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\pageno=135
\headline={\ifnum\pageno=1 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{\hfil Large solutions and metasolutions
\hfil\folio}
\def\leftheadline{\folio\hfil Julian L\'opez-G\'omez
\hfil}
\def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt %
Nonlinear Differential Equations,\hfill\break Electronic Journal
of Differential Equations, Conf. 05, 2000, pp 135--171.\hfill\break
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfill\break
ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp) \bigskip
} }
\topmatter
\title
Large solutions, metasolutions, and asymptotic behaviour of the
regular positive solutions of sublinear parabolic problems
\endtitle
\thanks
{\it Mathematics Subject Classifications:} 35K57, 35K60, 35D05.
\hfil\break\indent
{\it Key words:} Large solutions. Metasolutions. Asymptotic behavior.
\hfil\break\indent
\copyright 2000 Southwest Texas State University.\hfil\break\indent
Published October 25, 2000. \hfil\break\indent
Supported by the Spanish Grant DGES PB96-0621.
\endthanks
\author Juli\'an L\'opez-G\'omez \\ \\
{\it Dedicated to Alan C. Lazer on his 60th birthday}
\endauthor
\address
Juli\'an L\'opez-G\'omez \hfill\break
Departamento de Matem\'atica Aplicada \hfill\break
Universidad Complutense de Madrid \hfill\break
28040-MADRID, Spain. \hfill\break
Phone (34) 91-394-4194, Fax (34) 91-394-4102
\endaddress
\email Lopez\_Gomez\@mat.ucm.es
\endemail
\abstract
In this paper we analyze the existence of regular and large
positive solutions for a class of non-linear elliptic boundary
value problems of logistic type in the presence of refuges. These
solutions describe the asymptotic behaviour of the regular
positive solutions of the associated parabolic model. The main
tool in our analysis is an extension of the interior estimates
found by J. B. Keller in \cite{Ke57} and R. Osserman in
\cite{Os57} to cover the case of changing sign nonlinearities
combined with the construction of adequate sub and supersolutions.
The supersolutions are far from obvious since the nonlinearity
vanishes in finitely many regions of the underlying support
domain.
\endabstract
\endtopmatter
\document
\head 1. Introduction \endhead
In this paper we analyze the asymptotic behaviour of the positive
solutions of
$$\gathered
{\partial u \over \partial t} -\Delta u=\lambda u -a(x)f(x,u)u
\quad \hbox{in } \Omega\times (0,\infty)\,, \cr
u(x,t)=0 \quad \hbox{on } \partial\Omega\times(0,\infty)\,,\cr
u(\cdot,0)=u_0\geq 0 \quad \hbox{in } \Omega\,,\endgathered
\eqno (1.1)
$$
where $\Omega$ is a bounded domain of ${\Bbb R}^N$, $N\geq 1$,
with boundary $\partial\Omega$ of class $C^3$, $\lambda\in{\Bbb
R}$, and $a\geq 0$, $a\neq 0$, is a function of class
$C^{\mu}(\bar\Omega)$, for some $\mu\in (0,1)$, satisfying the
following assumptions:
\roster
\item"(Ha1)" The open set
$$
\Omega_+:= \{\,x\in\Omega\,:\;a(x)>0\,\}
$$
is connected with boundary $\partial\Omega_+$ of class $C^3$.
Moreover, if $\Gamma_+$ and $\Gamma$ are components of $\partial
\Omega_+$ and $\partial \Omega$, respectively, such that
$\Gamma_+\cap \Gamma\neq \emptyset$, then $\Gamma_+=\Gamma$ and
$a(x)$ is bounded away from zero on $\Gamma_+$.
\smallskip
\item"(Ha2)" If $\Gamma_+$ is a component of $\partial \Omega_+$ such that
$\Gamma_+ \cap \partial \Omega =\emptyset$, then
$$
a(x)=o(\hbox{dist}(x,\Gamma_+)) \quad \hbox{as }
\hbox{dist}(x,\Gamma_+)\downarrow 0\,,\quad x\in \Omega_+\,.
\eqno (1.2)
$$
\smallskip
\item"(Ha3)" The open set
$ \Omega_0:=\Omega\setminus \bar \Omega_+$ possesses a finite
number of components, say $\Omega^i_{0,j}$, $1\leq i \leq m$,
$1\leq j \leq n_i$, such that
$$
\bar \Omega^i_{0,j} \cap \bar \Omega^{\hat i}_{0,\hat j} = \emptyset
\quad \hbox{if } (i,j)\neq (\hat i,\hat j)\,.
$$
In the sequel given a regular subdomain $D$ of $\Omega$ and $V\in
C^\mu(\bar D)$, we denote by $\sigma^D[-\Delta+V]$ the principal
eigenvalue of $-\Delta+V$ in $D$ under homogeneous Dirichlet
boundary conditions, and set $\sigma^D:=\sigma^D[-\Delta]$.
Without loss of generality we can label the components of
$\Omega_0$ so that
$$
\sigma^{\Omega^i_{0,j}}=\sigma^{\Omega^i_{0,j+1}}\,, \quad
1\leq i \leq m\,, \;\; 1 \leq j \leq n_i-1\,,
\eqno (1.3a)
$$
and
$$
\sigma^{\Omega^i_{0,1}} < \sigma^{\Omega^{i+1}_{0,1}}\,,
\quad 1\leq i \leq m-1\,.
\eqno (1.3b)
$$
\endroster
As for the nonlinearity, we suppose the following:
\smallskip
\roster
\item"(Hf)" The function $f:\bar \Omega\times [0,\infty)\to{\Bbb R}$
is continuous and of class
$C^{\mu,1+\mu}(\bar\Omega\times(0,\infty))$ and it satisfies
$f(x,0)=0$, $f(x,u)>0$, and $\partial_u f(x,u)>0$ for all $u>0$
and $x\in\Omega$. Moreover,
$$
\lim_{u\uparrow\infty}f(x,u)=\infty
$$
uniformly on compact subsets of $\bar \Omega_+$.
\endroster
\smallskip
\noindent Problem (1.1) provides us with the evolution of
a single species obeying a generalized logistic growth law
\cite{Mu93}, \cite{Ok80}. Typically, $u$
is the density of the species, $\Omega$ is the inhabiting region,
$\lambda$ is the net growth rate of $u$, and the coefficient
$a(x)$ measures the saturation effect responses to the population
stress in $\Omega_+$. In $\Omega_0$ the individuals of the
population are free from other effects than diffusion and so each
of the components of $\Omega_0$ can be regarded as a {\it refuge}.
The refuges have been ordered accordingly to the size of the
principal eigenvalue of $-\Delta$ under homogeneous Dirichlet
boundary conditions. The refuges $\Omega_{0,j}^1$, $1\leq j \leq
n_i$, will be called the lower order refuges. Throughout this
paper we use the following notation
$$
\sigma_0:=\sigma^\Omega\,,\quad \sigma_i:=
\sigma^{\Omega^i_{0,1}}\,, \quad 1\leq i \leq m\,.
\eqno (1.4)
$$
Thanks to (1.3b),
$$
\sigma_0<\sigma_1<\sigma_2<\cdots<\sigma_m\,.
$$
In case $m=1$ one should simply write $\sigma_0<\sigma_1$.
\par
The classical results for the case when $a(x)$ is bounded away
from zero in $\bar\Omega$ strongly suggest that the dynamics of
the positive solutions of (1.1) should be regulated by its
non-negative steady states, which are the non-negative solutions
of
$$
-\Delta u=\lambda u -a(x)f(x,u)u \quad \hbox{in }\;\Omega\,, \quad
u|_{\partial \Omega} =0\,.
\eqno (1.5)
$$
Under the assumptions above, any non-negative solution of (1.5)
lies in the Banach space
$$
U:=\{\,u\in C^{2+\mu}(\bar\Omega)\;:\;u|_{\partial\Omega}=0\,\}\,.
$$
Moreover, if $u\in U\setminus\{0\}$ is a non-negative solution of
(1.5), then it follows from the strong maximum principle that
$u(x) > 0$ for all $x\in \Omega$ and $\frac{\partial u}{\partial
n}(x) < 0$ for all $x\in \partial \Omega$, where $n$ is the
outward unit normal to $\Omega$ at $x$, i.e. $u$ lies in the
interior of the cone $U^+$ of non-negative functions of $U$.
Furthermore, for any $p>{N\over 2}$ and $u_0\in
U_0:=W^{2,p}_0(\Omega)$, $u_0\geq 0$, the parabolic problem (1.1)
has a unique global regular solution
$u_{[\lambda,a,\Omega]}(x,t;u_0)$ (cf. Proposition 24.9 of
\cite{DK92}). By a global regular solution we mean that
$$
u_{[\lambda,a,\Omega]}\in C(\bar\Omega\times[0,\infty))
\cap C^{2+\mu,1+{\mu\over 2}}(\bar\Omega\times (0,\infty))\,.
\eqno (1.6)
$$
Indeed, for any non-negative solution $u$ of (1.1) we have
$$
{\partial u \over \partial t}
-\Delta u=\lambda u -a(x)f(x,u)u \leq \lambda u\,,
$$
and hence
$$
u(x,t;u_0)\leq T(t)u_0\,,
$$
where $T(t)$ is the $L_p$-evolution operator associated with
$\Delta+\lambda$ under homogeneous Dirichlet boundary conditions.
Therefore, the solutions are global in time. If the initial data
has less regularity, e.g. $u_0\in C(\bar\Omega)$ instead of
$u_0\in U_0$, then (1.6) might fail. Nevertheless, by parabolic
regularity, for any $u_0\in C(\bar \Omega)$, $u_0\geq 0$, we have
$$
u_{[\lambda,a,\Omega]}\in C^{2+\mu,1+{\mu\over 2}}(\bar\Omega\times
(0,\infty))\,.
$$
Some pioneer results about the dynamics of the positive solutions
of (1.1) in the presence of refuges were found in \cite{FKLM96},
where assuming $m=1$ and $n_1=1$ it was shown that problem (1.5)
possesses a positive solution if, and only if,
$\sigma_0<\lambda<\sigma_1$ and that within this range of values
of the parameter $\lambda$ the unique positive solution of (1.5)
is a global attractor for the positive solutions of (1.1), whereas
the species is driven to extinction if $\lambda\leq \sigma_0$.
Some pioneer results about the existence of positive solutions of
(1.5) had already been given in \cite{BO86} and \cite{Ou92}.
\par
In \cite{GGLS98} problem (1.5) was analyzed under more restrictive
assumptions than those imposed here in. Precisely, it was assumed
that $m=n_1=1$, that $D:=\Omega\setminus \bar \Omega_{0,1}^1$
satisfies $\bar D \subset \Omega$, and that there exists $p>0$
such that
$$
f(x,u)=|u|^p\,,\quad (x,u)\in \bar\Omega\times {\Bbb R}\,.
\eqno (1.7)
$$
One of the many results found in \cite{GGLS98} shows that if
$u_\lambda$ stands for the maximal regular non-negative
steady-state of (1.5), then
$$
\lim_{\lambda\uparrow\sigma_1}u_\lambda = \infty \quad \hbox{uniformly in }
\bar \Omega_{0,1}^1\,,
$$
while in $D:=\Omega\setminus \bar \Omega_{0,1}^1$ the positive
solutions $u_\lambda$ stabilize as $\lambda\uparrow\sigma_1$ to a
large regular positive solution of the problem
$$\gathered
-\Delta u = \lambda u - a |u|^p u \quad \hbox{in }D\,, \cr
u=\infty \quad \hbox{on }\partial D\,, \endgathered
\eqno (1.8)
$$
for the value of the parameter $\lambda=\sigma_1$. This result is
of great interest by itself because it implies the existence of a
large solution in a problem where the weight function $a(x)$
vanishes on a component of the boundary, whereas most of the
previous results about the existence of large solutions had been
given in the simplest case when the nonlinearity is bounded away
from zero (cf. \cite{Ke57},
\cite{Os57}, \cite{BM91}, \cite{Ve92}, \cite{LM93},
\cite{LM94}, \cite{MV97}, and the references therein),
and in addition it shows how a uniform interior Harnack inequality
might fail when the nonlinearity vanishes in some region of the
support domain. It should be pointed out that the classical
interior estimates of \cite{Ke57} and \cite{Os57} can not be
applied straight away to show the existence of a large positive
solution of (1.8), since our nonlinearities change of sign, and
that this fact provokes the existence of some unimportant gaps in
some of the proofs given in references (e.g. the proof of Lemma
1.3 in \cite{MV97}). Nevertheless, the results of
\cite{LM93}, \cite{LM94} and \cite{GGLS98} can be used to
fill in these gaps, since the large solutions in balls provide us
with those interior estimates.
\par
Thanks to these results a rather natural question arises. How does
behave the population as times grows to infinity when $\lambda\geq
\sigma_1$? It can not approach to a regular positive solution, of
course. In fact, for any $u_0>0$ and $\varepsilon>0$ we have that
$$
u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\geq
u_{[\sigma_1-\varepsilon,a,\Omega]}(\cdot,t;u_0)\,,
$$
and hence
$$
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)
\geq u_{\sigma_1-\varepsilon}\,.
$$
On the other hand, thanks to Theorem 3.1 of \cite{GGLS98},
$$
\lim_{\varepsilon\downarrow 0}
u_{\sigma_1-\varepsilon}=\infty \quad \hbox{uniformly
in } \bar \Omega_{0,1}^1\,.
$$
Therefore, for each $\lambda\geq \sigma_1$ it follows that
$$
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)=\infty \quad
\hbox{uniformly in } \bar \Omega_{0,1}^1\,,
\eqno (1.9)
$$
and hence the population grows arbitrarily in the refuge as times
passes by. How does behave the population in the complement of the
refuge? The answer to this question in the very special case when
$m=1$, $n_1=1$, $\bar \Omega_{0,1}^1\subset \Omega$ and (1.7) are
satisfied was given very recently in \cite{DH99}, where it was
shown that the limiting population in $D$ as $t\uparrow\infty$
lies in between the minimal and the maximal large solution of the
following problem
$$\gathered
-\Delta u = \lambda u-a(x)u^{p+1} \quad \hbox{in }D\,, \cr
u=\infty \quad \hbox{on } \partial D\cap\Omega\,, \cr
u=0 \quad \hbox{on } \partial D \cap \partial\Omega\,. \endgathered
\eqno (1.10)
$$
For this special location of the refuge one can consider a general
boundary operator on $\partial D \cap \partial\Omega$ of mixed
tipe (cf. \cite{FKLM96} and \cite{DH99}). Note that in (1.10)
$a(x)$ is bounded away from zero in any compact subset of $D$.
\par
The case when the species possesses an arbitrarily large number of
refuges within its inhabiting region and (1.7) is satisfied has
been already analyzed in \cite{GL98} and \cite{G99}, where the
concept of {\it metasolution} was introduced in order to
characterize all the possible limiting profiles of the population
as $t\uparrow\infty$. Setting
$$
\Omega_k := \Omega\setminus \cup_{i=1}^k \cup_{j=1}^{n_i}
\bar \Omega_{0,j}^i\,, \quad 1\leq k \leq m\,,
$$
a function
$$
\Cal M :\Omega\to [0,\infty]
$$
is said to be a metasolution of order $k$ of (1.5) supported in
$\Omega_k$ if $\Cal M|_{\Omega_k}$ is a large regular solution of
$$\gathered
-\Delta u = \lambda u-a(x)u f(x,u) \quad \hbox{in }\Omega_k\,, \cr
u=\infty \quad \hbox{on } \partial \Omega_k \setminus \partial \Omega\,, \cr
u=0 \quad \hbox{on }\partial\Omega_k\cap \partial\Omega\,, \endgathered
\eqno (1.11)
$$
and
$$
\Cal M=\infty \quad \hbox{in }
(\Omega\setminus\Omega_k)\cup(\partial\Omega_k\setminus\partial\Omega)\,.
$$
In other words, a metasolution is the continuous extension by
infinity to the totality of $\Omega$ of a large solution of
(1.11). The main analytical result obtained in \cite{GL98} shows
that (1.5) possesses a metasolution of order $k$ supported in
$\Omega_k$ provided
$$
\sigma_k\leq \lambda <\sigma_{k+1}
$$
if $k\leq m-1$, and provided $\lambda\geq \sigma_m$ if $k=m$. This
result was obtained for the special choice (1.7), where the
interior estimates of \cite{GGLS98} work out.
