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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 151, pp. 1--11.\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-2011/151\hfil Oscillation via Picone formulas]
{Oscillation criteria for damped quasilinear second-order
elliptic equations}

\author[Tadie\hfil EJDE-2011/151\hfilneg]
{Tadie}

\address{Tadie \newline
Mathematics Institut \\
Universitetsparken 5 \\
2100  Copenhagen, Denmark}
\email{tadietadie@yahoo.com, tad@math.ku.dk}

\dedicatory{Dedicated to my mother Meguem Ghomsi Mabou
and all her Meguems}

\thanks{Submitted July 21, 2011. Published November 8, 2011.}
\subjclass[2000]{34C10, 34K15, 35J70}
\keywords{Picone; oscillation criteria for half-linear elliptic
equations}

\begin{abstract}
 In 2010, Yoshida \cite{yo1} stated that oscillation
 criteria for the superlinear-sublinear elliptic equation 
 equation
 \[
 \nabla \cdot \big(A(x)\Phi(\nabla v)\big)
 + (\alpha+1)B(x)\cdot\Phi(\nabla v) + C(x) \phi_\beta(v)
 + D(x) \phi_\gamma (v)=f(x)
 \]
 were not known.
 In this article, we provide some answers to this question
 using boundedness conditions on the coefficients of half-linear
 quasilinear elliptic  equations. This is obtained by using some
 comparison methods and Picone-type formulas.
\end{abstract}

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

\section{Introduction}

 In \cite{yo1}, for  $A\in C^1(\mathbb{R}^n,  \mathbb{R})$,
$ C,D, f \in C(\mathbb{R}^n,  \mathbb{R})$ and
$ B\in C(\mathbb{R}^n, \mathbb{R}^n)$,  the equation
\begin{equation} \label{e1.1}
\nabla \cdot \big(A(x)\Phi(\nabla v)\big)
+ (\alpha+1)B(x)\cdot\Phi(\nabla v) + C(x) \phi_\beta(v)
 + D(x) \phi_\gamma (v)=f(x)
\end{equation}
was given.
Here the central dot denotes the Euclidean scalar
product between elements of $\mathbb{R}^n$.
Let $\alpha$ be a positive fixed number. We define the
following functions for
$(t, \zeta)\in \mathbb{R}\times \mathbb{R}^n$ and $\nu>0$,
\[
\phi(t):=|t|^{\alpha-1}t; \quad
\Phi(\zeta):=|\zeta|^{\alpha-1}\zeta,\quad
\phi_\nu(t):=|t|^{\nu-1}t; \quad
\Phi_\nu(\zeta):=|\zeta|^{\nu-1}\zeta .
\]
Recall that for any $ \alpha>0$, the function
$\phi = \phi_\alpha \;$ has the following properties:
\begin{gather*}
\forall t,s \in \mathbb{R}, \quad
\phi(t)\phi(s)=\phi(ts),  \quad t\phi'(t) = \alpha \phi(t),\quad
t\phi(t)=|t|^{\alpha+1};\\
\forall  (s, \zeta)\in \mathbb{R}\times \mathbb{R}^n ,\quad
\phi(s)\Phi(\zeta)=\Phi(s\zeta); \quad
\zeta \Phi(\zeta)=|\zeta|^{\alpha+1}.
\end{gather*}
The quest is to investigate oscillation criteria for
equations similar to \eqref{e1.1},
following some different process but still based on Picone-type
formulae.


\section{One-dimensional and radially symmetric equations}

First, we consider the simple equation
\begin{equation} \label{e2.1}
\{ a(t) \phi( y')\}' + c(t) \phi(y) + h(t,y,y')=0
\end{equation}
where $ \{.\}'$ denotes the derivative with respect to
the  variable $t$. In the sequel, we assume
\begin{itemize}
\item[(H0)] $a$ is a positive constant or
$a\in C^1 (\mathbb{R},  (0, \infty ))$ and is non decreasing;
the other coefficients  are continuous in all their arguments.
\end{itemize}
Also we need some definitions:

A function $u$ will be said to be a (regular) solution of \eqref{e2.1}
if there exists $T>0$ such that $u$ is  locally piecewise $C^2$
and $u ,\phi(u')$ are $C^1$  in $ D_T:= (T, \infty)$.

This indicates that our focus is on the behaviour of the solutions
in exterior domains.

\begin{definition} \label{def2.1} \rm
 Let $ u\in C(\mathbb{R}, \mathbb{R})$.
\begin{itemize}
\item[(1)] A  nodal set of $u$, is any bounded open and
connected set  $ D=D(u) \neq \emptyset$ such that
$u|_{\partial D}=0$ and $u\neq 0$ in $D$.

\item[(2)] A function $u$ is said to be (weakly)  oscillatory
(in $\mathbb{R}$) if it has a zero in any $D_T$ and is
 strongly oscillatory if it has a nodal set in any $D_T$.

\item[(3)] An equation will be said to be oscillatory if any
of its non-trivial solutions is oscillatory.

