Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp. 223–235. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp 147.26.103.110 or 129.120.3.113 (login: ftp)

PERSISTENCE OF CRANDALL–RABINOWITZ TYPE BIFURCATIONS UNDER SMALL PERTURBATIONS

BETTINA E. SCHMIDT

Abstract 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.

INTRODUCTION

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 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 ∈ U | 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_{ϵ}) should be somehow “close” to the solution set of the unperturbed problem (P), although (P_{ϵ}) has no trivial solutions, due to the spatial inhomogeneity. It is rather obvious that near 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 bifurcation points (or more specifically, near 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)?

1991 Subject Classification: 34B15, 34C23, 46N20.

Key words and phrases: Neumann problem, nonlinear eigenvalue problem,

bifurcation from simple eigenvalues, Crandall-Rabinowitz theorem, regular-

singular points, perturbed bifurcation theory.

◯^{c}1998 Southwest Texas State University and University of North Texas.

Published November 12, 1998.

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 functionalanalytic 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.

1. PERTURBED CRANDALL–RABINOWITZ BIFURCATION

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,  (1.1)

where H ∈ C^{2}(ℝ^{2}, ℝ). Suppose that (x_{0}, y_{0}) is a solution of Equation (1.1). We call (x_{0}, y_{0}) a 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 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 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 Ω in a Banach space W into a Banach space Y. Let w_{0} be a point in Ω and suppose that the Fréchet 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 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 singular point of H. Specifically, w_{0} is called a 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_{1}w_{1} belongs to R(H′(w_{0})), while the mixed derivative H′′(w_{0})w_{1}w_{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 {w̃_{1}, w̃_{2}} of N(H′(w_{0})) such that both quadratic derivatives, H′′(w_{0})w̃_{1}w̃_{1} and H′′(w_{0})w̃_{2}w̃_{2}, belong to R(H′(w_{0})), while the mixed derivative H′′(w_{0})w̃_{1}w̃_{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 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 family of mappings H(ϵ, ·) from an open subset Ω of a Banach space W into a Banach space Y, where ϵ varies over an open interval J containing 0. We think of H(ϵ, ·), for ϵ ∈ J, as a perturbation of the mapping H(0, ·) and wish to describe the structure of the set Σ_{ϵ} := {w ∈ Ω | H(ϵ, w) = 0}, for ϵ close to 0, in the vicinity of a point w_{0} ∈ Ω with H(0, w_{0}) = 0. Throughout we assume that H ∈ C^{2}(J × Ω, Y) and that the (partial) Fréchet derivative H_{w}(0, w_{0}) = \frac{∂H}{∂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 regular point of H(0, ·), then not only for ϵ = 0, but for every ϵ sufficiently close to 0, the set Σ_{ϵ} is locally, near w_{0}, a simple C^{1}-arc (which varies continuously with ϵ). For the case where w_{0} is a regular-singular point of H, we have the following generalization of the Crandall-Rabinowitz theorem (see [7, Theorem 4.6]).

Theorem 1.1. Suppose w_{0} ∈ Ω is a regular-singular point of H(0, ·). Then there exist

• an open interval I ⊂ J with 0 ∈ I,

• an open neighborhood U ⊂ Ω of w_{0},

• a continuous mapping \overline{w} : D → U, defined on an open set D ⊂ ℝ × ℝ^{2} that contains (ϵ, 0, 0) for every ϵ ∈ I, with \overline{w}(0, 0, 0) = w_{0}, and

• a continuously differentiable mapping ρ : I → ℝ with ρ(0) = 0,

such that for every ϵ ∈ I, the mapping \overline{w}(ϵ, ·, ·) is one-to-one and continuously differentiable, with linearly independent derivatives \overline{w}_{σ}(ϵ, σ, τ) and \overline{w}_{τ} (ϵ, σ, τ) for all (σ, τ) ∈ ℝ^{2} with (ϵ, σ, τ) ∈ D, and we have

\{w ∈ U | H(ϵ, w) = 0\} = \{\overline{w}(ϵ, σ, τ) | στ + ρ(ϵ) = 0 and (ϵ, σ, τ) ∈ D\} .

