\Big(\frac{105}{384}\Big)^{2}\pi^{2}.
$$
If
$$
   C_m \geq \Big(\frac{384}{105}\Big)^{2}
              \Big(2\sqrt{2}\Big(1+\frac{1}{e}\Big)\Big)^{e^{2}}A_m,
$$
then we observe that $M[u_m;G_m] \leq 0$, and consequently
it follows from Theorem \ref{thm1} that
every solution  $v \in \mathcal{D}_{Q}(\Omega)$ of {\rm \eqref{eq1}}
is oscillatory in $\Omega$.
\end{example}

\begin{example}\label{ex2}\rm
We consider the case where $\Omega$ contains a sequence of
annular domains
$$
   G_k = \{ x \in \mathbb{R}^n;\ a_k < | x| < b_k\}
   \quad (k=1,2,\dots),
$$
where $0 < b_k-a_k \leq 1$ and $\lim_{k\to\infty} a_k = \infty$.
It is assumed that
$$
   A(x) \leq A_k, \quad C(x) \geq C_k \quad \text{on } G_k
$$
for some constants $A_k > 0$, $C_k > 0$, and let
\begin{gather*}
    \alpha(x) = \alpha_k(| x|)
                 = e^{(b_k-| x|)(| x|-a_k)+1}-1, \\
\begin{aligned}  \frac{B(x)}{A(x)} 
&= \frac{B_k(x)}{A_k(x)}
 = \tilde{B}_k(| x|)\frac{x}{| x|} \\
 &= -e^{(b_k-| x|)(| x|-a_k)+1} \bigl(a_k+b_k-2| x|\bigr)
      \frac{x}{| x|}
\end{aligned}
\end{gather*}
on each $G_k$.  For any $r > 0$ there exists an integer
$m = m(r) \in \mathbb{N}$ such that 
$\overline{G_m} \subset \Omega_{r}$, because
$\lim_{k\to\infty} a_k = \infty$.
Let
$$
   u_m = \bigl(b_m-| x|\bigr)\bigl(| x|-a_m\bigr),
$$
we find that $u_m \in C^{1}(\overline{G_m};\mathbb{R})$,
$u_m = 0$ on $\partial G_m$, $u_m$ has no zero in $G_m$,
and that there exists a function $F_m \in C(\overline{G_m}; \mathbb{R})
\cap C^{1}(G_m; \mathbb{R})$ satisfying \eqref{eq5}. In fact, we derive
\begin{align*}
   & \frac{\log | u_m|}
         {\alpha_m(x)+1}\nabla\alpha_m(x)
         - \frac{B_m(x)}{(\alpha_m(x)+1)A_m(x)} \\
   &= \frac{\log\bigl((b_m-| x|)(| x|-a_m)\bigr)}
              {e^{(b_m-| x|)(| x|-a_m)+1}}
         \Big(\alpha_m'(r)\Bigl|_{r=| x|}\Big)
         \frac{x}{| x|}
   + (a_m+b_m-2| x|) \frac{x}{| x|} \\
   &= \Bigl[ (a_m+b_m-2| x|)
         \log\bigl((b_m-| x|)(| x|-a_m)\bigr)
         + a_m+b_m-2| x| \Bigr]\frac{x}{| x|} \\
   &= \nabla F_m \quad \text{in }G_m,
\end{align*}
where
$$
   F_m = f_m(| x|),
$$
$f_m(r)$ being the function defined by
$$
   f_m(r) = (b_m-r)(r-a_m)\log\bigl((b_m-r)(r-a_m)\bigr).
$$
Moreover, we see that $F_m \in C(\overline{G_m};\mathbb{R})$.
