\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 214, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/214\hfil Sturm-Picone type theorems] {Sturm-Picone type theorems for second-order nonlinear elliptic differential equations} \author[A. T\.iryak\.i \hfil EJDE-2014/214\hfilneg] {Aydin T\.iryak\.i} % in alphabetical order \address{Aydin T\.iryak\.i \newline Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Izmir University, 35350 Uckuyular, Izmir, Turkey} \email{aydin.tiryaki@izmir.edu.tr} \thanks{Submitted July 16, 2014. Published October 14, 2014.} \subjclass[2000]{35B05} \keywords{Comparison theorem; Sturm-Picone theorem; half-linear equations, \hfill\break\indent variational lemma; elliptic equations; oscillation} \begin{abstract} The aim of this article is to give Sturm-Picone type theorems for the pair of second order nonlinear elliptic differential equations \begin{gather*} \operatorname{div}(p_1(x)|\nabla u|^{\alpha-1}\nabla u ) +q_1(x)f_1(u)+r_1(x)g_1(u)=0,\\ \operatorname{div}(p_2(x)|\nabla v|^{\alpha-1}\nabla v ) +q_2(x)f_2(v)+r_2(x)g_2(v)=0, \end{gather*} where $|\cdot|$ denotes the Euclidean length and $\nabla= (\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_{n}} )^{T}$ (the superscript $T$ denotes the transpose). Our results include some earlier results and generalize to n-dimensions well-known comparison theorems given by Sturm, Picone and Leighton \cite{3Kreith1,Swanson} which play a key role in the qualitative behavior of solutions. By using generalization of $n$ dimensional Leigton's comparison theorem, an oscillation result is given as an application. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In the qualitative theory of ordinary differential equations, the celebrated Sturm-Picone theorem plays a crucial role. In 1836, the first important comparison theorem was established by Sturm \cite{1Sturm}. In 1909, Picone \cite{2Picone} modified Sturm's theorem. For a detailed study and earlier developments of this subject, we refer the reader to the books \cite{3Kreith1,Swanson}. Sturm-Picone theorem is extended in several directions, see \cite{Ahmad} and \cite{Ahmad2} for linear systems, \cite{Muller} for nonself adjoint differential equations, \cite{TyagiandRaghavendra} for implicit differential equations, \cite{JarosandKusano, Li} for half linear equations, \cite{6Allegretto1} for degenerate elliptic equations,\cite{Zhang} for linear equations on time scales and \cite{Tiryaki2014, Tyagi2013} for a pair of nonlinear differential equations. On the other hand, we emphasize that the classical proof of Sturm-Picone theorem heavily depends on the Leighton's variational lemma \cite{Leighton} (see \cite{Swanson} also). Since when it was proved, it has been extended in different contexts, see, for instance \cite{Dosly2003, JarosKusanoYoshida2000, Komkov1972}. There is also a good amount of interest in the qualitative theory of differential equations to determine whether the given equation is oscillatory or not and Sturm-Picone theorem also plays an important role in this direction. For earlier developments, we