Roughly speaking, the assertion of Theorem 1.1 is that for every ϵ sufficiently close to 0, the solution set Σ_{ϵ} of the equation H(ϵ, w) = 0 is locally, near w_{0}, a homeomorphic image of the solution set, near (0, 0), of a simple, algebraic equation in ℝ^{2}, namely, στ +ρ(ϵ) = 0. Thus, whenever we have ρ(ϵ) = 0, the set Σ_{ϵ} is locally, near w_{0}, the union of two simple C^{1}-arcs that intersect transversally at the point \overline{w}(ϵ, 0, 0). In particular, we recover the Crandall-Rabinowitz theorem, since ρ(0) = 0 and \overline{w}(0, 0, 0) = w_{0}. But whenever ρ(ϵ) ≠ 0, the set Σ_{ϵ} 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, ·), located on the level-0 set of H(0, ·). For ϵ close to 0, the perturbed mapping H(ϵ, ·) would still have a (unique) saddle point w_{ϵ} close to w_{0}, but at a level ρ(ϵ) in general different from 0. Only if ρ(ϵ) = 0, would the level-0 set of H(ϵ, ·) 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]).

Theorem 1.2. Suppose w_{0} ∈ Ω is a regular-singular point of H(0, ·) and y_{0} ∈ Y is a vector that does not belong to the range of H_{w}(0, w_{0}). Then there exist an open interval I ⊂ J with 0 ∈ I and an open neighborhood U ⊂ Ω of w_{0} such that for every ϵ ∈ I, the mapping H(ϵ, ·) has a unique singular point w_{ϵ} ∈ U whose image H(ϵ, w_{ϵ}) is a constant multiple of y_{0}. The mapping ϵ ↦ w_{ϵ} is continuously differentiable, and for every ϵ ∈ I, the point w_{ϵ} is a regular-singular point (and thus, a Crandall-Rabinowitz point) of H(ϵ, ·). Only if H(ϵ, w_{ϵ}) = 0, does the set {w ∈ U | H(ϵ, w) = 0} contain a singular point of H(ϵ, ·).

Now let w_{0} ∈ Ω be a regular-singular point (and thus, a Crandall-Rabinowitz point) of H(0, ·). We say that the bifurcation at w_{0} is persistent (or nonpersistent, respectively) if there exists a neighborhood U ⊂ Ω of w_{0} such that for every ϵ ∈ J \ {0} sufficiently close to 0, the set {w ∈ U | H(ϵ, w) = 0} contains a regularsingular point (or does not contain a singular point, respectively) of H(ϵ, ·).

It is easy to derive a simple, sufficient condition for nonpersistence. To that end, pick a vector y_{0} ∈ Y \ R(H_{w}(0, w_{0})) and let y_{0}^{∗} ∈ 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_{ϵ} according to Theorem 1.2 and define ρ ∈ C^{1}(I) by ρ(ϵ) := <y_{0}^{∗}, H(ϵ, w_{ϵ})>. Then we have H(ϵ, w_{ϵ}) = ρ(ϵ)y_{0}, for all ϵ ∈ I, and the set {w ∈ U | H(ϵ, w) = 0} contains a singular point of H(ϵ, ·) if and only if ρ(ϵ) = 0. Since ρ(0) = 0, it follows that ρ′(0) ≠ 0 is a sufficient condition for nonpersistence. Using the fact that y_{0} ∉ R(H_{w}(0, w_{0})), we readily show that ρ′(0) = <y_{0}^{∗}, H_{ϵ}(0, w_{0})> and conclude that the bifurcation at w_{0} is nonpersistent provided that

H_{ϵ}(0, w_{0}) ∉ R(H_{w}(0, w_{0})),  (1.2)

that is, provided that the partial derivative H_{ϵ}(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 persistence (obviously, H_{ϵ}(0, w_{0}) ∈ R(H_{w}(0, w_{0})) is necessary, but not sufficient). However, in concrete applications it is often possible to show directly that ρ(ϵ) must vanish for all ϵ 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.

2. A SPATIALLY HOMOGENEOUS NEUMANN PROBLEM

Consider the boundary-value problem

−\frac{d}{dx}(k(x)\frac{du}{dx}) = μf(u) − g(u) in (−1, 1),  u′(±1) = 0.  (P)

Here, μ 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 ℝ_{+}, and there exists a number c > 0 such that f(y) = 0 for all y ∈ [0, c], while f is twice continuously differentiable on (c, ∞), with f > 0, f′ ≥ 0, and f′′ ≤ 0. The function g is continuous on ℝ_{+}, with g(0) = 0, and twice continuously differentiable on (0, ∞), with g > 0, g′ ≥ 0, and g′′ ≥ 0. Finally we assume that \frac{g(y)}{f(y)} → ∞ as y → ∞. (Note that due to our earlier assumptions, f(y) grows at most linearly, g(y) 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 μf(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) ∼ u^{4} if the process is governed by the Stefan-Boltzmann law).

