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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 25, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/25\hfil Positive solutions]
{Positive solutions of a nonlinear problem involving the
$p$-Laplacian with nonhomogeneous boundary conditions}

\author[A. Lakmeche, A. Lakmeche, M. Yebdri\hfil EJDE-2007/25\hfilneg]
{Ahmed Lakmeche, Abdelkader Lakmeche, Mustapha Yebdri}  % in alphabetical order

\address{Ahmed Lakmeche \newline
 D\'epartement de math\'ematiques,
 Facult\'e des sciences,
 Universit\'e Djillali Liabes,
 BP. 89, 22000 Sidi Bel Abbes, Algeria}
\email{lakahmed2000@yahoo.fr}

\address{Abdelkader Lakmeche \newline
$^{\dag}$ D\'epartement de math\'ematiques,
 Facult\'e des sciences,
 Universit\'e Djillali Liabes,
 BP. 89, 22000 Sidi Bel Abbes, Algeria}
\email{lakmeche@yahoo.fr}

\address{Mustapha Yebdri \newline
D\'epartement de math\'ematiques,
 Facult\'e des sciences, Universit\'e Aboubakr Belkaid,
BP. 119, 13000 Tlemcen, Algeria}
\email{yebdri@yahoo.com}

\thanks{Submitted July 13, 2006. Published February 12, 2007.}
\subjclass[2000]{35J65}
\keywords{$p$-Laplacian; positive solutions;  positone problem;
\hfill\break\indent  quadrature method, nonhomogeneous boundary conditions}

\begin{abstract}
 In this work we consider a boundary-value problem involving
 the p-Laplacian with nonhomogeneous boundary conditions.
 We prove the existence of multiple solutions using the
 quadrature method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

The $p$-Laplacian operator arises in the
modelling of physical and natural phenomena
\cite{d2,e1,g1,l4,l5,o1,o2}, and has has been considered
in many papers; see for example
\cite{a1,a3,a5,b1,d1,d2,d3,d4,d5,e1,g1,j1,l2,l5,r1}.
 In this work we consider the boundary-value problem
\begin{gather}
-(|u'(x)|^{p-2}u'(x))'=\lambda f(u(x)), \quad\text{a.e. } 0< x< 1, \label{e1}\\
u(0)=u(1)+k(u(1))u'(1)=0 \label{e2}
\end{gather}
where $\lambda \geq 0$, $p\in (1,2]$, $k:\mathbb{R}_+\to
\mathbb{R}_{+}^{*}$, and  $f:\mathbb{R}_+\to
\mathbb{R}_{+}^{*}$ smooth enough.

 Problem \eqref{e1}--\eqref{e2} was considered by Lakmeche and Hammoudi
\cite{l2} for $k$ constant, in Anuradha et al. \cite{a2}[2] and
Lakmeche \cite{l3}
for $p=2$ and $k$ constant. In this work we generalize \cite{l2} by
considering the nonhomogeneous boundary conditions.
 Our aim in this work is to prove existence of
solutions of \eqref{e1}, \eqref{e2} and their multiplicity, using the
quadrature method \cite{b3,c1,d3,l1}.
In section 2, we give some preliminaries and definitions,
in section 3 we give our main results, and we conclude by some
remarks in the last section.

\section{Preliminaries}

In this section we give some definitions and preliminaries.

\begin{definition}\label{def1} \rm
A pair $(u,\lambda)\in \mathrm{C^{1}}([0,1];\mathbb{R}_+)\times
[0,+\infty[$ is called a solution of \eqref{e1}-\eqref{e2}, if
\begin{itemize}
\item $(|u'|^{p-2}u')$ is absolutely continuous, and
\item $-(|u'|^{p-2}u')'=\lambda f(u)$ a.e. in $(0,1)$, and
$u(0)=u(1)+k(u(1))u'(1)=0$.
\end{itemize}
\end{definition}

Note that the pair $(0,0)$ is a solution of \eqref{e1}, \eqref{e2}.
 