Introducing hyperspherical coordinates $(r,\theta)$, and letting
$u_m(x) = z_m(| x|)$, we have
\begin{equation}
\begin{aligned}
  & M[u_m;G_m]  \\
  & \leq  \int_{a_m}^{b_m}\int_{S_1}
      \Bigl[A_m\Bigl| z_m'(r) + \frac{z_m(r)
      \log | z_m(r)|}
                 {\alpha_m(r)+1}\alpha_m'(r)  \\
  &\quad  - \frac{z_m(r)}{(\alpha_m(r)+1)}\tilde{B}_m(r)
                  \Bigr|^{\alpha_m(r)+1}
  - C_m| z_m(r) |^{\alpha_m(r)+1}  \Bigr]r^{n-1}drd\omega,
\end{aligned}  \label{eq11}
\end{equation}
where $\omega$ denotes the measure on the unit sphere $S_1$.
It is readily seen that
\begin{align*}
 & z_m'(r) + \frac{z_m(r)
         \log | z_m(r)|}{\alpha_m(r)+1}\alpha_m'(r)
         - \frac{z_m(r)}{(\alpha_m(r)+1)}\tilde{B}_m(r)  \\
 &= z_m'(r) + z_m(r)f_m'(r) \\
 &= \bigl(a_m+b_m-2r\bigr)
  \Bigl[ 1 + (b_m-r)(r-a_m)\log\bigl((b_m-r)(r-a_m)\bigr)\\
  &\quad    + (b_m-r)(r-a_m)\Bigr].
\end{align*}
Simple computations show that
\begin{gather*}
 | a_m+b_m-2r | \leq b_m-a_m, \\
| (b_m-r)(r-a_m)\log\bigl((b_m-r)(r-a_m)\bigr)| \leq \frac{2}{e}(b_m-a_m), \\
| (b_m-r)(r-a_m)| \leq  \frac{(b_m-a_m)^{2}}{4} \\
 e \leq \alpha_m(r)+1 = e^{(b_m-r)(r-a_m)+1} \leq e^{(b_m-a_m)^{2}/4 +1},
\end{gather*}
on $[a_m, b_m]$, and therefore we arrive at
\begin{eqnarray}
   &\Big|
            z_m'(r) + \frac{z_m(r)
            \log | z_m(r)|}{\alpha_m(r)+1}\alpha_m'(r)
            - \frac{z_m(r)}{(\alpha_m(r)+1)}\tilde{B}_m(r)
            \Big|^{\alpha_m(r)+1}
            \\
   & \leq  \bigl(b_m-a_m\bigr)^{e}
            \Big[ 1 + \frac{2}{e}(b_m-a_m) +
                   \frac{(b_m-a_m)^{2}}{4} \Big]^{\tilde{\alpha}_m},
            \label{eq12}
\end{eqnarray}
where $\tilde{\alpha}_m = e^{(b_m-a_m)^{2}/4 +1}$.
It is easily verified that
\begin{equation}
\begin{aligned}
   &  \int_{a_m}^{b_m} | z_m(r) |^{\alpha_m(r)+1} r^{n-1}dr
         \\
   &= \int_{a_m}^{b_m} \left| (b_m-r)(r-a_m)
         \right|^{\alpha_m(r)+1} r^{n-1}dr  \\
   & \geq  a_m^{n-1} \int_{a_m}^{b_m}
            (b_m-r)^{\alpha_m(r)+1}(r-a_m)^{\alpha_m(r)+1}dr.
\end{aligned} \label{eq13}
\end{equation}
By making a change of variable $r = a_m+(b_m-a_m)t$, we obtain
\begin{equation}
\begin{aligned}
   &  \int_{a_m}^{b_m}  (b_m-r)^{\alpha_m(r)+1}(r-a_m)^{\alpha_m(r)+1}dr
         \\
   & \geq  (b_m-a_m)^{2\tilde{\alpha}_m+3}\int_{0}^{1}
            (1-t)^{\tilde{\alpha}_m+1}t^{\tilde{\alpha}_m+1}dt
            \\
   &= (b_m-a_m)^{2\tilde{\alpha}_m+3}B(\tilde{\alpha}_m+2,
         \tilde{\alpha}_m+2),
 \end{aligned}  \label{eq14}
\end{equation}
where $B(s,t)$ denotes the Bessel function.
