\documentclass[twoside]{article} \usepackage{amsmath, amsfonts, amsthm} \pagestyle{myheadings} \markboth{\hfil Oscillation criteria for nonlinear PDE \hfil EJDE--2002/28} {EJDE--2002/28\hfil Robert Ma\v r\'\i k \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 28, pp. 1--10. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Oscillation criteria for a class of nonlinear partial differential equations % \thanks{ {\em Mathematics Subject Classifications:} 35B05. \hfil\break\indent {\em Key words:} Oscillation criteria, nonlinear oscillation, unbounded domains. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted May 24, 2001. Published March 8, 2002.} } \date{} % \author{Robert Ma\v r\'\i k} \maketitle \begin{abstract} This paper presents sufficient conditions on the function $c(x)$ to ensure that every solution of partial differential equation $$\sum_{i=1}^{n}{\partial \over \partial x_i} \Phi_{p}({\partial u \over \partial x_i})+B(x,u)=0, \quad \Phi_p(u):=|u|^{p-1}\mathop{\rm sgn} u. \quad p>1$$ is weakly oscillatory, i.e. has zero outside of every ball in $\mathbb{R}^n$. The main tool is modified Riccati technique developed for Schr\"odinger operator by Noussair and Swanson \cite{N-S}. \end{abstract} \newcommand{\dxi}{{\partial\over\partial x_i}} \newcommand{\duxi}{\frac{\partial u}{\partial x_i}} \newcommand{\dx}{\,\textrm{d}x} \newcommand{\ds}{\,\textrm{d}s} \newcommand{\dS}{\,\textrm{d}S} \newcommand{\pnorm}[1]{\|#1\|_p } \newcommand{\qnorm}[1]{\|#1\|_q } \newcommand{\sgn}{\mathop{\rm sgn}} \newcommand\diver{\mathop{\rm div}} \newtheorem{lemma}{Lemma}%[section] \newtheorem{theorem}{Theorem}%[section] \newtheorem{corollary}[theorem]{Corollary}%[section] \newtheorem{definition}{Definition} \newtheorem{remark}{Remark} \newtheorem{example}{Example} \section{Introduction} In the oscillation theory of linear second order ordinary differential equation $$y''+q(x)y=0 \label{lin:e}$$ plays an important role the associated Riccati equation $$v'+c(x)+v^2=0$$ which can be obtained from \eqref{lin:e} by substitution $v(x)=\frac{y'(x)}{y(x)}$, $y(x)$ being a nonzero solution of \eqref{lin:e}, see e.g. \cite{S66}. The use of this substitution, the so-called Riccati technique, has been later developed also for various types of equations, namely discrete, half--linear, Schr\"odinger and also equations with $p-$Laplacian, see \cite{D2,D-L,D-M,BCHK,M,N-S,Sch89}. In this paper we will study the partial differential equation $$\sum_{i=1}^{n}\dxi \Phi_{p}\Bigl(\duxi\Bigr)+B(x,u)=0, \label{E}$$ where $\Phi_p(u):=|u|^{p-1}\sgn u$, $p>1$. The nonlinearity $B(x,u):\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$ is continuous function odd with respect to the second variable, i.e. \begin{enumerate} \item[(i)] $B(x,-u)=-B(x,u)$ for all $x\in\mathbb{R}^n$ and $u\in\mathbb{R}$. \end{enumerate} Hence if the function $u(x)$ solves \eqref{E}, then the function $-u(x)$ is also solution of \eqref{E}. Futhermore we suppose that there exist real-valued functions $c(x)\in C(\mathbb{R}^n)$, $\varphi(u)\in C^1(\mathbb{R})$ such that the following conditions hold \begin{enumerate} \item[(ii)] $B(x,u)\geq c(x)\varphi(u)$ for all $u>0$ \item[(iii)] $\varphi(u)>0$ for $u>0$, \item[(iv)] there exists $k>0$ such that $\varphi^{q-2}(u)\varphi'(u)\geq k$ for $u>0$, where $q$ is the conjugate number to $p$, i.e., $q=\frac{p}{p-1}>1$ \end{enumerate} A significant particular case of \eqref{E} we obtain for $B(x,u)=c(x)\Phi_{p}(u)$. In this case $k=p-1$ holds in (iv) and \eqref{E} has the form $$\sum_{i=1}^{n}\dxi \Phi_{p}\Bigl(\duxi\Bigr)+c(x)\Phi_p(u)=0, \label{E*}$$ The study of this equation is motivated by the fact that it is Euler--Lagrange equation for the $p-$degree functional \begin{align*} {\mathcal F}_p(u;\Omega ):=& \int_{\Omega}\Big[\sum_{i=1}^n \left|\frac{\partial u}{\partial x_i}\right|^p -c(x)|u(x)|^p\Big]\,dx \\ =& \int_{\Omega }\left[\|\nabla u\|^p_p-c(x)|u|^p\right]\,dx. \end{align*} Equation \eqref{E*} has been investigated in a series of papers of G.~Bogn\'ar \cite{G-B-1,G-B-2,G-B-3} where the basic properties of the eigenvalue problem have been established. The Picone--type identity and Riccati--type substitution for \eqref{E*} has been recently introduced by O. Do\v sl\'y \cite{D}. If $p=2$ then \eqref{E*} is linear Schr\"odinger partial differential equation $$\Delta u+c(x)u=0.$$ Oscillation properties of this equation are deeply studied in the literature. \smallskip The aim of this paper is to study oscillation properties of equation \eqref{E} via modified Riccati technique and derive oscillation criteria for this equation. The following notation will be used throughout the paper: the $p$ and $q$-norms in $\mathbb{R}^n$ $$\pnorm x =\Bigl(\sum_{i=1}^n |x_i|^p\Bigr)^{1/p},\quad \qnorm x =\Bigl(\sum_{i=1}^n |x_i|^q\Bigr)^{1/q}\quad\text{ for }x\in\mathbb{R}^n,$$ and the sets \begin{gather*} \Omega(a,b)=\{x\in\mathbb{R}^n:a\leq \qnorm x \leq b\},\\ \Omega(a)=\lim_{b\to\infty}\Omega(a,b)=\{x\in\mathbb{R}^n:a\leq \qnorm x \},\\ S(a)=\partial \Omega(a)=\{x\in\mathbb{R}^n:a=\qnorm x \}. \end{gather*} The norm $\|\cdot\|$ is the usual Euclidean norm in $\mathbb{R}^n$ and $\omega_{n,q} :=\int_{S(1)}\dS$ is the surface of the unit sphere (with respect to the $q$-norm) in $\mathbb{R}^n$. Motivated by the terminology in \cite{N-S}, we define an oscillation of \eqref{E} as follows \begin{definition}[Weak oscillation] \label{def} \rm A function $f:\Omega\to\mathbb{R}$ is called \textit{(weakly) oscillatory}, if and only if $f(x)$ has zero in $\Omega\cap \Omega(a)$ for every $a>0$. Equation \eqref{E} is called \textit{(weekly) oscillatory} in $\Omega$ whenever every solution $u$ of \eqref{E} is oscillatory in $\Omega$. \end{definition} Since we will not deal with another definition of oscillation, we will refer weak oscillation simply as \textit{oscillation}. The paper is organized as follows. The next section contains the presentation of the main results. In Section 3 we prove some auxiliary results used in the proofs, which are contained in Section 4. \section{Main results} \begin{theorem} \label{th:1} Let $a_0\in\mathbb{R}^+$, $\alpha\in C^1((a_0,\infty),\mathbb{R}^+)$ and $l>1$. If \begin{gather} \lim_{r\to\infty}\int_{\Omega(a_0,r)}\Bigl[ \alpha(\qnorm x)c(x)-\frac 1p \Bigl(\frac{l}{kq} \Bigr)^{p-1}\alpha^{1-p}(\qnorm x)|\alpha'(\qnorm x)|^p\Bigr]\dx=+\infty \label{c1} \\ \intertext{and} \lim_{r\to\infty}\int_{a_0}^r{1\over \bigl(r^{n-1}\alpha(r)\bigr)^{1\over p-1}}=+\infty,\label{c2} \end{gather} then \eqref{E} is oscillatory in $\mathbb{R}^n$. \end{theorem} \begin{remark} \rm Remark that Theorem \ref{th:1} does not deal with the existence of solution. In other words it states that if there exists a solution, then this solution is oscillatory function (in the sense of Definition \ref{def}). \end{remark} A suitable choice of the function $\alpha$ in Theorem \ref{th:1} leads to effective oscillation criteria for equations \eqref{E} and \eqref{E*}. This is the content of the following corollaries. The first one is a Leighton--type oscillation criterion (see \cite[Th.~2.24, p.~70]{S66}). \begin{corollary}\label{cor1} Suppose that $p\geq n$ and $$\lim_{r\to\infty}\int_{\Omega (1,r)}c(x)\dx=+\infty. \label{Leighton}$$ Then \eqref{E} is oscillatory in $\mathbb{R}^n$. \end{corollary} We remark that the condition $p\geq n$ cannot be removed, which is known already from the study of Schr\"odinger equation (for $p=2$). Another choice of the function $\alpha$ improves this criterion criterion, if $p>2$. \begin{corollary}\label{cor3} Let $p\geq n$, $p>2$ and $$\lim_{r\to\infty} \int_{\Omega(1,r)}\ln(\qnorm x)c(x)\dx=+\infty \label{Leighton-improved}.$$ Then \eqref{E} is oscillatory in $\mathbb{R}^n$. \end{corollary} The following theorem covers also the case when $p \omega_{n,q} {|p-n|^p\over p(kq)^{p-1}}. \label{podm:c2} Then \eqref{E} is oscillatory in$\mathbb{R}^n$. \end{corollary} \begin{corollary}\label{cor2spec} Let $$\liminf_{r\to\infty}{1\over \ln r}\int_{\Omega(1,r)}\qnorm x^{p-n}c(x)\dx> \omega_{n,q} \Bigl|{p-n\over p}\Bigr|^p. \label{podm:c2:spec}$$ Then \eqref{E*} is oscillatory in$\mathbb{R}^n$. \end{corollary} \begin{remark} \rm The constant$\omega_{n,q} \bigl|{p-n\over p}\bigr|^p$in \eqref{podm:c2:spec} is optimal and cannot be decreased. This follows from the example of equation $$\sum_{i=1}^{n}\dxi \Phi_{p}\Bigl(\duxi\Bigr)+\Bigl|{p-n\over p}\Bigr|^p \qnorm x^{-p} \Phi_p(u)=0.$$ This equation is not oscillatory, since it has nonoscillatory solution$u(x)=\qnorm x^{p-n\over p}$and the function$c(x)=\bigl|{p-n\over p}\bigr|^p \qnorm x^{-p}$produces equality in condition \eqref{podm:c2:spec}. \end{remark} \begin{remark} \rm We have already mention that the function$\Phi_p(u):=|u|^{p-1}\sgn u$satisfies hypothesis (iii) and (iv) with$k=p-1$. On the other hand in most real applications we claim$B(x,0)=0$for all$x$and consequently$\varphi(0)=0$. In this case integration of (iv) implies$\varphi(u)\geq \Bigl({k\over p-1}\Bigr)^{p-1} u^{p-1}$and the function$\varphi(u)$must satisfy this growth condition. \end{remark} \begin{example} \rm Let us consider perturbed equation \eqref{E*} $$\sum_{i=1}^{n}\dxi \Phi_{p}\Bigl(\duxi\Bigr) + c(x)\Phi_p(u) +\sum_{i=1}^m q_i(x)\psi_i(u)= 0, \quad p\in(1,2] \label{ctverec}$$ where$c(x), q_i(x)\in C(\mathbb{R}^n)$,$\psi_i(u)\in