Let Σ denote the set of all pairs (μ, u) with μ ∈ ℝ_{+} and u ∈ C^{2}([−1, 1]) a nonnegative solution of (P). With slight abuse of language, we call the pairs (μ, u) ∈ Σ solutions of Problem (P). A solution (μ, u) is called trivial if the function u is constant. Clearly, Σ contains exactly two maximal continua of trivial solutions, namely, ℝ_{+} × \{0\} and the trace Σ^{∗} of the curve μ = \frac{g(y)}{f(y)} with y ∈ (c, ∞).

To describe the trivial solution branch Σ^{∗} in more detail, we need to analyze the function μ̄ := \frac{g}{f} |_{(c,∞)}. Clearly, we have μ̄(c+) = ∞ = μ̄(∞). Moreover, μ̄′ = \frac{σ}{f^{2}} with σ := g′f −gf′. The function σ is nondecreasing on (c, ∞) (in fact, σ′ = g′′f −gf′′ ≥ 0), negative near c (since f(c) = 0 and f′(c+) ∈ (0, ∞]), and positive near ∞ (else, μ̄ would be nonincreasing throughout, contradicting the fact that μ̄(∞) = ∞). This implies the existence of two numbers \underline{y}_{0} and \overline{y}_{0} with c < \underline{y}_{0} ≤ \overline{y}_{0} < ∞ such that σ is negative on (c, \underline{y}_{0}), zero on [\underline{y}_{0}, \overline{y}_{0}], and positive on (\overline{y}_{0}, ∞). The same then holds for μ̄′, and it follows that μ̄ is strictly decreasing on (c, \underline{y}_{0}) (with values between ∞ and μ_{0} := μ̄(\underline{y}_{0})), constant on [\underline{y}_{0}, \overline{y}_{0}] (with value μ_{0}), and strictly increasing on (\overline{y}_{0}, ∞) (with values between μ_{0} and ∞). (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 Σ^{∗} consists of the graphs of two functions, namely, \underline{u} := (μ̄|_{(c,\underline{y}_{0})})^{−1} and \overline{u} := (μ̄|_{(\overline{y}_{0},∞)})^{−1}, connected by the vertical segment \{μ_{0}\} × [\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 (μ_{0}, ∞); the former is strictly decreasing with range (c, \underline{y}_{0}), the latter is strictly increasing with range (\overline{y}_{0}, ∞). 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 Σ^{∗}, we compute the eigenvalues of the linearization of Problem (P), with respect to u, at a point (μ, y) ∈ Σ^{∗}, that is, the eigenvalues of

−\frac{d}{dx}(k(x)\frac{dv}{dx}) = (μf′(y) − g′(y))v + λv in (−1, 1),  v′(±1) = 0.  (2.1)

If we enumerate the eigenvalues of −\frac{d}{dx}(k \frac{d}{dx}) (under Neumann boundary conditions) as a strictly increasing sequence 0 = ℓ_{0} < ℓ_{1} < ℓ_{2} < …, then the eigenvalues of (2.1) are simply given by λ_{j}(μ, y) = ℓ_{j} + g′(y) − μf′(y), for j ∈ ℤ_{+}. In particular, λ_{0}(μ, y) = g′(y) − μf′(y) and

λ_{j}(μ, y) = λ_{0}(μ, y) + ℓ_{j} ,  (2.2)

for j ∈ ℕ. Also, since (μ, y) ∈ Σ^{∗}, we have μ = μ̄(y) = \frac{g(y)}{f(y)} and thus,

λ_{0}(μ, y) = g′(y) − μ̄(y)f′(y) = μ̄′(y)f(y) .  (2.3)

This implies that λ_{0} is positive on the graph of \overline{u} (the upper branch of Σ^{∗}, where μ̄′ > 0), zero on the vertical segment \{μ_{0}\}×[\underline{y}_{0}, \overline{y}_{0}] (the possibly degenerate turning point of Σ^{∗}, where μ̄′ = 0), and negative on the graph of \underline{u} (the lower branch of Σ^{∗}, where μ̄′ < 0). Thus, the solutions on the upper branch are stable, while those on the lower branch are unstable, and bifurcations can occur only on the lower branch, at points where one of the eigenvalues λ_{j} with j ≥ 1 vanishes (that is, at points (μ, y) ∈ Σ^{∗} where λ_{0}(μ, y) = −ℓ_{j} for some j ≥ 1).

