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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 174, pp. 1--8.\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-2012/174\hfil Picone identity]
{Picone-type identity for pseudo p-Laplacian with variable power}

\author[G. Bogn\'ar, O. Do\v{s}l\'y\hfil EJDE-2012/174\hfilneg]
{Gabriella Bogn\'ar, Ond\v{r}ej Do\v{s}l\'y}  % in alphabetical order

\address{Gabriella Bogn\'ar \newline
Department of Analysis, University of Miskolc,
H-3515 Miskolc-Egytemv\'aros, Hungary}
\email{matvbg@uni-miskolc.hu}

\address{Ond\v{r}ej Do\v{s}l\'y \newline
Department of Mathematics and Statistics,
Masaryk University,
Kotl\'a\v{r}sk\'a 2, CZ-611 37 Brno, Czech Republic}
\email{dosly@math.muni.cz}

\thanks{Submitted September 12 2012. Published October 12, 2012.}
\subjclass[2000]{35B06, 35J15, 35J92}
\keywords{Pseudo $p(x)$-Laplacian; Picone-type inequality;
\hfill\break\indent Sturmian comparison theorem}

\begin{abstract}
 A Picone type identity is established for homogeneous differential
 operators involving the pseudo $p$-Laplacian with variable exponent
 $p=p(x)$. Using this identity, it is shown that the classical Sturmian
 theory extends to the associated partial differential equations.
\end{abstract}

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

\section{Introduction}

A Picone type identity for some specific class of equations usually indicates
that one can elaborate a ``reasonable'' Sturmian
oscillation and comparison theory for this class of equations.
The classical form of this identity concerns the second
order linear differential equation
$$
(r(t)x')'+c(t)x=0
$$
and its origin goes back to the beginning of the previous century \cite{Pic}. Since
that time, Picone type identity has been established for various types
of equations and operators. We refer at least to the papers
\cite{A,C-Sw,H-W,J-NA,J-K,J-K-Y-JIA,Kr-2,M-Pf,Sw-3} for more details.


Our research follows this line and it is motivated by the recent papers
of  Yoshida \cite{Y-1,Y-2,Y-3}. These papers concern the partial
differential operators involving the $p$-Laplacian
$\operatorname{div}(a(x)|\nabla u|^{p(x)-2}\nabla u)$ with the variable
power $p(x)$.
Operators of this form find applications in various fields as shown
e.g. in \cite{Ruz}.
The case of the constant power $p(x)\equiv p\in (1,\infty)$ is relatively
deeply developed and the Sturmian theory for these operators was elaborated
in the papers \cite{A-H-1,A-H-2,Dun,J-K-Y, Y-book}. More precisely, in these
papers, the equation of the form
\begin{equation} \label{ac}
\operatorname{div}(a(x)\Vert u\Vert^{p-2}\nabla u) + c(x)|u|^{p-2}u=0,
\end{equation}
is investigated and an important role was played by the fact that
the solution space of this equation is {\it homogeneous} in the sense that
if $u$ is a solution of \eqref{ac} and $k$ is a real constant, then $ku$ is
a solution of \eqref{ac} as well.

However, if the power $p$ in \eqref{ac} is not a constant, as observed in the
above papers \cite{Y-1,Y-2,Y-3}, the solution space of \eqref{ac}
is no longer homogeneous. This ``drawback'' was removed by introducing
an extra term in \eqref{ac} and after this modification the
$p(x)$-Laplacian operator became homogeneous and this enabled to
introduce a Riccati type equation and to establish Picone's identity
and Sturmian theory.

Another motivation are the results in \cite{B-D,Dos} where
the equation with the so-called pseudo $p$-Laplacian
\begin{equation} \label{pseudo}
\sum_{i=1}^n \frac{\partial }{\partial x_i}
\Big(a_i(x)\Phi\bigl(\frac{\partial u}{\partial x_i}\bigr)\Big)+ c(x)\Phi(u)=0,
\quad \Phi(u)=|u|^{p-2}u,
\end{equation}
was investigated and it was shown that similarly to the ``classical''
$p$-Laplacian equation \eqref{ac}, concepts like Riccati type equation,
 Picone's identity, and, in turn, the Sturmian theory, can be established also
for this equation.


