\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 171, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2015/171\hfil Existence and asymptotic behavior of solutions] {Existence and asymptotic behavior of solutions to nonlinear radial $p$-Laplacian equations} \author[S. Masmoudi, S. Zermani \hfil EJDE-2015/171\hfilneg] {Syrine Masmoudi, Samia Zermani} \address{Syrine Masmoudi \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia} \email{syrine.sassi@fst.rnu.tn} \address{Samia Zermani \newline D\'epartement de Math\'ematiques, Facult\'e des Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia} \email{zermani.samia@yahoo.fr} \thanks{Submitted February 2, 2015. Published June 22, 2015.} \subjclass[2010]{34B18, 35J66} \keywords{$p$-Laplacian problem; positive solution; boundary behavior; \hfil\break\indent Schauder fixed point theorem} \begin{abstract} This article concerns the existence, uniqueness and boundary behavior of positive solutions to the nonlinear problem \begin{gather*} \frac{1}{A}(A\Phi _p(u'))'+a_1(x)u^{\alpha_1}+a_2(x)u^{\alpha_2}=0, \quad \text{in } (0,1), \\ \lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0, \end{gather*} where $p>1$, $\alpha _1,\alpha _2\in (1-p,p-1)$, $\Phi_p(t)=t|t|^{p-2}$, $t\in \mathbb{R}$, $A$ is a positive differentiable function and $a_1,a_2$ are two positive measurable functions in $(0,1)$ satisfying some assumptions related to Karamata regular variation theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In recent years, the existence of positive solutions for elliptic problems involving the $p$-Laplacian has found considerable interest and different approaches have been developed. This is due to their signification in various areas of pure and applied mathematics including geological sciences, fluid dynamics, electrostatics, cosmology (see \cite{AM,ADT,CN,FOP,KYY}), as well as in relation to inequalities of Poincar\'e, Writinger, Sobolev type and isoperimetric inequalities (see \cite{BFK,DM,DT,K,MGS}). Motivations for studying radial solutions can be found in \cite{BM,BD,G,K,RW} and references therein. M\^{a}agli et al \cite{MGS} considered the quasilinear elliptic problem \begin{equation} \begin{gathered} -\Delta _pu:=-\operatorname{div}( |\nabla u|^{p-2}\nabla u) =q( x) u^{\alpha } \quad \text{in }\Omega \\ u=0\quad \text{on }\partial \Omega , \end{gathered} \label{1.1} \end{equation} where $\Omega $ is a $C^{2}$ bounded domain of $\mathbb{R}^{n}$ $( n\geq 2) $, $p>1$, the exponent $\alpha \in (-1,p-1) $ and $q\in C( \Omega ) $ is a positive function having singular behavior near the boundary $\partial \Omega $. More precisely, let $d(x)$ be the Euclidean distance of $x\in \Omega $ to $\partial \Omega $ then $q( x) =d( x) ^{-\beta }L(d(x)) $, with $0<\beta
\operatorname{diam}( \Omega ) )$ by \[ \mathcal{K}:=\big\{ t\to L(t):=c\exp \Big(\int_{t}^{\eta }\frac{z(s) }{s}ds\Big): c>0 \text{ and }z\in C([0,\eta ]),\;\;z(0)=0\big\} . \] For the convenience of the readers, we briefly describe the result proved in \cite{MGS}. \begin{theorem}[\cite{MGS}] \label{thm1} Let $L\in \mathcal{K\cap C}^{2}( ( 0,\eta ]) $ such that $\int_{0}^{\eta }t^{\frac{1-\beta }{p-1}}L(t)^{\frac{1}{ p-1}}dt<\infty $. Then problem \eqref{1.1} has a unique positive and continuous solution $u$ satisfying, for $x\in (0,1)$, \begin{equation} u(x)\approx \begin{cases} \Big( \int_{0}^{d(x)}s^{-1}L(s)^{\frac{1}{p-1}}ds\Big) ^{\frac{p-1}{ p-1-\alpha }} & \text{if }\beta =p\text{,} \\[4pt] d(x)^{\frac{p-\beta}{p-1-\alpha }}L( d(x)) ^{\frac{1}{p-1-\alpha }} & \text{if }1+\alpha \leq \beta
0$ such that $\frac{1}{c}g(x)\leq f(x)\leq cg(x)$, for each $x\in
S $.