Combining \eqref{eq11}--\eqref{eq14}, we observe that
\begin{align*}
   &\  M[u_m;G_m]  \\
   & \leq  \omega_nA_mb_m^{n-1}(b_m-a_m)
            \bigl(b_m-a_m\bigr)^{e}
            \Big[ 1 + \frac{2}{e}(b_m-a_m) +
                   \frac{(b_m-a_m)^{2}}{4} \Big]^{\tilde{\alpha}_m} \\
   &\quad   - \omega_nC_ma_m^{n-1}(b_m-a_m)^{2\tilde{\alpha}_m+3}
            B(\tilde{\alpha}_m+2, \tilde{\alpha}_m+2),
\end{align*}
and hence $M[u_m;G_m] \leq 0$ if
\begin{align*}
   &  C_ma_m^{n-1}(b_m-a_m)^{2\tilde{\alpha}_m+2}
            B(\tilde{\alpha}_m+2, \tilde{\alpha}_m+2) \\
   & \geq  A_mb_m^{n-1}
            \bigl(b_m-a_m\bigr)^{e}
            \left[ 1 + \frac{2}{e}(b_m-a_m) +
                   \frac{(b_m-a_m)^{2}}{4} \right]^{\tilde{\alpha}_m}.
\end{align*}
Then, Theorem \ref{thm1} implies that
every solution  $v \in \mathcal{D}_{Q}(\Omega)$ of {\rm \eqref{eq1}}
is oscillatory in $\Omega$.
\end{example}

\begin{example}\label{ex3}\rm
Suppose that $\Omega$ contains a sequence of balls
$$
   G_k = \{ x \in \mathbb{R}^n;\ | x - a_k| < b_k \}
   \quad (k=1,2,\dots),
$$
where $0 < b_k \leq 1$ and $\lim_{k\to\infty} | a_k| = \infty$.
We assume that
$$
   A(x) \leq A_k, \quad C(x) \geq C_k \quad \text{on } G_k
$$
for some constants $A_k > 0$, $C_k > 0$, and let
\begin{gather*}
    \alpha(x) = \alpha_k(| x- a_k |)  = e^{| x- a_k |+1}-1, \\
\frac{B(x)}{A(x)} = \frac{B_k(x)}{A_k(x)}
              = \tilde{B}_k(| x- a_k|)
                           \frac{x-a_k}{| x-a_k|} 
    = -e^{| x- a_k |+1}  \frac{x-a_k}{| x-a_k|}
\end{gather*}
on each $G_k$.  For any $r > 0$ there is an integer
$m = m(r) \in \mathbb{N}$ such that $\overline{G_m} \subset
\Omega_{r}$, because $\lim_{k\to\infty} | a_k| = \infty$.
Letting
$$
   u_m = b_m - | x - a_m |,
$$
we observe that $u_m \in C^{1}(\overline{G_m};\mathbb{R})$,
$u_m = 0$ on $\partial G_m$, $u_m$ has no zero in $G_m$.
A simple calculation yields
\begin{align*}
   &\nabla u_m + \frac{u_m\log | u_m|}
                 {\alpha_m(x)+1}\nabla\alpha_m(x)
         - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x) \\
   &= - \frac{x-a_m}{| x-a_m|}
         + \bigl(b_m-| x-a_m |\bigr)
           \left(\frac{x-a_m}{| x-a_m|}\right)
           \log\bigl| b_m-| x-a_m |\bigr| \\
   &\quad  + \bigl(b_m-| x-a_m |\bigr)
         \left(\frac{x-a_m}{| x-a_m|}\right),
\end{align*}
and therefore
\begin{align*}
   &  \Bigl|
         \nabla u_m + \frac{u_m\log | u_m|}
                 {\alpha_m(x)+1}\nabla\alpha_m(x)
         - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
         \Bigr|  \\
   &\leq  1 + \bigl(b_m-| x-a_m |\bigr)
            \log\bigl( b_m-| x-a_m |\bigr)
            + b_m  \\
   &\leq  1 + \frac{1}{e} + b_m,
\end{align*}
in view of the inequality
$0 \leq b_m-| x-a_m | \leq b_m$ and $$
   \bigl(b_m-| x-a_m |\bigr)
            \log\bigl( b_m-| x-a_m |\bigr) \leq \frac{1}{e}.