\item[(4)]  An equation will be said to be homogeneous if whenever
 $u$ is a solution so is also $\lambda u$ for all
 $\lambda \in \mathbb{R} \setminus \{0\}$.
 When this holds only for $\lambda=-1 $ or $1$ the
 equation is said to be odd.
\end{itemize}
\end{definition}

\begin{remark} \label{rmk2.2} \rm
When $h\equiv 0 $,  equation \eqref{e2.1} is homogeneous and odd.
From the definitions above, a function $u$ would be non-oscillatory
if it is eventually non zero;
i.e., there exists $T>0$ and $u(t)\neq 0$ for all $t\in D_T$.
If such a non-oscillatory  function happens to be a solution of
an odd equation, we can freely chose it to be eventually positive
or eventually negative.
\end{remark}

The main strategy in this work is to use some comparison methods
via Picone-type formulae to
obtain oscillation criteria of some  general equations.
Of course for some of the simpler equations, the oscillation
criteria will be obtained through direct investigations as
in \cite{ra}.
As examples of simple strongly oscillatory equations,
for $ \alpha>0 $, we have
\begin{equation} \label{e2.2}
\{\phi_\alpha(u')\}' + \alpha \phi_\alpha(u)=0
\end{equation}
whose solutions are the  generalized sine functions
$ S:=S_\alpha $ \cite{ra,j2} with the following properties:
\begin{equation} \label{e2.3}
\begin{gathered}
|S_\alpha(t)|^{\alpha+1} + |S'_\alpha(t)|^{\alpha+1}=1, \quad
S_\alpha(t + \pi_\alpha)=-S_\alpha(t), \\
\text{where }  \pi_\alpha
= \frac{2\pi}{(\alpha+1)\sin\{\frac{\pi}{\alpha+1}\}}.
\end{gathered}
\end{equation}
When $\alpha=1$ the above functions are the usual trigonometric
functions.

Easy calculations show  that for $k\in \mathbb{R}\setminus\{0\}$,
the function $W(t):= S_\alpha( e^{kt})$ satisfies
\begin{equation} \label{e2.4}
\{ e^{-k\alpha t} \phi_\alpha(W')\}' + |k|^{\alpha+1} \alpha e^{kt}
\phi_\alpha(W)=0,
\end{equation}
and the function $Y(t):= S_\alpha(t^k)$ with $t\geq 0$ satisfies
\begin{equation} \label{e2.4b}
\{ t^{(1-k)\alpha} \phi_\alpha(Y')\}' + |k|^{\alpha+1} t^{k-1}
 \alpha \phi_\alpha(Y) =0.
\end{equation}

A one-dimensional equation associated to \eqref{e1.1},
for some $\beta, \gamma >0$, is
\begin{equation} \label{e2.5}
\{a(t)\phi(y')\}' + c(t)\phi_\beta(z) + d(t)\phi_\gamma(z)=f(t)
\end{equation}
where the coefficients satisfy (H0).
Also assume that
\begin{itemize}
\item[(H1)] there exists $T>0$ such that $c,d >0$ and  $f\leq 0$
 on $D_T$.
\end{itemize}

\begin{lemma} \label{lem2.3}
 Assume that {\rm (H1)} holds and there is a bounded non-trivial
solution $z$ of \eqref{e2.5}.

(1)  If  the coefficient $a$ is a positive constant
and  $z$ is  eventually positive, then the derivative $z'$ is
also eventually positive and decreases to $0$ at $\infty$.

(2)  If $a'>0$ and decreases to 0 at $\infty$, and $c$ is unbounded,
 then the conclusion in still (1) holds  if
 $ f \not\equiv 0 $ in some $D_T$.
 However, if  $ f(t) \equiv 0 $ in some  $D_T$, then conclusion in (1)
 holds  provided that  the solution $z$ is eventually greater
 than a positive constant.
\end{lemma}

\begin{proof}
(1) If $a\equiv 1$, from \eqref{e2.5} for a large $T$ and  $t>s>T$,
$$
\phi(z'(t)) - \phi(z'(\tau))= -\int_\tau^t\{ c(s)\phi_\beta(z)
+ d(s)\phi_\gamma(z) - f(s)\}ds
$$
whose second member is  strictly negative. So $z'$ is eventually
decreasing and tends to $0$  since  $z$ is bounded.

(2) Also from \eqref{e2.5},
\begin{align*}
a'(t)\phi(z')+ a(t)\frac{z''}{z'} z'\phi'(z')
&=a'(t)\phi(z')+ a(t) \alpha z'' \frac{\phi(z')}{z'} \\
&= -\{ c(t)\phi_\beta(z) + d(t)\phi_\gamma(z) - f(t)\}.
\end{align*}
As $a'$ decays to $0$, the last member is eventually negative while
the one before last has the same sign as $z''$
eventually, if $\phi(z)>m>0$ eventually.
We then have the same conclusion as in (1).
\end{proof}


\begin{theorem} \label{thm2.4}
Let $z$ be a bounded  non-trivial solution of \eqref{e2.5}.

Under the hypotheses of (1), and (H1) of Lemma \ref{lem2.3},
$z$ is oscillatory in $\mathbb{R}$.

Under the hypotheses of (2), and (H1) of lemma \ref{lem2.3},
$z$ is oscillatory  if $f \not\equiv 0$ in any $D_T$;
otherwise it will be oscillatory  unless
\begin{equation} \label{e2.6}
\lim \inf_{t\nearrow \infty} |z(t)|=0\,.
\end{equation}
\end{theorem}

It is easy to verify that under the conditions that $|f(t)|$
is eventually bounded  and  the functions $c$ and $d$ are
eventually positive and unbounded, the conclusions of the theorem
still  hold.

