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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 53, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/53\hfil Comparison theorems]
{Comparison theorems for Riccati inequalities arising in
the theory of PDE's with $p$-Laplacian}

\author[O. Do\v sl\'y, S. Fi\v snarov\'a, R. Ma\v r\'ik\hfil
EJDE-2011/53\hfilneg]
{Ond\v rej Do\v sl\'y, Simona Fi\v snarov\'a, Robert Ma\v r\'ik}
 % in alphabetical order

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

\address{Simona Fi\v snarov\'a \newline
Department of Mathematics, Mendel University in
Brno, Zem\v ed\v elsk\' a 1, CZ-613 00 Brno, Czech Republic}
\email{fisnarov@mendelu.cz}

\address{Robert Ma\v r\'ik \newline
Department of Mathematics, Mendel University in
Brno, Zem\v ed\v elsk\' a 1, CZ-613 00 Brno, Czech Republic}
\email{marik@mendelu.cz}

\thanks{Submitted February 18, 2011. Published April 13, 2011.}
\thanks{Supported by grants P201/10/1032 and P201/11/0768 from the Czech
Science Foundation.}
\subjclass[2000]{35J92, 35B05, 35B09}
\keywords{PDE with p-Laplacian; power comparison theorem;
\hfill\break\indent Riccati inequality; half-linear ODE}

\begin{abstract}
  In this article, we study a Riccati inequality that appears
  in the theory of partial differential equations with $p$-Laplacian.
  Our results allow to compare existence and nonexistence of positive
  solutions for Riccati type inequalities which are associated with
  equations with different powers $p$.
\end{abstract}

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

\newcommand{\norm}[1]{\|#1\|}
\newcommand{\inner}[2]{\langle#1,#2\rangle}

\section{Introduction}

In this article, we consider a problem which is closely related to the
second-order elliptic half-linear differential operator; i. e., the
operator with the $p$-Laplacian
\[
  \Delta_p u(x)=\operatorname{div}(\norm{\nabla u(x)}^{p-2}\nabla u(x))
\]
and signed power-type nonlinearity of degree $p-1$:
\begin{equation}   \label{eq:L[u]}
  L[u](x):=\Delta_p u(x)+c(x)|u(x)|^{p-2}u(x),
\end{equation}
where $p> 1$ and the norm $\norm{\cdot}$ is the Euclidean norm.
The corresponding differential equation
\begin{equation}
  L[u]=0
  \label{eq:E}
\end{equation}
attracted a considerable attention in the last years because of its
applications in physics, glaceology, and biology, see the recent book
\cite{D-R} which summarizes results related to this equation up to
2005 and also the book \cite{Ru} which deals with the application
aspects of the problem.

Many problems related to the theory of equation \eqref{eq:E} can
be studied using the corresponding Riccati type operator
\begin{equation}
  \label{eq:R[w]}
  R[ w](x):=\operatorname{div}  w(x)+c(x)+(p-1)\norm{ w(x)} ^q,
\end{equation}
where $q=\frac{p}{p-1}$ is the conjugate number to the number $p$. In
this paper we study the Riccati type partial differential inequality
\begin{equation}
  R[ w]\leq 0\label{eq:R[w]leq}.
\end{equation}
Throughout the paper we suppose for simplicity that $c(x)$ is a
H\"older continuous function on a domain with piecewise smooth
boundary
$\Omega\subseteq \mathbb{R}^n$ and the domain of the operators
\eqref{eq:L[u]} and \eqref{eq:R[w]} is the set of $C^2(\Omega,\mathbb{R})$ and
$C^1(\Omega,\mathbb{R}^n)$ functions,
respectively.