refer to \cite{3Kreith1, 2Picone, 1Sturm, Swanson} and for recent developments, we refer to Yoshida's book \cite{Yoshidakitap}. Sturm comparison theorems for half linear elliptic equations and Picone type identities have been studied in, for example \cite{7Allegretto2, 8Allegretto3, 6Allegretto1, Allegretto2,9Bognar, Dosly2002,Fisnarova,JarosKusanoYoshida2001,JarosKusanoYoshida2,118Jaros, 11Kusano2000,Tadie,13Yoshida2,12Yoshida1,Yoshidakitap,13Yoshida,34Yoshida,35Yoshida}. Recently, Tyagi \cite{Tyagi2013} studied a pair of second order nonlinear elliptic partial differential equations \begin{gather} -\Delta u =q_1(x)f_1(u)+b_1(x)r_1(u), \label{11}\\ -\Delta v =q_2(x)f_2(v)+b_2(x)r_2(v), \label{12} \end{gather} under suitable conditions. By establishing a nonlinear version of Leighton's variational lemma, he gave the generalization of Sturm-Picone theorem for \eqref{11} and \eqref{12}. But it is obvious that this result does not work for the half linear elliptic case. A natural question now arises: Is it possible to generalize the Sturm comparison results to the nonlinear elliptic partial differential equations that contain the half linear case by using a nonlinear version of Leighton's variational lemma? Motivated by the ideas in \cite{11Kusano2000, Tiryaki2014, Tyagi2013}, extending Tyagi's results, we prove a nonlinear analogue for n-dimensional Leighton's theorem and we give a generalization of n-dimensional Sturm-Picone theorem by establishing a suitable nonlinear version of Leighton's variational lemma which contain the half linear and also linear elliptic equations. \section{Main results} Let us consider a pair of second-order nonlinear elliptic type partial differential operators: \begin{gather} \ell u := \nabla \cdot ( p_1(x)|\nabla u|^{\alpha-1}\nabla u ) +q_1(x)f_1(u)+r_1(x)g_1(u), \label{21} \\ L v := \nabla \cdot ( p_2(x)|\nabla v|^{\alpha-1}\nabla v ) +q_2(x)f_2(u)+r_2(x)g_2(u), \label{22} \end{gather} where $|\cdot|$ denotes the Euclidean length and $\nabla= \big(\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_{n}} \big)^{T}$ (the superscript $T$ denotes the transpose). In this section, by establishing a nonlinear version of Leighton's variational Lemma, we focused on obtaining a generalization of n-dimensional Sturm-Picone theorem for \eqref{21} and \eqref{22}. Let $G$ be a bounded domain in $\mathbb{R}^{n}$ with boundary $\partial G$ having a piecewise continuous unit normal. Let also $p_i \in C(\bar{G}, \mathbb{R})$, $q_i$, $r_i \in C^{\mu}(\bar{G}, \mathbb{R})$, $f_1 \in C^{1}(\mathbb{R}, \mathbb{R})$, $f_2 \in C(\mathbb{R}, \mathbb{R})$, $g_i \in C(\mathbb{R}, \mathbb{R})$, for $i=1,2$ where $0<\mu \leq 1$, $q_i$'s are of indefinite sign for $i=1,2$ and $p_i(x)> 0$, $r_i(x)\geq 0$ for all $x \in \bar{G}$ and $\alpha$ is a positive real constant. The domain $D_{\ell}(G)$ of $\ell$ is defined to be the set of all functions $u$ of class $C^{1}(\bar{G},\mathbb{R})$ with the property that $p_1(x)|\nabla u|^{\alpha-1} \nabla u \in C^{1}(G; \mathbb{R})\cap C(\bar{G},\mathbb{R})$. The domain $D_{L}(G)$ of $L$ is defined