\documentclass[reqno]{amsart}
\usepackage{graphicx}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 12, pp. 1--11.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/12\hfil Existence of positive solutions]
{Existence of positive solutions for semi-positone systems}
\author[ A. K. Dram\'e, D. G. Costa\hfil EJDE-2008/12\hfilneg]
{Abdou K. Dram\'e, David G. Costa} % in alphabetical order
\dedicatory{Dedicated to Professor Claude Lobry on the occasion of
his 60-th birthday}
\address{Abdou K. Dram\'e \newline
Department of Mathematical Sciences,
University of Nevada Las Vegas, Las Vegas, NV, USA}
\email{abdoukhadry.drame@unlv.edu}
\address{David G. Costa \newline
Department of Mathematical Sciences,
University of Nevada Las Vegas, Las Vegas, NV, USA}
\email{costa@unlv.nevada.edu}
\thanks{Submitted July 18, 2007. Published January 24, 2008.}
\subjclass[2000]{34B18, 47H10}
\keywords{Positive solution; semi-positone system;
phase plane analysis; \hfill\break\indent Schauder fixed point theorem}
\begin{abstract}
This paper studies semi-positone systems of equations by using phase
plane analysis and fixed point theory.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{pro}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction}
This paper concerns the existence of positive solutions for the
system
\begin{equation} \label{eP}
\begin{gathered}
-u''=\lambda f(u,v)\\
-v''=\mu\, g(u,v)\\
u(0)=u(1)=0=v(0)=v(1).
\end{gathered}
\end{equation}
where $\lambda, \mu>0$ are positive parameters and the nonlinear
functions $f,\; g: [0, \infty)\to \mathbb{R}$ are
continuous and satisfy the assumptions:
\begin{itemize}
\item[(A1)] For $v$ fixed,
\begin{gather*}
f(s,v)<0,\quad\text{for } 0< s< s_{0}\quad \text{or}\quad
s>s_{1}\\
f(s,v)>0,\quad \text{for } s_{0}< s< s_{1},\\
\int_{0}^{s_1}f(s,v)ds>0\,;
\end{gather*}
\item[(A2)] For $u$ fixed, $g$ satisfies hypotheses similar
to those of $f$ in (A1);
\item[(A3)]
\begin{itemize}
\item[(i)] $f$ (resp. $g$) is monotone increasing in $v$ (resp. in $u$)
or $f$ (resp. $g$) is monotone decreasing in $v$ (resp. in $u$)
or
\item[(ii)]$\int_{s_{0}}^{s_{1}}\min_{0\leq v\leq
s_{1}}f(u,v)du>-\int_{0}^{s_{0}}\min_{0\leq v\leq s_{1}}f(u,v)du$
(resp. $g$).
\end{itemize}
\end{itemize}
We point out that assumptions (A1) and
(A2) on the nonlinearities $f$ and $g$ are natural and
necessary for existence of positive solutions when $f(.,v)$ and
$g(u,.)$ are negative at zero, as shown by the results of
\cite{Dan87} and \cite{Cle87} in the case of a single equation.
Systems of type \eqref{eP} are referred to as semi-positone systems. Such
problems arise in many situations, for instance, in steady state
problems of population models with nonlinear sources. In section 4
we provide an application to a model leading to a system of type
\eqref{eP}. We refer the reader to \cite{Cas88}, where Castro and Shivaji
initially called such problems (with $f(0)<0$) \emph{non-positone
problems}, in contrast with the terminology \emph{positone problems},
coined by Cohen and Keller in \cite{Coh67}, when the nonlinearity
$f$ was \emph{positive} and \emph{monotone}. In fact, P. L. Lions
provided in \cite{Lio82} an extensive review on positive solutions
of scalar positone equations and already remarked that the case when
the nonlinearity is negative near the origin was mathematically
challenging and important for applications.