The oscillation theory for \eqref{eq:E} is similar to the classical
oscillation theory of linear second order differential equations.
Among others, it turns out that the Sturm type comparison theorems
extend to \eqref{eq:E} and equations can be classified as oscillatory
or nonoscillatory. There are many results which guarantee that
equation \eqref{eq:E} is oscillatory; i. e., it possesses no positive
solution on any exterior domain in $\mathbb{R}^n$. This is partly due to
the fact that each known oscillation criterion for the half-linear
ordinary differential equation
\begin{equation}
  \label{eq:E_ODE}
  \bigl(r(t)|u'|^{p-2}u'\bigr)'+c(t)|u|^{p-2}u=0,
  \quad ({}'=\frac{\mathrm{d}}{\mathrm{d}t}),
\end{equation}
can be extended easily to equation \eqref{eq:E} using results of
\cite{D-M} and \cite{JKY2000}. Roughly speaking, oscillation of
\eqref{eq:E} can be deduced if \eqref{eq:E} is a majorant (in the
sense of integral average over spheres) of some radially symmetric
equation, which can be reduced into an oscillatory ordinary
differential equation. There is also an alternative approach (see
 \cite{M2}), which is based on the fact that the substitution
$ w=\frac{\norm{\nabla u}^{p-2}\nabla u}{|u|^{p-2}u}$ transforms a
nonzero solution of \eqref{eq:E} into a solution of the Riccati type
equation
\begin{equation}
  R[ w]=0.
  \label{eq:R[w]=}
\end{equation}
Then one can employ integration over balls to convert \eqref{eq:R[w]=}
into an inequality in one variable and finally to follow known methods from
the one-dimensional case, to finish the proof of the corresponding
oscillation criteria. This approach allows (among others) to deal with
more general unbounded domains than exterior ones. In the application of
the Riccati technique in proofs of oscillation criteria, we prove in
fact that the Riccati equation \eqref{eq:R[w]=} has no solution on the
domain under consideration (usually the complement of a ball
centered at the origin with arbitrarily large radius). Criteria
for the nonexistence of solutions of \eqref{eq:R[w]leq} and
\eqref{eq:R[w]=} have been derived in \cite{MEJDE}.

In contrast to a voluminous literature devoted to oscillation
criteria, there are only a few nonoscillation criteria, even for the
linear operator
\begin{equation}
  L_2[u]:=\Delta u+c(x)u\label{eq:L_2}
\end{equation}
and the linear equation $L_2[u]=0$, which is a special case $p=2$ in
\eqref{eq:E}.  Neglecting some trivial results based on a comparison
with radially symmetric nonoscillatory majorant, we have only a few
results based on the investigation of positive solutions of the inequality
$L_2[u]\leq 0$ (see \cite{A-T86,E-W}) or of the Riccati equation
$R_2[ w]=0$ with the operator
\begin{equation}
  R_2[ w]=\operatorname{div}  w+c(x)+\norm{ w}^2\label{eq:R_2}
\end{equation}
and the inequality $R_2[ w]\leq 0$ {(see \cite{E-W})}.

The approach based on the investigation of the inequality $L_2[u]\leq 0$
can be also used
for half-linear equations. Indeed, Allegretto and Huang \cite{A-H}
used Picone's identity and Harnack's inequality to prove the following
theorem ($g$ and $g_1$ are supposed to belong to $L^{n/p}(\Omega)\cap
L^{\infty}_{\mathrm{loc}}(\Omega)$).

\begin{theorem}   \label{th:Allegretto}
  Suppose that the inequality $-\Delta_pu\geq g_1|u|^{p-2}u$ has a
  positive solution in $\Omega$. If $g\leq g_1$ in $\Omega$, then so does
  the inequality $-\Delta_pu=g|u|^{p-2}u$.
\end{theorem}

Eliason and White \cite{E-W} proved the following theorem for linear
operator $L_2$ defined on $\mathbb{R}^2$. As mentioned in \cite{E-W}, this
result extends to the more general operator
\[
  \operatorname{div}(r(x)\nabla u)+c(x)u
\]
with an elliptic matrix $r(x)$ and $x\in\mathbb{R}^n$.

\begin{theorem}\label{th:eliason}
  The inequality $R_2[ w]\leq 0$ has a conservative $C^1(G)$ vector field solution
  $ w$ on a subdomain $G\subseteq \Omega\subseteq \mathbb{R}^2$ if and
  only if the equation $R_2[ w]=0$ has one; and this holds if and only if
  the equation $L_2[u]=0$ has a positive $C^2(G)$ solution on $G$.
\end{theorem}

The aim of this article is to extend Theorem \ref{th:eliason} to
half-linear equations. This extension shows that the associated
Riccati inequality \eqref{eq:R[w]leq} plays an important role not only
in oscillation criteria, but also in problems related to the existence
of (eventually) positive solutions, which are closely related to
nonoscillation criteria.

