% \documentclass[oneside,draft,reqno,11pt]{amsart}
\documentclass[10pt]{elsarticle}


\usepackage{amsfonts,amsmath,amsthm}
% \usepackage{amsthm}
% \usepackage{a4wide}
% \usepackage{cite}
% \usepackage[notcite,notref]{showkeys}
% \usepackage[showlabels,textmath,displaymath]{preview}
%\usepackage{syntonly}\syntaxonly
%\usepackage[IL2]{fontenc} \usepackage{times}
%\usepackage[IL2]{fontenc} \usepackage{bookman}

% \newenvironment{proof}{\begin{pf}}{\end{pf}}
% \newenvironment{remark}{\begin{rem}}{\end{rem}}
% \newenvironment{definition}{\begin{defn}}{\end{defn}}
% \newenvironment{example}{\begin{exmp}}{\end{exmp}}
% \newenvironment{theorem}{\begin{thm}}{\end{thm}}

\let\hat\widehat
\let\~\tilde
\numberwithin{equation}{section}
\theoremstyle{plain}
 \newtheorem{theorem}{Theorem}[section]
 \newtheorem{lemma}{Lemma}[section]
% \newtheorem{Theorem}{Theorem}[section]
% \def\theTheorem{\Alph{Theorem}}
 \newtheorem{corollary}{Corollary}[section]
 \theoremstyle{definition}
 \newtheorem{definition}{Definition}[section]
% \newtheorem{proof}{Proof}[section]
 \newtheorem{remark}{Remark}[section]
 \newtheorem{example}{Example}[section]

\pagestyle{plain}
%\interdisplaylinepenalty 10000
% \def\ead{}
% \usepackage{refcheck}
%\usepackage[dvipdfm]{hyperref}


\makeatletter
\let\over\@@over
\makeatother

\def\theenumi{\roman{enumi}}
\def\labelenumi{\textrm{\upshape{(\theenumi)}}}
\def\konst{\textrm{const}}
\def\div{\mathop{\hbox{\rm div}}}
\def\mess{\mathop{\hbox{\rm mess}}}
\def\sgn{\mathop{\hbox{\rm sgn}}}
\def\laplac{\Delta}
\def\R{\mathbb{R}}
\def\dxi{\,\hbox{\rm d}\xi\,}
\def\dx{\,\hbox{\rm d}x\,}
\def\dS{\,\hbox{\rm dS}\,}
\def\dt{\,\hbox{\rm d}t\,}
\def\dT{\,\hbox{\rm d}T\,}
\def\du{\,\hbox{\rm d}u\,}
\def\ds{\,\hbox{\rm d}s\,}
\def\dr{\,\hbox{\rm d}r\,}
\def\dphi{\,\hbox{\rm d}\phi\,}
\let\phi\varphi

\def\ss#1#2{\left\langle{#1}\,,\,{#2}\right\rangle}
% \def\vec#1{\boldsymbol{#1}}

\def\norm#1{||#1||}
\def\Norm#1{\Bigl|\Bigl|#1\Bigr|\Bigr|}
\def\x{\norm{x}}
\def\w{\norm{\vec{w}}}
\def\a{{\alpha}}
\def\aa{{\alpha-1}}
\def\at{{a\leq\x\leq t}}
\def\o{\omega_n}
\def\O{\Omega}
\def\c{\cdot}
\def\const{\hbox{const}}
\def\mes{\mathop{\hbox{\rm mes}}}
\def\eps{\varepsilon}
%\interdisplaylinepenalty 50
%\overfullrule 2mm
%\arraycolsep 0pt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{hyperref}
\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ss#1#2{\langle{#1},{#2}\rangle}

%\begin{frontmatter}
% \title{Oscillation of the half-linear partial differential equation
%   with damping}
\title{Riccati--type partial differential inequality\\ and oscillation\\
  of half--linear differential equation\\ with damping}
 \author{Robert Ma\v r\'\i k}
 \address{Mendel University, Department of Mathematics,
         Zem\v ed\v elsk\'a 3, 613 00 Brno, Czech Republic}
          \ead{marik\symbol{64}mendelu.cz}
 
% \author{Robert Ma\v r\'\i k}
% \address{ Mendel University, Department of Mathematics\\
%          Zem\v ed\v elsk\'a 3, 619 00 Brno, Czech Republic}
% \ead{marik\symbol{64}mendelu.cz}
\ead[url]{www.mendelu.cz/user/marik}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
  In the paper the Riccati--type partial differential inequality is
  studied. Various integral criteria for the nonexistence of solution
  on the unbounded domains are given. As an application of these
  results also some oscillation criteria for the second order damped
  half--linear partial differential equation are derived. These
  criteria improve and complete the earlier results concerning the
  oscillation of the partial differential equations. The main
  difference lies in the fact that the obtained oscillation criteria
  are not radially symmetric and do not depend only on the mean value
  of the coefficients. General types of unbounded domains are
  considered and effective oscillation criteria for a conic domain in
  the plain are given.
\end{abstract}
% \maketitle
\footnotetext{Supported by the Grant of the Czech Grant Agency}
%\end{frontmatter}

\textbf{Keywords.} $p-$Laplacian, oscillatory solution, Riccati
equation, partial differential equation, half--linear equation, damped
equation 


\textbf{AMS Class.} 35J60, 35B05

% \textbf{Running head:} Integral averages and PDE


\section{Introduction}

It is well known the fact that the Riccati differential equation
\begin{equation}
  w'+w^2+c(x)=0\label{R}
\end{equation}
plays an important role in the study of the second order linear
differential equation
\begin{equation}
  u''+c(x)u=0.\label{SL}
\end{equation}
Really, if \eqref{SL} has a positive solution $u$ on the interval $I$,
then the function $w=u'/u$ is a solution of \eqref{R}, defined on
$I$. Conversely, if the Riccati inequality 
\begin{equation}
  \label{RN}
  w'+w^2+c(x)\leq 0
\end{equation}
has a solution $w$, defined on $I$, then \eqref{SL} has a 
positive solution on $I$. It is also well known that this property can
be extended also to several other types of second order differential
equations and inequalities, which include the selfadjoint second order
differential equation
\begin{equation*}
  \Bigl(r(x)u'\Bigr)'+c(x)u=0
\end{equation*}
half-linear equation
\begin{equation}\label{PL}
  \Bigl(r(x)|u'|^{p-2}u'\Bigr)+c(x)|u|^{p-2}u=0, \qquad p>1
\end{equation}
and Schr\"odinger equation
\begin{equation}\label{LN} 
 \sum_{i,j=1}^n\frac{\partial}{\partial x_i}\Bigl(a_{ij}(x)\frac{\partial
 u}{\partial x_j}\Bigr)+c(x)u=0
\end{equation}
see e.g. \cite{B,S66,S79,N-S,Sch89}

Another importance of the Riccati equation and Riccati-type
substitution $w=u'/u$ lies in the fact, that it is embedded into the
Picone identity, which forms the link between the so-called Riccati
technique and variational technique in the oscillation theory of equation
\eqref{SL} (see also Section 3 for a short discussion concerning the
Picone identity).

In the paper we will study the partial Riccati-type differential
inequality
\begin{equation}
  \label{eq:R1}
  \div\vec w+||\vec w||^q+c(x)\leq 0
\end{equation}
and some generalizations of this inequality in the form
\begin{equation}
  \label{eq:R2}
  \div(\alpha(x)\vec w)+K\alpha(x)||\vec w||^q+\alpha(x)c(x)\leq0
\end{equation}
and
\begin{equation}
  \label{eq:R3}
  \div\vec w+K||\vec w||^q+c(x)+\ss{\vec w}{\vec b}\leq0,
\end{equation}
where $K\in\R$, $q>1$ and the assumptions on the functions $\alpha$,
$b$ and $c$ are stated bellow. The operator $\div (\cdot)$ is the
usual divergence operator, i.e. for $\vec w=(w_1,\dots, w_n)$ it holds
$\div \vec w=\sum_{i=1}^{n}\frac{\partial w_i}{\partial x_i}$, the norm
$||\cdot||$ is the usual Euclidean norm in $\R^n$ and
$\ss{\cdot}{\cdot}$ is the usual scalar product in $\R^n$.

As an application of these results new oscillation criteria for the
half-linear partial differential equation with damping are derived.
The main difference between the obtained criteria and similar results
in the literature lies in the fact, that our criteria are not
``radially symmetric'', see the discussion in Section 3, bellow.

The paper is organized as follows.  In the next section the
Riccati--type inequality is studied. The results of this section are
applied in the third section, which contains the results concerning
the oscillation of damped half-liner PDE. The last
section is devoted to the some examples and comments.

\section{Riccati inequality}

\textbf{Notation}: Let $\Omega(a)$, $\Omega(a,b)$ and $S(a)$
be the sets in $\R^n$ defined as follows:
\begin{align*}
\Omega(a)&=\{x\in\R^n:a\leq ||x||\},\\
\Omega(a,b)&=\{x\in\R^n:a\leq||x||\leq b\},\\
S(a)&=\{x\in\R^n: ||x||=a\}.
% \\
% D&=\{(t,x)\in\R\times\R^n:t\geq||x||\geq t_0\},\\
% D_0&=\{(t,x)\in\R\times\R^n:t>||x||\geq t_0\},
\end{align*}

The numbers $p>1$ and $q>1$ be mutually conjugate numbers, i.e.
$1/p+1/q=1$, $\omega_n$ be the surface of the unit sphere in $\R^n$.
For $M \subseteq \R^n$ the symbols $\overline M$ and $M^0$ denotes the
closure and the interior of $M$, respectively.


