\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 30, 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 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/30\hfil Exact number of solutions] {Exact number of solutions for a Neumann problem involving the $p$-Laplacian} \author[J. S\'anchez, V. Vergara\hfil EJDE-2014/30\hfilneg] {Justino S\'anchez, Vicente Vergara} % in alphabetical order \address{Justino S\'anchez \newline Departamento de Matem\'aticas, Universidad de La Serena, Avda. Cisternas 1200, La Serena, Chile} \email{jsanchez@userena.cl} \address{Vicente Vergara \newline Instituto de Alta Investigaci\'on, Universidad de Tarapac\'a, Antofagasta No. 1520, Arica, Chile} \email{vvergaraa@uta.cl} \thanks{Submitted September 26, 2013. Published January 27, 2014.} \thanks{V. Vergara partially was supported by Grant FONDECYT 1110033.} \subjclass[2000]{34B15, 35J60} \keywords{Neumann boundary value problem; $p$-Laplacian; \hfill\break\indent lower-upper solutions; exact multiplicity} \begin{abstract} We study the exact number of solutions of the quasilinear Neumann boundary-value problem \begin{gather*} (\varphi_p(u'(t)))'+g(u(t))=h(t)\quad\text{in } (a,b),\\ u'(a)=u'(b)=0, \end{gather*} where $\varphi_p(s)=|s|^{p-2}s$ denotes the one-dimensional $p$-Laplacian. Under appropriate hypotheses on $g$ and $h$, we obtain existence, multiplicity, exactness and non existence results. The existence of solutions is proved using the method of upper and lower solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The $p$-Laplacian operator appears in the study of non-Newtonian fluids in which the quantity $p$ is a characteristic of the medium. In particular, media with $1
1$, denotes the one-dimensional $p$-Laplacian and the functions $g$ and $h$ satisfy suitable conditions. Our goal is to obtain the exact number of solutions to problem \eqref{eN} which belongs to a certain set, depending only on the nonlinearity $g$. For this, we mainly apply the method of lower and upper solutions. This method allows us to establish the existence of at least one solution of the problem considered. We consider two cases, the first one when the lower and the upper solutions are well ordered; i.e., the lower solution is less than the upper one, and the second (less common) case when the lower and the upper solution are reversely ordered. In the last case, even when $p=2$, the existence of solutions to problem \eqref{eN} is not certain, in general. A typical example is given by the problem \[ u''+u=\cos t,\quad u'(0)=u'(\pi)=0, \] which has no solution even though $\alpha(t)\equiv 1$ and $\beta(t)\equiv -1$ are lower and upper solutions, respectively. In this direction, when $p=2,\, a=0$ and $b=1$, by using Sobolev and Wirtinger inequalities, lower and upper solutions method and fixed-point techniques for completely continuous and increasing operators in ordered Banach spaces, results of exact number of solutions and positive solutions for this problem are established in \cite{F1}. One of the main results reads as follows (see \cite[Theorem 4.1]{F1}): let $g\in C^1(\mathbb{R})$. If $g'(x)<\pi^2/4$, $x\in\mathbb{R}$, $g'$ is a strictly increasing function and $\lim_{x\to \pm\infty}g(x)=+\infty$, then there exists an $M\in\mathbb{R}$ such that \begin{itemize} \item [(a)] if $h(t)\leq M$, with strict inequality on a set of positive measure, then problem \eqref{eN} has no solution. \item [(b)] if $h(t)\equiv M$, problem \eqref{eN} has exactly one solution. \item [(c)] if $h(t)\geq M$, with strict inequality on a set of positive measure, then problem \eqref{eN} has exactly two solutions. \end{itemize} We point out that to obtain the exact number of solutions in part (c) it is necessary to show that the solutions of problem \eqref{eN} may be ordered. For this, the author of \cite{F1} studies a homogeneous problem associated with \eqref{eN}, where $g(u(t))$ is replaced by $q(t)u(t)$ with $q\in L^r(0,1)$ for some $r\in [1,+\infty)$; i.e., the problem \begin{equation} \label{eH} \begin{gathered} u''(t)+q(t)u(t)=0\quad\text{in }(0,1),\\ u'(0)=u'(1)=0. \end{gathered} \end{equation} In that case the linear nature of the underlying equation is used in an essential way. In this article, since we deal with the nonlinear case, to get accurate results with respect to the number of solutions (multiple solutions) we need to impose a restriction on the range of values of $p$ to the interval $(1,2]$ (such restriction is necessary in view of the counterexample given in Section 5 of \cite{CHP}) and apply \cite[Theorem 4.1]{CHP}. For this, the following property of the $p$-Laplacian ($1
0$
(depending on $p$ if $p\neq 2$) such that for all $u,v\in [k_1,k_2]$
\begin{equation}\label{P}
(\varphi_p(u)-\varphi_p(v))(u-v)\geq K(u-v)^2.
