\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 91, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/91\hfil Harnack inequality] {Harnack inequality for quasilinear elliptic equations with $(p,q)$ growth conditions and absorption lower order term} \author[K. Buryachenko \hfil EJDE-2018/91\hfilneg] {Kateryna Buryachenko} \address{Kateryna Buryachenko \newline Vasyl' Stus Donetsk National University, 600-richa Str., 21, Vinnytsia, 21021, Ukraine} \email{katarzyna\_@ukr.net} \dedicatory{Communicated by Marco Squassina} \thanks{Submitted June 14, 2017. Published April 16, 2018.} \subjclass[2010]{35J15, 35J60, 35J62} \keywords{Harnack inequality; quasilinear elliptic equation; \hfill\break\indent Keller-Osserman type estimate; absorption lower term} \begin{abstract} In this article we study the quasilinear elliptic equation with absorption lower term $$ -\operatorname{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0, \quad u\geq 0. $$ Despite of the lack of comparison principle, we prove a priori estimate of Keller-Osserman type. Particularly, under some natural assumptions on the functions $g,f$ for nonnegative solutions we prove an estimate of the form $$ \int_0^{u(x)} f(s)\,ds\leq c\frac{u(x)}{r}g\big(\frac{u(x)}{r}\big),\quad x\in\Omega, B_{8r}(x)\subset\Omega, $$ with constant $c$, independent on $u(x)$. Using this estimate we give a simple proof of the Harnack inequality. \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 this article we consider nonnegative solutions of the quasilinear elliptic equation \begin{equation}\label{e1.1} -\operatorname{div} A(x, \nabla u)+a_0(u)= 0, x\in\Omega, \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^n, n\geq 2$. We suppose that the functions $A=(a_1, a_2,\dots,a_n)$ and $a_0$ satisfy the Caratheodory conditions and the following structural conditions \begin{equation}\label{e1.2} \begin{gathered} A(x,\xi)\xi\geq\nu_1g(|\xi|)|\xi|,\quad |A(x,\xi)|\leq\nu_2g(|\xi|),\\ \nu_1f(u)\leq a_0(u)\leq\nu_2f(u), \end{gathered} \end{equation} where $\nu_1, \nu_2$ are positive constants and $g$ is positive function satisfying conditions \begin{equation}\label{e1.3} g\in C(\mathbb{R}^1_+), \quad \big(\frac{t}{\tau}\big)^{p-1}\leq\frac{g(t)}{g(\tau)} \leq\big(\frac{t}{\tau}\big)^{q-1}, \quad t\geq\tau>0,\; 10$ we set \begin{gather*} F(u)=\int_0^uf(s)\,ds, \delta(u)=\frac{F(u)}{f(u)}, M(\rho) =\sup_{B_{\rho}(x_0)} u, \\ \delta(\rho)=\sup_{B_{\rho}(x_0)} \delta(u), \quad F(\rho)=\sup_{B_{\rho}(x_0)} F(u), \end{gather*} where $B_{\rho}(x_0)$ is ball $\{x: |x-x_0|<\rho\}$. The next theorem is an a priori estimate of Keller-Osserman type, which is interesting in itself and which can be used in the theory of ``large'' solutions (see for example \cite{Shish,Veron}, \cite{Kon1}--\cite{Kon3}). \begin{theorem} \label{thm1} Let conditions \eqref{e1.2}, \eqref{e1.3} be fulfilled and $u$ be