Semipositone problems with scalar equations have motivated many
researchers in the last fifteen years. We refer the reader to the
survey paper by Castro-Maya-Shivaji \cite{Cas002} and references
therein for a review. On the other hand, semi-positone systems have
been studied in \cite{Anu94}, \cite{Hai00} and \cite{Cas001}, where
the authors used the method of sub- and super-solutions. However, in
their results the nonlinear functions $f$ and $g$ are usually
monotonic in both variables $u$ and $v$. In \cite{Cas001}, the
construction of sub- and super-solutions was based on the structure
$f=f_{1}(u,v)-h_1(x)$ and $g=g_1(u,v) -h_2(x)$ of the nonlinear
functions $f$ and $g$, with the functions $f_1$ and $g_1$ having
nonnegative partial derivative in the variables $u$ and $v$. Also,
in \cite{Dan87} and \cite{Cle87}, semi-positone problems for a
scalar equation have been studied under general assumptions on the
nonlinearity $f$. As mentioned earlier, it was shown (among other
things) that if $f(0)<0$, then the condition
$\int_{0}^{s_{1}}f(s)ds>0$ is a necessary condition for existence of
positive solutions (sufficiency was also shown in \cite{Cle87}).
More recently, the authors in \cite{Cos06} studied semi-positone
problems in the ODE case under very weak conditions on $f$ by using
phase plane analysis for conservative systems.
In the present paper, we prove existence of positive solutions for
semi-positone systems under general conditions on the
nonlinearities. We prove that the condition given in \cite{Dan87}
and \cite{Cle87} is a sufficient condition in the case of systems,
and we extend one of the results of \cite{Cos06} to systems of
equations. Our approach combines the theory of conservative systems
and tools of functional analysis, like fixed point theory.
For the rest of this article, we let $C_{0}[0, 1]$ denote the space
of continuous
functions $v$ on $[0,1]$ satisfying the boundary conditions
$v(0)=v(1)=0$, and endowed with the sup norm. Also, we let
$\mathcal{K}$ denote the closed, convex cone of nonnegative
functions in $C_{0}[0, 1]$, namely, $\mathcal{K}=\{v\in
C_{0}[0, 1: v\geq 0\}$. In addition, for a given $v$ in
$\mathcal{K}$, we define the continuous function
$$
f_{v}(s,u)=f(u,v(s)),\quad \text{for } 0\leq s\leq 1,\;u\geq 0,
$$
and consider the auxiliary problem
\begin{equation} \label{ePv}
\begin{gathered}
-u''=\lambda f_{v}(s,u)\\
u(0)=u(1)=0
\end{gathered}
\end{equation}
\section{Existence result for \eqref{ePv}}
The proof of existence of positive solutions for \eqref{ePv} is based
on the method of sub and super-solutions. From the function $f_{v}$
we construct the functions $F_{v}$ and $G_{v}$ below, where we shall
omit the subscript $v$ in the latter functions for simplicity of
notation:
$$
F(u)=\min_{0\leq s\leq 1}f_{v}(s,u) \quad \text{and}\quad
G(u)=\max_{0\leq s\leq 1}f_{v}(s,u).
$$
Since $f$ is continuous in $(u,v)$ and $v$ is in $\mathcal{K}$, the
functions $F$ and $G$ are well-defined, continuous, and satisfy
the following properties:
\begin{itemize}
\item[(i)]
\begin{gather*}
F(s), G(s)<0,\quad \text{for } 0\leq s< s_{0}, \text{ or } s>s_{1} \\
F(s), G(s)>0,\quad \text{for } s_{0}~~0,\quad \int_{0}^{s_{1}}G(s)ds>0
\end{gather*}
\item[(ii)] $F(u)\leq f_{v}(s,u)\leq G(u)$, for
$0\leq s\leq 1$, and $u\geq 0$.
\end{itemize}
Note that solutions of the problems
\begin{equation} \label{ePF}
\begin{gathered}
-u''=\lambda F(u)\\
u(0)=u(1)=0
\end{gathered}
\end{equation}
and
\begin{equation} \label{ePG}
\begin{gathered}
-u''=\lambda G(u)\\
u(0)=u(1)=0
\end{gathered}
\end{equation}
are, respectively, sub-solutions and super-solutions of \eqref{ePv}.
