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\markboth{ Nonlinear eigenvalue problems }
{ A. Castro, M. Chhetri, \& R. Shivaji}
\begin{document}
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
Nonlinear Differential Equations, \newline
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 33--49\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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Nonlinear eigenvalue problems with semipositone structure
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\thanks{ {\em Mathematics Subject Classifications:} 35J45, 35J55, 35J60, 34B18.
\hfil\break\indent
{\em Key words:} elliptic systems, semipositone problems, positive solutions.
\hfil\break\indent
\copyright 2000 Southwest Texas State University.
\hfil\break\indent Published October 24, 2000. } }
\date{}
\author{ Alfonso Castro, c. Maya , \& R. Shivaji \\[12pt]
{\em Dedicated to Alan Lazer} \\ {\em on his 60th birthday }}
\maketitle
\begin{abstract}
In this paper we summarize the developments of semipositone problems to date,
including very recent results on semipositone systems. We also discuss
applications and open problems.
\end{abstract}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\section{Introduction}
Many problems in physics, chemistry, engineering, and biology lead
to the study of reaction diffusion processes. A simple example of
a diffusion process is the heat conduction in a solid.
Let $u(x,t)$ be the temperature at position $ x $ and time $ t $, $
k(x) $ the ``heat conductivity" and let $ H(t) $ be the amount of
heat contained in a region $ \Omega $ in the solid. If $ c, \,
\rho $ are constants, and if there is an external source (or sink)
$ f(x,u,t) $, then the general inhomogeneous diffusion equation
takes the form
\[
c \rho u_{t}= \nabla .(k \nabla u)+ f(x,u,t).
\]
Now, if we assume that the external source (or sink) is independent of $ t
$, in particular $ f(x,u,t)=f(x,u) $, then the ``steady state"
(time independent state) of the diffusion equation is
\[
\nabla .( k \nabla u)+ f(x,u)=0\,.
\]
For the past forty years or so, the study of such steady states of
diffusion problems subject to Dirichlet ($ u $ is specified on $
\partial \Omega $) boundary conditions has been of considerable
interest. In particular, when $ k $ is a constant and $
f(x,u)=f(u) $, leads to nonlinear eigenvalue problems of the form
\begin{eqnarray}
&-\Delta u = \lambda f(u); \quad \Omega &\\
& u = 0; \quad \partial \Omega\,,&
\end{eqnarray}
where $ \lambda > 0 $ is a parameter, $ \Omega $ is a bounded
domain in $ {\mathbb R}^n; \, n\geq 1 $ with smooth boundary $ \partial
\Omega $ and $ \Delta $ is the Laplacian operator. Cohen and
Keller in \cite{Cohen-Keller} initiated the study of such problems
when $ f $ was \underbar{ posi}tive and mono\underbar{tone}, and
also introduced the terminology ``positone" problems. Their
motivation to study such reaction terms $ f $ is the fact that the
resistance increases with temperature. For an excellent review on
positone problems see \cite{Lions}. For important results on
positone problems see \cite{Amann},\cite{Brown-Ibrahim-Shivaji},
\cite{Crandall-Rabinowitz}, \cite{Djairo-Lions-Nussbaum},
\cite{G-N-N}, \cite{Joseph-Lundgren},\cite{Kazdan-Warner},
\cite{Laetsch}, \cite{Rabinowitz} and \cite{Sattinger}.
We next describe briefly a population dynamics model which leads
to the study of steady states different from positone problems.
Let $ N(x,t) $ denote the population of a species which is
harvested at a constant rate. Assuming that the logistic growth
model fits the normal growth of the population (without
harvesting) and supposing that the quantity harvested per unit
time is independent of time and is denoted by $ H(x) $, the
resulting population model is of the form:
$$\displaylines{
\frac{\partial N}{\partial t} = c \Delta N+(B-SN)N - H;\quad
\Omega \times (0,\infty) \cr
N(x,0) = A; \quad \Omega \cr
N(x,t) = 0; \quad \partial \Omega \times [0,\infty),
}$$
where $ \Omega $ is a bounded domain in $ {\mathbb R}^3$, $c, B, S $
are positive constants, and $ A $ denotes the initial population.
The natural question is whether the population exists after a long
time. This question is equivalent to establishing the existence of
a positive solution to the steady state problem
$$\displaylines{
c \Delta N+(B-SN)N - H = 0;\quad \Omega \cr
N = 0; \quad \partial \Omega\,.
}$$
Here one needs to find not only nonnegative solutions but also
solutions that are pointwise larger than $ H $, the amount being
harvested. It is worthwhile mentioning that from a practical point
of view, constant effort harvesting is favored over density
dependent ($ f(x,u)= \tilde f(u) - H(x)u $) harvesting (see
\cite{Selgrade}). These observations lead to the study of {\it
positive} solutions to the problems of the form $ (1.1)-(1.2) $
with $f(0)<0 $. Such problems are referred in the literature as
``semipositone" problems and are the main focus of this review
paper.