\end{equation}
Note that this property is not verified by the $p$-Laplacian, with $p>2$.
Although the condition \eqref{P} restricts the range of values of $p$,
this condition is optimal for generating the approximation of solutions
between lower and upper solutions in the reversed order by means of
monotone iterative techniques and anti-maximum principles.
We point out that this kind of nonlinear elliptic problems has been
the object of intensive research in recent years, mostly in the linear
case $p=2$, see for example \cite{JYCO',L1,SL,SCO',WZ,WCZ,WZY}.
Methods used in the cited literature include fixed point theorems in
cones and degree arguments. However, there have not been many results
in the nonlinear case $p\neq 2$. Further, all these results deal with
a single solution, or the least number of solutions. The reason for this
is that exact multiplicity results are usually difficult to establish.
As mentioned here we use mainly the very important technique of lower
and upper solutions. For a survey of this technique,
see \cite{CH1,CH}. We refer the reader to \cite{C} for a recent review
on the formidable literature about this method. Our results are inspired
by those of \cite{F1}, for the corresponding second-order Neumann boundary
value problem.
To state our main result we impose the following two hypotheses:
\begin{itemize}
\item[(H1)] $g$ belongs to $C^1(\mathbb{R})$ with
$g'(x)$ strictly increasing and $\lim_{x\to \pm\infty} g(x)=+\infty$.
\item[(H2)] The function $h$ belongs to $C([a,b])$.
\end{itemize}
By hypothesis (H1) there exists $\theta\in\mathbb{R}$ such that
\[
g(\theta) = \min_{x\in\mathbb{R}}g(x),\quad g'(\theta) = 0.
\]
Let $m = g(\theta)$. Then $g(x)\geq m$ for all $x\in\mathbb{R}$.
Since $\lim_{x\to\pm\infty}g(x)=+\infty$,
there exist constants $c_1$ and $c_2$ such that $c_1 <\theta 1$.
On the other hand, in (3) we seek solutions of problem \eqref{eN}
in the set
$\{u:c_1\leq u(t)\leq c_2,\text{ for all } t\in [a,b]\}$.
When $u\in [\theta, c_2]$, we have the a priori bound $k$ over the
derivatives of these solutions. So, if the lower and the upper solution
are reversely ordered, then the behavior of $\varphi_p$ outside of a
compact interval plays no role. This is precisely the meaning of \eqref{P}.
(ii) Note that when $p=2$ we can take $K_k=1$ (independently of $k$).
Also in this case $K_c=1$ (see Remark \ref{recover}). Thus, we recover
\cite[Theorem 4.1]{F1}.
\end{remark}
This article is organized as follows.
In Section 2, we establish some notation, as well
as some basic facts, and we prove the Lemmas \ref{order} and \ref{atleast}
that will be used in Section 3 to prove our main result, Theorem \ref{mainthm}.
Finally, in Section 4, we give an example to illustrate our results.
\section{Preliminaries}
We say that $u$ is a solution of \eqref{eN} if
$u\in C^1([a,b])$, $|u'|^{p-2}u'\in W^{1,1}((a,b))$, $u'(a)=u'(b)=0$, and
$(\varphi_p(u'(t)))'+g(u(t))=h(t)$
for almost all $t\in (a,b)$. Here $W^{1,1}((a,b))$ denotes the Banach
space of absolutely continuous functions on $(a,b)$. For later use,
it is convenient to define $f_+(t,u):=g(u)-h(t)$ and $f_-(t,u):=h(t)-g(u)$.
We use the following symbols. Let $I=[a,b]$ and $q\geq 1$.
For $u\in L^q(I)$, we write
\[
\|u\|_q=\Big(\int_{a}^{b}|u(s)|^{q}ds\Big)^{1/q}
\]
and for $u\in C(I)$,
\[
\|u\|_{\infty}=\sup_{t\in I}|u(t)|.
\]
Let us first recall the following classical integral inequality.
\begin{lemma}\label{ineqC1}
Let $u\in C^1(I)$. If $u(a)=0$ or $u(b)=0$, then
\[
\frac{\pi}{2(b-a)}\,\|u\|_{2}\leq \|u'\|_{2}\,.