Another aim of this article is to prove some comparison results for the
existence of a solution of the Riccati type inequality.  It is a well
known fact from the theory of ordinary half-linear equations
\eqref{eq:E_ODE}, that bigger $p$ speeds up oscillation of the
equation, see \cite{Rehak, Japonci}.  Another approach which allows to
compare oscillatory properties of half-linear differential equations
with different power in nonlinearity appeared in works
\cite{DF,DP, E-S}. More precisely, the oscillation properties of
half-linear equations are studied within the framework of the linear
equations, as the following theorem shows.

\begin{theorem}[{\cite[Theorem 1 and Theorem 2]{DF}}]\label{th:DF}
  Denote $\Phi(x)=|x|^{p-2}x$ and suppose that the equation
  \begin{equation}
    \label{eq:DF0}
    \bigl(\widetilde r(t)\Phi(x')\bigr)'+\widetilde c(t)\Phi(x)=0
  \end{equation}
  is nonoscillatory and possesses a positive solution $h(t)$ such that
  $h'(t)\neq 0$ for large $t$. Consider the equations
  \begin{equation}
    \label{eq:DF1}
    \bigl(r(t)\Phi(x')\bigr)'+c(t)\Phi(x)=0
  \end{equation}
  and
  \begin{equation}
    \label{eq:DF2}
    (R(t)y')'+\frac p2 C(t) y=0,
  \end{equation}
  where
  $$
  C(t)=h(t)\Bigl[\Bigl((r(t)-\widetilde r(t))\Phi(h'(t))\Bigr)'+(c(t)-\widetilde c(t))\Phi(h(t))\Bigr]
  $$
  and $R(t)=r(t)h^2(t)|h'(t)|^{p-2}$.
  \begin{enumerate}
  \item If $p\geq 2$ and \eqref{eq:DF2} is nonoscillatory, then
    \eqref{eq:DF1} is also nonoscillatory.
  \item If $p\in(1, 2]$ and \eqref{eq:DF2} is oscillatory, then
    \eqref{eq:DF1} is also oscillatory.
  \end{enumerate}
\end{theorem}

The second aim of this paper is to provide a version of Theorem
\ref{th:DF} suitable for a differential inequality which appears in
the theory of \eqref{eq:E}. In addition to the fact that we introduce a
multidimensional version, we also provide more freedom in comparison.
More precisely, we follow the idea suggested in \cite{F-M} and the
equation which is used as a replacement for \eqref{eq:DF2} need not to
be linear. However, we do not formulate the comparison theorems
directly for the second order PDE's, but for the corresponding Riccati
type inequalities. For an explanation and more details see Remarks
\ref{rem} and \ref{rem2} at the end of the paper.


\section{Preliminary results}

In this section we present some technical lemmas which allow us to
formulate our main results in the last section.

First of all, we
find (in Lemma~\ref{lemma:P} below) an upper and lower estimate for a
function
\begin{equation}
  \label{eq:P}
  P( u_1, u_2)=\frac{\norm{ u_1}^p}p-\inner{ u_1}{ u_2}+\frac{\norm{ u_2}^q}{q}
\end{equation}
which appears frequently in the qualitative theory of equations with
$p$-Laplacian.  In the proof of Lemma \ref{lemma:P} we show that the
problem can be reduced to an inequality for a function in one
variable, which is studied in Lemma \ref{L1} below. Further, we derive
an inequality between two Riccati type operators. This inequality is
used to prove our main results.

\begin{lemma}    \label{L1}
Let $q=\frac{p}{p-1}$ and consider the function
$$
  f(t)=\frac{|t|^q}{q}-t+\frac{1}{p}-\frac{4}{\alpha 2^\alpha}|t-1|^\alpha,
$$
where $\alpha\in [2,q]$ in the case $1<p\leq 2$ and $\alpha\in [q,2]$
for $p\geq 2$.
Then $f(t)\geq 0$ for $1<p\leq 2$ and $f(t)\leq 0$ for $p\geq 2$.
\end{lemma}