similarly. Note that such a function $u \in D_{\ell}(G)$ (and $v \in D_{L}(G)$) exists for \eqref{21} (and \eqref{22}) \cite{Bergerkitap, Renardy}. The principal part of \eqref{21} (and \eqref{22}) is reduced to the p-Laplacian $\Delta_p u:=\nabla \cdot (|\nabla u|^{p-2} \nabla u)$ ($p=\alpha+1$, $p_1(x)\equiv 1$). We know that a variety of physical phenomena are modelled by equations involving the p-Laplacian \cite{1Ahmed, 2Aris, 3Astarita, Diaz, 13Oden, 14Pelissier, 16Schoenauer}. In what follows, we make the following hypotheses on $f_i$ and $g_i$. \begin{itemize} \item[(H1)] Let $f_1 \in C^{1}(\mathbb{R},\mathbb{R})$ and there exist $\alpha_0$, $\alpha_1 \in (0, \infty)$ such that $\alpha_{0}|u|^{\alpha-1} \leq f'_1(u)$ and $\alpha_1|u|^{\alpha-1}u \geq f_1(u)\neq 0$ for all $0\neq u \in R$. \item[(H1*)] Let $f_1 \in C^{1}(\mathbb{R},\mathbb{R})$ and there exists a $k>0$ such that $\frac{f'_1(u)}{|f_1(u)|^{\frac{\alpha-1}{\alpha}}}\geq k$ for all $0\neq u \in R$. \item[(H2)] Let $g_1 \in C(\mathbb{R},\mathbb{R})$ and there exists a $\beta \geq 0$ such that $\frac{g_1(u)}{f_1(u)}\geq \beta$ for all $0\neq u \in R$. \item[(H3)] Let $f_2, g_2 \in C(\mathbb{R},\mathbb{R})$ and there exists $\alpha_2, \alpha_3, \alpha_4 \in (0, \infty)$ such that \\ $\alpha_3|v|^{\alpha+1}\leq f_2(v)v \leq \alpha_2|v|^{\alpha+1}$ and $g_2(v)v \leq \alpha_4|v|^{\alpha+1}$ \end{itemize} \begin{remark} \label{rmk2.1}\rm Assumption (H1) motivates us to study nonlinearities of the form $$ f_1(u)=|u|^{\alpha-1} u (1\mp \text{ a nonlinear part}) $$ where nonlinear part is decays at $\infty$. \end{remark} \begin{remark} \label{rmk2.2}\rm Assumption (H3) simply says that $\frac{f_2(v)}{|v|^{\alpha-1}v}$ is bounded for all $0\neq v \in R$. \end{remark} \begin{remark} \label{rmk2.3}\rm Assumption (H1*) is a very common condition in the literature for half linear equations. \end{remark} We begin with a lemma and the definition of some concepts needed in this article. \begin{lemma}[\cite{11Kusano2000}] \label{lem2.4} Define $\Phi (\xi)=|\xi|^{\alpha-1}\xi$, $\xi \in \mathbb{R}^{n}$, $\alpha>0$. If $X, Y in \mathbb{R}^{n}$, then \begin{equation} X\Phi(X)+\alpha Y \Phi(Y)-(\alpha+1)X \cdot \Phi(Y)\geq 0. \label{23} \end{equation} where the equality holds if and only if $X=Y$. \end{lemma} Let $U$ be the set of all real valued continuous functions defined on $\bar{G}$ which vanish on $\partial G$ and have uniformly continuous firs partial derivatives on $G$. Also define the functions $j$, $j^{*}$ and $J$: $U\to \mathbb{R}$ by \begin{equation} \label{24} \begin{gathered} j(\eta)=\int_{G}\{p_1(x)|\nabla \eta|^{\alpha+1}-C_1\big(q_1(x) +\beta r_1(x) \big) |\eta|^{\alpha+1}\}dx, \\ j^{*}(\eta)=\int_{G}\{p_1(x)|\nabla \eta|^{\alpha+1} -C_2\big(q_1(x)+\beta r_1(x) \big) |\eta|^{\alpha+1}\}dx,\\ J(\eta)= \int_{G}\{p_2(x)|\nabla \eta|^{\alpha+1}-(\alpha_2q_2^{+}(x)-\alpha_3q_2^{-}(x)+\alpha_4 r_2(x))|\eta|^{\alpha+1}\} dx \end{gathered} \end{equation} where $C_1=(\frac{\alpha_{0}}{\alpha_1 \alpha})^{\alpha}\alpha_1$, $C_2=(\frac{k}{\alpha})^{\alpha}$, $q_2^{+}=max\{q_2, 0\}$ and $q_2^{-}=max\{-q_2, 