In this paper we combine ideas of the above mentioned papers. We allow
the exponent $p$ in the pseudo $p$-Laplacian operator to be a differentiable
function of $x$. After that, this operator is no longer homogeneous
and one needs again to introduce an extra term which cares about
homogeneity of the resulting operator.


To describe our idea in more detail, let $\Omega \in {\mathbb R}^n$ be a
bounded domain with piecewise
smooth boundary $\partial \Omega $
and $p:\mathbb{R}^{n}\to (1,\infty)$  be a differentiable function.
Further, let $u\in C^1(\Omega)$
and let us denote
\[
\Psi(\nabla u):=\Big( \Phi \big( \frac{\partial u}{\partial x_{i}}\big)
,\ldots ,\Phi
\big( \frac{\partial u}{\partial x_{n}}\big) \Big) ,
\]
where now and later on $\Phi (u) :=| u| ^{p(x)-2}u$.
Our aim is to establish Picone type identity
for the pair of operators $q$ and $Q$ defined by
\begin{equation} \label{q}
\begin{split}
q[u] &:=\operatorname{div}\bigl(a(x)\Psi(\nabla u)\bigr)-a(x)\log |
u|  \langle \nabla p(x),\Psi(\nabla u)\rangle +
\langle b(x),\Psi(\nabla u)\rangle \\
&\quad +c(x)\Phi (u)   
\end{split} 
\end{equation}
and
\begin{equation} \label{Q}
\begin{split}
Q[u] &:= \operatorname{div}\bigl(A(x)\Psi(\nabla u)\bigr)
-A(x)\log |
u|  \langle \nabla p(x),\Psi(\nabla u)\rangle +
\langle B(x),\Psi(\nabla u)\rangle \\
&\quad +C(x)\Phi (u),
\end{split}
\end{equation}
where $a,A,b,B,c,C$ are continuous functions and $a(x)>0$,
$A(x)>0$ in $\overline{\Omega}$.
Applying Picone's identity we derive then Leighton type
Sturmian comparison theorem for $Q$ and $q$.


Note that we are not concerned with the existence of a solution of
equations $q(u)=0$ and $Q(u)=0$ in our paper. We refer the reader to
\cite{Ch-F,F-Z,S-Z},
where the variational methods of solvability of nonlinear problems
are demonstrated. These methods are based on the functional analytic
background established in \cite{K-R}.

\section{Homogeneity and Riccati type equation}

First we show that the operator $q$ is homogeneous in the sense that if
$u$ is a solution of $q[u]=0$ then $ku$ is also a solution of
this equation for any
constant $k\ne 0$. So, let $u$ be any solution to $q[u]=0$ and $k\neq 0$ be
any constant. One gets
\begin{align*}
q[ku]&= \operatorname{div}\bigl(a\Psi(k\nabla u\bigr)- a\log|ku|
\langle \Psi(k\nabla u),\nabla p\rangle
+ \langle b,\Psi(k\nabla u)+c \Phi(ku)
\\[2mm]
&= \nabla\Phi(k)a\Psi(\nabla u)+\Phi(k)\operatorname{div}\bigl(a\Psi(\nabla u)\bigr)
-a\Phi(k)\log|k|\langle \Psi(\nabla u),\nabla p\rangle
\\[2mm]
&\quad - a\Phi(k)\log|u|\langle \Psi(\nabla u),\nabla p\rangle
+\Phi(k)\langle b,\Psi(\nabla u)\rangle+\Phi(k)c\Phi(u).
\end{align*}
Since
\begin{equation*}
\nabla \Phi(k)=\nabla \big(| k| ^{p(x)-2}k\big)
 =|k| ^{p(x)-2}k\log | k| \nabla p(x)
\end{equation*}
we see that the terms containing $\log |k|$ cancel, hence
with \eqref{q} we obtain that
\[
q[ku] =\Phi(k) q[u]=0
\]
for any constant $k\ne 0$. Therefore, $q$  is really homogeneous.
The same property is
valid for $Q$.

