\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 119, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/119\hfil Estimates and uniqueness]
{Estimates and uniqueness for boundary blow-up solutions of
p-Laplace equations}

\author[M. Marras, G. Porru\hfil EJDE-2011/119\hfilneg]
{Monica Marras, Giovanni Porru}  % in alphabetical order

\address{Monica Marras \newline
 Dipartimento di Matematica e Informatica,
 Universit\'a di Cagliari,
 Via Ospedale 72, 09124 Cagliari, Italy}
\email{mmarras@unica.it}

\address{Giovanni Porru \newline
 Dipartimento di Matematica e Informatica,
 Universit\'a di Cagliari,
 Via Ospedale 72, 09124 Cagliari, Italy}
\email{porru@unica.it}

\thanks{Submitted November 21, 2010. Published September 15, 2011.}
\subjclass[2000]{35B40, 35B44, 35J92}
\keywords{p-Laplace equations; large equations; uniqueness;
\hfill\break\indent second order boundary approximation}

\begin{abstract}
 We investigate boundary blow-up solutions of the  p-Laplace
 equation $\Delta_p u=f(u)$, $p>1$, in a bounded smooth
 domain $\Omega\subset R^N$. Under appropriate conditions on the
 growth of $f(t)$ as $t$ approaches infinity, we find an estimate
 of the solution $u(x)$ as $x$ approaches $\partial\Omega$, and
 a uniqueness result.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

Let $f(t)$ be a $C^1(0,\infty)$ function, positive, non decreasing,
satisfying $f(0)=0$ and the condition
\begin{equation}\label{e1}
\lim_{t\to\infty}\frac{t\bigl(f^\frac{1}{p-1}(t)
\bigr)'}{f^\frac{1}{p-1}(t)}=\alpha,
\end{equation}
with $p>1$ and $\alpha>1$. It is well known
(see \cite[page 282]{CR2}) that a smooth
function $f$ which satisfies \eqref{e1} has the following
representation
\begin{equation}\label{e2}
f^\frac{1}{p-1}(t)=Ct^\alpha\exp\Bigl(\int_{t_0}^t
\frac{g(\tau)}{\tau}d\tau\Bigr),
\end{equation}
where $C$ and $t_0$ are positive constants and $g(t)\to 0$ as
$t\to\infty$. Functions
which have this representation are said to be normalized regularly
varying at $\infty$. More precisely, $f^\frac{1}{p-1}(t)$ is
regularly varying of index $\alpha$, and $f(t)$ is regularly varying
of index $\alpha(p-1)$. Since
$$
\Bigl(\frac{f^\frac{1}{p-1}(t)}{t^\beta}\Bigr)'
=t^{-\beta-1}f^\frac{1}{p-1}(t)
\Bigl[\frac{t\bigl(f^\frac{1}{p-1}(t)
\bigr)'}{f^\frac{1}{p-1}(t)}-\beta\Bigr],
$$
if $f$ satisfies \eqref{e1} then the function $\frac{f^\frac{1}{p-1}(t)}{t^\beta}$ is
increasing for large $t$ whenever $\beta<\alpha$. In particular,
since $\alpha>1$, the function $\frac{f(t)}{t^{p-1}}$ is increasing
for large $t$. Furthermore, condition \eqref{e1} implies the
generalized Keller-Osserman condition
\begin{equation}\label{e3}
\int_1^\infty\frac{dt}{\bigl(F(t)\bigr)^{1/p}}<\infty,\quad
F(t)=\int_0^tf(\tau)d\tau.
\end{equation}

Consider the Dirichlet problem
\begin{equation}\label{e4}
 \Delta_p u=f(u)\quad \text{in }\Omega,\ \ \ \
u(x)\to\infty\quad\text{as }x\to\partial\Omega.
\end{equation}
It is well known that when $f$ satisfies condition \eqref{e3},
problem \eqref{e4} has a solution (see for example \cite{GP}). In the
present paper, assuming condition \eqref{e1}, we find a quite precise
estimate for a solution near the boundary $\partial\Omega$, and we
derive a result of uniqueness.

In case of $p=2$, problems about the existence of boundary blow-up
solutions have been investigated for a long time, see the classical
papers \cite{Ke,Os}, and the recent survey \cite{Ra}. We
refer to the paper \cite{Lo3} for a description of spatial
heterogeneity models, including historical hints. For the
investigation of the boundary behaviour of blow-up solutions we
refer to
\cite{AP,BM,BP,CR1,CR2,LM}. The case of weighted semilinear
equations has been discussed in \cite{Lo1,Lo2,Za}. The case $p>1$, has
been treated in \cite{GP,HT,Mo}. In the present
paper, assuming condition \eqref{e1}, we find an estimate of the
solution up to the second order.