a nonnegative weak solution to the equation \eqref{e1.1} in $\Omega$. Let $x_0\in\Omega$. Fix $\sigma\in(0, 1)$. Then there exist a positive numbers $c_1, c_2$, depending only on $n, p, q, \nu_1, \nu_2$ such that \begin{equation}\label{e1.5} F(\sigma\rho)\leq c_1(1-\sigma)^{-c_2}\frac{\delta(\rho)}{\rho} \Big(g\big(\frac{M(\rho)}{\rho}\big)+g\big(\frac{\delta(\rho)}{\rho}\big)\Big), \end{equation} for all $B_{8\rho}(x_0)\subset\Omega$. \end{theorem} \begin{remark} \label{rmk1.1} \rm Conditions \eqref{e1.1}, \eqref{e1.3} imply the local boundedness and H\"older continuity of solutions (see, for example \cite{Lieberman2}). \end{remark} \begin{remark} \label{rmk1.2}\rm For the case $p=q$ inequality \eqref{e1.5} was proved in \cite{SkSh}. In the case $p=q$, using the comparison theorem an radial type solutions, inequality of the type \eqref{e1.5} was proved in \cite{Kon1}. \end{remark} To prove the Harnack inequality for equations with absorption lower terms we need the following condition. \begin{definition}\label{de4} \rm We say that a continuous function $\psi$ satisfies condition (A) if there exists $\mu>0$ such that \begin{equation}\label{e1.6} \frac{\psi(t)}{\psi(\tau)}\leq \big(\frac{t}{\tau}\big)^{\mu}, \end{equation} for all $t\geq\tau>0$. \end{definition} Condition (A) arises due to presence of absorption lower order terms in the equation \eqref{e1.1}: this condition was not presented in \cite{Lieberman2}, but it is closely connected with analogous conditions in the works \cite{PucciSerrin2}--\cite{PucciSerrin3}. \begin{theorem}\label{thm2} Let $G^{-1}$ be the inverse function to the function $G(t)=tg(t)$, and let conditions \eqref{e1.2}, \eqref{e1.3} be fulfilled. Let also $u$ be a nonnegative weak solution to the equation \eqref{e1.1}, function $f(u)$ be nondecreasing and $\psi(u)=u^{-1}G^{-1}(F(u))$ satisfies condition {\rm (A)}.Then there exists positive number $c_3$, depending only on $n, p, q, \nu_1, \nu_2,$\\$c$ such that \begin{equation}\label{e1.9} F(u(x))\leq c_3\frac{u(x)}{\rho}g\left(\frac{u(x)}{\rho}\right), \end{equation} for almost all $x\in B_{\rho}(x_0)$ and for any $x_0\in \Omega$, such that $B_{8\rho}(x_0)\subset\Omega$. \end{theorem} The following theorem is Harnack inequality for the nonnegative weak solutions to the equation \eqref{e1.1}, which is simple consequence of the Theorem \ref{thm2}. \begin{theorem}\label{thm3} Let $u$ be a nonnegative weak solution to the equation \eqref{e1.1}, let conditions \eqref{e1.2}, \eqref{e1.3} be fulfilled. Assume that function $f(u)$ is nondecreasing and $\psi(u)=u^{-1}G^{-1}(F(u))$ satisfies condition $(A)$.Then there exists positive number $c_4$, depending only on $n, p, q, \nu_1, \nu_2$, such that \begin{equation}\label{e1.10} \sup_{B_{\rho}(x_0)} u(x)\leq c_4\inf_{B_{\rho}(x_0)} u(x), \end{equation} for almost all $x\in B_{\rho}(x_0)$, and for any $x_0\in \Omega$, such that $B_{8\rho}(x_0)\subset\Omega$. \end{theorem} \begin{remark} \rm The formulation of the Theorem \ref{thm2} is the same as in \cite{Lieberman2}, however due to presence of absorption lower order term, the results of \cite{Lieberman2} cannot be used. The main novelty of our result that the constant $c_4$ is independent on $u$. \end{remark} \begin{remark} \rm If $f(u)=g(u)f_1(u)$, where function $f_1(u)$ satisfies condition (A) with $\mu_1>q-p$, then the function $u^{-1}G^{-1}(F(u))$ satisfies condition (A) with $\mu=\frac{\mu_1-q+p}{q}>0$. A simple example of the function $f_1(u)$, which satisfies condition (A) for $\mu_1=1$ is a function $f_1\in C^1(\mathbb{R}_+^1, f_1$ is nondecreasing and $f_1(0)=0$. \end{remark} \begin{remark} \rm If $f(u)=g^s(u)f_1(u)$, where $f_1$ is nondecreasing and $s>\frac{q-1}{p-1}$, then the function $u^{-1}G^{-1}(F(u))$ satisfies condition (A) with $\mu=\frac{(p-1)s-q+1}{q}$. \end{remark} \section{Keller-Osserman a priori sub-estimate. Proof of Theorem \ref{thm1}} \subsection{Auxiliary statements and local energy estimates} First of all we prove the following auxiliary statements, which will be used for further investigations. \begin{lemma}\label{lem2.1} Let $\{y_j\}_{j\in N}$ be a sequence of nonnegative numbers such that the following inequalities $$ y_{j+1}\leq Cb^jy_j^{1+\varepsilon} $$ hold for $j=0, 1, 2, \dots$ with positive constants $\varepsilon, C>0, b>1$. Then $$ y_j\leq C^{\frac{(1+\varepsilon)^j-1}{\varepsilon}} b^{\frac{(1+\varepsilon)^j-1}{\varepsilon^2}-\frac{j}{\varepsilon}} y_0^{(1+\varepsilon)^j}. $$ In particular, if $y_0\leq C^{-\frac{1}{\varepsilon}} b^{-\frac{1}{\varepsilon^2}}$, then $\lim_{j\to\infty} y_j=0$. \end{lemma} We denote by the $\gamma$ some constant depending only on $n, p, q, \nu_1, \nu_2$ which may vary from line to line. Let $B_r(\bar x)\subset\Omega$ be a ball in $\Omega$, then we denote by the $\zeta$ some nonnegative piecewise smooth truncated function vanishing on the boundary of the ball $B_r(\bar x)$. \begin{lemma}\label{lem2.2} Let $u$ be a nonnegative weak solution to the equation \eqref{e1.1} and let conditions \eqref{e1.2} and \eqref{e1.3} hold. Then for every $B_r(\bar x)\subset\Omega$ and for every $k>0$ \begin{equation} \label{e2.1} \begin{aligned} &\int_{A_{k,r}}f(u)\,G(|\nabla u|)\zeta^q\,dx +\int_{A_{k,r}}(F(u)-k)_+f(u)\zeta^q\,dx\\ &\leq\gamma\int_{A_{k,r}}(F(u)-k)_+g(\delta(u)|\nabla \zeta|)|\nabla\zeta|dx, \end{aligned} \end{equation} where $A_{k,r}=\{x\in B_r(\bar x): F(u)>k\}$. \end{lemma} \begin{proof} Testing integral equality \eqref{e1.4} by the $\varphi=(F(u)-k)_+\zeta^q$. Using conditions \eqref{e1.2} and \eqref{e1.3} we obtain \begin{align*} &\int_{A_{k,r}}f(u)\,G(|\nabla u|)\zeta^q\,dx+\int_{A_{k,r}}(F(u)-k)_+ f(u)\zeta^q\,dx\\ &\leq\gamma\int_{A_{k,r}}(F(u)-k)_+g(|\nabla u|)|\nabla \zeta|\zeta^{q-1}dx. \end{align*} Let