$$
Since $\alpha_m(x)+1 \leq e^{b_m+1}$, we obtain
\begin{equation}
\begin{aligned}
   &  \Bigl| \nabla u_m + \frac{u_m\log | u_m|}
                 {\alpha_m(x)+1}\nabla\alpha_m(x)
         - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
         \Bigr|^{\alpha_m(x)+1}  \\
   &\leq  \Big(1 + \frac{1}{e} + b_m\Big)^{e^{b_m+1}}.
 \end{aligned} \label{eq15}
\end{equation}
Introducing hyperspherical coordinates in the ball $G_m$, we find that
\begin{equation}
\begin{aligned}
   \int_{G_m} | u_m|^{\alpha_m(x)+1}dx
   &= \int_{G_m}
         \bigl(b_m-| x-a_m |\bigr)^{\alpha_m(x)+1}dx
         \\
   &= \int_{0}^{b_m}\int_{S_1}(b_m-r)^{e^{r}+1}r^{n-1}drd\omega
         \\
   &= \omega_n\int_{0}^{b_m}(b_m-r)^{e^{r}+1}r^{n-1}dr.
\end{aligned} \label{eq16}
\end{equation}
By making a change of variable $r/b_m = s$, we obtain
\begin{equation}
\begin{aligned}
   \int_{0}^{b_m}(b_m-r)^{e^{r}+1}r^{n-1}dr
   &= \int_{0}^{1} \bigl(b_m-b_ms\bigr)^{e^{b_ms+1}}(b_ms)^{n-1}
         b_mds  \\
   & \geq  b_m^{e^{b_m+1}+n}\int_{0}^{1}(1-s)^{e^{b_m+1}}s^{n-1}ds
            \\
   &= b_m^{e^{b_m+1}+n}B(e^{b_m+1}+1,n),
\end{aligned}   \label{eq17}
\end{equation}
in light of $0 < b_m \leq 1$.
Combining \eqref{eq15}--\eqref{eq17}, we have
\begin{align*}
   M[u_m;G_m]  
  &\leq  \int_{G_m}
      \Bigl[A_m\Bigl| \nabla u_m + \frac{u_m\log | u_m|}
                 {\alpha_m(x)+1}\nabla\alpha_m(x) \\
  &\quad  - \frac{u_m}{(\alpha_m(x)+1)A_m(x)}B_m(x)
                  \Bigr|^{\alpha_m(x)+1}
  - C_m| u_m |^{\alpha_m(x)+1}      \Bigr]dx   \\
  &\leq  A_m\Big(1 + \frac{1}{e} + b_m\Big)^{e^{b_m+1}}
           \frac{\omega_n}{n}b_m^n
           - C_m\omega_nb_m^{e^{b_m+1}+n}B(e^{b_m+1}+1,n),
\end{align*}
and hence
$$
   M[u_m;G_m] < 0
$$
if
$$
   C_mb_m^{e^{b_m+1}}nB(e^{b_m+1}+1,n)
   > A_m\Big(1 + \frac{1}{e} + b_m\Big)^{e^{b_m+1}}.
$$
It follows from Theorem \ref{thm1} that
every solution  $v \in \mathcal{D}_{Q}(\Omega)$ of  \eqref{eq1}
is oscillatory in $\Omega$.
