\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,\; 1
0$ 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), 0