To find out whether such points exist, we must trace the smallest eigenvalue, λ_{0}, along the lower branch of Σ^{∗}. To that end, define λ̄ : (c, \underline{y}_{0}) → ℝ by λ̄(y) := λ_{0}(μ̄(y), y). By (2.3), λ̄ = g′ − μ̄f′ = μ̄′f, and we know that this is strictly negative on (c, \underline{y}_{0}), with λ̄(\underline{y}_{0}−) = 0. Moreover, λ̄′ = g′′ − μ̄′f′ − μ̄f′′ ≥ −μ̄′f′, since g′′ ≥ 0, f′′ ≤ 0, and μ̄ > 0. Also, μ̄′ < 0 on (c, \underline{y}_{0}), that is, σ = g′f − gf′ < 0, and thus,

f′ > \frac{g′f}{g} > 0 (our assumptions on g imply g′ > 0 on (0, ∞)). This proves that λ̄′ > 0

on (c, \underline{y}_{0}). Finally, we observe that λ̄(c+) = −∞, since λ̄ = g′ − μ̄f′ = g′ − g \frac{f′}{f}

and \frac{f′(y)}{f(y)} → ∞ as y → c+ (note that g′(c) > 0, g(c) > 0, f′(c+) ∈ (0, ∞], and f(c+) = 0+).

Summarizing, we showed that λ̄ is strictly increasing on (c, \underline{y}_{0}), with range (−∞, 0) and with a strictly positive derivative. But this means that the function μ ↦ λ_{0}(μ, \underline{u}(μ)) is strictly decreasing on (μ_{0}, ∞), with range (−∞, 0) and with a strictly negative derivative. Because of (2.2), it follows that in fact all the eigenvalues λ_{j} are strictly decreasing and eventually negative along the lower branch of Σ^{∗}, with \frac{d}{dμ}λ_{j}(μ, \underline{u}(μ)) < 0 for all μ ∈ (μ_{0}, ∞). In particular, each of the eigenvalues λ_{j} with j ≥ 1 has a unique nondegenerate zero on the lower branch of Σ^{∗}.

The following lemma gathers our findings about the trivial solutions of Problem (P).

Lemma 2.1. (a) There are exactly two maximal continua of trivial solutions of Problem (P), namely ℝ_{+}×\{0\} and the trace Σ^{∗} of the curve μ = \frac{g(y)}{f(y)} with y ∈ (c, ∞). The set Σ^{∗} consists of the graphs of two functions \underline{u}, \overline{u} ∈ C^{2}((μ_{0}, ∞)) and a vertical segment \{μ_{0}\} × [\underline{y}_{0}, \overline{y}_{0}], where μ_{0} > 0 and c < \underline{y}_{0} ≤ \overline{y}_{0} < ∞. 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}, ∞).

(b) Denoting by (λ_{j}(μ, y))_{j∈ℤ_{+}} the strictly increasing enumeration of the eigenvalues of the linearization of Problem (P) at (μ, y) ∈ Σ^{∗}, we have λ_{0} > 0 on the graph of \overline{u}, λ_{0} = 0 on the vertical segment \{μ_{0}\} × [\underline{y}_{0}, \overline{y}_{0}], and λ_{0} < 0 on the graph of \underline{u}. All the eigenvalues λ_{j} are strictly decreasing and eventually negative along the graph of \underline{u}, with \frac{d}{dμ}λ_{j}(μ, \underline{u}(μ)) < 0 for all μ ∈ (μ_{0}, ∞). In particular, each of the eigenvalues λ_{j} with j ≥ 1 has a unique zero (μ_{j}, u_{j}) on Σ^{∗}, with μ_{0} < μ_{1} < μ_{2} < … and u_{j} = \underline{u}(μ_{j}) for j ∈ ℕ.

The figure at the end of the paper depicts the trivial solution branch Σ^{∗} 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 (μ_{j}, u_{j}), j = 1, 2, 3 …; those were found by solving the equation λ̄(u_{j}) = −ℓ_{j}, under the assumption that k ≡ 1 (so that ℓ_{j} = \frac{π^{2}}{4} j^{2}, for j ∈ ℕ).

Next we collect some a-priori information about possible nontrivial solutions of Problem (P).

