\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 $$\label{e1.1} \nabla \cdot \big(A(x)\Phi(\nabla v)\big) + (\alpha+1)B(x)\cdot\Phi(\nabla v) + C(x) \phi_\beta(v) + D(x) \phi_\gamma (v)=f(x)$$ 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 $$\label{e2.1} \{ a(t) \phi( y')\}' + c(t) \phi(y) + h(t,y,y')=0$$ 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 $$\label{e2.2} \{\phi_\alpha(u')\}' + \alpha \phi_\alpha(u)=0$$ whose solutions are the generalized sine functions $S:=S_\alpha$ \cite{ra,j2} with the following properties: $$\label{e2.3} \begin{gathered} |S_\alpha(t)|^{\alpha+1} + |S'_\alpha(t)|^{\alpha+1}=1, \quad S_\alpha(t + \pi_\alpha)=-S_\alpha(t), \\ \text{where } \pi_\alpha = \frac{2\pi}{(\alpha+1)\sin\{\frac{\pi}{\alpha+1}\}}. \end{gathered}$$ 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 $$\label{e2.4} \{ e^{-k\alpha t} \phi_\alpha(W')\}' + |k|^{\alpha+1} \alpha e^{kt} \phi_\alpha(W)=0,$$ and the function $Y(t):= S_\alpha(t^k)$ with $t\geq 0$ satisfies $$\label{e2.4b} \{ t^{(1-k)\alpha} \phi_\alpha(Y')\}' + |k|^{\alpha+1} t^{k-1} \alpha \phi_\alpha(Y) =0.$$ A one-dimensional equation associated to \eqref{e1.1}, for some $\beta, \gamma >0$, is $$\label{e2.5} \{a(t)\phi(y')\}' + c(t)\phi_\beta(z) + d(t)\phi_\gamma(z)=f(t)$$ 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 $$\label{e2.6} \lim \inf_{t\nearrow \infty} |z(t)|=0\,.$$ \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, $$\label{e2.7} H'(t)=- \{ c(t)|z|^{\beta-\alpha} + d(t)|z|^{\gamma -\alpha} \} + \frac {f(t)}{\phi(z)} - \frac{\alpha a(t)}{\phi(z)} |z'|^{\alpha+1} \leq \frac {f(t)}{\phi(z)}.$$ 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 $$\label{e3.1} \begin{gathered} \{ a(t) \phi( y')\}' + c(t) \phi(y) =0, \\ \{ a_1(t) \phi( z')\}' + c_1(t) \phi(z) + h(t,z,z')=0 \end{gathered}$$ 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 $$\label{e3.2} \zeta_\gamma(u,v):= |u'|^{\gamma+1} - (\gamma+1)\phi_\gamma(\frac uv v')u' + \gamma |\frac {u'}v v'|^{\gamma+1}$$ 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 $z\neq 0$, $$\label{P1} \begin{split} \{ y a(t)\phi(y') - y \phi (\frac yz )a_1(t) \phi(z') \}' &= a_1(t)\zeta_\alpha(y,z) + [a(t)-a_1(t)]|y'|^{\alpha+1}\\ &\quad+ [ c_1(t)- c(t)]|y|^{\alpha+1} + |y|^{\alpha+1}\frac{h(t,z,z')}{\phi(z)} \end{split}$$ 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 $$\label{e3.3} \{ a(t) \phi_\alpha( z')\}' + c(t) \phi_\alpha(z)=0, \quad t>0$$ is strongly oscillatory. Moreover, if $M$ is a positive constant, then $$\label{e3.4} \{ a(t) \phi_\alpha( u')\}' + c(t) \phi_\alpha(u) + a(t)M =0, \quad t>0$$ 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 $$\label{e3.5} \{ a(t) \phi( z')\}' + c(t) \phi(z) + q(t) \phi(z')= f(t); \quad t>0.$$ \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 $$\label{e3.6} \lim\inf_{t\nearrow \infty}\; |z(t)|=0 .