Observe, that a (sufficiently smooth) function $u$ for which 
$u(\bar x)=0$ at some point $\bar x\in \Omega$ {\it is not} in the domain of
the operators $q$ and $Q$ because the logarithmic term is not
defined for such a function. However, in Picone-type identities
presented later the operator $q$ (and also $Q$) appears in the form
$uq[u]$ only. In this case we define $u\log |u|=0$ for $u=0$ and
hence a function having a zero point in $\Omega$ is in the domain of
the operator $\widetilde q[u]:=uq[u]$ (and also of $\widetilde
Q[u]:=uQ[u]$). With this convention concern in $u=0$ the operators
$\widetilde q$ and $\widetilde Q$ are homogeneous also for $k=0$;
i.e. if $u$ is a solution of $\widetilde q[u]=0$ then $\widetilde
q[ku]=0$ for any $k\in \mathbb{R}$, the same holds for $\widetilde Q$.


Once a homogeneity of a differential operator is established, a natural
idea is to look for an associated Riccati type differential equation.
We denote the
``$p(x)$-norm'' in ${\mathbb R}^n$ by
$\| x\|_{p(x)}:=\big(\sum_{i=1}^{n}| x_{i}| ^{p(x)}\big) ^{1/p(x)}$,
and the ``$q(x)$-norm'' in ${\mathbb R}^n$ by
$\| x\| _{q(x)}=\big(\sum_{i=1}^{n}|x_{i}| ^{q(x)}\big) ^{1/q(x)}$, where
$q(x):=\frac{p(x)}{p(x)-1}$ is the conjugate exponent of $p(x)$.

Introducing the Riccati variable $w:= \frac{a\Psi(\nabla u)}{\Phi(u)}$,
we have
\begin{align*}
\operatorname{div}w
&= \frac{1}{\Phi (u)}\operatorname{div} \bigl(a\Psi(\nabla u)\bigr)+
\big\langle \nabla \big(\frac{1}{\Phi (u)}\big), a\Psi(\nabla u) \big\rangle
\\
&= \frac{1}{\Phi (u)} \operatorname{div} \bigl(a\Psi(\nabla u)\bigr)
-\frac{\log | u|}{\Phi(u)}\langle \nabla p, a\Psi(\nabla u)\rangle -(p-1)
\frac{\langle \nabla u,a\Psi(\nabla u)\rangle }{|u|^p}
\\
&= \frac{1}{\Phi (u)}\operatorname{div} \bigl(a\Psi(\nabla u)\bigr)
-\frac{\log | u|}{\Phi(u)} \langle \nabla p(x), a\Psi(\nabla u)\rangle
-(p-1)a^{1-q}\Vert w\Vert^{q(x)}_{q(x)}.
\end{align*}
Together with \eqref{q} we have the identity
\begin{equation} \label{R-q}
|u|^p\Big[\operatorname{div}w+c(x)+ \langle \frac{b(x)}{a(x)},w\rangle
+(p(x)-1)(a(x))^{1-q(x)} \Vert w\Vert_{q(x)}^{q(x)}\Big]=uq[u].
\end{equation}
Consequently, if $u$ is a solution of $q[u]=0$ then
from \eqref{R-q} one gets the
Riccati type equation
\begin{equation*}                                     %\label{RE}
\operatorname{div} w+c(x)+\langle \frac{b(x)}{a(x)},w\rangle +
(p(x)-1)
(a(x))^{1-q(x)}\| w\| _{q(x)}^{q(x)}=0.            %\label{2}
\end{equation*}

\section{Picone's identity}


First we establish a Picone type identity in its simpler form,
for one operator only. Before formulating it, we recall the modified
Young inequality as given e.g. in \cite{Dos}.

\begin{lemma} \label{L:Young}
For any $\alpha,\beta\in \mathbb{R}^n$ and $p>1$, $q=\frac{p}{p-1}$
we have the inequality
\begin{equation} \label{Young}
G(\alpha,\beta):=\frac{\Vert\alpha\Vert_p^p}{p}-\langle \alpha,\beta\rangle
+\frac{\Vert \beta\Vert_q^q}{q}\geq 0
\end{equation}
with equality if and only if $\Psi(\alpha)=\beta$ for
$\alpha=(\alpha_1,\dots,\alpha_n)$ and
$\Psi(\alpha)=\bigl(\Phi(\alpha_1),\dots,\\ \Phi(\alpha_n)\bigr)$.
\end{lemma}