\subsection*{Existence of a sub-solution for \eqref{ePv}}
We are interested in the existence of a positive solution for
problem \eqref{ePF}. We have the following result.
\begin{lemma}\label{lem1}
Assume that $f$ satisfies assumptions {\rm (A1)} and {\rm (A3)}.
Then, there exists $\Lambda_{1}>0$ such that the problem \eqref{ePF}
has a positive solution if and only if $\lambda\geq\Lambda_{1}$.
\end{lemma}
\begin{proof}
We prove this lemma by using phase-plane analysis on
the conservative system associated to \eqref{ePF}. Let us introduce
the parameter $L=\sqrt{\lambda}$, and the new function $w(Lt)=u(t)$.
Then problem \eqref{ePF} becomes
\begin{gather*}
-w''=F(w)\\
w(0)=w(L)=0\,.
\end{gather*}
Next, by defining $u_{1}=w$ and $u_{2}=u_{1}'$, the above problem
is equivalent to the system
\begin{equation} \label{eP1}
\begin{gathered}
u_{1}'= u_{2}\\
u_{2}'= -F(u_{1})\\
u_{1}(0)= u_{1}(L)=0\,.
\end{gathered}
\end{equation}
Therefore, we study the solutions of the conservative system
\eqref{eP1}. It is known that the energy is constant along the
trajectories of \eqref{eP1} in the phase plane $(u_{1}, u_{2})$;
i.~e., we have
\begin{equation}\label{energy}
E=\frac{u_{2}^{2}}{2}+U(u_{1})=\mathrm{constant}\,,
\end{equation}
where $U$ is the potential energy function given by
$$
U(x)=\int_{0}^{x}F(\xi)d\xi\,.
$$
We represent in Figure 1 and 2, respectively, the potential energy
function $U$ and the phase-portrait of the system \eqref{eP1}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} % epotentialU.eps
\caption{Potential energy function $U$ }
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} % {sp-portrait.eps
\caption{Phase-portrait of system \eqref{eP1}}
\end{center}
\end{figure}
From equation \eqref{energy}, we define the time map
$T:\,[0, k_{0})\to (0, \infty)$,
$$
T(k) = \int_{0}^{u(k)} \frac{d\xi}{\sqrt{2(E(k)-U(\xi))}}
$$
for initial conditions $u_{1}(0)=0$ and $u_{2}(0)=k$, where $u(k)$
is the distance reached by $u_{1}(t)$, from the origin, the first
time when $u_{2}(t)=0$ in the $u_{1}u_{2}$-plane. Therefore,
problem \eqref{eP1} has a positive solution if and only if there is
$k\geq 0$ such that $L=2T(k)$.
Let $a$ be the first zero of $U$ in $(0, s_{1})$ (see figure 1),
$M=U(s_{1})>0$, $m=U(s_{0})$, and $k_{0}=\sqrt{2M}$. Then we have
$a\leq u(k)$ for all $0\leq k0$ such that \eqref{ePG} has a
positive solution if and only if $\lambda\geq\Lambda_{2}$.
\end{lemma}
\begin{proof} The proof is similar to that of Lemma 2.1. We recall
the problem \eqref{ePG} is
\begin{gather*} % \label{ePG}
-u''=\lambda G(u)\\
u(0)=u(1)=0
\end{gather*}
Similarly, the potential energy function $V$ and the time map $S$
are defined by
$$
V(x)=\int_{0}^{x}G(\xi)d\xi,\quad
S(k)=\int_{0}^{u(k)}\frac{d\xi}{\sqrt{2(E(k)-V(\xi))}}
$$
and the associated conservative system is
\begin{equation} \label{eP2}
\begin{gathered}
x_{1}'=x_{2}\\
x_{2}'=-G(x_{1})\\
x_{1}(0)=x_{2}(L)=0\,.