In case of $p=2$, condition \eqref{e1} appears in the paper
\cite{Gm}, where the author proves a uniqueness result for problem
\eqref{e4}.  We emphasize that the method used in \cite{Gm} is not
applicable in the present case because of the nonlinearity of the
p-Laplacian.

For $s>0$, define the function $\phi(s)$ as
\begin{equation}\label{e5}
\int_{\phi(s)}^{\infty}\frac{dt}{(qF(t))^{1/p}}=s,
\end{equation}
 where $q=\frac{p}{p-1}$. If $u$ is a solution to problem \eqref{e4},
we prove the estimate
\begin{equation}\label{e6}
u(x)=\phi(\delta)[1+O(1)\delta],
\end{equation}
where $\delta=\delta(x)=operatorname{dist}(x,\partial\Omega)$
and $O(1)$ denotes
a bounded quantity. Estimate \eqref{e6} implies, in particular, that
if $u_1$ and $u_2$ are two solutions of problem \eqref{e4} then
$$
\lim_{x\to\partial\Omega}\frac{u_1(x)}{u_2(x)}=1.
$$
By using this result, the monotonicity of $f(t)$ for $t>0$ and the
monotonicity of $\frac{f(t)}{t^{p-1}}$ for large  $t$ we prove the
uniqueness of the solution to problem \eqref{e4}.


\section{Main results}

We have already noticed that if $f(t)$ satisfies \eqref{e1}
then the representation \eqref{e2} holds. By \eqref{e2} it follows
that, for $\epsilon>0$, we can find positive constants $C_1$ and
$C_2$ such that for $t$ large we have
\begin{equation}\label{e7}
C_1t^{\alpha(p-1)+1-\epsilon}<F(t)<C_2t^{\alpha(p-1)+1+\epsilon},
\end{equation}
where $F$ is defined as in \eqref{e3}. Furthermore, the function
$\phi$ defined in \eqref{e5}, for $s$ small satisfies
\begin{equation}\label{e8}
C_1\Bigl(\frac{1}{s}\Bigr)^\frac{p-\epsilon}{(p-1)(\alpha-1)}
 <\phi(s)<
C_2\Bigl(\frac{1}{s}\Bigr)^\frac{p+\epsilon}{(p-1)(\alpha-1)}.
\end{equation}

\begin{lemma} \label{lem2.1}
Let $A(\rho,R)\subset \mathbb{R}^N$, $N\ge 2$, be the annulus with
radii $\rho$ and $R$ centered at the origin. Let $f(t)>0$ be smooth,
increasing for $t>0$ and such that $\eqref{e1}$ holds with
$\alpha>1$. If $u(x)$ is a radial solution to problem \eqref{e4} in
$\Omega=A(\rho,R)$ and $v(r)=u(x)$ for $r=|x|$, then
\begin{equation}\label{e9}
v(r)<\phi(R-r)[1+C(R-r)], \quad \tilde r<r<R,
\end{equation}
and,
\begin{equation}\label{e10}
v(r)>\phi(r-\rho)[1- C(r-\rho)], \quad  \rho<r<\tilde
r,\end{equation}
where $\phi$ is defined as in \eqref{e5},
$\rho<\tilde r<R$ and $C$ is a suitable positive constant.
\end{lemma}