Integration over the domain $\Omega(a,b)$ is performed introducing
hyperspherical coordinates $(r,\theta)$, i.e.
\begin{equation*}
  \int_{\Omega(a,b)}f(x)\dx=\int_a^b\int_{S(r)}f(x(r,\theta))\dS\dr,
\end{equation*}
where $\dS$ is the element of the surface of the sphere $S(r)$.

We will study the Riccati inequality on the unbounded domains. Two
types of unbounded domains in $\R^n$ will be considered: the exterior
of a ball, centered in the origin and also a general unbounded domain
$\Omega$. In the latter case we will use the following assumption:
\begin{description}
\item[(A1)] the set $\Omega$ is an unbounded domain in $\R^n$, simply
  connected with a piecewise smooth boundary $\partial \Omega$ and
  $\mess (\Omega\cap S(t))>0$ for $t>1$.
\end{description}

\begin{theorem}\label{L2}
  Let $\Omega$ satisfy assumption \textup{(A1)} and $c\in
  C(\Omega,\R)$. Suppose $\alpha$ is a function satisfying the
  conditions
  \begin{gather}
    \alpha\in C^1(\Omega\cap\Omega(a_0),\R^+)\cap
    C_0(\overline\Omega, \R)\label{eq:P100a}\\
%     \label{eq:P0a}
%     0<\int_{\Omega \cap S(t)}\alpha(x)\dS<\infty, \qquad \text{for
%       every $t\geq a_0$,}\\
    \intertext{and }
    \label{eq:P1a}
    \int_{a_0}^\infty\Bigl(\int _{\Omega\cap S(t)}\alpha
    (x)\dS\Bigr)^{1-q}\dt=\infty
  \end{gather}
  Finally, suppose that there exists $a\geq a_0$, real constant $K>0$
  and a real--valued differentiable vector function $\vec w(x)$ which
  is bounded (in the sense of traces, if necessary) on every
  compact subset of $\overline{\Omega\cap \Omega(a)}$ and satisfies
  the differential inequality \eqref{eq:R2}
%   \begin{equation}
%     \label{eq:RicNer}
%     \div\vec w+c(x)+ K\norm{\vec w}^q+ \ss{\vec
%     w}{\frac{\nabla\alpha}{\alpha}}\leq 0 
%   \end{equation}
  on $\Omega\cap\Omega(a)$. Then 
  \begin{equation}
    \label{eq:P2a}
    \liminf_{t\to\infty}\int_{\Omega\cap\Omega(a_0,t)}\alpha(x)c(x)\dx<\infty.
  \end{equation}
\end{theorem}
\begin{proof}\let\tilde\widetilde
  For simplicity let us denote $\tilde\Omega(a)=\Omega(a)\cap\Omega$,
  $\tilde S(a)=S(a)\cap\Omega$, $\tilde\Omega(a,b)=\Omega(a,b)\cap
  \Omega$.  Suppose, by contradiction, that %\eqref{eq:P0a},
  \eqref{eq:P1a} and \eqref{eq:R2}
  are fulfilled and 
  \begin{equation}
    \lim_{t\to\infty}\int_{\tilde\Omega(a_0,t)}\alpha(x)c(x)\dx=
    \infty. \label{eq:*} 
  \end{equation}
  Integration of \eqref{eq:R2} over
  the domain $\tilde\Omega(a,t)$ and application of the
  Gauss--Ostrogradski divergence theorem gives
  \begin{multline}
    \label{L2:0}
    \int_{\tilde S(t)}\alpha(x)\ss{\vec w(x)}{\vec\nu(x)}\dS-\int_{\tilde
      S(a)}\alpha(x)\ss{\vec w(x)}{\vec \nu(x)}\dS \\
    +\int_{\tilde \Omega(a,t)}\alpha(x)c(x)\dx+K\int_{\tilde
      \Omega(a,t)} \alpha(x)\norm{\vec w(x)}^q\dx\leq 0,
  \end{multline}
  where $\vec\nu(x)$ is the outside normal unit vector to the sphere
  $S(\norm x)$ in the point $x$ (note that the product $\alpha (x)\vec
  w(x)$ vanishes on the boundary $\partial\Omega$ since
  $\alpha\in C_0(\overline \Omega,\R)$ and $\vec w$ is bounded near the
  boundary).
  In view of \eqref{eq:*} there exists $t_0\geq a$ such that
  \begin{equation}
    \label{L2:1}
    \int_{\tilde \Omega(a,t)}\alpha(x)c(x)\dx-\int_{\tilde
    S(a)}\alpha(x)\ss{\vec w(x)}{\vec\nu(x)}\dS \geq 0 
  \end{equation}
  for every $t\geq t_0$. Further Schwarz and H\"older inequality give
  \begin{align}\label{L2:2}
  -\int_{\tilde S(t)}\alpha(x)&\ss{\vec w(x)}{\vec\nu(x)}\dS\leq
   \int_{\tilde  S(t)}\alpha(x)\norm{\vec 
   w(x)}\dS \nonumber\\& \leq \Bigl(\int_{\tilde S(t)}\alpha(x)\norm{\vec
   w(x)}^q\dS\Bigr)^{\frac{1}{q}}
   \Bigl(\int_{\tilde S(t)}\alpha(x)\dS\Bigr)^{\frac1p}. 
  \end{align}
  Combination of the inequality \eqref{L2:0} with inequalities
  \eqref{L2:1} and \eqref{L2:2} gives
  \begin{equation*}
    K\int_{\tilde \Omega(a,t)}\alpha(x)\norm{\vec w (x)}^q\dx\leq
    \Bigl(\int_{\tilde S(t)}\alpha(x)\norm{\vec w(x)}^q\dS
    \Bigr)^{\frac1q}\Bigl( \int_{\tilde S(t)}\alpha
    (x)\dS\Bigr)^{\frac 1p}  
  \end{equation*}
  for every $t\geq t_0$. Denote 
  \begin{equation*}
    g(t)=\int_{\tilde \Omega(a,t)}\alpha(x)\norm{\vec w(x)}^q\dx. 
  \end{equation*}
  Then the last inequality can be written in the form
  \begin{equation*}
    Kg(t)\leq\Bigl(g'(t)\Bigr)^{\frac
    1q}\Bigl(\int_{\tilde S(t)}\alpha(x)\dS\Bigr)^{\frac 1p}.
  \end{equation*}
  From here we conclude for every $t\geq t_0$
  \begin{equation*}
    K^qg^q(t)\leq g'(t)\Bigl(\int_{\tilde
    S(t)}\alpha(x)\dS\Bigr)^{\frac qp} 
  \end{equation*}
  and equivalently
  \begin{equation*}
    K^q\Bigl(\int_{\tilde S(t)}\alpha(x)\dS\Bigr)^{1-q}\leq
    \frac{g'(t)}{g^q(t)}. 
  \end{equation*}
  This inequality shows that the integral on the left-hand side of
  \eqref{eq:P1a} has an integrable majorant on $[t_0,\infty)$ and
  hence it is convergent as well, a contradiction to \eqref{eq:P1a}.
\end{proof}

An often considered case is $\Omega=\R^n$ or $\Omega=\Omega(a_0)$. In
this case the preceding lemma gives

\begin{theorem}\label{L3}
  Let $\alpha\in C^1(\Omega(a_0),\R^+)$, $c\in C(\Omega(a_0),
  \R)$. Suppose that 
  \begin{equation}
    \label{eq:P1}
    \int_{a_0}^\infty\Bigl(\int _{S(t)}\alpha
    (x)\dS\Bigr)^{1-q}\dt=\infty .
  \end{equation}
  Further suppose, that there exists $a\geq a_0$, real constant $K>0$
  and real--valued differentiable vector function $\vec w(x)$ defined
  on $\Omega(a)$ which satisfies the differential inequality
  \eqref{eq:R2} on $\Omega(a)$.
  Then 
  \begin{equation}
    \label{eq:P2}
    \liminf_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)c(x)\dx<\infty.
  \end{equation}
\end{theorem}
\begin{proof}
  The proof is a modification and simplification of the proof of
  Theorem \ref{L2}.
\end{proof}

The following Theorem employs the two-parametric weighting function
$H(t,x)$ defined on the closed domain 
\begin{equation*}
  D=\{(t,x)\in\R\times\R^n: a_0\leq ||x||\leq t\}
\end{equation*}
instead of $\alpha(x)$. Further denote $D_0=\{(t,x)\in\R\times\R^n:
a_0<||x||< t\}$ and suppose that $H(t,x)\in C(D,\R_0^+)\cap C^1(D_0,
\R_0^+)$. Additional assumptions to the function $H$ are stated bellow.

Remark that this technique (called integral averaging technique) is
due to Philos \cite{Ph}, where the linear ordinary differential
equation is considered. This technique has been later extended in
several directions, let us remind at least the result of Wang
\cite{Wang}, where the usual condition on monotonicity of the function
$H(t,x)$ with respect to the second variable is improved. The
application of this method to the partial differential equation is due
to \cite{M-AMC}, where $\vec b\equiv 0$ is considered. 