 Let $F:\mathbb{R}_+\to \mathbb{R}_+$ be defined by
$F(u)=\int_{0}^{\rho}f(s)ds,$ and $g:\mathbb{R_{+}}\to
\mathbb{R_{+}}$ be defined by
$$
g(\rho)=\begin{cases}
2\big(\frac{p-1}{p}\big)^{1/p}
\int_{0}^{\rho} \frac{ds}{[F(\rho)-F(s)]^{1/p}}, &\text{for }\rho>0,\\
0 &\text{for }\rho=0.
\end{cases}
$$
 Let $n\geq 0$. Define $h_n:[n,+\infty)\to \mathbb{R}_{+}^{*}$, by
$$
h_n(\rho)=
\big(\frac{p-1}{p}\big)^{1/p} \Big[\int_{0}^{\rho}
\frac{ds}{[F(\rho)-F(s)]^{1/p}}+\int_{n}^{\rho}
\frac{ds}{[F(\rho)-F(s)]^{1/p}}\Big].
$$
Note that $g\equiv h_0$. % Remark 2

\begin{lemma} \label{lem1}
The functions $g$ and $h_n$ are continuous, and $g(\rho)\leq
2h_n(\rho)\leq 2g(\rho)$, for all $\rho \geq n\geq 0$.
\end{lemma}

The proof of the lemma above can be found in  \cite[Theorem 7]{b2}.

For $u\in \mathrm{C^{1}}([0,1];\mathbb{R}_+)$, we define
$\|u\|:=\sup \{ u(s); s\in (0,1)\}$.

\begin{lemma} \label{lem2}
If $(u,\lambda)$ is a solution of \eqref{e1}, \eqref{e2} with $\lambda>0$,
then
\begin{enumerate}
\item $u'(1)<0$, $u(1)>0$, and
\item $\lambda^{1/p}=h_n(\|u\|)$, where $n=u(1)$.
\end{enumerate}
\end{lemma}

\begin{proof}
 Let $(u,\lambda)$ be a positive
solution of \eqref{e1}, \eqref{e2} with $\lambda>0$, then $u\neq 0$. Using the
maximum principle \cite{v1}, we obtain $u>0$ in $(0,1)$, then
$u(1)\geq 0$, which implies
$$
u'(1)=-\frac{u(1)}{k(u(1))}\leq0.
$$
 Since $f(0)>0$, then $u'(1)<0$ and $u(1)>0$.
 Also there exists a unique $x_0\in (0,1)$ such
that $u'(x_0)=0$, $u(x_0)=\|u\|$, $u'(x)>0$ for $x\in(0,x_0)$, and
$u'(x)<0$ for $x\in (x_0,1)$.

 Let $(u,\lambda )$ be a solution of \eqref{e1},
\eqref{e2}, and $u(1)=n$ with $0<n<\rho$, then we have
$u(x_0)=\max_{x\in [0,1]}|u(x)|=\rho$.
 Multiplying \eqref{e1} by $u'(x)$, and integrate it
for $x\in [0,x_0]$ and $x_0$, we obtain
\begin{equation} \label{e3}
-\int_{x}^{x_0}(|u'(t)|^{p-2}u'(t))'u'(t)dt=\int_{x}^{x_0}\lambda
f(u(t))u'(t)dt.
\end{equation}
We have in one hand
\begin{equation} \label{e4}
\int_{x}^{x_0}\lambda f(u(t))u'(t)dt=\lambda \int_{u(x)}^{u(
x_0)}f(y)dy=\lambda (F(\rho)-F(u(x))),
\end{equation}
and in the other hand
\begin{equation} \label{e5}
-\int_{x}^{x_0}(|u'(t)|^{p-2}u'(t))'u'(t)dt
=\frac{(p-1)}{p}(u'(x))^{p}.
\end{equation}
 From \eqref{e3}, \eqref{e4} and \eqref{e5}, we have
\begin{equation} \label{e6}
\frac{(p-1)}{p}(u^{\prime }(x))^{p}=\lambda (F(\rho)-F(u(x))).
\end{equation}
Then for all $x\in ( 0,x_0) $, we have
\begin{equation} \label{e7}
\left(u^{\prime }(x)\right)^{p}=\big(\frac{p}{p-1}\big)\lambda
(F(\rho)-F(u(x))),
\end{equation}
which implies
\begin{equation} \label{e8}
u'(x)=\big(\frac{p}{p-1}\big)^{1/p}
[\lambda(F(\rho)-F(u(x)))]^{1/p} \quad \mbox{for}\ x\in
[0,x_0],
\end{equation}
and by symmetry
\begin{equation} \label{e9}
u'(x)=-\big(\frac{p}{p-1}\big)^{1/p}
[\lambda(F(\rho)-F(u(x)))]^{1/p} \quad  \mbox{for}\ x\in
[x_0,1].
\end{equation}
Integrate \eqref{e8} between $0$ and  $x_0$, we obtain
\begin{equation} \label{e10}
\lambda ^{1/p}x_0=\big(\frac{p-1}{p}\big)^{1/p}
\int_{0}^{\rho}\frac{ds }{[F(\rho)-F(s)]^{1/p}}.
\end{equation}
Similarly, by integration of \eqref{e9} between $1$ and $x_0$, we obtain
\begin{equation} \label{e11}
\lambda
^{1/p}(1-x_0)=\big(\frac{p-1}{p}\big)^{1/p}\int^{\rho}_{n}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}.
\end{equation}
 From \eqref{e10} and \eqref{e11}, we deduce that
\begin{equation} \label{e12}
\lambda
^{1/p}=\big(\frac{p-1}{p}\big)^{1/p}\Big[\int_{0}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}+\int_{n}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}\Big].
\end{equation}
 From equation this equation, we deduce the results of lemma \ref{lem2}.
\end{proof}