\end{example}


\section{Generalizations}

In this section we treat the more general elliptic operator $\hat{Q}$
defined by
\begin{align*}
   \hat{Q}[v] & :=  \nabla\cdot\bigl(A(x)|\nabla v|^
               {\alpha(x)-1}\nabla v\bigr)
               - A(x)(\log | v|)|\nabla v|^
               {\alpha(x)-1}\nabla\alpha(x)\cdot\nabla v \\
        &\quad  + |\nabla v|^
               {\alpha(x)-1}B(x)\cdot\nabla v + C(x)| v|^{\alpha(x)-1}v
               \\
        &\quad  + \sum_{i=1}^{\ell} D_{i}(x)| v|^{\beta_{i}(x)-1}v
               + \sum_{j=1}^{L} E_{j}(x)| v|^{\gamma_{j}(x)-1}v,
\end{align*}
where $\beta_{i}(x) > \alpha(x) > \gamma_{j}(x) > 0$, and
$D_{i}(x), E_{j}(x) \in C(\overline{G}; [0,\infty))$ 
($i=1,2,\dots,\ell$; $j=1,2,\dots,L$). The domain $\mathcal{D}_{\hat{Q}}(G)$ 
of $\hat{Q}$ is defined as the same as $\mathcal{D}_{Q}(G)$.
Let $N = \min \{\ell, L\}$ and we define
$$
   \hat{C}(x) =
   \sum_{i=1}^{N} H(\beta_{i}(x),\alpha(x),\gamma_{i}(x); D_{i}(x),E_{i}(x)),
$$
where
\begin{align*}
   &  H(\beta(x),\alpha(x),\gamma(x); D(x),E(x)) \\
   &= \Big(\frac{\beta(x)-\gamma(x)}{\alpha(x)-\gamma(x)}\Big)
      \Big(\frac{\beta(x) - \alpha(x)}
               {\alpha(x)-\gamma(x)}\Big)^{\frac{\alpha(x)-\beta(x)}
               {\beta(x)-\gamma(x)}}
               D(x)^{\frac{\alpha(x)-\gamma(x)}{\beta(x)-\gamma(x)}}
               E(x)^{\frac{\beta(x)-\alpha(x)}{\beta(x)-\gamma(x)}}.
\end{align*}
Then we have the following Lemma which is analogous to Lemma \ref{lem1}
(see \cite{yos12}).

\begin{lemma}[Picone-type inequality for $\hat{Q}$]\label{lem5}
If $v \in \mathcal{D}_{\hat{Q}}(G)$ and $v$ has no zero in $G$,
then we obtain the following Picone-type inequality for any
$u \in C^{1}(G; \mathbb{R})${\rm :}
\begin{align*}
   & - \nabla \cdot \Big(u\varphi(u)
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}\Big) \\
   & \geq  - A(x)\Big| \nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x)\Big|^{\alpha(x)+1} \\
   &\quad  + \bigl(C(x) + \tilde{C}(x)\bigr)| u |^{\alpha(x)+1}\\
   &\quad  + A(x)\Bigl[\Big|\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)
         - \frac{u}{(\alpha(x)+1)A(x)}B(x)\Big|^{\alpha(x)+1} \\
   &\quad   + \alpha(x) \big|\frac{u}{v}\nabla v \big|^{\alpha(x)+1}
   - (\alpha(x)+1)\big|\frac{u}{v}\nabla v
         \big|^{\alpha(x)-1}\Big(\nabla u + \frac{u\log | u|}
         {\alpha(x)+1}\nabla\alpha(x)   \\
   &\quad   - \frac{u}{(\alpha(x)+1)A(x)}B(x)
         \Big)\cdot\big(\frac{u}{v}\nabla v
         \big)\Bigr]
   - \frac{| u|^{\alpha(x)+1}}{| v|^{\alpha(x)+1}}
         \bigl(v \hat{Q}[v]\bigr) \quad \text{in }G.