\begin{theorem} \label{T:Picone1}
Let $u\in C^1(\Omega)$ be such that $a\Psi(\nabla u)\in C^1(\Omega)$
and $u(x)\ne 0$ in $\Omega$. Then for any $y\in C^{1}(\Omega)$
we have the identity (suppressing the argument $x$)
\begin{equation}
\begin{aligned}
&\operatorname{div}\Big(\frac{|y|^p a \Psi(\nabla u)}{\Phi(u)}\Big)\\
&=-c|y|^p+ a\|\nabla y+\frac{y\log|y|}{p}\nabla p-\frac{y}{ap}b\|_{p}^{p}
+\frac{|y|^pq[u]}{\Phi(u)}
\\
&\quad -pa^{1-q}G(a^{q-1}(\nabla y+
\frac{y\log|y|}{p}\nabla p-\frac{y}{ap}b),
\frac{\Phi(y)a\Psi(\nabla u)}{\Phi(u)}),
\end{aligned} \label{div-formula}
\end{equation}
where the function $G$ is given in \eqref{Young}.
\end{theorem}

\begin{proof}
Using \eqref{R-q} and the fact that
$$
\nabla(|y|^{p(x)})=|y|^{p(x)} \Big(\log |y| \nabla p(x)+
p(x)\frac{\nabla y}{y}\Big),
$$
we have
\begin{align*}
&\operatorname{div}\Big(\frac{|y|^pa\Psi(\nabla u)}{\Phi (u)}\Big) \\
&= |y|^p  
\Big[-c-\langle \frac{b}{a},\frac{a\Psi(\nabla u)}{\Phi(u)}\rangle -
(p-1)
a^{1-q}\|\frac{a\Psi(\nabla u)}{\Phi(u)} \|_{q(x)}^{q(x)}
+\frac{q[u] }{\Phi (u)}\Big]  
\\
&\quad +|y|^p\big\langle \big(\log | y| \nabla p
+ p\frac{\nabla y}{y}\big), \frac{a\Psi(\nabla u)}{\Phi(u)}\big\rangle
\\
&= \frac{|y|^p}{\Phi(u)} q[u]-|y|^pc-(p-1) |y|^pa^{1-q}
\| \frac{a\Psi(\nabla u)}{\Phi(u)}\| _{q}^{q} 
\\
&\quad +p\big\langle \big(\nabla y+\frac{y\log | y| }{p}\nabla p
-\frac{y}{ap}b\big),
\frac{a\Phi(y)\Psi(\nabla u)}{\Phi(u)}\big\rangle .
\end{align*}
In the last two terms we factor out $pa^{1-q}$ and
the remaining terms we take as two terms in Young's inequality
with
$$
\alpha= a^{q-1}\Big(\nabla y+\frac{y\log|y|}{p}-\frac{y}{ap}b\Big),
\quad \beta =\frac{a\Phi(y)\Psi(\nabla u)}{\Phi(u)}.
$$
so we add and subtract the term $\frac{\Vert\alpha\Vert_p^p}{p}$. The resulting
identity is
\begin{align*}
&\operatorname{div}\Big(\frac{|y|^pa\Psi(\nabla u)}{\Phi (u)}\Big)\\
&= -c|y|^p+
a\|\nabla y+\frac{y\log|y|}{p}\nabla p-\frac{y}{ap}b\|_{p}^{p}
+\frac{|y|^pq[u]}{\Phi(u)}
\\
&\quad - pa^{1-q}\Big[\frac{a^q}{p}\| \nabla y+
\frac{y\log|y|}{p}\nabla p-\frac{y}{ap}b \|^{p}_{p}
\\
&\quad -\Big\langle a^{q-1}\Big(\nabla y+\frac{y\log|y|}{p}\nabla p-
\frac{y}{ap}b\Big),
\frac{a\Phi(y)\Psi(\nabla u)}{\Phi(u)}\Big\rangle
+\frac{1}{q}\| \frac{a\Phi(y)\Psi(\nabla u)}{\Phi(u)}\|_{q}^{q}\Big]
\end{align*}
which is what we need to prove.
\end{proof}