 Consider the  boundary-value problem consisting of
 \eqref{e1} and the Dirichlet boundary conditions
\begin{equation} \label{e13}
u(0)=u(1)=0\,.
\end{equation}


\begin{lemma}\cite{l2} \label{lem3}
 We have
\begin{enumerate}
\item If $\lim_{s\to +\infty}\frac{f(s)}{s^{p-1}}=0$, then
$\lim_{s\to +\infty} g(s)=+\infty$
\item If $\lim_{s\to +\infty}\frac{f(s)}{s^{p-1}}=+\infty$, then
$\lim_{s\to +\infty} g(s)=0$
\end{enumerate}
\end{lemma}

For the proof of the lemma above, see  \cite{l2}.

\begin{theorem}[\cite{l2}] \label{thm1}
 If $\lim_{s\to +\infty}f(s)/s^{p-1}=0$, then
problem \eqref{e1}, \eqref{e13} has
at least one positive solution for all $\lambda>0$.
\end{theorem}

\begin{proof} From lemma \ref{lem3} we have
$\lim_{s\to +\infty} g(s)=+\infty$ and $
g(0)=0$.
\end{proof}

\begin{theorem}[\cite{l2}] \label{thm2}
 If  $\lim_{s\to +\infty}f(s)/s^{p-1}=+\infty$, then there exist
$\lambda^{*}>0$ such that the problem \eqref{e1}, \eqref{e13}
has at least two positive solutions for $\lambda\in (0,\lambda^{*})$,
 and no positive solution for $\lambda>\lambda^{*}$.
\end{theorem}

\begin{proof} From lemma \ref{lem3}, we have $\lim_{s\to +\infty}g(s)=g(0)=0$.
Then $g$ is bounded and reaches its maximum at some point $\rho_0>0$.
Further $\lambda^{*}=(g(\rho_0))^{p}$.
\end{proof}

\section{Main results}

 Let $( u,\lambda ) $ be a solution of \eqref{e1},
\eqref{e2}, and $u(1)=n$ with $0<n<\rho$, then we have
$u(x_0)=\max_{x\in [0,1]}|u(x)|=\rho$.
Substituting $x$ by $1$ in \eqref{e9}, we obtain
\begin{equation} \label{e14}
\frac{n}{k(n)}=\big(\frac{p}{p-1}\big)^{1/p}
\left[\lambda (F(\rho)-F(s))\right]^{1/p}.
\end{equation}
Hence
\begin{equation} \label{e15}
\lambda^{1/p}=\big(\frac{p-1}{p}\big)^{1/p}\frac{n}{k(n)
\left[F(\rho)-F(n)\right]^{1/p}}
\end{equation}
Then from \eqref{e12} and \eqref{e15}, we have
\begin{equation} \label{e16}
\int_{0}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}+\int_{n}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}=\frac{n}{k(n)\left[
F(\rho)-F(n)\right]^{1/p}}.
\end{equation}

\begin{theorem} \label{thm3}
Assume that $k\in\mathrm{C(\mathbb{R}_+;\mathbb{R}_{+}^{*})}$.
Let $\rho>0$, then
\begin{enumerate}
\item there exist at least $n^* \in(0,\rho)$, such that \eqref{e16} is satisfied
for $n=n^{*}$;
\item for each $n^{*}$ satisfying \eqref{e16}, there is a unique
$\lambda=\lambda(\rho,n^{*})$ given by \eqref{e12} or \eqref{e15}
 such that \eqref{e1}, \eqref{e2} has exactly one solution $(u,\lambda)$,
with $\|u\|=\rho$,
$u(1)=n^{*}$, $u'(1)=-\frac{n^{*}}{h(n^{*})}$ and
$$
x_0=\big(\frac{p-1}{p}\big)^{1/p}\lambda^{-\frac{1}{p}}
\int_{0}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}};
$$
\item if $k$ is decreasing, $n^*$ is unique.
\end{enumerate}
\end{theorem}