\begin{proof}
  If $p=2$, then $f\equiv 0$.  Consider the case $p< 2$; i. e.,
  $q> 2$, the case $p> 2$ can be treated analogically. We have
$$
  f'(t)=\Phi_q(t)-1-\frac{4}{2^\alpha}\Phi_\alpha(t-1),\quad
  f''(t)= (q-1)|t|^{q-2}-\frac{4(\alpha-1)}{2^\alpha}|t-1|^{\alpha-2},
$$
where $\Phi_q(t)=|t|^{q-2}t$, $\Phi_{\alpha}$ is defined analogically.
Hence $f'(-1)=0=f'(1)$, $f(-1)=2-\frac{4}{\alpha}\geq 0$, and
$f''(-1)=q-\alpha\geq 0$. Drawing the graphs of the functions $|t|$
and
$\left(\frac{4(\alpha-1)}{2^\alpha(q-1)}\right)^{\frac{1}{q-2}}|t-1|^{\frac{\alpha-2}{q-2}}$
shows that the equation $f''(t)=0$ has exactly 2 roots, one positive
in the interval $(0,1)$, and one negative in $[-1,0)$. Hence $f''$ is
positive outside of the interval determined by these roots and
negative inside of it.  This means that $f$ has at both stationary
points $t=\pm 1$ nonnegative local minima. This also implies that the
equation $f'(t)=0$ may have at most one zero in $(-1,1)$, where the
function $f$ attains a positive local maximum.  Consequently,
summarizing these facts about the graph of the functions $f$ we obtain
that $f(t)\geq 0$ for $t\in \mathbb{R}$.
\end{proof}

Observe also that substituting $t\to-t$ gives for $q\geq 2$
the inequality
\begin{equation}   \label{inequality+}
\frac{|t|^q}{q}+t+\frac{1}{p}-\frac{4}{\alpha 2^\alpha}|t+1|^\alpha\geq 0,
\quad t\in \mathbb{R}
\end{equation}
and the opposite inequality for $q\in (1,2]$.


The following lemma is an extension of \cite[Lemma 2.4]{DE} which
deals with the scalar case and $\alpha=2$.

\begin{lemma}   \label{lemma:P}
  \begin{itemize}
\item[(i)]
    Let $p\geq 2$ and $\norm{ u_1}\neq 0$. Then for every
    $\alpha\in[q,2]$ there exists a number $\gamma(\alpha,p)$ such
    that
    \begin{equation}  \label{eq:ner-vec1}
      P( u_1, u_2)\leq
      \gamma(\alpha,p)\norm{ u_1}^{(p-1)(q-\alpha)}\norm{ u_2-\norm{ u_1}^{p-2} u_1}^\alpha.
    \end{equation}

\item[(ii)] Let $p\in(1,2]$. Then for every $\alpha\in[2,q]$
there exists    a number $\gamma(\alpha,p)$ such that
    \begin{equation}
      \label{eq:ner-vec2}
      P( u_1, u_2)\geq
      \gamma(\alpha,p)\norm{u_1}^{(p-1)(q-\alpha)}
\norm{ u_2-\norm{ u_1}^{p-2} u_1}^\alpha.
    \end{equation}
  \end{itemize}
\end{lemma}

\begin{remark} \label{rmk1} \rm
In the proof we will show that we can take
$\gamma(\alpha,p)=4/\alpha 2^\alpha$. However, the numerical
computations show that this constant is not optimal and can be
improved. To find this optimal constant is a subject of the present
investigation.
\end{remark}


\begin{proof}[Proof of Lemma \ref{lemma:P}]
Observe that \eqref{eq:ner-vec2} trivially holds for $\norm{ u_1}=0$.
Therefore, in the remaining part of the proof we suppose
$\norm{ u_1}\neq  0$. We will prove the first statement of lemma
($p\geq 2$), the proof of the second part is analogical.
By dividing both sides of \eqref{eq:ner-vec1}
with the factor $\norm{u_1}^p$ we get the inequality
\begin{equation}   \label{uu-inequality}
     \frac 1p-\big\langle\frac{u_1}{\norm{u_1}},
     \frac{ u_2}{\norm{ u_1}^{p-1}}\big\rangle
     + \frac{\norm{\frac{ u_2}{\norm{ u_1}^{p-1}}}^q}{q}\leq
      \gamma(\alpha,p)\norm{\frac{ u_2}{\norm{
            u_1}^{p-1}}-\frac{ u_1}{\norm{ u_1}}}^\alpha.%\\
  \end{equation}
Define $x=\frac{ u_2}{\norm{ u_1}^{p-1}}$ and
   $a=\frac{ u_1}{\norm{ u_1}}$. Then $\Vert a\Vert=1$ and
  \eqref{uu-inequality}
  can be written in the form
\begin{equation}   \label{inequality}
   \frac{\Vert x\Vert^q}{q}-\langle a,x\rangle +\frac{1}{p}\leq
   \gamma(\alpha,p)\Vert x-a\Vert^{\alpha}.
\end{equation}
As mentioned above, we show that this inequality holds with
$\gamma(\alpha,p)=\frac{4}{\alpha 2^\alpha}$.