0\}$. The variation $V(\eta)$ and $V^{*}(\eta)$ are defined as \begin{equation} \label{26} \begin{gathered} V(\eta)=J(\eta)-j(\eta) ,\\ V^{*}(\eta)=J(\eta)-j^{*}(\eta) \end{gathered} \end{equation} with domain $D:=D_{j}\cap D_{J}= D_{j^{*}}\cap D_{J}$. To prove a nonlinear analogue of Leighton's theorem we first establish a nonlinear version of Leighton's variational lemma (Generalization of n-dimensional Leighton's variational type lemma). \begin{lemma} \label{lem2.5} Assume that there exists a nontrivial function $\eta \in U$ such that $j(\eta)\leq 0$ (or $j^{*}(\eta)\leq 0$ ). Then under the hypotheses {\rm (H1)} (or {\rm (H1*)}) and {\rm (H2)}, every solution $u\in D_{j}$ of $\ell(u)=0$ vanishes at some points of $\bar{G}$. \end{lemma} \begin{proof} Let us give the proof under the conditions $j(\eta)\leq 0$, (H1) and (H2). Similarly proof holds for $j^{*}(\eta)\leq 0$, (H1*) and (H2). Assume on the contrary that the statement is false. Suppose that there exists a solution $u \in D_{\ell}(G)$ of $\ell(u)=0$ satisfying $u\neq 0$ on $\bar{G}$. By (H1), we have $f_1(u(x))\neq 0$, $\forall x \in \bar{G}$. Then for $\eta \in U$, the following equality is valid in $G$: \begin{align*} &\nabla \cdot \Big(\frac{\alpha \eta \Phi(\eta)}{f_1(u(x))}p_1(x)|\nabla u|^{\alpha-1}\nabla u\Big)\\ &=\sum_{i=1}^{n} \{ \frac{\partial}{\partial x_i}(\alpha \eta \Phi(\eta))\frac{p_1(x)|\nabla u|^{\alpha-1}\nabla u}{f_1(u(x))}\\ &\quad + \alpha \eta \Phi(\eta) (\frac{\partial}{\partial x_i} \frac{1}{f_1(u(x))})p_1(x)|\nabla u|^{\alpha-1}\nabla u \\ &\quad + \frac{\alpha \eta \Phi(\eta)}{f_1(u(x))} \frac{\partial}{\partial x_i} (p_1(x)|\nabla u|^{\alpha-1}\nabla u)\} \\ &= p_1(x)\frac{|f_1(u(x))|^{\alpha-1}}{(f'_1(u(x)))^{\alpha}}|\alpha \nabla \eta|^{\alpha+1}-\alpha q_1(x)|\eta|^{\alpha+1}-\alpha r_1(x) \frac{g_1(u(x))}{f_1(u(x))}|u|^{\alpha+1} \\ &\quad -p_1(x)\frac{|f_1(u(x))|^{\alpha-1}}{(f'_1(u(x)))^{\alpha}} F \Big(\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))}, \alpha \nabla \eta \Big), \end{align*} where \begin{align*} & F (\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))}, \alpha \nabla \eta )\\ & =|\alpha \nabla \eta|^{\alpha+1} +\alpha \big|\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))} \big|^{\alpha+1} -(\alpha+1)\alpha \nabla\eta \cdot \Phi \Big(\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))} \Big) \end{align*} By (H1) and (H2), we obtain \begin{equation} \begin{aligned} &p_1(x)|\nabla \eta|^{\alpha+1}-C_1(q_1(x)+\beta r_1(x))| \eta|^{\alpha+1} \\ &\geq C_1 \nabla \cdot \Big(\frac{\eta \Phi (\eta)}{f_1(u(x))}p_1(x)|\nabla u|^{\alpha-1}\nabla u \Big)\\ &\quad +\frac{C_1}{\alpha}p_1(x)\frac{|f_1(u(x))|^{\alpha-1}}{f'_1(u(x))^{\alpha}} F\Big(\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))}, \alpha \nabla \eta \Big). \end{aligned} \label{27} \end{equation} We integrate \eqref{27} over $G$ and then apply the divergence theorem to obtain $$ j(\eta)\geq \frac{C_1}{\alpha}\int_{G} p_1(x) \frac{|f_1(u(x))|^{\alpha-1}}{f'_1(u(x))^{\alpha}}F \Big(\frac{\alpha \nabla u f'_1(u(x))}{f_1(u(x))}, \alpha \nabla \eta \Big)dx \geq 0 \,. $$ Therefore, $$ \int_{G}p_1(x)\frac{|f_1(u(x))|^{\alpha-1}}{f'_1(u(x))^{\alpha}}F \Big(\frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))}, \alpha \nabla \eta \Big)=0. $$ From Lemma \ref{lem2.4}, we see that $$ \frac{\eta \nabla u f'_1(u(x))}{f_1(u(x))}=\alpha \nabla \eta \quad \text{or}\quad \nabla ( \frac{|\eta (x)|^{\alpha}}{|f_1(u(x))|} )=0\quad\text{in } G. $$ Since $\eta \in U$, there exists a nonzero constant $K$ such that $$ |\eta(x)|^{\alpha}=|K f_1(u(x))| $$ in $G$ and hence on $\bar{G}$ by continuity. This is not possible because $\eta(x)=0$ on $\partial G$ but $f_1(u(x)) \neq 0$ on $\partial G$ ($u(x)\neq 0$ on $\partial G$). This implies that $j(\eta)>0$, which is a contradiction and hence every solution $u$ of $\ell u=0$ vanishes at some point of $\bar{G}$. This completes the proof. \end{proof} Lemma \ref{lem2.5} plays a crucial role to establish the following Generalization of $n$-dimensional Leighton's theorem. \begin{theorem} \label{thm2.6} Let {\rm (H1)} (or {\rm (H1*)}), {\rm (H2)} and {\rm (H3)} hold. If there exists a nontrivial solution $v \in D$ of $L v=0$ in $\bar{G}$ such that $v=0$ on $\partial G$ and $V(v)\geq 0$ (or $V^{*}(v)\geq 0$), then every solution $u$ of $\ell u=0$ vanishes at some point of $\bar{G}$. \end{theorem} \begin{proof} As in the proof of Lemma \ref{lem2.5} , let us give the proof under the conditions (H1), (H2) and $V(v)\geq 0$. Since $v$ is a solution of $L v=0$ and $v=0$ on $\partial G$ so by an application of Green's theorem we have \begin{equation} \label{28} \begin{aligned} &\int_{G} \Big(q_2(x)f_2(v)v+r_2(x)g_2(v) v \Big)dx\\ &=-\int_{G} v \nabla \cdot (p_2(x)|\nabla v|^{\alpha-1}\nabla v)dx \\ &=-v (p_2(x) |\nabla v|^{\alpha-1}\nabla v)\mid_{\partial G} + \int_{G} p_2(x)|\nabla v|^{\alpha+1}dx \\ &=\int_{G} p_2(x)|\nabla v|^{\alpha+1}dx. \end{aligned} \end{equation} In view of (H3), one can see that \begin{equation} \int_{G} \Big(q_2(x)f_2(v)v+r_2(x)g_2(v) v \Big)dx \leq \int_{G} [(\alpha_2 q^{+}_2(x)-\alpha_3 q^{-}_2(x)) + \alpha_4r_2(x)]|v|^{\alpha+1} dx. \label{2828} \end{equation} By \eqref{28} and \eqref{2828}, we have $J(v)\leq 0$. Since $V(u)\geq 0$, this implies $$ j(v)\leq J(v)\leq 0 $$ and hence by application of Lemma \ref{lem2.5} every nontrivial solution $u$ of $\ell u=0$ vanishes at some point of $\bar{G}$. This completes the proof. \end{proof} \begin{remark} \label{rmk2.7} \rm If the condition $V(v)\geq 0$ (or $V^{*}(v)\geq 0$) is strengthened to $V(v)>0$ (or $V^{*}(v)>0$), the conclusion of Theorem \ref{thm2.6} holds also in the domain $G$. \end{remark} From Theorem \ref{thm2.6}. we immediately have the following Corollary which is an $n$-dimensional extension of Sturm-Picone comparison theorem for the operators \eqref{21} and \eqref{22}. \begin{corollary} \label{coro2.8} Let {\rm (H1)} (or {\rm (H1*)}), {\rm (H2)} and {\rm (H3)} hold. Suppose there exists a nontrivial solution $v$ of $L v=0$ in $\bar{G}$ such that $v=0$ on $\partial G$. If $p_2(x)\geq p_1(x)$ and \begin{gather*} C_1(q_1(x)+\beta r_1(x))\geq [\alpha_2q_2(x) -(\alpha_3-\alpha_2)q^{-}_2(x)+\alpha_4r_2(x)],\\ \Big(\text{or } C_2(q_1(x)+\beta r_1(x))\geq [\alpha_2q_2(x) -(\alpha_3-\alpha_2)q^{-}_2(x)+\alpha_4r_2(x)] \Big), \end{gather*} for every $x \in \bar{G} $. Then every nontrivial solution $u$ of $\ell u=0$ vanishes at some point of $\bar{G}$. \end{corollary} From Lemma \ref{lem2.5}, Theorem \ref{thm2.6} and Corollary \ref{coro2.8} we easily obtain the following results which are straightforward extensions of the variational Lemma, Leighton's theorem and the celebrated Sturm-Picone theorem from \cite{3Kreith1, Swanson} valid for linear second order ordinary differential equations to half linear elliptic partial differential equations that contain linear case. \begin{corollary} \label{coro2.9} Let $f_1(u)=|u|^{\alpha-1}u$ and either $r_1(x)\equiv 0$ or $g_1(u)\equiv 0$ in \eqref{21}. If there exists a nontrivial function $\eta \in U$ such that \begin{eqnarray} \int_{G} \{p_1(x)|\nabla \eta|^{\alpha+1}-q_1(x)|\eta|^{\alpha+1} \} dx \leq 0 \label{210} \end{eqnarray} then every nontrivial solution $u$ of half linear elliptic equation \begin{equation} \nabla \cdot (p_1(x)|\nabla u|^{\alpha-1}\nabla u )+q_1(x)|u|^{\alpha-1}u=0 \label{211} \end{equation} vanishes at some point in $\bar{G}$. \end{corollary} \begin{corollary} \label{coro2.10} Suppose that there exists a nontrivial solution $v$ of \begin{equation} \nabla \cdot (p_2(x)|\nabla v|^{\alpha-1}\nabla v )+q_2(x)|v|^{\alpha-1}v=0 \label{212} \end{equation} in $\bar{G}$ such that $v=0$ on $\partial G$. If \begin{equation} \int_{G} \{ (p_2(x)-p_1(x))|\nabla v|^{\alpha+1} +(q_1(x)-q_2(x))|v|^{\alpha+1} \} dx\geq 0 \label{213} \end{equation} then every nontrivial solution $u$ of \eqref{211} vanishes at some point of $\bar{G}$. \end{corollary} \begin{corollary} \label{coro2.11} Let $p_2(x)\geq p_1(x)$ and $q_1(x)\geq q_2(x)$ for every $x \in \bar{G}$. If there exists a nontrivial solution $v$ of \eqref{212} in $\bar{G}$ such that $v=0$ on $\partial G$, then any nontrivial solution $u$ of \eqref{211} vanishes at some point of $\bar{G}$. \end{corollary} Note that the Corollaries 2.9--2.11 were also obtained in \cite{Dosly2002, 10Dunninger, JarosKusanoYoshida2, 11Kusano2000, Yoshidakitap}. But their proofs depend on the Picone-type and Wirtinger type inequalities. Recently Bal \cite{K.Bal} gave a nonlinear version of the Sturmian comparison principle for a special case of \eqref{21} and \eqref{22} as the follows. \begin{theorem}[\cite{K.Bal}] \label{thm2.12} Let $q_1$ and $q_2$ be the two weight functions such that $q_20$ but for $u<0$ $f_1(u)$ is not defined. This result can be corrected by using Corollary \ref{coro2.8}, for the bounded domain $\bar{G}$ in $\mathbb{R}^{n}$ and we can give the following Sturmian comparison result for the equations \eqref{*1} and \eqref{*2} as follows: \begin{corollary} \label{coro2.13} Let {\rm (H1*)} hold with $k=\alpha=p-1$. If there exists a nontrivial solution $v \in D$ of \eqref{*1} in $\bar{G}$ such that $v=0$ on $\partial G$ and $q_1(x)\geq q_2(x)$, then every solution $u$ of \eqref{*2} vanishes at some point of $\bar{G}$. \end{corollary} \section{An