Semipositone problems arise in buckling of mechanical systems,
design of suspension bridges, chemical reactions, astrophysics
(thermal equilibrium of plasmas), combustion and management of
natural resources (constant effort harvesting as derived above).
As pointed out by P.L. Lions in \cite{Lions}, semipositone
problems are mathematically very challenging, and from the point
of view of many important natural applications, interesting,
particularly for positive solutions. In fact, during the last ten
years finding positive solutions to problems of the form
(1.1)-(1.2) with $ f(0)< 0 $ has been actively pursued. The
difficulty of studying positive solutions to such problems was
first encountered by Brown and Shivaji in \cite{Brown-Shivaji}
when they studied the perturbed bifurcation problem
$$\displaylines{
-\Delta u = \lambda(u-u^3) - \epsilon;\quad \Omega \cr
u = 0; \quad \partial \Omega
}$$
with $ \epsilon > 0 $. However, the study of semipositone problems was
formally introduced by Castro and Shivaji in
\cite{Castro-Shivaji}. Also semipositone problems lead to symmetry
breaking phenomena (see \cite{Smoller-Wasserman}). In
\cite{Smoller-Wasserman}, the authors proved that $ f(0) < 0 $ is
a necessary condition for symmetry breaking of positive solutions
in a ball in ${\mathbb R}^n$. Significant progress on semipositone
problems has taken place in the last ten years; in particular, due
to the pioneering efforts by Castro and Shivaji.
In general, studying positive solutions for semipositone problems
is more difficult compared to that of positone problems. The
difficulty is due to the fact that in the semipositone case,
solutions have to live in regions where the reaction term is
negative as well as positive. We will briefly introduce the method
of sub-super solutions (see \cite{Sattinger}), which has been a
very successful method in handling positone problems, to
demonstrate the difficulty of studying positive solutions for
semipositone problems over positone problems.
A super solution to (1.1)-(1.2) is defined as a function $ \psi
\in C^{2}(\overline \Omega) $ such that
$$\displaylines{
-\Delta \psi \geq \lambda f(\psi ); \quad \Omega \cr
\psi \geq 0; \quad \partial \Omega\,.
}$$
Sub solutions are defined similarly with the inequalities reversed
and it is well known that if there exists a sub solution $ \phi $
and a super solution $ \psi $ to (1.1)-(1.2) such that $
\phi(x)\leq \psi(x) $ for $ x \in \bar \Omega $, then (1.1)-(1.2)
has a solution $ u $ such that $ \phi(x) \leq u(x) \leq \psi(x) $
for $ x \in \bar \Omega $. Further note that if $ \phi(x) \geq 0 $
for $ x \in \Omega $ then $ u \geq 0 $ for $ x \in \Omega $.
In the positone case, $ \phi \equiv 0 $ is a sub solution to
(1.1)-(1.2). Thus to ensure the existence of a positive solution
it is enough to find a positive super solution. On the other hand,
in the semipositone case $ \psi \equiv 0 $ is a super solution.
Suppose we try to find a nonnegative sub solution $ \phi $ such
that
$$\displaylines{
-\Delta \phi = h; \quad \Omega \cr
\phi = 0; \quad \partial \Omega,
}$$
then we have to choose $ h $ to be negative near the boundary of $
\Omega $ since $ f(0) < 0 $ while to ensure that $ \phi $ is
nonnegative it is necessary that $ h $ is sufficiently positive in
the interior of $ \Omega $. Further $ h $ must be such that $ h
\leq f(\phi) $ for each $ x $ in $ \Omega $. In short, finding a
nonnegative solution is not an easy task in the case of
semipositone problems. We encounter similar problems when we try
to use other methods that have been successful in the case of
positone problems.
However, mathematicians have found their way via sub-super
solution methods, variational methods, degree theory, fixed point
theory, shooting methods, reflection arguments, Maximum
principles, bifurcation theory etc. to establish a rich collection
of results for the single equations case(see Section 2).
The study of semipositone systems for positive solutions is even
more challenging since not only one has to deal with the systems,
but also needs to establish the positivity of the solution
componentwise. In the single equations case a popular plan was to
find a solution with ``big" enough supremum norm and use this to
establish that the solution is positive. In systems, knowing that
the supremum norm of $ u = (u_{1}, \dots, u_{m}) $ (say) is large
does not necessarily mean that the supremum norm of each $ u_{i} $
is large. Thus establishing that each component $ u_{i} $ of the
solution is positive is an additional challenge.
Semipositone systems occur naturally in important applications;
for example, predator - prey systems with constant effort
harvesting. In particular, with diffusion included, such systems
in steady states will be of the form:
$$\displaylines{
-\Delta u = \lambda[f(u,v) - K]; \quad \Omega \cr
-\Delta v = \lambda[g(u,v) - H]; \quad \Omega \cr
u = v = 0; \quad \partial\Omega\,,
}$$
where $ \Omega $ is a smooth bounded region in $ {\mathbb R}^n$, and $ K $ and
$ H $ represent harvesting (or stocking if they are negative)
densities of the predator $ u $ and prey $ v $ respectively. See
\cite{Brauer-Soudack}, \cite{Brauer-Soudack1},
\cite{Brauer-Soudack2}, \cite{Dai-Tang}, \cite{M-G-H-N} and
\cite{Selgrade} for examples of $ f $'s and $ g $'s where the
authors study various predator-prey systems with constant effort
harvesting but without diffusion. Thus the study when diffusion is
included (i.e., the study of semipositone systems) will greatly
enhance the understanding of these problems in population
dynamics.