\end{gathered}
\end{equation}
The time map $S(k)$ has the same shape of the time map $T(k)$ in the
proof of Lemma \ref{lem1}. Also, by using arguments similar to those
in Lemma \ref{lem1}, we conclude that \eqref{ePG} has a positive
solution if and only if $\lambda\geq \Lambda_{2}$, where
$\Lambda_{2}=4(\min_{0\leq k< k_{1}}S(k))^{2}$ and $k_{1}=\sqrt{2
V(s_{1})}$.
\end{proof}
\subsection*{Existence of a positive solution for \eqref{ePv}}
Before giving the main result in this section, we prove, in the next
lemma, existence of a particular ordered pair $(w_{1}, w_{2})$
formed by a positive solution of \eqref{ePF} and a positive solution
of \eqref{ePG}.
\begin{lemma}\label{lem3}
Assume that $f$ satisfies assumptions {\rm (A1)} and {\rm (A3)}.
Then, there exists $\lambda_{0}>0$ such that for any $\lambda\geq
\lambda_{0}$, there exists a positive solution $w_{1}$ of \eqref{ePF}
and a positive solution $w_{2}$ of \eqref{ePG} with
$$
w_{1}\leq w_{2}\quad \text{everywhere in } [0, 1].
$$
\end{lemma}
\begin{proof}
Let $\lambda_{0}=\max(\Lambda_{1}, \;\Lambda_{2})$,
where $\Lambda_{1}$ and $\Lambda_{2}$ are given in Lemmas \ref{lem1}
and \ref{lem2}, and let us consider the potential energy functions $U$
and $V$ corresponding for the systems \eqref{eP1} and \eqref{eP2}.
These can be represented together in the same figure as follows (see
figure 4).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig4} % UandV.eps
\caption{Potential energy functions for \eqref{eP1} and \eqref{eP2}}
\end{center} %\label{qpetit}
\end{figure}
Let $a_{F}$ and $a_{G}$ be, respectively, the first zero of $U$ and
$V$ in $(0, s_{1})$ (cf.~figure 4). Then
$$
T(0)=\int_{0}^{a_{F}}\frac{d\xi}{\sqrt{-2U(\xi)}},\quad
S(0)=\int_{0}^{a_{G}}\frac{d\xi}{\sqrt{-2V(\xi)}},
\quad \text{ and } \quad a_{G}\leq a_{F}0$, we define $\bar
k=\sqrt{2V(a_{F})}>0$, and denote by $(u_{1}, u_{2})$ the solution
of \eqref{eP1} corresponding to $T(0)$ and by $(x_{1},x_{2})$ the
solution of \eqref{eP2} corresponding to $S(\bar k)$. Clearly,
$T(0)=S(\bar k)$ and $u_{1}(T(0))=a_{F}=x_{1}(S(\bar k))$. The
solutions $(u_{1}, u_{2})$ and $(x_{1}, x_{2})$ are represented
together in figure 5.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig5} % sol1and2.eps
\caption{Special solutions of \eqref{eP1} and \eqref{eP2}}
\end{center}%\label{qpetit}
\end{figure}
Now, we claim that $u_{1}(t)\leq x_{1}(t)$ for all $0\leq t\leq L$.
The proof of this claim will be done by integrating backward the
systems \eqref{eP1} and \eqref{eP2}. Indeed, we first introduce the
functions $\varphi_{1}(t)=u_{1}(\frac{L}{2}-t)$ and
$\varphi_{2}(t)=u_{2}(\frac{L}{2}-t)$, so that we have
\begin{equation}\label{e1}
\begin{gathered}
\varphi_{1}'=-\varphi_{2}\\
\varphi_{2}'=F(\varphi_{1})\\
\varphi_{1}(0)=a_{F},\quad \varphi_{2}(0)=0\,.
\end{gathered}
\end{equation}
Similarly, by introducing $\psi_{1}(t)=x_{1}(\frac{L}{2}-t)$ and
$\psi_{2}(t)=x_{2}(\frac{L}{2}-t)$, we have
\begin{equation}\label{e2}
\begin{gathered}
\psi_{1}'=-\psi_{2}\\
\psi_{2}'= F(\psi_{1})\\
\psi_{1}(0)= a_{F},\quad \psi_{2}(0)=0\,.