Lemma 2.2. Let (μ, u) be a nontrivial solution of Problem (P). Then we have μ > μ_{0} and 0 < u(x) < \overline{u}(μ) for all x ∈ [−1, 1]. More precisely, every local minimum of u lies between 0 and \underline{u}(μ), while every local maximum of u lies between \underline{u}(μ) and \overline{u}(μ). The function u − \underline{u}(μ) has a positive finite number of zeros, all of which are simple and occur in the open interval (−1, 1).

Proof. Let (μ, u) ∈ Σ with u ≠ const. Suppose that u attains a local maximum at x_{0} ∈ [−1, 1]. Then u′(x_{0}) = 0 and u′′(x_{0}) ≤ 0 (even if x_{0} is a boundary point). In fact, we must have u′′(x_{0}) < 0, for otherwise −\frac{d}{dx}(k(x)\frac{du}{dx})|_{x=x_{0}} = 0 and thus, μf(u(x_{0})) − g(u(x_{0})) = 0, which would mean that u(x_{0}) is a constant solution of Problem (P). But then, the uniqueness theorem for second-order ordinary differential equations would imply that u(x) = u(x_{0}) for all x ∈ [−1, 1]. (The fact that f may not be Lipschitz continuous near c does not interfere with this argument, since necessarily u(x_{0}) ≠ c.) So we have indeed u′′(x_{0}) < 0, hence −\frac{d}{dx}(k(x)\frac{du}{dx})|_{x=x_{0}} > 0, and thus, μf(u(x_{0})) − g(u(x_{0})) > 0. But this implies u(x_{0}) > c and μ > μ̄(u(x_{0})) = \frac{g(u(x_{0}))}{f(u(x_{0}))}, or equivalently, μ > μ_{0} and \underline{u}(μ) < u(x_{0}) < \overline{u}(μ).

Similarly, one shows that if u attains a local minimum at x_{1} ∈ [−1, 1], then 0 < u(x_{1}) < \underline{u}(μ). This proves the first two assertions of the lemma and the fact that u − \underline{u}(μ) has at least one zero. Another invocation of the uniqueness theorem yields that all zeros of u − \underline{u}(μ) are simple and located in the open interval (−1, 1).

3. BIFURCATION ANALYSIS OF PROBLEM (P)

To make the abstract theory of Section 1 applicable, we need to reformulate Problem (P) as an operator equation in a suitable Banach or Hilbert space setting. To that end, let L denote the selfadjoint linear operator in Y := L^{2}((−1, 1)) induced by the differential expression −\frac{d}{dx}(k \frac{d}{dx}) under Neumann boundary conditions. Having a compact resolvent, L is a Fredholm operator of index 0 in Y. Let X denote the domain of L, endowed with the graph norm. Then X is a Banach space that embeds compactly into C([−1, 1]).

The functions f and g induce substitution operators u ↦ f ∘ u and u ↦ g ∘ u in C([−1, 1]), defined and continuous on the cone of nonnegative functions and twice continuously differentiable on the open set {u ∈ C([−1, 1]) | u(x) > c ∀x ∈ [−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 : ℝ_{+} × X_{+} → Y, defined by

N(μ, u) := μf ∘ u − g ∘ u,

is completely continuous (with respect to the norms of ℝ × X and Y) and twice continuously differentiable on Ω := (0, ∞) × {u ∈ X | u(x) > c ∀x ∈ [−1, 1]} (an open subset of ℝ × X).

With these definitions, Problem (P) is equivalent to the equation Lu = N(μ, u), which we can also write as

H(μ, u) = 0,  (P̃)

where H : ℝ_{+} × X_{+} → Y is defined by

H(μ, u) := Lu − N(μ, u).

More precisely, the solution set Σ of Problem (P) coincides with that of (P̃):

Σ = {(μ, u) ∈ ℝ_{+} × X_{+} | H(μ, u) = 0}.

We consider Σ a metric subspace of ℝ × X; as such it is closed and locally compact. Moreover, routine arguments show that the topology of Σ as a metric subspace of ℝ × X coincides with the metric topologies it inherits from ℝ × 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 (μ, y) ∈ Σ^{∗}, is equivalent to the abstract eigenvalue problem for the operator

H_{u}(μ, y) = L − N_{u}(μ, y) = L + λ_{0}(μ, 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.

Lemma 3.1. For every (μ, y) ∈ Σ^{∗}, the Fréchet derivative H′(μ, y) is a Fredholm operator of index 1. The only singular points of H on Σ^{∗} are the zeros (μ_{j}, u_{j}) of the eigenvalues λ_{j} with j ≥ 1, as given in Lemma 2.1(b). All of those points are in fact regular-singular points of H.