C^1(\mathbb{R})$,$\psi_i(-u)=-\psi_i(u)$for all$i=1..m$and all$u\in\mathbb{R}$, and$\psi_i(u)$are positive and nondecreasing functions for$u>0$and all$i=1..m$. Define \begin{gather*} q(x)=\min\{c(x), q_1(x), q_2(x), \dots, q_m(x)\}\\ \intertext{and} \varphi(u)=\Phi_p(u)+\sum_{i=1}^m \psi_i(u). \end{gather*} Then \begin{equation*} c(x)\Phi_p(u) +\sum_{i=1}^m q_i(x)\psi_i(u) \geq q(x)\varphi(u) \quad \varphi'(u)\varphi^{q-2}(u)\geq p-1 \end{equation*} and hence Theorem \ref{th:1} can be applied. Remark that we suppose no sign restrictions for the functions$q_i$and so \eqref{ctverec} needs not to be majorant for \eqref{E*} in the sense of Sturmian theory. \end{example} \section{Auxiliary results} A modification of Riccati substitution from \cite{D} is presented in the following lemma. \begin{lemma} \label{l:ric} Let$a_0\in\mathbb{R}^+$,$\alpha\in C^1((a_0,\infty),\mathbb{R}^+)$. If$u\in C^2(\mathbb{R}^n,\mathbb{R})$is a solution of \eqref{E} on$\Omega(a_0)$such that$u(x)\neq 0$for$x\in\Omega(a_0)$, then the vector function$\vec w(x)$is well-defined on$\Omega(a_0)$by $$\vec w(x)=\bigl(w_i(x)\bigr)_{i=1}^n,\quad w_i(x)=-{\alpha(\qnorm x )\over \varphi (u(x))}\Phi_p\Bigl(\duxi\Bigr) \label{ric:w}$$ and satisfies the inequality $$\diver{\vec w}\geq \alpha(\qnorm x )c(x)+k \alpha^{1-q}(\qnorm x ) \qnorm {\vec w}^q +{\alpha'(\qnorm x)\over \alpha(\qnorm x)}\sum_{i=1}^n w_i\nu_i, \label{ric:in}$$ where$\nu_i=\Phi_q\Bigl({x_i\over \qnorm x}\Bigr)$. \end{lemma} \begin{proof} In view of (i), without loss of generality let us consider that$u(x)>0$on$\Omega(a_0). It holds \begin{align*} {\partial w_i\over \partial x_i} =&-{\alpha (\qnorm x)\over \varphi (u)} \dxi\Bigl(\Phi_p\Bigl(\duxi\Bigr)\Bigr)-\Phi_p\Bigl(\duxi\Bigr){\alpha'( \qnorm x)\over \varphi(u)}{\partial \qnorm x\over \partial x_i}\\ &+\alpha(\qnorm x)\Bigl|\duxi\Bigr|^p{\varphi'(u)\over \varphi^2(u)}. \end{align*} Since{\partial \qnorm x \over \partial x_i}=\Phi_q\Bigl({x_i\over \qnorm x}\Bigr)=\nu_i$, we get \begin{multline*} {\partial w_i\over \partial x_i}\\ =-{\alpha (\qnorm x)\over \varphi(u)} \dxi\Bigl(\Phi_p\Bigl(\duxi \Bigr)\Bigr)+ {\alpha'(\qnorm x)\over \alpha(\qnorm x)} w_i \nu_i + \varphi'(u)\varphi^{q-2}(u)\alpha^{1-q}(\qnorm x)|w_i|^q. \end{multline*} From this equation and from \eqref{E} it follows \begin{multline*} \diver \vec w\\= \alpha(\qnorm x){B(x,u)\over \varphi(u)}+ \varphi'(u)\varphi^{q-2}(u)\alpha^{1-q}(\qnorm x)\qnorm {\vec w}^q +{\alpha'(\qnorm x)\over \alpha(\qnorm x)}\sum_{i=1}^n w_i \nu_i. \end{multline*} Taking into account conditions (ii), (iii) and (iv) we obtain inequality \eqref{ric:in}. \end{proof} \begin{lemma}\label{l:ner} It holds $${\pnorm x^p\over p}+\sum_{i=1}^n x_i y_i+{\qnorm y^q\over q}\geq 0$$ for every$x,y\in\mathbb{R}^n$,$x=(x_i)_{i=1}^n$,$y=(y_i)_{i=1}^n$. \end{lemma} For the proof of this lemma, see \cite{D}. \section{Proofs of the main results} \begin{proof}[Proof of Theorem \ref{th:1}] Suppose, by contradiction, that$u$is a solution of \eqref{E} which is positive on$\Omega(a_0)$for some$a_0>0$. Then$\vec w$is defined on$\Omega(a_0)$. From inequality \eqref{ric:in}, using integration over the domain$\Omega(a_0,r)$and the Gauss--Ostrogradski divergence theorem, follows \begin{multline} \label{gauss-ostr} \int_{S(r)} \vec w\vec n\dS- \int_{S(a_0)} \vec w\vec n\dS\geq\\ \geq\int_{\Omega(a_0,r)}\biggl(\alpha(\qnorm x )c(x)+k \alpha^{1-q}(\qnorm x ) \qnorm {\vec w}^q +{\alpha'(\qnorm x)\over \alpha(\qnorm x)}\sum_{i=1}^n w_i\nu_i\biggr)\dx, \end{multline} where$\vec n$is the outward normal unit vector to %$S(r)$($S(a_0)$),$\Omega(a_0,r)$i.e.$\vec n=\pm{\vec \nu\over \|\vec \nu\|}$,$\vec \nu=(\nu_i)_{i=1}^n$and$\nu_i$is defined in Lemma \ref{l:ric}. Observe that$\pnorm {\vec \nu}=1$. Now, let$l^*={l\over l-1}>1$be the conjugate number to the number$l. Then \begin{multline*} k \alpha^{1-q}(\qnorm x ) \qnorm {\vec w}^q +{\alpha'(\qnorm x)\over \alpha(\qnorm x)}\sum_{i=1}^n w_i\nu_i=\\ ={kq\over l}\alpha^{1-q}(\qnorm x)\biggl( {\qnorm {\vec w}^q\over q}+ {l\alpha'(\qnorm x)\alpha^{q-2}(\qnorm x)\over qk} \sum_{i=1}^n w_i\nu_i \biggr)+\\ +{k\over l^*}\alpha^{1-q}(\qnorm x)\qnorm {\vec w}^q. \end{multline*} Using Lemma \ref{l:ner} we obtain \begin{align*} k \alpha^{1-q}&(\qnorm x ) \qnorm {\vec w}^q +{\alpha'(\qnorm x)\over \alpha(\qnorm x)}\sum_{i=1}^n w_i\nu_i\\ \geq & -{qk\over lp}\alpha^{1-q}(\qnorm x)\left|\left| {l\alpha'(\qnorm x)\alpha^{q-2}(\qnorm x)\over qk}\vec\nu \right|\right|_p^p+{k\over l^*}\alpha^{1-q}(\qnorm x)\qnorm{\vec w}^q\\ =&- \frac 1p \Bigl({l\over kq}\Bigr)^{p-1}\alpha^{1-p}(\qnorm x)|\alpha'(\qnorm x)|^p+{k\over l^*}\alpha^{1-q}(\qnorm x)\qnorm{\vec w}^q. \end{align*} This inequality together with \eqref{gauss-ostr} yields \begin{multline} \int_{S(r)} \vec w\vec n\dS- \int_{S(a_0)} \vec w\vec n\dS\\ \geq \int_{\Omega(a_0,r)}\Bigl[ \alpha(\qnorm x)c(x)-\frac 1p \Bigl({l\over kq}\Bigr)^{p-1}\alpha^{1-p}(\qnorm x)|\alpha'(\qnorm x)|^p\Bigr]\dx\\+{k\over l^*}\int_{\Omega(a_0,r)}\alpha^{1-q}(\qnorm x) \qnorm{\vec w}^q\dx. \label{..} \end{multline} In view of \eqref{c1}, there existsr_0>a_0$such that \begin{multline*} \int_{\Omega(a_0,r)}\Bigl[ \alpha(\qnorm x)c(x)-\frac 1p \Bigl({l\over kq}\Bigr)^{p-1}\alpha^{1-p}(\qnorm x)|\alpha'(\qnorm x)|^p\Bigr]\dx+\\ + \int_{S(a_0)} \vec w\vec n\dS\geq 0 \end{multline*} and now \eqref{..} implies $$\int_{S(r)}\vec w\vec n\dS\geq {k\over l^*}\int_{\Omega(a_0,r)}\alpha^{1-q}(\qnorm{x})\qnorm{\vec w}^q\dx \label{in1}$$ for$r>r_0$. Application of the H\"older inequality in$\mathbb{R}^n$yields $$\int_{S(r)}\vec w\vec n\dS\leq \int_{S(r)}\qnorm{\vec w}\pnorm{\vec n}\dS.$$ Since$\|\cdot\|$and$\pnorm \cdot$are equivalent norms in$\mathbb{R}^n$, there exists$K>0$such that$\pnorm{\vec n}\leq K\|\vec n\|=K$. This fact and another application of H\"older inequality gives $$\int_{S(r)}\vec w\vec n\dS\leq K \Bigl(\omega_{n,q} r^{n-1}\Bigr)^{1/p} \Bigl(\int_{S(r)}\qnorm{\vec w}^q\dS \Bigr)^{1/q} \label{...}$$ Denote $$g(r)=\int_{\Omega(a_0,r)}\alpha^{1-q}(\qnorm{x})\qnorm{\vec w}^q\dx.$$ Then it holds $$g'(r)=\alpha^{1-q}(r)\int_{S(r)} \qnorm{\vec w}^q\dS.$$ and \eqref{...