In summary, while strengthening the result for single equations it
would be challenging and important to extend the theory for single
equations to systems in the following two directions:\\ \noindent
[A] To study systems arising in applications such as
predator-prey, cooperative and competitive models with constant
effort harvesting. \\ \noindent [B] To identify and study systems
that exhibit qualitative properties similar to that of the single
equations case. \\ To date, no results are known in the direction
[A], while some developments have occured recently in the
direction [B] which we will outline in Section 3.
We now conclude this introduction by outlining the rest of the
paper. Namely, in section 2 we provide the known literature on
semipositone single equations to date and open problems. In
section 3, we discuss the very recent developments on semipositone
systems.
\section{Survey of semipositone problems for single equations case}
Semipositone problems were formally introduced by Castro and
Shivaji in \cite{Castro-Shivaji} where the authors studied two
point boundary value problems of the form
$$\displaylines{
-u'' = \lambda f(u); \quad 0< x <1 \cr
u(0) = u(1) = 0
}$$
and obtained detailed results via a quadrature method for various
classes of reaction terms $ f $. In particular they considered
classes of superlinear nonlinearities (eg: $ f(u)= u^{p} -
\epsilon; \ \epsilon > 0, \ p > 1,\quad f(u) = e^{u}-2 $ etc.) for
which they proved that there exists $ \lambda_{1}> 0 $ such that
for $ 0 < \lambda \leq \lambda_{1} $ there is a unique positive
solution while for $ \lambda > \lambda_{1} $ there are no positive
solution. For classes of sublinear nonlinearities (eg: $ f(u)=
(1+u)^{1/3}-3, \quad f(u)=au-bu^{2}-c; a>0,b>0,c>0 $ etc.) they
proved that there exists $ 0 < \mu < \lambda_{1} $ such that for $
\lambda < \mu $ there are no positive solutions, for $ \mu <
\lambda \leq \lambda_{1} $ there are at least two positive
solutions, and for $ \lambda = \mu $ and $ \lambda > \lambda_{1} $
there is exactly one positive solution. They also studied classes
of superlinear nonlinearities which were initially concave and
later convex (eg: $ f(u)=u^{3}! -au^{2}+bu -c; \, a>0, b>0, c>0 $
and $ b> (32/81)a^{2}, a^{3} > 54c $) and established that there
are ranges of $ \lambda $ for which there are at least three
positive solutions. Further, in all cases, they established a
sequence $ \{\lambda_{n}\}; n=1,2,\dots $ such that for $ \lambda
= \lambda_{n} $, the problem had a unique nonnegative solution
with $ (n-1) $ interior zeros, which satisfy both the Dirichlet as
well as Neumann boundary conditions. Note that such solutions are
possible only if $ f(0)< 0 $.
Many mathematicians during the past ten years or so have
successfully extended these results to higher dimensions. The
first major breakthrough came in \cite{Castro-Shivaji1} when the
authors proved that all nonnegative solutions for
$$\displaylines{
-\Delta u = \lambda f(u); \quad \Omega \cr
u = 0; \quad \partial \Omega
}$$
with $ \lambda > 0$, $f(0) < 0 $ and $ \Omega = B_{n} $ a ball in
$ {\mathbb R}^n$; $n > 1 $ are in fact positive. Since positivity
implies radial symmetry (see \cite{G-N-N}), various results
appeared in the literature following this result for positive
(radial) solutions for semipositone problems.
This positivity result when $ \Omega $ is a ball in $ {\mathbb R}^n;
\, n > 1 $ is unlike the case when $ n=1 $, and authors in
\cite{Castro-Shivaji1} used the fact that the boundary is
connected. In \cite{G-N-N}, due to the result for $ n = 1 $, it
was conjectured that the problem may have nonnegative solutions
with interior zeros in higher dimensions as well, which we see is
false when at least $ \Omega $ is a ball. However, the conjecture
remains to be proven/disproven in general bounded regions.
\subsection{Superlinear nonlinearities}
In this section we discuss results on superlinear nonlinearities.
See \cite{Castro-Shivaji2} and \cite{Smoller-Wasserman1} where the
authors establish the existence of positive solutions for $
\lambda $ small for classes of superlinear nonlinearities when $
\Omega $ is a ball. In \cite{Smoller-Wasserman1} authors use
shooting methods to prove this existence result for nonlinearities
of the form $ f(u) = u^{p} - \epsilon; \ \epsilon > 0,\ 1 < p <
\frac{n}{n-2} $. In \cite{Castro-Shivaji2} authors do better by
combining shooting methods with Pohozaev's identity. In fact,
their theorem gives this existence result for $ f(u) = u^{p} -
\epsilon; \ \epsilon > 0,\ 1 < p < \frac{n+2}{n-2} $ which is an
optimal result since it can be proven that positive solution do
not exist if $ p \geq \frac{n+2}{n-2} $ (the critical exponent).