\par
In this paper we obtain some substantial generalizations of all
these results to cover the general case when conditions (Ha1-3)
and (Hf) are satisfied, and in particular we complete the analysis
already begun in
\cite{GL98} where it was conjectured that the metasolutions
of order $k\leq m-1$ (resp $k=m$) should give the limiting profile
of the population as $t\uparrow\infty$ if $\sigma_k\leq
\lambda<\sigma_{k+1}$ (resp. $\lambda \geq \sigma_m$).
\par
The distribution of this paper is as follows. In Section 2 we
characterize the existence of regular positive solutions for (1.5)
and show their global asymptotic stability with respect to the
positive solutions of (1.1).
\par
In Section 3 we generalize the interior estimates of J. B. Keller
\cite{Ke57} and R. Osserman \cite{Os57} to cover the case when the
nonlinearity changes of sign. Some previous results for changing
sign nonlinearities were found by A. C. Lazer and P. J. McKenna in
\cite{LM94}. In the special case when $a$, $\lambda\in{\Bbb R}$
and the nonlinearity has the form
$$
h(u) := a u f(u)-\lambda u
\eqno (1.12)
$$
it was assumed in \cite{LM94} that $h\in C^1([f^{-1}({\lambda\over
a}),\infty)$ with $h'\geq 0$ and that in addition $h'(u)$ is
nondecreasing for $u$ large, and
$$
\liminf_{u\uparrow\infty}{h'(u)\over
\sqrt{\int_{f^{-1}({\lambda\over a})}^u h(z)\,dz}}>0\,.
\eqno (1.13)
$$
Instead of these conditions, for the special choice (1.12) we only
need to assume that there exists $u_*>f^{-1}({\lambda\over a})$
such that
$$
\int_{u_*}^\infty \left[ \int_{u_*}^u h(z)\,dz \right]^{-
{1\over 2}} du <\infty\,.
\eqno (1.14)
$$
Note that for the choice (1.7), (1.14) is satisfied for any $p>0$,
while (1.13) is only satisfied for $p\geq 2$. It should be pointed
out that if $f(u)$ satisfies (Hf), then
$$
\int_{f^{-1}({\lambda\over a})}^\infty
\left[ \int_{f^{-1}({\lambda\over a})}^u h(z)\,dz \right]^{-
{1\over 2}} du =\infty\,,
$$
and hence condition (2) of \cite{Ke57} fails. Therefore, the
interior estimates of \cite{Ke57} can not be applied straight away
to our problem.
\par
Once obtained these interior estimates, we shall use them to get
some very general results about the existence of large solutions
going to infinity on some of the components of the boundary where
$a(x)$ is bounded away from zero. These results are completely
new, since $a(x)$ can vanish on a finite number of interior
subdomains. This degenerate situation was dealt with by the first
time in \cite{GL98}. These results provide us with the first step
to obtain the existence of metasolutions of order $1\leq k \leq m$
by slightly modifying the domain $\Omega_k$ along the components
of its boundary where $a(x)$ vanishes, but this analysis will be
done in Section 5, not in Section 3.
\par
In Section 4 we use the results of Section 3 to characterize the
limiting behaviour of the regular positive solution as
$\lambda\uparrow\sigma_1$. It will be shown that the regular
positive solution is point-wise increasing towards the minimal
metasolution of order one of (1.5) supported in $\Omega_1$ (for
$\lambda=\sigma_1$).
\par
In Section 5 we use the theory of Section 3 to show that if $k\leq
m-1$, then (1.5) exhibits a metasolution of order $k$ supported in
$\Omega_k$ if, and only if, $\lambda<\sigma_{k+1}$, and that in
case $k=m$ (1.5) possesses a metasolution of order $m$ supported
in $\Omega_m=\Omega_+$ for each $\lambda\in{\Bbb R}$, as well as
to prove that the point-wise limit of the minimal metasolution of
order $k\leq m-1$ as $\lambda\uparrow\sigma_{k+1}$ provides us
with a metasolution of order $k+1$ supported in $\Omega_{k+1}$ for
the value of the parameter $\lambda=\sigma_{k+1}$.
\par
In Section 6 we combine the previous results with the parabolic
interior estimates obtained in \cite{Re82} and
\cite{Re86} to show that for any
$1\leq k\leq m-1$, $\lambda\in[\sigma_k,\sigma_{k+1})$, and
$u_0\in U_0$, $u_0>0$, the restriction of the orbit of
$u_{[\lambda,a,\Omega]}(\cdot,t;u_0)$, $t\geq 0$, to any compact
subset $K$ of $\Omega_k$ is relatively compact in $C^2(K)$ with
its $\omega$-limit set contained in the closed interval whose ends
are the minimal and the maximal regular positive solutions of
(1.11), whereas
$$
\lim_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)=\infty\quad
\hbox{uniformly in }\cup_{i=1}^k \cup_{j=1}^{n_i}
\bar \Omega_{0,j}^i\,.
$$
Therefore, if (1.11) possesses a unique regular solution, say
$\theta$, then $u_{[\lambda,a,\Omega]}(\cdot,t;u_0)$ stabilizes to
the metasolution defined by $\theta$ as $t\uparrow\infty$.
\par
Most of the available results about the uniqueness of large
solutions were obtained in domains where $a(x)$ is bounded away
from zero and for rather special classes of nonlinearities (cf.
\cite{BM91}, \cite{LM93},
\cite{LM94}, \cite{MV97}, and the references there in).
One of the main results of
\cite{DH99} establishes that if $m=1$, $n_1=1$,
$\bar \Omega_{0,1}^1\subset \Omega$, and there exist positive
constants $\alpha$ and $c$ such that
$$
\lim_{d(x)\downarrow 0}{a(x)\over d^\alpha(x)}=c\,, \quad
d(x):=\hbox{dist}(x,\partial\Omega_{0,1}^1)\,,
$$
then (1.10) possesses a unique regular solution. The idea of the
proof consists in combining an extension of Theorem I in
\cite{Ke57} with a very well known uniqueness device
coming from \cite{BO86}. This seems to be the sole uniqueness
result for the case when the coefficient is not bounded away from
zero in the totality of $\Omega$. We shall analyze the uniqueness
problem in a forthcoming paper.
\par
It should be pointed out that in order to prove most of the
results of this paper we need assuming that in any compact subset
$K$ of $\Omega_+$ the following estimate holds
$$
a(x) u f(x,u)-\lambda u \geq h_K(u)
$$
for some function $h_K(u)$ satisfying (1.14). This assumption is
far from being of technical nature. In fact, if it fails one would
need to impose some additional condition on the size of the domain
to get the existence of large solutions even in the most simple
situations, but this analysis is out of the scope of this work.
\par
The results contained in this paper were communicated in a talk
given by the author in Miami during the celebration of the
Conference honoring Alan C Lazer (January/9th/1999). The paper was
concluded and sent to the journal on August 1999.
\head
2. Regular positive steady states.
\endhead
\noindent In this section we characterize the existence,
prove the uniqueness and analyze the point-wise growth of the
regular positive solutions of (1.5), as well as their global
attractive character with respect to the positive solutions of
(1.1).
\par
In the sequel, given a function $w\in C(\bar\Omega)$ we say that
$w>0$ if $w\geq 0$ and $w\neq 0$. Given $w \in U$ we say that
$w\gg 0$ if $w$ lies in the interior of $U^+$.
\proclaim{Theorem 2.1} Assume $\hbox{\rm (Ha1-3)}$ and
$\hbox{\rm (Hf)}$. Then, the following assertions are true:
\smallskip
\roster
\item "(i)" The problem $(1.5)$ possesses a
regular positive solution if, and only if,
$$
\sigma_0 < \lambda < \sigma_1\,.
\eqno (2.1)
$$
Moreover, it is unique if it exists. In the sequel it will be
denoted by $\theta_{[\lambda,a,\Omega]}$.
\smallskip
\item"(ii)" The map
$$
\matrix (\sigma_0,\sigma_1)& \longmapsto & U \cr
\lambda & \to & \theta_{[\lambda,a,\Omega]} \endmatrix
$$
is point-wise increasing and differentiable. Moreover,
$\partial_\lambda \theta_{[\lambda,a,\Omega]}\in U^+$.
\smallskip
\item"(iii)" Suppose $(2.1)$. Then,
$$
\lim_{\lambda \downarrow \sigma_0}
\|\theta_{[\lambda,a,\Omega]}\|_{C(\bar\Omega)} = 0\,,
\eqno (2.2)
$$
and
$$
\lim_{\lambda \uparrow \sigma_1}\theta_{[\lambda,a,\Omega]} =\infty
\quad \hbox{\rm uniformly in }\;
\cup_{j=1}^{n_1}\bar \Omega^1_{0,j}\setminus \partial \Omega\,.
\eqno (2.3)
$$
\endroster
\endproclaim
\demo{Proof} (i) Let $u$ be a regular
positive solution of (1.5). Then, $u|_{\partial\Omega}=0$ and
$$
(-\Delta + a f(\cdot,u))u=\lambda u\,.
$$
Hence, $u$ is a positive eigenfunction associated with the
eigenvalue $\lambda$ of the operator $-\Delta+a f(\cdot,u_0)$
under homogeneous Dirichlet boundary conditions. Thus, by the
uniqueness of the principal eigenvalue (e.g. \cite{Am76}),
$$
\lambda = \sigma^\Omega[-\Delta+a f(\cdot,u)]\,,
\eqno (2.4)
$$
and hence, by the monotonicity of the principal eigenvalue with
respect to the potential
$$
\lambda >\sigma^\Omega[-\Delta]=\sigma_0\,,
$$
because $a>0$, $u\gg 0$ and $f(x,u(x))>0$ for all $x\in\Omega$.
Moreover, since $a=0$ in $\Omega^1_{0,1}$, we find from (2.4) that
$$
\lambda=\sigma_1^\Omega[-\Delta+a f(\cdot,u)]<
\sigma^{\Omega^1_{0,1}}=\sigma_1\,,
$$
by the monotonicity of the principal eigenvalue with respect to
the domain. Therefore, (2.1) is necessary for the existence of a
regular positive solution of (1.5). In order to show that
condition (2.1) is sufficient for the existence of a regular
positive solution we use the method of sub and supersolutions.
Suppose (2.1) and let $\varphi$ denote the principal eigenfunction
associated with $\sigma_0$. Then, thanks to (Hf), for any
$\varepsilon>0$ sufficiently small the function
$\varepsilon\varphi$ provides us with a positive subsolution of
(1.5). To complete the proof of the existence it remains to show
the existence of a supersolution $\bar u$ of (1.5) such that
$\varepsilon \varphi \leq \bar u$. For any $\delta>0$ sufficiently
small we consider the $\delta$-neighborhood of $\Omega_0$ in
$\Omega$
$$
\Omega_0^\delta := \{\,x\in \Omega\,:\;\hbox{dist}(x,\Omega_0)<\delta\,\}\,.
$$
Then,
$$
\Omega_0^\delta = \cup_{i=1}^m \cup_{j=1}^{n_i} \Omega_{0,j}^{i,\delta}\,,
\eqno (2.5)
$$
where
$$
\Omega_{0,j}^{i,\delta} := \{\,x\in \Omega \,:\;
\hbox{dist}(x,\Omega_{0,j}^i)<\delta\,\}\,, \quad
1\leq i \leq m\,,\quad 1\leq j \leq n_i\,.
$$
Thanks to (Ha3), $\bar \Omega^i_{0,j} \cap \bar \Omega^{\hat
i}_{0,\hat j} = \emptyset$ if $(i,j)\neq (\hat i,\hat j)$, and
hence $\delta >0$ can be chosen sufficiently small so that
$$
\bar \Omega^{i,\delta}_{0,j} \cap \bar \Omega^{\hat i,\delta}_{0,\hat j}
= \emptyset \quad \hbox{if }\; (i,j)\neq (\hat i,\hat j)\,.
\eqno (2.6)
$$
In the sequel we shall assume that $\delta>0$ has been chosen in
this way.
\par
Since $a>0$, and hence $\Omega_+\neq \emptyset$, for each $1\leq i
\leq m$ and $1\leq j \leq n_i$ the component $\Omega^i_{0,j}$ is a
proper subdomain of $\Omega$. Thus, $\partial\Omega_{0,j}^i \cap
\Omega \neq \emptyset$, and so $\Omega_{0,j}^i$ is a proper
subdomain of $\Omega_{0,j}^{i,\delta/2}$ and
$\Omega_{0,j}^{i,\delta/2}$ is a proper subdomain of
$\Omega_{0,j}^{i,\delta}$. Therefore, by the monotonicity of the
principal eigenvalue with respect to the domain, we find that
$$
\sigma^{\Omega_{0,j}^{i,\delta}}<\sigma^{\Omega_{0,j}^{i,\delta/2}}<
\sigma^{\Omega_{0,j}^i}\,, \quad 1\leq i \leq m\,,\;\;
1 \leq j \leq n_i\,.
\eqno (2.7)
$$
On the other hand, thanks to (Ha1), if $\Gamma_0$ and $\Gamma$ are
components of $\partial \Omega_0$ and $\partial \Omega$,
respectively, such that $\Gamma_0\cap \Gamma\neq \emptyset$, then
$\Gamma_0=\Gamma$. Thus, it follows from (2.5) and (2.6) that
$$
\partial \Omega_0^\delta \setminus \partial \Omega \subset \Omega_+\,.
\eqno (2.8)
$$
Moreover, for each $1\leq i \leq m$ and $1\leq j \leq n_i$ we have
that
$$
\lim_{\delta\downarrow 0} \Omega_{0,j}^{i,\delta}=\Omega_{0,j}^i\,,
$$
e.g. in the sense of Definition 4.1 of \cite{Lo96}. Thus, it
follows from Theorem 4.2 of \cite{Lo96} that
$$
\lim_{\delta \downarrow 0} \sigma^{\Omega_{0,j}^{i,\delta}}= \sigma^{\Omega_{0,j}^i}\,,
\quad 1\leq i \leq m\,, \quad 1\leq j \leq n_i\,.
\eqno (2.9)
$$
Therefore, thanks to (1.3), (2.1), (2.7) and (2.9), we find that
for each $\delta>0$ sufficiently small
$$
\lambda < \sigma^{\Omega_{0,j}^{1,\delta}}<\sigma^{\Omega_{0,\hat j}^{i,\delta}}\,,
\quad 1\leq j \leq n_1\,, \;\; 2\leq i \leq m\,,\;\;
1\leq \hat j \leq n_i\,.
\eqno (2.10)
$$
Now, for any $1\leq i \leq m$ and $1\leq j \leq n_i$ let
$\varphi_{0,j}^{i,\delta}\gg 0$ denote the principal eigenfunction
associated with $\sigma^{\Omega_{0,j}^{i,\delta}}$, and consider
the function
$$
\Phi(x):= \left\{ \matrix \varphi_{0,j}^{i,\delta}(x) \quad
\hbox{if }\; x\in \Omega_{0,j}^{i,\delta/2} \;\;\hbox{for some }\;
1\leq i \leq m\,, \;\; 1\leq j \leq n_i\,,\cr \cr
\psi(x) \quad \;\;\; \hbox{if } x \in \bar \Omega
\setminus \Omega_0^{\delta/2}\,, \hfill \cr \endmatrix \right.