If $\Omega $ is the unit ball, similar result was shown in \cite{BM}
for radial solution of problem \eqref{1.1} which becomes in the radial form
\begin{equation}
\begin{gathered}
\frac{1}{A}(A\Phi _p(u'))'+q(x)u^{\alpha}=0,\quad \text{in } (0,1), \\
\lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0,
\end{gathered} \label{1.4}
\end{equation}
where $\Phi _p(t)=t|t|^{p-2}$, $t\in \mathbb{R}$ and $A(t)=t^{n-1}$.
Indeed, by using Karamata variation theory, the authors in \cite{BM}
established existence and asymptotic behavior of a unique positive
continuous solution to \eqref{1.4} for $\alpha 1)$ such that
\[
\int_{0}^{\eta }t^{\frac{1-\beta _i}{p-1}}L_i(t)^{\frac{1}{p-1}}dt<\infty .
\]
\end{itemize}
As it turns out, estimates \eqref{1.5} depend closely on
$\min {(\frac{ p-\beta }{p-1-\alpha },\frac{p-1-\mu }{p-1})}$.
Also, as it will be seen, the numbers
\begin{gather*}
\delta _1=\min{(\frac{p-\beta _1}{p-1-\alpha _1},\frac{p-1-\mu }{p-1})}, \\
\delta _2=\min{(\frac{p-\beta _2}{p-1-\alpha _2},\frac{p-1-\mu }{p-1})}
\end{gather*}
play a crucial role in the combined effects of singular and sublinear
nonlinearities in problem \eqref{1.7} and lead to a competition. However,
without loss of generality, we can suppose that
$\frac{p-\beta _1}{p-1-\alpha _1}\leq \frac{p-\beta _2}{p-1-\alpha _2}$
and we introduce the function $\theta $ defined on $(0,1)$ by
\begin{equation}
\theta ( x) :=\begin{cases}
(1-x)^{\delta _1}\Psi _{L_1,\beta _1,\alpha _1}(x) &
\text{if }\delta _1<\delta _2, \\
(1-x)^{\delta _1}( \Psi _{L_1,\beta _1,\alpha
_1}(x)+\Psi _{L_2,\beta _2,\alpha _2}(x)) & \text{if }\delta
_1=\delta _2,
\end{cases} \label{1.9}
\end{equation}
where for $i\in \{1,2\}$, $\Psi _{L_i,\beta _i,\alpha _i}$ is the
function defined by \eqref{1.6}.
Now, we are ready to state our main result.
\begin{theorem} \label{thm3}
Assume {\rm (H0)} and {\rm (H2)} hold and suppose that
$\alpha _1,\alpha _2\in (1-p,p-1)$. Then problem \eqref{1.7} has a unique
positive and continuous solution $u$ satisfying, for each $x\in (0,1)$,
\begin{equation}
u(x)\approx \theta (x), \label{1.10}
\end{equation}
where $\theta $ is the function defined by \eqref{1.9}.
\end{theorem}
The outline of this article is as follows.
In Section 2, we give some already known results on functions in
$\mathcal{K}$ useful for our study and
we give estimates of some potential functions.
In Section 3, we prove our main result.
\section{Preliminary results}
Our arguments combine a method of fixed point theorem with Karamata
regular variation theory. So, we are quoting some properties of functions in
$\mathcal{K}$ useful for our study.
\subsection{The Karamata class $\mathcal{K}$}
It is obvious to see that a function $L$ is in $\mathcal{K}$ if and only if
$L$ is a positive function in $C^{1}( (0,\eta ]) $ such that
\[
\lim_{t\to 0}\frac{tL'(t)}{L(t)}=0.