Proof. The Fréchet derivative of H at (μ, y) ∈ Σ^{∗} is given by

H′(μ, y)(ν, v) = νH_{μ}(μ, y) + H_{u}(μ, y)v ,

for (ν, v) ∈ ℝ × X. Moreover, H_{μ}(μ, y) is the constant function −f(y), and H_{u}(μ, y) is the operator L + λ_{0}(μ, y)Id, a Fredholm operator of index 0. It follows that H′(μ, y) is a Fredholm operator of index 1. Also, since the eigenvalues of H_{u}(μ, y) are simple, the nullspace of H′(μ, y) is at most two-dimensional. It is indeed twodimensional if and only if 0 is an eigenvalue of H_{u}(μ, y) and H_{μ}(μ, y) belongs to the range of H_{u}(μ, y). Consequently, (μ, y) is a singular point of H (in the sense of Section 1) if and only if one of the eigenvalues λ_{j}(μ, y), with j ∈ ℤ_{+}, vanishes and the nonzero constant f(y) belongs to the range of L+λ_{0}(μ, y)Id = L+(λ_{j}(μ, y)−ℓ_{j})Id = L−ℓ_{j}Id. But R(L−ℓ_{j}Id) is the orthogonal complement (in Y) of the j-th normalized eigenfunction ϕ_{j} of L and contains nonzero constants if and only if j ≥ 1. It follows that the singular points of H on Σ^{∗} coincide with the zeros (μ_{j}, u_{j}) of the eigenvalues λ_{j} with j ≥ 1.

We claim that all those points are in fact regular-singular points of H. To verify this, let j ∈ ℕ and let ϕ_{j}, as before, denote the j-th normalized eigenfunction of L. We know already that N(H′(μ_{j}, u_{j})) is two-dimensional, and it clearly contains the vector (0, ϕ_{j}). To find a second, linearly independent member of N(H′(μ_{j}, u_{j})), we differentiate the equation H(μ, \underline{u}(μ)) = 0, valid for all μ > μ_{0}, with respect to μ and obtain

H′(μ, \underline{u}(μ))(1, \underline{u}′(μ)) = 0 ,  (3.1)

for all μ > μ_{0}. Setting μ = μ_{j}, we see that (1, \underline{u}′(μ_{j})) ∈ N(H′(μ_{j}, u_{j})). Differentiating (3.1) once again, we obtain

H′′(μ, \underline{u}(μ))(1, \underline{u}′(μ))(1, \underline{u}′(μ)) + H′(μ, \underline{u}(μ))(0, \underline{u}′′(μ)) = 0 ,

for μ > μ_{0}. With μ = μ_{j}, it follows that H′′(μ_{j}, u_{j})(1, \underline{u}′(μ_{j}))(1, \underline{u}′(μ_{j})) belongs to R(H′(μ_{j}, u_{j})). Now it just remains to be shown that the mixed derivative H′′(μ_{j}, u_{j})(1, \underline{u}′(μ_{j}))(0, ϕ_{j}) does not belong to R(H′(μ_{j}, u_{j})). To that end, observe that for all μ > μ_{0}, we have H′(μ, \underline{u}(μ))(0, ϕ_{j}) = H_{u}(μ, \underline{u}(μ))ϕ_{j} = λ_{j}(μ, \underline{u}(μ))ϕ_{j}. Differentiating this with respect to μ, we get

H′′(μ, \underline{u}(μ))(1, \underline{u}′(μ))(0, ϕ_{j}) = \frac{d}{dμ}λ_{j}(μ, \underline{u}(μ))ϕ_{j} ,

for μ > μ_{0}, and thus

H′′(μ_{j}, u_{j})(1, \underline{u}′(μ_{j}))(0, ϕ_{j}) = (\frac{d}{dμ}λ_{j}(μ, \underline{u}(μ))|_{μ=μ_{j}})ϕ_{j} .

Since \frac{d}{dμ}λ_{j}(μ, \underline{u}(μ))|_{μ=μ_{j}} ≠ 0 (see Lemma 2.1(b)) and ϕ_{j} is not in R(H′(μ_{j}, u_{j})) = R(H_{u}(μ_{j}, u_{j})) = R(L − ℓ_{j}Id), neither is H′′(μ_{j}, u_{j})(1, \underline{u}′(μ_{j}))(0, ϕ_{j}). This finishes the proof.

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).