\eqno (2.11)
$$
where $\psi(x)$ is any regular extension of the functions
$\varphi_{0,j}^{i,\delta}$ outside $\Omega_0^{\delta/2}$. Thanks
to (2.8), we can assume that $\psi$ is positive and bounded away
from zero in $\bar\Omega\setminus\Omega_0^{\delta/2}$.
\par
We claim that if $\kappa >1$ is sufficiently large, then the
function
$$
\bar u:=\kappa \Phi
$$
is a supersolution of (1.5) satisfying $\varepsilon \varphi \leq
\bar u$ for $\varepsilon>0$ small enough. Indeed, in the set
$$
\bar\Omega \setminus \Omega_0^{\delta/2} \subset \bar \Omega_+
$$
the function $\psi$ is positive and bounded away from zero, as
well as $a(x)$, by (Ha1). Thus, for $\kappa$ large
$$
-\Delta \psi \geq \lambda\,\psi - a\psi f(\cdot,\kappa\psi) \quad
\hbox{in }\bar \Omega \setminus \Omega_0^{\delta/2}\,,
$$
since for each $x\in\bar\Omega\setminus\Omega_0^{\delta/2}$
$$
f(x,\kappa\psi(x))\geq
f(x,\kappa\inf_{\bar\Omega\setminus\Omega_0^{\delta/2}}\psi)
$$
and
$$
\lim_{\kappa\uparrow\infty}f(\cdot,\kappa
\inf_{\bar\Omega\setminus\Omega_0^{\delta/2}}
\psi)=\infty
$$
uniformly in $\bar\Omega\setminus\Omega_0^{\delta/2}$. Moreover,
thanks to (2.10), in each of the components of
$\Omega_0^{\delta/2}$, $\Omega_{0,j}^{i,\delta/2}$, $1\leq i \leq
m$, $1\leq j \leq n_i$, we have that for any $\kappa >0$
$$
-\Delta (\kappa \Phi)=\kappa\,\sigma^{\Omega_{0,j}^{i,\delta}}
\varphi_{0,j}^{i,\delta}
> \lambda\,\kappa\, \varphi_{0,j}^{i,\delta}\geq \lambda\,\kappa\,
\varphi_{0,j}^{i,\delta}
-a \,\kappa\,\varphi_{0,j}^{i,\delta} f(\cdot,\kappa\,
\varphi_{0,j}^{i,\delta})\,.
$$
Therefore, $k\Phi$ provides us with a positive supersolution of
(1.5) if $\kappa$ is sufficiently large.
\par
Let $\Gamma$ be a component of $\partial\Omega$. If $\Gamma$ is a
component of $\Omega_+$, then by the construction itself we have
that $\Phi$ is positive and bounded away from zero on $\Gamma$,
while if $\Gamma \cap \partial \Omega_+ =\emptyset$, then $\Phi=0$
on $\Gamma$ and ${\partial \Phi \over \partial n}(x)<0$ for all
$x\in \Gamma$, where $n$ is the outward unit normal. Therefore,
$\varepsilon\varphi \leq \kappa \Phi$ provided $\varepsilon>0$ is
sufficiently small. This shows that (2.1) is sufficient for the
existence of a regular positive solution of (1.5).
\par
We now show the uniqueness of the positive solution. Suppose (2.1)
and let $u$, $v$ be two positive solutions of (1.5), $u\neq v$.
Then,
$$
(-\Delta+a g -\lambda)(u-v)=0\quad \hbox{in }\Omega\,,\quad
u-v=0 \quad \hbox{on }\partial\Omega\,,
\eqno (2.12)
$$
where
$$
g(x):=\left\{ \matrix {u(x)f(x,u(x))-v(x)f(x,v(x))\over
u(x)-v(x)} &\quad \hbox{if }u(x)\neq v(x)\,, \cr &\cr
f(x,u(x))\hfill & \quad \hbox{if }u(x)= v(x)\,,
\endmatrix \right. \quad x\in \bar \Omega\,.
$$
By the monotonicity of $f$ on its second argument, it is easily
seen that
$$
g > f(\cdot,u)\quad \hbox{in }\Omega\,,
$$
since $u\neq v$. Thus, it follows from (2.4) and the monotonicity
of the principal eigenvalue with respect to the potential that
$$
\sigma^\Omega[-\Delta+a\,g -\lambda]\geq
\sigma^\Omega[-\Delta+a f(\cdot,u) -\lambda]=0\,.
$$
Note that it might happen $g=f(\cdot,u)$ in $\Omega_+$, and hence
in the previous inequality we should not substitute $\geq$ by $>$
without some additional work. Assume
$$
\sigma^\Omega[-\Delta+a\,g -\lambda]> 0\,.
$$
Then, zero can not be an eigenvalue of $-\Delta+a\,g-\lambda$, and
hence we find from (2.12) that $u=v$, which is impossible. Thus,
$$
\sigma^\Omega[-\Delta+a\,g -\lambda]=0\,.
$$
Moreover, $u\neq v$. Hence, it follows from (2.12) that there
exists $\kappa \in{\Bbb R}\setminus\{0\}$ such that
$$
u-v=\kappa \varphi\,,
$$
where $\varphi\gg 0$ stands for the principal eigenfunction
associated with $\sigma^\Omega[-\Delta+a\,g-\lambda]=0$.
Therefore, either $u(x)v(x)$
for all $x\in\Omega$. In any of these situations we have that $g >
f(\cdot,u)$ in $\Omega_+$ and therefore
$$
\sigma^\Omega[-\Delta+a\,g-\lambda]>0\,,
$$
which implies $u=v$. This contradiction shows the uniqueness and
concludes the proof of Part (i).
\par
(ii) Consider the operator $\Cal F :{\Bbb R}\times U^+\to
C^\mu(\bar\Omega)$ defined by
$$
\Cal F(\lambda,u):=-\Delta u -\lambda u + a u F(\cdot,u)\,,\quad
(\lambda,u)\in{\Bbb R}\times U^+\,,
$$
where $F$ stands for the substitution operator induced by $f$, and
pick up $\lambda\in (\sigma_0,\sigma_1)$. Then, $\Cal F$ is an
operator of class $C^1$ in the interior of $U^+$ such that
$$
\Cal F(\lambda,\theta_{[\lambda,a,\Omega]})=0\,,
$$
and
$$
D_u \Cal F(\lambda,\theta_{[\lambda,a,\Omega]})= -\Delta -\lambda +
a f(\cdot,\theta_{[\lambda,a,\Omega]})+a \theta_{[\lambda,a,\Omega]}
\partial_u f(\cdot,\theta_{[\lambda,a,\Omega]})\,.
$$
By (Hf), (2.4) and the monotonicity of the principal eigenvalue
with respect to the potential we find that
$$
\sigma^\Omega[ D_u \Cal F(\lambda,\theta_{[\lambda,a,\Omega]})]>
\sigma^\Omega[-\Delta -\lambda + a f(\cdot,\theta_{[\lambda,a,\Omega]})]=0\,.
\eqno (2.13)
$$
Therefore, $D_u\Cal F(\lambda,\theta_{[\lambda,a,\Omega]})$ is an
isomorphism and it follows from the implicit function theorem that
there exist $\varepsilon>0$ and a map of class $C^1$,
$u:(\lambda-\varepsilon,\lambda+\varepsilon)\to U^+$, such that
$u(\lambda)=\theta_{[\lambda,a,\Omega]}$ and for each $s\in
(\lambda-\varepsilon,\lambda+\varepsilon)$
$$
\Cal F(s,u(s))=0\,.
$$
Moreover, those are the unique zeroes of $\Cal F$ in a
neighborhood of $(\lambda,\theta_{[\lambda,a,\Omega]})$ in ${\Bbb
R}\times U^+$, and $u(s)\gg 0$, since $u(\lambda)\gg 0$. Thus,
thanks to the uniqueness of the positive solution,
$$
u(s)=\theta_{[s,a,\Omega]}\,,\quad s \simeq \lambda\,.
$$
Furthermore, by implicit differentiation we find that
$$
D_u \Cal F(\lambda,\theta_{[\lambda,a,\Omega]})
\partial_\lambda \theta_{[\lambda,a,\Omega]} = \theta_{[\lambda,a,\Omega]}\gg 0\,.
\eqno (2.14)
$$
Thanks to (2.13) the differential operator on the left hand side
of (2.14) satisfies the strong maximum principle. Therefore,
(2.14) implies
$$
\partial_\lambda \theta_{[\lambda,a,\Omega]}\gg 0\,.
$$
This completes the proof of Part (ii).
\par
(iii) Relation (2.2) follows from the uniqueness of the positive
solution taking into account that thanks to the main theorem of
\cite{CR71} $\lambda=\sigma_0$ is a bifurcation value to positive
solutions of (1.5) from $u=0$. We now show (2.3). Set
$$
\Omega_0^1 := \cup_{j=1}^{n_1} \Omega_{0,j}^1\,.
$$
Since $a=0$ in $\Omega_0^1$, (2.14) gives
$$
(-\Delta-\lambda) \partial_\lambda \theta_{[\lambda,a,\Omega]}
= \theta_{[\lambda,a,\Omega]}
\quad \hbox{in }\; \Omega_0^1\,,\quad \lambda\in(\sigma_0,\sigma_1)\,.
$$
Now, pick $ \hat \lambda\in(\sigma_0,\sigma_1)$ and consider
$c>0$ such that for each $1\leq j \leq n_1$,
$$
\theta_{[\hat\lambda,a,\Omega]} > c \varphi_{0,j}^1
\quad \hbox{in }\Omega_{0,j}^1\,.
$$
Recall that $\varphi_{0,j}^1$ is the principal eigenfunction
associated with $\sigma^{\Omega_{0,j}^1}$. Thanks to Part (ii),
for each $\lambda\in(\hat\lambda,\sigma_1)$ we have that
$$
\theta_{[\lambda,a,\Omega]} >\theta_{[\hat\lambda,a,\Omega]}
> c \varphi_{0,j}^1
\quad \hbox{in }\Omega_{0,j}^1\,,\quad 1\leq j \leq n_1\,.
$$
Moreover, for each $\lambda\in(\hat\lambda,\sigma_1)$ and $1\leq
j \leq n_1$ the operator $-\Delta-\lambda$ satisfies the strong
maximum principle in $\Omega_{0,j}^1$. Hence,
$$
\partial_\lambda \theta_{[\lambda,a,\Omega]}> c\,
(-\Delta-\lambda)^{-1}\varphi_{0,j}^1 =
{c\over \sigma_1-\lambda}\varphi_{0,j}^1 \quad
\hbox{in }\;\Omega_{0,j}^1\,.
$$
Note that, thanks to (Ha3), $\sigma_1=\sigma^{\Omega_{0,j}^1}$ for
each $1\leq j \leq n_1$. On the other hand, for each $1\leq j \leq
n_1$, the function $\varphi_{0,j}^1$ is bounded away from zero on
any compact subset of $\Omega^1_{0,j}$. Thus,
$$
\lim_{\lambda\uparrow\sigma_1}\partial_\lambda \theta_{[\lambda,a,\Omega]}= \infty
\quad \hbox{uniformly in compact subsets of }\; \Omega_0^1\,,
$$
and therefore
$$
\lim_{\lambda\uparrow\sigma_1} \theta_{[\lambda,a,\Omega]}= \infty
\quad \hbox{uniformly in compact subsets of }\;\Omega_0^1\,.
$$
It remains to show that
$$
\lim_{\lambda\uparrow\sigma_1} \theta_{[\lambda,a,\Omega]}(x)= \infty
\quad \hbox{for all } x\in \partial\Omega_0^1\setminus\partial\Omega\,.
\eqno (2.15)
$$
For this, consider $\delta >0$ sufficiently small, pick $\lambda$
satisfying
$$
\sigma^{\Omega_{0,j}^{1,\delta}}<\sigma^{\Omega_{0,j}^{1,\delta/2}}<\lambda<
\sigma^{\Omega_{0,j}^1}=\sigma_1\,, \quad 1\leq j \leq n_1\,,
$$
and introduce the function $u_\delta\in C(\bar\Omega)$ defined by
$$
u_\delta(x):=\left\{ \matrix C\,\varphi_{0,j}^{1,\delta}(x) \quad
& \hbox{if }\; x\in \bar\Omega_{0,j}^{1,\delta/2} \;\hbox{for some }\;
1\leq j \leq n_1\,,\cr & \cr
0 \quad \hfill & \hbox{if } x \in \bar\Omega \setminus
\cup_{j=1}^{n_1} \Omega_{0,j}^{1,\delta/2}\,, \hfill \cr \endmatrix \right.
\eqno (2.16)
$$
where $C>0$ is a positive constant. Then, the argument of the
proof of Theorem 4.3 in \cite{LS98} can be easily adapted to show
that under condition (Ha2) there exists $C=C(\delta)>0$ such that
$u_\delta$ is a subsolution of (1.5) satisfying
$$
\lim_{\delta\downarrow 0}u_\delta(x)=\infty
$$
for each $x\in \partial\Omega_0^1\setminus\partial\Omega$. By the
uniqueness of the positive solution, necessarily $u_\delta \leq
\theta_{[\lambda,a,\Omega]}$ and hence, (2.15) is satisfied. The
uniform divergence in $\partial \Omega_0^1$ follows from the
point-wise monotonicity in $\lambda$ as an immediate consequence
from Dini's theorem. This concludes the proof of the theorem.
\qed
\enddemo
\noindent We shall say that a non-negative positive steady-state
$u$ of (1.1), i.e. a non-negative solution of (1.5), is {\it
globally asymptotically stable} if
$$
\lim_{t\uparrow\infty}\|u_{[\lambda,a,\Omega]}(\cdot,t;u_0)-u\|_{C(\bar\Omega)}=0
$$
for each $u_0\in U_0$, $u_0>0$. The main result on the longtime
behaviour of the positive solutions of (1.1) reads as follows.
\proclaim{Theorem 2.2} Under the assumptions of Theorem 2.1,
the following assertions are true:
\smallskip
\roster
\item"(i)" If $\lambda\leq \sigma_0$, then $u=0$ is globally
asymptotically stable.
\smallskip
\item"(ii)" If $\sigma_0<\lambda<\sigma_1$, then $\theta_{[\lambda,a,\Omega]}$ is
globally asymptotically stable.
\smallskip
\item"(iii)" Set
$$
\Omega_0^i := \cup_{j=1}^{n_i}\Omega_{0,j}^i\,, \quad
1\leq i \leq m\,,
\eqno (2.17)
$$
and assume $\sigma_i \leq \lambda$ for some $1\leq i \leq m$.
Then, for any $u_0\in U_0$ with $u_0>0$ we have
$$
\lim_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)=\infty \quad
\hbox{\rm uniformly in } \cup_{k=1}^i \bar \Omega_0^k
\setminus\partial\Omega\,.
\eqno (2.18)
$$
\endroster
\endproclaim
\demo{Proof} Recall that in the proof of Theorem 2.1(i) we have
shown that (1.5) possesses arbitrarily large supersolutions in the
interior of $U^+$ for any $\lambda < \sigma_1$.
\par
(i) Let $u_0\in U_0$ such that $u_0>0$ and consider the solution
$u(x,t;u_0)$ of (1.1). By the parabolic maximum principle for any
$t>0$ the function $u(\cdot,t;u_0)$ lies in the interior of $U^+$.
Fix $t_1>0$ and consider a supersolution $\bar u \gg 0$ of (1.5)
such that
$$
u(\cdot,t_1;u_0)\ll \bar u\,.
$$
Then, for all $t\geq t_1$ we have that
$$
u(\cdot,t;u_0)\ll u(\cdot,t-t_1;\bar u)\,.