\end{gathered}
\end{equation}
Now, assume that there exists $t_{0}\in (0, \frac{L}{2})$ such that
$\varphi_{1}(t_{0})= \psi_{1}(t_{0})$ and $\varphi_{2}(t_{0})=
\psi_{2}(t_{0})$ (equivalently, $u_{1}(\frac{L}{2}-t_{0})=
x_{1}(\frac{L}{2}-t_{0})$ and $u_{2}(\frac{L}{2}-t_{0})=
x_{2}(\frac{L}{2}-t_{0})$, or in other words the solution's curves
in the phase plane touch each other). Then, we have
$$
\varphi_{2}'(t_{0})=F(\varphi_{1}(t_{0}))\leq
G(\varphi_{1}(t_{0}))=G(\varphi_{1}(t_{0}))=\psi_{2}'(t_{0}).
$$
Therefore, as $t$ increases from $t_{0}$, $\varphi_{1}'(t)$ will
remain greater than $\psi_{1}'(t)$, that is
$u_{1}'(\frac{L}{2}-t)$ remains less than
$x_{1}'(\frac{L}{2}-t)$ for all $t$ in $[t_{0},
t_{0}+\varepsilon]$ and for small $\varepsilon>0$. Hence,
$u_{1}(\frac{L}{2}-t)\leq x_{1}(\frac{L}{2}-t)$, for all $t\in
[t_{0}, t_{0}+\varepsilon]$. And repeating this reasoning from
$t_{0}+\varepsilon$, we finally get $u_{1}(t)\leq x_{1}(t)$ for all
$0\leq t\leq \frac{L}{2}$. By symmetry of the periodic solutions
$(u_{1}, u_{2})$ and $(x_{1}, x_{2})$ about the horizontal axis in
the phase plane, we have $u_{1}(t)\leq x_{1}(t)$ for all
$\frac{L}{2}\leq t\leq L$. To complete the proof, we let
$w_{1}(t)=u_{1}(L t)$ and $w_{2}(t)=x_{1}(L t)$.
\end{proof}
Based on the above lemmas, we state and prove the main
result of this section.
\begin{theorem}\label{th1}
Assume that $f$ satisfies assumptions {\rm (A1)} and {\rm (A3)}.
Then, for any $\lambda\geq \lambda_{0}$, problem \eqref{ePv} has a
positive solution $u$ satisfying
$$
0\leq w_{1}\leq u\leq w_{2}\leq a_{F}
$$
where $\lambda_{0}$, $w_{1}$ and $w_{2}$ are given in Lemma $2.3$.
Moreover, $u$ is uniquely defined by the conditions
$u(\frac{1}{2})=a_{F}$ and $u'(\frac{1}{2})=0$. We denote the
mapping $v\mapsto u$ by $\mathcal{F}_{1}$.
\end{theorem}
\begin{proof}
The existence of $u$ is guaranteed by the existence
of the particular sub- and super-solutions $w_{1}$ and $w_{2}$ given
in Lemma \ref{lem3}. Therefore, we have to prove that $u$ is uniquely
defined by the conditions stated above.
Indeed, since $w_{1}\leq u\leq w_{2}$ and $w_{1}(\frac{1}{2})=
w_{2}(\frac{1}{2})=a_{F}$, it follows that $u$ satisfies the Cauchy
Problem
\begin{equation}\label{e3}
\begin{gathered}
-u''=F(u)\\
u(\frac{1}{2})=a_{F},\quad u'(\frac{1}{2})=0
\end{gathered}
\end{equation}
Therefore, by integrating equation (\ref{e3}) forward and backward,
we see that $u$ is uniquely defined on both intervals
$[\frac{1}{2}, 1]$ and
$[0, \frac{1}{2}]$. This complete the proof.