\begin{proof}[Proof of Theorem \ref{thm2.4}]
Assume that there is such a non-oscillatory solution $z$;
i.e., $z>0$ in some $D_T$.
Then the non-negative function
\[
H(t):= \frac{a(t)\phi(z')}{\phi(z)}
 =a(t) \phi(\frac {z'}z)
\]
satisfies, eventually,
\begin{equation} \label{e2.7}
H'(t)=- \{ c(t)|z|^{\beta-\alpha}
+ d(t)|z|^{\gamma -\alpha} \}
+ \frac {f(t)}{\phi(z)} - \frac{\alpha a(t)}{\phi(z)} |z'|^{\alpha+1}
\leq \frac {f(t)}{\phi(z)}.
\end{equation}
Therefore, $H(t) \leq H(T) + \int_T^t \frac {f(s)}{\phi(z)} ds $
which is invalid for large $T>0$ as the  right hand side
is eventually negative. Such a solution cannot
be non-oscillatory unless \eqref{e2.6} holds  for the case $2$.
\end{proof}

\section{Some Picone-type formulae and results in one-dimensional
 equations}

We consider the equations
\begin{equation} \label{e3.1}
\begin{gathered}
\{ a(t) \phi( y')\}' + c(t) \phi(y) =0,  \\
\{ a_1(t) \phi( z')\}' + c_1(t) \phi(z) + h(t,z,z')=0
\end{gathered}
\end{equation}
and define the two-form $\zeta $  on $ C^1(\mathbb{R}, \mathbb{R})$
for $\gamma >0$ and $u,v \in C^1(\mathbb{R})$,
by
\begin{equation} \label{e3.2}
\zeta_\gamma(u,v):= |u'|^{\gamma+1}
- (\gamma+1)\phi_\gamma(\frac uv v')u'
+ \gamma |\frac {u'}v v'|^{\gamma+1}
\end{equation}
which is non negative and null only if there exists
$k\in \mathbb{R}$ such that $u=kv$. (see e.g.  \cite{j2}).
Easy verifications show that if $y$ and $z$ are solutions
of \eqref{e3.1}, then  wherever $z\neq 0$,
\begin{equation} \label{P1}
\begin{split}
\{ y a(t)\phi(y') - y \phi (\frac yz )a_1(t) \phi(z') \}'
&= a_1(t)\zeta_\alpha(y,z)  + [a(t)-a_1(t)]|y'|^{\alpha+1}\\
&\quad+ [ c_1(t)- c(t)]|y|^{\alpha+1}
+ |y|^{\alpha+1}\frac{h(t,z,z')}{\phi(z)}
\end{split}
\end{equation}
Given the importance of the half-linear equations
(when $h\equiv 0$  in \eqref{e3.1})  in our investigation,
we have the following result.

\begin{theorem} \label{thm3.1}
 Assume that $a \in C^1(\mathbb{R}, (0,\infty))$ is  non-decreasing
and $ c\in C(\mathbb{R},\mathbb{R})$  is strictly
positive in some $D_T$.
Then  for any  $\alpha>0$  the half-linear equation
\begin{equation} \label{e3.3}
\{ a(t) \phi_\alpha( z')\}' + c(t) \phi_\alpha(z)=0, \quad t>0
\end{equation}
is  strongly oscillatory.
Moreover, if $M$ is a positive constant, then
\begin{equation} \label{e3.4}
 \{ a(t) \phi_\alpha( u')\}' + c(t) \phi_\alpha(u) + a(t)M =0,
\quad t>0
\end{equation}
is strongly oscillatory.
\end{theorem}

In case where $M<0$ but large enough, we have the same conclusion
unless
$$
\lim \inf_{t\nearrow \infty} |u(t)| =0 .
$$

\begin{proof}[Proof of Theorem \ref{thm3.1}]
Assume that there is a solution $z$ of \eqref{e3.3} which is
strictly positive in $D_T$.
From \eqref{e2.4}, $\{a_0(t) \phi_\alpha( y')\}'
+ c_0(t) \phi_\alpha(y)=0 $  is strongly oscillatory
where for  some $k_0\leq 0$,
$a(t) \leq \exp\{- k_0  \alpha t\}:= a_0(t) $ and
$  c(t) \geq |k_0|^{\alpha+1} \alpha \exp\{ k_0t\}:= c_0(t)$
in some $D_T$.

substituting $a$ and $c$ in \eqref{P1} (where $h\equiv 0$),
we obtain, in $D_T$,
\begin{align*}
&\{ y a_0(t)\phi(y') - y \phi (\; \frac yz \; )a(t) \phi(z') \}'\\
&= a(t)\zeta_\alpha(y,z)  + [a_0(t)-a(t)]|y'|^{\alpha+1}
+ [ c(t)- c_0(t)]|y|^{\alpha+1}>0.
\end{align*}
The integration over any nodal set $ D(y)\subset D_T$
of the above equation leads to an absurdity as the
right-hand side   will be strictly positive.
The solution $z$ cannot be eventually positive.

Let $u$ be a bounded and non-trivial solution of \eqref{e3.4}.
Wherever it is non-null, \eqref{P1} applied to \eqref{e3.3}
and \eqref{e3.4} gives
$$
\{ a z\phi(z') - z \phi(\frac zu) a \phi(u') \}'
 = a(t)\zeta_\alpha(z,u) + |z|^{\alpha+1} a(t) \frac M{\phi(u)}
$$
and the conclusion follows as before if $M \geq 0$.
If $M$ is a negative but large enough and $ u>\mu>0$ in $D_T$
for some $\mu>0$ then in  some $D_T$,
$a(t)\{\zeta_\alpha(z,u) + |z|^{\alpha+1} \frac M{\phi(u)} \}<0 $
 and we reach the same conclusion.
\end{proof}

\begin{remark} \label{rmk3.2}\rm
The result of Theorem \ref{thm3.1}  includes the case where
$ a(t) \equiv c(t)$ is  an
increasing function and positive in some $D_T$.

