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\nopagenumbers
\pageno=223
\input amssym.def % The R for Real nunbers.
\font\eightrm=cmr8 \font\eighti=cmti8 \font\eightbf=cmbx8
\headline={\ifnum\pageno=223 \hfill\else%
{\tenrm\ifodd\pageno\rightheadline \else
\leftheadline\fi}\fi}
\def\rightheadline{\eightrm\hfil Crandall--Rabinowitz Type Bifurcations
\hfil\folio}
\def\leftheadline{\eightrm\folio\hfil Bettina E.~Schmidt \hfil}
\voffset=2\baselineskip
\vbox {\eightrm\noindent\baselineskip 9pt %
Differential Equations and Computational Simulations III\hfill\break
J.~Graef, R.~Shivaji, B.~Soni, \& J.~Zhu (Editors)\hfill\break
Electronic Journal of Differential Equations, Conference~01, 1997,
pp.~223--235.\hfill\break
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\hfil\break ftp 147.26.103.110 or 129.120.3.113 (login: ftp)}
\footnote{}{\vbox{\hsize=10cm\eightrm\noindent\baselineskip 9pt %
1991 {\eighti Subject Classification:} 34B15, 34C23, 46N20.
\hfil\break
{\eighti Key words and phrases:} Neumann problem, nonlinear eigenvalue problem,
bifurcation from simple eigenvalues, Crandall-Rabinowitz theorem,
regular-singular points, perturbed bifurcation theory.
\hfil\break
\copyright 1998 Southwest Texas State University and
University of North Texas. \hfil\break
Published November 12, 1998.} }
\bigskip\bigskip
\centerline{PERSISTENCE OF CRANDALL--RABINOWITZ TYPE BIFURCATIONS}
\centerline{UNDER SMALL PERTURBATIONS}
\bigskip
\centerline{BETTINA E.~SCHMIDT}
\bigskip\bigskip
{\eightrm\baselineskip=10pt \narrower
\centerline{\eightbf Abstract}
\noindent
We discuss a class of nonlinear operator equations in a Banach space setting
and present a generalization of the Crandall-Rabinowitz bifurcation theorem
that describes the effect of small perturbations of the operators involved on
the local structure of the solution set in the vicinity of a bifurcation
point of the unperturbed equation. The result is applied to a
parameter-dependent Neumann boundary-value problem with spatially homogeneous
source terms that exhibits infinitely many bifurcation points.
We obtain conditions for the persistence or nonpersistence of these
bifurcations under small, spatially inhomogeneous perturbations of the
source terms.\par}
\bigskip\bigskip
\bigbreak
\centerline{\smallrm INTRODUCTION}
\nobreak\noindent
One of the most frequently quoted works in local bifurcation theory is
the 1971 paper [2], by M.~G.~Crandall and P.~H.~Rabinowitz, on bifurcation from
simple eigenvalues. Roughly speaking, the paper's main result asserts that if
$H$ is a $C^2$-mapping between Banach spaces and if $w_0$ is a {\it
regular-singular point\/} of $H$ (see Section~1 for the definition), then there
exists a neighborhood $U$ of $w_0$ such that the set $\{w\in U\mid
H(w)=H(w_0)\}$ is the union of two simple $C^1$-arcs that intersect
transversally at $w_0$.
In countless applications, this result has been used, in one way or other, to
establish the existence of bifurcation points on given solution curves of
so-called nonlinear eigenvalue problems. Continuing the tradition, we use it
here to prove the occurrence of infinitely many bifurcations from a branch of
trivial (that is, constant) solutions of a parameter-dependent Neumann
boundary-value problem (P) with spatially homogeneous source terms. Our main
concern, however, is the effect of small, spatially inhomogeneous perturbations
of the source terms on the local structure of the solution set of Problem~(P).
If the perturbation is ``small,'' the solution set of the perturbed problem
(P$_\epsilon$) should be somehow ``close'' to the solution set of the
unperturbed problem (P), although (P$_\epsilon$) has no {\it trivial\/}
solutions, due to the spatial inhomogeneity. It is rather obvious that near
{\it regular points\/} on a given solution curve of (P), the effect of the
perturbation is simply a continuous deformation of the curve. What happens near
{\it bifurcation points\/} (or more specifically, near {\it regular-singular
points\/}) is much less obvious: Will the bifurcations persist or will they
``unfold''? Does the letter $X$, under small perturbations, become a slightly
distorted letter $X$, or will it turn into something like the union of two
letters $V$ (not intersecting and facing each other tip to tip)?
We will argue that in the abstract setting of the Crandall-Rabinowitz theorem,
either scenario is possible. Nonpersistence (or ``unfolding'') of bifurcations
is, in a sense, the generic case, but at least if the operators involved exhibit
certain symmetries, bifurcations may very well persist. In our Neumann problem
(P), for example, every other one of infinitely many bifurcations is persistent
under small spatially inhomogeneous perturbations with a certain symmetry
property.
The paper is organized as follows. In Section~1 we present a general
functional-analytic framework for ``perturbed Crandall--Rabinowitz
bifurcation.''
We do not claim the two theorems in this section to be entirely new, but we are
not aware of any directly quotable reference. In Sections~2 and 3 we study the
trivial solution curve of our Neumann problem and, using the
Crandall--Rabinowitz
theorem, prove the existence of infinitely many bifurcation points. In Section~4
we apply the abstract results of Section~1 to describe possible effects of
small, spatially inhomogeneous perturbations on the local structure of the
solution set of (P) in the vicinity of a bifurcation point.
\bigskip
\bigbreak
\centerline{1. \smallrm PERTURBED CRANDALL--RABINOWITZ BIFURCATION}
\nobreak\noindent
In this section, we briefly discuss some general results on the local
structure of the solution sets of certain types of nonlinear operator equations
in Banach spaces. Our main concern is the effect of a small perturbation of such
an equation on the local structure of the solution set in the vicinity of a
Crandall--Rabinowitz type bifurcation point. The proofs of our main results,
which rely on the Ljapunov--Schmidt method, the implicit function theorem, and
the Morse lemma, are rather straightforward, but quite technical and lengthy.
For details, the reader is referred to [7].
By way of motivation, consider a simple, algebraic equation in two variables,
$$
H(x,y)=0,\eqno(1.1)
$$
where $H\in C^2({\Bbb R}^2,{\Bbb R})$. Suppose that $(x_0,y_0)$ is a solution of
Equation~(1.1). We call $(x_0,y_0)$ a {\it regular point\/} of $H$ if the
gradient of $H$ does not vanish at $(x_0,y_0)$, that is, if at least one of the
two partial derivatives is nonzero. By the implicit function theorem, it is then
possible to solve Equation~(1.1) for either $x$ or $y$. In any case, the
solution set is locally a simple $C^1$-arc.
We call $(x_0,y_0)$ a {\it regular-singular point\/} of $H$ if the gradient of
$H$ vanishes at $(x_0,y_0)$, while the Hessian of $H$ at $(x_0,y_0)$ has one
positive and one negative eigenvalue. In this case, the Morse Lemma implies that
$(x_0,y_0)$ is a {\it saddle point\/} of $H$. That is, the solution set of
Equation~(1.1) is locally, near $(x_0,y_0)$, the union of two simple $C^1$-arcs,
transversally intersecting at $(x_0,y_0)$.