Let $g(x)=\langle x,a\rangle +\frac{4}{\alpha 2^\alpha}
\Vert x-a\Vert^{\alpha}$. We will examine
the minimal value of this function over the sphere $\Vert x\Vert=t$,
$t\geq 0$. Any $x\in \mathbb{R}^n$ can be written in the form
$x=\mu a+ \nu a^{\perp}$ for some unit vector $a^{\perp}$
with $\langle a,a^{\perp}\rangle=0$. Then
$$
t^2=\Vert x\Vert ^2=\langle \mu a+\nu a^\perp,\mu a+\nu a^\perp \rangle
=\mu^2 + \nu^2.
$$
We have
\begin{align*}
g(x)=&\langle \mu a+\nu a^\perp, a\rangle  + \frac{4}{\alpha 2^\alpha}
\langle \mu a + \nu a^\perp - a,
\mu a + \nu a^\perp - a\rangle^{\alpha/2}
\\
=& \mu +\frac{4}{\alpha 2^\alpha}(t^2-2\mu +1)^{\alpha/2}.
\end{align*}
Now we solve the extremal problem $g(x)\to \min$, $\Vert x \Vert=t$
which can be written in the form
$$
\mu +\frac{4}{\alpha 2^\alpha} (t^2-2\mu  +1)^{\alpha/2}\to \min,
\quad \mu \in [-t,t].
$$
Since $\alpha/2\leq 1$, the minimized function is concave
and hence it attains its minimum over
$[-t,t]$ at the boundary point of this interval; i. e.,
$$
g(x)\bigr|_{\Vert x\Vert=t}\geq \min\big\{-t+\frac{4}{\alpha
2^\alpha}|t+1|^{\alpha}, t+\frac{4}{\alpha 2^\alpha}|t-1|^\alpha\big\}.
$$
Consequently, inequality \eqref{inequality} holds if
$$
  \frac{|t|^q}{q}+\frac{1}{p}\mp t-\frac{4}{\alpha 2^\alpha}|t\mp 1|^2
\leq 0.
$$
But this is just the inequality from Lemma \ref{L1} for the sign ``$-$'' or
its equivalent reformulation after the substitution $t\to -t$
(see \eqref{inequality+} in case $q\in (1,2]$) . The Lemma is proved.
\end{proof}

The next lemma presents a link between two Riccati type operators,
namely the operator which corresponds to half-linear equation
\eqref{eq:E} (the power at the dependent variable is $q$) and the
Riccati operator with $\alpha$-degree nonlinearity, where $\alpha\in
[\min\{q,2\},\max\{q,2\}]$.

\begin{lemma}\label{lemma:ineq}
  Let $h\in C^2(\Omega, \mathbb{R}^+)$.  Define $ G=h\norm{\nabla
    h}^{p-2}\nabla h$ and $ v=h^p w- G$. Further, let
  $\alpha\in [\min\{q,2\},\max\{q,2\}]$ and $\gamma(\alpha,p)$ be the
  number from Lemma \ref{lemma:P}.

\begin{itemize}
\item[(i)]
  If $1< p\leq 2$, then
  \begin{equation}
    \label{eq:ner1}
    h^pR[ w]\geq \operatorname{div}  v+hL[h]+\gamma(\alpha,p)
    p h^{-\alpha}\norm{\nabla h}^{(p-1)(q-\alpha)}\norm{ v}^\alpha
  \end{equation}
  holds on $\Omega$.

\item[(ii)]
  If $p\geq 2$ and $\norm {\nabla h}\neq 0$ on $\Omega$, then
  \begin{equation}
    \label{eq:ner2}
    h^pR[ w]\leq \operatorname{div}  v+hL[h]+\gamma(\alpha,p)
 p h^{-\alpha}\norm{\nabla h}^{(p-1)(q-\alpha)}\norm{ v}^\alpha
  \end{equation}
  holds on $\Omega$.
\end{itemize}
\end{lemma}