First let us remind the well-known Young inequality
\begin{lemma}[Young inequality]
  For $\vec a,\vec b\in\R^n$
  \begin{equation}\label{eq:YoungIneq}
    \frac{\norm{\vec a}^p}p\pm\ss{\vec a}{\vec b}+\frac{\norm{\vec
        b}^q}q\geq 0
  \end{equation}
  holds.  
\end{lemma}

\begin{theorem}\label{L4}
  Let $\Omega$ be an unbounded domain in $\R^n$ which satisfy the
  hypothesis \textup{(A1)}, $c\in C(\Omega,\R)$ and $\vec b\in
  C(\Omega, \R^n)$. Suppose that the nonnegative continuous function
  $H(t,x)$ defined on $D$ has partial derivatives with respect to
  $x_i$ for $i=1..n$ and satisfies
  \begin{enumerate}
  \item $H(t,x)\equiv 0$ for $x\not\in \overline\Omega$.
  \item If $x\in\partial\Omega$, then $H(t,x)=0$ and $\norm {\nabla
      H(t,x)}=0$ for every $t\geq x$.
  \item If $x\in \Omega^0$, then $H(t,x)=0$ if and only if $||x||=t$.
  \item The vector function $\vec h(x)$ defined on $D_0$ with the
    relation
    \begin{equation}
      \label{eq:h}
      \vec h(t,x)=-\nabla H(t,x)+\vec b(x)H(t,x)
    \end{equation}
    satisfies
    \begin{equation}
      \label{eq:konv-h}
      \int_{\Omega(a_0,t)\cap\Omega}H^{1-p}(t,x)\norm{\vec
      h(t,x)}^p\dx<\infty. 
    \end{equation}
  \item There exists a continuous function $k(r)\in C([a_0,\infty),
    \R^+)$ such that the function $\Phi(r):= k(r)
    \int_{S(r)\cap\Omega}H(t,x)\dx$ is positive and nonincreasing on
    $[a_0,t)$ with respect to the variable $r$ for every $t$, $t>r$.
  \end{enumerate}
  Finally, suppose that there exists a real numbers $a\geq a_0$, $K>0$
  and differentiable vector function $\vec w(x)$ defined on $\Omega$
  which is bounded on every compact subset of
  $\overline{\Omega\cap\Omega(a)}$ and satisfies the Riccati inequality
  \eqref{eq:R3} on $\Omega\cap\Omega(a)$.
  Then   
  \begin{equation}
    \label{eq:limsup}
    \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\dS\Bigr)^{-1}
    \int_{\Omega(a_0,t)\cap\Omega}\Bigl[H(t,x)c(x)-{\norm{\vec
    h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx<\infty 
  \end{equation}
\end{theorem}
\begin{remark}
  Let us mention that nabla operator in $\nabla H(t,x)$ relates
  only to the variables of $x$, i.e. $\nabla
  H(t,x)=(\frac{\partial}{\partial x_1}, \dots,
  \frac{\partial}{\partial x_n})H(t,x)$, and does not relate to the
  variable $t$.
\end{remark}
\begin{proof}[Proof of Theorem \ref{L4}] For simplicity let us
  introduce the notation $\tilde \Omega(a)$, $\tilde S(a)$ and $\tilde
  \Omega(a,b)$ as in the proof of Lemma \ref{L2}.  Suppose that the
  assumptions of theorem are fulfilled. Multiplication
  of \eqref{eq:R3} by the function $H(t,x)$ gives
  \begin{multline*}
    H(t,x)\div \vec w(x)+H(t,x)c(x)\\+KH(t,x)\norm{\vec
    w(x)}^q+H(t,x)\ss{\vec w(x)}{\vec b(x)}\leq 0
  \end{multline*}
  and equivalently
  \begin{multline*}
    \div (H(t,x)\vec w(x))+H(t,x)c(x)\\+KH(t,x)\norm{\vec
    w(x)}^q+\ss{\vec 
    w(x)}{H(t,x)\vec b(x)-\nabla H(t,x)}\leq 0
  \end{multline*}
  for $x\in\tilde\Omega(a)$ and $t\geq \norm x$.  This and Young
  inequality \eqref{eq:YoungIneq} implies
  \begin{equation*}
    \div (H(t,x)\vec w(x))+H(t,x)c(x)-{\norm{H(t,x)\vec b(x)-\nabla
    H(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\leq 0.
  \end{equation*}
  Integration of this inequality over the domain $\tilde\Omega(a,t)$
  and the Gauss-Ostrogradski divergence theorem give
  \begin{multline*}
    -\int_{\tilde S(a)}H(t,x)\ss{\vec w(x)}{\vec
     \nu(x)}\dS\\+\int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x) - {\norm{\vec
     h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx\leq 0
  \end{multline*} 
  and hence
  \begin{equation*}
    \int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x) - {\norm{\vec
     h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx\leq \int_{\tilde
     S(a)}H(t,x)\norm{\vec  w(x)}\dS
  \end{equation*} 
  holds for $t>a$. This bound we will use to estimate the integral from
  the condition \eqref{eq:limsup}
  \begin{align*}
    \int_{\tilde \Omega(a_0,t)}&\Bigl[H(t,x)c(x) - {\norm{\vec
        h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx\\
    =&\int_{\tilde \Omega(a_0,a)}\Bigl[H(t,x)c(x) - {\norm{\vec
        h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx\\&+
    \int_{\tilde \Omega(a,t)}\Bigl[H(t,x)c(x) - {\norm{\vec
        h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx\\
    \leq&
    \int_{\tilde\Omega(a_0,a)}H(t,x)c(x)\dx+\int_{\tilde
        S(a)}H(t,x)\norm{\vec w(x)}\dS. 
  \end{align*}
  Denote the maximal functions $c^*(r)=\max\{|c(x)|:x\in S(r)\}$ and
  $w^*(r)=\max\{\norm{\vec w(x)}:x\in S(r)\}$. Then it holds 
  \begin{align*}
    \int_{\tilde\Omega(a_0,t)}\Bigl[H(t,x)c(x) &- {\norm{\vec
        h(t,x)}^p\over (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx
    \\
    &\leq
    \int_{a_0}^{a}\Bigl[ k(r)\int_{\tilde S(r)}H(t,x)\dS\Bigr]{c^*(r)\over
        k(r)}\dr\\  &+k(a){w^*(a)\over k(a)}\int_{\tilde S(a)}H(t,x)\dS \\
     & \leq
      k(a_0)\int_{\tilde S(a_0)}H(t,x)\dS\left[\int_{a_0}^a {c^*(r)\over
        k(r)}\dr +{w^*(a)\over k(a)}\right]
  \end{align*}
  for every $t\geq a_0$. From here we conclude that the expression 
  \begin{equation*}
    \Bigl(\int_{\tilde S(a_0)}H(t,x)\dS\Bigr)^{-1}
    \int_{\tilde\Omega(a_0,t)}\Bigl[H(t,x)c(x)-{\norm{\vec h(t,x)}^p\over
      (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx
  \end{equation*}
  is bounded for all $t\geq a_0$. Hence \eqref{eq:limsup} follows. The
  proof is complete.
\end{proof}

As in Theorem \ref{L3}, we specify the result of Theorem \ref{L4} also
for the case $\Omega=\R^n$.

\begin{theorem}\label{L5}
  Let $c\in C(\Omega(a_0))$, $\vec b\in C(\Omega, \R^n)$. Suppose that
  there exists nonnegative differentiable function $H(t,x)$ defined on
  $D$ which satisfies
  \begin{enumerate}
  \item $H(t,x)=0$ if and only if $||x||=t$
  \item The vector function $\vec h(x)$ defined on $D_0$ with the
    relation \eqref{eq:h}
                                %   \begin{equation}
                                %     \label{eq:h}
                                %     \vec h(t,x)=-\nabla H(t,x)+\vec b(x)
                                %   \end{equation}
    satisfies
    \begin{equation}
      \label{eq:konv-h-2}
      \int_{\Omega(a_0,t)}H^{1-p}(t,x)\norm{\vec h(t,x)}^p\dx<\infty
    \end{equation}
  \item There exists a continuous function $k(r)\in C([a_0,\infty),
    \R^+)$ such that the function $\Phi(r):=k(r)\int_{S(r)}H(t,x)\dx$
    is positive and nonincreasing on $[a_0,t)$ with respect to the
    variable $r$ for every $t$, $t>r$.
  \end{enumerate}
Further suppose that there exist a real numbers $a\geq a_0$, $K>0$
and differentiable vector function $\vec w(x)$ defined on $\Omega(a)$
which satisfies the Riccati inequality \eqref{eq:R3} on $\Omega(a)$.
Then
\begin{equation}
  \label{eq:limsup-2}
  \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\dS\Bigr)^{-1}
  \int_{\Omega(a_0,t)}\Bigl[H(t,x)c(x)-{\norm{\vec h(t,x)}^p\over
    (Kq)^{p-1}pH^{p-1}(t,x)}\Bigr]\dx<\infty 
\end{equation}



\end{theorem}
\begin{proof}
  The proof is a simplification of the proof of Theorem  \ref{L4}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\section{Oscillation of half--linear equation}

In this section we will employ the results concerning the Riccati
inequality to derive oscillation criteria for the second order partial
differential equation
\begin{equation}
  \label{eq:E}\tag{E}
  \div(\norm{\nabla u}^{p-2}\nabla u)+\ss{\vec b(x)}{\norm {\nabla
  u}^{p-2}\nabla u}+c(x)|u|^{p-2}u=0,
\end{equation}
where $p>1$.  The second order differential operator $\div(\norm
{\nabla u}^{p-2}\nabla u)$ is called the $p$-Laplacian and
this operator is important in various technical applications and
physical problems -- see \cite{Diaz}. The functions $c$ and $\vec b$
are assumed to be H\"older continuous functions on the domain
$\Omega(1)$. The solution of \eqref{eq:E} is every function defined on
$\Omega(1)$ which satisfies \eqref{eq:E} everywhere on $\Omega(1)$.