This existence result has been extended to the general bounded
regions (see \cite{Allegretto-Nistri-Zecca},
\cite{Ambrosetti-Arcoya-Buffoni} and \cite{Sumalee}). In
\cite{Allegretto-Nistri-Zecca} the authors use degree theory, in
\cite{Sumalee} the authors use variational methods while in
\cite{Ambrosetti-Arcoya-Buffoni} the result is obtained via
bifurcation from infinity. The often successful technique has been
to obtain a solution with big enough supremum norm and use this
fact to show that the solution is positive. See also
\cite{Anuradha-Hai-Shivaji} and \cite{Arcoya-Zertiti} for
extensions of this existence result for $ \lambda $ small.
For classes of superlinear problems non-existence result for $
\lambda $ large has been proven in \cite{Allegretto-Nistri-Zecca},
\cite{Arcoya-Zertiti} and \cite{Brown-Castro-Shivaji}. Here the
authors prove the result by analyzing the qualitative behavior of
solutions (if they exist) near the boundary and obtaining a
contradiction using an energy analysis or on the positivity of the
solution.
The instability of the solution for convex monotone nonlinearities
was first established in \cite{Brown-Shivaji} where the authors
use Green's identities to prove that the first eigenvalue of the
linearized equation about the solution has the appropriate sign.
See \cite{Tertikas} and \cite{Maya-Shivaji} for an extension of
this result for non-monotone functions.
The uniqueness result for superlinear nonlinearities for $ \lambda
$ small for the case when $ \Omega $ is a ball was established in
\cite{Ali-Castro-Shivaji} and \cite{Castro-Gadam-Shivaji} using
the Implicit function theorem, variations with respect to
parameters, the Pohozaev's identity and comparison arguments. In
\cite{Castro-Gadam-Shivaji}, it was further established in the
ball that if $ f(u)= u^{p} - \epsilon; \ \epsilon > 0, \ 1< p <
\frac{n+2}{n-2} $ then the problem has at most one positive
solution for any $ \lambda $. However, this uniqueness result
remains an open question in general bounded regions even in convex
regions other than a ball. Also the case when the nonlinearity is
concave first and then convex needs to be studied beyond the $ n=1
$ case discussed in \cite{Castro-Shivaji}.
\subsection{Sublinear nonlinearities}
In this section we discuss results on sublinear semipositone
problems. For classes of sublinear concave nonlinearities when $
\Omega $ is a general bounded region in ${\mathbb R}^n; \, n > 1
$, existence results were established in
\cite{Ambrosetti-Arcoya-Buffoni}, \cite{Anuradha-Dickens-Shivaji},
\cite{Castro-Garner-Shivaji} and \cite{Clement-Sweers}. In
\cite{Clement-Sweers} for nonlinearities with falling zeros,
variational methods were used to prove an existence result for $
\lambda $ large. In \cite{Anuradha-Dickens-Shivaji} and
\cite{Castro-Garner-Shivaji}, the method of sub-super solutions
was employed to establish existence results, one for $ \lambda $
large, and the other near the first eigenvalue of the Laplacian
operator with Dirichlet boundary conditions. For this latter
existence result, the Anti-maximum principle (see
\cite{Clement-Peletier}) was used to create a nonnegative sub
solution. Further in \cite{Anuradha-Dickens-Shivaji} nonexistence
of positive solutio! ns for $ \lambda $ small was established via
Green's identities using the fact that $ f(0)< 0 $ and the
sublinearity assumption. In \cite{Ambrosetti-Arcoya-Buffoni}, an
existence result for $ \lambda $ large was established via
bifurcation from infinity. See also \cite{Maya-Shivaji1} where
existence result from semipositone problems is used to establish
existence and multiplicity results for classes of sublinear
nonlinearities which vanish at the origin but negative near the
origin.
When $ \Omega $ is a ball, for classes of sublinear nonlinearities
complete details are known. In \cite{Castro-Gadam} and
\cite{Castro-Shivaji3} they established the exact bifurcation
diagram. In particular, there exists $ 0 < \mu < \lambda_{1} $
such that for $ \lambda < \mu $ no solution, for $ \mu < \lambda
\leq \lambda_{1} $ exactly two solution, and for $ \lambda \geq
\lambda_{1} $ as well as for $ \lambda= \mu $ exactly one
solution. Further, the upper branch of the solution is stable
while the lower branch including at $ \lambda = \mu $ is unstable.
