\magnification = \magstephalf \hsize=14truecm \hoffset=1truecm \parskip=5pt \overfullrule=0pt \font\smallrm=cmr9 \font\smallerrm=cmr8 \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 $00$, 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