$$ \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 $z>0$, $$\label{e3.7} \begin{split} &\{ y a(t)\phi(y') - y \phi ( \frac yz )a(t) \phi(z') \}'\\ &=a(t)\zeta_\alpha(y,z) + a(t) y\{ M + q(t)\phi(\frac yz z')\} -|y|^{\alpha+1}\frac{f(t)}{\phi(z)}. \end{split}$$ If we integrate over a nodal set $D(y)\subset D_T$, where elements are positive, the above equation yields $$\label{e3.8} 0= \int_{D(y)} \{ a(t)\zeta_\alpha (y,z) + a(t) y\{ M + q(t) \phi(\frac yz z')\} -|y|^{\alpha+1}\frac{f(t)}{\phi(z)}\}dt.$$ (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 $$\label{e3.9} \{ a(t) \phi(y')\}' + c(t)\phi(y) -a(t)M=0 .$$ Consider the equation $$\label{e3.10} \{ a(t) \phi(z')\}' + c(t)\phi(z) + q(t)\phi(z')=0$$ 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 $z\neq 0$, \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} 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 $$\label{e3.12} \{ a(t) \phi(z')\}' + c(t)\phi(z) + q(t)\phi(z')=0$$ (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 \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} 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 $$\label{e4.3} \begin{split} R_1(U)&=\{ r^{n-1} \; a(r)\phi(U') \}' + r^{n-1}\; c(r) \phi(U) \\ &\quad +r^{n-1}\Big( B_2(x)\cdot\frac X{|X|} \phi(U') + F (x) \Big). \end{split}$$ If the regular functions $u$ and $v$ satisfy $Pu=Rv=0$ in $\mathbb{R}^n$, then a Picone formula reads $$\label{e4.4} \begin{split} &\nabla \cdot\{ u A(x) \Phi(\nabla u) -u\phi( \frac uv ) a(r) \Phi(\nabla v) \} \\ &=a(r)Z_\alpha(u,v) + \big(A(x)-a(r) \big)|\nabla u|^{\alpha+1} + \big( c(r) - C(x)\big)|u|^{\alpha+1} \\ &\quad + |u|^{\alpha+1}[ B_2(x)\cdot\Phi(\frac {\nabla v}v ) - B_1(x)\cdot\Phi( \frac{\nabla u}u )] + |u|^{\alpha+1}\frac{F(x)}{\phi(v)} \end{split}$$ 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 $H\in C(\mathbb{R}^n, \mathbb{R}^n)$ $$\label{e4.5} \begin{gathered} h^+(r) := \max_{|x|=r} h(x) , \quad H^+(r):= \max_{|x|=r} H(x)\cdot\frac X{|X|}, \\ h^-(r) := \min_{|x|=r} h(x), \quad H^-(r):= \min_{|x|=r} H(x)\cdot\frac X{|X|}. \end{gathered}$$ 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 $$\label{e4.6} \nabla\cdot\big(\Phi_\alpha(\nabla v) \big) + \phi_\beta(v) + \phi_\gamma (v) + B(x)\cdot\Phi_\alpha(\nabla v) + F(x)=0$$ 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 $$\label{e4.7} \Big( r^{n-1}\phi_\alpha( z')\Big)' + r^{n-1}\{ \phi_\beta(z)+ \phi_\gamma(z) + B(x)\cdot\frac X{|X|} \phi_\alpha (z') + F(x) \}=0.$$ 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 $$\label{e4.8} \begin{split} &\{yr^{n-1} \phi_\alpha(y') - r^{n-1} y \phi_\alpha( \frac yz ) \phi_\alpha(z') \}' \\ &=r^{n-1}[ \zeta_\alpha(y,z)+ |y|^{\alpha+1} \Big(|z|^{\beta-\alpha} + |z|^{\gamma-\alpha} - 1 \Big) \\ &\quad +y \{ M + B(x)\cdot\frac{X}{|X|}\phi_\alpha(\frac{yz'}z ) + F(x)\phi(\frac yz ) \}]\\ &=r^{n-1}\big[ \zeta_\alpha(y,z)+ |y|^{\alpha+1}\Big(|z|^{\beta-\alpha} + |z|^{\gamma-\alpha} \Big) \\ &\quad +y \{ M - \phi_\alpha(y) + B(x)\cdot\frac{X}{|X|} \phi_\alpha(\frac{yz'}z ) + F(x)\phi(\frac yz ) \}\big]. \end{split}$$ 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 $$\label{e4.9} \nabla\cdot\big(\Phi_\alpha(\nabla z) \big) + \phi_\beta(z) + \phi_\gamma (z) + B(x)\cdot\Phi_\alpha(\nabla z) + F(x)=0$$ is oscillatory, unless $$\label{e4.10} \lim \inf_{r\nearrow \infty} |z(r)|=0.