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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 01, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/01\hfil Picone-type inequality]
{Picone-type inequality and Sturmian comparison theorems for
quasilinear elliptic operators with $p(x)$-Laplacians}

\author[N. Yoshida\hfil EJDE-2012/01\hfilneg]
{Norio Yoshida}

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

\thanks{Submitted November 2, 2011. Published January 2, 2012.}
\thanks{Supported by Grant-in-Aid for Scientific Research (20540159),
from the Ministry of \hfill\break\indent
Education, Culture, Sports, Science and
Technology, Japan}
\subjclass[2000]{35B05, 35J92}
\keywords{$p(x)$-Laplacian;  Picone-type inequality;
 quasilinear elliptic operators; \hfill\break\indent
 Sturmian comparison theorem; mixed nonlinearities}

\begin{abstract}
 A Picone-type inequality for quasilinear elliptic operators
 with mixed nonlinearities and with $p(x)$-Laplacian is established,
 and Sturmian comparison theorems are derived on the basis of
 the Picone-type inequality. Generalizations to more general
 quasilinear elliptic operators with $p(x)$-Laplacians and
 specializations to quasilinear ordinary differential operators with
 $p(t)$-Laplacians are shown.
\end{abstract}

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

\section{Introduction}

Recently Picone identities and Sturmian comparison theorems for
$p$-Laplacian equations have been
developed; see for example,
Allegretto \cite{all01},
Allegretto and Huang \cite{allhua98, allhua99},
Bogn\'ar and Do\v{s}l\'y \cite{bogdos03},
Do\v{s}l\'y and \v{R}eh\'ak \cite{dosreh05},
Dunninger \cite{dun95}, Kusano, Jaro\v{s} and Yoshida
\cite{kusjaryos00}, Yoshida \cite{yos09, yos009}.
Picone identities or inequalities play an important role in
establishing Sturmian comparison theorems and oscillation results.

Much current interest has been focused on
various mathematical problems with variable exponent growth condition
(see \cite{harhas10}).
The study of such problems arise from nonlinear elasticity theory,
electrorheological fluids (cf. \cite{ruz00, zhi87}).

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
$p(x) = p$ (constant), where the central dot
denotes the scalar product,
$\nabla =(\partial/\partial x_1,\dots,\partial/\partial x_{n})$ and
$| x|$ denotes the Euclidean length of $x \in \mathbb{R}^{n}$.

Since the pioneering work of Zhang \cite{zha07}, there has been
an increasing interest in studying oscillation problems for
$p(x)$-Laplacian equations (cf. Yoshida \cite{yos11,yos10}).
The $p(t)$-Laplacian ($p(t)>1$) equation
$$
   \bigl(| u'|^{p(t)-2}u'\bigr)'
   + t^{-\theta(t)}g(t,u) = 0,
   \quad t > 0
$$
was treated by Zhang \cite{zha07}, and the half-linear
elliptic inequality with $p(x)$-Laplacian
($p(x)=\alpha(x)+1$, $\alpha(x)>0$)
$$
   vQ[v] \leq 0
$$
was investigated by Yoshida \cite{yos11} via Riccati method, where
\begin{align*}
   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{align*}
We note that $vQ[v] \leq 0$ is {\it half-linear} in the sense that
any constant multiple of a solution $v$ of $vQ[v] \leq 0$
is also a solution of $vQ[v] \leq 0$.  In fact, it can be shown that
$$
   (kv)Q[kv] = | k|^{\alpha(x)+1}vQ[v]\quad (k \in \mathbb{R})
$$
(cf. Yoshida \cite[Proposition 2.1]{yos11}).
For Sturmian comparison theorems for half-linear
elliptic inequalities with $p(x)$-Laplacians we refer to
Yoshida \cite{yos10}.
We mention, in particular, the paper \cite{all07}
by Allegretto in which Picone identity arguments are used.