\eqno (2.19)
$$
Thanks to the results of \cite{Sa73}, the function $t\to
u(\cdot,t-t_1;\bar u)$ is decreasing and converges to a
non-negative solution of (1.5). Due to Theorem 2.1(i) it must
converge to zero. Therefore, (2.19) completes the proof of Part
(i).
\par
(ii) Let $u_0\in U_0$ be such that $u_0>0$ and pick $t_1>0$.
Since $\sigma_0<\lambda<\sigma_1$, there exist a subsolution
$\underline u$ and a supersolution $\bar u$ of (1.5), both in the
interior of $U^+$, such that
$$
\underline u \ll u(\cdot,t_1;u_0)\ll \bar u\,.
$$
Then, for all $t\geq t_1$ we have that
$$
u(\cdot,t-t_1;\underline u)\ll u(\cdot,t;u_0)\ll
u(\cdot,t-t_1;\bar u)\,.
\eqno (2.20)
$$
Thanks to the results of \cite{Sa73}, the function $t\to
u(\cdot,t-t_1;\bar u)$ is decreasing and converges to a
non-negative solution of (1.5), whereas $t\to
u(\cdot,t-t_1;\underline u)$ is increasing and converges to a
positive solution of (1.5). By the uniqueness of the positive
solution, both functions must converge to
$\theta_{[\lambda,a,\Omega]}$. Combining these features with
(2.20) concludes the proof of this part.
\par
(iii) Assume that $\lambda\geq \sigma_i$ for some $1\leq i \leq m$
and consider $u_0\in U_0$ with $u_0>0$. Let
$u=u_{[\lambda,a,\Omega]}(x,t;u_0)$ be the unique global regular
solution of (1.1). For all $\varepsilon>0$ we have
$\lambda>\sigma_1-\varepsilon$, and hence
$$
\partial_t u -\Delta u =\lambda u-auf(x,u)>
(\sigma_1-\varepsilon)u-auf(x,u)\,.
$$
Thus, for each $t>0$
$$
u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\gg
u_{[\sigma_1-\varepsilon,a,\Omega]}(\cdot,t;u_0)\,.
\eqno (2.21)
$$
By Part (ii), we already know that
$$
\lim_{t\uparrow\infty}\|u_{[\sigma_1-\varepsilon,a,\Omega]}(\cdot,t;u_0)-
\theta_{[\lambda,a,\Omega]}\|_{C(\bar\Omega)}=0\,.
$$
Hence, it follows from (2.21) that for each $\varepsilon>0$
$$
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\geq
\theta_{[\sigma_1-\varepsilon,a,\Omega]}\,.
\eqno (2.22)
$$
Thanks to Theorem 2.1(iii),
$$
\lim_{\varepsilon \downarrow 0}\theta_{[\sigma_1-\varepsilon,a,\Omega]} =\infty
\quad \hbox{\rm uniformly in }\;
\bar \Omega^1_0\setminus \partial \Omega\,.
$$
Therefore, we find from (2.22) that
$$
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)=\infty
\quad \hbox{\rm uniformly in }\;
\bar \Omega^1_0\setminus \partial \Omega\,.
$$
This completes the proof if $i=1$.
\par
Assume $i\geq 2$ and
$$
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)=\infty
\quad \hbox{\rm uniformly in }\;\cup_{k=1}^{i-1}
\bar \Omega^k_0\setminus \partial \Omega\,.
$$
Let $\hat a\in C^\mu(\bar\Omega)$ any weight function such that
$\hat a > a$ and $\hat a(x)>0$ if, and only if, $x\in \Omega_+\cup
\cup_{k=1}^{i-1} \bar \Omega_0^k$. Then, the lower order refuges
of $\hat a$ are $\Omega^i_{0,j}$, $1\leq j\leq n_i$. Recall that
we are assuming
$$
\lambda \geq \sigma_i=\sigma^{\Omega_{0,1}^i}\,.
$$
Since $\hat a > a$, we have
$$
u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\gg u_{[\lambda,\hat a,\Omega]}(\cdot,t;u_0)\,.
$$
Moreover, by the corresponding result for the case $i=1$,
$$
\liminf_{t\uparrow\infty}u_{[\lambda,\hat a,\Omega]}(\cdot,t;u_0)=\infty
\quad \hbox{\rm uniformly in }\;\bar \Omega^i_0\setminus \partial \Omega\,.
$$
This completes the proof of the theorem.
\qed
\enddemo
\medskip
\head
3. A priori interior estimates and the existence of large regular
solutions.
\endhead
Beside their own interest, the following results are crucial to
show the stabilization in $\Omega\setminus \bar \Omega_0^1$ of the
regular positive solutions of (1.5) as $\lambda \uparrow
\sigma_1$. They also show the existence of large regular solutions
of
$$
-\Delta u =\lambda u - a u f(\cdot,u)
$$
in $\Omega$. By a large regular solution we mean a solution of
class $C^{2+\mu}$ which grows arbitrarily when the spatial
variable approaches to some of the components of $\partial\Omega$.
Those solutions will provide us with the limiting behaviour of the
population as times grows to infinity for any
$\lambda\geq\sigma_1$.
\proclaim{Theorem 3.1} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and
$$
\partial \Omega\cap\partial\Omega_+\neq \emptyset\,.
$$
Let $\Gamma^j_+$, $1\leq j \leq q$, be $q$ arbitrary components of
$\partial \Omega\cap\partial\Omega_+$, and $\alpha_j >0$, $1\leq j
\leq q$, $q$ arbitrary constants, and consider the nonlinear
boundary value problem
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in
}\;\Omega\,, \cr
u|_{\Gamma_+^j} =\alpha_j>0\,, \quad 1\leq j\leq q\,, \cr
u=0 \quad \hbox{\rm on } \partial\Omega\setminus
\cup_{j=1}^q\Gamma_+^j\,. \hfill \endgathered
\eqno (3.1)
$$
The following assertions are true:
\smallskip
\roster
\item "(i)" Problem $(3.1)$ possesses a
regular positive solution if, and only if, $\lambda < \sigma_1$.
Moreover, it is unique if it exists. In the sequel we shall denote
it by $\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$ where
$$
\Gamma_+:=(\Gamma_+^1,\dots,\Gamma_+^q)\,,\quad
\alpha := (\alpha_1,\dots,\alpha_q)\,.
$$
\smallskip
\item "(ii)" The map $(-\infty,\sigma_1)\to C(\bar \Omega)$,
$\lambda \to \Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$ is
point-wise increasing, as well as the map $(0,\infty)^q\to C(\bar
\Omega)$, $\alpha=(\alpha_1,\dots,\alpha_q) \to
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$. Moreover,
$$
\lim_{\lambda \uparrow \sigma_1}
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]} =\infty
\quad \hbox{\rm uniformly in }\;
\cup_{j=1}^{n_1}\bar\Omega^1_{0,j}\setminus \partial \Omega\,.
\eqno (3.2)
$$
\endroster
\endproclaim
\demo{Proof} (i) Suppose (3.1) possesses a regular
positive solution, say $u$. Then,
$$
(-\Delta+af(\cdot,u)-\lambda)u=0 \quad \hbox{in }\;\Omega\,,
$$
and $u|_{\partial\Omega}>0$. Thus, $u$ provides us with a positive
strict supersolution of $-\Delta+a f(\cdot,u)-\lambda$ in $\Omega$
under homogeneous Dirichlet boundary conditions. Hence, it follows
from the characterization of the strong maximum principle found in
\cite{LM94} (cf. \cite{Lo96} and \cite{AL98}) that
$$
\sigma^\Omega[-\Delta+a f(\cdot,u)-\lambda]>0\,.
\eqno (3.3)
$$
Therefore,
$$
\lambda<\sigma^\Omega[-\Delta+a f(\cdot,u)]<\sigma^{\Omega_{0,1}^1}=\sigma_1\,,
$$
since $a=0$ in $\Omega_{0,1}^1$.
\par
Suppose $\lambda<\sigma_1$. To show that (3.1) possesses a regular
positive solution we use the method of sub and supersolutions. The
function $\underline u:=0$ provides us with a subsolution of
(3.1). Moreover, we can use the same supersolutions constructed in
the proof of Theorem 2.1. Indeed, to show that $\kappa \Phi$
provides us with a positive supersolution of (3.1), where $\Phi$
is the function defined in (2.11), it remains to check that for
any $\kappa$ sufficiently large
$$
\kappa \Phi|_{\Gamma_+^j}>\alpha_j\,,\quad 1\leq j \leq q\,.
\eqno (3.4)
$$
By construction,
$$
\cup_{j=1}^q \Gamma_+^j \subset \bar \Omega\setminus \Omega_0^{\delta/2}\,,
$$
and hence
$$
\kappa \Phi|_{\Gamma_+^j}= \kappa \psi|_{\Gamma_+^j}\,,\quad 1\leq j \leq q\,.
$$
Moreover, $\psi$ is positive and bounded away from zero in $\bar
\Omega\setminus \Omega_0^{\delta/2}$. Therefore, (3.4) holds
provided $\kappa$ is large enough. This concludes the proof of the
existence.
\par
The uniqueness of the regular positive solution follows from (3.3)
with the same argument used to show the uniqueness in the proof of
Theorem 2.1. This completes the proof of Part (i).
\par
(ii) Let $\lambda_1$, $\lambda_2 \in (\sigma_0,\sigma_1)$ such
that $\lambda_1 < \lambda_2$. Then, setting
$$
\Theta_i:=\Theta_{[\lambda_i,a,\Omega,\Gamma_+,\alpha]}\,, \quad i=1\,,\,2\,,
$$
we find from their definition that
$$
(-\Delta+a \, g -\lambda_1)(\Theta_2-\Theta_1)>0
\quad \hbox{in }\Omega\,,\quad (\Theta_2-\Theta_1)|_{\partial\Omega}=0\,.
\eqno (3.5)
$$
where
$$
g(x):=\cases \dfrac{\Theta_2(x)f(x,\Theta_2(x))-\Theta_1(x)f(x,\Theta_1(x))}
{\Theta_2(x)-\Theta_1(x)} & \hbox{if }
\Theta_2(x)\neq \Theta_1(x)\,,\; x\in \bar \Omega\,, \cr
f(x,\Theta_2(x)) & \hbox{if }\Theta_2(x)=\Theta_1(x)\,,\;x\in \bar \Omega\,.
\endcases
$$
By the monotonicity of $f$ on its second argument, it is easily
seen that
$$
g > f(\cdot,\Theta_1)\quad \hbox{in }\Omega\,,
$$
since $\Theta_2\neq \Theta_1$. Thus, thanks to (3.3), it follows
from the monotonicity of the principal eigenvalue with respect to
the potential that
$$
\sigma^\Omega[-\Delta+a\,g -\lambda_1]\geq
\sigma^\Omega[-\Delta+a f(\cdot,\Theta_1) -\lambda_1]>0\,.
$$
Henceforth, the operator $-\Delta+a\,g-\lambda_1$ satisfies the
strong maximum principle in $\Omega$ under homogeneous Dirichlet
boundary conditions, and therefore we find from (3.5) that
$$
\Theta_2-\Theta_1 \gg 0\,.
$$
This completes the proof of the monotonicity in $\lambda$. The
same argument can be easily adapted to get the monotonicity in
$\alpha$.
\par
Relation (3.2) follows easily from Theorem 2.1 taking into account
that for any $\lambda\in (\sigma_0,\sigma_1)$
$$
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\gg \theta_{[\lambda,a,\Omega]}\,.
$$
The proof is completed.
\qed
\enddemo
\medskip
\noindent{\bf Remark 3.2:} (a) Theorem 3.1 also holds in
the absence of refuges, i.e. in case $\Omega=\Omega_+$. In this
special situation its proof is simpler, since large constants
provide us with supersolutions for any $\lambda\in{\Bbb R}$.
Therefore, in case $\Omega=\Omega_+$ the positive solution exists
for each $\lambda\in{\Bbb R}$. This is the case dealt with in most
of the references and in particular in \cite{MV97} and \cite{DH99}
for the special choice (1.7).
\smallskip
\noindent (b) By the uniqueness of the positive solution,
any couple $(\underline u,\overline u)$ formed by a nonnegative
subsolution $\underline u$ and a nonnegative supersolution
$\overline u$ of (3.1) must satisfy
$$
\underline u \leq \Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\leq \overline u\,.
$$
This estimate can be easily obtained with the same comparison
argument used to show the uniqueness and the monotonicities, and
so we will omit the details of its proof.
\medskip
\noindent In the sequel we shall assume the following:
\smallskip
\roster
\item"(Hfb)" There exists a continuous function
$f_b:[0,\infty)\to{\Bbb R}$ of class $C^{1+\mu}((0,\infty))$ such
that $f_b(0)=0$, $f_b(u)>0$ and $f_b'(u)>0$ for all $u>0$,
$$
\lim_{u\uparrow\infty}f_b(u)=\infty\,,
$$
and
$$
f(\cdot,u)\geq f_b(u)\,,\quad u\geq 0\,.
$$
\endroster
\smallskip
\noindent The following result is a substantial generalization
of the interior estimates found in \cite{Ke57} and
\cite{Os57}. It provides us with uniform interior estimates
in $\Omega_+$ for the solutions of (3.1) under assumption (Hfb),
and we shall we use it to show the stabilization in $\Omega_+$ of
$\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$ as either
$\alpha_j\uparrow\infty$, $1\leq j \leq q$, or $\lambda\uparrow
\sigma_1$, and to obtain very general existence results of large
regular solutions.
\proclaim{Theorem 3.3} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$. Let $D\subset \Omega_+$
be open and $K\subset \Omega_+$ compact such that
$D+B_\delta\subset K$ for some $\delta>0$, where $B_\delta$ stands
for the ball of radius $\delta$ centered at the origin. Set
$$
A:= \inf_{x\in K} a(x) >0\,,
$$
fix $\beta>0$ and let $u_K$ denote the unique positive zero of the
function
$$
h(u):=A u f_b(u)-\beta u \,.
\eqno (3.6)
$$
Thanks to $\hbox{\rm (Hfb)}$, $u_K$ is well defined. Assume in
addition that for each $u_*>u_K$
$$
I(u_*):=\int_{u_*}^\infty \left[ \int_{u_*}^u h(z)\,dz
\right]^{-1/2} \,du <\infty\,.
\eqno (3.7)
$$
Then, there exists a universal constant $M=M(\beta)$ such that for
any $\lambda\leq \beta$ and any regular positive solution
$u_\lambda$ of
$$
-\Delta u = \lambda u-a u f(\cdot,u) \quad \hbox{\rm in }\Omega_+\,,
\eqno (3.8)
$$
the following uniform estimate is satisfied
$$
\|u_\lambda\|_{C(\bar D)}\leq M\,.
\eqno (3.9)
$$
\endproclaim
\noindent{\bf Remarks 3.4:} (a) In (3.9) no condition on
the growth of $u_\lambda$ on $\partial\Omega_+$ is imposed.
Therefore, (3.9) provides us with a universal interior estimate
for the positive solutions of (3.8) which is independent of the
behaviour of the solutions on the boundary.
\smallskip
\noindent (b) We should point out that
$$
\int_{u_K}^\infty \left[ \int_{u_K}^u h(z)\,dz
\right]^{-1/2} \,du =\infty\,.
\eqno (3.10)
$$
Indeed, the auxiliary function
$$
g(u):= \int_{u_K}^u h(z)\,dz \,,\quad u >u_K\,,
$$
satisfies
$$
g(u_K)=0\,,\quad g'(u_K)=h(u_K)=0\,,
$$
and
$$
g''(u_K)=h'(u_K)=Af_b(u_K)+Au_Kf'_b(u_K)-\beta=
A u_K f'_b(u_K)>0\,.
$$
Therefore, the integral diverges.