Next we establish a Picone type identity for the pair of operators
$q$ and $Q$.
Similarly to the case of $p(x)$-Laplacian, we require an extra assumption
on the function $y$ appearing in this identity.
For the sake of later comparison, let us recall
Picone's identity as established in \cite{B-D} for $p(x)\equiv p$
and $b(x)=0=B(x)$ in \eqref{q} and \eqref{Q}.

\begin{proposition} \label{P1}
For sufficiently smooth functions $y,u$ with $u(x)\ne 0$ we have
the identity
\begin{align*}
&\operatorname{div}[y a(x)\Psi(\nabla y)-
|y|^p\frac{A(x)\Psi(\nabla u)}{\Phi(u)}]\\
&= (a(x)-A(x))\Vert \nabla y\Vert^p_p+(C(x)-c(x))|y|^p
 +yq[y]-\frac{|y|^p}{\Phi(u)}Q[u]
 -pA^{1-q}(x) \\
&\quad\times \Big[\frac{A^q(x)}{p}\Vert \nabla y\Vert^p_p
-\langle A^{q-1}(x)\nabla y,\Phi(y)\frac{A(x)\Psi(\nabla u)}{\Phi(u)}\rangle
+\frac{|y|^p}{q}\|\frac{A(x)\Psi(\nabla u)}{\Phi(u)}\|^q_q\Big].
\end{align*}
\end{proposition}

Our computations in the proof of the next theorem follow the general
idea of \cite{Y-1}, but the technical
realization is different because of the difference between $p$-Laplacian
and pseudo $p$-Laplacian.

\begin{theorem} \label{T:Picone}
Let $y$ be a  $C^1(\Omega)$ function
which has no zero in $\Omega$ and  the following hypothesis holds:
\begin{itemize}
\item[(H1)] There exists a function $f\in C^1(\overline{\Omega})$
such that
\begin{equation*}
\nabla f(x)=\frac{\log | y(x)| }{p(x)}\nabla p(x)-
\frac{1}{a(x)p(x)}b(x)\quad \text{in }\Omega.  
\end{equation*}
\end{itemize}
Then for any
$u\in C^{1}(\Omega)$ which has no zero in $\Omega $ and $y\in C^1(\Omega)$
we have the Picone's identity of the form
\begin{equation} \label{divv-formula}
\begin{aligned}
&\operatorname{div}[e^{(1-p(x))f}
ya(x) \Psi\bigl(\nabla(e^fy)\bigr) -\frac{|y|^{p(x)}}{\Phi(u)}
A(x)\Psi(\nabla u)]
\\
&= a(x)\| \nabla y+\frac{y\log | y|
}{p(x)}\nabla p(x)-\frac{y}{a(x)p(x)}b(x)\|_{p(x)}^{p(x)}
\\
&\quad -A(x)\|\nabla y+\frac{y\log | y| }{p(x)}\nabla p(x)-
\frac{y}{A(x)p(x)}B(x)\| _{p(x)}^{p(x)}
+[C(x)-c(x)] |y|^{p(x)}
\\
&\quad -p(x)(A(x))^{1-q(x)}G(\alpha(x),\beta(x))+e^{(1-p(x))f}yq[e^fy]-
\frac{|y|^{p(x)}Q[u]}{\Phi(u)},
\end{aligned}
\end{equation}
where the function $G$ is given in Lemma \ref{L:Young} and
\begin{equation} \label{alpha-beta}
\begin{gathered}
\alpha(x)= (A(x))^{q(x)-1}\Big(
\nabla y+\frac{y\log | y|}{p(x)}\nabla
p(x)-\frac{y}{A(x)p(x)}B(x)\Big),
\\
\beta(x)=\frac{A(x)\Phi(y)\Psi(\nabla u)}{\Phi(u)}.
\end{gathered}
\end{equation}
\end{theorem}