The objective of this paper is to establish Picone-type inequalities
for the half-linear elliptic operator $q$ defined by
\begin{align*}
   q[u] &:=  \nabla\cdot\bigl(a(x)|\nabla u|^
               {\alpha(x)-1}\nabla u\bigr)
               - a(x)(\log | u|)|\nabla u|^
               {\alpha(x)-1}\nabla\alpha(x)\cdot\nabla u  \\
        &\quad + |\nabla u|^
               {\alpha(x)-1}b(x)\cdot\nabla u
               + c(x)| u|^{\alpha(x)-1}u,
\end{align*}
and the quasilinear elliptic operator $\tilde{Q}$ defined by
\begin{equation}
\begin{split}
   \tilde{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 \\
        &\quad + C(x)| v|^{\alpha(x)-1}v
               + D(x)| v |^{\beta(x)-1}v
               + E(x)| v |^{\gamma(x)-1}v,
\end{split}   \label{eq1}
\end{equation}
and derive Sturmian comparison theorems for
$q$ and $\tilde{Q}$ by using
the Picone-type inequality obtained.
We remark that $\log | v|$ in \eqref{eq1} 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
$v\tilde{Q}[v]$ has no singularities and is
continuous
in $\Omega$. We give the same remarks about $q$.
We note that
$v\tilde{Q}[v] \leq 0$ is not half-linear.

In Section 2 Picone-type inequalities are established for
quasilinear elliptic operators with $p(x)$-Laplacians.
In Section 3 we obtain Sturmian comparison theorems by
utilizing the Picone-type inequality in Section 2.
In Section 4 we present extensions to more general quasilinear
elliptic operators with $p(x)$-Laplacians, and
specializations to quasilinear ordinary differential
operators with $p(t)$-Laplacians.

\section{Picone-type inequality}

In this section we establish Picone-type inequalities when
$Q[v]$ has the super-half-linear
term $D(x)| v |^{\beta(x)-1}v$ and the sub-half-linear term
$E(x)| v |^{\gamma(x)-1}v$, where
$\beta(x) > \alpha(x) > \gamma(x)>0$.

Let $G$ be a bounded domain in $\mathbb{R}^{n}$ with piecewise
smooth boundary $\partial G$.
It is assumed that $a(x), A(x) \in
C(\overline{G}; (0,\infty))$,
$b(x), B(x) \in C(\overline{G}; \mathbb{R}^{n})$,
$c(x), C(x) \in C(\overline{G}; \mathbb{R})$,
$D(x), E(x) \in C(\overline{G}; [0,\infty))$,
$\alpha(x) \in C^{1}(\overline{G}; (0,\infty))$ and
$\beta(x), \gamma(x)$ belong to $C(\overline{G}; (0,\infty))$.

The domain $\mathcal{D}_{q}(G)$ of $q$ is defined as the
set of all functions $u$ of class $C^{1}(\overline{G}; \mathbb{R})$
such that
$a(x)|\nabla u|^{\alpha(x)-1}\nabla u
\in C^{1}(G; \mathbb{R}^{n}) \cap C(\overline{G}; \mathbb{R}^{n})$.
The domain $\mathcal{D}_{\tilde{Q}}(G)$ of $\tilde{Q}$ is defined
similarly.

\begin{theorem}[Picone-type inequality for $\tilde{Q}$]\label{thm1}
If $v \in \mathcal{D}_{\tilde{Q}}(G)$ and $v$ has no zero in $G$,
then we obtain the following Picone-type inequality for any
$u \in C^{1}(G; \mathbb{R})$ which has no zeros in $G$:
\begin{equation}
\begin{split}
 &- \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) |\frac{u}{v}\nabla v |^{\alpha(x)+1}
    - (\alpha(x)+1)|\frac{u}{v}\nabla v
    |^{\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\tilde{Q}[v]\bigr) \quad \mbox{in } G,
\end{split}         \label{eq2}
\end{equation}
where $\varphi(u) = | u |^{\alpha(x)-1}u$ and
$$
   \tilde{C}(x) =
     \Big(\frac{\beta(x)-\gamma(x)}{\alpha(x)-\gamma(x)}\Big)
     \Big(\frac{\beta(x) - \alpha(x)}
               {\alpha(x)-\gamma(x)}\Big)^{\frac{\alpha(x)-\beta(x)}
               {\beta(x)-\gamma(x)}}
               D(x)^{\frac{\alpha(x)-\gamma(x)}{\beta(x)-\gamma(x)}}
               E(x)^{\frac{\beta(x)-\alpha(x)}{\beta(x)-\gamma(x)}}.
$$
\end{theorem}