\begin{proof}
  We start with the following obvious identities
 \begin{equation}
    \operatorname{div}  G=\norm{\nabla h}^p+h\Delta_ph
\end{equation}
and
\begin{equation}
 \begin{aligned}
      h^p\operatorname{div} w
&=h^p\operatorname{div}(h^{-p}( v+ G))\\
&=\operatorname{div}  v+\operatorname{div}  G-ph^{-1}\inner{ v+ G}{\nabla h}\\
&=\operatorname{div}  v+\norm{\nabla h}^p+h\Delta_p h-ph^{-1}
\inner{v+ G}{\nabla h}.
\end{aligned}
  \end{equation}
Now a direct computation shows
  \begin{align*}
    h^p R[ w]&=
    h^p\operatorname{div}  w+h^p c(x)+(p-1)h^p \norm{ w}^q\\
    &=\operatorname{div}  v+\norm{\nabla h}^p+h\Delta_p h-ph^{-1}\inner{ v+
      G}{\nabla h}+
    h^p c(x)+(p-1)h^{-q}\norm{h^p w}^q\\
    &=\operatorname{div}  v+hL[h]+ph^{-q}\Big(\frac{\norm{h^{q-1}\nabla
          h}^p}p-\inner{ v+ G}{h^{q-1}\nabla h}+\frac{\norm{
          v+ G}^q}q\Big)
    \\
    &=\operatorname{div}  v+hL[h]+ph^{-q}P(h^{q-1}\nabla h, v+ G).
  \end{align*}
  For $ u_1=h^{q-1}\nabla h$ and $ u_2= v+ G$ we have
  \begin{align*}
    \norm{ u_1}^{(p-1)(q-\alpha)}&\norm{ u_2-\norm{
        u_1}^{p-2} u_1}^\alpha\\
&=    h^{(q-1)(p-1)(q-\alpha)}\norm{\nabla
      h}^{(p-1)(q-\alpha)} \norm{ v+ G-h\norm{\nabla h}^{p-2}\nabla h}^\alpha\\
    &=h^{q-\alpha}\norm{\nabla
      h}^{(p-1)(q-\alpha)}\norm{ v}^\alpha.
\end{align*}
Now the lemma follows from the estimates in Lemma \ref{lemma:P}.
\end{proof}


\section{Main results}

In this section we introduce the main results of the paper. Since most
of the work has been already done in the previous section, the proofs are
short and straightforward. Our first theorem certifies the importance
of Riccati type inequality \eqref{eq:R[w]leq} in the theory of half-linear
differential equations \eqref{eq:E}.

\begin{theorem}\label{th1}
The following statements are equivalent:
\begin{itemize}
\item[(i)] The equation $L[u]=0$ has a positive $C^2$ solution on $\Omega$.
\item[(ii)] The inequality $L[u]\leq 0$ has a positive $C^2$ solution on $\Omega$.
\item[(iii)] The equation $R[ w]=0$ has a $C^1$ solution $ w$ on $\Omega$
  such that the vector field $\norm{ w}^{q-2} w$
  is conservative.
\item[(iv)] The inequality $R[ w]\leq 0$ has a $C^1$ solution $ w$ on
  $\Omega$ such that the vector field $\norm{ w}^{q-2} w$
  is conservative.
\end{itemize}
\end{theorem}

\begin{proof}
  Define
  \begin{equation}
     w=\frac{\norm{\nabla u}^{p-2}\nabla u}{|u|^{p-2}u}.\label{eq:def_w}
  \end{equation}
  By a direct calculation, the $i$-th component of the vector $ w$
  satisfies
  \[
    \frac{\partial w_i}{\partial x_i}= \frac{\frac{\partial }{\partial x_i}\left(\norm{\nabla u}^{p-2}\frac{\partial u}{\partial x_i}\right)}{|u|^{p-2}u}
    -(p-1)\frac{\norm{\nabla u}^{p-2}\frac{\partial u}{\partial x_i}}{|u|^{p}}\frac{\partial u}{\partial x_i}
  \]
  and summing up over all independent variables we get
  \begin{align*}
    \operatorname{div}  w&=\frac{\operatorname{div}\left(\norm{\nabla u}^{p-2}\nabla u\right)}{|u|^{p-2}u}
    -(p-1)\frac{\norm{\nabla u}^{p-2}}{|u|^{p}}\norm{\nabla u}^2\\&=
    \frac{\operatorname{div}\left(\norm{\nabla u}^{p-2}\nabla u\right)}{|u|^{p-2}u}
    -(p-1)\norm{ w}^q.
  \end{align*}
  Using this computation we easily observe that
  \begin{equation}
    \label{eq:RL}
    R[ w]=\frac{L[u]}{|u|^{p-2}u}
  \end{equation}
  holds.