\]
A standard function belonging to the class $\mathcal{K}$ is given by
\[
L(t):=\prod_{k=1}^{p}(\log _{k}(\frac{\omega }{t}))^{\lambda _{k}},
\]
where $p\in {\mathbb{N}}^{\ast }$,
$(\lambda _1,\lambda _2,\dots ,\lambda_p)\in {\mathbb{R}}^{p}$,
$\omega $ is a positive real number
sufficiently large and $\log _{k}(x)=\log o\log o\dots .o\log (x)$ ($k$ times).
\begin{lemma}[\cite{S}] \label{k1} \quad
\begin{itemize}
\item[(i)] Let $L\in \mathcal{K}$ and $\varepsilon >0$. Then
$\lim_{t\to 0}t^{\varepsilon }L(t)=0$.
\item[(ii)] Let $L_1,L_2\in \mathcal{K},p\in \mathbb{R}$. Then
$L_1+L_2\in \mathcal{K},L_1L_2\in \mathcal{K}$ and $L_1^{p}\in
\mathcal{K}$.
\end{itemize}
\end{lemma}
\begin{lemma}[Karamata's Theorem]\label{kar}
Let $L\in \mathcal{K}$ be defined on $(0,\eta ]$ and $\sigma \in \mathbb{R}$.
\begin{itemize}
\item[(i)] If $\sigma >-1$, then $\int_{0}^{\eta }t^{\sigma }L(t)dt$ converges and
\[
\int_{0}^{t}s^{\sigma }L(s)ds\sim \frac{t^{1+\sigma }L(t)}{\sigma +1}
\quad\text{as }t\to 0^{+}.
\]
\item[(ii)] If $\sigma <-1$, then $\int_{0}^{\eta }t^{\sigma }L(t)dt$ diverges and
\[
\int_{t}^{\eta }s^{\sigma }L(s)ds\sim -\frac{t^{1+\sigma }L(t)}{\sigma +1}
\quad\text{as }t\to 0^{+}.
\]
\end{itemize}
\end{lemma}
\begin{lemma}[\cite{CMMZ}] \label{k2}
Let $L\in \mathcal{K}$ be defined on $(0,\eta ]$. Then
\[
t\to \int_{t}^{\eta }\frac{L(s)}{s}ds\in \mathcal{K}.
\]
If further $\int_{0}^{\eta }\frac{L(t)}{t}dt$ converges, then
\[
t\to \int_{0}^{t}\frac{L(s)}{s}ds\in \mathcal{K}.
\]
\end{lemma}
\begin{lemma}[\cite{CMMZ}] \label{estim2}
For $i\in \{1,2\}$, let $\eta _i<1$ and $L_i\in \mathcal{K}$.
For $t\in (0,\eta )$, put
\[
J(t)=\Big( \int_{t}^{\eta }\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\eta_1}}
+\Big( \int_{t}^{\eta }\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\eta _2}}.
\]
Then, for $t\in (0,\eta )$, we have
\[
\int_{t}^{\eta }\frac{(J^{\eta _1}L_1+J^{\eta _2}L_2)(s)}{s}
ds\approx J(t).
\]
\end{lemma}
\begin{lemma}[\cite{CMMZ}] \label{estim1}
For $i\in \{1,2\}$, let
$\eta _i<1$ and $L_i\in \mathcal{K}$ such that
$\int_{0}^{\eta }\frac{ L_i(s)}{s}ds<\infty $. For $t\in (0,\eta )$, put
\[
I(t)=\Big( \int_{0}^{t}\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{1-\eta _1}
}+\Big( \int_{0}^{t}\frac{L_2(s)}{s}ds\Big) ^{\frac{1}{1-\eta _2}}.
\]
Then, for $t\in (0,\eta )$, we have
\[
\int_{0}^{t}\frac{(I^{\eta _1}L_1+I^{\eta _2}L_2)(s)}{s}ds\approx
I(t).