} gives $$\int_{S(r)}\vec w\vec n\dS\leq K \omega_{n,q}^{1/p} r^{n-1\over p} \Bigl(\alpha^{q-1}(r)g'(r)\Bigr)^{1\over q}. \label{in2}$$ Combining \eqref{in1} and \eqref{in2} we obtain the inequality $${k\over l^*}g(r)\leq K \omega_{n,q}^{1/p} r^{n-1\over p} \Bigl(\alpha^{q-1}(r)g'(r)\Bigr)^{1/q}$$ for$r>r_0$. Hence $$\Bigl({1\over r^{n-1}\alpha(r)}\Bigr)^{1\over p-1}\leq {l^*\omega_{n,q}^{q\over p}\over kK^q}{g'(r)\over g^{q}(r)}.$$ Integration of this inequality over$[r_0,\infty]$gives the divergent integral on the left hand side, according to the assumption \eqref{c2}, and the convergent integral on the right hand side. This contradiction completes the proof. \end{proof} The Proof of Corollary \ref{cor1} follows immediately from Theorem \ref{th:1} for$\alpha(r)\equiv 1$. \begin{proof}[Proof of Corollary \ref{cor3}] Let$a_0>e$be arbitrary and$\alpha(r)=\ln(r)$on$[a_0,\infty)$. Since \begin{equation*} \lim_{r\to\infty} \frac{\alpha^{\frac 1{1-p}}(r)r^{\frac{1-n}{p-1}}} {\frac{1}{r\ln r}}=\lim_{r\to\infty} r^{\frac{p-n}{p-1}}\ln^{\frac{p-2}{p-1}}r\geq 1, \end{equation*} the integral \eqref{c2} diverges by ratio-convergence test. Further, since \begin{multline*} \int_{\Omega(a_0,r)}|\alpha'(\qnorm x)|^{p}\alpha^{1-p}(\qnorm x)\dx= \omega_{n,q} \int_{e}^r \xi^{n-1-p} \ln^{1-p}\xi\,\textrm{d}\xi\\ \leq \omega_{n,q} \int_{a_0}^r \xi^{-1} \ln^{1-p}\xi\,\textrm{d}\xi= \omega_{n,q} \frac 1{p-2}[1-\ln^{2-p}r], \end{multline*} the limit$% \begin{equation*} \lim_{r\to\infty} \int_{\Omega(a_0,r)}|\alpha'(\|x\|)|^{p} \alpha^{1-p}(\|x\|)\dx $% \end{equation*} converges and \eqref{Leighton-improved} is equivalent to the condition \eqref{c1} of Theorem \ref{th:1}. All conditions of Theorem~\ref{th:1} are satisfied and the proof is complete. \end{proof} \begin{proof}[Proof of Corollary \ref{cor2}] Let$\alpha(r)=r^{p-n}$. Then \eqref{c2} holds and it is sufficient to prove that also \eqref{c1} holds, i.e. that there exists$l>1$such that $$\lim_{r\to\infty}\int_{\Omega(1,r)}\Bigl[ \qnorm x^{p-n}c(x)- \frac 1p \Bigl({l\over kq}\Bigr)^{p-1}|{p-n}|^p\qnorm x^{-n}\Bigr]\dx=+\infty. \label{2}$$ According to \eqref{podm:c2} there exists$m>1$,$\varepsilon>0$and$r_0>1$such that $$\int_{\Omega(1,r)}\qnorm x^{p-n} c(x)\dx> (m+\varepsilon) \omega_{n,q} {|{p-n}|^p\over p(kq)^{p-1}} \ln r \label{1}$$ for$r>r_0$. Since $$\int_{\Omega(1,r)} \qnorm x^{-n}\dx =\omega_{n,q}\int_1^r {1\over s}\ds =\omega_{n,q}\ln r,$$ can be \eqref{1} written in the form $$\int_{\Omega(1,r)}\Bigl[\qnorm x^{p-n} c(x) -m {|p-n|^p\over p(kq)^{p-1}}\qnorm x^{-n}\Bigr] \dx> \varepsilon \omega_{n,q}{|p-n|^p\over p(kq)^{p-1}} \ln r$$ which implies \eqref{2}. The proof is complete. \end{proof} The Proof of Corollary \ref{cor2spec} follows immediately from Corollary \ref{cor2}. \begin{thebibliography}{99} \frenchspacing \bibitem{G-B-1} {G. 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Schminke, The lower spectrum of Schr\"odinger operators, Arch. Rational Mech. Anal. \textbf{75} (1989), 147--155. \end{thebibliography} \noindent\textsc{Robert Ma\v r\'\i k }\\ Dept. of Mathematics, Mendel University \\ Zem\v ed\v elsk\'a 3\\ 613 00 Brno, Czech Republic \\ e-mail: marik@mendelu.cz \end{document}