us note that the next inequality is evident \begin{equation} \label{e2.2} g(a)b\leq\varepsilon g(a)a+g\Big(\frac{b}{\varepsilon}\Big)b,\quad a, b, \varepsilon>0. \end{equation} We use this inequality with $a=|\nabla u|, b=\gamma(F(u)-k)_+\frac{|\nabla\zeta|}{\zeta}, \varepsilon =\frac{1}{2}f(u)$ and arrive to the required inequality \eqref{e2.1} \end{proof} \subsection{Proof of Theorem \ref{thm1}} Consider a ball $B_{\rho}(x_0)$ and for fixed $\sigma\in (0, 1)$ let $\bar x$ be an arbitrary point in ball $B_{\sigma\rho}(x_0)$. Further we set \begin{gather*} \rho_j=\frac{1-\sigma}{4}\rho(1+2^{-j}),\quad B_j=B_{\rho_j}(\bar x), A_{k_j,j}=\{x\in B_j: F(u)>k_j\}, \quad j=0, 1, \dots \\ \zeta_j\in C_0^{\infty}(B_j), \quad 0\leq\zeta_j\leq 1,\quad |\nabla\zeta_j|\leq\gamma(1-\sigma)^{-1}2^{-j}\rho^{-1} \end{gather*} and $\zeta_j\equiv 1$ in $B_{j+1}$. By the embedding theorem and H\"older inequality we obtain \begin{equation} \label{e2.3} \begin{aligned} &\int_{A_{k_{j+1},j+1}}(F(u)-k_{j+1})_+dx \\ &\leq\Big(\int_{A_{k_{j+1},j}}((F(u)-k_{j+1})_+\zeta_j^q)^{\frac{n}{n-1}}dx \Big)^{\frac{n-1}{n}}|A_{k_{j+1},j+1}|^{1/n} \\ &\leq \gamma\int_{A_{k_{j+1},j}}|\nabla((F(u)-k_{j+1})_+\zeta_j^q)| |A_{k_{j+1},j}|^{1/n}\\ &\leq\gamma\Big(\int_{A_{k_{j+1},j}}f(u)|\nabla u|\zeta_j^q dx \\ &\quad +\int_{A_{k_{j+1},j}}(F(u)-k_{j+1})_+|\nabla\zeta_j|\zeta_j^{q-1}dx\Big) |A_{k_{j+1},j}|^{1/n}. \end{aligned} \end{equation} Let $\ell=\delta(\rho)/\rho$. Using inequality \eqref{e2.2} with $a=\ell, b=|\nabla u|, \varepsilon=1$ and the evident inequality $(F(u)-k_{j+1})_+\geq\frac{k}{2^{j+1}}$ on $A_{k_{j+1},j}$, we estimate the first term in the right-hand side of \eqref{e2.3} as follows \begin{equation} \label{e2.4} \begin{aligned} &\int_{A_{k_{j+1},j}}f(u)|\nabla u|\zeta_j^qdx \\ &=\frac{1}{g(\ell)}\int_{A_{k_{j+1},j}}f(u)g(\ell)|\nabla u|\zeta_j^qdx \\ &\leq\ell\int_{A_{k_{j+1},j}}f(u)\zeta_j^qdx +\frac{1}{g(\ell)}\int_{A_{k_{j+1},j}}f(u)G(|\nabla u|)\zeta_j^qdx \\ &\leq 2^j\frac{\ell}{k}\int_{A_{k_{j+1},j}}(F(u)-k_{j+1})_+f(u)\zeta_j^qdx +\frac{1}{g(\ell)}\int_{A_{k_{j+1},j}}f(u)G(|\nabla u|)\zeta_j^qdx. \end{aligned} \end{equation} From the previous inequality and Lemma \ref{lem2.2} it follows that \begin{equation} \label{e2.5} \begin{aligned} &\int_{A_{k_{j+1},j}}f(u)|\nabla u|\zeta_j^qdx \\ &\leq \gamma(1-\sigma)^{-\gamma}2^{j\gamma} \big(\frac{\ell}{k}+\frac{1}{g(\ell)}\big)\rho^{-1} g\big(\frac{\delta(\rho)}{\rho}\big)\int_{A_{k_{j},j}}(F(u)-k_{j})_+dx. \end{aligned} \end{equation} Choosing $k$ such that \begin{equation} \label{e2.6} k\geq G(\ell)=G\left(\frac{\delta(\rho)}{\rho}\right), \end{equation} from inequalities \eqref{e2.3} and \eqref{e2.4} we obtain \begin{equation} \label{e2.7} y_{j+1}=\int_{A_{k_{j+1},j+1}}(F(u)-k_{j+1})dx \leq\gamma(1-\sigma)^{-\gamma}2^{j\gamma}\rho^{-1} k^{-\frac{1}{n}}y_j^{1+\frac{1}{n}}. \end{equation} from Lemma \ref{lem2.1} it follows that $y_j\to 0$ as $j\to\infty$, provided $k$ is chosen to satisfy \begin{equation} \label{e2.8} k\geq \gamma(1-\sigma)^{-\gamma}\rho^{-n} \int_{B_{\frac{1-\sigma}{2}\rho}(\bar x)}F(u)dx. \end{equation} Inequalities \eqref{e2.5} and \eqref{e2.6} imply that \begin{equation} \label{e2.9} F(u(\bar x))\leq\gamma(1-\sigma)^{-\gamma}G\left(\frac{\delta(\rho)}{\rho}\right) + \gamma(1-\sigma)^{-\gamma}\rho^{-n}\int_{B_{\frac{1-\sigma}{2}\rho}(\bar x)} F(u)dx. \end{equation} Let $\xi\in C_0^{\infty}(B_{(1-\sigma)\rho}(\bar x)), 0\leq\xi\leq 1, \xi\equiv 1$ in $B_{\frac{1-\sigma}{2}\rho}(\bar x)$ and $|\nabla\xi|\leq 2(1-\sigma)^{-1}\rho^{-1}$. To estimate the integral in the right-hand side of the \eqref{e2.9} we test \eqref{e1.4} by $\varphi=\xi^q$. Using conditions \eqref{e1.2}, \eqref{e1.3} we obtain \begin{align*} \int_{B_{\frac{1-\sigma}{2}\rho}(\bar x)}F(u)dx &\leq\delta(\rho)\int_{B_{(1-\sigma)\rho}(\bar x)}f(u)\xi^qdx \\ &\leq\gamma(1-\sigma)^{-1}\frac{\delta(\rho)}{\rho} \int_{B_{(1-\sigma)\rho}(\bar x)}g(|\nabla u|)\xi^{q-1}dx. \end{align*} We use inequality \eqref{e2.2} with $a=|\nabla u|, b=\xi^{-1}$ to obtain \begin{equation} \label{e2.10} \begin{aligned} \int_{B_{\frac{1-\sigma}{2}\rho}(\bar x)}F(u)dx &\leq\gamma(1-\sigma)^{-1}\frac{\delta(\rho)}{M(\rho)} \int_{B_{(1-\sigma)\rho}(\bar x)}G(|\nabla u|)\xi^qdx \\ &\quad+ \gamma(1-\sigma)^{-1}\frac{\delta(\rho)}{\rho} g\left(\frac{M(\rho)}{\rho}\right)\rho^n. \end{aligned} \end{equation} Test \eqref{e1.4} by the function $\varphi=u\xi^q$. Using \eqref{e1.2} and \eqref{e1.3} we obtain \begin{equation} \label{e2.11} \int_{B_{(1-\sigma)\rho}(\bar x)}G(|\nabla u|)\xi^qdx \leq \gamma(1-\sigma)^{-\gamma}G\left(\frac{M(\rho)}{\rho}\right)\rho^n. \end{equation} Combining \eqref{e2.10} and \eqref{e2.11} we arrive at \begin{equation} \label{e2.12} \int_{B_{(1-\sigma)\rho}(\bar x)}F(u)dx \leq \gamma(1-\sigma)^{-\gamma}\frac{\delta(\rho)}{\rho} g\left(\frac{M(\rho)}{\rho}\right)\rho^n. \end{equation} Since $\bar x$ is an arbitrary point in $B_{\sigma\rho}(x_0)$, from \eqref{e2.8} and \eqref{e2.12} we obtain the required inequality \eqref{e1.5}. So, Theorem \ref{thm1} is proved. \subsection{Proof of Theorem \ref{thm2}} For $j=1, 2, \dots$, let us define the sequences $\{\sigma_j\}$, $\{\rho_j\}$, $\{M_j\}$ such that $$ \sigma_j=\frac{1-2^{-j-1}}{1-2^{-j-2}}, \quad \rho_j=\rho\big(1+\frac{1}{2}+\dots+\frac{1}{2^j}\big), \quad M_j=\sup_{B_{\rho_j(x_0)}} u. $$ Rewrite inequality \eqref{e1.5} for the pair of balls $B_{\rho_{j+1}}(x_0), B_{\rho_j}(x_0)$: $$ G^{-1}(F(M_j))\leq\gamma 2^{\gamma j}\rho^{-1}M_{j+1}. $$ If $\varepsilon>0$, we obtain \begin{align*} \psi(M_j)&\leq \psi(\varepsilon M_{j+1}) +\frac{1}{\varepsilon}\frac{\psi(M_j)M_j}{M_{j+1}} \\ &\leq\psi_a(\varepsilon M_{j+1})+\varepsilon^{-1}\gamma 2^{\gamma j}\rho^{-1}. \end{align*} Using condition (A) we