Theorem \ref{thm3.1} shows that besides some well known
oscillation criteria for half-linear elliptic equations
\cite{yo1,k2,t3}, (H0) and (H1) provide some
other important criteria.
\end{remark}

Now we consider the equation
\begin{equation} \label{e3.5}
\{ a(t) \phi( z')\}' + c(t) \phi(z) + q(t) \phi(z')= f(t);  \quad t>0.
\end{equation}

\begin{theorem} \label{thm3.3}
 Assume that
\begin{itemize}
\item[(i)]  $a \in C^1(\mathbb{R}, (0, \infty))$ is  non decreasing
with  decaying $a'$  and $ c\in C(\mathbb{R}, \mathbb{R}))$
is strictly   positive in some $D_T$;

\item[(ii)] $ q \in C(\mathbb{R})$ is  bounded and
$f \in C(\mathbb{R},  \mathbb{R})$ is non positive.
\end{itemize}
(a) If $q$ is eventually positive then
any non-trivial and bounded solution $z$  of \eqref{e3.5}
is oscillatory .
\\
(b)  But if $q$ is not eventually positive, $z$ is oscillatory
unless
\begin{equation} \label{e3.6}
\lim\inf_{t\nearrow \infty}\; |z(t)|=0 .
\end{equation}
\end{theorem}

\begin{proof}
As before, from the hypotheses, eventually from \eqref{e3.5},
\[
a'(t)\phi(z') + \alpha a(t) z'' \frac{\phi(z')}{z'}
= - \{  c(t) \phi(z) + q(t) \phi(z')- f(t)  \} <0
\]
with $ |a'(t)\phi(z')|$ decaying to $0$.
As for Lemma \ref{lem2.3}, $z$
and $z'$ are both positive with $z'$ decreasing to 0.
Let  $ M $ be a very large positive number   and $y$ an
oscillatory solution of
$$
\{ a(t) \phi_\alpha( y')\}' + c(t) \phi_\alpha(y) + a(t)M =0 .
$$
Assume that there is such a solution $z$ of \eqref{e3.5} which
is eventually positive. Then as in \eqref{P1},
wherever $z>0$,
\begin{equation} \label{e3.7}
\begin{split}
&\{ y a(t)\phi(y') - y \phi ( \frac yz  )a(t) \phi(z') \}'\\
&=a(t)\zeta_\alpha(y,z) + a(t) y\{ M + q(t)\phi(\frac yz z')\}
-|y|^{\alpha+1}\frac{f(t)}{\phi(z)}.
\end{split}
\end{equation}
If we integrate over a nodal set $ D(y)\subset D_T$,
where elements are positive, the above equation yields
\begin{equation} \label{e3.8}
0= \int_{D(y)} \{ a(t)\zeta_\alpha (y,z) + a(t) y\{ M + q(t)
\phi(\frac yz z')\}
 -|y|^{\alpha+1}\frac{f(t)}{\phi(z)}\}dt.
\end{equation}

(a)  As in the proof of Lemma \ref{lem2.3}, if $a$ is a positive constant
then if $z'$ is eventually positive, so is $z'$ and
 (even without the help of $M$) the right-hand side of \eqref{e3.7}
is strictly positive which is a contradiction.

(b)  In this case, if $M>0$ is large  enough, we obtain the same
conclusion provided that eventually  $z>\mu>0$ for some $\mu>0$;
in fact $ M + q(t)\phi(\frac yz z')$ needs to be positive for
a fixed large $M$.
\end{proof}

For $ a\in C^1(\mathbb{R}^n, (0, \infty)) $ and
$ c\in C(\mathbb{R}, \mathbb{R})$ such that for some $T>0$,
$c$, $a' >0$ in $D_T  $ and a large $M$,
we consider a strongly oscillatory solution $y$ of
\begin{equation} \label{e3.9}
\{ a(t) \phi(y')\}' + c(t)\phi(y) -a(t)M=0 .
\end{equation}
Consider the equation
\begin{equation} \label{e3.10}
\{ a(t) \phi(z')\}' + c(t)\phi(z) + q(t)\phi(z')=0
\end{equation}
where there exists $Q\in C^1(\mathbb{R},\mathbb{R})$ and
$ k\in C(\mathbb{R}, \mathbb{R});\quad Q'(t)=q(t)+k(t)$.

For a solution $z$ of \eqref{e3.10} and $y$ that of \eqref{e3.9},
wherever $ z\neq 0 $,
\begin{equation} \label{e3.11}
\begin{aligned}
&\big\{a(t) y \phi(y') - y\phi(\frac yz) a(t)\phi(z')
-y\phi(\frac yz)a(t)Q(t)\phi(z')\big\}' \\
&=a(t) \zeta_\alpha(y,z) +a(t) y\{ M - k(t) \phi(\frac yz z' ) \}
- Q(t)\Big( ya(t)\phi(\frac yz z') \Big)' .
\end{aligned}
\end{equation}

We then have the following result.

\begin{theorem} \label{thm3.4}
 Assume that  there are
\begin{itemize}
\item[(i)] $ Q\in C^1(\mathbb{R}, \mathbb{R});  q$,
$k \in C(\mathbb{R}, \mathbb{R}) $ such that
$Q'(t)= q(t)+k(t)$;