\smallskip
\noindent (c) If there exist $\eta>0$ and $p>0$
such that
$$
f_b(u)\geq \eta\, u^p\,,\quad u\geq 0\,,
$$
then condition (3.7) is satisfied for any $A>0$ and $\beta>0$, and
therefore the estimate (3.9) is valid in any open set $D$
satisfying $\bar D\subset \Omega_+$. Indeed, for any $u_*>u_K$ we
have that
$$
\eqalign{ I(u_*)= & \sqrt{2} \int_{u_*}^\infty
\left[ 2A \int_{u_*}^u z f_b(z)\,dz - \beta (u^2-u_*^2)
\right]^{-1/2}du \cr = &
\sqrt{2} \int_1^\infty \left[ 2A \int_1^\theta t f_b(u_* t)\,dt
-\beta (\theta^2-1)\right]^{-1/2}d\theta\,. \cr }
$$
In particular, $I(u_*)$ is decreasing and
$$
I(u_*)\leq \sqrt{2} \int_1^\infty \left[ {2A\eta\over p+2}
u_*^p(\theta^{p+2}-1)-\beta(\theta^2-1) \right]^{-1/2}d\theta
<\infty\,.
$$
\noindent (d) Condition (3.7) entails the existence
of a unique regular positive solution of
$$
-u''=\beta u - A u f_b(u)\quad \hbox{in }(-R,R)\,,\quad
u(-R)=u(R)=\infty\,,
\eqno (3.11)
$$
for each $R>0$. Indeed, multiplying the differential equation of
(3.11) by $u'$ and setting
$$
v=u'\,, \quad u(0)=u_*\,,
$$
it is easily seen that the solutions of (3.11) are given by the
$u_*$'s such that for each $x\in (0,R)$
$$
v^2(x)=2\int_{u_*}^{u(x)} h(z)\,dz
$$
and $\lim_{x\uparrow R}u(x)=\infty$. Necessarily $u_*>u_K$,
because otherwise $u\leq u_K$, since $u_K$ is an equilibrium of
the equation. Moreover, this occurs if and only if
$$
R = \sqrt{2}\int_{u_*}^\infty \left[ \int_{u_*}^u h(z)\,dz
\right]^{-1/2}du\,.
$$
Therefore, $u_*$ corresponds to a solution of (3.11) if and only
if $u_*>u_K$ and
$$
R=\sqrt{2}I(u_*)\,.
$$
We already know that $\lim_{u_*\downarrow u_K}I(u_*)=\infty$ and
that $I(u_*)$ is decreasing. Moreover, since $h$ is nondecreasing,
it is easily seen from (3.7) that
$$
\lim_{u_*\uparrow\infty}I(u_*)=0\,.
$$
Therefore, there exists a unique $u_*\in (u_K,\infty)$ such that
$R=\sqrt{2}I(u_*)$. Actually, this implies the existence of a
radially symmetric regular large solution for the problem in a
ball (cf. Lemma 6.2 of \cite{GGLS98}). The proof of Theorem 3.3
can be accomplished very easily from these facts, but in this
paper we prefer using the techniques introduced in \cite{Ke57},
since they are pioneer in the field. If in addition $h''(u)\geq 0$
for $u$ large and
$$
\lim_{u\uparrow\infty}{h'(u)\over \sqrt{\int_{u_K}^u h(z)\,dz}}
>0\,,
$$
then the regular large solution in the ball is unique,
\cite{LM94}.
\smallskip
\noindent (e) Even in the simplest case when there exists
$p>0$ such that $f(x,u) =f_b(u)=u^p$ the condition (2) of
\cite{Ke57} fails, because of (3.10).
This is because the nonlinearity $h(u)$ of \cite{Ke57} is
increasing and bounded away from zero. Therefore, the interior
estimates obtained in \cite{Ke57} do not guarantee straight away
the existence of interior estimates for the positive solutions of
(3.1). Among other things this implies that the proof of Lemma 1.3
of \cite{MV97} contains a gap, since the interior estimates of
\cite{Ke57} can not be used. Our Theorem 3.3 completes the proof
of Lemma 1.3 of \cite{MV97}.
\medskip
Although the proof of Theorem 3.3 is an easy consequence from
Theorem III of \cite{Ke57}, as pointed out to us by the referee,
we are going to give a complete selfcontained proof of it by means
of the technical tools introduced in the proof of Theorem I of
\cite{Ke57}. Note that in \cite{Ke57} Theorem III was obtained as
a consequence from Theorem I.
\demo{Proof of Theorem 3.3}
It suffices to show that for each $x_0\in \bar D$ there exists
$\eta>0$ and a constant $M>0$ such that for any $\lambda\leq
\beta$ and any regular positive solution $u_\lambda$ of (3.8)
$$
\|u_\lambda\|_{C(B_\eta(x_0))}\leq M\,,
$$
where $B_\eta(x_0)$ stands for the ball of radius $\eta$ centered
at $x_0$. Let $x_0\in\bar D$ and consider $R>0$ such that $\bar
B_R(x_0)\subset K$. Then, for each $\lambda\leq \beta$ the
following differential inequality holds
$$
\Delta u_\lambda \geq h(u_\lambda) \quad \hbox{in }B_R(x_0)\,,
\eqno (3.12)
$$
where $h(u)$ is the function defined by (3.6). Set
$$
\alpha_\lambda := \max\{u_K+1\,,\,\sup_{x\in\partial B_R(x_0)}u_\lambda\}\,,
\eqno (3.13)
$$
and let $\Theta_\lambda$ denote the unique positive solution of
$$
\Delta u = h(u) \quad \hbox{in }B_R(x_0)\,,\quad
u|_{\partial B_R(x_0)} = \alpha_\lambda \,,
\eqno (3.14)
$$
whose existence is guaranteed by Theorem 3.1 (cf. Remark 3.2(a)).
Thanks to (3.12) and (3.13), for each $\lambda\leq \beta$ any
solution $u_\lambda$ of (3.8) is a positive subsolution of (3.14)
in $B_R(x_0)$, and hence it follows from Remark 3.2(b) that
$$
u_\lambda \leq \Theta_\lambda \quad \hbox{in } B_R(x_0)\,.
$$
Thus, to complete the proof it suffices to show that there exist
$\eta\in (0,R)$ and a constant $M>0$ such that for any
$\alpha_\lambda>0$ sufficiently large
$$
\|\Theta_\lambda\|_{C(B_\eta(x_0))}\leq M\,.
\eqno (3.15)
$$
By definition, $h(u_K)=0$ and $h(u)>0$ for $u>u_K$. Thus, for any
$\alpha_\lambda>u_K$ the constant $u_K$ is a strict positive
subsolution of (3.14), and hence it follows from Remark 3.2(b)
that
$$
u_K\leq \Theta_\lambda\,.
$$
In fact, the strong maximum principle implies $u_K\ll
\Theta_\lambda$. On the other hand, by the uniqueness of the
positive solution of (3.14), $\Theta_\lambda$ must be radially
symmetric, since the problem is invariant by rotations. Thus,
$$
\Theta_\lambda(x)=\Psi_\lambda(|x-x_0|)\,,\quad x \in B_R(x_0)\,,
$$
where $\Psi_\lambda(r)$ is the unique positive solution of
$$
\psi''(r)+{N-1\over r}\psi'(r)=h(\psi(r))\,,\quad 00$ and $M>0$ such that for any $\alpha_\lambda > u_K$
$$
\|\Psi_\lambda\|_{C([0,\eta])}\leq M\,.
\eqno (3.17)
$$
We already know that $\Psi_\lambda(r)>u_K$ for each $r\in[0,R]$
and $\alpha_\lambda>u_K$. Hence,
$$
h(\Psi_\lambda(r))>0\,.
$$
Moreover, for each $u>u_K$ we have that
$$
h'(u)=A f_b(u)+A u f_b'(u)-\beta > A f_b(u_K)-\beta + A u f'_b(u)
= A u f'_b(u)>0\,,
$$
and hence $h$ is increasing. We now follow the proof of Theorem I
in page 506 of \cite{Ke57}. The functions $\Psi_\lambda$ satisfy
$$
(R^{N-1}\Psi_\lambda'(r))'=r^{N-1}h(\Psi_\lambda(r))\,,
\eqno (3.18)
$$
and hence integrating (3.18) from $0$ to $r$ yields
$$
\Psi_\lambda'(r)=r^{1-N}\int_0^r s^{N-1}h(\Psi_\lambda(s))\,ds >0\,.
\eqno (3.19)
$$
This shows that $r\to \Psi_\lambda(r)$ is increasing, as well as
$r\to h(\Psi_\lambda(r))$. Thus, we find from (3.19) that
$$
\Psi_\lambda'(r)\leq r^{1-N}h(\Psi_\lambda(r))\int_0^r s^{N-1}\,ds
= {r\over N}h(\Psi_\lambda(r))\,.
\eqno (3.20)
$$
Now, substituting (3.20) into (3.16) gives
$$
\Psi_\lambda'' \geq {h(\Psi_\lambda)\over N}\,.
$$
Moreover, since $\Psi_\lambda'\geq 0$, (3.16) gives
$\Psi_\lambda''\leq h(\Psi_\lambda)$. Hence,
$$
h(\Psi_\lambda)\geq \Psi_\lambda'' \geq {h(\Psi_\lambda)\over N}\,.
\eqno (3.21)
$$
We now multiply (3.21) by $\Psi_\lambda'$ and integrate from $0$
to $r$ to obtain
$$
2\int_{\Psi_\lambda(0)}^{\Psi_\lambda(r)}h(z)\,dz \geq
[\Psi_\lambda'(r)]^2 \geq {2\over N}
\int_{\Psi_\lambda(0)}^{\Psi_\lambda(r)}h(z)\,dz\,.
\eqno (3.22)
$$
Now, taking the square root of the reciprocal of (3.22) and
integrating again gives
$$
{1\over \sqrt{2}}\! \int_{\Psi_\lambda(0)}^{\Psi_\lambda(r)} \!
\left[ \int_{\Psi_\lambda(0)}^u h(z)\,dz\right]^{-1/2}\!du
\leq r \leq \sqrt{{N\over 2}}\!\int_{\Psi_\lambda(0)}^{\Psi_\lambda(r)}
\!\left[ \int_{\Psi_\lambda(0)}^u h(z)\,dz\right]^{-1/2}\!du\,.
\eqno (3.23)
$$
In particular,
$$
{1\over \sqrt{2}}\! \int_{\Psi_\lambda(0)}^{\alpha_\lambda} \!
\left[ \int_{\Psi_\lambda(0)}^u h(z)\,dz\right]^{-1/2}\!du
\leq R\,.
$$
Hence, thanks to (3.10), $\Psi_\lambda(0)$ must be uniformly
bounded away from $u_K$. Moreover,
$$
R \leq \sqrt{{N\over 2}}\!\int_{\Psi_\lambda(0)}^{\infty}
\!\left[ \int_{\Psi_\lambda(0)}^u h(z)\,dz\right]^{-1/2}\!du\,.
$$
So, since $h$ in increasing we find from (3.7) that
$\Psi_\lambda(0)$ must be uniformly bounded above. Therefore,
using Remark 3.2(b) gives
$$
\Psi_\lambda(0)< \limsup_{\alpha_\lambda\uparrow\infty}
\Psi_\lambda(0)=\Psi_0\in(u_K,\infty)\,,
$$
for any $\alpha_\lambda>u_K$. Let $\Psi_\infty$ denote the unique
positive solution of the Cauchy problem
$$
\psi''(r)+{N-1\over r}\psi'(r)=h(\psi(r))\quad 00$ sufficiently small so that
$$
K_{j,\delta}:=\{x\in\Omega\;:\;\hbox{dist}(x,\Gamma_+^j)\leq\delta\}
\subset \Omega_+\,, \quad 1 \leq j \leq q\,,
$$
and $ K_{i,\delta}\cap K_{j,\delta}=\emptyset$ if $i\neq j$, and
consider the open set
$$
D:= \Omega\setminus \cup_{j=1}^q K_{j,\delta}\,.
$$
By construction
$$
D_+:=\{x\in D \;:\;a(x)>0\}=\Omega_+\setminus
\cup_{j=1}^q K_{j,\delta}
$$
and
$$
\bar D_0:=\{x\in D\;:\;a(x)=0\}=\bar \Omega_0\,.
$$
Moreover,
$$
\partial D = \gamma_0\cup \gamma_\infty^D\,,
$$
where
$$
\gamma_\infty^D := \cup_{j=1}^q\{x\in\Omega_+\;:\;
\hbox{dist}(x,\Gamma_+^j)=\delta\}\,.
$$
By construction, $\gamma_\infty^D$ is a compact subset of
$\Omega_+$, and hence it follows from Theorem 3.3 that there
exists a universal constant $M>0$ such that
$$
\|\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\|_{C(\gamma_\infty^D)}\leq M
$$
for all $\alpha=(\alpha_1,\dots,\alpha_q)\in (0,\infty)^q$. Thus,
any solution of (3.1) provides us with a subsolution of
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in }\;D\,, \cr
u =M \quad \hbox{\rm on } \gamma_\infty^D\,, \cr
u=0 \quad \hbox{\rm on } \gamma_0\,. \endgathered
\eqno (3.30)
$$
Due to Remark 3.2(b) we have that for each $\alpha\in(0,\infty)^q$
$$
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\leq
\Theta_{[\lambda,a,D,\gamma_\infty^D,M]}
\quad \hbox{in }D\,,
\eqno (3.31)
$$
where $\Theta_{[\lambda,a,D,\gamma_\infty^D,M]}$ is the unique
positive solution of (3.30), whose existence is guaranteed by
Theorem 3.1.
Since $\delta$ can be taken arbitrarily small,
$$
\lim_{\delta\downarrow 0} D = \Omega
$$
in the sense of \cite{Lo96}, and the mapping $\alpha \to
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$ is increasing, we
find from (3.31) that the point-wise limit (3.25) is well defined.
\par
Let $\Cal O_1 \subset \Cal O$ two open subsets of $\Omega$ such
that $\bar\Cal O_1 \subset \Cal O$, $\bar \Cal O \subset \Omega$,
and choose $\delta>0$ sufficiently small so that $\bar \Cal O
\subset D$. Then, (3.31) implies
$$
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\leq
\Theta_{[\lambda,a,D,\gamma_\infty^D,M]}
\quad \hbox{in }\Cal O\,,
$$
for each $\alpha\in (0,\infty)^q$, and hence, by the
$L^p$-estimates of Agmon, Douglis \& Nirenberg, for each $p>1$
there exists a constant $M_1=M(p,\Cal O_1)$ such that
$$
\|\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\|_{W^{2,p}(\Cal O_1)} \leq M_1
$$
for each $\alpha\in (0,\infty)^q$. Thus, thanks to Morrey's
embedding theorem and Schauder's estimates, there exists a
constant $M_2=M(\Cal O_1)$ such that
$$
\|\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\|_{C^{2+\mu}(\bar\Cal O_1)}
\leq M_2\,, \quad \alpha \in (0,\infty)^q\,.
$$
Now, a rather standard compactness argument combined with the
uniqueness of the point-wise limit (3.25) shows that
$\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}\in C^{2+\mu}(\bar\Cal
O_1)$ is a regular solution of
$$
-\Delta u = \lambda u - a u f(\cdot,u)
$$
in $\Cal O_1$, and that
$$
\lim_{\alpha\uparrow\infty}\|\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}-
\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}\|_{C^{2+\mu}(\bar\Cal O_1)}=0\,.
\eqno (3.32)
$$
This completes the proof of Parts (i) and (ii).