\begin{proof}
The following relations are obtained by
Yoshida \cite[proof of Theorem 2.1]{yos10}:
\begin{equation}
\begin{split}
- \nabla \cdot \Big(u\varphi(u)
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}\Big)
& =  - \nabla(u\varphi(u))\cdot
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}  \\
&\quad- u\varphi(u)A(x)|\nabla v|^{\alpha(x)-1}
         \nabla\big(\frac{1}{\varphi(v)}\big)\cdot\nabla v
         \\
&\quad- \frac{u\varphi(u)}{\varphi(v)}
         \nabla\cdot\bigl(A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v\bigr),
\end{split} \label{eq3}
\end{equation}

\begin{equation}
\begin{split}
&\nabla(u\varphi(u))\cdot
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}\\
& =  (\alpha(x)+1)A(x)|\frac{u}{v}\nabla v|^{\alpha(x)-1}
     (\nabla u)\cdot\left(\frac{u}{v}\nabla v\right)
         \\
&\quad+ A(x)u(\log | u|)\frac{\varphi(u)}
         {\varphi(v)}|\nabla v|^{\alpha(x)-1}
         \nabla\alpha(x)\cdot\nabla v
\end{split} \label{eq4}
\end{equation}
and
\begin{equation}
\begin{split}
& u\varphi(u)A(x)|\nabla v|^{\alpha(x)-1}
         \nabla\big(\frac{1}{\varphi(v)}\big)\cdot\nabla v\\
& =  - A(x)\alpha(x)|\frac{u}{v}\nabla v|^{\alpha(x)+1}
- \frac{u\varphi(u)}{\varphi(v)}A(x)
 (\log | v|)|\nabla v|^{\alpha(x)-1}
  \nabla\alpha(x)\cdot\nabla v.
\end{split}   \label{eq5}
\end{equation}
From \eqref{eq1} it follows that
\begin{equation}
\begin{split}
& \frac{u\varphi(u)}{\varphi(v)}
         \nabla\cdot\bigl(A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v\bigr)  \\
& =  \frac{u\varphi(u)}{\varphi(v)}
         \Bigl(\tilde{Q}[v] + A(x)(\log | v|)|\nabla v|^
               {\alpha(x)-1}\nabla\alpha(x)\cdot\nabla v  \\
& - |\nabla v|^{\alpha(x)-1}B(x)\cdot  \nabla v
 - C(x)| v|^{\alpha(x)-1}v
         - D(x)| v |^{\beta(x)-1}v
         - E(x)| v |^{\gamma(x)-1}v \Bigr)    \\
& =  \frac{u\varphi(u)}{\varphi(v)}\tilde{Q}[v]
 + \frac{u\varphi(u)}{\varphi(v)}A(x)(\log | v|)
 |\nabla v|^{\alpha(x)-1}\nabla\alpha(x)\cdot\nabla v
         \\
&\quad- \frac{u\varphi(u)}{\varphi(v)}|\nabla v|^{\alpha(x)-1}
         B(x)\cdot\nabla v    \\
&\quad- C(x)| u|^{\alpha(x)+1}- | u|^{\alpha(x)+1}
    \Bigl(D(x)| v|^{\beta(x)-\alpha(x)}
        + \frac{E(x)}{| v|^{\alpha(x)-\gamma(x)}}\Bigr).
\end{split}    \label{eq6}
\end{equation}
We remark that for any fixed $x \in G$, Young's inequality
$$
   ab \leq \frac{a^{p(x)}}{p(x)} + \frac{b^{q(x)}}{q(x)}
   \quad \Big(p(x) > 1,\ \frac{1}{p(x)} + \frac{1}{q(x)} = 1\Big)
$$
holds for any $a \geq0$, $b \geq 0$.
Hence, the following inequality holds:
\begin{equation}
    D(x)| v|^{\beta(x)-\alpha(x)}
        + \frac{E(x)}{| v|^{\alpha(x)-\gamma(x)}}
     \geq \tilde{C}(x)  \label{eq7}
\end{equation}
(cf. Jaro\v{s}, Kusano and Yoshida \cite[p.717]{jarkusyos02}).
Combining \eqref{eq6} with \eqref{eq7} yields the
inequality
\begin{equation}
\begin{split}
&\frac{u\varphi(u)}{\varphi(v)}
         \nabla\cdot\bigl(A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v\bigr)  \\
& \leq  \frac{u\varphi(u)}{\varphi(v)}\tilde{Q}[v]
         + \frac{u\varphi(u)}{\varphi(v)}A(x)(\log | v|)
         |\nabla v|^{\alpha(x)-1}\nabla\alpha(x)\cdot\nabla v
         \\
&\quad- \frac{u\varphi(u)}{\varphi(v)}|\nabla v|^{\alpha(x)-1}
         B(x)\cdot\nabla v
 - \bigl(C(x) + \tilde{C}(x)\bigr)| u|^{\alpha(x)+1}.
\end{split}   \label{eq8}
\end{equation}
Combining \eqref{eq3}--\eqref{eq5} and
\eqref{eq8}, we derive
\begin{align*}
&- \nabla \cdot \Big(u\varphi(u)
         \frac{A(x)|\nabla v|^{\alpha(x)-1}
         \nabla v}{\varphi(v)}\Big) \\
& \geq  \bigl(C(x) + \tilde{C}(x)\bigr)| u|^{\alpha(x)+1}
 + A(x)\Bigl[\alpha(x)|\frac{u}{v}\nabla v |^{\alpha(x)+1}  \\
& - (\alpha(x)+1)|\frac{u}{v}\nabla v
|^{\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\varphi(u)}{v\varphi(v)}\bigl(v\tilde{Q}[v]\bigr),
\end{align*}
which is equivalent to \eqref{eq2}.
\end{proof}