\item[(ii)] $a \in C^1(\mathbb{R}, (0, \infty)) $ and
$ c\in C(\mathbb{R}, \mathbb{R}) $ such that
$ c, a' >0 $ in some $D_T$.
\end{itemize}
Then any non-trivial and bounded solution  $ z  $ of
\begin{equation} \label{e3.12}
\{ a(t) \phi(z')\}' + c(t)\phi(z) + q(t)\phi(z')=0
\end{equation}
(a)  is oscillatory if $ k\equiv 0$;\\
(b)  is oscillatory if $ k\not\equiv 0  $ and bounded in $D_T$, unless
 $\lim\inf_{t\nearrow \infty} \; |z(t)|=0$.
\end{theorem}

\begin{proof}
If in \eqref{e3.10} we replace $Q$ by $Q + \mu$ with
$\mu \in \mathbb{R}$,  \eqref{e3.11}
remains valid with $Q+\mu$. If there is a solution $z$
of \eqref{e3.12} which is positive in some $D_T$, the integration
of \eqref{e3.11} over any nodal set $ D(y^+)\subset D_T$  gives
\begin{equation} \label{e3.13}
\begin{aligned}
&0=\int_{D(y^+)} a(t) \Big( \zeta_\alpha(y,z) + y\{ M - k(t)
  \phi(\frac yz z' ) \} \Big) dt \\
&\quad  - \int_{D(y^+)}\big( Q(t)+\mu \big)
[ a(t)y\phi(\frac yz z') ]'  \,dt , \quad
\forall \mu \in  \mathbb{R}.
\end{aligned}
\end{equation}
The formula \eqref{e3.13} can only hold if each integrand in
the formula in null in $D_T$; in particular only if
\[
 a(t) [\zeta_\alpha(y,z) + y\{ M - k(t) \phi(\frac yz z' ) \}]=0
\]
in any  $ D(y^+)\subset D_T$.

(a) If $k\equiv 0$ this is absurd for $M\geq 0$. Therefore,
the assumption is false; $z$ cannot be eventually positive.

(b) If $ k\not\equiv 0$ but bounded with $z>\nu$ for some
$\nu>0$ in $D_T$, the same conclusion holds by choosing
a large enough $M>0$.
\end{proof}

\section{Multidimensional case}

If $ w\in C^1(\mathbb{R}^n , \mathbb{R})$ is  radially symmetric; i.e.,
 $w(x) :=W(r):=W(|x|)$  for some  $W\in C^1(\mathbb{R})$ then
easy but elaborate calculations show that
$$
\nabla w(x)= W'(r) \frac X{|X|} \quad \text{and} \quad
\nabla \cdot \{ a(r)\Phi(\nabla w)\}=
 \frac 1{r^{n-1}} \{ r^{n-1} a(r) \phi(W')\}'
$$
and for $ B\in C(\mathbb{R}^n, \mathbb{R}^n) $,
$B(x)\cdot\Phi(\nabla u)= B(x)\cdot\frac X{|X|} \phi(U')$,
where $a\in C^1(\mathbb{R})$ say, $X= ^t(x_1, x_2, \dots , x_n) $
denotes the position-vector and
$r:=|X|=\sqrt{\{\sum_{i=1}^n x_i^2 \}}  $ its module.

Consider the operators
\begin{gather}
P(u):=\nabla\cdot \{A(x)\Phi(\nabla u) \} + C(x)\phi(u)
 + B_1(x)\cdot\Phi(\nabla u); \label{e4.1} \\
R(u):=\nabla \cdot\{a(r)\Phi(\nabla u) \} + c(r)\phi(u)
 + B_2(x)\cdot\Phi(\nabla u)+ F(x)  \label{e4.2}
\end{gather}
where the real functions $a,A$ are positive and continuously
differentiable, $ c, C, F$ are continuous in all their arguments
 and $ B_i \in C(\mathbb{R}^n, \mathbb{R}^n )$.
If a  function  $u$ in \eqref{e4.2} is
radially symmetric; i.e.m  $u(x):= U(|x|)=U(r)$,
 then,  in terms of $U$, \eqref{e4.2} reads
\begin{equation} \label{e4.3}
\begin{split}
R_1(U)&=\{ r^{n-1} \; a(r)\phi(U') \}' + r^{n-1}\; c(r) \phi(U)  \\
&\quad +r^{n-1}\Big( B_2(x)\cdot\frac X{|X|} \phi(U') + F (x) \Big).
\end{split}
\end{equation}
If the regular functions $u$ and $v$ satisfy
$Pu=Rv=0$ in  $ \mathbb{R}^n$, then a Picone formula reads
\begin{equation} \label{e4.4}
\begin{split}
&\nabla \cdot\{ u A(x) \Phi(\nabla u) -u\phi( \frac uv ) a(r)
\Phi(\nabla v) \} \\
&=a(r)Z_\alpha(u,v) + \big(A(x)-a(r) \big)|\nabla u|^{\alpha+1}
 + \big( c(r) - C(x)\big)|u|^{\alpha+1}  \\
&\quad + |u|^{\alpha+1}[ B_2(x)\cdot\Phi(\frac {\nabla v}v )
- B_1(x)\cdot\Phi( \frac{\nabla u}u )] +
 |u|^{\alpha+1}\frac{F(x)}{\phi(v)}
\end{split}
\end{equation}
 where for all $\gamma>0$ and all $u,v \in C^1(\mathbb{R}^n)$,
\[
Z_\gamma (u,v):= |\nabla u|^{\gamma+1} - (\gamma+1)
\Phi_\gamma( \frac uv \nabla v)\cdot\nabla u + \gamma |\frac uv
 \nabla v|^{\gamma+1}.
\]
If the coefficients $a$ and $c$ were not radially symmetric, but
$a_1(x)$ and $c_1(x)$  are, then \eqref{e4.1} would be the same with
$a_1(x)$ and $c_1(x) $ replacing them.