\par
We now prove Part (iii). By construction, for each $\alpha\in
(0,\infty)^q$ we have that
$$
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}<\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}\,.
\eqno (3.33)
$$
Therefore, thanks to (3.2),
$$
\lim_{\lambda \uparrow \sigma_1}\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]} =\infty
\quad \hbox{\rm uniformly in }\;
\cup_{j=1}^{n_1}\bar\Omega^1_{0,j}\setminus \partial \Omega\,.
\eqno (3.34)
$$
We now show that the point-wise limit (3.28) is well defined.
Reduce $\delta>0$, if necessary, so that the open neighborhoods
$$
\Omega_{0,j}^{1,\delta}:=\{x\in\Omega\;:\;\hbox{dist}(x,\Omega_{0,j}^1)
<\delta\}\,,\quad 1 \leq j \leq n_1\,,
$$
satisfy
$$
\bar \Omega_{0,j}^{1,\delta}\cap \bar \Omega_{0,\ell}^{1,\delta}=\emptyset
$$
for any $j\neq \ell$, and consider the open set
$$
D_1:=D\setminus \cup_{j=1}^{n_1}\bar \Omega_{0,j}^{1,\delta}\,.
$$
Then,
$$
\partial D_1=\partial D \cup \cup_{j=1}^{n_1}
(\partial \Omega_{0,j}^{1,\delta}\setminus\partial\Omega)=
\gamma_0\cup \gamma_\infty^D\cup \cup_{j=1}^{n_1}
(\partial \Omega_{0,j}^{1,\delta}\setminus\partial\Omega)\,.
$$
By construction, the set
$$
K_1:= \gamma_\infty^D \cup \cup_{j=1}^{n_1}
(\partial \Omega_{0,j}^{1,\delta}\setminus\partial\Omega)
$$
is a compact subset of $\Omega_+$. Thus, thanks to Theorem 3.3,
there exists a universal constant $M_3>0$ such that for any
$\lambda<\sigma_1$
$$
\|\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}\|_{C(K_1)}\leq M_3\,.
$$
Hence, for each $\lambda<\sigma_1$ the function
$\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}$ provides us with a
subsolution of the problem
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in }\;D_1\,, \cr
u =M_3 \quad \hbox{\rm on } K_1\,,\cr
u=0 \quad \hbox{\rm on } \gamma_0=\partial D_1\setminus K_1\,.
\endgathered\eqno (3.35)
$$
The lower order refuges of $D_1$ are $\Omega_{0,j}^2$, $1\leq j
\leq n_2$. Therefore, thanks to Theorem 3.1 the problem (3.35)
possesses a regular positive solution if, and only if,
$\lambda<\sigma_2$. Moreover, it is unique if it exists. Let
$\Theta_{[\sigma_1,a,D_1,K_1,M_3]}$ denote the unique regular
positive solution of (3.35) for $\lambda=\sigma_1$. Thanks to
Remark 3.2(b), for each $\lambda<\sigma_1$ we have
$$
\Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}\leq
\Theta_{[\sigma_1,a,D_1,K_1,M_3]}\quad \hbox{in }D_1\,.
\eqno (3.36)
$$
Since $\delta$ can be taken arbitrarily small and the mapping
$\lambda \to \Theta_{[\lambda,a,\Omega,\Gamma_+,\infty]}$ is
nondecreasing, it follows from (3.36) that the point-wise limit
(3.28) is well defined. The same bootstrapping and compactness
arguments as above combined with (3.28) show that
$\Theta_{[\sigma_1,a,\Omega,\Gamma_+,\infty]}$ is a regular large
solution of (3.29). Its minimal character follows easily from the
construction itself. This completes the proof.
\qed
\enddemo
\medskip
\head
4. Stabilization of the regular positive solutions in
$\Omega\setminus\bar \Omega_0^1$ as $\lambda\uparrow \sigma_1$.
\endhead
Thanks to Theorem 2.1, the problem (1.5) possesses a regular
positive solution if, and only if,
$$
\sigma_0 <\lambda < \sigma_1\,.
$$
Moreover, it is unique if it exists. As in Section 2 we shall
denote it by $\theta_{[\lambda,a,\Omega]}$. The following result
shows that $\theta_{[\lambda,a,\Omega]}$ converges to a large
regular solution in $\Omega\setminus \bar \Omega_0^1$ as $\lambda
\uparrow \sigma_1$.
\proclaim{Theorem 4.1} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$, and assume in addition
that $(3.7)$ is satisfied for any compact subset $K$ of $\Omega_+$
and $\beta=\sigma_1$. Set
$$
\Omega_1:=\Omega\setminus \bar \Omega_0^1\,,\quad
\Omega_0^1 :=\cup_{j=1}^{n_1}\Omega_{0,j}^1\,.
$$
Then, the point-wise limit
$$
\Xi_{[\sigma_1,a,\Omega_1]}(x):= \lim_{\lambda\uparrow\sigma_1}\theta_{[\lambda,a,\Omega]}(x)
\quad x\in \Omega_1\,,
\eqno (4.1)
$$
is well defined. Moreover, $\Xi_{[\sigma_1,a,\Omega_1]}\in
C^{2+\mu}(\Omega_1)$,
$$
\lim_{\lambda\uparrow\sigma_1}\|\theta_{[\lambda,a,\Omega]}-
\Xi_{[\sigma_1,a,\Omega_1]}\|_{
C^{2+\mu}(\Omega_1)}=0\,,
\eqno (4.2)
$$
and $\Xi_{[\sigma_1,a,\Omega_1]}$ is the minimal large regular
solution of
$$\gathered
-\Delta u=\sigma_1 u -a(x)u f(x,u) \quad \hbox{\rm in
}\;\Omega_1\,, \cr
u =\infty \quad \hbox{\rm on } \partial\Omega_1\cap\Omega\,, \cr
u=0 \quad \hbox{\rm on } \partial\Omega_1\cap\partial\Omega\,,
\endgathered \eqno (4.3)
$$
\endproclaim
\demo{Proof} Basically this result is a
particular case of Theorem 3.5(iii), but not exactly, since
Theorem 3.5 dealt with the limiting behaviour of large regular
solutions of (3.1) going to infinity on some of the components of
$\partial\Omega_+\cap \partial\Omega$ and now it might occur
$\partial\Omega_+\cap \partial\Omega=\emptyset$. Nevertheless, the
proof of Theorem 3.5(iii) carries over almost mutatis mutandis to
prove Theorem 4.1. In the sequel the notations introduced in the
proof of Theorem 3.5 will be kept out.
\par
Pick up $\delta>0$ sufficiently small so that the open
neighborhoods
$$
\Omega_{0,j}^{1,\delta}:=\{x\in\Omega\;:\;\hbox{dist}(x,\Omega_{0,j}^1)
<\delta\}\,,\quad 1 \leq j \leq n_1\,,
$$
satisfy
$$
\bar \Omega_{0,j}^{1,\delta}\cap \bar \Omega_{0,\ell}^{1,\delta}=
\emptyset
$$
for any $j\neq \ell$, and consider the open set
$$
D:=\Omega\setminus \cup_{j=1}^{n_1}\bar \Omega_{0,j}^{1,\delta}\,.
$$
By definition, $D\subset \Omega_1$. Moreover, any component of
$\partial\Omega_{0,j}^{1,\delta}\setminus\partial\Omega$, $1\leq j
\leq n_1$, lies within $\Omega_+$. Let $\Gamma_+^j$, $1\leq j \leq
q$, denote all the components of $\partial D$ contained in
$\Omega_+$. Note that any component of $\partial D$ either it is
entirely contained in $\Omega_+$, or it is a component of
$\partial\Omega$. Then,
$$
K:= \cup_{j=1}^q \Gamma_+^j
$$
is a compact subset of $\Omega_+$. Thanks to Theorem 3.3, there
exists a universal constant $M>0$ such that
$$
\|\theta_{[\lambda,a,\Omega]}\|_{C(K)}\leq M
$$
for any $\lambda\in (\sigma_0,\sigma_1)$, and hence
$\theta_{[\lambda,a,\Omega]}$ provides us with a subsolution of
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in }\;D\,, \cr
u =M \quad \hbox{\rm on } K\,, \cr
u=0 \quad \hbox{\rm on } \partial D\setminus K\,.
\endgathered\eqno (4.4)
$$
The lower order refuges of $D$ are $\Omega_{0,j}^2$, $1\leq j \leq
n_2$. Therefore, thanks to Theorem 3.1 the problem (4.4) possesses
a regular positive solution if, and only if, $\lambda<\sigma_2$.
Moreover, it is unique if it exists. Let
$\Theta_{[\sigma_1,a,D,K,M]}$ denote the unique regular positive
solution of (4.4) for $\lambda=\sigma_1$. Thanks to Remark 3.2(b)
we have that
$$
\theta_{[\lambda,a,\Omega]}\leq
\Theta_{[\sigma_1,a,D,K,M]}\quad \hbox{in }D\,,
\eqno (4.5)
$$
for all $\lambda\in (\sigma_0,\sigma_1)$. Since $\delta$ can be
taken arbitrarily small,
$$
\lim_{\delta\downarrow 0}D = \Omega_1\,,
$$
and the mapping $\lambda \to \theta_{[\lambda,a,\Omega]}$ is
increasing, it follows from (4.5) that the point-wise limit (4.1)
is well defined.
\par
Let $\Cal O_1 \subset \Cal O$ two open subsets of $\Omega_1$ such
that $\bar\Cal O_1 \subset \Cal O$, $\bar \Cal O \subset
\Omega_1$, and choose $\delta>0$ sufficiently small so that $\bar
\Cal O \subset D$. Then, (4.5) implies
$$
\theta_{[\lambda,a,\Omega]}\leq
\Theta_{[\sigma_1,a,D,K,M]}\quad \hbox{in }\bar \Cal O\,,
\quad \lambda\in (\sigma_0,\sigma_1)\,.
$$
Thus, by the $L^p$-estimates of Agmon, Douglis \& Nirenberg, for
each $p>1$ there exists a constant $C_1=C(p,\Cal O_1)$ such that
$$
\|\theta_{[\lambda,a,\Omega]} \|_{W^{2,p}(\Cal O_1)} \leq C_1\,,
\quad \lambda \in (\sigma_0,\sigma_1)\,.
$$
Hence, thanks to Morrey's embedding theorem and Schauder's
estimates, there exists a constant $C_2=C(\Cal O_1)$ such that
$$
\|\theta_{[\lambda,a,\Omega]} \|_{C^{2+\mu}(\bar\Cal O_1)}
\leq C_2\,,\quad \lambda \in (\sigma_0,\sigma_1)\,.
$$
Now, a well known compactness argument combined with the
uniqueness of the point-wise limit (4.1) shows that
$\Xi_{[\sigma_1,a,\Omega_1]}\in C^{2+\mu}(\bar\Cal O_1)$ is a
regular solution of
$$
-\Delta u = \sigma_1 u - a u f(\cdot,u)
$$
in $\Cal O_1$, and that
$$
\lim_{\lambda \uparrow \sigma_1}\| \theta_{[\lambda,a,\Omega]}-
\Xi_{[\lambda_1,a,\Omega_1]} \|_{C^{2+\mu}(\bar\Cal O_1)}=0\,.
$$
This completes the proof of (4.2). By (2.3) and the construction
itself $\Xi_{[\sigma_1,a,\Omega_1]}$ must be the minimal regular
positive solution of (4.3). The proof is completed.
\qed
\enddemo
\medskip
\head
5. The existence of metasolutions for $\lambda \geq \sigma_1$.
\endhead
The following concept goes back to \cite{GL98} and
\cite{Go99}, where it was introduced in the special case when $f(x,u)=u^p$
for some $p>0$.
\proclaim{Definition 5.1} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf) }$ and set
$$
\Omega_k:= \Omega\setminus \cup_{i=1}^k \cup_{j=1}^{n_i}
\bar \Omega_{0,j}^i\,,\quad 1 \leq k \leq m\,.
$$
A function
$$
u: \Omega \to [0,\infty]
$$
is said to be a regular metasolution of order $k$ of $(1.5)$
supported in $\Omega_k$ if $u|_{\Omega_k}$ is a large regular
solution of
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad\hbox{\rm in }\Omega_k\,,
\cr u =\infty \quad \hbox{\rm on } \partial\Omega_k\cap\Omega\,,
\cr u=0 \quad \hbox{\rm on }
\partial\Omega_k\cap\partial\Omega\,, \endgathered
\eqno (5.1)
$$
in the sense of the statement of Theorem 3.5(i), while
$$
u=\infty\quad \hbox{\rm in } (\Omega\setminus\bar\Omega_k)\cup
(\partial\Omega_k\cap\Omega)\,.
$$
\endproclaim
Using this concept, the following result is an immediate
consequence from Theorem 4.1.
\proclaim{Corollary 5.2} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$, and assume in addition
that $(3.7)$ is satisfied for any compact subset $K$ of $\Omega_+$
and $\beta=\sigma_1$. Then, the function $\Cal
M_{[\sigma_1,a,\Omega]}:\Omega\to[0,\infty]$ defined by
$$
\Cal M_{[\sigma_1,a,\Omega]}
:=\cases \Xi_{[\sigma_1,a,\Omega_1]} & \hbox{\rm in } \Omega_1
\,,\\
\infty & \hbox{\rm in } (\Omega\setminus \bar\Omega_1 )
\cup (\partial \Omega_1\cap\Omega)\,,
\endcases
\eqno (5.2)
$$
is a regular metasolution of order one of $(1.5)$, with
$\lambda=\sigma_1$, supported in $\Omega_1$.
\endproclaim
The following result characterizes the range of values of the
parameter $\lambda$ for which (1.5) admits a regular metasolution
of order $k$ supported in $\Omega_k$, $1\leq k \leq m$. Thanks to
Theorem 2.2 those metasolutions are the candidates to describe the
limiting behaviour of the population as time passes by for any
$\lambda\geq \sigma_1$.
\proclaim{Theorem 5.3} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$, and assume in addition
that $(3.7)$ is satisfied for any compact subset $K$ of $\Omega_+$
and $\beta>0$. Fix $k \in\{1,\dots,m\}$. If $k0$ and consider the problem
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in }\Omega_k\,,
\cr
u =M \quad \hbox{\rm on } \partial\Omega_k\cap\Omega\,, \cr
u=0 \quad \hbox{\rm on } \partial\Omega_k\cap\partial\Omega\,. \endgathered
\eqno (5.3)
$$
This problem does not fit into the setting of Theorem 3.1, since
$a(x)=0$ on $\partial\Omega_k\setminus\partial\Omega$, but we can
slightly modify $\Omega_k$ so that the corresponding problem does
it. For each $\delta>0$ consider the open neighborhoods
$$
\Omega_{0,j}^{i,\delta}:=\{x\in\Omega\;:\;\hbox{dist}(x,\Omega_{0,j}^i)<\delta\}\,,
\quad 1\leq i\leq k\,,\quad 1\leq j\leq n_i\,.
$$
By (Ha1-3) there exists $\delta_0>0$ such that for each $\delta\in
(0,\delta_0)$
$$
\bar \Omega_{0,j}^{i,\delta}\cap \bar \Omega_{0,\hat j}^{\hat i,\delta}
= \emptyset
$$
if $(i,j)\neq (\hat i,\hat j)$. Now, consider the open set
$$
\Omega_k^\delta:= \Omega\setminus \cup_{i=1}^{k}\cup_{j=1}^{n_i}
\bar \Omega_{0,j}^{i,\delta}\,.
$$
By definition,
$$
\Omega_k^\delta \subset \Omega\setminus \cup_{i=1}^{k}
\cup_{j=1}^{n_i} \bar \Omega_{0,j}^i=\Omega_k\,,
$$
and
$$
\lim_{\delta\downarrow 0}\Omega_k^\delta=\Omega_k\,,
$$
in the sense of \cite{Lo96}. Moreover, any component of $\partial
\Omega_{0,j}^{i,\delta}\setminus\partial\Omega$, $1\leq i\leq k$,
$1\leq j \leq n_i$, lies within $\Omega_+$. Note that any
component of $\partial \Omega_k^\delta$ either it is entirely
contained in $\Omega_+$, or it is a component of $\partial\Omega$.