\begin{proof}
 We have
\begin{equation}\label{e11}
\bigl(|v'|^{p-2}v'\bigr)'+\frac{N-1}{r}|v'|^{p-2}v'=f(v),\ \
v(\rho)=v(R)=\infty.
\end{equation}
It is easy to show that there is
$r_0$ such that $v(r)$ is decreasing for $\rho<r<r_0$ and increasing
for $r_0<r<R,$ with $v'(r_0)=0$. For $r>r_0$ we have
$$
\bigl(|v'|^{p-2}v'\bigr)'=\bigl((v')^{p-1}\bigr)'=(p-1)(v')^{p-2}v''.
$$
Therefore, multiplying \eqref{e11} by $v'$ and integrating over
$(r_0,r)$ we find
\begin{equation}\label{e12}
\frac{(v')^p}{q}+(N-1)\int_{r_0}^r\frac{(v')^p}{s}ds=F(v)-F(v_0),\quad
v_0=v(r_0).
\end{equation}
 Since $F(v_0)>0$, \eqref{e12} implies that
\begin{equation}\label{e13}
v'<(qF(v))^{1/p},\quad r\in(r_0,R).
\end{equation}
As a consequence we have
\begin{equation}\label{e14}
\int_{r_0}^r\frac{(v')^p}{s}ds\le\frac{1}{r_0}
\int_{r_0}^r(qF(v))^{1/q}v'ds<
\frac{q^{1/q}}{r_0}\int_{0}^v(F(t))^{1/q}dt.
\end{equation}
On the other hand, by \eqref{e12} we find
$$
\frac{(v')^p}{qF(v)}=1-\frac{(N-1)\int_{r_0}^r
\frac{(v')^p}{s}ds+F(v_0)}{F(v)}.
$$
The above equation yields
\begin{equation}\label{e15}
\frac{v'}{(qF(v))^{1/p}}=1-\Gamma(r),
\end{equation}
where,
$$
\Gamma(r)=1-\Bigl(1-\frac{(N-1)\int_{r_0}^r\frac{(v')^p}{s}ds
+F(v_0)}{F(v)}\Bigr)^{1/p}.
$$
By using the inequality $1-(1-t)^{1/p}<t$ (true for $0<t<1$),
and \eqref{e14} we find, for some constant $M$,
$$
\Gamma(r)\le \frac{(N-1)\int_{r_0}^r\frac{(v')^p}{s}ds+F(v_0)}{F(v)}
\le M\frac{\int_{0}^v(F(t))^{1/q}dt}{F(v)}.
$$
Since
$$
\int_{0}^v(F(t))^{1/q}dt\le(F(v))^{1/q}v,
$$
we have
\begin{equation}\label{e16}
\Gamma(r)<\frac{Mv(r)}{(F(v(r)))^{1/p}}.
\end{equation}
By using \eqref{e7} (with $\epsilon$ small enough) one finds that
$\Gamma(r)\to 0$ as $r\to R$. Furthermore, using \eqref{e2} one
proves that
$$
\lim_{t\to\infty}\frac{F(t)}{tf(t)}=\frac{1}{\alpha(p-1)+1}.
$$
Hence, since
$$
\bigl(\frac{t}{(F(t))^{1/p}}\bigr)'
=\frac{tf(t)}{(F(t))^\frac{p+1}{p}}\bigl[\frac{F(t)}{tf(t)}-
\frac{1}{p}\bigr],
$$
and $\frac{1}{\alpha(p-1)+1}<\frac{1}{p}$, the
function $\frac{t}{(F(t))^{1/p}}$ is decreasing for large $t$.
As a consequence, the function $\frac{Mv(r)}{(F(v(r)))^{1/p}}$
tends to zero monotonically as $r$ tends to $R$.

The inverse function of $\phi$ is the following
$$
\psi(s)=\int_s^\infty\frac{1}{(qF(t))^{1/p}}dt.
$$
Integration of \eqref{e15} over $(r,R)$ yields
\begin{equation}\label{e17}
\psi(v)=R-r-\int_r^R\Gamma(s)ds,\end{equation} from which we find
\begin{equation}\label{e18}
v(r)=\phi(R-r)-\phi'(\omega)\int_r^R\Gamma(s)ds,
\end{equation}
with
$$
R-r>\omega>R-r-\int_r^R\Gamma(s)ds.
$$
Since
$$
-\phi'(\omega)=(qF(\phi(\omega))^{1/p},
$$
and since the function $t\to F(\phi(t))$ is decreasing we have
$$
-\phi'(\omega)<\Bigl(qF\bigl(\phi\bigl(R-r-\int_r^R\Gamma(s)ds\bigr)\bigr)\Bigr)^{1/p}
=(qF(v))^{1/p},
$$
where \eqref{e17} has been used in the last
step. Hence, by \eqref{e18} and \eqref{e16} we find
$$
v(r)<\phi(R-r)+(qF(v))^{1/p}\int_r^R\frac{Mv(s)}{(F(v(s)))^{1/p}}ds.
$$
Recalling that the function $\frac{Mv(r)}{(F(v(r)))^{1/p}}$ is
decreasing  for $r$ close to $R$, the latter estimate implies
$$
v(r)<\phi(R-r)+q^{1/p}Mv(r)(R-r),
$$
and
$$
v(r)<\frac{\phi(R-r)}{1-q^{1/p}M(R-r)},
$$
from which  inequality \eqref{e9} follows.