The special cases of equation \eqref{eq:E} are the linear equation
\begin{equation}
  \label{eq:L}
  \Delta u+\ss{\vec b}{\nabla u}+c(x)u=0
\end{equation}
which can be obtained for $p=2$, the Schr\"odinger equation
\begin{equation}
  \label{eq:Schr}
  \Delta u+c(x)u=0
\end{equation}
obtained for $p=2$ and $\vec b\equiv 0$
and the undamped half-linear equation
\begin{equation}
  \label{eq:ENetl}
  \div(\norm{\nabla u}^{p-2}\nabla u)+c(x)|u|^{p-2}u=0
\end{equation}
for $\vec b\equiv 0$.

Equation \eqref{eq:E} is called the \textit{half--linear equation},
since the operator on the left-hand side is homogeneous and hence a
constant multiple of every solution of \eqref{eq:E} is a solution of
\eqref{eq:E} as well. If $p=2$ then equation \eqref{eq:E} is linear
elliptic equation \eqref{eq:L}, however in the general case $p\neq 2$
is the linearity of the space of solutions lost and only homogenity
remains.

Concerning the linear equation two types of oscillation are studied
--- \textit{nodal} oscillation and \textit{strong} oscillation. The
equivalence between these two types of oscillation has been proved in
\cite{M-P78} for locally H\"older continuous function $c$, which is an
usual assumption concerning the smoothness of $c$, see also \cite{DM}
for short discussion concerning the general situation $p\neq 2$.  In
the connection to equation \eqref{eq:E} we will use the following concept
of oscillation.
\begin{definition}
  The function $u$ defined on $\Omega(1)$ is said to be
  \textit{oscillatory}, if the set of the zeros of the function $u$ is
  unbounded with respect to the norm.  Equation \eqref{eq:E} is said
  to be \textit{oscillatory} if every its solution defined on
  $\Omega(1)$ is oscillatory.
\end{definition}
\begin{definition}
  Let $\Omega$ be an unbounded domain in $\R^n$.  The function $u$
  defined on $\Omega(1)$ is said to be \textit{oscillatory in the
    domain $\Omega$}, if the set of the zeros of the function $u$,
  which lies in the closure $\overline \Omega$ is unbounded with
  respect to the norm. Equation \eqref{eq:E} is said to be
  \textit{oscillatory in the domain $\Omega$} if every its solution
  defined on $\Omega(1)$ is oscillatory in $\Omega$.  The equation is
  said to be \textit{nonoscillatory} (\textit{nonoscillatory in
    $\Omega$}) if it is not oscillatory (oscillatory in $\Omega)$.
\end{definition}
Due to 
the homogenity of the set of solutions, it follows from the
definition that the equation which possesses a solution on $\Omega(1)$
is nonoscillatory, if it has a solution $u$ which is positive on
$\Omega(T)$ for some $T>1$ and oscillatory otherwise. Further the
equation is nonoscillatory in $\Omega$ if it has a solution $u$ such
that $u$ is positive on $\overline\Omega\cap\Omega(T)$ for some $T>1$
and oscillatory otherwise.

Jaro\v s et. al. studied in \cite{JKY} the partial differential equation
\begin{equation}
  \label{eq:E-JKY}
  \div\bigl(a(x)\norm{\nabla u}^{p-2}\nabla u\bigr)+c(x)|u|^{p-2}u=0,
\end{equation}
where $a(x)$ is a positive smooth function and obtained the
Sturmian--types comparison theorems and oscillation criteria for
\eqref{eq:E-JKY}. The same results have been proved independently by
Do\v sl\'y and Ma\v r\'\i k in \cite{DM} for the case $a(x)\equiv 1$.
\begin{theorem}[\cite{DM}, \cite{JKY}]\label{th:JKY}
  Equation \eqref{eq:E-JKY} is oscillatory, if the ordinary
  differential equation
  \begin{equation*}
    (r^{n-1}\overline a(r)|y'|^{p-2}y')'+r^{n-1}\overline
    c(r)|y|^{p-2}y=0, \qquad {}'=\frac{\hbox{\rm d}}{\dr}
  \end{equation*}
  is oscillatory, where $\overline a(r)$ and $\overline c(r)$ denote
  the mean value of the function $a$ and $c$ over the sphere $S(r)$,
  respectively, i.e.
  \begin{equation*}
    \overline a(r)={1\over \omega_nr^{n-1}}\int_{S(r)}a(x)\dS, \qquad
    \overline c(r)={1\over \omega_nr^{n-1}}\int_{S(r)}c(x)\dS.
  \end{equation*}
\end{theorem}
The main tool in the proof of this theorem is a Picone identity for
equation \eqref{eq:E-JKY}.  Another application (not only to the
oscillation or comparison theory) of the Picone identity to the
equation with $p-$Laplacian can be found in \cite{AH}.

Concerning the Riccati-equation methods in the oscillation theory of
PDE, Noussair and Swanson used in \cite{N-S} the  transformation 
\begin{equation*}
  \vec w(x)=-\frac{\alpha(\norm x)}{\phi(u)}(A\nabla u)(x)
\end{equation*}
to detect nonexistence of eventually positive solution of the
semilinear inequality 
\begin{equation*}
  \sum _{i,j=1}^n \frac{\partial}{\partial
  x_i}\Bigl(a_{ij}(x)\frac{\partial u}{\partial 
  x_j}\Bigr)+p(x)\phi(u)\leq 0,
\end{equation*}
which seems to be one of the first papers concerning the
transformation of PDE into the Riccati type equation. 

In the paper of Schminke \cite{Sch89} is the Riccati technique used in
the proof of nonexistence of positive and eventually positive solution
of Schr\"odinger equation \eqref{eq:Schr}. The results are expressed
in the spectral terms, concerning the lower spectrum of Schr\"odinger
operator. 


Recently Kandelaki et. al. \cite{B} via the Riccati technique improved
the Nehari and Hille criteria for oscillation and nonoscillation of
linear second order equation \eqref{SL} and extended these criteria to
the half-linear equation \eqref{PL}.  The further extension of the
oscillatory results from \cite{B} to the case of equation
\eqref{eq:ENetl} can be found in \cite{M2}. One of the typical result
concerning the oscillation of equation \eqref{eq:ENetl} is the
following.
\begin{theorem}[Hartman--Wintner type criterion, \cite{M1}]\label{th:HW}
  Denote
  \begin{equation*}
    C(t)={p-1\over t^{p-1}}\int_1^t s^{p-2}\int_{\O(1,s)}  
    \x^{1-n}c(x)\dx\ds.  
  \end{equation*}
  If  
  \begin{equation*}
    -\infty<\liminf_{t\to\infty}C(t) < \limsup_{t\to\infty}C(t)  
    \leq \infty  
    \quad\text{ or if  }\quad
    \lim_{t\to\infty}C(t)=\infty,  
  \end{equation*}
  then equation \eqref{eq:ENetl} is oscillatory. 
\end{theorem}

A quick look at this condition and also at Theorem \ref{th:JKY} reveals
that the potential function $c(x)$ is in these criteria contained only
within the integral over the balls, centered in the origin. As a
consequence of this fact it follows that though the criteria are sharp
in the cases when the function $c(x)$ is radially symmetric, these
criteria cannot detect the contingent oscillation of the equation in
the cases when the mean value of the function $c(x)$ over the balls
centered in the origin is small. In order to remove this disadvantage
we will apply the theorems from the preceding section to the Riccati
equation obtained by the transformation of equation \eqref{eq:E}. As
a result we obtain the oscillation criteria which are applicable also
in such extreme cases when $\int_{S(r)} c(x)\dS=0$. The criteria can
detect also the oscillation over the more general exterior domains, than
the exterior of some ball. An application to the oscillation over the
conic domain is given in Section 4.

Remark that there are only few results in the literature concerning
the oscillation on another types of unbounded domain, than an exterior
of a ball. Let us mention the paper of Atakarryev and Toraev
\cite{A-T86}, where Kneser--type oscillation criteria for various
types of unbounded domains were derived for the linear equation
\begin{equation*}
  \sum _{i,j=1}^n a_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial
  x_j}+p(x)u=0.
\end{equation*}
In the paper \cite{JKY2} of Jaro\v s et. al. the forced superlinear
equation
\begin{equation*}
  \sum_{i,j=1}^n\frac{\partial}{\partial
  x_i}\Bigl(a_{ij}(x)\frac{\partial u}{\partial
  x_j}\Bigr)+c(x)|u|^{\beta-1}u=f(x),\qquad \beta>1
\end{equation*}
is studied via the Picone identity and the results concerning
oscillation on the domains with piecewise smooth boundary are established. 