application} This section deals with an application of Theorem \ref{thm2.6}. This theorem enables us to develop some oscillation criteria for the equation $\ell u=0$. Let $\Omega$ be an exterior domain in $\mathbb{R}^{n}$, that is, a domain such that $\Omega \supset \{x \in \mathbb{R}^{n}: |x|\geq r_{0} \}$ for some $r_{0}>0$, and consider the nonlinear elliptic equation \begin{equation} \nabla \cdot (p_1(x)|\nabla u|^{\alpha-1}\nabla u )+ q_1(x)f_1(u)=0 \label{31} \end{equation} in $\Omega$ where $\alpha>0$ is a constant, $p_1 \in C(\Omega, \mathbb{R}^{+})$, $q_1 \in C(\Omega, \mathbb{R})$ and $f_1$ satisfy the hypothesis (H1) (or (H1*)). A nontrivial solution of \eqref{31} is said to be oscillatory if it has a zero in $\Omega \cap \{x \in \mathbb{R}^{n}: |x|>r^{*} \}$ for any $r^{*}>r_{0}$. For brevity, \eqref{31} is called oscillatory if all of its nontrivial solutions are oscillatory. We will show that an explicit oscillation criterion for \eqref{31} can be obtained via the comparison principle proven in the preceding section. Our main idea is to compare \eqref{31} with suitably chosen equations with radial symmetry of the type \begin{equation} \nabla \cdot (\tilde{p_1}(|x|)|\nabla v|^{\alpha-1}\nabla v ) + \bar{q_1}(|x|)f_1(v)=0 \label{32} \end{equation} in $\{x \in \mathbb{R}^{n}: |x|\geq r_{0} \}$ and employ information about the oscillatory behavior of radially symmetric solutions of \eqref{32}. It is easily verified that if $v=y(|x|)$ is radially symmetric solution of \eqref{32}, then the function $y(r)$ satisfies the differential equation \begin{equation} (r^{n-1} \tilde{p_1}(r)|y'|^{\alpha-1}y' )' + r^{n-1}\bar{q_1}(r)f_1(y)=0,\quad r\geq r_{0} \label{33} \end{equation} We note that \eqref{33} is a special case of the equation \begin{equation} ( p(r)|y'|^{\alpha-1}y' )'+ q(r)f_1(y)=0,\;\;r\geq r_{0} \label{34} \end{equation} the oscillatory behavior of which has been intensively investigated in recent years by numerous authors \cite{Agarwal, Dosly2005}. Suppose that $p(r)$ and $q(r)$ are continuous functions defined on $[r_{0}, \infty)$ such that $p(r)>0$ on $[r_{0}, \infty)$. A solution of \eqref{34} is a function $y:[r_{0}, \infty)\to \mathbb{R}$ which is continuously differentiable on $[r_{0}, \infty)$ together with $p|y'|^{\alpha-1}y'$ and satisfies \eqref{34} at every point of $[r_{0}, \infty)$. A nontrivial solution is said to be oscillatory if it has a sequence of zeros clustering ar $r=\infty$, and nonoscillatory otherwise. Now we give an oscillation criterion for \eqref{34}. Its proof can be found, for example, in \cite{Agarwal}. \begin{lemma} \label{lem3.1} Let {\rm(H1)} (or {\rm (H1*)}) hold. Suppose that $p \in C([r_{0},\infty), \mathbb{R}^{+})$ and $q \in C([r_{0},\infty), \mathbb{R})$ satisfies $$ \int_{r_1}^{r}\Big( \int_{r_0}^{s} p(u) du \Big)^{-1/ \alpha} ds=\infty $$ and $$ \lim _{r \to \infty } {\frac{1}{r} \int_{r_{0}}^{r} \Big( \int_{r_0}^{s} q(u) du \Big)}ds=\infty. $$ Then \eqref{34} is oscillatory. \end{lemma} We first establish a principle which enables us to deduce the oscillation of \eqref{31} from the one-dimensional oscillation of \eqref{33}. \begin{theorem} \label{thm3.2} If there exist functions $\tilde{p_1} \in C([r_{0},\infty), \mathbb{R}^{+})$ and $\tilde{q_1} \in C([r_{0},\infty), \mathbb{R})$ such that $$ \tilde{p_1}(r) \geq \max_{|x|=r} {p_1(x)} $$ and \begin{equation} \alpha_2\tilde{q_1}^{+}(r)-\alpha_3\tilde{q_1}(r) \leq C_1 \min_{|x|=r} {q_1(x)} \label{311} \end{equation} \[ \Big(\text{or }\alpha_2\tilde{q_1}^{+}(r)-\alpha_3\tilde{q_1}(r) \leq C_2 \min_{|x|=r} {q_1(x)}\Big) \] where $\alpha_2$, $\alpha_3$, $C_1$ and $C_2$ are defined as before, and the ordinary differential equation \eqref{33} is oscillatory, then \eqref{31} is oscillatory in $\Omega$. \end{theorem} \begin{proof} By hypothesis there exists an oscillatory solution $y(r)$ of \eqref{33} on $[r_0, \infty)$. Let $\{r_i\}$ be the set of all zeros of $y(r)$ such that $r_0\leq r_1 < r_2 <\dots< r_i<\dots$, $\lim_{i \to \infty} {r_i}=\infty $. Then the function $v(x)=y(|x|)$ is a radially symmetric solution of \eqref{32} which is defined in $\{x \in \mathbb{R}^{n}: |x|\geq r_{0} \}$ and has the spherical nodes $|x|=r_i$, $i=1,2,\dots$. Let us compare \eqref{31} with \eqref{32} in the annular domains $G_i= \{ x \in \mathbb{R}^{n}: r_i <|x|< r_{i+1} \}$, $i=1,2,\dots$. For each $i$, $v$ is a solution of \eqref{32} in $G_i$ such that $v\neq 0$ in $G_i$ and $v=0$ on $\partial G_i$. Since \eqref{311} implies $$ \tilde{p_1}(|x|)\geq p_1(x) $$ and $$ \alpha_2{\tilde{q_1}}^{+}(|x|)-\alpha_3{\tilde{q_1}}^{-}(|x|)\leq C_1 q_1(x) $$ $$ \Big(\text{or }\alpha_2{\tilde{q_1}}^{+}(|x|)-\alpha_3{\tilde{q_1}}^{-}(|x|)\leq C_2 q_1(x)\Big) $$ in $\{x \in \mathbb{R}^{n}: |x|\geq r_{0} \}$, we obtain \begin{align*} V(v) &\equiv \int_{G_i} \{ (\tilde{p_1}(|x|)-p_1(x))|\nabla v|^{\alpha+1}+[C_1 q_1(x)\\ &\quad -( \alpha_2 \tilde{q_1}^{+}(|x|) -\alpha_3\tilde{q_1}^{-}(|x|)) ] |v|^{\alpha+1} \}dx \geq 0 \\ \Big(\text{or } V^{*}(v) &\equiv \int_{G_i} \{ (\tilde{p_1}(|x|) -p_1(x))|\nabla v|^{\alpha+1}+[C_2 q_1(x)\\ &\quad -( \alpha_2 \tilde{q_1}^{+}(|x|) -\alpha_3\tilde{q_1}^{-}(|x|)) ] |v|^{\alpha+1} \}dx \geq 0\Big). \end{align*} Consequently from Theorem \ref{thm2.6}, it follows that every solution $u$ of \eqref{31} has a zero in $G_i$, $i=1,2,\dots,$ which shows that $u$ is oscillatory in $\Omega$. This completes the proof. \end{proof} \begin{remark} \label{rm3.3} \rm An immediate consequence of Theorem \ref{thm3.2} is that \eqref{32} with $\tilde{p_1} \in C([r_0, \infty), \mathbb{R}^{+})$ and $\tilde{a_1} \in C([r_0, \infty), \mathbb{R})$ is oscillatory in $\{x \in \mathbb{R}^{n}: |x|\geq r_{0} \}$ if it has one radially symmetric solution which is oscillatory there. \end{remark} Combining Theorem \ref{thm3.2} with Lemma \ref{lem3.1} applied to \eqref{33} gives the following oscillation criteria for \eqref{31}. \begin{theorem} \label{thm3.4} Let $\tilde{p_1} \in C([r_{0}, \infty), \mathbb{R}^{+})$ and $\tilde{q_1} \in C([r_{0}, \infty), \mathbb{R})$ be functions satisfying \eqref{311}. Let also {\rm (H1)} (or {\rm(H1*)}) hold. 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