\begin{theorem}[Picone-type inequality for $q$ and $\tilde{Q}$]
\label{thm2}
Assume that  $\alpha(x)$ belongs to $C^{2}(G; (0,\infty))$,
$b(x)/a(x) \in C^{1}(G; \mathbb{R}^{n})$,
and that $u \in C^{1}(G; \mathbb{R})$,
$u$ has no zero in $G$,
and the following hypothesis holds:
\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
$$
 \nabla f = \frac{\log | u|} {\alpha(x)+1}\nabla\alpha(x)
 - \frac{b(x)}{(\alpha(x)+1)a(x)} \quad \mbox{in } G.
$$
\end{itemize}
If $e^{f}u \in \mathcal{D}_{q}(G)$, $v \in \mathcal{D}_{Q}(G)$
and $v$ has no zero in $G$,
then we obtain the  Picone-type inequality:
\begin{equation}
\begin{split}
&\nabla \cdot \Bigl(e^{-(\alpha(x)+1)f}(e^{f}u)a(x)
         |\nabla(e^{f}u)|^{\alpha(x)-1}\nabla(e^{f}u)
  - \frac{u\varphi(u)}{\varphi(v)}A(x)|\nabla v
         |^{\alpha(x)-1}\nabla v\Bigr) \\
& \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 - 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) - 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}
         \\
&\quad - (\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]
         \\
&\quad+ e^{-(\alpha(x)+1)f}(e^{f}u)q[e^{f}u]
 - \frac{| u|^{\alpha(x)+1}}{| v|^{\alpha(x)+1}}
 \bigl(v\tilde{Q}[v]\bigr) \quad \mbox{in } G.
\end{split} \label{eq9}
\end{equation}
\end{theorem}

\begin{proof} The following identity holds:
\begin{equation}
\begin{split}
& \nabla \cdot \Big(e^{-(\alpha(x)+1)f}(e^{f}u)a(x)
        |\nabla(e^{f}u)|^{\alpha(x)-1}\nabla(e^{f}u)\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} \\
&\quad- c(x)| u |^{\alpha(x)+1}+ e^{-(\alpha(x)+1)f}(e^{f}u)q[e^{f}u]
\end{split}  \label{eq10}
\end{equation}
(see Yoshida \cite[proof of Theorem 2.2]{yos10}).
The Picone-type inequality \eqref{eq9} follows by
combining \eqref{eq2} with \eqref{eq10}.
\end{proof}