Both scenarios have natural generalizations in a Banach space setting. Consider
a $C^2$-mapping $H$ from an open set $\Omega$ in a Banach space $W$ into a
Banach space $Y$. Let $w_0$ be a point in $\Omega$ and suppose that the
Fr\'echet derivative $H'(w_0)$ is a Fredholm operator of index~1 (that is, the
range $R(H'(w_0))$ of $H'(w_0)$ is closed and of finite codimension in $Y$, the
nullspace $N(H'(w_0))$ of $H'(w_0)$ is finite-dimensional, and $\dim
N(H'(w_0))-\dim Y/R(H'(w_0))=1$). The point $w_0$ is called a {\it regular
point\/} of $H$ if $\dim N(H'(w_0))=1$ (or equivalently, if $R(H'(w_0))=Y$).
Otherwise, $w_0$ is called a {\it singular point\/} of $H$. Specifically, $w_0$
is called a {\it regular-singular point\/} of $H$ if $\dim N(H'(w_0))=2$ and
there exists a basis $\{w_1,w_2\}$ of $N(H'(w_0))$ such that the quadratic
derivative $H''(w_0)w_1w_1$ belongs to $R(H'(w_0))$, while the mixed derivative
$H''(w_0)w_1w_2$ does not. (Note that if $N(H'(w_0))$ is two-dimensional, then
$R(H'(w_0))$ has codimension~1 in $Y$.) We mention that whenever $w_0$ is
regular-singular in the above sense, there exists in fact a basis
$\{\tilde w_1,\tilde w_2\}$ of $N(H'(w_0))$ such that {\it both\/} quadratic
derivatives, $H''(w_0)\tilde w_1\tilde w_1$ and $H''(w_0)\tilde w_2\tilde w_2$,
belong to $R(H'(w_0))$, while the mixed derivative $H''(w_0)\tilde w_1\tilde
w_2$ does not.
Just as in the case of Equation~(1.1), the implicit function theorem guarantees
that if $w_0$ is a regular point of $H$, then the solution set of the equation
$H(w)=H(w_0)$ is locally, near $w_0$, a simple $C^1$-arc. On the other hand, if
$w_0$ is a regular-singular point of $H$, then the Crandall-Rabinowitz
bifurcation theorem (see [2, Theorem~1]) applies and asserts that the solution
set of $H(w)=H(w_0)$ is locally, near $w_0$, the union of two simple $C^1$-arcs
that intersect transversally at $w_0$. In the sequel, a point $w_0$ with this
latter property will be referred to as a {\it Crandall-Rabinowitz point\/} of
$H$.
A question of interest is how the local structure of the solution set changes
when the mapping $H$ is perturbed. To study this question, let us now consider a
{\it family\/} of mappings $H(\epsilon,\cdot)$ from an open subset $\Omega$ of a
Banach space $W$ into a Banach space $Y$, where $\epsilon$ varies over an open
interval $J$ containing 0. We think of $H(\epsilon,\cdot)$, for $\epsilon\in J$,
as a perturbation of the mapping $H(0,\cdot)$ and wish to describe the structure
of the set $\Sigma_\epsilon:=\{w\in\Omega\mid H(\epsilon,w)=0\}$, for $\epsilon$
close to 0, in the vicinity of a point $w_0\in\Omega$ with $H(0,w_0)=0$.
Throughout we assume that $H\in C^2(J\times\Omega,Y)$ and that the (partial)
Fr\'echet derivative $H_w(0,w_0)={\partial H\over\partial w}(0,w_0)$ is a
Fredholm operator of index~1.
A straightforward generalization of the implicit function theorem shows that if
$w_0$ is a {\it regular point\/} of $H(0,\cdot)$, then not only for
$\epsilon=0$, but for every $\epsilon$ sufficiently close to 0, the set
$\Sigma_\epsilon$ is locally, near $w_0$, a simple $C^1$-arc (which varies
continuously with $\epsilon$). For the case where $w_0$ is a {\it
regular-singular point\/} of $H$, we have the following generalization of the
Crandall-Rabinowitz theorem (see [7, Theorem~4.6]).
%\medskip
\noindent{\bf Theorem~1.1.} Suppose $w_0\in\Omega$ is a regular-singular point
of $H(0,\cdot)$. Then there exist
\item{$\bullet$} an open interval $I\subset J$ with $0\in I$,\parskip=0pt
\item{$\bullet$} an open neighborhood $U\subset\Omega$ of $w_0$,
\item{$\bullet$} a continuous mapping $\overline w:D\to U$, defined on an open
set $D\subset{\Bbb R}\times{\Bbb R}^2$ that contains $(\epsilon,0,0)$ for every
$\epsilon\in I$, with $\overline w(0,0,0)=w_0$, and
\item{$\bullet$} a continuously differentiable mapping $\rho:I\to{\Bbb R}$ with
$\rho(0)=0$,\parskip=5pt\par\noindent
such that for every $\epsilon\in I$, the mapping $\overline
w(\epsilon,\cdot,\cdot)$ is one-to-one and continuously differentiable, with
linearly independent derivatives $\overline w_\sigma(\epsilon,\sigma,\tau)$ and
$\overline w_\tau(\epsilon,\sigma,\tau)$ for all $(\sigma,\tau)\in{\Bbb R}^2$
with $(\epsilon,\sigma,\tau)\in D$, and we have
$$
\{w\in U\mid H(\epsilon,w)=0\}=\{\overline w(\epsilon,\sigma,\tau)\mid
\sigma\tau+\rho(\epsilon)=0\hbox{ and }(\epsilon,\sigma,\tau)\in D\}\,.
$$
%\medskip
Roughly speaking, the assertion of Theorem~1.1 is that for every $\epsilon$
sufficiently close to 0, the solution set $\Sigma_\epsilon$ of the equation
$H(\epsilon,w)=0$ is locally, near $w_0$, a homeomorphic image of the solution
set, near $(0,0)$, of a simple, algebraic equation in ${\Bbb R}^2$, namely,
$\sigma\tau+\rho(\epsilon)=0$. Thus, whenever we have $\rho(\epsilon)=0$, the
set $\Sigma_\epsilon$ is locally, near $w_0$, the union of two simple $C^1$-arcs
that intersect transversally at the point $\overline w(\epsilon,0,0)$. In
particular, we recover the Crandall-Rabinowitz theorem, since $\rho(0)=0$ and
$\overline w(0,0,0)=w_0$. But whenever $\rho(\epsilon)\ne0$, the set
$\Sigma_\epsilon$ is locally, near $w_0$, the union of two disjoint, simple
$C^1$-arcs.
Intuitively, this result is not very surprising. Recall that in the case of an
algebraic equation in the plane, $w_0$ would be a saddle point of the
unperturbed mapping $H(0,\cdot)$, located on the level-0 set of $H(0,\cdot)$.
For $\epsilon$ close to 0, the perturbed mapping $H(\epsilon,\cdot)$ would still
have a (unique) saddle point $w_\epsilon$ close to $w_0$, but at a level
$\rho(\epsilon)$ in general different from 0. Only if $\rho(\epsilon)=0$, would
the level-0 set of $H(\epsilon,\cdot)$ be ``cross-shaped'' near $w_0$.
These last observations, too, can be naturally extended to our abstract Banach
space setting (see [7, Theorem~4.16]).