Our main tool will be the following Lemma \ref{Le:3.1} which presents
the relationship between positive solution of \eqref{eq:E} and a
solution of the Riccati--type equation.
\begin{lemma}\label{Le:3.1}
  Let $u$ be solution of \eqref{eq:E} positive on the domain $\Omega$.
  Let $\alpha \in C^1(\Omega,\R^+)$. The vector function $\vec w(x)$
  defined by
  \begin{equation}
    \label{eq:RicSub}
    \vec w(x)=\frac{\norm {\nabla u(x)}^{p-2}\nabla
    u(x)}{|u(x)|^{p-2}u(x)} 
  \end{equation}
  is well defined on $\Omega$ and satisfies the Riccati equation
  \begin{equation}
    \label{eq:RicRce}
    \div\vec w+c(x)+(p-1)\norm {\vec w}^q+\ss{\vec w}{\vec b(x)}=0
  \end{equation}
  for every $x\in\Omega$.
\end{lemma}
\begin{proof} From \eqref{eq:RicSub} it follows (the dependence on the
  variable $x$ is suppressed in the notation)
  \begin{equation*}
    \div \vec w=\frac{\div(\norm{\nabla u}^{p-2}\nabla
    u)}{|u|^{p-2}u}- (p-1)\frac{\norm{\nabla
    u}^{p}}{|u|^p}
  \end{equation*}
  on the domain $\Omega$.  Since $u$ is a positive solution of
  \eqref{eq:E} on $\Omega$ it follows
  \begin{align*}
    \div \vec w&= -c- \ss{\vec b}{\frac{\norm{\nabla
          u}^{p-2}\nabla u}{|u|^{p-2}u}}-(p-1)
    \frac{\norm{\nabla u}^{p}}{|u|^p}\\
    &=- c-(p-1)\frac{\norm {\nabla
        u}^{p}}{|u|^p}-\ss{\vec
      b}{{\frac{\norm{\nabla
            u}^{p-2}\nabla u}{|u|^{p-2}u}}}.
  \end{align*}
  Application of \eqref{eq:RicSub} gives
  \begin{equation*}
    \div \vec w=- c-(p-1)\norm{\vec w}^{q}-\ss{\vec
      b}{\vec w}
  \end{equation*}
  on $\Omega$. Hence \eqref{eq:RicRce} follows.
\end{proof}

The first theorem concerns the case in which left-hand sides of
\eqref{eq:RicRce} and \eqref{eq:R2} differ only in a multiple
by the function $\alpha$.

\begin{theorem}\label{Th1}
  Suppose that there exists function $\alpha\in C^1(\Omega(a_0),
  \R^+)$ which satisfies
  \begin{enumerate}
  \item  for $x\in\Omega(a_0)$
    \begin{equation}
      \label{eq:P3}
      \nabla \alpha(x)=\vec b(x)\alpha(x)
    \end{equation}
  \item the condition \eqref{eq:P1} holds and
  \item 
    \begin{equation}
      \label{eq:P2n}
      \lim_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)c(x)\dx=\infty.
    \end{equation}
  \end{enumerate}
  Then equation \eqref{eq:E} is oscillatory in $\Omega(a_0)$.
\end{theorem}
\begin{proof}
  Suppose, by contradiction, that \eqref{eq:P1}, \eqref{eq:P3} and
  \eqref{eq:P2n} hold and \eqref{eq:E} is not oscillatory in
  $\Omega(a_0)$. Then there exists a real number $a\geq a_0$ such that
  equation \eqref{eq:E} possesses a solution $u$ positive on
  $\overline\Omega(a)$. The function $\vec w(x)$ defined on
  $\Omega(a)$ by \eqref{eq:RicSub} is well-defined, satisfies
  \eqref{eq:RicRce} on $\Omega(a)$ and is bounded on every compact
  subset of $\overline\Omega(a)$. In view of the condition
  \eqref{eq:P3} equation \eqref{eq:RicRce} can be written in the form 
  \begin{equation*}
    \alpha \div \vec w+\alpha c+(p-1)\alpha \norm{\vec w}^q+\ss{\vec
    w}{\nabla \alpha}=0
  \end{equation*}
  which implies \eqref{eq:R2} with $K=p-1$.  Theorem \ref{L3} shows
  that \eqref{eq:P2} holds, a contradiction to \eqref{eq:P2n}.
\end{proof}

The following theorem concerns the linear case $p=2$.
\begin{theorem}\label{th:lin}
  Let $\alpha\in C(\Omega(a_0),\R^+)$ Denote
  \begin{equation}
    \label{eq:C(x)}
    C_1(x)=c(x)-\frac 1{4\alpha^2(x)}\norm{\alpha(x)\vec
    b(x)-\nabla\alpha(x)}^2 -\frac 1{2\alpha(x)}\div\Bigl(\alpha(x)\vec
    b(x)-\nabla\alpha(x)\Bigr).
  \end{equation}
  Suppose that
  \begin{align}
    \label{eq:P6}
    \int_{a_0}^\infty\Bigl(\int_{S(t)}\alpha(x)\dS\Bigr)^{-1}\dt=\infty\\
    \noalign{and}
    \label{eq:P7}
    \lim_{t\to\infty}\int_{\Omega(a_0,t)}\alpha(x)C_1(x)\dx=\infty.
  \end{align}
  Then equation \eqref{eq:L} is oscillatory in $\Omega(a_0)$.
\end{theorem}
\begin{proof}%[Proof of Theorem \ref{th:lin}]
  Suppose, by contradiction, that \eqref{eq:L} is nonoscillatory. As
  in the proof of Theorem \ref{Th1}, there exists $a\geq a_0$ such
  that \eqref{eq:RicRce} with $p=2$ has a solution $\vec w(x)$ defined
  on $\Omega (a)$.  Denote $\vec W(x)=\vec w(x)+\frac12\Bigl(\vec
  b-\frac{\nabla \alpha}{\alpha}\Bigr)$. Direct computation shows that
  the function $\vec W$ satisfies the differential equation
  \begin{equation*}
    \div\vec W+C_1(x)+\norm{\vec W}^2+\Bigl\langle\frac{\nabla
    \alpha}\alpha, \vec W\Bigr\rangle=0
  \end{equation*}
  on $\Omega(a)$.  
  From here we conclude that the function $\vec W$ satisfies
  \begin{equation*}
    \div(\alpha \vec W)+C_1\alpha+\alpha||\vec W||^2=0
  \end{equation*}
  on $\Omega (a)$. However by Theorem \ref{L3} inequality
  \eqref{eq:P2} with $C_1$ instead of $c$ holds, a contradiction to
  \eqref{eq:P7}.
\end{proof}

The next theorem concerns the general case $p>1$. 
% The term from
% \eqref{eq:RicRce} with the scalar product is eliminated by the Young
% inequality \eqref{eq:YoungIneq} instead of the conversion to the
% squares, as in the preceding theorem.
In this case we also allow also another types of unbounded domains,
than $\Omega(a_0)$.

\begin{theorem}\label{T3}
  Let $\Omega$ be an unbounded domain which satisfies hypothesis
  \textup{(A1)}. Suppose that $k\in(1,\infty)$ is a real number and
  $\alpha\in C^1(\Omega(a_0), \R_0^+)$ is a function defined on
  $\Omega(a_0)$ such that
  \begin{enumerate}
  \item $\alpha(x)=0$ if and only if $x\not\in\Omega\cap\Omega(a_0)$ and
  \item \eqref{eq:P1a} holds.
\end{enumerate}
  For $x\in\Omega\cap\Omega(a_0)$ denote
  \begin{equation}
    \label{eq:C2}
    C_2(x)=c(x)-\frac k{(p\alpha(x))^p}\norm{\alpha(x)\vec b(x)-
    \nabla\alpha(x)}^p. 
  \end{equation}
  If 
  \begin{equation}
    \label{eq:P8}
    \lim_{t\to\infty} \int_{\Omega\cap\Omega(a_0,t)}\alpha(x)C_2(x)\dx
    =\infty 
  \end{equation}
  holds, then \eqref{eq:E} is oscillatory in $\Omega$.
\end{theorem}
\begin{remark}
  Under \eqref{eq:P8} we understand that the integral 
  \begin{equation*}
    f(t)=\int_{\Omega\cap S(t)}\alpha(x)C_2(x)\dS
  \end{equation*}
  which may have singularity near the boundary $\partial\Omega$, is
  convergent for large $t$'s and the function $f$ satisfy $\int^\infty
  f(t)\dt=\infty.$
\end{remark}
\begin{proof}[Proof of Theorem \ref{T3}]
  Suppose, by contradiction, that \eqref{eq:E} is not oscillatory.
  Then there exists a number $a\geq a_0$ and a function $u$ defined on
  $\Omega(a)$ which is positive on $\overline{\Omega\cap\Omega(a)}$
  and satisfies \eqref{eq:E} on $\Omega\cap\Omega(a)$. The vector
  function $\vec w(x)$ defined by \eqref{eq:RicSub} satisfies
  \eqref{eq:RicRce} on $\Omega\cap\Omega(a)$ and is bounded on every
  compact subset of $\overline{\Omega\cap\Omega(a)}$. Denote
  $l=k^{1\over p-1}$ and let $l^*$ be a conjugate number to the number
  $l$, i.e.  ${1\over l}+{1\over l^*}=1$ holds.  Clearly $l>1$ and
  $l^*>1$. The Riccati equation \eqref{eq:RicRce} can be written in
  the form
  \begin{equation*}
    \div\vec w+c(x)+{p-1\over l}\norm {\vec w}^q+\ss{\vec w}{\vec
    b(x)-\frac{\nabla \alpha}\alpha}+
    {p-1\over l^*}\norm {\vec w}^q+\ss{\vec w}{\frac{\nabla\alpha}{\alpha}}=0  
  \end{equation*}
  for $x\in\Omega\cap\Omega(a)$.
  From inequality \eqref{eq:YoungIneq} it follows
  \begin{align*}
    {p-1\over l}\norm{\vec w}^q+\ss{\vec w}{\vec
      b-\frac{\nabla\alpha}{\alpha}}&= {(p-1)q\over l}\left\{{\norm{\vec
          w}^q\over q}+\ss{\vec w} {{l\over (p-1)q}\Bigl(\vec
        b-\frac{\nabla
          \alpha}{\alpha} \Bigr)}\right\}\\
    &\geq -{(p-1)q\over l} {l^p\over [(p-1)q]^p}\Norm {\vec
      b-\frac{\nabla\alpha}{\alpha}}^p {1\over p}\\
    &=-{l^{p-1}\over p^p}\Norm{\vec
      b-\frac{\nabla\alpha}{\alpha}}^p\\
    &=-{k\over p^p}\Norm{\vec b-\frac{\nabla\alpha}{\alpha}}^p
  \end{align*}
  Hence the function $\vec w$ is a solution of the inequality
  \begin{equation*}
    \div \vec w+C_2(x)+{p-1\over l^*}\norm{\vec w}^q+\ss{\vec
      w}{\frac{\nabla\alpha}{\alpha}}\leq 0 
  \end{equation*}
  on $\Omega\cap\Omega(a)$. This last inequality is equivalent to
  \begin{equation*}
    \div(\alpha \vec w)+\alpha C_2+\frac{p-1}{l^*}\alpha||\vec w||^q\leq 0.
  \end{equation*}
  By Theorem \ref{L2} inequality \eqref{eq:P2a} with $C_2$ instead of
  $c$ holds, a contradiction
  to \eqref{eq:P8}. The proof is complete.
\end{proof}