\section{Sturmian comparison theorems}

On the basis of the Picone-type inequality \eqref{eq9}, we can
establish Sturmian comparison theorems for $q$ and $\tilde{Q}$.

\begin{lemma}\label{lem1} 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}

\begin{theorem}[Sturmian comparison theorem]\label{thm3}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$ and \\
$b(x)/a(x), B(x)/A(x) \in C^{1}(G; \mathbb{R}^{n})$.
Assume that there exists
$u \in C^{1}(\overline{G}; \mathbb{R})$ such that
$u = 0$ on $\partial G$, $u$ has no zero in $G$,
the hypothesis {\rm (H1)} of Theorem  \ref{thm2} holds
and that
\begin{itemize}
\item[(H2)] there is a function
 $F \in C(\overline{G}; \mathbb{R})$ such that
 $F \in C^{1}(G; \mathbb{R})$ and
 $$
 \nabla F = \frac{\log | u|} {\alpha(x)+1}\nabla\alpha(x)
  - \frac{B(x)}{(\alpha(x)+1)A(x)} \quad \mbox{in } G.
$$
\end{itemize}
If the following conditions are satisfied
\begin{itemize}
   \item[(i)] $e^{f}u \in \mathcal{D}_{q}(G)$ and
                $(e^{f}u)q[e^{f}u] \geq 0$ in $G$;
   \item[(ii)]
   \begin{align*}
    V_{G}[u]
 &:=  \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- 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) - c(x)\bigr)
    | u|^{\alpha(x)+1}\Bigr]dx \geq 0,
    \end{align*}
\end{itemize}
then every solution $v \in \mathcal{D}_{\tilde{Q}}(G)$ of
$v\tilde{Q}[v] \leq 0$ must
vanish at some point of $\overline{G}$.
\end{theorem}

\begin{proof}
 Suppose to the contrary that there exists a solution
$v \in \mathcal{D}_{\tilde{Q}}(G)$ of $v\tilde{Q}[v] \leq 0$ such that
$v$ has no zero on $\overline{G}$.
Integrating the Picone-type inequality \eqref{eq9} over $G$
and using the divergence theorem, we obtain
$$
   0  \geq  V_{G}[u] + \int_{G}W(u,v)\,dx \geq 0,
$$
from which we observe
$$
   \int_{G}W(u,v)\,dx = 0,
$$
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*}
From Lemma \ref{lem1} we see that
$$
\nabla u + \frac{u\log | u|}
{\alpha(x)+1}\nabla\alpha(x)
- \frac{u}{(\alpha(x)+1)A(x)}B(x) \equiv \frac{u}{v}\nabla v
\quad \mbox{in } G;
$$
that is,
$$
   \nabla u + u\nabla F \equiv \frac{u}{v}\nabla v
   \quad \mbox{in } G,
$$
which is equivalent to
$$
   e^{-F}v\nabla\big(e^{F}\frac{u}{v}\big) \equiv 0 \quad \mbox{in } G.
$$
Therefore, there exists a constant
$k_0$ such that $e^{F}u/v = k_0$ in $G$ and hence on
$\overline{G}$ by continuity.
Since $u = 0$ on $\partial G$, we see that $k_0 = 0$,
which contradicts the hypothesis that $u$ is nontrivial.
The proof is complete.
\end{proof}

\begin{corollary}\label{cor1}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$,
$b(x)/a(x), B(x)/A(x) \in C^{1}(G; \mathbb{R}^{n})$.
Assume that
\begin{itemize}
   \item[(i)] $\displaystyle \frac{b(x)}{a(x)} = \frac{B(x)}{A(x)}$
  in  G;