%\medskip
\noindent{\bf Theorem~1.2.} Suppose $w_0\in\Omega$ is a regular-singular point
of $H(0,\cdot)$ and $y_0\in Y$ is a vector that does not belong to the range of
$H_w(0,w_0)$. Then there exist an open interval $I\subset J$ with $0\in I$ and
an open neighborhood $U\subset\Omega$ of $w_0$ such that for every $\epsilon\in
I$, the mapping $H(\epsilon,\cdot)$ has a unique singular point $w_\epsilon\in
U$ whose image $H(\epsilon,w_\epsilon)$ is a constant multiple of $y_0$. The
mapping $\epsilon\mapsto w_\epsilon$ is continuously differentiable, and for
every $\epsilon\in I$, the point $w_\epsilon$ is a regular-singular point (and
thus, a Crandall-Rabinowitz point) of $H(\epsilon,\cdot)$. Only if
$H(\epsilon,w_\epsilon)=0$, does the set $\{w\in U\mid H(\epsilon,w)=0\}$
contain a singular point of $H(\epsilon,\cdot)$.
%\medskip
Now let $w_0\in\Omega$ be a regular-singular point (and thus, a
Crandall-Rabinowitz point) of $H(0,\cdot)$. We say that the bifurcation at $w_0$
is {\it persistent\/} (or {\it nonpersistent\/}, respectively) if there exists a
neighborhood $U\subset\Omega$ of $w_0$ such that for every
$\epsilon\in J\setminus\{0\}$ sufficiently close to 0, the set
$\{w\in U\mid H(\epsilon,w)=0\}$ contains a regular-singular point
(or does not contain a singular point, respectively) of $H(\epsilon,\cdot)$.
It is easy to derive a simple, sufficient condition for {\it nonpersistence\/}.
To that end, pick a vector $y_0\in Y\setminus R(H_w(0,w_0))$ and let $y_0^*\in
Y^*$ denote the (unique) functional with ${<}y_0^*,y_0{>}=1$ and
$N(y_0^*)=R(H_w(0,w_0))$. Choose an interval $I$, a neighborhood $U$, and points
$w_\epsilon$ according to Theorem~1.2 and define $\rho\in C^1(I)$ by
$\rho(\epsilon):={<}y_0^*,H(\epsilon,w_\epsilon){>}$. Then we have
$H(\epsilon,w_\epsilon)=\rho(\epsilon)y_0$, for all $\epsilon\in I$, and the set
$\{w\in U\mid H(\epsilon,w)=0\}$ contains a singular point of
$H(\epsilon,\cdot)$ if and only if $\rho(\epsilon)=0$. Since $\rho(0)=0$, it
follows that $\rho'(0)\ne0$ is a sufficient condition for nonpersistence. Using
the fact that $y_0\notin R(H_w(0,w_0))$, we readily show that
$\rho'(0)={<}y_0^*,H_\epsilon(0,w_0){>}$ and conclude that the bifurcation at
$w_0$ is nonpersistent provided that
$$
H_\epsilon(0,w_0)\notin R(H_w(0,w_0)),\eqno(1.2)
$$
that is, provided that the partial derivative $H_\epsilon(0,w_0)$ does not
belong to the
hyperplane $R(H_w(0,w_0))$. This is, in a sense, the generic case: In the
absence of special symmetries, the condition (1.2) is likely to be satisfied.
We do not have an equally simple, sufficient condition for {\it persistence\/}
(obviously, $H_\epsilon(0,w_0)\in R(H_w(0,w_0))$ is {\it necessary\/}, but not
{\it sufficient\/}). However, in concrete applications it is often possible to
show directly that $\rho(\epsilon)$ must vanish for all $\epsilon$ near 0. Not
surprisingly, the argument is usually based on special symmetry properties of
the mapping $H$. A specific example will be discussed in Section~4.
\bigskip
\bigbreak
\centerline{2. \smallrm A SPATIALLY HOMOGENEOUS NEUMANN PROBLEM}
\nobreak\noindent
Consider the boundary-value problem
$$
-{d\over dx}\Bigl(k(x){du\over dx}\Bigr)=\mu f(u)-g(u)\quad\hbox{in }(-1,1),
\qquad u'(\pm1)=0.\eqno({\rm P})
$$
Here, $\mu$ is a nonnegative parameter, and we seek nonnegative classical
solutions $u$ in $C^2([-1,1])$. Our assumptions on the data are as follows: The
coefficient $k$ is a positive, continuously differentiable function on $[-1,1]$.
The function $f$ is continuous on ${\Bbb R}_+$, and there exists a number $c>0$
such that $f(y)=0$ for all $y\in[0,c]$, while $f$ is twice continuously
differentiable on $(c,\infty)$, with $f>0$, $f'\ge0$, and $f''\le0$. The
function $g$ is continuous on ${\Bbb R}_+$, with $g(0)=0$, and twice
continuously
differentiable on $(0,\infty)$, with $g>0$, $g'\ge0$, and $g''\ge0$. Finally we
assume that ${g(y)\over f(y)}\to\infty$ as $y\to\infty$. (Note that due to our
earlier assumptions, $f(y)$ grows {\it at most\/} linearly, $g(y)$ {\it at
least\/} linearly. Thus, the last assumption just excludes the case of both
functions being asymptotically linear.)
We can interpret Problem~(P) as a simple equilibrium model for heat conduction
in a thin rod with insulated ends. In this model, $k$ would be the rod's thermal
conductivity. The term $\mu f(u)$ would represent a parameter-dependent heat
source that kicks in as soon as the (absolute) temperature $u$ exceeds a certain
threshold $c$. The term $g(u)$ would represent radiative cooling ($g(u)\sim u^4$
if the process is governed by the Stefan-Boltzmann law).
Let $\Sigma$ denote the set of all pairs $(\mu,u)$ with $\mu\in{\Bbb R}_+$ and
$u\in C^2([-1,1])$ a nonnegative solution of~(P). With slight abuse of language,
we call the pairs $(\mu,u)\in\Sigma$ solutions of Problem~(P). A solution
$(\mu,u)$ is called {\it trivial\/} if the function $u$ is {\it constant\/}.
Clearly, $\Sigma$ contains exactly two maximal continua of trivial solutions,
namely, ${\Bbb R}_+\times\{0\}$ and the trace $\Sigma^*$ of the curve
$\mu={g(y)\over f(y)}$ with $y\in(c,\infty)$.
To describe the trivial solution branch $\Sigma^*$ in more detail, we need to
analyze the function $\bar\mu:={g\over f}\big|_{(c,\infty)}$. Clearly, we have
$\bar\mu(c+)=\infty=\bar\mu(\infty)$. Moreover, $\bar\mu'={\sigma\over f^2}$
with $\sigma:=g'f-gf'$. The function $\sigma$ is nondecreasing on $(c,\infty)$
(in fact, $\sigma'=g''f-gf''\ge0$), negative near $c$ (since $f(c)=0$ and
$f'(c+)\in(0,\infty]$), and positive near $\infty$ (else, $\bar\mu$ would be
nonincreasing throughout, contradicting the fact that $\bar\mu(\infty)=\infty$).