We recall that for all $\gamma>0 $ the two-form
$Z_\gamma (u,v) \geq 0 $ and is null only if either $uv=0$ or
there exist $k\in \mathbb{R}$ with $u=kv$.
(see e.g. \cite{k2,t1,t3}).

For easy writing we define for $h \in C(\mathbb{R}^n, \mathbb{R})$
and $ H\in C(\mathbb{R}^n, \mathbb{R}^n)$
\begin{equation} \label{e4.5}
\begin{gathered}
h^+(r) :=  \max_{|x|=r} h(x) , \quad
H^+(r):=  \max_{|x|=r} H(x)\cdot\frac X{|X|}, \\
h^-(r) :=  \min_{|x|=r}  h(x),  \quad
H^-(r):=  \min_{|x|=r} H(x)\cdot\frac X{|X|}.
\end{gathered}
\end{equation}
In \cite{yo1}, we have the  equation
$$
\nabla\cdot\big(\Phi_\alpha(\nabla v) \big) + \phi_\beta(v)
+ \phi_\gamma (v) =0
$$
where $0<\gamma < \alpha<\beta$. Here we consider the more general
 equation
\begin{equation} \label{e4.6}
\nabla\cdot\big(\Phi_\alpha(\nabla v) \big) + \phi_\beta(v)
+ \phi_\gamma (v) + B(x)\cdot\Phi_\alpha(\nabla v) + F(x)=0
\end{equation}
where $ B\in C(\mathbb{R}^n, \mathbb{R}^n)$.
If $v(x):= z(r)$ is a radially symmetric solution of \eqref{e4.6},
 then \begin{equation} \label{e4.7}
\Big( r^{n-1}\phi_\alpha( z')\Big)' + r^{n-1}\{ \phi_\beta(z)+
\phi_\gamma(z) +
  B(x)\cdot\frac X{|X|} \phi_\alpha (z') + F(x) \}=0.
\end{equation}
Let $y$ be a strongly oscillatory solution of
(see  Remark \ref{rmk3.2} and Theorem \ref{thm3.1})
$$
 \Big( r^{n-1}\phi_\alpha( y')\Big)' + r^{n-1} \big(\phi_\alpha(y)
- M\big)=0.
$$
Then
\begin{equation} \label{e4.8}
\begin{split}
&\{yr^{n-1} \phi_\alpha(y') - r^{n-1} y \phi_\alpha( \frac yz )
 \phi_\alpha(z') \}' \\
&=r^{n-1}[ \zeta_\alpha(y,z)+  |y|^{\alpha+1}
 \Big(|z|^{\beta-\alpha} + |z|^{\gamma-\alpha} - 1 \Big) \\
&\quad +y \{ M + B(x)\cdot\frac{X}{|X|}\phi_\alpha(\frac{yz'}z )
 + F(x)\phi(\frac yz ) \}]\\
&=r^{n-1}\big[ \zeta_\alpha(y,z)+  |y|^{\alpha+1}\Big(|z|^{\beta-\alpha} + |z|^{\gamma-\alpha}  \Big) \\
&\quad +y \{ M  - \phi_\alpha(y) + B(x)\cdot\frac{X}{|X|}
\phi_\alpha(\frac{yz'}z ) + F(x)\phi(\frac yz ) \}\big].
\end{split}
\end{equation}
For $R>0$, define $\Omega_R:=\{ x \in \mathbb{R}^n: |x|>R \}$.

\begin{theorem} \label{thm4.1}
Assume that The functions $B(x)\cdot X/|X|$ and
$F(x)$  are radially symmetric and bounded
in some  $\Omega_R$.
Then  any non-trivial and bounded radially symmetric solution $z$ of
\begin{equation} \label{e4.9}
\nabla\cdot\big(\Phi_\alpha(\nabla z) \big) + \phi_\beta(z)
+ \phi_\gamma (z) + B(x)\cdot\Phi_\alpha(\nabla z) + F(x)=0
\end{equation}
is oscillatory, unless
\begin{equation} \label{e4.10}
\lim \inf_{r\nearrow \infty} |z(r)|=0.
\end{equation}
\end{theorem}

\begin{proof}
It is sufficient to note that in \eqref{e4.8},
if $|z|>\mu>0$ in $\Omega_R$, as
\[
 |\{  B(x)\cdot\frac{X}{|X|}\phi_\alpha(\frac{yz'}z )
+ F(x)\phi(\frac yz ) - \phi(y) \}|
\]
is uniformly bounded under the hypotheses, for $M>0$ large enough,
\[
\{ M  - \phi(y) + B(x)\cdot\frac{X}{|X|}\phi_\alpha(\frac{yz'}z )
+ F(x)\phi(\frac yz ) \}>0 .
\]
So such a solution $z$ cannot be eventually positive,
unless \eqref{e4.10} holds.
\end{proof}

\section{Main results}

We  start with  an important link between multi-dimensional and
one-dimensional oscillation criterion for half-linear operators,
and  some oscillation criteria by means of the comparison method.

\begin{theorem} \label{thm5.1}
(1)   For any regular functions  $a, c \in C(\mathbb{R}^n, \mathbb{R})$,
 if the equation
\[
\{ r^{n-1} \; a^+(r)\phi(y')\}' + r^{n-1}  c^-(r) \phi(y)=0
\]
is oscillatory in $\mathbb{R}$,
then so is
$\nabla \cdot\{a(x)\Phi(\nabla u )\} + c(x) \phi(u)=0 $
in $\mathbb{R}^n$.
(see \cite[Theorem 3.1]{k2})