\end{align*}
\end{lemma}

For a nonempty piecewise smooth bounded domain $G \subset \Omega$,
we define the functional
\begin{align*}
   \hat{M}[u;G] & :=  \int_{G}\Bigl[
                 A(x)\Big| \nabla u + \frac{u\log | u|}
                 {\alpha(x)+1}\nabla\alpha(x)
                 - \frac{u}{(\alpha(x)+1)A(x)}B(x)
                  \Big|^{\alpha(x)+1} \\
          &\quad  - \bigl(C(x)+\hat{C}(x)\bigr)| u |^{\alpha(x)+1}
                \Bigr]dx
\end{align*}
with domain consisting of all real-valued piecewise $C^{1}$-functions
$u$ on $\overline{G}$.

By the same arguments as were used in Lemmas \ref{lem3},
\ref{lem4} and Theorem \ref{thm1} we obtain the following Lemma  and
Theorem.

\begin{lemma}\label{lem6}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$ and
$B(x)/A(x) \in C^{1}(G; \mathbb{R}^n)$.
If there exists a function
$u \in C^{1}(\overline{G}; \mathbb{R})$ such that
$u = 0$ on $\partial G$ and $u$ has no zero in $G$,
$\hat{M}[u;G] \leq 0$ and the hypothesis {\rm (H1)}
of Lemma {\rm \ref{lem3}} holds,
then every solution $v \in \mathcal{D}_{\hat{Q}}(G)$ of
$v\hat{Q}[v] \leq 0$ must
vanish at some point of $\overline{G}$. Moreover,
If there exists a nontrivial function
$u \in C^{1}(\overline{G}; \mathbb{R})$ such that
$u = 0$ on $\partial G$ and $\hat{M}[u;G] < 0$,
then every solution $v \in \mathcal{D}_{\hat{Q}}(G)$ of
$v\hat{Q}[v] \leq 0$ must
vanish at some point of $\overline{G}$.
\end{lemma}

\begin{theorem}\label{thm2}
Assume that $\Omega$ contains a sequence of domains
$G_k$ $(k=1,2,\dots)$ such that $G_k$ are nonempty
bounded domains with piecewise smooth boundaries $\partial G_k$.
If for any $r > 0$
there exist a natural number $m=m(r) \in \mathbb{N}$
such that $\overline{G_m} \subset \Omega_{r}$ and
a nontrivial piecewise $C^{1}$-function
$u_m$ on $\overline{G_m}$ with the
following properties:
\begin{itemize}
   \item[(i)] $\alpha(x) = \alpha_m(x) \in C^{2}(G_m; (0,\infty))$
      and 
\[
B(x)/A(x) = B_m(x)/A_m(x)  \in C^{1}(G_m; \mathbb{R}^n)\quad
\text{on }G_m;
\]
   \item[(ii)] $u_m = 0$ on $\partial G_m$;
   \item[(iii)] $u_m$ has no zero in $G_m$;
   \item[(iv)] $\hat{M}[u_m;G_m] \leq 0$;
   \item[(v)] there exists a function
               $F_m \in C(\overline{G_m}; \mathbb{R})$ such that
                    $F_m \in C^{1}(G_m; \mathbb{R})$ and
                    $$
                       \nabla F_m = \frac{\log | u_m|}
                       {\alpha_m(x)+1}\nabla\alpha_m(x)
                        - \frac{B_m(x)}{(\alpha_m(x)+1)A_m(x)} \quad
                       \text{in }G_m,
                    $$
\end{itemize}
then every solution $v \in \mathcal{D}_{\hat{Q}}(\Omega)$ of
$v\hat{Q}[v] \leq 0$
is oscillatory in $\Omega$.  Furthermore, if $\hat{M}[u_m;G_m] < 0$ in the
hypothesis {\rm (iv)},
then the conclusion is valid without the hypotheses {\rm (iii), (v)}.
\end{theorem}


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