\]
\end{lemma}
\subsection{Potential estimates}
For a nonnegative measurable function $f$ in $(0,1)$, let
\[
G_pf(x):=\int_{x}^{1}( \frac{1}{A(t)}
\int_{0}^{t}A(s)f(s)ds) ^{\frac{1}{p-1}}dt.
\]
We point out that if $f$ is a nonnegative measurable function such that the
mapping $x\to A(x)f(x)$ is integrable in $\left[ 0,1\right] $, then $
G_pf$ is the solution of the problem
\begin{equation}
\begin{gathered}
\frac{1}{A}(A\Phi _p(u'))'+f=0,\quad \text{in } (0,1), \\
\lim_{x\to 0}A\Phi _p(u')(x)=0,\quad u(1)=0.
\end{gathered} \label{potentiel}
\end{equation}
In what follows, we aim to prove Proposition \ref{compasymp}. To this end,
we need the following two lemmas which are proved in \cite{BM} and
\cite{CMMZ}.
\begin{lemma}[\cite{BM}] \label{estimarad}
Assume {\rm (H0)} and {\rm (H1)} hold. Then for $x\in (0,1)$, we have
\[
G_p(q)(x)\approx (1-x)^{\min {(\frac{p-\beta }{p-1},\frac{p-1-\mu }{p-1})}}
\begin{cases}
\int_{0}^{1-x}\frac{L(s)^{\frac{1}{p-1}}}{s}ds & \text{if }\beta =p, \\[4pt]
L(1-x)^{\frac{1}{p-1}} & \text{if }\mu +1<\beta 0$, $\eta _1<1$ and $\eta _2<1$, we have
\[
2^{-max(1-\eta _1,1-\eta _2)}(t+s)\leq t^{1-\eta _1}(t+s)^{\eta
_1}+s^{1-\eta _2}(t+s)^{\eta _2}\leq 2(t+s).
\]
\end{lemma}
\begin{proposition}\label{compasymp}
Assume {\rm (H0)} and {\rm (H2)} hold. Let $\theta $ be the function given
by \eqref{1.9}. Then for $x\in (0,1)$,
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
\theta (x).
\]
\end{proposition}
\begin{proof}
For $t\in(0,1)$, let
\begin{gather*}
K(t):=\Big(L_1^{\frac{1}{p-1-\alpha_1}}+L_2^{\frac{1}{p-1-\alpha_2}}\Big)(t),\\
N(t):=\Big(\int_{t}^{\eta}\frac{L_1(s)}{s}ds\Big)^{\frac{1 }{p-1-\alpha_1}}
+\Big(\int_{t}^{\eta}\frac{L_2(s)}{s}ds\Big)^{\frac{1}{p-1-\alpha_2}},\\
M(t):=\Big( \int_{0}^{t}\frac{(L_1(s))^{\frac{1}{p-1}}}{s}
ds\Big) ^{\frac{p-1}{p-1-\alpha _1}}+\Big( \int_{0}^{t}
\frac{(L_2(s))^{\frac{1}{p-1}}}{s}ds\Big) ^{\frac{p-1}{p-1-\alpha _2}}.
\end{gather*}
Since $\delta _1<\delta _2$ is equivalent to
$\frac{p-\beta _1}{p-1-\alpha _1}<\frac{p-\beta _2}{p-1-\alpha _2}$ and
$\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}<\beta _1 0, \label{st}
\end{equation}
applying Lemma \ref{estim} for
$t=L_1(x)^{\frac{1}{p-1-\alpha _1}}$,
$s=L_2(x)^{\frac{1}{p-1-\alpha _2}}$ and
$\eta _i=\frac{\alpha _i}{p-1}$, $(i\in \{1,2\})$, we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)
\approx (1-x)^{\frac{p-\beta _1}{p-1-\alpha _1}}K(1-x)
=\theta (x).
\]
\smallskip
\noindent\textbf{Case 4.} $\beta _1=\frac{(\mu
+1)(p-1-\alpha _1)+p\alpha _1}{p-1}$ and $\beta _2=\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.