(2)  If $ \nabla \cdot\{a(x)\Phi(\nabla u )\} + c(x) \phi(u)=0 $
is strongly  oscillatory, then any bounded solution $v$ of
the equation
\begin{equation} \label{e5.1}
\nabla \cdot\{a(x)\Phi(\nabla v ) \} + c(x) \phi(v) +M=0
\end{equation}
is oscillatory:
(i) for all $M\geq 0 $;
(ii) for all $M<0$,  provided that it is large enough,
 unless $\lim\inf_{|x|\nearrow \infty} |v(x)|=0$.
\end{theorem}


\begin{proof}
(1) As in \eqref{e4.4},
\begin{equation} \label{e5.2}
\begin{split}
&\nabla\cdot\{ a^+(r) y \Phi(\nabla y)
 - y \phi(\frac yu) a(x)\Phi(\nabla u)\}\\
&= a(x)Z(y,u) + ( a^+ - a) |\nabla y|^{\alpha+1}
 + ( c - c^- ) |y|^{\alpha+1}
\end{split}
\end{equation}
which for  non-null and distinct $u$ and $y$ is strictly positive.
If $u$ is assumed to  be eventually strictly positive in some
$\Omega_T$, then the integration of \eqref{e5.2} over any nodal set
$D(y)\subset \Omega_T$ would lead to a contradiction.
Thus $u$ cannot be eventually positive.

(2) In this case, with $ \mu \in \{ M, -M  \}$, if we assume that
$\nabla \cdot\{a(x)\Phi(\nabla v )\} + c(x) \phi(v) + \mu =0 $
has a non-trivial and bounded solution $v$
which is strictly positive in some $\Omega_T$ then in application
of \eqref{e4.4},
$$
 \nabla \cdot \{ u a(x) \Phi(\nabla u) -u\phi( \frac uv ) a(x)
\Phi(\nabla v) \}= a(x)Z(u,v) + \mu u \phi(\frac uv).
$$
If $\mu \geq 0$  then the right-hand side of the above equation
is strictly positive;
but if $\mu<0 $ but very large,
$ a(x)Z(u,v) + \mu u \phi(\frac uv)<0 $  provided that $v>\nu$
in $\Omega_T$ for some $\nu>0$.
In both cases, integration over any nodal set $D(u)\subset \Omega_T$
leads to a contradiction.
\end{proof}

It is important to mention that for the above result, $M$ can be
replaced by $a(x)M$, $a$ being that in
the concerning equation
$\nabla \cdot\{a(x)\Phi(\nabla u )\} + c(x) \phi(u)=0 $
which is  assumed bounded below in some $\Omega_T$ by
a positive constant in the case where the result is based on
``big $M$''. In fact in this case the right hand side of the equation
 above reads $ a(x)\{ Z(u,v) + M u \phi(\frac uv)\}$.

Now we go back to the equation
\begin{equation} \label{e5.3}
\nabla\cdot\Big(A(x)\Phi(\nabla v)\Big) + A(x) B(x)\cdot\Phi(\nabla v)
+ C(x) \phi_\beta(v)  + D(x) \phi_\gamma (v)+f(x)=0
\end{equation}
where as said before,
$A \in C^1(\mathbb{R}^n,(0, \infty) )$,
$ f,  C , D  \in C(\mathbb{R}^n, \mathbb{R}); B\in
 C(\mathbb{R}^n, \mathbb{R}^n) $
and $\beta, \gamma >0$.
We suppose that there exists $b \in C^1(\mathbb{R}^n, \mathbb{R})$
such that
\begin{equation} \label{e5.4}
\nabla b(x)= B(x) + K(x),
\end{equation}
where  $K\in C(\mathbb{R}^n, \mathbb{R}^n) $  is bounded.

Let $u$ be a strongly oscillatory solution of
\begin{equation} \label{e5.5}
 \nabla\cdot\big(A(x)\Phi(\nabla u)\big)
+ C_1(x) \phi_\beta(u) - A(x)M=0
\end{equation}
where $M>0$  and $ \frac {C_1}A  $  bounded.