arrive at following recursive inequalities $$ \psi(M_j)\leq\varepsilon^{\mu}\psi( M_{j+1})+\varepsilon^{-1}\gamma 2^{\gamma j}\rho^{-1}, $$ $j=0, 1, 2, \dots$, or $$ \psi(M_0)\leq\varepsilon^{j\mu}\psi( M_{j})+\varepsilon^{-1}\gamma \rho^{-1}\sum_{k=0}^{j-1}\varepsilon^{k\mu}2^{kj}. $$ We chose $\varepsilon^{\mu}=2^{-\gamma-1}$ so that the sum on the previous inequality can be majorized by convergent series. Let $j\to\infty$. Then $$ \psi(u(x_0))\leq\psi(M_0)\leq\gamma\rho^{-1}. $$ This proves the Theorem \ref{thm2}. \section{Harnack inequality. Proof of Theorem \ref{thm3}} Let $x_0$ be some inner point in $\Omega$ and $B_{8\rho}(x_0)\subset\Omega$. Fix $\bar x\in B_{\rho}(x_0), \sigma\in (0,1), 00\}$. \end{lemma} \begin{proof} Testing \eqref{e1.4} by $\varphi=(u-k)_+\xi^q$ and using \eqref{e1.2}, \eqref{e1.3}, \eqref{e2.2} we arrive at \eqref{e3.1}. To prove \eqref{e3.2} we test \eqref{e1.4} by the function $\varphi=(k-u)_+\xi^q$, using \eqref{e2.2} we obtain $$ \int_{B_r(\bar x)}G(|\nabla (k-u)_+|)\xi^qdx \leq\gamma G\Big(\frac{k}{(1-\sigma)r}\Big)|A_{k,r}^- |+\gamma\int_{B_r(\bar x)}f(u) (k-u)_+\xi^qdx. $$ The last term of the previous inequality can be estimated in the following way \begin{align*} \int_{B_r(\bar x)}f(u) (k-u)_+\xi^qdx &=\int_{B_r(\bar x)}f(u)\chi(u0, N>0$ and some $\alpha\in(0,1)$, $$ |\{x\in B_r(\bar x): u(x)\leq N\}|\leq (1-\alpha)|B_r(\bar x)|. $$ Then for any $\varepsilon\in (0, 1)$ there exists constant $\delta\in (0, 1/2)$ depending only on $n, p, q, \nu_1, \nu_2, \alpha$ and $\varepsilon$ such that $$ |\{x\in B_{4r}(\bar x): u(x)\leq 2\delta N\}|\leq \varepsilon|B_{4r}(\bar x)|, $$ and furthermore $u(x)\geq\delta N$ for almost all $x\in B_{2r}(\bar x)$. \end{lemma} The next lemma is a De Giorgi type lemma, the proof of which is similar to that of \cite[Lemma 6.4]{Lieberman2}. \begin{lemma} \label{lem3.3} Let the conditions of Theorem \ref{thm3} be fulfilled, $\bar x\in\Omega$, fix $r>0, \xi, a\in (0, 1)$. There exists number $\varepsilon_0\in (0, 1)$ depending only on $n, p, q, \nu_1, \nu_2, \xi$ and $a$ such that if $$ |\{x\in B_r(\bar x): u(x)\leq M(1-\xi)\}|\leq \varepsilon_0|B_r(\bar x)|, $$ with some $M\geq\underset{B_r(\bar x)} {\rm sup} u$, then $$ u(x)\leq M(1-a\xi)\, for\, a.a.\, x\in B_{2r}(\bar x). $$ \end{lemma} Because of Lemmas \ref{lem3.2} and \ref{lem3.3}, the rest of the arguments do not differ from the corresponding result in \cite{Giusti} and \cite{Lieberman2}. This completes the proof Theorem \ref{thm3}. \subsection*{Conclusion} In the paper there was studied quasilinear double-phase elliptic equations with absorption term $$ -\operatorname{div} \Big(g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\Big)+f(u)= 0, \quad u\geq 0. $$ Despite the lack of comparison principle, we proved a priori estimate of Keller-Osserman type. 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