Using (H2) we have
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-(1+\mu )}(L_1N^{\alpha _1}+L_2N^{\alpha _2})(1-x).
\]
Furthermore, the function $x\to (L_1N^{\alpha _1}+L_2N^{\alpha
_2})(1-x)\in \mathcal{K}$. So applying Lemma \ref{estimarad} for $\beta
=1+\mu $ and $L=L_1N^{\alpha _1}+L_2N^{\alpha _2}$, we obtain that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big( \int_{1-x}^{\eta }\frac{
(L_1N^{\alpha _1}+L_2N^{\alpha _2})(t)}{t}dt\Big) ^{\frac{1}{p-1}}.
\]
Since
\[
N^{p-1}(t)\approx \Big( \int_{t}^{\eta }\frac{L_1(s)}{s}
ds\Big) ^{\frac{p-1}{p-1-\alpha _1}}+\Big( \int_{t}^{\eta }
\frac{L_2(s)}{s}ds\Big) ^{\frac{p-1}{p-1-\alpha _2}},
\]
it follows that $N^{p-1}\approx J$, where $J$ is the function given in Lemma
\ref{estim2}, for $\eta _i=\frac{\alpha _i}{p-1},i\in \{1,2\}$. So, we have
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big( \int_{1-x}^{\eta }\frac{
(L_1J^{\eta _1}+L_2J^{\eta _2})(t)}{t}dt\Big) ^{\frac{1}{p-1}}.
\]
Hence, since $\eta _i<1$ for $i\in \{1,2\}$, by Lemma
\ref{estim2}, it follows that
\begin{align*}
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)
&\approx (1-x)^{\frac{p-1-\mu }{p-1}}J^{\frac{1}{p-1}}(1-x) \\
&\approx (1-x)^{\frac{p-1-\mu }{p-1}}N(1-x)=\theta (x).
\end{align*}
\smallskip
\noindent\textbf{Case\thinspace \thinspace 5.}
$\beta _1=\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}$ and
$\beta _2<\frac{(\mu +1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.
Let $b(1-x)=\Big( \int_{1-x}^{\eta }\frac{L_1(s)}{s}ds\Big) ^{\frac{1}{
p-1-\alpha _1}}$. Using (H2), we have
\begin{align*}
&a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\\
&\approx (1-x)^{-(1+\mu )}(L_1b^{\alpha _1})(1-x)+(1-x)^{-\beta _2+\alpha _2
\frac{p-1-\mu }{p-1}}(L_2b^{\alpha _2})(1-x).
\end{align*}
Now, since for $i\in \{1,2\}$, the function
$x\to (L_ib^{\alpha _i})(1-x)\in \mathcal{K}$ and
$-(1+\mu )<-\beta _2+\alpha _2\frac{p-1-\mu }{p-1}$, it follows from
Lemma \ref{k1}(i) that
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-(1+\mu )}(L_1b^{\alpha _1})(1-x).
\]
Applying again Lemma \ref{estimarad}, for $\beta =1+\mu $ and
$L=L_1b^{\alpha _1}$, we obtain
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}\Big(\int_{1-x}^{\eta }\frac{(L_1b^{\alpha
_1})(s)}{s}ds\Big) ^{\frac{1}{p-1-\alpha _1}}\approx \theta (x).
\]
\smallskip
\noindent\textbf{Case 6.}
$\beta _1<\frac{(\mu +1)(p-1-\alpha _1)+p\alpha _1}{p-1}$.
Using (H2), we have
\[
a_1(x)\theta ^{\alpha _1}(x)\approx (1-x)^{-\beta _1+\alpha _1\frac{
p-1-\mu }{p-1}}L_1(1-x).
\]
Applying Lemma \ref{estimarad} for $\beta =\beta _1-\alpha _1\frac{
p-1-\mu }{p-1}<1+\mu $ and $L=L_1$, we deduce that
\[
G_p(a_1\theta ^{\alpha _1})(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.