In \cite{Castro-Gadam}, the authors study increasing
nonlinearities (eg: $ f(u)=(u+1)^{1/3}-2 $) while in
\cite{Castro-Shivaji3} nonlinearities with falling zeros (eg: $
f(u) = au^{2} -bu -c; \ a,b,c > 0 $) are discussed. The Implicit
Function Theorem, variation with respect to parameters, the above
mentioned existence and nonexistence results, and the uniqueness
and stability results established in \cite{Ali-Castro-Shivaji}!
were useful in proving this exact bifurcation diagram. In
\cite{Ali-Castro-Shivaji}, the authors use again the Implicit
function theorem, variations with respect to parameters and Sturm
comparison theorem to establish the uniqueness and stability
results. See also \cite{Castro-Gadam-Shivaji1} and
\cite{Castro-Gadam-Shivaji2} where evolution of bifurcation curves
for positive solution are studied with concave nonlinearities.
However to date, the uniqueness result for $ \lambda $ large, in
general bounded regions, has been proven only for classes of
increasing nonlinearities and when the outer boundary of the
region is convex (see \cite{Castro-Hassanpour-Shivaji}). Here the
authors prove their results by first establishing qualitative
properties of the solution near the boundary, namely they
establish that for large $ \lambda $ the solution $ u(x) \geq k\,
dist(x,\partial \Omega) $ where $ k > 0 $ is a constant. Uniqueness result for
$ \lambda $ large when the outer boundary of the region is
non-convex is open. For the case when the nonlinearity has a
falling zero, this uniqueness result has been proven only in the
case when $ \Omega $ is a ball (see \cite{Castro-Shivaji3}).
Finally, the multiplicity result is also open in regions other
than a ball.
\subsection{Quasilinear equations}
In this section we discus existence results for radial solutions
to quasilinear equations of the form
$$\displaylines{
-\mathop{\rm div}(\alpha(|\nabla u|^{2})\nabla u) = \lambda f(|x|,u);
\quad x \in \Omega \cr
u = 0;\quad x \in \partial \Omega\,,
}$$
where $ \lambda > 0 $ and $ \Omega $ is an annulus. Here $
\alpha(s^{2})s $ is an odd increasing homeomorphism on the real
line. The special case when $ \alpha(|\nabla u|^{2}) = |\nabla
u|^{p-2} , \ p > 1 $ corresponds to the p-Laplacian case. Some
existence results has been established in
\cite{Hai-Schmitt-Shivaji} via fixed point theory in a cone for
such systems. For classes of superlinear functions they prove the
existence result for $ \lambda $ small and for classes of
sublinear functions they prove the existence result for $ \lambda
$ large. The existence result for $ \lambda $ small for classes of
superlinear functions has been extended to the case when $ \Omega
$ is a ball in ${\mathbb R}^n$ in \cite{Hai} via degree theory.
To our knowledge these are the only two papers in this direction
and thus many questions on uniqueness, non-existence, multiplicity
all remain open even for radial solutions when $ \Omega $ is a
ball/annulus. Further, the study of such quasilinear equations
with semipositone structure is open in the case of general bounded
regions.
\subsection{Remarks}
Here we summarize the known results to date for semipositone
problems with Neumann/Robin boundary conditions and in unbounded
regions. For results on positive solutions for semipositone
problems with Neumann boundary conditions see
\cite{Allegretto-Nistri} and \cite{Agnes-Shivaji}. In
\cite{Agnes-Shivaji} the authors study two point boundary value
problems via a quadrature method. In \cite{Allegretto-Nistri} the
authors study classes of superlinear problems via degree theory.\\
See also \cite{Anuradha-Maya-Shivaji} and \cite{Anuradha-Shivaji}
where classes of two point boundary value problems with Robin
boundary conditions are discussed via quadrature methods. Further
note that the results in \cite{Allegretto-Nistri},
\cite{Allegretto-Nistri-Zecca}, \cite{Anuradha-Dickens-Shivaji},
\cite{Brown-Shivaji}, \cite{Maya-Shivaji} and \cite{Tertikas}
holds for Robin boundary conditions as well.\\ For results on
positive solutions in unbounded regions see
\cite{Allegretto-Odiobala}. Here the authors discuss existence
result for $ \lambda $ small for classes of superlinear
nonlinearities via variational methods.\\ Finally, for the study
of sign-changing solutions see \cite{Anuradha-Shivaji1} and
\cite{Castro-Gadam-Shivaji}. In \cite{Anuradha-Shivaji1}, a two
point boundary value problem is discussed via a quadrature method
while in \cite{Castro-Gadam-Shivaji}, a thorough study is carried
out for all branches of solutions when $ \Omega $ is a ball in $
{\mathbb R}^n; \, n > 1 $ via the Implicit function theorem and
variations with respect to parameters.
\section{Recent developments on semipositone systems}
\setcounter{equation}{0}
Here we describe recent developments on
semipositone systems, namely results in
\cite{Castro-Maya-Shivaji}, \cite{Castro-Maya-Shivaji1},
\cite{Hai}, \cite{Hai-Maya-Shivaji}, \cite{Hai-Shivaji} and
\cite{Hai-Shivaji-Shobha}. These results give the complete picture to
date in this direction.