This implies the existence of two numbers $\underline y_0$ and $\overline y_0$
with $c<\underline y_0\leq\overline y_0<\infty$ such that $\sigma$ is negative
on $(c,\underline y_0)$, zero on $[\underline y_0,\overline y_0]$, and positive
on $(\overline y_0,\infty)$. The same then holds for $\bar\mu'$, and it follows
that $\bar\mu$ is strictly decreasing on $(c,\underline y_0)$ (with values
between $\infty$ and $\mu_0:=\bar\mu(\underline y_0)$), constant on
$[\underline y_0,\overline y_0]$ (with value $\mu_0$), and strictly increasing
on $(\overline y_0,\infty)$ (with values between $\mu_0$ and $\infty$). (Of
course, we will usually have $\underline y_0=\overline y_0$, except in
degenerate cases where the graphs of $f$ and $g$ contain parallel line
segments.)
We conclude that $\Sigma^*$ consists of the graphs of two functions, namely,
$\underline u:=(\bar\mu|_{(c,\underline y_0)})^{-1}$ and $\overline
u:=(\bar\mu|_{(\overline y_0,\infty)})^{-1}$, connected by the
vertical segment $\{\mu_0\}\times[\underline y_0,\overline y_0]$ (a turning
point if $\underline y_0=\overline y_0$). Both $\underline u$ and $\overline u$
are defined and twice continuously differentiable on $(\mu_0,\infty)$; the
former is strictly decreasing with range $(c,\underline y_0)$, the latter is
strictly increasing with range $(\overline y_0,\infty)$. The figure at the end
of the paper shows a typical example (see the discussion following Lemma~2.1
for details).
To investigate the possibility of bifurcations of nontrivial solutions from the
trivial solution branch $\Sigma^*$, we compute the eigenvalues of the
linearization of Problem~(P), with respect to $u$, at a point
$(\mu,y)\in\Sigma^*$, that is, the eigenvalues of
$$
-{d\over dx}\Bigl(k(x){dv\over dx}\Bigr)=\bigl(\mu f'(y)-g'(y)\bigr)v+\lambda
v\quad\hbox{in }(-1,1),\qquad v'(\pm1)=0.\eqno(2.1)
$$
If we enumerate the eigenvalues of $-{d\over dx}\bigl(k{d\over dx}\bigr)$
(under Neumann boundary conditions) as a strictly increasing sequence
$0=\ell_0<\ell_1<\ell_2<\dots\,$, then the eigenvalues of (2.1) are simply
given by $\lambda_j(\mu,y)=\ell_j+g'(y)-\mu f'(y)$, for $j\in{\Bbb Z}_+$. In
particular, $\lambda_0(\mu,y)=g'(y)-\mu f'(y)$ and
$$
\lambda_j(\mu,y)=\lambda_0(\mu,y)+\ell_j\,,\eqno(2.2)
$$
for $j\in{\Bbb N}$. Also, since $(\mu,y)\in\Sigma^*$, we have
$\mu=\bar\mu(y)={g(y)\over f(y)}$ and thus,
$$
\lambda_0(\mu,y)=g'(y)-\bar\mu(y)f'(y)=\bar\mu'(y)f(y)\,.\eqno(2.3)
$$
This implies that $\lambda_0$ is {\it positive\/} on the graph of $\overline u$
(the upper branch of $\Sigma^*$, where $\bar\mu'>0$), {\it zero\/} on the
vertical segment $\{\mu_0\}\times[\underline y_0,\overline y_0]$ (the possibly
degenerate turning point of $\Sigma^*$, where $\bar\mu'=0$), and {\it
negative\/} on the graph of $\underline u$ (the lower branch of $\Sigma^*$,
where $\bar\mu'<0$). Thus, the solutions on the upper branch are {\it stable\/},
while those on the lower branch are {\it unstable\/}, and bifurcations can
occur only on the lower branch, at points where one of the eigenvalues
$\lambda_j$ with $j\ge1$ vanishes (that is, at points $(\mu,y)\in\Sigma^*$
where $\lambda_0(\mu,y)=-\ell_j$ for some $j\ge1$).
To find out whether such points exist, we must trace the smallest eigenvalue,
$\lambda_0$, along the lower branch of $\Sigma^*$. To that end, define
$\bar\lambda:(c,\underline y_0)\to{\Bbb R}$ by
$\bar\lambda(y):=\lambda_0(\bar\mu(y),y)$. By (2.3), $\bar\lambda=g'-\bar\mu
f'=\bar\mu'f$, and we know that this is strictly negative on $(c,\underline
y_0)$, with $\bar\lambda(\underline y_0-)=0$. Moreover,
$\bar\lambda'=g''-\bar\mu'f'-\bar\mu f''\ge-\bar\mu'f'$, since $g''\ge0$,
$f''\le0$, and $\bar\mu>0$. Also, $\bar\mu'<0$ on $(c,\underline y_0)$, that is,
$\sigma=g'f-gf'<0$, and thus, $f'>{g'f\over g}>0$ (our assumptions on $g$ imply
$g'>0$ on $(0,\infty)$). This proves that $\bar\lambda'>0$ on $(c,\underline
y_0)$. Finally, we observe that $\bar\lambda(c+)=-\infty$, since
$\bar\lambda=g'-\bar\mu f'=g'-g{f'\over f}$ and ${f'(y)\over f(y)}\to\infty$ as
$y\to c+$ (note that $g'(c)>0$, $g(c)>0$, $f'(c+)\in(0,\infty]$, and
$f(c+)=0+$).
Summarizing, we showed that $\bar\lambda$ is strictly increasing on
$(c,\underline y_0)$, with range $(-\infty,0)$ and with a strictly positive
derivative.
But this means that the function $\mu\mapsto\lambda_0(\mu,\underline u(\mu))$ is
strictly decreasing on $(\mu_0,\infty)$, with range $(-\infty,0)$ and with a
strictly
negative derivative. Because of (2.2), it follows that in fact {\it all\/} the
eigenvalues $\lambda_j$ are strictly decreasing and eventually negative along
the lower branch of $\Sigma^*$, with ${d\over d\mu}\lambda_j(\mu,\underline
u(\mu))<0$ for all $\mu\in(\mu_0,\infty)$. In particular, each of the
eigenvalues $\lambda_j$ with $j\ge1$ has a unique nondegenerate zero on the
lower branch of $\Sigma^*$.
The following lemma gathers our findings about the trivial solutions of
Problem~(P).
%\medskip
\noindent{\bf Lemma 2.1.} (a) There are exactly two maximal continua of trivial
solutions of Problem~(P), namely ${\Bbb R}_+\times\{0\}$ and the trace
$\Sigma^*$
of the curve $\mu={g(y)\over f(y)}$ with $y\in(c,\infty)$. The set $\Sigma^*$
consists of the graphs of two functions $\underline u$, $\overline u\in
C^2((\mu_0,\infty))$ and a vertical segment $\{\mu_0\}\times[\underline
y_0,\overline y_0]$, where $\mu_0>0$ and $c<\underline y_0\le\overline
y_0<\infty$. The function $\underline u$ is strictly decreasing with range
$(c,\underline y_0)$, while the function $\overline u$ is strictly increasing
with range $(\overline y_0,\infty)$.