\begin{proof}
The divergence of the second term in the formula \eqref{divv-formula}
(with $a$ replaced by $A$ and $q$ by $Q$) is computed
in the previous theorem. As for the first term, using
the assumption (H1) and the form of $q$, we have
\begin{align*}
&\operatorname{div}[e^{-pf}\bigl(e^fy\bigr)a\Psi\bigl(\nabla
(e^fy)\bigr)]
\\
&=  \langle \nabla\bigl[e^{-pf}\bigl(e^fy\bigr)\bigr],
a\Psi\bigl(\nabla (e^fy)\bigr)\rangle +
e^{(1-p)f}y\operatorname{div}\bigl(a\Psi\bigl(\nabla(e^fy)\bigr)\bigr)
\\
&=\langle e^{-pf}[-\nabla p f-p\nabla f](e^fy)
+ e^{-pf}\nabla(e^fy),a\Psi\bigl(\nabla(e^fy)\bigr)\rangle
\\
&\quad +e^{(1-p)f}y\operatorname{div}\bigl(a\Psi\bigl(\nabla(e^fy)\bigr)\bigr)
\\
&= \Big\langle e^{(1-p)f}y[-f\nabla p-\log|y|\nabla p+
\frac{1}{a}b],a\Psi\bigl(\nabla(e^fy)\bigr)\Big\rangle
\\
&\quad + e^{-pf}a\sum_{i=1}^n \big|\frac{\partial(e^fy)}{\partial x_i}\big|^p
 +e^{(1-p)f}y\operatorname{div}\bigl(a\Psi\bigl(e^fy)\bigr)\bigr)
\\
&=  e^{(1-p)f}y\langle [-\nabla p\log |e^fy|+
\frac{1}{a}b],a\Psi\bigr(\nabla (e^fy)\bigr)\rangle
\\
&\quad +a\sum_{i=1}^n\big|e^{-f}\frac{\partial(e^fy)}{\partial x_i}\big|^p+
e^{(1-p)f}y\operatorname{div}\bigl(a\Psi\bigl(\nabla(e^fy)\bigr)\bigr)
\\
&=  e^{(1-p)f}y\log|e^fy|\langle -\nabla p,
a\Psi\bigr(\nabla(e^fy)\bigr)\rangle+
\langle b,\Psi\bigr(\nabla(e^fy)\bigr)\rangle
\\
&\quad + a\| y\nabla f+\nabla y\|^p_p+
e^{(1-p)f}y\operatorname{div}\bigl(a\Psi\bigl(e^fy\bigr)\bigr)
\\
&=  e^{(1-p)f}yq\bigl[e^fy\bigr]-c|y|^p
+a\| \nabla y+\frac{y\log|y|}{p}\nabla p -\frac{y}{ap}b\|_p^p.
\end{align*}
Therefore,
\[
\operatorname{div}[e^{(1-p)f} ya\Psi\bigl(\nabla(e^fy)\bigr)]
=a\| \nabla y+\frac{y\log | y|}{p}
\nabla p-\frac{y}{ap}b\| _{p}^{p} +e^{(1-p)f} q[e^{f}y]-c|y|^p.
\]
Altogether, we  obtain
\begin{align*}
&\operatorname{div}\Big[e^{(1-p)f}ya\Psi\bigl(\nabla(e^fy)\bigr) -
\frac{A|y|^p\Psi(\nabla u)}{\Phi(u)}\Big]
\\
&= a\| \nabla y+\frac{y\log | y|
}{p}\nabla p-\frac{y}{ap}b\|_{p}^{p} 
 -A\|\nabla y+\frac{y\log | y| }{p}\nabla p-
\frac{y}{Ap}B\| _{p}^{p} \\
&\quad +[C(x)-c(x)] |y|^p  +A \| \nabla y+
\frac{y\log | y| }{p}\nabla p-\frac{y}{Ap}
B(x)\| _{p}^{p}
\\
&\quad -p \big\langle \big(\nabla y+\frac{y\log |y|}{p}\nabla p-
\frac{y}{Ap}B\big),\frac{A\Phi(y)\Psi(\nabla u)}{\Phi(u)}\big\rangle
\\
&\quad +(p-1) A^{1-q}\|
\frac{A\Phi(y)\Psi(\nabla u)}{\Phi(u)}\| _{q}^{q}
+e^{(1-p)f}yq\bigl[e^fy\bigr]-\frac{|y|^pQ[u]}{\Phi(u)}
\end{align*}
which is the required identity.
\end{proof}

\section{Comparison theorem}

Using the previous statement we can now prove the following Leighton
type comparison theorem. For its original version we refer to
\cite{L} and to \cite{Kr-1,M-Pf,Sw-1,Sw-2} for its extension to
elliptic PDE's.