   \item[(ii)] $a(x) \geq A(x)$,
   $C(x)+\tilde{C}(x) \geq c(x)$  in $G$.
\end{itemize}
If there exists a function $u \in C^{1}(\overline{G}; \mathbb{R})$
with the properties that $u = 0$ on $\partial G$,
$u$ has no zero in $G$,
the hypothesis {\rm (H1)} of
Theorem \ref{thm2} holds and {\rm (i)} of
Theorem {\rm \ref{thm3}} holds,
then every solution $v \in \mathcal{D}_{\tilde{Q}}(G)$ of
$v\tilde{Q}[v] \leq 0$ must
vanish at some point of $\overline{G}$.
\end{corollary}

\begin{proof}
 Conditions (i), (ii) imply that $V_{G}[u] \geq 0$ for
any $u \in C^{1}(\overline{G}; \mathbb{R})$ and (H2) is
the same as (H1).  The conclusion follows from
Theorem \ref{thm3}.
\end{proof}

\begin{remark}\label{rmk1}\rm
When specialized to the case where $D(x) = E(x) \equiv 0$,
our results reduce to those of Yoshida
\cite[Theorem 3.1 and Corollary 3.1]{yos10}.
\end{remark}


\section{Generalizations and specializations}

First we derive extensions to more general quasilinear elliptic
operators with $p(x)$-Laplacians.  Let the quasilinear elliptic
operator $\hat{Q}$ be 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}^{m} E_{j}(x)| v|^{\gamma_{j}(x)-1}v,
\end{align*}
where $\beta_{i}(x) > \alpha(x) > \gamma_{j}(x) > 0$, and
$D_{i}(x), E_{j}(x) \in C(\overline{G}; [0,\infty))$
($i=1,2,\dots,\ell$; $j=1,2,\dots,m$). The domain
$\mathcal{D}_{\hat{Q}}(G)$ of $\hat{Q}$ is defined as the same
as $\mathcal{D}_{\tilde{Q}}(G)$.
Let $N = \min \{\ell, m\}$ 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*}
Applying Young's inequality, we obtain
\begin{align*}
   &\sum_{i=1}^{\ell} D_{i}(x)| v|^{\beta_{i}(x)-\alpha(x)}
         + \sum_{j=1}^{m} E_{j}(x)| v|^{\gamma_{j}(x)-\alpha(x)}
         \\
   & \geq  \sum_{i=1}^{N} \Big( D_{i}(x)| v|^{\beta_{i}(x)-\alpha(x)}
            +\frac{E_{i}(x)}{| v|^{\alpha(x)-\gamma_{i}(x)}}\Big)
            \\
   & \geq  \sum_{i=1}^{N}
    H(\beta_{i}(x),\alpha(x),\gamma_{i}(x); D_{i}(x), E_{i}(x))
    = \hat{C}(x).
\end{align*}
In the proof of Theorem \ref{thm1}, we use the above
inequality instead of \eqref{eq7}, and
observe that the Picone-type inequality
\eqref{eq2} holds  for $\tilde{Q}[v]$
and $\tilde{C}(x)$ replaced by $\hat{Q}[v]$ and $\hat{C}(x)$,
respectively.
Therefore we conclude that Theorems \ref{thm1}--\ref{thm3},
Corollary \ref{cor1} remain true if
we replace $\tilde{Q}[v]$ and $\tilde{C}(x)$ by $\hat{Q}[v]$ and
$\hat{C}(x)$, respectively.

For example, we state the analogue of Corollary \ref{cor1}.

\begin{corollary}\label{cor2}
Let $\alpha(x) \in C^{2}(G; (0,\infty))$,
$b(x)/a(x), B(x)/A(x) \in C^{1}(G; \mathbb{R}^{n})$.
Assume that
\begin{itemize}
  \item[{\rm (i)}]
   $\displaystyle\frac{b(x)}{a(x)} = \frac{B(x)}{A(x)}$ in $G$;

  \item[{\rm (ii)}] $a(x) \geq A(x)$,  $C(x)+\hat{C}(x) \geq c(x)$
        in $G$.
\end{itemize}
If there exists a function $u \in C^{1}(\overline{G}; \mathbb{R})$
with the properties that $u = 0$ on $\partial G$,
$u$ has no zero in $G$,
the hypothesis {\rm (H1)} of
Theorem  \ref{thm2} holds and {\rm (i)} of
Theorem  \ref{thm3} holds,
then every solution $v \in \mathcal{D}_{\hat{Q}}(G)$ of
$v\hat{Q}[v] \leq 0$ must
vanish at some point of $\overline{G}$.
\end{corollary}