Corollary 3.2. The only bifurcation points on the trivial solution branch Σ^{∗} of Problem (P) are the zeros (μ_{j}, u_{j}) of the eigenvalues λ_{j} with j ≥ 1, as given in Lemma 2.1(b). Locally, near any of the points (μ_{j}, u_{j}), the solution set Σ of Problem (P) is the union of two simple C^{1}-arcs that intersect transversally at (μ_{j}, u_{j}).

Following Rabinowitz [4], we can show that the bifurcations occurring at the points (μ_{j}, u_{j}) are in fact global rather than local phenomena. Since this is not our primary concern here, we sketch the argument only briefly.

For j ∈ ℕ, let C_{j} denote the connected component of (μ_{j}, u_{j}) in \overline{Σ \ Σ^{∗}}, which we consider a metric subspace of ℝ × X (or equivalently, of ℝ × C([−1, 1]) or of ℝ × C^{1}([−1, 1])). The continuum C_{j} obeys the “Rabinowitz alternative”: Either C_{j} is a compact subset of the interior of ℝ_{+} × X_{+} (the domain of H) and contains a point of Σ^{∗} different from (μ_{j}, u_{j}). Or C_{j} is unbounded or intersects the boundary of ℝ_{+} × 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 ℝ_{+} × X_{+}. Moreover, C_{j} can intersect Σ^{∗} only at a zero of the eigenvalue λ_{j}, that is, only at the point (μ_{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.

Moreover, C_{j} \ {(μ_{j}, u_{j})} is entirely contained in the set Σ_{j} consisting of all nontrivial solutions (μ, u) of Problem (P) for which u − \underline{u}(μ) has exactly j zeros (necessarily simple and located in the open interval (−1, 1)). We can decompose Σ_{j} into subsets Σ_{j}^{+} and Σ_{j}^{−} consisting of those pairs (μ, u) ∈ Σ_{j} for which u − \underline{u}(μ) is positive or negative, respectively, at x = 1. The bifurcation point (μ_{j}, u_{j}) belongs to \overline{Σ_{j}^{+}} ∩ \overline{Σ_{j}^{−}}, and the Rabinowitz alternative holds, separately, for the connected

components C_{j}^{±} of (μ_{j}, u_{j}) in \overline{Σ_{j}^{±}} (see the references quoted above for similar reasoning). It follows that both C_{j}^{+} and C_{j}^{−} are unbounded with C_{j}^{+} ∩ C_{j}^{−} = {(μ_{j}, u_{j})} and C_{j}^{+} ∪ 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 (μ, u) with arbitrarily large μ. Summarizing all our results, we have the following theorem.

Theorem 3.3. There is a sequence ((μ_{j}, u_{j}))_{j∈ℕ}, with (μ_{j})_{j∈ℕ} strictly increasing, of Crandall-Rabinowitz type bifurcation points on the lower part of the trivial solution branch Σ^{∗} of Problem (P), and these are the only points where nontrivial solutions bifurcate from the trivial solution set Σ^{∗} ∪ (ℝ_{+} × {0}). At each of the points (μ_{j}, u_{j}), an unbounded continuum C_{j} of solutions bifurcates from Σ^{∗}, which does not contain any trivial solution other than (μ_{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 (μ, u) with arbitrarily large μ. (b) If (μ, u) ∈ C_{j}^{+} (C_{j}^{−}) and (μ, u) ≠ (μ_{j}, u_{j}), then μ > μ_{0} and the function u − \underline{u}(μ) has exactly j zeros and is positive (negative) at x = 1.

4. SPATIALLY INHOMOGENEOUS PERTURBATIONS OF PROBLEM (P)

We will now embed Problem (P) into a family (P_{ϵ}) of problems with spatially inhomogeneous source terms. Due to the spatial inhomogeneity of the right-hand side, Problem (P_{ϵ}) will not have trivial solutions (μ, u) with u ≠ 0. Still, the abstract results of Section 1 will allow us to prove the existence of solutions of Problem (P_{ϵ}) close to the trivial solution branch Σ^{∗} 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 J̄ × [−1, 1] with q(0, x) = 1 for all x ∈ [−1, 1]. For ϵ ∈ J̄, consider the boundaryvalue problem

−\frac{d}{dx}(k(x)\frac{du}{dx}) = μq(ϵ, x)f(u) − g(u) in (−1, 1),  u′(±1) = 0, (P_{ϵ})

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_{ϵ}) as an operator equation of the form Lu = N(ϵ, μ, u), with N : J̄ × ℝ_{+} × X_{+} → Y defined by

N(ϵ, μ, u) := μq(ϵ, ·)f ∘ u − g ∘ u,

or equivalently, as

H(ϵ, μ, u) = 0,  (P̃_{ϵ})

with H : J̄ × ℝ_{+} × X_{+} → Y defined by

H(ϵ, μ, u) := Lu − N(ϵ, μ, u).