The last theorem makes use of the two-parametric weighting function
$H(t,x)$ from Theorem \ref{L4} to prove the nonexistence of the
solution of Riccati equation.

\begin{theorem}\label{T4}
  Let $\Omega$ be an unbounded domain in $\R^n$ which satisfy
  hypothesis \textup{(A1)}. Let $H(t,x)$ be the function defined on
  the domain $D$ with the properties \textup{(i)}--\textup{(v)} of
  Theorem \ref{L4}. If 
  \begin{equation}
    \label{eq:limsup...}
    \limsup_{t\to\infty}\Bigl(\int_{S(a_0)}H(t,x)\dS\Bigr)^{-1}
    \int_{\Omega(a_0,t)\cap\Omega}\Bigl[H(t,x)c(x)-{\norm{\vec
    h(t,x)}^p\over p^pH^{p-1}(t,x)}\Bigr]\dx=\infty,
  \end{equation}
  then equation \eqref{eq:E} is oscillatory in $\Omega$.
\end{theorem}
\begin{proof}
  Suppose that the equation is nonoscillatory. Then the Riccati
  equation \eqref{eq:RicRce} has a solution defined on
  $\Omega\cap\Omega(T)$ for some $T>1$, which is bounded near the
  boundary $\partial\Omega$. Hence \eqref{eq:limsup} of Theorem
  \ref{L4} with $K=p-1$ holds, a contradiction to
  \eqref{eq:limsup...}. Hence the theorem follows.
\end{proof}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Examples}

In the last part of the paper we will concretize the general ideas
from the preceding section.

The specification of the function $\alpha$ in Theorem \ref{T3} leads
to the following oscillation criterion for a conic domain on the
plane. In this case the function $\alpha$ is only the function of a
polar coordinate $\phi$.
\begin{corollary}\label{Co:4.1}
  Let us consider equation \eqref{eq:ENetl} on the
  plane (i.e. $n=2$) with polar coordinates $(r, \phi)$ and let
  \begin{equation}\label{eq:OM}
    \Omega=\{(x,y)\in\R^2: \phi_1< \phi(x,y)<\phi_2\},
  \end{equation}
  where $0\leq\phi_1<\phi_2\leq 2\pi$ and $\phi(x,y)$ is a polar
  coordinate of the point $(x,y)\in\R^2$. Further suppose that the
  smooth function $\alpha\in C^1(\Omega(1),\R^+_0)$ does not depend on
  $r$, i.e.  $\alpha=\alpha(\phi)$. Finally, suppose that
  \begin{enumerate}
  \item $\alpha(\phi)\neq0$ if and only if $\phi\in(\phi_1,\phi_2)$ 
  \item 
    \begin{equation}
      \label{eq:intI1}
      I_1:=\int_{\phi_1}^{\phi_2}
      \frac{|\alpha'_\phi(\phi)|^p}{4\alpha^{p-1}(\phi)} < \infty,
    \end{equation}
    where $\alpha'_\phi=\frac{\partial \alpha}{\partial \phi}$.
  \end{enumerate}
  Every of the following conditions is sufficient for oscillation of
  \eqref{eq:ENetl} on the domain $\Omega$:
  \begin{enumerate}
  \item $p>2$ and
    \begin{equation}
%       \lim_{t\to\infty}
%       \int_{\Omega\cap\Omega(1,t)}c(r,\phi)\,\alpha(\phi)\dx = 
      \lim_{t\to\infty}
      \int_1^t r\int_{\phi_1}^{\phi_2} c(r,\phi)\,\alpha(\phi)\dphi\dr = 
      \infty\label{eq:Cor1}  
    \end{equation}
  \item $p=2$ and 
    \begin{equation}
      \liminf_{t\to\infty}\frac{1}{\ln t}
      \int_1^t r\int_{\phi_1}^{\phi_2} c(r,\phi)\,\alpha(\phi)\dphi\dr 
%        \int_{\Omega\cap \Omega(1,t)}c(r,\phi)\,\alpha(\phi)\dx>\frac
%        14\int_{\phi_1}^{\phi_2}{||\alpha'_\phi(\phi)||^2}
%        {\alpha^{-1}(\phi)}\dphi
      >I_1,\label{eq:Cor2}  
    \end{equation} 
  \end{enumerate}
  where $c(r,\phi)$ is the potential $c(x)$ transformed into the polar
  coordinates.
\end{corollary}
\begin{proof}
  First let us remind that in the polar coordinates $\dx=r\dr\dphi$
  and $\dS=r\dphi$ holds. Direct computation shows that
  \begin{equation*}
    \int^\infty\Bigl(\int_{\Omega \cap S(t)}\alpha(\phi)\dS\Bigr)^{1-q}\dt=
    \int_{\phi_1}^{\phi_2}\alpha(\phi)\dphi\cdot\int^\infty t^{1-q}\dt.
  \end{equation*}
  and the integral diverges, since $p\geq 2$ is equivalent to $q\leq
  2$. Hence \eqref{eq:P1a} holds.
  
  Transformation of the nabla operator in the polar coordinates gives
  $\nabla \alpha=(0,r^{-1} \alpha'_\phi(\phi))$. Hence, according to
  Theorem \ref{T3}, it is sufficient to show that there exists $k>1$
  such that
  \begin{equation}\label{eq:PrCol11}
    \lim_{t\to\infty} \int_{\Omega\cap\Omega(1,t)}
    \Bigl[c(r,\phi)\alpha(\phi)-\frac k{p^p} \frac{|
    \alpha'_\phi(\phi)|^p}{r^p\alpha^{p-1}(\phi)}\Bigr]\dx = \infty.
  \end{equation}
  Since for $p>2$
  \begin{equation*}
    \lim_{t\to\infty}\int_{\Omega\cap\Omega(1,t)}\frac{|\alpha'_\phi(\phi)|^p}
    {r^p\alpha^{p-1}(\phi)}\dx
    =
    \int_{\phi_1}^{\phi_2}\frac{|\alpha'_\phi(\phi)|^p}
    {\alpha^{p-1}(\phi)}\dphi\lim_{t\to\infty}    \int_1^t r^{1-p}\dr<\infty,
  \end{equation*}
  the conditions \eqref{eq:PrCol11} and \eqref{eq:Cor1} are
  equivalent.
  