\end{proof}
\section{Existence of a positive solution for \eqref{eP}}
Let us consider the solution $u$ of \eqref{ePv} given in
Theorem \ref{th1}. Observing that $u$ belongs to $\mathcal{K}$, we
consider (by symmetry) the problem
\begin{equation} \label{ePu}
\begin{gathered}
-v''=\mu g_{u}(s,v)\\
v(0)=v(1)=0
\end{gathered}
\end{equation}
where $g_{u}(s,v)=g(u(s),v)$ for all $0\leq s\leq 1$ and $v\geq 0$.
Then, similarly to problem \eqref{ePv}, we obtain the following result
on existence of a positive solution for \eqref{ePu}.
\begin{corollary}\label{co1}
Assume that $g$ satisfies assumptions {\rm (A2)--(A3)}. Then, there
exists $\mu_{0}>0$ such that, for any $\mu\geq \mu_{0}$, problem
\eqref{ePu} has a positive solution $v$ uniquely defined by
$v(\frac{1}{2})=a_{G}$ and $v'(\frac{1}{2})=0$ and such that
$$
0\leq v\leq s_{1}.
$$
We denote the mapping $u\mapsto v$ by $\mathcal{F}_{2}$.
\end{corollary}
We shall use Schauder's Fixed Point Theorem \cite[p
126]{Mar76} to prove existence of a positive solution for \eqref{eP}. For
that, we define the mapping
$$
\mathcal{F}=\mathcal{F}_{2}\circ\mathcal{F}_{1}: \mathcal{K}
\to \mathcal{K}
$$
where $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are given by Theorem
\ref{th1} and Corollary \ref{co1}, respectively. Then, we have the
following result.
\begin{theorem}\label{th2}
Assume assumptions {\rm (A1)--(A3)}. Then, there exist $\lambda_{0}>0,
\,\mu_{0}>0$ such that for any pair $(\lambda,\mu)$ with
$\lambda\geq \lambda_{0}$, $\mu\geq \mu_{0}$, problem \eqref{eP} has a
positive solution $(u,v)$. Furthermore, $v$ is a fixed point of the
mapping $\mathcal{F}$.
\end{theorem}
\begin{proof}
Let $\mathcal{K}_{0}$ be the bounded, closed, convex subset of $C_0
[0,1]$ consisting of those functions $v\in\mathcal{K}$ satisfying
$0\leq v\leq s_1$; i.~e.,
$$
\mathcal{K}_{0} = \{ v\in \mathcal{K}: 0\leq v\leq s_1\}\,.
$$
On the other hand, by Theorem \ref{th1} and Corollary \ref{co1}, we
note that $\mathcal{F} : \mathcal{K} \to \mathcal{K}$
is well-defined and continuous (by continuous dependence of
solutions of differential equations on parameters) and, furthermore,
$\mathcal{F}$ maps $\mathcal{K}_{0}$ into $\mathcal{K}_{0}$. Also,
by standard regularity, we have that $\mathcal{F}(v)\in C^2
[0, 1]$, so that $\mathcal{F}$ is completely continuous. Therefore,
by Schauder's Fixed Point Theorem, $\mathcal{F}$ has a fixed point
$v\in\mathcal{K}_{0}$. And, if we denote by $u$ the solution of
problem \eqref{ePv} given in Theorem \ref{th1} with $v$ being the
fixed point of $\mathcal{F}$, then $u\in\mathcal{K}_{0}$ and the
pair $(u,v)$ is a positive solution of \eqref{eP}.
\end{proof}
\section{An example}
We shall consider the following reaction-diffusion system that
describes the dynamics of two competing species in an environment
$\Omega$ that we assume to be one-dimensional. Let $u(t,x)$ and
$v(t,x)$ represent the concentrations or population densities of the
two species. With the standard non-dimensionalization process, we
may assume the diffusion coefficient equal to $1$, and $\Omega=
(0, 1 )$. The boundary of the environment is considered to be
hostile to the species (so, one has zero Dirichlet boundary
conditions) and we assume that the reproduction or growth of the
species follows the law $\rho_{1}(u,v)=\lambda\frac{mu}
{K+u+\frac{u^2}{\alpha}}(v+1)$ for the species $u$ and
$\rho_{2}(u,v)=\mu\frac{mv}
{K+v+\frac{v^2}{\alpha}}(u+1)$ for the species $v$. Then, the
dynamics of the two species is described by
\begin{equation}\label{expl1}
\begin{gathered}
\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial
x^2}+\lambda(\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha)(v+1)\\
\frac{\partial v}{\partial t}=\frac{\partial^2 v}{\partial
x^2}+\mu(\frac{mv}{K+v+\frac{v^2}{\alpha}}-\alpha)(u+1)\\
u(0) = u(1) = 0,\quad v(0) = v(1) = 0\,.