Next we consider the special case where $n = 1$,
$b(x) = B(x) \equiv 0$.
We let $x_1 = t$, $G = (t_1,t_{2})$, and define
$q_0$ and $Q_0$ by
\begin{gather}
   q_0[y] :=
      \big(a(t)| y'|^{\alpha(t)-1}y'\big)'
      - a(t)(\log\,| y|) | y' |^{\alpha(t)-1}\alpha'(t)y'
      + c(t)| y|^{\alpha(t)-1}y,  \label{eq11}\\
\begin{aligned}
   Q_0[z] &:=
      \big(A(t)| z'      |^{\alpha(t)-1}z'\big)'
      - A(t)(\log\,| z|) | z' |^{\alpha(t)-1}\alpha'(t)z' \\
  &\quad + C(t)| z|^{\alpha(t)-1}z
      + D(t)| z|^{\beta(t)-1}z
      + E(t)| z|^{\gamma(t)-1}z,
 \end{aligned}     \label{eq12}
\end{gather}
where the coefficients appearing in \eqref{eq11} and \eqref{eq12}
are supposed to satisfy the same conditions as in Section 2.
The domains $\mathcal{D}_{q_0}(I)$, $\mathcal{D}_{Q_0}(I)$ are
defined as in Section 2, where $I = (t_1,t_{2})$.

\begin{theorem}\label{thm4}
Let $\alpha(t) \in C^{2}(I; (0,\infty)) \cap C^{1}(\overline{I};
(0,\infty))$.
Assume that there exists a function
$y \in C^{1}(\overline{I}; \mathbb{R})$ such that
$y(t_1) = y(t_{2}) = 0$, $y$ has no zero in $I$,
and the following hypothesis is satisfied:
\begin{itemize}
   \item[(H1')] there is a function
 $f \in C(\overline{I}; \mathbb{R})$ such that
 $f \in C^{1}(I; \mathbb{R})$ and
$$
 f'(t) = \frac{\log | y|}{\alpha(t)+1}\alpha'(t)
 \quad \mbox{in } I.
$$
\end{itemize}
If $e^{f}y \in \mathcal{D}_{q_0}(I)$,
$(e^{f}y)q_0[e^{f}y] \geq 0 \quad \mbox{in } I$,
and
\begin{align*}
 V_{I}[u] & =  \int_{I}\Bigl[
   \bigl(a(t) - A(t)\bigr)\big| y'
   + \frac{y\log | y|}{\alpha(t)+1}\alpha'(t)
   \big|^{\alpha(t)+1}\\
&\quad  +\bigl(C(t)+\tilde{C}(t) - c(t)\bigr)
     | y|^{\alpha(t)+1}\Bigr]dt \geq 0,
\end{align*}
then every solution $z \in \mathcal{D}_{Q_0}(I)$
of $zQ_0[z] \leq 0$ must
vanish at some point of $\overline{I}$.
\end{theorem}

The proof of the above theorem follows from Theorem \ref{thm3}.


\begin{corollary}\label{cor3}
Let $\alpha(t) \in C^{2}(I; (0,\infty)) \cap C^{1}(\overline{I};
(0,\infty))$.
Assume that there is a function
$y \in C^{1}(\overline{I}; \mathbb{R})$ such that
$y(t_1) = y(t_{2}) = 0$, $y$ has no zero in $I$,
and the hypothesis {\rm (H1')} of
Theorem \ref{thm4} holds.
If $e^{f}y \in \mathcal{D}_{q_0}(I)$,
$(e^{f}y)q_0[e^{f}y] \geq 0$ in  $I$,
and
$$
   a(t) \geq A(t),\ \  C(t)+\tilde{C}(t) \geq c(t) \quad \mbox{in } I,
$$
then every solution $z \in \mathcal{D}_{Q_0}(I)$
of $zQ_0[z] \leq 0$ must
vanish at some point of $\overline{I}$.
\end{corollary}


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\end{document}