\paragraph{Result 1:}
In\cite{Castro-Maya-Shivaji}, the authors study cooperative
semipositone systems in a ball. In particular, they consider a
classical nonnegative solution $ u:=(u_{1}\geq 0,u_{2}\geq 0,
\dots,u_{m}\geq 0) $ for the system
$$\displaylines{
-\Delta u_{i} = f_{i}(u_{1},u_{2}, \dots , u_{m});\quad \Omega
\quad 1\leq i\leq m,\cr
u_{i} = 0\quad \partial \Omega,
}$$
where $ \Omega $ is a ball in $ {\mathbb R}^n; \, n > 1 $ and $
f_{i}:\underbrace{[0,\infty)\times [0,\infty)\dots, [0,\infty)}_{m
\ \mbox{times}} \to {\mathbb R} $ are $ C^{1} $ functions
satisfying
$$\displaylines{
f_{i}(0,0, \dots, 0)<0, \qquad i=1,2, \dots, m \quad \mbox{
(semipositone system) } \quad \mbox{ and }\cr
\frac{\partial f_{i}}{\partial u_{j}} \geq 0, \qquad i \neq j \quad
\mbox{(cooperative system) }.
}$$
Then they prove that $ u_{i} > 0 $ for each $ i=1,2,\dots, m $
i.e., nonnegative solutions are componentwise positive. This
result is of great importance since positivity implies that the
solutions are radially symmetric and radially decreasing (see
\cite{Troy}). They prove the result by combining Lemma 4.2 of
\cite{Troy}, Maximum principle/reflection arguments and analysis
of solutions near the boundary. This result holds even if $ \Omega
$ is a region between two balls or the union of balls. However,
the question of positivity of nonnegative solutions in general
bounded region remains open including in the single equation
case. On the other hand, in unbounded regions there are
nonnegative solutions with interior zeros. Indeed, consider the
two point boundary value problem
\begin{eqnarray}
-u'' = \lambda f(u); \quad 0 0 $ and $ \lim\limits_{u \to
\infty}f(u)=\infty $. Then from \cite{Castro-Shivaji} it follows
that there exists an increasing sequence of positive numbers $
\lambda_{n} $ such that $ (3.3)-(3.4) $ has a nonnegative solution
$ u_{n}(x) $ with $ n $ interior zeros $ x_{n} $ in $ (0,1) $. Now
consider
\begin{eqnarray}
&-\Delta w = \lambda f(w); \quad \Omega:=\{ (x,y): 0 0 $ is a parameter, $ f,g : [0,\infty) \times
[0,\infty) \to {\mathbb R}$ are continuous and there exists
$ M > 0 $ such that $ f(u,v) \geq - \frac{M}{2}, \ g(u,v) \geq
-\frac{M}{2} $ for every $ (u,v) \in [0,\infty) \times [0,\infty)
$. They first consider the case when $ f, g $ further satisfy:
\begin{description}
\item{\bf (A1)} $ \lim\limits_{v \to \infty}f(u,v)=\infty, \lim\limits_{u
\to \infty}g(u,v)=\infty, $ where each limit is uniform
with respect to the other variable and $ \lim\limits_{z
\to \infty}\frac{h^{*}(z)}{z}=\infty $, where \\
$h^{*}(z):=\inf\limits_{u,v \geq z}\left\{\min(f(u,v),g(u,v))\right\}$.
\end{description}
\noindent and prove that the
system has a positive solution $ (u_{\lambda},v_{\lambda}) $ for $
\lambda $ small with \\$
|(u_{\lambda}(t),v_{\lambda}(t))|_{\infty} \to \infty $ as
$ \lambda \to 0 $ uniformly for $ t $ in compact intervals
of $ (a,b) $.
\noindent Next, they consider the case when $ f, g $ satisfy:
\begin{description}
\item{\bf (A2)}
$ \lim\limits_{v \to \infty}f(u,v)=\infty, \lim\limits_{u
\to \infty}g(u,v)=\infty, $ where each limit is uniform
with respect to the other variable and $ \lim\limits_{z
\to \infty}\frac{\tilde h(z)}{z}=0 $, where \\
$ \tilde h(z):=\sup\limits_{0 \leq u,v \leq z
}\left\{\max(f(u,v),g(u,v))\right\} $
\end{description}
\noindent and prove that the
system has a positive solution $(u_{\lambda},v_{\lambda})$ for $
\lambda $ large with \\$ \lambda^{-1} \max
(u_{\lambda}(t),v_{\lambda}(t)) \to \infty $ as $ \lambda
\to \infty $ uniformly for $ t $ in compact intervals of $
(a,b) $.
\noindent Finally, they consider the case when $ f(u,v)=f(v),
g(u,v)=g(u) $ satisfy $ (A1) $ and
\begin{description}
\item{\bf (A3)}
there exists $ r > 0 $ and $ 0< \alpha < 1 $ such that $ h(x)
\geq(\frac{x}{r})^{\alpha}h(r) $ for $ x \in [0,r] $ where $ h(x)=
\min \{f(x)-f(0), g(x)-g(0)\} $,
\end{description}
\noindent and establish the existence
of at least two positive solutions for certain ranges of $ \lambda
$ under some additional conditions.