\end{gathered}
\end{equation}
In the above system the the constants $K$, $\alpha$, $\lambda$ and
$\mu$ are positive and the terms $-\lambda(v+1)$ and $-\mu(u+1)$
represent a mortality or harvesting of the species occurring in the
interior of the environment. The steady state problem associated
with \eqref{expl1} is given by
\begin{equation}\label{expl}
\begin{gathered}
-u''=\lambda f(u,v)\\
-v''=\mu g(u,v)\\
u(0)=u(1)=0=v(0)=v(1)\,,
\end{gathered}
\end{equation}
where
\begin{gather*}
f(u,v)=\Big(\frac{mu}{K+u+\frac{u^2}{\alpha}} -\alpha\Big)(v+1)\,,\\
g(u,v)=\Big(\frac{mv}{K+v+\frac{v^2}{\alpha}} -\alpha\Big)(u+1)
\end{gather*}
In what follows, we show that our assumptions (A1)--(A3) are
satisfied for this system. We only check the assumptions for the
function $f$ since the situation for $g$ is similar.
\noindent (A1) is satisfied:
Let $\alpha_{0}:=m+2K-2\sqrt{K(m+K)}$. Then,
for any $0<\alpha<\alpha_{0}$, we have $(m-\alpha)^{2}>4\alpha K$
and the function $f$ satisfies $f(s_{0},v)=f(s_{1},v)=0$ for any
$v\geq 0$, where
$$
s_{0}=\frac{m-\alpha-\sqrt{(m-\alpha)^{2}-4\alpha K}}{2},\quad
s_{1}=\frac{m-\alpha+\sqrt{(m-\alpha)^{2}-4\alpha K}}{2}.
$$
In addition, $f(s,v)<0$ for $0~~~~s_{1}$, and $f(s,v)>0$
for $s_{0}~~~~0$ such that $\varphi(\alpha)>0$ for all $\alpha\leq
\alpha_{1}$, hence $\int_{0}^{s_{1}}f(u,v)du>0$ for
all $0<\alpha\leq \alpha_{1}$. This shows that assumption
(A1) is satisfied. Similarly, we see that assumption
(A2) is satisfied.
\noindent (A3) part (ii) is satisfied:
Let us consider the integral
\begin{align*}
I&=\int_{0}^{s_{0}}\min_{0\leq v\leq
s_{1}}f(u,v)du+\int_{s_{0}}^{s_{1}}\min_{0\leq v\leq
s_{1}}f(u,v)du \\
&=\int_{0}^{s_{0}}\min_{0\leq v\leq
s_{1}}\Big(\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\Big)(v+1)du\\
&\quad + \int_{s_{0}}^{s_{1}}\min_{0\leq v\leq s_{1}}\Big(
\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\Big)(v+1)du\,.
\end{align*}
Since
$$
\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\leq 0 \quad
\text{for all } u \in [0, s_{0}]
$$
and
$$
\frac{mu}{K+u+\frac{u^2}{\alpha}}-\alpha\geq 0 \quad
\text{for all } u\in [s_{0}, s_{1}]\,,
$$
we have
\begin{align*}
I&=\int_{0}^{s_{0}}\Big(\frac{mu}
{K+u+\frac{u^2}{\alpha}}-\alpha\Big)(s_{1}+1)du+
\int_{s_{0}}^{s_{1}}\Big(\frac{mu}
{K+u+\frac{u^2}{\alpha}}-\alpha\Big)du\\
&= s_{1}\int_{0}^{s_{0}}\Big(\frac{mu}
{K+u+\frac{u^2}{\alpha}}-\alpha\Big)du
+\int_{0}^{s_{1}}\Big(\frac{mu}{K+u+\frac{u^2}
{\alpha}}-\alpha\Big)du\,.