Now, consider the modified problem
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad\hbox{\rm in }\Omega_k^\delta\,, \cr
u =M \quad \hbox{\rm on } \partial\Omega_k^\delta\cap\Omega\,, \cr
u=0 \quad \hbox{\rm on } \partial\Omega_k^\delta\cap\partial\Omega\,.
\endgathered \eqno (5.4)
$$
Thanks to Theorem 3.1, (5.4) possesses a unique positive solution,
say $\Theta_{[\lambda,a,\delta,M]}$. Let $\Cal O_1\subset \Cal O$
be two open subdomains of $\Omega_k$ such that $ \bar \Cal O_1
\subset \Cal O$, $\bar \Cal O \subset \Omega_k$. By construction,
there exists $\delta_1\in (0,\delta_0)$ such that for any
$\delta\in (0,\delta_1)$
$$
\bar \Cal O \subset \Omega_k^{\delta_1} \subset \Omega_k^\delta
\subset \Omega_k\,,
$$
Set
$$
K:= \partial\Omega_k^{\delta_1}\setminus\partial\Omega\,.
$$
By construction, for each $\delta\in (0,\delta_1)$ $K$ is a
compact subset of
$$
\{x\in \Omega_k^\delta\;:\; a(x)>0\}\,.
$$
Thus, thanks to Theorem 3.3, there exists a constant $M_1>0$ such
that for each $\delta\in (0,\delta_1)$
$$
\|\Theta_{[\lambda,a,\delta,M]}\|_{C(K)}\leq M_1\,.
$$
Hence, for each $0<\delta<\delta_1$ the function
$\Theta_{[\lambda,a,\delta,M]}$ provides us with a positive
subsolution of
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad\hbox{\rm in
}\Omega_k^{\delta_1}\,, \cr
u =M_1 \quad \hbox{\rm on } \partial\Omega_k^{\delta_1}\cap\Omega\,,\cr
u=0 \quad \hbox{\rm on }\partial\Omega_k^{\delta_1}\cap\partial\Omega\,.
\endgathered \eqno (5.5)
$$
Let $\Theta_{[\lambda,a,\delta_1,M_1]}$ denote the unique positive
solution of (5.5), whose existence is guaranteed by Theorem 3.1.
Thanks to Remark 3.2(b), for each $\delta\in (0,\delta_1)$ we have
$$
\Theta_{[\lambda,a,\delta,M]}\leq \Theta_{[\lambda,a,\delta_1,M_1]} \quad
\hbox{in } \Omega_k^{\delta_1}\,,
$$
and therefore, there exists a constant $M_2>0$ such that for each
$0<\delta<\delta_1$
$$
\|\Theta_{[\lambda,a,\delta,M]}\|_{C(\bar \Cal O)}\leq M_2\,.
$$
By the same bootstrapping argument used in the proof of Theorem
3.5 and Theorem 4.1, there exists a constant $M_3>0$ such that for
each $\delta \in (0,\delta_1)$
$$
\|\Theta_{[\lambda,a,\delta,M]}\|_{C^{2+\mu}(\bar \Cal O_1)}\leq M_3\,.
$$
Now, by a standard compactness argument combined with a diagonal
procedure it is clear that we can substract a subsequence
$\delta_n\downarrow 0$ such that
$$
\lim_{n\to\infty}\|\Theta_{[\lambda,a,\delta_n,M]}-\Theta_{[\lambda,a,M]}\|_
{C^{2+\mu}(\Omega_k)}=0
$$
for some $\Theta_{[\lambda,a,M]}\in C^{2+\mu}(\Omega_k)$.
Necessarily,
$$
\lim_{x\to[\partial\Omega_k\setminus\partial\Omega]}\Theta_{[\lambda,a,M]}(x)=M\,.
$$
Therefore, $\Theta_{[\lambda,a,M]}$ provides us with a solution of
(5.3). The same argument of the proof of Theorem 3.1 shows that in
fact $\Theta_{[\lambda,a,M]}$ is the unique regular positive
solution of (5.3), and that $M\to \Theta_{[\lambda,a,M]}$ is
point-wise increasing.
\par
Thanks to Theorem 3.3, there exists a constant $M_4>0$ such that
for any $M>0$
$$
\|\Theta_{[\lambda,a,M]}\|_{C(K)}\leq M_4\,.
$$
Hence, for each $M>0$ the function $\Theta_{[\lambda,a,M]}$
provides us with a subsolution of
$$\gathered
-\Delta u=\lambda u -a(x)u f(x,u) \quad \hbox{\rm in }\Omega_k^{\delta_1}\,,\cr
u =M_4 \quad \hbox{\rm on }\partial\Omega_k^{\delta_1}\cap\Omega\,,\cr
u=0 \quad \hbox{\rm on }\partial\Omega_k^{\delta_1}\cap\partial\Omega\,,
\endgathered \eqno (5.6)
$$
and so
$$
\Theta_{[\lambda,a,M]}\leq \Theta_{[\lambda,a,\delta_1,M_4]}\quad \hbox{in }
\Omega_k^{\delta_1}\,.
$$
Therefore, the point-wise limit
$$
\Theta_{[\lambda,a,\infty]}(x):=\lim_{M\uparrow\infty}
\Theta_{[\lambda,a,M]}(x)\,,
\quad x\in \Omega_k\,,
$$
is well defined. Finally, the same regularity and compactness
argument given before shows that $\Theta_{[\lambda,a,\infty]}$ is
a large solution of (5.1). Its minimality follows easily from the
construction itself.
\par
In case $k=m$, by Remark 3.2(a) we do not have any limitation on
the size of $\lambda$ in order to apply Theorem 3.1, and therefore
the previous procedure provides us with a minimal large solution
of (5.1) for each $\lambda\in{\Bbb R}$. This completes the proof.
\qed
\enddemo
The following result shows the point-wise behavior of the
metasolutions of order $k\leq m-1$ supported in $\Omega_k$ as
$\lambda \uparrow \sigma_{k+1}$. They stabilize in $\Omega_{k+1}$,
whereas they grow to infinity in
$$
\cup_{j=1}^{n_{k+1}}\bar \Omega_{0,j}^{k+1}\setminus \partial \Omega\,.
$$
Therefore, they provide us with metasolutions of order $k+1$
supported in $\Omega_{k+1}$.
\proclaim{Theorem 5.4} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$, and assume in addition
that $(3.7)$ is satisfied for any compact subset $K$ of $\Omega_+$
and $\beta>0$. Fix $k \in\{1,\dots,m-1\}$ and consider a sequence
$$
\lambda_n \in (\sigma_k,\sigma_{k+1})\,,\quad n\geq 1\,,
$$
such that
$$
\lim_{n\to\infty}\lambda_n = \sigma_{k+1}\,.
$$
For each $n\geq 1$, let $u_n$ be a metasolution of order $k$ of
$(1.5)$, with $\lambda = \lambda_n$, supported in
$$
\Omega_k=\Omega\setminus \cup_{i=1}^k \cup_{j=1}^{n_i}
\bar \Omega_{0,j}^i\,,
$$
whose existence is guaranteed by Theorem 5.3. Then,
$$
\lim_{n\to\infty}u_n=\infty \quad
\hbox{\rm uniformly in }
\cup_{j=1}^{n_{k+1}}\bar\Omega_{0,j}^{k+1}\setminus \partial \Omega\,.
\eqno (5.7)
$$
Moreover, for any open subdomain $\Cal O$ of $\Omega_{k+1}$ with
$\bar \Cal O\subset \Omega_{k+1}$ there exists a subsequence of
$(\lambda_n,u_n)$, $n\geq 1$, relabeled again by $n$, and a
regular solution of
$$
-\Delta u = \sigma_{k+1} u - a u f(\cdot,u)
$$
in $\Cal O$, say $u_\omega\in C^{2+\mu}(\bar\Cal O)$, such that
$$
\lim_{n\to\infty}\|u_n-u_\omega\|_{C^{2+\mu}(\bar\Cal O)}=0\,.
\eqno (5.8)
$$
Furthermore, if $u_\lambda$ stands for the minimal metasolution of
order $k$ of $(1.5)$ supported in $\Omega_k$, then the point-wise
limit
$$
u_\omega(x)=\lim_{\lambda\uparrow\sigma_{k+1}}u_\lambda(x)\,,
\quad x\in\Omega_{k+1}\,,
$$
is well defined and its extension by $\infty$ to $\Omega$ provides
us with a metasolution of order $k+1$ of $(1.5)$ supported in
$\Omega_{k+1}$ (for the value of the parameter
$\lambda=\sigma_{k+1}$).
\endproclaim
\demo{Proof} By definition, for each $n\geq 1$
the function $u_n|_{\Omega_k}$ is a large regular solution of
$$\gathered
-\Delta u=\lambda_n u -a(x)u f(x,u) \quad \hbox{\rm in
}\;\Omega_{k}\,, \cr
u =\infty \quad \hbox{\rm on }\partial\Omega_{k}\cap\Omega\,, \cr
u=0 \quad \hbox{\rm on }\partial\Omega_{k}\cap\partial\Omega\,.
\endgathered
$$
In particular, $u_n|_{\Omega_k}$ is a positive strict
supersolution of
$$\gathered
-\Delta u = \lambda_n u-a(x) u^{p+1} \quad \hbox{in }\Omega_k\,,\cr
u =0 \quad \hbox{on }\partial \Omega_k\,,
\endgathered
\eqno (5.9)
$$
and hence,
$$
u_n|_{\Omega_k} \geq \theta_{[\lambda_n,a,\Omega_k]}\,,
\eqno (5.10)
$$
where $\theta_{[\lambda_n,a,\Omega_k]}\gg 0$ is the unique
positive solution of (5.9), whose existence and uniqueness is
guaranteed by Theorem 2.1(i). Note that the lower order refuges of
$\Omega_k$ are $\Omega_{0,j}^{k+1}$, $1\leq j \leq n_{k+1}$, and
that
$$
\lambda_n<\sigma_{k+1}=\sigma^{\Omega_{0,j}^{k+1}}\,.
$$
Thanks to Theorem 2.1(iii), we have
$$
\lim_{n\to\infty}\theta_{[\lambda_n,a,\Omega_k]} =\infty \quad
\hbox{uniformly in } \cup_{j=1}^{n_{k+1}}
\bar \Omega_{0,j}^{k+1} \setminus \partial \Omega_k\,,
$$
and hence we find from (5.10) that
$$
\lim_{n\to\infty}u_n =\infty \quad
\hbox{uniformly in } \cup_{j=1}^{n_{k+1}}
\bar \Omega_{0,j}^{k+1} \setminus \partial \Omega_k\,.
\eqno (5.11)
$$
Moreover, $u_n = \infty$ in $(\Omega\setminus\bar \Omega_k)
\cup (\partial\Omega_k\cap\Omega)$, since $u_n$ is
a metasolution of order $k$ supported in $\Omega_k$. Thus, (5.11)
implies (5.7).
\par
Now, for each $\delta>0$ sufficiently small consider the open set
$$
D:= \Omega\setminus \cup_{i=1}^{k+1}\cup_{j=1}^{n_i}
\bar \Omega_{0,j}^{i,\delta}\,,
$$
where
$$
\Omega_{0,j}^{1,\delta}:=\{x\in\Omega\;:\;\hbox{dist}(x,\Omega_{0,j}^i)<\delta\}\,,
\quad 1\leq i\leq k+1\,,\quad 1\leq j\leq n_i\,.
$$
By definition,
$$
D \subset \Omega\setminus \cup_{i=1}^{k+1}\cup_{j=1}^{n_i}
\bar \Omega_{0,j}^i=\Omega_{k+1}\,,
$$
and
$$
\lim_{\delta\downarrow 0}D=\Omega_{k+1}\,,
$$
in the sense of \cite{Lo96}. Moreover, any component of $\partial
\Omega_{0,j}^{i,\delta}\setminus\partial\Omega$, $1\leq i\leq
k+1$, $1\leq j \leq n_i$, lies within $\Omega_+$. Let
$\Gamma_+^j$, $1\leq j \leq q$, denote all the components of
$\partial D$ contained in $\Omega_+$. Note that any component of
$\partial D$ either it is entirely contained in $\Omega_+$, or it
is a component of $\partial\Omega$. Then,
$$
K:= \cup_{j=1}^q \Gamma_+^j
$$
is a compact subset of $\Omega_+$. Thanks to Theorem 3.3, there
exists a constant $M>0$ such that for any $n\geq 1$,
$$
\|u_n\|_{C(K)}\leq M\,,
$$
and hence $u_n$ provides us with a positive subsolution of
$$\gathered
-\Delta u=\sigma_{k+1} u -a(x)u f(x,u) \quad \hbox{\rm in }\;D\,,
\cr u =M \quad \hbox{\rm on } K\,,\cr u=0 \quad \hbox{\rm on
}\partial D\setminus K\,. \endgathered
\eqno (5.12)
$$
The lower order refuges of (5.12) are $\Omega_{0,j}^{k+2}$, $1\leq
j \leq n_{k+2}$, if $k\leq m-2$, and $a(x)$ is bounded away from
zero if $k=m-1$. Moreover, $\sigma_{k+1}<\sigma_{k+2}$. Therefore,
thanks to Theorem 3.1 the problem (5.12) possesses a unique
regular positive solution. Let $\Theta$ denote it. By Remark
3.2(b) we find that for each $n\geq 1$
$$
u_n \leq \Theta \quad \hbox{in } D\,.
\eqno (5.13)
$$
Let $\Cal O$ be an open subset of $\Omega_{k+1}$ with $\bar \Cal
O\subset \Omega_{k+1}$ and choose $\delta>0$ sufficiently small so
that
$$
\Cal O \subset D\,.
$$
The same bootstrapping argument of the proof of Theorem 4.1 shows
that there exists a constant $M_1>0$ such that
$$
\|u_n\|_{C^{2+\mu}(\bar \Cal O)}\leq M_1\,,\quad
n\geq 1\,.
$$
Therefore, we can substract a subsequence of $(\lambda_n,u_n)$,
again labeled by $n$, such that
$$
\lim_{n\to\infty}\|u_n-u_\omega\|_{C^{2+\mu}(\bar \Cal O)} =0
$$
for some $u_\omega\in C^{2+\mu}(\bar\Cal O)$. Necessarily,
$u_\omega$ is a solution of
$$
-\Delta u = \sigma_{k+1} u -a u f(\cdot,u)
$$
in $\Cal O$.
\par
The last assertion follows easily from the fact that $\lambda \to
u_\lambda(x)$ is nondecreasing for each $x\in\bar\Omega$. This
completes the proof of the theorem.
\qed
\enddemo
\medskip
\head
6. The asymptotic behaviour of the population for $\lambda \geq
\sigma_1$.
\endhead
\noindent In this section we characterize the asymptotic
behaviour of the population as $t\uparrow\infty$ for any value of
the parameter $\lambda\geq \sigma_1$.