For $r<r_0$ we have $v'<0$ and, instead of equation \eqref{e12}, we
find
\begin{equation}\label{e19}
\frac{|v'|^p}{q}=F(v)-F(v_0)+(N-1)\int_r^{r_0}\frac{|v'|^p}{s}ds,
\end{equation}
with $\rho<r<r_0$. Note that, since $|v'(r)|^p\to\infty$ as
$r\to\rho$ and $v''>0$, we have (Lemma 2.1 of \cite{LM})
$$
\lim_{r\to\rho}\frac{\int_r^{r_0}\frac{|v'|^p}{t}dt}{|v'|^p}=0.
$$
Hence, \eqref{e19} implies $|v'|<q(F(v))^{1/p}$ for $r$ near to
$\rho$. Using equation \eqref{e19} again we find
$$
\frac{|v'|^p}{qF(v)}=1+\frac{(N-1)\int_r^{r_0}\frac{|v'|^p}{s}ds
-F(v_0)}{F(v)}.
$$
The above equation yields
\begin{equation}\label{e20}
\frac{-v'}{(qF(v))^{1/p}}=1+\tilde\Gamma(r),
\end{equation}
where
$$
\tilde\Gamma(r)=\Bigl(1+\frac{(N-1)\int_r^{r_0}\frac{|v'|^p}{s}ds
-F(v_0)}{F(v)}\Bigr)^{1/p}-1.
$$
Since $(1+t)^{1/p}-1<t$ (true for $t>0$), we have
$$
\tilde\Gamma(r)<\frac{(N-1)\int_r^{r_0}\frac{|v'|^p}{s}ds-F(v_0)}{F(v)}.
$$
Using the estimate $|v'|<q(F(v))^{1/p}$ we find
$|v'|^p<q^{p-1}(F(v))^\frac{p-1}{p}(-v')$. Therefore,
$\tilde\Gamma(r)$ satisfies
\begin{equation}\label{e21}
\tilde\Gamma(r)\le\frac{M v(r)}{(F(v(r)))^{1/p}},
\end{equation}
where $M$ is a suitable constant (possible different from that of
\eqref{e16}). It follows that $\tilde\Gamma(r)\to 0$ as $r\to\rho$.

Integration of \eqref{e20} over $(\rho,r)$ yields
$$
\psi(v)=r-\rho+\int_\rho^r\tilde\Gamma (s)ds,
$$
from which we find
\begin{equation}\label{e22}
v(r)=\phi(r-\rho)+\phi'(\omega_1)\int_\rho^r\tilde\Gamma(s)ds,\end{equation}
with
$$
r-\rho<\omega_1<r-\rho+\int_\rho^r\tilde\Gamma(s)ds.
$$
Since $\phi'(s)$ is increasing we have
$$
\phi'(\omega_1)> \phi'(r-\rho)=-\bigl(qF(\phi(r-\rho))\bigr)^{1/p}.
$$
This estimate, \eqref{e21} and \eqref{e22} imply
$$
v(r)>\phi(r-\rho)-\bigl(qF(\phi(r-\rho))\bigr)^{1/p}
\int_\rho^r\frac{Mv(s)}{
\bigl(F(v(s))\bigr)^{1/p}}ds.
$$
Since the function $\frac{t}{(F(t))^{1/p}}$ is decreasing for $t$
large and the function $v(r)$ is decreasing for $r$ close to $\rho$,
it follows that $\frac{v(r)}{(F(v(r)))^{1/p}}$ is increasing.
Therefore,
\begin{equation}\label{e23}
v(r)>\phi(r-\rho)-\bigl(qF(\phi(r-\rho))\bigr)^{1/p}\frac{Mv(r)}{
\bigl(F(v(r))\bigr)^{1/p}}(r-\rho).
\end{equation}
On the other hand, by \eqref{e20} we have
$$
\frac{-v'}{(qF(v))^{1/p}}<2,\quad \rho<r<\tilde r.
$$
Integrating over $(\rho,r)$ we find
$$
\psi(v)<2(r-\rho),
$$
whence,
\begin{equation}\label{e24}
v(r)>\phi(2(r-\rho)).
\end{equation}