\]
\end{lemma}
We need the following version of the Gronwall's lemma for showing
uniqueness of solutions of an initial value problem for the $p$-Laplacian.
\begin{lemma}[Gronwall's lemma]
Suppose that $a < b$, and let $z, v$ be nonnegative continuous
functions defined on $[a, b]$. Furthermore, suppose that $C$ is a nonnegative
constant. If
\[
v(t)\leq C+\int_{a}^{t}z(s)v(s)ds,\quad t\in [a,b],
\]
then
\[
v(t)\leq C\,e^{\int_{a}^{t}z(s)ds},\quad t\in [a,b].
\]
\end{lemma}
\begin{remark}\label{remagron} \rm
In particular, if $C=0$, we have $v\equiv 0$ on $[a,b]$.
\end{remark}
\begin{remark}\label{rema1} \rm
(i) Note that, for every $R>0$, we have
\[
|f_+(t,u)|\leq h_R(t)\text{ for all } t\in [a,b] \text{ and all } u \text{ with }
|u|\leq R,
\]
where $h_R(t):=\max_{|s|\leq R} |g(s)|+|h(t)|$.
(ii) Note that we have an a priori estimate over the derivatives
of the solutions of problem \eqref{eN}. Indeed, let $u(t)$ be a
solution of \eqref{eN}. Define $\overline{h}(t):=h(t)-m$, then
$\varphi_p(|u'(t)|)\leq \|\overline{h}\|_1\text{ for all } t\in [a,b]$.
Therefore, if $u(t)$ is a solution of \eqref{eN}, then
\begin{equation}\label{c}
\|u'\|_\infty\leq\|\overline{h}\|_1^{\frac{1}{p-1}}=:c\quad
(\text{only depending on $ g, h, p$}).
\end{equation}
\end{remark}
We shall say that $\alpha\in C^1([a,b])$ is a {\it lower solution} of
\eqref{eN} if $\varphi_{p}\circ\alpha'\in W^{1,1}((a,b))$ and
\[
-(\varphi_p(\alpha'(t)))'\leq f_+(t,\alpha(t)),\quad
\alpha'(a)\geq 0\geq \alpha'(b).
\]
An {\it upper solution} is defined by reversing inequalities in the previous
definition. Let $\alpha$ and $\beta\in C^1([a,b])$ be such that
$\beta(t)\leq \alpha(t)$ on $[a,b]$. We write
\[
[\beta,\alpha]:=\{v\in C^1([a,b]): \beta(t)\leq v(t)\leq \alpha(t)\text{ on }
[a,b]\}.
\]
By Remark \ref{rema1}, part (i), we can find a continuous function
$\widetilde{h}$ such that $|f_+(t, u)|\leq\widetilde{h}(t)$ for all
$t\in [a,b]$ and all $u\in [\beta(t), \alpha(t)]$.
We define
\[
k(\alpha, \beta):=\|\widetilde{h}\|_1^{\frac{1}{p-1}}.
\]
If further $\alpha$ and $\beta$ are lower and upper solutions,
it is easy to check that for all $t\in [a,b]$,
\[
\alpha'(t), \beta'(t)\in [-k(\alpha, \beta), k(\alpha, \beta)].
\]
Note that the constant $\theta$ is an upper solution while $c_1$ and $c_2$
are lower solutions of problem \eqref{eN}. Moreover, since $g$ is increasing
for $u\in[\theta,c_2]$, we have
\[
|f_+(t,u)|\leq |g(u)|+|h(t)|\leq |g(c_2)|+|h(t)|
=|\widetilde{m}|+|h(t)|=:\widetilde{h}(t).
\]
Using the previous notation we define
\begin{equation}\label{k}
k:=k(c_2,\theta)=\|\widetilde{h}\|_1^{\frac{1}{p-1}}
=\left(|\widetilde{m}|(b-a)+\|h\|_1\right)^{\frac{1}{p-1}}.
\end{equation}
The next result is key to study the exact number of solutions of
\eqref{eN}.
\begin{lemma}\label{order}
Let $1 0$, for all
$t\in [a,b]$. Therefore, there exist at most two solutions to \eqref{eN}
in the set $S$.
\end{proof}
\begin{remark}\label{exact} \rm
It follows from the above proof that, if there exist at least two solutions
of \eqref{eN} in $S$, then there exist exactly two solutions of
\eqref{eN} in $S$, one to the left of $\theta$ and the other to the right
of $\theta$.
\end{remark}
\section{Existence and exact number of solutions}
This section is devoted to prove our main result, Theorem \ref{mainthm}.
\begin{proof}[Proof of the Theorem \ref{mainthm}]
(1) If $h(t)\leq m$, with strict inequality on a subinterval of $[a,b]$, then
\[
\int_{a}^{b}h(t) dt c$, we have $K_{k}=(p-1)^{p-2}k^{p-2}