Note that no sign conditions are required on the reaction terms at
the origin and thus allowing the semipositone structure. Also no
monotonicity assumptions are required for these results to hold.
They establish these result by using fixed point theory in a
cone.
\paragraph{Result 3:} In \cite{Castro-Maya-Shivaji1},
the authors establish an existence result for classes of sublinear
cooperative semipositone systems in general bounded regions. In
particular, they consider the existence of positive solutions to
the system
$$\displaylines{
-\Delta u_{i} = \lambda [f_{i}(u_{1},u_{2}, \dots
u_{m})-h_{i}];\quad \Omega \cr
u_{i}= 0; \quad \partial\Omega
}$$
where $ \lambda > 0 $ is a parameter, $\Omega $ is a bounded
domain in $ {\mathbb R}^n; \, n \geq 1 $ with a smooth boundary $
\partial \Omega,\, h_{i} $ are nonnegative continuous functions in
$ \Omega $ for $ i=1,2,\dots, m $ and $ f_{i}
:\underbrace{[0,\infty)\times[0,\infty)\times \dots \times
[0,\infty)}_{\mbox{m times}} \to {\mathbb R} $ are $ C^1 $
functions for $ i=1,2,\dots, m $. Further, we assume that
for each $ i=1,2, \dots, m $, we have
$$\displaylines{
f_{i}(0,0,\dots,0)=0 \cr
\frac{\partial f_{i}}{\partial u_{j}}(z_{1},z_{2},\dots,z_{m})
\geq 0,\quad i \neq j \ z_{1},z_{2}, \dots, z_{m} \in {\mathbb R} \cr
\frac{\partial f_{i}}{\partial u_{i}}(z,z,\dots,z) \geq 0\,, \quad
\forall z \in {\mathbb R} \cr
\sum\limits_{j=1}^{m}{\frac{\partial f_{i}}{\partial
u_{j}}(0,\dots,0)}> 0 \cr
\lim\limits_{z \to \infty}\frac{f_{i}(z,\dots,z)}{z}=0\,;
\quad\mbox{and}\quad
\lim\limits_{z \to \infty}f_{i}(z, \dots,z)= \infty\,.
}$$
Then they establish that there exists $ \tilde \lambda > 0 $ such
that for $ \lambda > \tilde \lambda $, the system has a positive
solution $ (u_{1},u_{2},\dots,u_{m}) $. Further, $
u_{i}(x)/\mathop{\rm dist}(x,\partial \Omega) =O(\lambda) $ as $ \lambda
\to \infty $ for $ i=1,2,\dots, m $. They prove this
result by producing a nonnegative sub solution and then applying
the method of sub-super solutions. As pointed out earlier,
producing nonnegative sub solution is non-trivial in semipositone
problems, and this is the important step in the proof of this
result. See \cite{Anuradha-Dickens-Shivaji} and
\cite{Castro-Garner-Shivaji} where the single equation case of
this problem was studied using the method of sub-super solutions.
Sub-super solutions are in general hard to apply in the
semipositone case since it is hard to construct a nonnegative
sub-solution. In fact, in \cite{Anuradha-Dickens-Shivaji} and
\cite{Castro-Garner-Shivaji}, a non-trivial existence result
proved in \cite{Clement-Sweers} for a class of semipositone
problem with reaction term having ``falling zeros", played a
crucial role in the construction of the nonnegative sub solution.
However here authors provide a direct method of constructing
sub-super solutions.
We note here that semipositone sublinear systems have also been
studied in the past in \cite{Anuradha-Castro-Shivaji}. However, in
\cite{Anuradha-Castro-Shivaji} the coupling was weak so that one
could use existence results from the study of single equations
case in the construction of the nonnegative sub solution.
\paragraph{Result 4:} In \cite{Hai-Maya-Shivaji}, the authors
prove an existence result for radial solutions in an annulus for
classes of quasilinear (including p-Laplacian) systems with
superlinear reaction terms. In particular, consider the existence
of positive radial solutions for the system
$$\displaylines{
-\mathop{\rm div}(\alpha(|\nabla u|^{2})\nabla u) = \lambda f(v)
;\quad a<|x|**0 $ there exist a constant $ A_{c}>0 $ such that
$\phi^{-1}(cx)\geq A_{c}\phi^{-1}(x)$ for all $x\geq 0$
and $A_{c} \to \infty$ as $c \to\infty$ (which implies the existence
of a constant $B_{c}:=1/A_{1/c}>0 $ such that
$\phi^{-1}(cx)\leq B_{c}\phi^{-1}(x)$ for all $x\geq 0$
and $B_{c} \to 0$ as $c \to 0$).