We claim that, for some $M>1$ and $\delta$ small, we have
\begin{equation}\label{e25}
\frac{1}{M}\phi(\delta)\le \phi(2\delta).
\end{equation}
Indeed, putting $\phi(\delta)=t$, we can write \eqref{e25} as
$$
\frac{t}{M}\le \phi(2\psi(t)),
$$
or
$$
\psi(t)\le \frac{1}{2}\psi\bigl(\frac{t}{M}\bigr)
$$
for $t$ large. To prove this inequality, we write
$$
\psi(t)=\int_t^\infty(qF(\tau))^{-1/p}d\tau=M\int_\frac{t}{M}^\infty
(qF(M\tau))^{-1/p}d\tau.
$$
 Since $f(t)$ is regularly varying
with index $\alpha(p-1)$, $F(t)$ is regularly varying with index
$\alpha(p-1)+1$, and (see \cite{CR2})
$$
\lim_{t\to\infty}\frac{F(M t)}{F(t)}=M^{\alpha(p-1)+1}.
$$
Therefore, for $t$ large we have
$$
(F(M\tau))^{-1/p}\le \frac{(F(\tau))^{-1/p}}
{M^\frac{\alpha(p-1)+1}{p}-1}.
$$
Hence,
$$
\psi(t)\le \frac{M}{M^\frac{\alpha(p-1)+1}{p}-1}
\int_\frac{t}{M}^\infty (qF(\tau))^{-1/p}d\tau
= \frac{M}{M^\frac{\alpha(p-1)+1}{p}-1}\psi\bigl(\frac{t}{M}\bigr).
$$
The claim follows with $M$ such that
$$
\frac{M}{M^\frac{\alpha(p-1)+1}{p}-1}=\frac{1}{2}.
$$
Using \eqref{e24}, \eqref{e25}, and recalling that $F(t)$ is regularly
varying with index $\alpha(p-1)+1$ we find, for $r$ close to $\rho$,
$$
\frac{F(\phi(r-\rho))}{F(v(r))}
\le \frac{F(\phi(r-\rho))}{F(\phi(2(r-\rho)))}
\le \frac{F(\phi(r-\rho))}{F\Bigl(\frac{1}{M}\phi(r-\rho)\Bigr)}
<M^{\alpha(p-1)+1}+2.
$$
Insertion of the latter estimate into \eqref{e23} yields
$$
v(r)>\phi(r-\rho)-\tilde M v(r)(r-\rho),
$$
from which \eqref{e10} follows. The lemma is proved.
\end{proof}

\begin{theorem} \label{est}
Let $\Omega\subset \mathbb{R}^N$, $N\ge 2$, be a
bounded smooth domain and let $f(t)>0$ be smooth, increasing and
satisfying $\eqref{e1}$ with $\alpha>1$. If $u(x)$ is a solution to
problem \eqref{e4} then we have
\begin{equation}\label{e26}
\phi(\delta)\bigl[1-C\delta\bigr]<u(x)
<\phi(\delta)\bigl[1+C\delta\bigr],
\end{equation}
where $\phi$ is defined as in $\eqref{e5}$, $\delta$ denotes the
distance from $x$ to $\partial\Omega$ and $C$ is a suitable positive
constant.
\end{theorem}

\begin{proof}
 If $P\in\partial\Omega$ we consider a suitable
annulus of radii $\rho$ and $R$ contained in $\Omega$ and such that
its external boundary is tangent to $\partial\Omega$ in $P$. If
$v(x)$ is the solution of problem \eqref{e4} in this annulus, by
using the comparison principle for elliptic equations
\cite[Theorem 10.1]{GT}
we have $u(x)\le v(x)$ for $x$ belonging to the
annulus. Choose the origin in the center of the annulus and put
$v(x)=v(r)$ for $r=|x|$. By \eqref{e9}, for $r$ near to $R$ we have
$$
v(r)<\phi(\delta)\bigl[1+C\delta\bigr].
$$
The latter estimate together with the inequality $u(x)\le v(x)$
yield the right hand side of \eqref{e26}.

Consider a new annulus of radii $\rho$ and $R$ containing $\Omega$
and such that its internal boundary is tangent to $\partial\Omega$
in $P$. If $v(x)$ is the solution of problem \eqref{e4} in this
annulus, by using the comparison principle for elliptic equations we
have $u(x)\ge v(x)$ for $x$ belonging to $\Omega$. Choose the origin
in the center of the annulus and put again $v(x)=v(r)$ for $r=|x|$.
By \eqref{e10}, for $r$ near to $\rho$ we have
$$
v(r)>\phi(\delta)\bigl[1-C\delta\bigr].
$$
The latter estimate together with the inequality $u(x)\ge v(x)$
yield the left hand side of \eqref{e26}. The theorem is proved.
\end{proof}

\begin{theorem}\label{grad}
Let $\Omega\subset \mathbb{R}^N$, $N\ge 2$, be a
bounded smooth domain and let $f(t)>0$ be smooth, increasing and
satisfying \eqref{e1} with $\alpha>1$. If $u(x)$ is a solution to
problem \eqref{e4} then, $|\nabla u|\to\infty$ as
$x\to\partial\Omega$.
\end{theorem}