(b) Denoting by $\bigl(\lambda_j(\mu,y)\bigr)_{j\in{\Bbb Z}_+}$ the strictly
increasing enumeration of the eigenvalues of the linearization of Problem~(P) at
$(\mu,y)\in\Sigma^*$, we have $\lambda_0>0$ on the graph of $\overline u$,
$\lambda_0=0$ on the vertical segment $\{\mu_0\}\times[\underline y_0,\overline
y_0]$, and $\lambda_0<0$ on the graph of $\underline u$. All the eigenvalues
$\lambda_j$ are strictly decreasing and eventually negative along the graph of
$\underline u$, with ${d\over d\mu}\lambda_j(\mu,\underline u(\mu))<0$ for all
$\mu\in(\mu_0,\infty)$. In particular, each of the eigenvalues $\lambda_j$ with
$j\geq1$ has a unique zero $(\mu_j,u_j)$ on $\Sigma^*$, with
$\mu_0<\mu_1<\mu_2<\dots$ and $u_j=\underline u(\mu_j)$ for $j\in{\Bbb N}$.
%\goodbreak\medskip
The figure at the end of the paper depicts the trivial solution branch
$\Sigma^*$
in a typical situation, where $f(y):=\sqrt{y-1}$, for $y>c:=1$, and $g(y)=y^4$,
for $y>0$. Open circles mark the location of the first few of the potential
bifurcation points $(\mu_j,u_j)$, $j=1,2,3\ldots$; those were found by
solving the equation $\bar\lambda(u_j)=-\ell_j$, under the assumption that
$k\equiv1$ (so that $\ell_j={\pi^2\over4}j^2$, for $j\in{\Bbb N}$).
Next we collect some a-priori information about possible nontrivial solutions of
Problem~(P).
%\medskip
\noindent{\bf Lemma 2.2.} Let $(\mu,u)$ be a nontrivial solution of Problem~(P).
Then we have $\mu>\mu_0$ and $0__0$, and thus, $\mu
f(u(x_0))-g(u(x_0))>0$. But this implies $u(x_0)>c$ and
$\mu>\bar\mu(u(x_0))={g(u(x_0))\over f(u(x_0))}$, or equivalently, $\mu>\mu_0$
and $\underline u(\mu)____c\;\forall x\in[-1,1]\}$}. Let $X_+$ denote the
cone
of nonnegative functions in $X$. Since $X$ embeds compactly into $C([-1,1])$
and $C([-1,1])$ embeds continuously into $Y$, the nonlinear operator
$N:{\Bbb R}_+\times X_+\to Y$, defined by
$$
N(\mu,u):=\mu f\circ u - g\circ u,
$$
is completely continuous (with respect to the norms of ${\Bbb R}\times X$ and
$Y$) and twice continuously differentiable on $\Omega:=(0,\infty)\times\{u\in
X\mid u(x)>c\;\forall x\in[-1,1]\}$ (an open subset of ${\Bbb R}\times X$).
With these definitions, Problem~(P) is equivalent to the equation $Lu=N(\mu,u)$,
which we can also write as
$$
H(\mu,u)=0,\eqno(\tilde{\rm P})
$$
where $H:{\Bbb R}_+\times X_+\to Y$ is defined by
$$
H(\mu,u):=Lu-N(\mu,u).
$$
More precisely, the solution set $\Sigma$ of Problem~(P) coincides with that of
($\tilde{\rm P}$):
$$
\Sigma=\{(\mu,u)\in{\Bbb R}_+\times X_+\mid H(\mu,u)=0\}.
$$
We consider $\Sigma$ a metric subspace of ${\Bbb R}\times X$; as such it is
closed and locally compact. Moreover, routine arguments show that the topology
of $\Sigma$ as a metric subspace of ${\Bbb R}\times X$ coincides with the metric
topologies it inherits from ${\Bbb R}\times C^m([-1,1])$ with $m=0$, $1$, or
$2$.
Finally, we observe that the eigenvalue problem (2.1), obtained by linearizing
Problem~(P) at a point $(\mu,y)\in\Sigma^*$, is equivalent to the abstract
eigenvalue problem for the operator
$$
H_u(\mu,y)=L-N_u(\mu,y)=L+\lambda_0(\mu,y)Id\,,
$$
which is selfadjoint as an unbounded operator in the Hilbert space $Y$.
The following lemma provides the basis for applying the abstract results of
Section~1.
%\medskip
\noindent{\bf Lemma 3.1.} For every $(\mu,y)\in\Sigma^*$, the Fr\'echet
derivative $H'(\mu,y)$ is a Fredholm operator of index~1. The only singular
points of $H$ on $\Sigma^*$ are the zeros $(\mu_j,u_j)$ of the eigenvalues
$\lambda_j$ with $j\ge1$, as given in Lemma~2.1(b). All of those points are in
fact regular-singular points of $H$.
%\medskip
\noindent{\it Proof.\/} The Fr\'echet derivative of $H$ at $(\mu,y)\in\Sigma^*$
is given by
$$
H'(\mu,y)(\nu,v)=\nu H_{\mu}(\mu,y)+H_u(\mu,y)v\,,
$$
for $(\nu,v)\in{\Bbb R}\times X$. Moreover, $H_\mu(\mu,y)$ is the constant
function $-f(y)$, and $H_u(\mu,y)$ is the operator $L+\lambda_0(\mu,y)Id$, a
Fredholm operator of index~0. It follows that $H'(\mu,y)$ is a Fredholm operator
of index~1. Also, since the eigenvalues of $H_u(\mu,y)$ are simple, the
nullspace of $H'(\mu,y)$ is at most two-dimensional. It is indeed
two-dimensional if and only if 0 is an eigenvalue of $H_u(\mu,y)$ and
$H_\mu(\mu,y)$ belongs to the range of $H_u(\mu,y)$. Consequently, $(\mu,y)$ is
a singular point of $H$ (in the sense of Section~1) if and only if one of the
eigenvalues $\lambda_j(\mu,y)$, with $j\in {\Bbb Z}_+$, vanishes and the nonzero
constant $f(y)$ belongs to the range of
$L+\lambda_0(\mu,y)Id=L+\bigl(\lambda_j(\mu,y)-\ell_j\bigr)Id=L-\ell_j Id$. But
$R(L-\ell_j Id)$ is the orthogonal complement (in $Y$) of the $j$-th normalized
eigenfunction $\phi_j$ of $L$ and contains nonzero constants if and only if
$j\ge1$. It follows that the singular points of $H$ on $\Sigma^*$ coincide with
the zeros $(\mu_j,u_j)$ of the eigenvalues $\lambda_j$ with $j\ge1$.