\proclaim{Theorem 6.1} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and
$$
\partial \Omega\cap\partial\Omega_+\neq \emptyset\,.
$$
Let $\Gamma^j_+$, $1\leq j \leq q$, be $q$ arbitrary components of
$\partial \Omega\cap\partial\Omega_+$, and $\alpha_j >0$, $1\leq j
\leq q$, $q$ arbitrary constants, and consider the evolution
problem
$$\gathered
{\partial u\over \partial t} -\Delta u=\lambda u -a(x)u f(x,u)
\quad \hbox{\rm in }\Omega\times (0,\infty)\,,\cr
u\big|_{\Gamma_+^j} =\alpha_j>0\,, \quad 1\leq j\leq q\quad
t>0\cr u=0 \hfill \quad \hbox{\rm on }
(\partial\Omega\setminus\cup_{j=1}^q\Gamma_+^j)\times (0,\infty)\,,\cr
u(\cdot,0)=u_0\,, \quad \hbox{\rm in }\Omega\,,\endgathered
\eqno (6.1)
$$
The following assertions are true:
\smallskip
\roster
\item"(i)" For each $u_0\in U_0$, $u_0\geq 0$, the problem
$(6.1)$ possesses a unique global regular solution
$u_{[\lambda,a,\Omega,\Gamma_+,\alpha]}(\cdot,\cdot;u_0)
\in C^{2+\mu,1+{\mu\over 2}}(\bar\Omega\times (0,\infty))$.
\smallskip
\item"(ii)"
For each $u_0\in U_0$, $u_0\geq 0$,
$$
\lim_{t\to\infty}\|u_{[\lambda,a,\Omega,\Gamma_+,\alpha]}(\cdot,t;u_0)-
\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}\|_{C(\bar \Omega)}=0\,,
\eqno (6.2)
$$
where $\Theta_{[\lambda,a,\Omega,\Gamma_+,\alpha]}$ is the unique
regular positive solution of $(3.1)$.
\endroster
\endproclaim
\demo{Proof}
The existence of a unique regular solution follows from the
results of \cite{DK92}. The global existence of these solutions is
easily obtained from the estimate
$$
{\partial u \over \partial t}-\Delta u = \lambda u-a u f(\cdot,u)\leq
\lambda u\,.
$$
This completes the proof of Part (i). Adapting the proof of
Theorem 2.2 it is easily seen that for any $u_0\in U_0$, $u_0\geq
0$, condition (6.2) holds.
\qed
\enddemo
\noindent From this result it easily follows the main theorem
of this section.
\proclaim{Theorem 6.2} Suppose $\hbox{\rm (Ha1-3)}$,
$\hbox{\rm (Hf)}$ and $\hbox{\rm (Hfb)}$, and assume in addition
that $(3.7)$ is satisfied for any compact subset $K$ of $\Omega_+$
and $\beta>0$ and that either $1\leq k \leq m-1$ and $\sigma_k\leq
\lambda<\sigma_{k+1}$, or $k=m$ and $\lambda \geq \sigma_m$. Then,
for any $u_0\in U_0$ there exists $\nu>0$ such that for any
compact subset $K$ of $\Omega_k$ the restriction of the orbit
$$
\Gamma(u_0):=\{u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\;:\;t\geq 0\}
$$
to $K$ is relatively compact in $C^{2+\nu}(K)$. Moreover,
$$
\Theta^{\hbox{\rm min}}_{[\lambda,a,\infty]}\leq
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0) \leq
\limsup_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\leq
\Theta^{\hbox{\rm max}}_{[\lambda,a,\Omega]}\,,
$$
where $\Theta^{\hbox{\rm min}}_{[\lambda,a,\infty]}$ and
$\Theta^{\hbox{\rm max}}_{[\lambda,a,\infty]}$ stand for the
minimal and maximal large solutions of problem $(5.1)$,
respectively. Thus, if $(5.1)$ possesses a unique large solution,
say $\Theta_{[\lambda,a,\Omega]}$, then
$$
\lim_{t\uparrow\infty}\|u_{[\lambda,a,\Omega]}(\cdot,t;u_0)-
\Theta_{[\lambda,a,\Omega]}\|_{C^{2+\nu}(K)}=0
\eqno (6.3)
$$
for any compact subset $K$ of $\Omega_k$, and therefore due to
Theorem 2.2(iii) the solution
$u_{[\lambda,a,\Omega]}(\cdot,t;u_0)$ is point-wise convergent as
$t\uparrow\infty$ to the metasolution of order $k$ supported in
$\Omega_k$ associated with $\Theta_{[\lambda,a,\Omega]}$.
\endproclaim
\demo{Proof} The notations introduced in the proof of
Theorem 5.3 will be kept out. Consider $1\leq k \leq m$ and
$\lambda$ within its corresponding range of values. Let $K\subset
\Omega_k$ compact and consider $\delta>0$ sufficiently small so
that
$$
K \subset \Omega_k^\delta\subset \Omega_k\,.
$$
Set
$$
M_L:= \inf_{\partial\Omega_k^\delta\cap\Omega}u_{[\lambda,a,\Omega]}(\cdot,1;u_0)\,,
\quad
M_S:= \sup_{\partial\Omega_k^\delta\cap\Omega}u_{[\lambda,a,\Omega]}(\cdot,1;u_0)
$$
and consider the auxiliary evolution problems
$$\gathered
{\partial u\over \partial t} -\Delta u=\lambda u -a(x)u f(x,u)
\quad \hbox{\rm in }\Omega_k^\delta\times (0,\infty)\,, \cr
u =M>0\,, \quad
\hbox{\rm on }(\partial\Omega_k^\delta\cap\Omega)\times(0,\infty) \,,\cr
u=0 \quad \hbox{\rm on }
(\partial\Omega_k^\delta\cap \partial\Omega)\times (0,\infty)\,,\hfill \cr
u(\cdot,0)=u_1\,, \quad \hbox{\rm in }\bar
\Omega_k^\delta\,,\endgathered
\eqno (6.4)
$$
where
$$
M\in\{M_L,M_s\}\,,\quad u_1:=u_{[\lambda,a,\Omega]}(\cdot,1;u_0)\,.
$$
Thanks to the parabolic maximum principle, for each $t\geq 1$ we
have that
$$
u_{[\lambda,a,\delta,M_L]}(\cdot,t-1;u_1)
\leq u_{[\lambda,a,\Omega]}(\cdot,t;u_0)
\leq u_{[\lambda,a,\delta,M_S]}(\cdot,t-1;u_1) \quad
\hbox{in }\Omega_k^\delta\,,
$$
where $u_{[\lambda,a,\delta,M]}(\cdot,t-1;u_1)$ is the unique
global regular solution of (6.4), whose existence is guaranteed by
Theorem 6.1(i). Moreover, it follows from Theorem 6.1(ii) that
$$
\lim_{t\uparrow\infty}\|u_{[\lambda,a,\delta,M]}(\cdot,t-1;u_1)-
\Theta_{[\lambda,a,\delta,M]}\|_{C^1(\bar \Omega_k^\delta)}=0\,,
\eqno (6.5)
$$
where $\Theta_{[\lambda,a,\delta,M]}$ is the unique regular
positive steady-state of (6.4), whose existence is guaranteed by
Theorem 3.1. In particular, for each $\delta>0$ we have that
$$
\Theta_{[\lambda,a,\delta,M_L]}\leq \liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}
(\cdot,t;u_0)\leq \limsup_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,
t;u_0)\leq \Theta_{[\lambda,a,\delta,M_S]}
\eqno (6.6)
$$
in $\bar \Omega_k^\delta$. On the other hand, thanks to Theorem
2.2(iii) and the parabolic estimates of \cite{Re82} and
\cite{Re86} it is easily seen that
$$
\lim_{\delta\downarrow 0}M_L=\infty\,.
$$
Thus, thanks to the analysis already done in the proof of Theorem
5.3,
$$
\Theta^{\hbox{\rm min}}_{[\lambda,a,\Omega]}\!\leq\!
\liminf_{t\uparrow\infty}u_{[\lambda,a,\Omega]}
(\cdot,t;u_0)\!\leq\!\limsup_{t\uparrow\infty}u_{[\lambda,a,\Omega]}(\cdot,
t;u_0)\!\leq\!\inf_{\delta>0}\Theta_{[\lambda,a,\delta,\infty]}\!=\!
\Theta^{\hbox{\rm max}}_{[\lambda,a,\Omega]}\,.
$$
This completes the proof of (6.3). Moreover, it shows that there
exists a constant $M>0$ such that
$$
\|u_{[\lambda,a,\Omega]}(\cdot,t;u_0)\|_{C(K)}\leq M\,.
$$
Hence, by the results of \cite{Re86}, the restriction of
$\Gamma(u_0)$ to $K$, say $\Gamma_K(u_0)$, is relatively compact
in $C^{2-}(K)$. Therefore, by the parabolic Schauder estimates,
\cite{LSU68}, there exists $\nu>0$ such that $\Gamma_K(u_0)$ is
relatively compact in $C^{2+\nu}(K)$. This completes the proof.
\qed
\enddemo
\bigskip
\noindent {\bf References}
\bigskip
\refindentwd 1.8cm
\ref \key {Am76} \by H Amann \paper Fixed point equations
and nonlinear eigenvalue problems in ordered Banach spaces
\jour SIAM Review \vol 18 \yr 1976 \pages 620--709 \endref
\smallskip
\ref \key {AL97} \by H Amann and J L\'opez-G\'omez \paper
A priori bounds and multiple solutions for superlinear indefinited
elliptic problems \jour J. Diff. Eqns. \yr 1998
\vol 146 \pages 336-374 \endref
\smallskip
\ref \key {BM91} \by C Bandle and M Marcus \paper Large
solutions of semilinear elliptic equations: Existence, uniqueness
and asymptotic behavior \jour J. D'Analysis Math.
\vol 58 \yr 1991 \pages 9-24 \endref
\smallskip
\ref \key {BO86} \by H Brezis and L Oswald \paper Remarks
on sublinear elliptic equations \jour Nonl. Anal. TMA \yr 1986
\pages 55-64 \endref
\smallskip
\ref \key {CR71} \by M G Crandall and P H Rabinowitz \paper
Bifurcation from simple eigenvalues \jour J. Funct. Anal.
\vol 8 \yr 1971 \pages 321--340 \endref
\smallskip
\ref \key {DK92} \by D Daners and P Koch-Medina \book
Abstract Evolution Equations, Periodic Problems and Applications
\publ Pitman R.N.M.S. 279, Pitman-Longman
\publaddr Essex \yr 1992 \endref
\smallskip
\ref \key {DH99} \by Y Du and Q Huang \paper Blow-up
solutions for a class of semilinear elliptic and parabolic
equations \jour Preprint \yr 1999 \endref
\smallskip
\ref \key {FKLM96} \by J M Fraile, P Koch-Medina, J L\'opez-G\'omez
and S Merino \paper Elliptic eigenvalue problems and unbounded
continua of positive solutions of a semilinear equation \jour J.
Diff. Eqns. \vol 127 \yr 1996 \pages 295-319 \endref
\smallskip
\ref \key {GGLS98} \by J Garc\'{\i}a-Meli\'an,
R G\'omez-Re\~nasco, J L\'opez-G\'omez and J C Sabina de Lis
\paper Pointwise growth and uniqueness of
positive solutions for a class of sublinear elliptic problems
where bifurcation from infinity occurs
\jour Arch. Rat. Mech. Anal. \vol 145
\yr 1998 \pages 261-289 \endref
\smallskip
\ref \key {GT83} \by D Gilbarg and N Trudinger \book Elliptic
Partial Differential Equations of Second Order \yr 1983\vol
\publ Springer Verlag, Berlin/New York \endref
\smallskip
\ref \key {Go99} \by R G\'omez-Re\~nasco \book The
effect of varying coefficients in semilinear elliptic boundary
value problems. From classical solutions to metasolutions \publ Ph
D Dissertation \publaddr Universidad de La Laguna, Tenerife \yr
March 1999\endref
\smallskip
\ref \key {GL98} \by R G\'omez-Re\~nasco and
J L\'opez-G\'omez \paper On the existence and numerical
computation of classical and non-classical solutions for a family
of elliptic boundary value problems
\jour Non. Anal. TMA, In press \yr 1998 \endref
\smallskip
\ref \key {Ke57} \by J B Keller \paper On solutions
of $\Delta u =f(u)$ \jour Comm. Pure Appl. Maths. \vol X
\pages 503-510 \yr 1957 \endref
\smallskip
\ref \key {LSU68} \by O A Ladyzenskaja, V A Solonnikov and
N N Uraltzeva \book Linear and Quasilinear Equations of Parabolic
type \publ Amer. Math. Soc. \publaddr Providence
\yr 1968 \endref
\ref \key {LM93} \by A C Lazer and P L McKenna \paper
On a problem of L Bieberbach and H Rademacher \jour Nonl. Anal.
\vol 21 \pages 327-335 \yr 1993 \endref
\smallskip
\ref \key {LM94} \by A C Lazer and P L McKenna \paper
A singular elliptic boundary value problem \jour Appl. Math. Comp.
\vol 65 \pages 183-194 \yr 1994 \endref
\smallskip
\ref \key {Lo96} \by J L\'opez-G\'omez \paper The maximum
principle and the existence of principal eigenvalues for some
linear weighted boundary value problems \jour J. Diff. Eqns. \vol
127 \yr 1996 \pages 263-294 \endref
\smallskip
\ref \key {LM94} \by J L\'opez-G\'omez and M Molina-Meyer
\paper The maximum principle for cooperative weakly
coupled elliptic systems and some applications \jour Diff. Int.
Equns. \vol 7 \yr 1994 \pages 383-398 \endref
\smallskip
\ref \key {LS98} \by J L\'opez-G\'omez and J C Sabina \paper
First variations of principal eigenvalues with respect to the
domain and point-wise growth of positive solutions for problems
where bifurcation from infinity occurs
\jour J. Diff. Eqns. \vol 148 \yr 1998 \pages 47-64 \endref
\smallskip
\ref \key {MV97} \by M Marcus and L V\'eron \paper Uniqueness
and asymptotic behavior of solutions with boundary blow-up for a
class of nonlinear elliptic equations \jour Ann. Inst. Henri
Poincar\'e \vol 14 \yr 1997 \pages 237-274 \endref
\smallskip
\ref \key {Mu93} \by J D Murray \book Mathematical Biology
\publ Springer Biomathematics Texts \vol 18 \yr 1993
\endref
\smallskip
\ref \key {Ok80} \by A Okubo \jour Diffusion and Ecological Problems:
Mathematical Models. Springer, New York 1980 \endref
\smallskip
\ref \key {Os57} \by R Osserman \paper On the inequality
$\Delta u \geq f(u)$ \jour Pacific J. Maths. \vol 7 \pages
1641-1647 \yr 1957 \endref
\smallskip
\ref \key {Ou92} \by T Ouyang \paper On the positive solutions of
semilinear equations $\Delta u + \lambda u - h u^p=0$ on the
compact manifolds
\jour Trans. Amer. Math. Soc. \vol 331 \yr 1992 \pages 503--527 \endref
\smallskip
\ref \key {Re82} \by R Redlinger \paper \"Uber die
$C^2$-Kompacktheit der Bahn von L\"osungen semilinearer
parabolischer Systeme \jour Proc. Roy. Soc. Edinburgh A
\vol 93 \yr 1982 \pages 99-103 \endref
\smallskip
\ref \key {Re86} \by R Redlinger \paper Compactness
results for time dependent parabolic systems \jour J. Diff. Eqns.
\vol 64 \yr 1986 \pages 133-153 \endref
\smallskip
\ref \key {Sa73} \by D Sattinger \book Topics in Stability
and Bifurcation Theory \publ Lectures Notes in Mathematics 309,
Springer \publaddr Berlin/New York \yr 1973 \endref
\smallskip
\ref \key {Ve92} \by L Veron \paper Semilinear elliptic
equations with uniform blow up on the boundary \jour J. D'Analyse
Math. \vol 59 \yr 1992 \pages 231-250 \endref
\enddocument