\begin{proof}
 By Theorem \ref{est} we have
$$
\lim_{x\to\partial\Omega} \frac{u(x)}{\phi(\delta(x))}=1.
$$
In particular, for $\delta<\delta_0$, $\delta_0$ small, we have
$$
\frac{1}{2}<\frac{u(x)}{\phi(\delta(x))}<2.
$$
Now we follow the argument described in \cite[page 105]{BE}, using
the same notation (with $\beta=\rho$ and $\rho<\rho_0$). For $\xi\in
\check D(\rho)$, define
$$
v(\xi)=\frac{u(\rho\xi)}{\phi(\rho)}.
$$
For $\xi\in \check D(\rho)$ we have
\begin{equation}\label{e27}
\frac{1}{2}\le v(\xi)\le 2.
\end{equation}
We find
$$
\nabla v=\frac{\rho}{\phi(\rho)}\nabla u(\rho\xi),
$$
and
$$
\Delta_p v=\frac{\rho^p}{(\phi(\rho))^{p-1}}\Delta_p
u(\rho\xi)=\frac{\rho^p}{(\phi(\rho))^{p-1}}f(u(\rho\xi))
=\frac{\rho^p}{(\phi(\rho))^{p-1}}f(v(\xi)\phi(\rho)).
$$
With $\psi(t)=\rho$ we have
\begin{equation}\label{e28}
\Delta_p v=\frac{(\psi(t))^p}{t^{p-1}}f(v(\xi)t)=
\Bigl(\frac{\psi(t)}{t^\frac{p-1}{p}(f(t))^{-1/p}}\Bigr)^p
\frac{f(v(\xi)t)}{f(t)}.
\end{equation}
Since $f(t)$ is regularly varying with index
$\alpha(p-1)$ we have
\begin{equation}\label{e29}
\lim_{t\to\infty}\frac{f(v(\xi)t)}{f(t)}=(v(\xi))^{\alpha(p-1)}.
\end{equation}
Furthermore, we have
$$
\frac{\psi(t)}{t^\frac{p-1}{p}(f(t))^{-1/p}}=\frac{\psi(t)}{t(F(t))^
{-\frac{1}{p}}}\Bigl(\frac{tf(t)}{F(t)}\Bigr)^{1/p}.
$$
We have already observed that \eqref{e2} implies
$$
\lim_{t\to\infty}\frac{tf(t)}{F(t)}=\alpha(p-1)+1.
$$
Using de l'Hospital rule
and the latter estimate we get
$$
\lim_{t\to\infty}\frac{\psi(t)}{t(F(t))^
{-\frac{1}{p}}}=\frac{q^{1/q}}{\alpha-1}.
$$
Hence,
\begin{equation}\label{e30}
\lim_{t\to\infty}\frac{\psi(t)}{t^\frac{p-1}{p}(f(t))^{-1/p}}=
\frac{q^{1/q}}{\alpha-1}\bigl(\alpha(p-1)+1\bigr)^{1/p}.
\end{equation}
By \eqref{e30}, \eqref{e29} and \eqref{e27}, \eqref{e28} implies that
\begin{equation}\label{e31}
C_1\le\Delta_p v\le C_2,\ \ \xi\in \check D(\rho)
\end{equation}
where $C_1$ and $C_2$ are suitable positive constants independent of
$\rho$.

Let $x_i\in\Omega$, $x_i\to\partial \Omega$, and let
$\rho_i=\operatorname{dist}(x_i,\partial\Omega)$.
 By \eqref{e31} with
$v_i(\xi)=\frac{u(\rho_i\xi)}{\phi(\rho_i)}$, and standard
regularity results (see \cite{To}), we find that the
$C^{1,\beta}(\check D(\rho_i))$ norm of the sequence $v_i(\xi)$ is
bounded far from zero. In particular,
$$
|\nabla v_i(\xi)|\ge c,
$$
with $c>0$ independent of $i$. Hence,
$$
|\nabla u(x_i)|=|\nabla v_i(\xi)|\frac{\phi(\rho_i)}{\rho_i}\ge
c\frac{\phi(\rho_i)}{\rho_i}.
$$
Since
$\frac{\phi(\rho_i)}{\rho_i}\to \infty$ as $i\to\infty$, the theorem
follows.
\end{proof}

Let us discuss now the uniqueness of problem \eqref{e4}. Observe
that if $\alpha>1+\frac{p}{p-1}$ then
$$
\lim_{\delta\to 0}\phi(\delta)\delta=\lim_{t\to \infty}t\psi(t)
=\lim_{t\to \infty}\frac{t^2}
{(qF(t))^{1/p}}=0,
$$
where \eqref{e7} with
$\epsilon<(\alpha-1)(p-1)-p$ has been used in the last step. Hence,
if $u(x)$ and $v(x)$ are solutions to problem $\eqref{e4}$ in case of
$\alpha>1+\frac{p}{p-1}$, by Theorem \ref{est} we have
$$
\lim_{x\to\partial\Omega}[u(x)-v(x)]=0.
$$
Since $f(t)$ is non decreasing, the comparison principle yields
$u(x)=v(x)$ in $\Omega$.