\item{\bf (B2)}
The functions $f, \, g :[0,\infty) \to {\mathbb R} $ are continuous,
and there exists $ M>0 $ such that $ f(z)\geq -M/2 $ and $
g(z)\geq -M/2$ for $ z \in [0,\infty) $
\item{\bf (B3)}
$ \lim\limits_{z \to \infty} \frac{h^{*}(z)}{\phi(z)}= \infty$
and $\lim\limits_{z \to \infty}\frac{A_{h^{*}(z)}}{z}= \infty $
where $ A_{c} $ is as defined in (B1) and
$h^{*}(z)=\inf\limits_{w \geq z}\left\{\min (f(w),g(w))\right\}$.
\end{description}
\noindent Then they establish that there exists
$\lambda^{*}>0 $ such that for $ 0<\lambda<\lambda^{*}
$, the system has a positive solution. This result is an extension
to classes of systems including p-Laplacian systems
($\alpha(s^2)=|s|^{p-2},\, p>1 $) of existence results for
superlinear problems discussed in \cite{Allegretto-Nistri-Zecca},
\cite{Ambrosetti-Arcoya-Buffoni}, \cite{Anuradha-Hai-Shivaji},
\cite{Castro-Shivaji}, \cite{Castro-Shivaji2}, \cite{Garazier},
\cite{Hai-Schmitt}, \cite{Hai-Schmitt-Shivaji} and
\cite{Hai-Shivaji}. In particular, in \cite{Hai-Schmitt-Shivaji}
the authors study the single equation case of this problem for
both sublinear and superlinear reaction terms. As noted earlier
(see Result 2), in \cite{Hai-Shivaji}, authors again establish
such existence results for radial solutions in an annulus but for
semilinear elliptic systems. Here authors succeed in establishing
an existence result for quasilinear systems, for a class of
superlinear reaction terms. Again here (as in Result 2) no
monotonicity assumptions are required on the reaction terms and
the semipositone structure is allowed. The result is established
via degree theory.
\paragraph{Result 5:} In \cite{Hai}, the
authors extend the result in \cite{Hai-Shivaji} for the
superlinear case when the region is a ball. In particular, they
consider a system of the form
$$\displaylines{
-(r^{n-1}u')'= \lambda r^{n-1}f(v); \quad 0 0 $ is a parameter, $ f,g : [0,\infty)
\to {\mathbb R}$ are continuous and
\begin{description}
\item{\bf (C1)}
$ \lim\limits_{s \to \infty}\frac{f(s)}{s}=\infty,\lim\limits_{s
\to \infty}\frac{g(s)}{s}=\infty $
\item{\bf (C2)}
there exists nonnegative numbers $ \alpha, \beta $ with $ \alpha +
\beta = n-2$, and a positive number $ \epsilon $ such that $
nF(s)-\alpha s f(s) \geq \epsilon s f(s) $ and $ nG(s)-\beta sf(s)
\geq \epsilon s g(s) $ for $ s $ large. Here $ F(s)=
\int_{0}^{s}f(t)\,dt$, $G(s)= \int_{0}^{s}g(t)\,dt $.
\end{description}
\noindent They prove that the system has a positive solution $
(u_{\lambda},v_{\lambda}) $ for $ \lambda $ small with
$|(u_{\lambda},v_{\lambda})|_{\infty} \to \infty $ as
$\lambda \to 0 $ via degree theory.
We note here that the common feature of Results 2, 4 and 5 is that
solutions of large supremum norm are obtained and then prove
positivity of such solutions (both in single equation and systems
case).
\paragraph{Result 6:} In \cite{Hai-Shivaji-Shobha}, again
for the superlinear case a non-existence result is proven. In
particular, they consider the system
$$\displaylines{
-\Delta u = \lambda f(v); \quad \Omega \cr
-\Delta v = \lambda g(u); \quad \Omega \cr
u=0 = v; \quad \partial \Omega\,,
}$$
where $ \Omega $ is a ball or an annulus in ${\mathbb R}^n$,
$ \lambda> 0 $ is a parameter and $ f, g : [0,\infty) \to {\mathbb R}$
are continuous, $ f(0) < 0$, $g(0) < 0$; $f' \geq 0$, $g' \geq 0 $ and
\begin{description}
\item{\bf (D1)}
There exists $ K_{i}> 0$, $M_{i}> 0$; $i=1,2 $ such that
$ f(z) \geq K_{1}z - M_{1} $ and $ g(z) \geq K_{2}z - M_{2} $ for
$ z \geq 0$.
\end{description}
\noindent Then they prove that the system has no nonnegative
solutions for $ \lambda $ large. They prove the result by
analyzing the solutions near the outer boundary and using
comparison principles.
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}\end{thebibliography}
\noindent{\sc Alfonso Castro} \\
Division of Mathematics and Statistics \\
The University of Texas at San Antonio \\
San Antonio TX 78249-0664 USA \\
e-mail: castro@math.utsa.edu \medskip
\noindent{\sc Maya Chhetri} \\
Department of Mathematical Sciences \\
University of North Carolina at Greensboro, NC 27402 USA \\
e-mail: maya@uncg.edu \medskip
\noindent{\sc R. Shivaji} \\
Department of Mathematics and Statistics \\
Mississippi State University, MS 39762 USA \\
e-mail: shivaji@math.msstate.edu
\end{document}
**