\begin{proof}
Equation \eqref{e16} is equivalent to
\begin{equation} \label{e17}
k(n)=\Big(\int_{0}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}+\int_{n}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}\Big)^{-1}
\frac{n}{[F(\rho)-F(n)]^{1/p}}.
\end{equation}
Let $\gamma:[0,\rho)\to \mathbb{R}_+$ be defined by
\begin{equation} \label{e18}
\gamma(n):=\Big(\int_{0}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}+\int_{n}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}\Big)^{-1}
\frac{n}{[F(\rho)-F(n)]^{1/p}}.
\end{equation}
We have $\gamma(0)=0$,  $\lim_{n\to
\rho^{-}}\gamma(n)=+\infty$ and $\gamma$ is differentiable on
$(0,\rho)$, with
\begin{align*}
\gamma'(n)
&:=\frac{p[F(\rho)-F(n)]+nf(n)}{p[F(\rho)-F(n)]^{1+\frac{1}{p}}
\big(\int_{0}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}+
\int_{n}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}\big)}
\\
&\quad +\frac{n}{[F(\rho)-F(n)]^{1/p}
\big(\int_{0}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}
+\int_{n}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}\big)^{2}}>0.
\end{align*} % 19
Then $\gamma$ increases from $0$ to $+\infty$ on $(0,\rho)$. Since
$k:\mathbb{R}_+\to \mathbb{R_+^*}$ is continuous and
$k(0)>0$, $k(\rho)<\infty$, then there exist at least $n^{*}\in
(0,\rho)$ such that $k(n^{*})=\gamma(n^{*})$.


 If $k$ decreases, we have $0<k(n)\leq k(0)$ for all
$n\in[0,\rho]$, $\gamma(0)=0$ and $\lim_{s\to \rho^-}\gamma(s)=+\infty$.

 Let $n_0\in(0,\rho)$ such that $k(n_0)=\gamma(n_0)$.
 Suppose that there exists $n_1\in(0,\rho)$ such that
$k(n_1)=\gamma(n_1)$. Then we have
$k(n_0)=\gamma(n_0)<\gamma(n_1)=k(n_1)$ for $n_0 <n_1$ and
$k(n_0)\geq k(n_1)$, which is a contradiction. Similarly we find a
contradiction if $n_0>n_1$. Finally,  we deduce that $n_0=n_1$.
\end{proof}

\begin{corollary} \label{coro1}
Assume that $k\in \mathrm{C(\mathbb{R}_+;\mathbb{R_+^*})}$.
Let $\rho>0$, then the bifurcation diagram $(\lambda,\rho)$ of the
positive solutions of \eqref{e1}, \eqref{e2} is given by
$$
\lambda(\rho)^{1/p}=\big(\frac{p-1}{p}\big)^{1/p}
\Big[\int_{0}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}
+\int_{n^{*}}^{\rho}\frac{ds}{[F(\rho)-F(s)]^{1/p}}\Big],
$$
where $n^*$ is the solution of \eqref{e16}.
\end{corollary}


\begin{remark} \label{rmk3} \rm
If $k$ is not decreasing, then the solution $n^{*}$ of \eqref{e16} is not
necessarily unique, in some cases it could be infinite. This is one
of different results with respect to precedent works \cite{a2,l1,l2}.
\end{remark}

\begin{theorem} \label{thm4}
Assume that $k\in \mathrm{C^{1}(\mathbb{R}_+;\mathbb{R_+^*})}$.
 Let $\rho>0$, and $k$ decreasing, then there exists a unique
$n^{*}(\rho)\in(0,\rho)$ such that $k(n^{*})=\gamma(n^{*})$. Further
$n^*$ is continuously differentiable.
\end{theorem}

\begin{proof} Because $k$ is decreasing,  $k'\leq 0$;
further $\gamma'>0$, hence $k'-\gamma'<0$. From the implicit
function theorem, there exists a unique $n^{*}(\rho)\in(0,\rho)$
such that $k(n^{*})=\gamma(n^{*})$, and $n^*(\rho)$ is continuously
differentiable.
\end{proof}