We claim that all those points are in fact regular-singular points of $H$. To
verify this, let $j\in{\Bbb N}$ and let $\phi_j$, as before, denote the $j$-th
normalized eigenfunction of $L$. We know already that $N(H'(\mu_j,u_j))$ is
two-dimensional, and it clearly contains the vector $(0,\phi_j)$. To find a
second, linearly independent member of $N(H'(\mu_j,u_j))$, we differentiate the
equation $H(\mu,\underline u(\mu))=0$, valid for all $\mu>\mu_0$, with respect
to $\mu$ and obtain
$$
H'(\mu,\underline u(\mu))(1,\underline u'(\mu))=0\,,\eqno(3.1)
$$
for all $\mu>\mu_0$. Setting $\mu=\mu_j$, we see that $(1,\underline
u'(\mu_j))\in N(H'(\mu_j,u_j))$. Differentiating (3.1) once again, we obtain
$$
H''(\mu,\underline u(\mu))(1,\underline u'(\mu))(1,\underline
u'(\mu))+H'(\mu,\underline u(\mu))(0,\underline u''(\mu))=0\,,
$$
for $\mu>\mu_0$. With $\mu=\mu_j$, it follows that $H''(\mu_j,u_j)(1,\underline
u'(\mu_j))(1,\underline u'(\mu_j))$ belongs to $R(H'(\mu_j,u_j))$. Now it just
remains to
be shown that the mixed derivative $H''(\mu_j,u_j)(1,\underline
u'(\mu_j))(0,\phi_j)$ does {\it not\/} belong to $R(H'(\mu_j,u_j))$. To that
end, observe that for all $\mu>\mu_0$, we have $H'(\mu,\underline
u(\mu))(0,\phi_j)=H_u(\mu,\underline u(\mu))\phi_j=\lambda_j(\mu,\underline
u(\mu))\phi_j$. Differentiating this with respect to $\mu$, we get
$$
H''(\mu,\underline u(\mu))(1,\underline u'(\mu))(0,\phi_j)={d\over
d\mu}\lambda_j(\mu,\underline u(\mu))\phi_j\,,
$$
for $\mu>\mu_0$, and thus
$$
H''(\mu_j,u_j)(1,\underline u'(\mu_j))(0,\phi_j)=\Bigl({d\over
d\mu}\lambda_j(\mu,\underline u(\mu))\Big|_{\mu=\mu_j}\Bigr)\phi_j\,.
$$
Since ${d\over d\mu}\lambda_j(\mu,\underline u(\mu))\big|_{\mu=\mu_j}\ne0$ (see
Lemma~2.1(b)) and $\phi_j$ is not in
$R(H'(\mu_j,u_j))=R(H_u(\mu_j,u_j))=R(L-\ell_j Id)$, neither is
$H''(\mu_j,u_j)(1,\underline u'(\mu_j))(0,\phi_j)$. This finishes the proof.
%\medskip
The following is an immediate consequence of Lemma~3.1 and the fact that
regular-singular points of $H$ are Crandall-Rabinowitz points of $H$ (see
Section~1).
%\medskip
\noindent{\bf Corollary 3.2.} The only bifurcation points on the trivial
solution branch $\Sigma^*$ of Problem~(P) are the zeros $(\mu_j,u_j)$ of the
eigenvalues $\lambda_j$ with $j\ge1$, as given in Lemma~2.1(b). Locally, near
any
of the points $(\mu_j,u_j)$, the solution set $\Sigma$ of Problem~(P) is the
union of two simple $C^1$-arcs that intersect transversally at $(\mu_j,u_j)$.
%\medskip
Following Rabinowitz [4], we can show that the bifurcations occurring at the
points $(\mu_j,u_j)$ are in fact {\it global\/} rather than {\it local\/}
phenomena. Since this is not our primary concern here, we sketch the argument
only briefly.
For $j\in{\Bbb N}$, let $C_j$ denote the connected component of $(\mu_j,u_j)$ in
$\overline{\Sigma\setminus\Sigma^*}$, which we consider a metric subspace of
${\Bbb R}\times X$ (or equivalently, of ${\Bbb R}\times C([-1,1])$ or of ${\Bbb
R}\times C^1([-1,1])$). The continuum $C_j$ obeys the ``Rabinowitz
alternative'': Either $C_j$ is a compact subset of the interior of ${\Bbb
R}_+\times X_+$ (the domain of $H$) and contains a point of $\Sigma^*$ different
from $(\mu_j,u_j)$. Or $C_j$ is unbounded or intersects the boundary of ${\Bbb
R}_+\times X_+$. (See [7, Theorem~3.5] for a version of the Rabinowitz
bifurcation theorem that is directly applicable in the situation at hand.) In
view of our a-priori bounds for nontrivial solutions of Problem~(P)
(see Lemma~2.2), $C_j$ cannot reach the boundary of ${\Bbb R}_+\times X_+$.
Moreover, $C_j$ can intersect $\Sigma^*$ only at a zero of the eigenvalue
$\lambda_j$, that is, only at the point $(\mu_j,u_j)$. The latter is a
consequence of the nodal properties of the nontrivial solutions of Problem~(P)
(see Lemma~2.2 again) and of analogous nodal properties of the eigenfunctions of
$L$. (We refer to [7,~Chapter~3] or [1, 3, 4, 5, 6, 8] for details of the
argument, which is by now routine.) It follows that $C_j$ is unbounded.
%\goodbreak
Moreover, $C_j\setminus\{(\mu_j,u_j)\}$ is entirely contained in the set
$\Sigma_j$ consisting of all nontrivial solutions $(\mu,u)$ of Problem~(P) for
which $u-\underline u(\mu)$ has exactly $j$ zeros (necessarily simple and
located in the open interval $(-1,1)$). We can decompose $\Sigma_j$ into subsets
$\Sigma_j^+$ and $\Sigma_j^-$ consisting of those pairs $(\mu,u)\in\Sigma_j$ for
which $u-\underline u(\mu)$ is positive or negative, respectively, at $x=1$. The
bifurcation point $(\mu_j,u_j)$ belongs to
$\overline{\Sigma_j^+}\cap\overline{\Sigma_j^-}$, and the Rabinowitz alternative
holds, separately, for the connected components $C_j^\pm$ of $(\mu_j,u_j)$ in
$\overline{\Sigma_j^\pm}$ (see the references quoted above for similar
reasoning). It follows that both $C_j^+$ and $C_j^-$ are unbounded with
$C_j^+\cap C_j^-=\{(\mu_j,u_j)\}$ and $C_j^+\cup C_j^-=C_j$.
Finally, we note that thanks to the a-priori bounds of Lemma~2.2, every
unbounded set of nontrivial solutions of Problem~(P) must contain solutions
$(\mu,u)$ with arbitrarily large $\mu$. Summarizing all our results, we have the
following theorem.
%\medskip
\noindent{\bf Theorem 3.3.} There is a sequence $((\mu_j,u_j))_{j\in{\Bbb N}}$,
with $(\mu_j)_{j\in{\Bbb N}}$ strictly increasing, of Crandall-Rabinowitz type
bifurcation points on the lower part of the trivial solution branch $\Sigma^*$
of Problem~(P), and these are the only points where nontrivial solutions
bifurcate from the trivial solution set $\Sigma^*\cup\big({\Bbb
R}_+\times\{0\}\bigr)$. At each of the points $(\mu_j,u_j)$, an unbounded
continuum $C_j$ of solutions bifurcates from $\Sigma^*$, which does not contain
any trivial solution other than $(\mu_j,u_j)$. This continuum $C_j$ is the union
of two subcontinua $C_j^+$ and $C_j^-$ with the following properties: (a)~Both
$C_j^+$ and $C_j^-$ contain solutions $(\mu,u)$ with arbitrarily large $\mu$.
(b)~If $(\mu,u)\in C_j^+$ ($C_j^-$) and $(\mu,u)\ne(\mu_j,u_j)$, then
$\mu>\mu_0$ and the function $u-\underline u(\mu)$ has exactly $j$ zeros and is
positive (negative) at $x=1$.