  (i)$\implies$(iii). Follows from \eqref{eq:RL} and from the fact
  that if $ w$ is defined by \eqref{eq:def_w}, then
$\norm{w}^{q-2} w=\frac{\nabla u}{u}=\nabla(\ln u)$ and $\ln u$ is a
  scalar potential to $\norm{ w}^{q-2} w$.

  (iii)$\implies$(iv). Clearly holds.

  (iv)$\implies$(ii). Since $\norm{ w}^{q-2} w$ has a scalar
  potential, there exists a scalar function $\varphi$, such that
  $\nabla \varphi= \norm{ w}^{q-2} w$. Define function
  $u=e^\varphi$. The function $u$ satisfies \eqref{eq:def_w} and in
  view of \eqref{eq:RL} the implication holds.

  (ii)$\implies$(i). Follows from Theorem \ref{th:Allegretto}.
\end{proof}

Our second theorem relates two Riccati type inequalities. One of them
is inequality \eqref{eq:R[w]leq} which is associated to the
half-linear equation with $p$-Laplacian \eqref{eq:E} (the dependent
variable appears in the inequality in the power $q$), while the second
one contains the dependent variable in the power $\alpha$; i. e., the
equation is associated with a half-linear PDE with $\beta$-degree
Laplacian, where $\beta$ is the conjugate number to the number
$\alpha$ (see also Remark \ref{rem} below).

\begin{theorem}\label{th2}
Let $h\in C^2(\Omega,\mathbb{R}^+)$.
  \begin{itemize}
\item [(i)]
    Let $p\in(1,2]$, \eqref{eq:R[w]leq} has a $C^1$ solution on
    $\Omega$, $\alpha\in[2,q]$ be arbitrary number and $\gamma(\alpha,p)$
    be the number from Lemma \ref{lemma:P}. Then
    \begin{equation}                          \label{eq:lin_rce}
      \operatorname{div}  v+h(x)L[h(x)]+p\gamma(\alpha,p)
      h^{-\alpha}(x)\norm{\nabla h(x)}^{(p-1)(q-\alpha)}\norm{ v}^\alpha
      \leq 0
    \end{equation}
    has also a $C^1$ solution on $\Omega$.

  \item[(ii)] Let $p\geq 2$, $\alpha\in[q,2]$ be arbitrary number,
    $\gamma(\alpha,p)$ be the number from Lemma \ref{lemma:P} and let
    $h$ satisfy $\norm{\nabla h}\neq 0$ on $\Omega$. If
    \eqref{eq:lin_rce} has a $C^1$ solution on $\Omega$, then
    \eqref{eq:R[w]leq} has also a $C^1$ solution on $\Omega$.
  \end{itemize}
\end{theorem}

The proof of the above theorem is a direct consequence of
the inequalities from Lemma \ref{lemma:ineq}.

\begin{remark}\label{rem} \rm
  Suppose that both $h$ and $\norm{\nabla h}$ do not vanish in
  $\Omega$.  The equation
  \[
    \operatorname{div}  v+h(x)L[h](x)+p\gamma(\alpha,p)
h^{-\alpha}(x)\norm{\nabla h(x)}^{(p-1)(q-\alpha)}\norm{ v}^\alpha
    =0
  \]
  is the Riccati equation for the second order partial differential
equation
  \begin{equation}
    \operatorname{div}\Bigl(A(x)\norm{\nabla u}^{\beta-2}\nabla u\Bigr)+
    h(x)L[h](x)|u|^{\beta -2}u=0,
    \label{eq:Emod}
  \end{equation}
where $\beta=\frac{\alpha}{\alpha-1}$ and
  $A(x)=\left[\frac{p\gamma(\alpha,p)}{\beta-1}\right]^{1-\beta}h^{\beta}
  (x)\norm{\nabla h(x)}^{p-\beta}$. Thus if \eqref{eq:Emod} has a
  positive solution on $\Omega$, then \eqref{eq:lin_rce} has a
  solution. Conversely, if \eqref{eq:lin_rce} has a $C^1$ solution
  $ v$ on $\Omega$ and $\norm{ v}^{\alpha-2} v$ is
  conservative, then \eqref{eq:Emod} has a positive $C^2$ solution
  on $\Omega$. If $\alpha=2$, then  \eqref{eq:Emod} becomes the
  linear partial differential equation
  \[
    \operatorname{div}\Bigl(h^{2}(x)\norm{\nabla h(x)}^{p-2}\nabla u\Bigr)+ \frac p2
    h(x)L[h](x)u=0.
  \]
  In this case, Theorem \ref{th2} allows us to transfer results
  from the linear theory to half-linear equations.
\end{remark}