\begin{theorem} \label{T:Sturm}
Suppose that there exists a function $y\in C^1(\Omega)$ with $y=0$
on the boundary $\partial \Omega$ such that {\rm (H1)} holds and
$yq\bigl[e^fy\bigr]\geq 0$ in $\Omega$. Moreover, assume that
\begin{itemize}
\item[(H2)] there exists a function $F\in C^1(\overline{\Omega})$
such that
\begin{equation*}
\nabla F(x)=\frac{\log | y(x)| }{p(x)}\nabla p(x)-
\frac{1}{A(x)p(x)}B(x)\text{ \ in \ }\Omega,  %\label{H1}
\end{equation*}
\end{itemize}
If
\begin{align*} \label{inequality}
&\mathcal{F} (y)\\
&:= \int_{\Omega}\Big\{
a(x)\| \nabla y+\frac{y\log | y|}{p}\nabla p(x)
-\frac{y}{a(x)p(x)}b(x)\|_{p(x)}^{p(x)} 
\\
&\quad -  A(x)\|
\nabla y+\frac{y\log | y| }{p(x)}\nabla p(x)-
\frac{y}{A(x)p(x)}B(x)\| _{p(x)}^{p(x)}
+[C(x)-c(x)] |y|^p \Big\}\,dx
\geq 0,
\end{align*}
then every solution of the equation $Q[u]=0$ has a zero in
$\overline{\Omega}$.
\end{theorem}

\begin{proof}
By contradiction, suppose that there exists a solution $u$ of
$Q[u]=0$ such that $u(x)\ne 0$ in $\overline{\Omega}$. 
By the divergence theorem, using the fact that
$y|_{\partial \Omega}=0$, we have
$$
\int_{\Omega} \operatorname{div}\Big[e^{(1-p(x))f}
a(x) \Psi\bigl(\nabla(e^fy)\bigr) -\frac{|y|^p}{\Phi(u)}
A(x)\Psi(\nabla u)\Big]\,dx=0.
$$
Hence
$$
0\geq \mathcal{F} (y)+\int_{\Omega} p(x)(A(x))^{1-q(x)}
G(\alpha(x),\beta(x))\,dx,
$$
where $\alpha,\beta$ are given by \eqref{alpha-beta} with $a,b$ 
replaced by
$A,B$. Consequently, $G(\alpha,\beta)=0$; i.e.,
$\alpha=\Psi^{-1}(\beta):=(\Phi^{-1}(\beta_1),\dots,\Phi^{-1}(\beta_n))$,
which means that
$$
\nabla y+\frac{y\log | y| }{p(x)}\nabla p(x)-
\frac{y}{A(x)p(x)}B(x)=\frac{y}{u}\nabla u\quad \text{in }\Omega.
$$
Using (H2), $\nabla y+y\nabla F=\frac{y}{u}\nabla u$ which implies
$$
e^{-F(x)}u(x)\nabla \Big[\frac{e^{F(x)}y(x)}{u(x)}\Big]=0,\quad x\in \Omega.
$$
This, together with $y|_{\partial \Omega}=0$ implies that
$y\equiv 0$, which contradicts the assumption that $y$ has no zero
in $\Omega$.
\end{proof}

As a consequence of the previous theorem we have the following
Sturmian comparison result.

\begin{corollary} \label{C1}
Suppose that $a(x)\geq A(x)$, $C(x)\geq c(x)$, and 
$\frac{b(x)}{a(x)}=\frac{B(x)}{A(x)}$ in $\Omega$.
Further, suppose that there exists a differentiable function $y$
such that $y(x)\ne 0$ in $\Omega$, $y|_{\partial \Omega}=0$,
the hypothesis {\rm (H1)} holds, and $yq[e^fy]\geq 0$ in
$\Omega$. Then any solution $u$ of the equation $Q[u]=0$ has a zero
in $\Omega$.
\end{corollary}

\subsection*{Acknowledgments}
This research was carried out as part of the
TAMOP-4.2.1.B-10/2/KONV-2010-0001 project
with support by the European Union, co-financed by the European Social Fund.
The second author is supported
by the Grant GAP 201/11/0768 of the Czech Grant Agency.

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\end{document}