\]
Moreover, since $\frac{p-1-\mu }{p-1}<\frac{p-\beta _1}{p-1-\alpha _1}
\leq \frac{p-\beta _2}{p-1-\alpha _2}$, it follows that $\beta _2<\frac{(\mu
+1)(p-1-\alpha _2)+p\alpha _2}{p-1}$.
So, in the same manner we obtain
\[
G_p(a_2\theta ^{\alpha _2})(x)\approx (1-x)^{\frac{p-1-\mu }{p-1}}.
\]
Then, we conclude that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
(1-x)^{\frac{p-1-\mu }{p-1}}=\theta (x).
\]
\smallskip
\noindent\textbf{Case 7.} $\beta _1=\beta _2=p$.
Using (H2), we have
\[
a_1(x)\theta ^{\alpha _1}(x)+a_2(x)\theta ^{\alpha _2}(x)\approx
(1-x)^{-p}(L_1M^{\alpha _1}+L_2M^{\alpha _2})(1-x).
\]
Furthermore, the function $x\to (L_1M^{\alpha _1}+L_2M^{\alpha
_2})(1-x)\in \mathcal{K}$.
Now, since $\frac{\alpha _i}{p-1}<1$ and
$\int_{0}^{\eta }\frac{ (L_i(s))^{\frac{1}{p-1}}}{s}ds<\infty $, for
$i\in \{1,2\}$,
applying \eqref{st} and Lemma \ref{estim1} for
$\eta _i=\frac{\alpha _i}{p-1}$, we obtain
\begin{align*}
\int_{0}^{\eta }\frac{(L_1M^{\alpha _1}+L_2M^{\alpha _2})^{\frac{1}{
p-1}}(t)}{t}dt
&\approx \int_{0}^{\eta }\frac{(L_1^{\frac{1}{p-1}}M^{
\frac{\alpha _1}{p-1}}+L_2^{\frac{1}{p-1}}M^{\frac{\alpha _2}{p-1}})(t)
}{t}dt \\
&\approx M(\eta )<\infty .
\end{align*}
So applying Lemma \ref{estimarad} for $\beta =p$ and $L=L_1M^{\alpha
_1}+L_2M^{\alpha _2}$, by \eqref{st} we deduce that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
\int_{0}^{1-x}\frac{((L_1M^{\alpha _1})^{\frac{1}{p-1}
}+(L_2M^{\alpha _2})^{\frac{1}{p-1}})(t)}{t}dt.
\]
Then, using again Lemma \ref{estim1} we conclude that
\[
G_p(a_1\theta ^{\alpha _1}+a_2\theta ^{\alpha _2})(x)\approx
M(1-x)=\theta (x).
\]
\end{proof}
\begin{proposition}\label{equicont}
Assume {\rm (H0)--(H1)} hold. Then
the family of functions
\[
\mathcal{F}_{q}=\{x\to G_pf(x);f\in B((0,1)),|f|\leq q\}
\]
is uniformly bounded and equicontinuous in $[0,1]$. Consequently
$\mathcal{F} _{q}$ is relatively compact in $C([0,1])$.
\end{proposition}
\begin{proof}
Let $f$ be a measurable function such that $|f(x)|\leq q(x),x\in (0,1)$. By
Proposition \ref{estimarad}, we have
\[
|G_pf(x)|\leq G_pq(x)\approx (1-x)^{\min {(\frac{p-\beta }{p-1},
\frac{p-1-\mu }{p-1})}}\Psi _{L,\beta ,0}(x).
\]
From Lemma \ref{k1} and Lemma \ref{k2}, the function
$x\to (1-x)^{\min {(\frac{p-\beta }{p-1},\frac{p-1-\mu }{p-1})}}\Psi _{L,\beta
,0}(x)$ is continuous on $[0,1)$ and tends to zero as $x\to 1$.
Then, we prove that $\mathcal{F}_{q}$ is uniformly bounded and
$\lim_{x\to 1}G_p(f)(x)=0$, uniformly on $f$.
Moreover, let $x,x'\in \lbrack 0,1]$ such that $x