\begin{remark}\label{rem2} \rm
  Note that we are not able to guarantee that the condition on the
  existence of scalar potential from (iv) part of Theorem \ref{th1}
  holds. For this reason we are not able yet to formulate the results from Theorem
  \ref{th2} in terms of second order half-linear differential
  equations, like in Theorem \ref{th:DF}.
\end{remark}

\begin{thebibliography}{00}

\bibitem{A-H} W. Allegretto, Y. X. Huang;
\emph{Principal eigenvalues and   Sturm Comparison via Picone's
identity}, J. Differential Equations
  \textbf{156} (1999), 427--438.

\bibitem{A-T86} M. Atakarryev, A. Toraev;
\emph{Oscillation and
  non-oscillation criteria of Knezer type for elliptic nondivergent
  equations in unlimited areas}, Proc. Acad.  Sci. Turkmen. SSR
  \textbf{6} (1986), 3--10 (Russian).

\bibitem{DE} O. Do\v sl\'y, A. Elbert;
\emph{Integral characterization of
  the principal solution of half-linear differential equations}, Studia
  Sci. Math. Hungar. \textbf{36} (2000), No. 3--4, 455--469.

\bibitem{DF} O. Do\v sl\'y, S. Fi\v snarov\'a;
\emph{Half-linear oscillation
  criteria: Perturbation in term involving derivative}, Nonlinear Anal.
  \textbf{73} (2010), 3756--3766.

\bibitem{D-M} O. Do\v sl\'y, R. Ma\v r\'\i k;
\emph{Nonexistence of the
    positive solutions of partial differential equations with
    $p-$Laplacian}, Acta Math. Hungar. \textbf{90 (1--2)} (2001),
  89--107.

\bibitem{DP} O. Do\v sl\'y, S. Pe\v na;
\emph{A linearization method in
  oscillation theory of half-linear second-order differential
  equations}, J. Inequal. Appl. vol. 2005 (2005), No. 5, 535--545.


\bibitem{D-R} O. Do\v sl\'y, P. \v Reh\' ak;
\emph{Half-linear Differential  Equations},
North-Holland Mathematics Studies 202., Amsterdam:
  Elsevier Science (2005), 517 p.


\bibitem{E-W} S. B. Eliason, L. W. White;
\emph{On positive solutions of  second order elliptic partial
differential equations},
Hiroshima  Math. J. \textbf{12} (1982), 469--484.

\bibitem{E-S} \'A. Elbert, A. Schneider;
\emph{Perturbations of the  half-linear Euler differential equation},
Results Math. \textbf{37}  (2000), 56--83.

\bibitem{F-M} S. Fi\v snarov\'a, R. Ma\v r\'ik;
\emph{Half-linear ODE and  modified Riccati equation:
Comparison theorems, integral characterization of principal solution},
submitted.

\bibitem{JKY2000} J. Jaro\v s, T. Kusano, N. Yoshida;
\emph{A Picone-type
  identity and Sturmian comparison and oscillation theorems for a
  class of half-linear partial differential equations of second
  order}, Nonlinear Anal. TMA \textbf{40} (2000), 381--395.

\bibitem{M2} R. Ma\v r\'\i k;
\emph{Oscillation criteria for PDE with
  $p$-Laplacian via the Riccati technique}, J. Math. Anal. Appl.
  \textbf{248} (2000), 290--308.

\bibitem{MEJDE} R. Ma\v r\'\i k;
\emph{Riccati-type inequality and
  oscillation of half--linear PDE with damping}, Electron. J. Diff.
  Eqns.  Vol. 2004(2004), no. 11,  1--17.


\bibitem{Rehak} P. \v Reh\' ak;
\emph{Comparison of nonlinearities in
  oscillation theory of half-linear differential equations}, Acta Math.
  Hungar. \textbf{121} (1--2) (2008), 93--105.

\bibitem{Ru} M. R\accent23u\v{z}i\v{c}ka;
\emph{Electroreheological Fluids:
  Modelling and Mathematical Theory}, Lecture Nores in Mathematics 1748,
  Springer Verlag, Berlin 2000.

\bibitem{Japonci} J. Sugie, N. Yamamoka;
\emph{Comparison theorems for oscillation of second-order
half-linear differential equations}, Acta
  Math. Hungar. \textbf{111} (2006), 165-179.

\end{thebibliography}

\end{document}