\bigskip
\bigbreak
\centerline{4. \smallrm SPATIALLY INHOMOGENEOUS PERTURBATIONS OF PROBLEM~(P)}
\nobreak \noindent
We will now embed Problem~(P) into a family (P$_\epsilon$) of problems
with spatially inhomogeneous source terms. Due to the spatial inhomogeneity of
the right-hand side, Problem (P$_\epsilon$) will not have trivial solutions
$(\mu,u)$ with $u\ne0$. Still, the abstract results of Section~1 will allow us
to prove the existence of solutions of Problem~(P$_\epsilon$) close to the
trivial solution branch $\Sigma^*$ of Problem~(P)=(P$_0$) and to describe the
local structure of this set of solutions.
Let $J$ be an open interval containing 0 and let $q$ denote a $C^2$-function on
$\bar J\times [-1,1]$ with $q(0,x)=1$ for all $x\in[-1,1]$. For
$\epsilon\in\bar J$, consider the boundary-value problem
$$
-{d\over dx}\Bigl(k(x){du\over dx}\Bigr)=\mu q(\epsilon,x)f(u)-g(u)\quad
\hbox{in }(-1,1),\qquad u'(\pm1)=0,\eqno({\rm P}_\epsilon)
$$
under the same assumptions on $k$, $f$, and $g$ as in Section~2. Clearly,
(P$_0$) coincides with our earlier problem, (P). Using the same
functional-analytic setting as in Section~3, we can write Problem~(P$_\epsilon$)
as an operator equation of the form $Lu=N(\epsilon,\mu,u)$, with $N:\bar
J\times{\Bbb R}_+\times X_+\to Y$ defined by
$$
N(\epsilon,\mu,u):=\mu q(\epsilon,\cdot)f\circ u-g\circ u,
$$
or equivalently, as
$$
H(\epsilon,\mu,u)=0,\eqno(\tilde{\rm P}_\epsilon)
$$
with $H:\bar J\times{\Bbb R}_+\times X_+\to Y$ defined by
$$
H(\epsilon,\mu,u):=Lu-N(\epsilon,\mu,u).
$$
The mapping $N$ is completely continuous (with respect to the norms of ${\Bbb
R}\times{\Bbb R}\times X$ and $Y$) and twice continuously differentiable on the
open set $J\times\Omega$ (where, as before,
\hbox{$\Omega=(0,\infty)\times\{u\in X\mid u(x)>c\;\forall x\in[-1,1]\}$}). In
conjunction with the well-known properties of $L$, this
implies that for every $(\epsilon,\mu,u)\in\Omega$, the partial Fr\'echet
derivative $H_{(\mu,u)}(\epsilon,\mu,u)$ is a Fredholm operator of index~1.
%\goodbreak
For $\epsilon\in\bar J$, let $\Sigma_\epsilon$ denote the solution set of
Problem~(P$_\epsilon$), or equivalently, of Problem~($\tilde{\rm P}_\epsilon$):
$$
\Sigma_\epsilon= \{(\mu,u)\in{\Bbb R}_+\times X_+\mid H(\epsilon,\mu,u)=0\}.
$$
For any $\epsilon\in\bar J$, the set $\Sigma_\epsilon$ contains the trivial
solutions $(\mu,0)$ with $\mu\in{\Bbb R}_+$, but no further trivial solutions
unless $q(\epsilon,\cdot)$ is constant (as in the case $\epsilon=0$). Of course,
$\Sigma_0$ coincides with the solution set $\Sigma$ of Problem~(P), which we
analyzed in Sections~2 and~3. We are interested in how the spatially
inhomogeneous perturbations affect the trivial solution branch $\Sigma^*$ of
Problem~(P).
Specifically, what is the local structure of $\Sigma_\epsilon$, for $\epsilon$
close to 0, near a point $(\mu^*,u^*)$ on $\Sigma^*$? Applying the results
of Section~1, we first note that if $(\mu^*,u^*)$ is a regular point of
$H(0,\cdot,\cdot)$, then $\Sigma_\epsilon$ is locally, near $(\mu^*,u^*)$, just
a continuous deformation of the curve $\Sigma^*$ and in particular, a simple
$C^1$-arc. If $(\mu^*,u^*)$ is one of the bifurcation points $(\mu_j,u_j)$ (and
thus, a regular-singular point of $H(0,\cdot,\cdot)$), then $\Sigma_\epsilon$ is
locally, near $(\mu^*,u^*)$, a homeomorphic image of the set
$\{(\sigma,\tau)\in{\Bbb R}^2\mid\sigma\tau+\rho(\epsilon)=0\}$, where $\rho$ is
a $C^1$-function with $\rho(0)=0$ (see Theorem~1.1 for a much more precise
statement). In particular, $\Sigma_\epsilon$ contains a bifurcation point near
$(\mu^*,u^*)$ if and only if $\rho(\epsilon)=0$.
At the end of Section~1, we derived a simple criterion for {\it
nonpersistence\/} of the bifurcations occurring at regular-singular points.
Applying this criterion in the present situation, we infer the following: If
$H_\epsilon(0,\mu^*,u^*)$ does not belong to the range of
$H_{(\mu,u)}(0,\mu^*,u^*)$, then $\rho(\epsilon)\ne0$ for all $\epsilon\ne0$
close to 0, and then, none of the corresponding sets $\Sigma_\epsilon$ contains
a bifurcation point near $(\mu^*,u^*)$. Now, if $(\mu^*,u^*)=(\mu_j,u_j)$ for
some $j\in{\Bbb N}$, then $H_{\epsilon}(0,\mu^*,u^*)=-\mu_j f(u_j){\partial
q\over\partial\epsilon}(0,\cdot)$, which is a nonzero constant multiple of
${\partial q\over\partial\epsilon}(0,\cdot)$. Moreover,
$R(H_{(\mu,u)}(0,\mu^*,u^*))=R(H_u(0,\mu^*,u^*))$ and $H_u(0,\mu^*,u^*)
=L-\ell_j Id$ (see the proof of Lemma 3.1), so that
$R(H_{(\mu,u)}(0,\mu^*,u^*))$ is nothing but the orthogonal complement, in
$Y=L^2((-1,1))$, of the $j$-th normalized eigenfunction $\phi_j$ of $L$. We
conclude that the bifurcation at $(\mu^*,u^*)$ is {\it nonpersistent\/} provided
that
$$
\int_{-1}^1{\partial q\over\partial\epsilon}(0,x)\phi_j(x)\,dx\ne0\,.
$$
In the absence of special symmetries, this condition is generic. But let us now
assume that the diffusion coefficient $k$ as well as the functions
$q(\epsilon,\cdot)$ are {\it even\/}:
$$
k(x)=k(-x)\quad\hbox{and}\quad q(\epsilon,x)=q(\epsilon,-x)
$$
for all $x\in[-1,1]$ and $\epsilon\in\bar J$. For every function
$u:[-1,1]\to{\Bbb R}$, let $\bar u$ denote its reflection at $x=0$, that is,
$\bar u(x):=u(-x)$ for $x\in[-1,1]$. A simple calculation shows that $L\bar
u=\overline{Lu}$ for every $u\in X$. As a consequence, the eigenfunctions of $L$
are either even or odd; in fact $\overline\phi_j=(-1)^j\phi_j$, for $j\in{\Bbb
Z}_+$. It follows that if $(\mu,u)\in{\Bbb R}_+\times X_+$ satisfies
$H(\epsilon,\mu,u)=r\phi_j$, for some $\epsilon\in\bar J$ and $r\in{\Bbb R}$,
then $(\mu,\bar u)$ satisfies $H(\epsilon,\mu,\bar u)=(-1)^j r\phi_j$.