\begin{theorem}  \label{thm5}
Assume that $k\in\mathrm{C(\mathbb{R}_+;\mathbb{R}_{+}^{*})}$.
 Let $\rho>0$ and $k$ decreasing. Then there exists a unique
$n^{*}(\rho)\in(0,\rho)$, such that \eqref{e16} is satisfied for
$n=n^{*}$. Also there exists a unique $\lambda=\lambda(\rho)$ given
by \eqref{e12} or \eqref{e15} for which \eqref{e1}, \eqref{e2}
has a unique solution
$(u,\lambda)$, with $\|u\|=\rho$, $u(1)=n^{*}(\rho)$,
$$
u'(1)=-\frac{n^{*}(\rho)}{h(n^{*}(\rho))}\quad{and}\quad
x_0=\big(\frac{p-1}{p}\big)^{1/p}\lambda^{-\frac{1}{p}}
\int_{0}^{\rho}\frac{ds }{[F(\rho)-F(s)]^{1/p}}.
$$
\end{theorem}

 The result is easily deduced from Theorems \ref{thm3} and \ref{thm4}.

\begin{corollary} \label{coro2}
Assume that $k\in\mathrm{C(\mathbb{R}_+;\mathbb{R_+^*})}$.
If $k$ is decreasing, then the bifurcation diagram
$(\lambda,\rho)$ of positive solutions of \eqref{e1}, \eqref{e2}
is given by
$$
\lambda(\rho)^{1/p}=\big(\frac{p-1}{p}\big)^{1/p}
\Big[\int_{0}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}+\int_{n^{*}(\rho)}^{\rho}\frac{ds
}{[F(\rho)-F(s)]^{1/p}}\Big],
$$
 where $n^*(\rho)$ is the unique solution of \eqref{e16}.
\end{corollary}

\begin{theorem} Assume that
$k\in\mathrm{C(\mathbb{R}_+;\mathbb{R_+^*})}$. If $k$ is
decreasing, then \begin{itemize}
\item[(1)] when
$\lim_{s\to +\infty} f(s)/s^{p-1}=0$, \eqref{e1}, \eqref{e2} has at
least one positive
solution for all $\lambda  >0$; and
\item[(2)] when  $\lim_{s\to +\infty}f(s)/ s^{p-1}=+\infty$,
 there exist
$$
\lambda _0^* = \left(\sup \{ h_{n^*(s)}( s ); s\in ( 0, +\infty)\}\right) ^p
$$
such that \eqref{e1}, \eqref{e2} has at least two
positive solutions for $\lambda\in ( 0, \lambda _0^*)$,
and zero positive solution for $\lambda > \lambda _0^*$.
\end{itemize}
\end{theorem}


\begin{proof} We have
$g(\rho )\le 2h_{n^{*}(\rho) }(\rho )\le 2g(\rho )$, for all
 $\rho > 0$. From theorems \ref{thm1} and \ref{thm2}, we deduce the results.
\end{proof}

\section*{Concluding remarks}

 In this work we have studied a boundary value problem of
the one-dimensional $p$-Laplacian with nonhomogeneous boundary
conditions. We have proved existence of positive solutions using
quadrature method, also we have proved the multiplicity of the
solutions for $\lim_{s\to +\infty}\frac{f(s)}
{s^{p-1}}=+\infty$. In the case where the nonhomogeneous term $k$ is
a decreasing function, we proved the uniqueness of the solution
$(u,\lambda)$ for each $\|u\|=\rho >0$. Our results generalize the
works  \cite{a2,l2}. When $k$ is not a decreasing function we can
find, some examples in which the solution $n^*$ of \eqref{e17} is not
unique, for example for $k$ given as it follows
$$
k(n)=\begin{cases}
\gamma (\rho_1),& \text{for } 0\leq n<\rho_1,\\
\gamma (n),&\text{for } \rho_1 \leq n<\rho_2,\\
\gamma (\rho_2),& \text{for } \rho_2 \leq n,
\end{cases}
$$
where $0<\rho_1<\rho_2<\rho$, we have an infinite number
of solutions of equation \eqref{e17}
 ($k(n)=\gamma (n)$) which
constitutes exactly the interval $[\rho_1 , \rho_2]$.

It will be interesting to analyze the ramification of
solutions for concrete and simple examples with boundary conditions
cited above.

 In this work we have considered un autonomous
problem, it will be interesting to consider the non-autonomous
problem using the fixed point method as the Avery and Peterson fixed
point theorem \cite{a4}.

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