For general $\alpha>1$, we have the following result.

\begin{theorem}\label{euniq}
Let $\Omega\subset \mathbb{R}^N$, $N\ge 2$, be a
bounded smooth domain and let $f(t)>0$ be smooth, increasing and
satisfying $\eqref{e1}$ with $\alpha>1$. If $u(x)$ and $v(x)$ are
positive large solutions to problem $\eqref{e4}$ then $u(x)=v(x)$.
\end{theorem}

\begin{proof}
 Theorem \ref{est} implies
$$
\lim_{x\to\partial\Omega}\frac{u(x)}{v(x)}=1.
$$
Let $t_0$ large enough so that $\frac{f(t)}{t^{p-1}}$ is increasing
for $t>t_0$, and let $\eta>0$ such that $u(x)>t_0$ in
$\Omega_\eta=\{x\in\Omega:\delta(x)<\eta\}$. For $\epsilon>0$ define
$$
D_{\epsilon,\eta}=\{x\in\Omega_\eta:(1+\epsilon)u(x)<v(x)\}.
$$
If $D_{\epsilon,\eta}$ is empty for any $\epsilon>0$ then we have
$u(x)\ge v(x)$ in $\Omega_\eta$. Define
$\Omega^\eta=\{x\in\Omega:\delta(x)>\eta\}$. Using the equations for
$u$ and $v$ in $\Omega^\eta$ and the monotonicity of $f(t)$ one
proves that $u(x)\ge v(x)$ in $\Omega^\eta$. Hence, in this case,
$u(x)\ge v(x)$ in $\Omega$. Changing the roles of $u$ and $v$ we get
$u(x)=v(x)$.

Suppose $D_{\epsilon,\eta}$ is not empty for $\epsilon<\epsilon_0$.
In this open set, since $\frac{f(t)}{t^{p-1}}$ is increasing for
large $t$, we have
\begin{gather*}
\Delta_p\bigl((1+\epsilon)u\bigr)=(1+\epsilon)^{p-1}f(u)\le
f\bigl((1+\epsilon)u\bigr), \\
\Delta_p v=f(v).
\end{gather*}
By the comparison principle we have
$$
v(x)-(1+\epsilon)u(x)\le
\max_{\delta(x)=\eta}[v(x)-(1+\epsilon)u(x)]\quad \text{in }
D_{\epsilon,\eta}.
$$
Letting $\epsilon\to 0$ we find
$$
v(x)-u(x)\le \max_{\delta(x)=\eta}[v(x)-u(x)]\quad \text{in }
 \Omega_{\eta}.
$$
Put
$$
\max_{\delta(x)=\eta}[v(x)-u(x)]=v(x_1)-u(x_1)=C.
$$
Using the equations for $u$ and $v$ in $\Omega^\eta$ and the
monotonicity of $f(t)$ one proves that $v(x)-u(x)\le C$ in
$\Omega^\eta$. Then, $v(x)-u(x)\le C$ in $\Omega$. We observe that
decreasing $\eta$ and arguing as before we find
$x_\eta\to\partial\Omega$ such that
$$
v(x)-u(x)\le v(x_\eta)-u(x_\eta)\quad \text{in } \Omega,
$$
with $v(x_\eta)-u(x_\eta)= constant$. In other words, $v(x)-u(x)$ attains
its maximum value in the set described by $x_\eta$ (which approaches
$\partial\Omega$). By Theorem \ref{grad}, $\nabla u$ and $\nabla v$
do not vanish in $\Omega_\eta$ for $\eta$ small. Hence, the strong
comparison principle applies (see \cite{GT}) and we must have
$v(x)-u(x)= C$ in $\Omega_\eta$.