$$ \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 $$\label{e5.1} \nabla \cdot\{a(x)\Phi(\nabla v ) \} + c(x) \phi(v) +M=0$$ 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}, $$\label{e5.2} \begin{split} &\nabla\cdot\{ a^+(r) y \Phi(\nabla y) - y \phi(\frac yu) a(x)\Phi(\nabla u)\}\\ &= a(x)Z(y,u) + ( a^+ - a) |\nabla y|^{\alpha+1} + ( c - c^- ) |y|^{\alpha+1} \end{split}$$ 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 $$\label{e5.3} \nabla\cdot\Big(A(x)\Phi(\nabla v)\Big) + A(x) B(x)\cdot\Phi(\nabla v) + C(x) \phi_\beta(v) + D(x) \phi_\gamma (v)+f(x)=0$$ 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 $$\label{e5.4} \nabla b(x)= B(x) + K(x),$$ where $K\in C(\mathbb{R}^n, \mathbb{R}^n)$ is bounded. Let $u$ be a strongly oscillatory solution of $$\label{e5.5} \nabla\cdot\big(A(x)\Phi(\nabla u)\big) + C_1(x) \phi_\beta(u) - A(x)M=0$$ 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 $$\label{e5.6} \begin{split} & \nabla \cdot \{ u A(x) \Phi(\nabla u) -u\phi( \frac uv ) A(x) \Phi(\nabla v) - u\phi(\frac uv) b(x) A(x)\Phi(\nabla v) \}\\ &=A(x) Z_\alpha(u,v) + |u|^{\alpha+1}[C(x) |v|^{\beta - \alpha} + D(x)|v|^{\gamma - \alpha}]\\ &\quad +A(x)u\Big(M - K(x)\cdot\Phi(\frac{u\nabla v}v) - \frac{C_1(x)}{A(x)} \phi_\beta(u) \Big) + uf(x)\phi(\frac uv)\\ &\quad + b(x)\{ A(x)[ Z_\alpha(u,v) - |\nabla u|^{\alpha+1} ] + uA(x) B(x)\cdot\Phi(\frac{u\nabla v}v)\\ &\quad + uf(x)\phi(\frac uv) + |u|^{\alpha+1}\{ C(x)|v|^{\beta-\alpha} + D(x)|v|^{\gamma- \alpha} \} \}. \end{split}$$ \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 $\mu \in \mathbb{R}$, $$\label{e5.7} \begin{split} 0&=\int_{D(u^+)}[A(x) Z_\alpha(u,v) + |u|^{\alpha+1} \Big(C(x) |v|^{\beta - \alpha} + D(x)|v|^{\gamma - \alpha}\Big)\\ &\quad +A(x)u\Big(M - K(x)\cdot\Phi(\frac{u\nabla v}v) - \frac{C_1(x)}{A(x)} \phi_\beta(u) \Big) + uf(x)\phi(\frac uv)] dx \\ &\quad + \int_{D(u^+)}\{ b(x) + \mu \}\{ A(x)[ Z_\alpha(u,v) - |\nabla u|^{\alpha+1} \; ]+ uA(x) B(x)\cdot\Phi(\frac{u\nabla v}v)\\ &\quad + uf(x)\phi(\frac uv) + |u|^{\alpha+1}\{ C(x)|v|^{\beta-\alpha} + D(x)|v|^{\gamma- \alpha} \} \}dx \end{split}$$ 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} \begin{thebibliography}{00} \bibitem{ra} Ravi P. Agarwal, Said R. Grace, Donal O'Regan; \emph{Oscillation Theory for Second Order Linear, Half-linear, Superlinear and Sublinear Dynamic Equations}, Kluwer Academic Publishers, Dordrecht, 2002. \bibitem{j1} J. Jaros, T. Kusano, N. Yosida; \emph{Picone-type Inequalities for Nonlinear Elliptic Equations and their Applications} J. of Inequal. \& Appl. (2001), vol. 6, 387-404. \bibitem{j2} J. Jaros, T. Kusano; \emph{A Picone-type identity for second order half-linear differential equations} Acta Math. Univ. 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