Again, let $(\mu^*,u^*)$ be one of the bifurcation points of Problem~(P) on
$\Sigma^*$, that is, $(\mu^*,u^*)=(\mu_j,u_j)$ for some $j\in{\Bbb N}$. Applying
Theorem~1.2, with the eigenfunction $\phi_j$ playing the role of the vector
$y_0\in Y\setminus R(H_{(\mu,u)}(0,\mu^*,u^*))$, we obtain a neighborhood
$U\subset\Omega$ of $(\mu^*,u^*)$ and an interval $I\subset J$ containing 0 such
that for every $\epsilon\in I$, the mapping $H(\epsilon,\cdot,\cdot)$ has a
unique singular point $(\mu^*_\epsilon,u^*_\epsilon)\in U$ whose image
$H(\epsilon,\mu^*_\epsilon,u^*_\epsilon)$ is a constant multiple of $\phi_j$.
The point $(\mu^*_\epsilon,u^*_\epsilon)$ is a regular-singular point of
$H(\epsilon,\cdot,\cdot)$, and there is a function $\rho\in C^1(I)$ such that
$H(\epsilon,\mu^*_\epsilon,u^*_\epsilon)=\rho(\epsilon)\phi_j$.
Being a regular-singular point of $H(\epsilon,\cdot,\cdot)$, the point
$(\mu^*_\epsilon,u^*_\epsilon)$ is a bifurcation point for the equation
$$
H(\epsilon,\mu,u)=\rho(\epsilon)\phi_j\,,
$$
and with our earlier observation regarding the symmetries of $H$ and $\phi_j$,
it follows that $(\mu^*_\epsilon,\overline u^*_\epsilon)$ is a bifurcation
point for
$$
H(\epsilon,\mu,u)=(-1)^j\rho(\epsilon)\phi_j\,.
$$
Also, since $(\mu^*_\epsilon,u^*_\epsilon)$ is close to $(\mu^*,u^*)$, so is
$(\mu^*_\epsilon,\overline u^*_\epsilon)$ (recall that $u^*$ is a constant).
That is, the neighborhood $U$ of $(\mu^*,u^*)$ provided by Theorem 1.2 can be
chosen so that both points, $(\mu^*_\epsilon,u^*_\epsilon)$ and
$(\mu^*_\epsilon,\overline u^*_\epsilon)$, belong to $U$. But $U$ contains only
{\it one\/} singular point of $H(\epsilon,\cdot,\cdot)$ whose image is a
constant multiple of $\phi_j\,$. We conclude that $\overline
u^*_\epsilon=u^*_\epsilon$ and $(-1)^j\rho(\epsilon)=\rho(\epsilon)$. If $j$ is
{\it odd\/}, we must have $\rho(\epsilon)=0$, and then
$(\mu^*_\epsilon,u^*_\epsilon)$ is a bifurcation point for
Problem~(P$_\epsilon$). In other words, at least every other one of the
bifurcations occurring on the trivial solution branch of Problem~(P) is
persistent!
The following theorem summarizes our results.
\noindent{\bf Theorem 4.1.} Let $((\mu_j,u_j))_{j\in{\Bbb N}}$ denote the
sequence of bifurcation points on the trivial solution branch $\Sigma^*$ of
Problem~(P), as described in Theorem~3.3, and let $(\phi_j)_{j\in{\Bbb Z}_+}$ be
the sequence of normalized eigenfunctions of the operator $L$.
(a) For every $j\in{\Bbb N}$, if $\int_{-1}^1{\partial
q\over\partial\epsilon}(0,x)\phi_j(x)\,dx\ne0$, then the bifurcation at
$(\mu_j,u_j)$ is {\it nonpersistent\/}: None of the problems (P$_\epsilon$) with
$\epsilon$ close to but different from 0 has a bifurcation point near
$(\mu_j,u_j)$. In fact, the solution set of (P$_\epsilon$), for any such
$\epsilon$, is locally, near $(\mu_j,u_j)$, the union of two disjoint, simple
$C^1$-arcs.
(b) If the coefficient $k$ and the functions $q(\epsilon,\cdot)$, for
$\epsilon\in\bar J$, are {\it even\/}, then the bifurcations occurring at {\it
odd-numbered\/} points $(\mu_j,u_j)$ are {\it persistent\/}: Each of the
problems (P$_\epsilon$) with $\epsilon$ close to 0 has a unique bifurcation
point $(\mu_{j,\epsilon},u_{j,\epsilon})$ near $(\mu_j,u_j)$. In fact, the
solution set of (P$_\epsilon$), for any such $\epsilon$, is locally, near
$(\mu_j,u_j)$, the union of two simple $C^1$-arcs that intersect transversally
at $(\mu_{j,\epsilon},u_{j,\epsilon})$.
\goodbreak
\input epsf
\line{}\vskip-3\baselineskip
%\epsfxsize=10cm
\epsfysize=9cm
\centerline{\epsffile{fig1.ps}}
\centerline{\eightrm Example of a trivial solution branch $\scriptstyle\Sigma^*$
with bifurcation points $\scriptstyle(\mu_j,u_j)$.}
\bigskip\bigskip
\bigbreak
\centerline{\smallrm REFERENCES}
\nobreak
\item{1.} {\smallrm M.~G.~CRANDALL \& P.~H.~RABINOWITZ}, Nonlinear
Sturm-Liouville eigenvalue problems and topological degree, {\it
J.~Math.\ Mech.\/}~19 (1970), 1083--1102.
\item{2.} {\smallrm M.~G.~CRANDALL \& P.~H.~RABINOWITZ}, Bifurcation from
simple eigenvalues, {\it J.~Funct.~Anal.\/}~8 (1971), 321--340.
\item{3.} {\smallrm P.~H.~RABINOWITZ}, Nonlinear Sturm-Liouville problems
for second order ordinary differential equations, {\it Comm.\ Pure Appl.\
Math.\/}~23 (1970), 939--962.
\item{4.} {\smallrm P.~H.~RABINOWITZ}, Some global results for nonlinear
eigenvalue problems, {\it J.~Funct.\ Anal.\/}~7 (1971), 487--513.
\item{5.} {\smallrm R.~SCHAAF}, Global behaviour of solution branches for
some Neumann problems depending on one or several parameters, {\it
J.~Reine Angew.\ Math.\/}~346 (1984), \hbox{1--31}.
\item{6.} {\smallrm R.~SCHAAF}, Global solution branches of two-point
boundary value problems, Lecture Notes in Mathematics, Vol.~1458,
Springer-Verlag, Berlin, 1990.
\item{7.} {\smallrm B.~E.~SCHMIDT}, Bifurcation of stationary solutions
for Legendre-type boundary value problems arising from energy balance climate
models, Ph.D.~thesis, Auburn University, 1994.
\item{8.} {\smallrm B.~E.~SCHMIDT}, On a nonlinear eigenvalue problem
arising from climate modeling, {\it Nonlin.\ Anal.\/}~30 (1997), 3645--3656.
\bigskip
\vbox{\eightrm\baselineskip9pt
\noindent Bettina E.~Schmidt \hfill\break
Department of Mathematics \hfill\break
Auburn University at Montgomery \hfill\break
Montgomery, AL 36124-4023, USA \hfill\break
Email address: bes@strudel.aum.edu}
\bye
__