  Developments as those give for $v$ in \eqref{e5.3}  (formally) lead to
\begin{align*}
& \nabla \cdot\{ u A(x) \Phi(\nabla u)
 -u\phi( \frac uv ) A(x) \Phi(\nabla v) \} \\
&=A(x)Z_\alpha(u,v) + |u|^{\alpha+1}\Big( C(x) |v|^{\beta-\alpha}
- C_1(x)|u|^{\beta - \alpha} + D(x)|v|^{\gamma - \alpha} \Big) \\
&\quad+ u [ A(x) \{  B(x)\cdot\Phi(\frac uv \nabla v ) + M  \}
 + f(x)\phi(\frac uv)  ].
\end{align*}
As
\begin{align*}
&\nabla \cdot\{u\phi(\frac uv) b(x) A(x)\Phi(\nabla v)\}\\
&=b(x)\{ A(x) \{|\nabla u|^{\alpha+1} -Z_\alpha(u,v)\}
 -u\Big( A(x) B(x)\cdot\Phi(\frac{u\nabla v}v)
 + f(x)\phi(\frac uv)\Big)\\
&\quad -|u|^{\alpha+1}[C(x) |v|^{\beta-\alpha}
 + D(x)|v|^{\gamma -\alpha} ] \}
 +  uA(x)(B(x)+K(x) )\cdot\Phi(\frac {u \nabla v}v ),
\end{align*}
we have
\begin{align*}
& \nabla \cdot\{ u A(x) \Phi(\nabla u) -u\phi( \frac uv ) A(x)
 \Phi(\nabla v)  - u\phi(\frac uv) b(x) A(x)\Phi(\nabla v) \}\\
&=A(x) Z_\alpha(u,v) + |u|^{\alpha+1}
 \big[C(x) |v|^{\beta - \alpha} -  C_1(x)|u|^{\beta- \alpha}
 + D(x)|v|^{\gamma - \alpha}\big]\\
&\quad +A(x)u\Big(M - K(x).\Phi(\frac{u\nabla v}v) \Big)
  + uf(x)\phi(\frac uv)\\
&+ b(x)\{ A(x)[ Z_\alpha(u,v) - |\nabla u|^{\alpha+1} ]
 + uA(x) B(x)\cdot\Phi(\frac{u\nabla v}v)\\
&\quad + uf(x)\phi(\frac uv) + |u|^{\alpha+1}
 \{ C(x)|v|^{\beta-\alpha} + D(x)|v|^{\gamma- \alpha} \} \}
\end{align*}
and
\begin{equation} \label{e5.6}
\begin{split}
& \nabla \cdot \{ u A(x) \Phi(\nabla u) -u\phi( \frac uv ) A(x) \Phi(\nabla v)  -
 u\phi(\frac uv) b(x) A(x)\Phi(\nabla v)  \}\\
&=A(x) Z_\alpha(u,v) + |u|^{\alpha+1}[C(x) |v|^{\beta - \alpha}
 + D(x)|v|^{\gamma - \alpha}]\\
&\quad +A(x)u\Big(M - K(x)\cdot\Phi(\frac{u\nabla v}v) - \frac{C_1(x)}{A(x)} \phi_\beta(u) \Big) + uf(x)\phi(\frac uv)\\
&\quad + b(x)\{ A(x)[ Z_\alpha(u,v) - |\nabla u|^{\alpha+1}  ]
 + uA(x) B(x)\cdot\Phi(\frac{u\nabla v}v)\\
&\quad + uf(x)\phi(\frac uv) + |u|^{\alpha+1}\{ C(x)|v|^{\beta-\alpha}
 + D(x)|v|^{\gamma- \alpha} \} \}.
\end{split}
\end{equation}

\begin{theorem} \label{thm5.2}
Consider the equation \eqref{e5.3} where
\begin{itemize}
\item[(i)]  $A\in C^1(\mathbb{R}^n, (0, \infty))$;

\item[(ii)] $ f, C, D \in C(\mathbb{R}^n, \mathbb{R})$ are
  positive in some $\Omega_R$;

\item[(iii)] there exist $b\in C^1(\mathbb{R}^n, \mathbb{R})$,
$K\in C(\mathbb{R}^n, \mathbb{R}^n)$ with
 $b(x):= B(x) + K(x) $
such that $K$ is eventually bounded.
\end{itemize}
If $v$ is a bounded non-trivial solution of \eqref{e5.3}, then
(a) \eqref{e5.3} is strongly oscillatory if $K\equiv 0$;
(b) \eqref{e5.3} is strongly oscillatory if $K\not\equiv 0$,  unless
$$
\lim\inf_{|x|\nearrow \infty} \; |v(x)|=0 .
$$
\end{theorem}

\begin{proof}
Assume that there is  a solution $v$ of \eqref{e5.3} which is not
oscillatory;  e.g.,
There exists $\rho \geq R $ such that $ v>0 $ in $\Omega_\rho$.

In \eqref{e5.5} the function $b$ can be replaced by a
$b_1:= b + \mu $, for any  constant $\mu \in \mathbb{R}$.
So  after such a  replacement the integration of the resulting
equation over any nodal set $ D(u^+)\subset \Omega_\rho  $
 ($u> 0$ in $D(u^+)$ and $u|_{\partial D(u^+)}=0$), we obtain that
for all $\mu \in \mathbb{R}$,
\begin{equation} \label{e5.7}
\begin{split}
0&=\int_{D(u^+)}[A(x) Z_\alpha(u,v) + |u|^{\alpha+1}
 \Big(C(x) |v|^{\beta - \alpha}  + D(x)|v|^{\gamma - \alpha}\Big)\\
&\quad +A(x)u\Big(M - K(x)\cdot\Phi(\frac{u\nabla v}v)
  - \frac{C_1(x)}{A(x)} \phi_\beta(u) \Big) + uf(x)\phi(\frac uv)] dx \\
&\quad + \int_{D(u^+)}\{ b(x) + \mu \}\{ A(x)[ Z_\alpha(u,v)
  - |\nabla u|^{\alpha+1} \; ]+ uA(x) B(x)\cdot\Phi(\frac{u\nabla v}v)\\
&\quad + uf(x)\phi(\frac uv) + |u|^{\alpha+1}\{ C(x)|v|^{\beta-\alpha}
  + D(x)|v|^{\gamma- \alpha} \} \}dx
\end{split}
\end{equation}
which could hold only if each integrand is null;
in particular that in the first integral.
In that integrand, all terms are non negative except for
\[
\Big(M - K(x)\cdot\Phi(\frac{u\nabla v}v) - \frac{C_1(x)}{A(x)}
 \phi_\beta(u) \Big).
\]
But from the hypotheses, this formula is positive.

(a) if $K\equiv 0$ and $M>0$ big enough. Then $v$ cannot be
eventually positive;

(b) if $K\not\equiv 0$ the same conclusion prevails unless,
because of the term $ \Phi(\frac{u\nabla v}v)$,
$\lim\inf_{|x|\nearrow \infty} |v(x)|=0 $.
\end{proof}

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\end{document}