\end{align*}
Therefore, as in the case of $\varphi (\alpha)$ above, there
exists $\alpha_{2}>0$ such that $I>0$ for $0<\alpha<\alpha_{2}$.
\subsection*{Conclusion} For any $\alpha$ satisfying $0<\alpha\leq
\min(\alpha_{0}, \alpha_{1}, \alpha_{2})$, the functions $f$ and
$g$ satisfy the assumptions (A1)--(A3).
\begin{remark} \label{rmk1} \rm
The existence result given in this paper is valid for a system of
$n$ equations (with $n>2$), the proof being similar to the one
presented here. Moreover, this idea can be extended to handle
systems of semi-positone PDEs.
\end{remark}
\begin{thebibliography}{99}
\bibitem{Anu94} V. Anuradha, A. Castro and R. Shivaji; Existence
results for semipositone systems, \emph{Dynamics Systems and
Applications} 5 (1996), 219-227.
\bibitem{Cas04} A. Castro, C. Maya and R. Shivaji; Stability analysis
of positive solutions to classes of reaction-diffusion systems, \emph{
Differential Integral Equations} 17 (2004), 391-406.
\bibitem{Cas001} A. Castro, C. Maya and R. Shivaji; An existence result
for a class of sublinear semipositone systems, \emph{Dynamics of
Continuous, Discrete and Impulsive Systems} 7 (2000) 533-540.
\bibitem{Cas002} A. Castro, C. Maya and R. Shivaji; Nonlinear
eigenvalue problems with semipositone structure, \emph{In
Proc. Nonlinear Differential Equations: A Conference Celebrating
the 60th Birthday of Alan C. Lazer.\ Electronic J. Diff. Eqns.
Conference} 05 (2000), 33-49.
\bibitem{Cas88} A. Castro and R. Shivaji; Nonnegative
solutions for a class of nonpositone problems, \emph{
Proc. Roy. Soc. Edin.} 108(A) (1988), 291-302.
\bibitem{Cle87} P. Clement and G. Sweers; Existence and multiplicity
results for a semilinear elliptic eigenvalue problem, \emph{
Ann. Scuola Norm. Sup. Pisa} 4(14) (1987), 97-121.
\bibitem{Coh67} D. S. Cohen and H. B. Keller; Some positone problems
suggested by nonlinear heat generation, \emph{J. Math. Mech.} 16
(1967), 1361-1376.
\bibitem{Cos06} D. G. Costa and H. Tehrani; Simple existence proofs
for one-dimensional semi-positone problems, \emph{Proc. Roy. Soc. Edin.}
136(A) (2006), 473-483.
\bibitem{Dan87} E. N. Dancer and K. Schmitt; On positive solutions of
semilinear elliptic equations, \emph{Proceedings of the AMS} 101 (1987),
445-452.
\bibitem{Dan84} E. N. Dancer; On positive solutions of some pairs of
differential equations, \emph{Transactions of the AMS} 284 (1984),
729-743.
\bibitem{Hai00} D. D. Hai and R. Shivaji; Positive solutions for
semipositone systems in an annulus, \emph{Rocky Mountain J. of
Math.} 29 (1999), 1285-1300.
\bibitem{Lio82} P. L. Lions; On the existence of positive solutions of
semilinear elliptic equations, \emph{Siam Review}, Vol. 24, No. 4,
October 1982.
\bibitem{Mar76} Robert H. Martin Jr.; Nonlinear operators and
differential equations in Banach spaces, \emph{John Wiley and Sons,}
1976.
\bibitem{Oru02} S. Oruganti, J. Shi and R. Shivaji; Diffusive logistic
equation with constant yield harvesting, I: Steady states, \emph{
Transactions of the AMS} 354 (2002), 3601-3619.
\end{thebibliography}
\end{document}
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