  Finally, suppose $p=2$. From \eqref{eq:Cor2} it follows that there
  exists $t_0>1$ and $\eps>0$ such that
  \begin{equation*}
    \frac{1}{\ln
      t}\int_{\Omega\cap \Omega(1,t)}c(r,\phi)\,\alpha(\phi)\dx>I_1+2\eps
  \end{equation*}
  for all $t\geq t_0$ and hence 
  \begin{equation*}
    \int_{\Omega\cap
    \Omega(1,t)}c(r,\phi)\,\alpha(\phi)\dx>\Bigl[kI_1+\eps\Bigr]\ln t
  \end{equation*}
  where $k=1+\eps I_1^{-1}$ holds for $t\geq t_0$.  Since
  \begin{align*}
    kI_1\ln t&=\frac {k  \ln t}4
    \int_{\phi_1}^{\phi_2}{|\alpha'_\phi(\phi)|^2} 
    {\alpha^{-1}(\phi)}\dphi\\&=\int_1^t\frac k{4r}\Bigl(\int_{\phi_1}^{\phi_2} {|\alpha'_\phi(\phi)|^2}
    {\alpha^{-1}(\phi)}\dphi\Bigr)\dr\\
    &=\int_{\Omega\cap\Omega(1,t)}\frac k{4r^2} {|\alpha'_\phi(\phi)|^2}
    {\alpha^{-1}(\phi)}\dx
  \end{align*}
  holds, the last inequality can be written in the form
  \begin{equation*}
    \int_{\Omega\cap \Omega(1,t)}\Bigl[c(r,\phi)\alpha(\phi)-\frac
    k4\,\frac{|\alpha'_\phi(\phi)|^2}{r^2\alpha(\phi)}\Bigr]\dx>\eps \ln t
  \end{equation*}
  and the limit process $t\to\infty$ shows that \eqref{eq:PrCol11}
  holds also for $p=2$. The proof is complete.
\end{proof}
\begin{example}
  For $n=2$ let us consider the Schr\"odinger equation
  \eqref{eq:Schr}, which in the polar coordinates $(r,\phi)$ reads as
  \begin{equation} \label{eq:SchrPol}
    \frac 1r\frac\partial{\partial r}\Bigl(r\frac{\partial u}{\partial
    r} \Bigr)+\frac 1{r^2}\,\frac{\partial^2u}{\partial \phi^2} +
    c(r,\phi)u=0.
  \end{equation}
  In Corollary \ref{Co:4.1} let us choose $\phi_1=0$, $\phi_2=\pi$,
  $\alpha(\phi)=\sin^2\phi$ for $\phi\in[0,\pi]$ and $\alpha(\phi)=0$
  otherwise. In this case the direct computation shows that the oscillation
  constant $I_1$ in \eqref{eq:Cor2} is $\frac \pi 2$, i.e. the
  equation is oscillatory on the half-plane $\Omega=\{(x_1,
  x_2)\in\R^2:x_2>0\}$ if 
  \begin{equation}
    \lim_{t\to\infty}\frac{1}{\ln
        t}\int_{1}^t r\int_0^\pi c(r,\phi)\,\sin^2(\phi)\dphi\dr>\frac
        \pi2. \label{eq:ex-rf}
  \end{equation}
  Similarly, the choice $\alpha(\phi)=\sin^3\phi$ gives an oscillation
  constant $\frac 32$. 
\end{example}

\begin{remark}
  It is easy to see that the condition \eqref{eq:ex-rf} can be
  fulfilled also for the function $c$ which satisfy
  $\int_0^{2\pi}c(r,\phi)\dphi=0$ and hence the criteria from
  Theorems \ref{th:JKY} and \ref{th:HW} fails to detect the
  oscillation.
\end{remark}

Another specification of the function $\alpha(x)$ leads to the
following corollary.
\begin{corollary}\label{Co:4.2}
  Let $\Omega$ be an unbounded domain in $\R^2$ specified in Corollary
  \ref{Co:4.1}. Let $A\in C^1([0,2\pi], \R^+_0)$ be a smooth function
  satisfying 
  \begin{enumerate}
  \item $A(\phi)\neq 0 $ if and only in $\phi\in(\phi_1,\phi_2)$
  \item $A(0)=A(2\pi)$ and $A'(0+)=A'(2\pi-)$
  \item the following integral converges
    \begin{equation}
      \label{eq:intI2}
      I_2:=\int_{\phi_1}^{\phi_2}\frac{[A^2(\phi)(p-2)^2+(A'(\phi))^2]^{\frac
      p2}}{p^p A^{p-1}(\phi)}\dphi<\infty. 
    \end{equation}
  \end{enumerate}
  If 
  \begin{equation}
    \label{eq:podm-I2}
    \liminf_{t\to\infty}\frac 1{\ln t}\int_1^t
    r^{p-1}\int_{\phi_1}^{\phi_2} c(r,\phi)A(\phi)\dphi\dr>I_2,
  \end{equation}
  then \eqref{eq:ENetl} is oscillatory in $\Omega$.
\end{corollary}
\begin{proof}
  Let $\alpha$ be defined in polar coordinates by the relation 
  \begin{equation*}
    \alpha(x(r,\phi))=r^{p-2}A(\phi).
  \end{equation*}
  Computation in the polar coordinates gives
  \begin{align*}
    \int^\infty\Bigl(\int_{\Omega\cap S(t)}\alpha(x)\dS\Bigr)^{1-q}\dt =&
    \int^\infty
    \Bigl(r^{p-1}\Bigr)^{1-q}\dr\int_{\phi_1}^{\phi_2}A(\phi)\dphi \\
    =&\int^\infty\frac1r\dr \int_{\phi_1}^{\phi_2}A(\phi)\dphi =
    \infty 
  \end{align*}
  and hence \eqref{eq:P1a} holds. The application of the nabla
  operator in polar coordinates yields
  \begin{equation*}
    \nabla\alpha(x(r,\phi))=\Bigl(\frac{\partial\alpha(x(r,\phi))}{\partial r},\frac 1r\,
    \frac{\partial \alpha(x(r,\phi))}{\partial \phi}\Bigr)=r^{p-3}((p-2)A(\phi), A'(\phi))
  \end{equation*}
  and hence on $\Omega$
  \begin{align*}
    \frac{\norm{\nabla\alpha(x(r,\phi))}^p}{\alpha^{p-1}(x(r,\phi))}= &
    \frac{r^{p(p-3)}\Bigl[(p-2)^2 A^2(\phi)+A'^2(\phi)\Bigr]^{\frac
        p2}}
    {r^{(p-1)(p-2)}A^{p-1}(\phi)} \\
    &= r^{-2}\frac{\Bigl[(p-2)^2 A^2(\phi)+A'^2(\phi)\Bigr]^{\frac p2}}
    {A^{p-1}(\phi)}
  \end{align*}
  holds. Integration over the part $\Omega\cap S(r)$ of the sphere
  $S(r)$ in polar coordinates gives (in view of \eqref{eq:intI2})
  \begin{equation*}
    \int_{\Omega\cap S(r)}\frac{\norm{\nabla
    \alpha(x(r,\phi))}^p}{p^p\alpha^{p-1}(x(r,\phi))}\dS=r^{-1} I_2. 
  \end{equation*}
  From \eqref{eq:podm-I2} it follows that there exist a real numbers
  $\eps>0$ and $t_0>1$ such that
  \begin{equation}\label{...}
    \frac 1{\ln t} \int_1^t r^{p-1} \int_{\phi_1}^{\phi_2}
    c(r,\phi)A(\phi) \dphi\dr >I_2+2\eps=I_2(1+\eps I_2^{-1})+\eps
  \end{equation}
  holds for $t>t_0$. Denote $k=1+\eps I_2^{-1}$. Clearly $k>1$. From
  \eqref{...} it follows that for $t>t_0$
  \begin{equation*}
    \int_1^t r^{p-1}\int_{\phi_1}^{\phi_2}c(r,\phi)
    A(\phi)\dphi\dr>kI_2\ln t +\eps\ln t
  \end{equation*}
  holds. This inequality can be written in the form
  \begin{equation*}
    \int_1^t\Bigl[r^{p-1}\int_{\phi_1}^{\phi_2}
    c(r,\phi)A(\phi)\dphi -r^{-1} kI_2\Bigr]\dr>\eps\ln t
  \end{equation*}
  which is equivalent to 
  \begin{equation*}
    \int_{\Omega \cap \Omega(1,t)}\Bigl[c(r,\phi)\alpha(r,\phi) -
    k \frac{\norm{\nabla \alpha(r,\phi)}^p}{p^p\alpha^{p-1}(r,\phi)}
    \Bigr] \dx >\eps \ln t,
  \end{equation*}
  where $\dx=r\dr\dphi$. Now the limit process $t\to\infty$ shows that
  \eqref{eq:P8} holds and hence \eqref{eq:ENetl} is oscillatory in
  $\Omega$ by Theorem \ref{T3}.
\end{proof}

\begin{example}
  An example of the function $A$ which for $p>1$, $\phi_1=0$
  and $\phi_2=\pi$ satisfies the conditions from Corollary
  \ref{Co:4.2} is $ A(\phi)= \sin^p\phi$ for ${\phi\in(0,\pi)}$ and
  $A(\phi)=0$ otherwise.  In this case the condition
  \begin{multline*}
    \liminf_{t\to\infty}\frac 1{\ln t}\int_1^tr^{p-1}\left( \int_0^\pi
    c(r,\phi) \sin^p\phi\dphi\right)\dr\\>
    \int_0^\pi\frac{\bigl[ (p-2)^2\sin^{2p}\phi
    + p^2\sin^{2p-2}\phi\cos^2\phi
    \bigr]^{p/2}}{p^p\sin^{p(p-1)}\phi}\dphi 
  \end{multline*}
  is sufficient for oscillation of \eqref{eq:ENetl} (with $n=2$) over
  the domain $\Omega$ specified in \eqref{eq:OM}. Here $c(r,\phi)$ if
  the potential $c(x)$ transformed into the polar coordinates
  $(r,\phi)$, i.e. $c(r,\phi)=c(x(r,\phi))$.
\end{example}