Since
$$
\Delta_pv=f(v)=f(u+C)
$$
and
$$
\Delta_pv=\Delta_pu=f(u),
$$
we must have $f(u)=f(u+C)$ in
$\Omega_\eta$. Since $f(t)$ is strictly increasing for $t$ large, we
find $C=0$. The theorem follows.
\end{proof}


\begin{thebibliography}{00}

\bibitem{AP}  C. Anedda and G. Porru;
Higher order boundary estimates for blow-up solutions of elliptic
equations, \emph{Differential and Integral Equations,}  19: 345--360
(2006).

\bibitem{BE} C. Bandle and M. Ess\'en;
  On the solutions of quasilinear elliptic
problems with boundary blow-up, \emph{Symposia Mathematica,} Volume
XXXV:  93--111 (1994).

\bibitem{BM} C. Bandle and M. Marcus;
  On second order effects in the boundary
behaviour of large solutions of semilinear elliptic problems,
\emph{Differential and Integral Equations,} 11:  23--34 (1998).

\bibitem{BP} S. Berhanu and G. Porru;
 Qualitative and quantitative estimates for large solutions to
semilinear equations, \emph{Communications in Applied Analysis,} 4:
121--131 (2000).

\bibitem{CR1} F. Cirstea and V. Radulescu;
  Uniqueness of the blow-up boundary
solution of logistic equations with absorbtion, \emph{C.R. Acad. Sc.
Paris, S\'er. I}, 335: 447--452 (2002).

\bibitem{CR2} F. Cirstea and V. Radulescu;
 Nonlinear problems with boundary
blow-up: a Karamata regular variation approach, \emph{Asymptotic
Analysis}, 46: 275--298 (2006).

\bibitem{Gm} J. Garc\'ia-Meli\'an;
Uniqueness of positive solutions for a boundary blow-up problem,
 \emph{J. Math. Anal. Appl.} 360: 530-–536 (2009).

\bibitem{GT} D. Gilbarg and N. S. Trudinger;
 Elliptic Partial Differential Equations of Second Order,
Springer Verlag, Berlin, 1977.

\bibitem{GP} F. Gladiali and G. Porru;
 Estimates for explosive solutions to p-Laplace equations,
Progress in partial differential equations, Vol. 1 (Pont-à-Mousson,
1997), Pitman Res. Notes Math. Ser.,  Longman, Harlow, 383: 117--127
(1998).

\bibitem{HT} Shuibo Huang and Qiaoyu Tian;
Asymptotic behaviour of large solutions to p-Laplacian of
Bieberbach-Rademacher type, \emph{Nonlinear Analysis}, 71:
5773--5780 (2009).

\bibitem{Ke} J. B. Keller, On solutions of $\Delta u=f(u)$,
\emph{Comm. Pure Appl. Math.}, 10 503--510 (1957).

\bibitem{LM} A. C. Lazer and P. J. McKenna;
Asymptotic behaviour of solutions of boundary blow-up problems,
 \emph{Differential and Integral Equations,}  7: 1001--1019 (1994).

\bibitem{Lo1} J. L\'opez-G\'omez;
The boundary blow-up rate of large solutions, \emph{J. Diff. Eqns.}
195: 25--45 (2003).

\bibitem{Lo3} J. L\'opez-G\'omez;
Metasolutions: Malthus versus Verhulst in population dynamics. A
dream of Volterra. Stationary partial differential equatins. Vol.
II, 211-309, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam,
2005.

\bibitem{Lo2} J. L\'opez-G\'omez;
Optimal uniqueness theorems and exact blow-up rates of large
solutions, \emph{J. Diff. Eqns.} 224: 385--439 (2006).

\bibitem{Mo} A. Mohammed;
Existence and asymptotic behavior of blow-up solutions to weighted
quasilinear equations,  \emph{J. Math. Anal. Appl.} 298: 621--637
(2004).

\bibitem{Os}  R. Osserman;
 On the inequality $\Delta u\geq f(u)$, \emph{Pacific J. Math.},
7: 1641--1647 (1957).

\bibitem{Ra}  V. Radulescu;
  Singular phenomena in nonlinear elliptic problems: from
boundary blow-up solutions to equations with singular
nonlinearities, in "Handbook of Differential Equations: Stationary
Partial Differential Equations", Vol. 4 (Michel Chipot, Editor),
483--591 (2007).

\bibitem{To} P. Tolksdorf;
 Regularity for a more general class of quasilinear elliptic equations.
\emph{J. Differential Equations,}  51: 126--150 (1984).

\bibitem{Za} Z. Zhang;
 The asymptotic behaviour of solutions with blow-up at the boundary for
semilinear elliptic problems, \emph{J. Math. Anal. Appl.} 308:
532--540  (2005).

\end{thebibliography}

\end{document}