\begin{corollary}\label{Co:4.3}
  Let us consider the Schr\"odinger equation \eqref{eq:SchrPol} in the
  polar coordinates. Every of the following conditions is sufficient
  for the oscillation of the equation over the half-plane
  \begin{equation}
    \label{eq:OM-Co:4.3}
    \Omega=\{(x_1,x_2)\in\R^2:x_2>0\}.
  \end{equation}
  \begin{enumerate}
  \item There exists $\lambda>1$ such that
    \begin{equation}
      \label{eq:pod-Co4.3-1}
      \limsup_{t\to\infty} t^{-\lambda} \int_1^t
      (t-r)^\lambda\Bigl(r\int_0^\pi c(r,\phi)\sin^2\phi\dphi-\frac
      \pi{2r} \Bigr)\dr=\infty.
    \end{equation}
  \item There exists $\lambda>1$ and $\gamma<0$ such that
    \begin{equation}
      \label{eq:pod-Co4.3-2}
      \limsup_{t\to\infty} t^{-\lambda} \int_1^t r^{\gamma+1}
      (t-r)^\lambda \int_0^\pi c(r,\phi)\sin^2\phi\dphi\dr=\infty.
    \end{equation}
  \end{enumerate}
\end{corollary}
\begin{proof}
  For $\gamma\leq 0$ let us define 
  \begin{equation*}H(t,x)=
    \begin{cases}
      r^\gamma (t-r)^\lambda\sin^2\phi &\phi\in (0\pi)\\
      0&\text{otherwise,}
    \end{cases}
  \end{equation*}
  where $(r,\phi)$ are the polar coordinates of the point
  $x\in\R^2$. In the polar coordinates
  $\nabla=(\frac\partial{\partial r}, \frac 1r\,\frac
  \partial{\partial\phi})$. Hence 
  \begin{align*}
    \vec h(t,x(r,\phi))&=-\nabla H(t,x(r,\phi))\\
    &=-\Bigl(r^{\gamma-1}(t-r)^{\lambda-1}(\gamma (t-r)-\lambda
    r)\sin^2\phi ,\ 2r^{\gamma-1}(t-r)^\lambda\sin\phi\cos\phi\Bigr)
  \end{align*}
  and consequently
  \begin{align}
    \frac{\norm{\vec h(t,x(r,\phi))}^2}{H(t,x(r,\phi))} =&
    \gamma^2r^{\gamma-2}(t-r)^\lambda\sin^2\phi-2\lambda\gamma
    r^{\gamma-1} (t-r)^{\lambda-1}\sin^2\phi\nonumber\\
    &+\lambda^2r^\gamma (t-r)^{\lambda-2}\sin^2\phi+4r^{\gamma-2}
    (t-r)^\lambda \cos^2\phi.\label{eq:hH}
  \end{align}
  Now it is clear that for $\lambda>1$ inequality $\lambda-2>-1$ holds.
  Hence the integral over $\Omega\cap\Omega(1,t)$ converges and
  \eqref{eq:konv-h} for $p=2$ holds. Further
  \begin{equation*}
    \int_{S(r)\cap \Omega}H(t,x)\dS=r\int_0^\pi r^\gamma(t-r)^\lambda
    \sin^2\phi\dphi =\frac\pi 2 r^{\gamma+1}(t-r)^\lambda
  \end{equation*}
  and the condition (v) of Theorem \ref{L4} holds with
  $k(r)=r^{-1-\gamma}$. It remains to prove that the conditions
  \eqref{eq:pod-Co4.3-1} and \eqref{eq:pod-Co4.3-2} imply
  the condition \eqref{eq:limsup...}. Since
  $\int_0^\pi\sin^2\phi\dphi=\int_0^\pi \cos^2\phi\dphi=\frac\pi2$,
  it follows from \eqref{eq:hH} that
  \begin{align}
    \int_{S(r)\cap \Omega} \frac{\norm{\vec h(t,x(r,\phi))}^2}
    {H(t,x(r,\phi))}\dS =&
    \frac\pi2(\gamma^2+4)r^{\gamma-1}(t-r)^\lambda-
    \pi\lambda\gamma  r^{\gamma} (t-r)^{\lambda-1}\nonumber\\
    &+\frac\pi2\lambda^2r^{\gamma+1} (t-r)^{\lambda-2}. \label{al:konv}
  \end{align}
  Next we will show that 
  \begin{align}
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^\gamma(t-r)^{\lambda-1}\dr&<\infty \label{al:1}\\ 
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^{\gamma+1}(t-r)^{\lambda-2}\dr&<\infty \label{al:2}
    \intertext{and for $\gamma<0$ also}
    \lim_{t\to\infty}t^{-\lambda}\int_1^t
    r^{\gamma-1}(t-r)^{\lambda}\dr&<\infty \label{al:3}
   \end{align}
  holds. Inequality \eqref{al:1} follows from the estimate
  \begin{equation*}
    \int_1^t r^\gamma(t-r)^{\lambda-1}\dr\leq\int_1^t 1^\gamma
    (t-r)^{\lambda-1} \dr=\frac 1\lambda (t-1)^\lambda.
  \end{equation*}
  Computation by parts shows 
  \begin{equation*}
    \int_1^tr^{\gamma+1}(t-r)^{\lambda-2}\dr=
    \frac{(t-1)^{\lambda-1}}{\lambda-1}  +\frac{\gamma+1}{\lambda-1}
    \int_1^t r^\gamma(t-r)^{\lambda-1}\dr
  \end{equation*}
  and in view of \eqref{al:1} inequality \eqref{al:2} holds as
  well. Finally, for $\gamma<0$ integration by parts gives
  \begin{equation*}
    \int_1^t
    r^{\gamma-1}(t-r)^{\lambda}\dr=\frac{(t-1)^\lambda}{\gamma} +
    \frac \lambda\gamma\int_1^t r^\gamma(t-r)^{\lambda-1}\dr
  \end{equation*}
  and again the inequality \eqref{al:3} follows from \eqref{al:1}.
  Hence the terms from \eqref{al:konv} have no influence on the
  divergence of \eqref{eq:limsup...} (except the term
  $r^{-1}(t-r)^\lambda$ which appears for $\gamma=0$) and hence \eqref{eq:limsup...} follows from
  \eqref{eq:pod-Co4.3-1} and \eqref{eq:pod-Co4.3-2}, respectively.
  Consequently, the equation is oscillatory by Theorem \ref{T4}.
\end{proof}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}
\bibitem{AH} W. Allegretto, Y. X. Huang, A Picone's identity for the
  $p-$Laplacian and applications, Nonlin. Anal. TMA \textbf{32}
  No. 7 (1998), 819--830 .

\bibitem{A-T86} M. Atakarryev, A. Toraev, 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{Diaz} J. I. D\'\i az, \textit{Nonlinear Partial Differential
    Equations and Free Boundaries}, Vol. I Elliptic Equations, Pitman
  Publ., London 1985.

\bibitem{DM} O. Do\v sl\'y, R. Ma\v r\'\i k, Nonexistence of positive
  solutions for PDE's with $p-$Laplacian, Acta Math. Hungar.,
  \textbf{90 (1--2)} (2001), 89--107.

% \bibitem{F88} F. Fiedler, {Oscillation criteria of Nehari--type
%     for Sturm--Liouville operators and elliptic operators of second order
%     and the lower spectrum,} Proc. of Roy. Soc. of Edinb.
%   \textbf{109A} (1988), 127--144

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

\bibitem{JKY2} J. Jaro\v s, T. Kusano, N. Yoshida, Picone--type
  inequalities for nonlinear elliptic equations and their
  applications, J. Ineq. Appl. \textbf{6} (2001), 387--404.
  
\bibitem{B} N. Kandelaki, A. Lomtatidze, D. Ugulava, On oscillation
  and nonsocillation of a second order half--linear equation, Georgian
  Math. J. \textbf{7} No.2 (2000), 329-346.

\bibitem{M-AMC} R. Ma\v r\'\i k, Integral averages and oscillation
  criteria for half--linear differential equation, to appear in
  Appl. Math. Comp.
  
\bibitem{M2} R. Ma\v r\'\i k, Oscillation criteria for PDE with
  $p-$Laplacian via the Riccati technique, J. Math. Anal. Appl.
  \textbf{248} (2000), 290--308.

\bibitem{M1} R. Ma\v r\'\i k, Hartman--Wintner type theorem for PDE
  with $p-$Laplacian, EJQTDE, Proc. 6th. Coll. QTDE  \textbf{18} (2000),
  1--7.
  
\bibitem{M-P78} W. F. Moss, J. Piepenbrick, {Positive solutions of
    elliptic equations,} Pacific J. Math. \textbf{75} No. 1~(1978),
  219--226.
  
% \bibitem{N-S79} E. W. Noussair, C. A. Swanson, Positive solutions of
%     semilinear Schr\"odinger equations in exterior domains, Ind. Uni.
%   Math. J. \textbf{6} (1979), 993--1003.
  
\bibitem{N-S} E. W. Noussair, C. A. Swanson, Oscillation of semilinear
  elliptic inequalities by Riccati equation, Can. Math.  J.
  \textbf{22} No. 4 (1980), 908--923.
  
\bibitem{Ph} Ch. G. Philos, Oscillation theorems for linear
  differential equations of second order, Arch. Math (Basel)
  \textbf{53} (1989), 483--492.
  
\bibitem{Sch89} U.--W. Schminke, The lower spectrum of Schr\"odinger
  operators, Arch. Rational Mech. Anal. \textbf{75} (1989), 147--155.
  
\bibitem{S66} C. A. Swanson, \textit{Comparison and oscillation theory
    of linear differential equations}, Math. in Sci. and Eng. vol 48,
  Ac. Press (1968).
  
\bibitem{S79} C. A. Swanson, Semilinear second order elliptic
    oscillation, Canad. Math. Bull. \textbf{22} (1979), 139--157.

\bibitem{Wang} Q. R. Wang, Oscillation and asymptotics for second--order
  half--linear differential equations, Appl. Math. Comp. \textbf{122}
  (2001), 253--266.

\end{thebibliography}


\end{document}




% LocalWords:  Picone Schr odinger majorant homogenity semilinear Nehari Hile
% LocalWords:  nonoscillation Kandelaki Schminke Wintner Atakarryev Kneser
% LocalWords:  Toraev superlinear