The mapping N is completely continuous (with respect to the norms of ℝ × ℝ × X and Y) and twice continuously differentiable on the open set J ×Ω (where, as before, Ω = (0, ∞) × {u ∈ X | u(x) > c ∀x ∈ [−1, 1]}). In conjunction with the well-known properties of L, this implies that for every (ϵ, μ, u) ∈ Ω, the partial Fréchet derivative H_{(μ,u)}(ϵ, μ, u) is a Fredholm operator of index 1.

For ϵ ∈ J̄, let Σ_{ϵ} denote the solution set of Problem (P_{ϵ}), or equivalently, of Problem (P̃_{ϵ}):

Σ_{ϵ} = {(μ, u) ∈ ℝ_{+} × X_{+} | H(ϵ, μ, u) = 0}.

For any ϵ ∈ J̄, the set Σ_{ϵ} contains the trivial solutions (μ, 0) with μ ∈ ℝ_{+}, but no further trivial solutions unless q(ϵ, ·) is constant (as in the case ϵ = 0). Of course, Σ_{0} coincides with the solution set Σ 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 Σ^{∗} of Problem (P).

Specifically, what is the local structure of Σ_{ϵ}, for ϵ close to 0, near a point (μ^{∗}, u^{∗}) on Σ^{∗}? Applying the results of Section 1, we first note that if (μ^{∗}, u^{∗}) is a regular point of H(0, ·, ·), then Σ_{ϵ} is locally, near (μ^{∗}, u^{∗}), just a continuous deformation of the curve Σ^{∗} and in particular, a simple C^{1}-arc. If (μ^{∗}, u^{∗}) is one of the bifurcation points (μ_{j}, u_{j}) (and thus, a regular-singular point of H(0, ·, ·)), then Σ_{ϵ} is locally, near (μ^{∗}, u^{∗}), a homeomorphic image of the set {(σ, τ) ∈ ℝ^{2} | στ + ρ(ϵ) = 0}, where ρ is a C^{1}-function with ρ(0) = 0 (see Theorem 1.1 for a much more precise statement). In particular, Σ_{ϵ} contains a bifurcation point near (μ^{∗}, u^{∗}) if and only if ρ(ϵ) = 0.

At the end of Section 1, we derived a simple criterion for nonpersistence of the bifurcations occurring at regular-singular points. Applying this criterion in the present situation, we infer the following: If H_{ϵ}(0, μ^{∗}, u^{∗}) does not belong to the range of H_{(μ,u)}(0, μ^{∗}, u^{∗}), then ρ(ϵ) ≠ 0 for all ϵ ≠ 0 close to 0, and then, none of the corresponding sets Σ_{ϵ} contains a bifurcation point near (μ^{∗}, u^{∗}). Now, if (μ^{∗}, u^{∗}) = (μ_{j}, u_{j}) for some j ∈ ℕ, then H_{ϵ}(0, μ^{∗}, u^{∗}) = −μ_{j}f(u_{j})\frac{∂q}{∂ϵ} (0, ·), which is a nonzero constant multiple of \frac{∂q}{∂ϵ} (0, ·). Moreover, R(H_{(μ,u)}(0, μ^{∗}, u^{∗})) = R(H_{u}(0, μ^{∗}, u^{∗})) and H_{u}(0, μ^{∗}, u^{∗}) = L−ℓ_{j}Id (see the proof of Lemma 3.1), so that R(H_{(μ,u)}(0, μ^{∗}, u^{∗})) is nothing but the orthogonal complement, in Y = L^{2}((−1, 1)), of the j-th normalized eigenfunction ϕ_{j} of L. We conclude that the bifurcation at (μ^{∗}, u^{∗}) is nonpersistent provided that

∫_{−1}^{1} \frac{∂q}{∂ϵ} (0, x)ϕ_{j}(x) dx ≠ 0 .

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(ϵ, ·) are even:

k(x) = k(−x) and q(ϵ, x) = q(ϵ, −x)

for all x ∈ [−1, 1] and ϵ ∈ J̄. For every function u : [−1, 1] → ℝ, let ū denote its reflection at x = 0, that is, ū(x) := u(−x) for x ∈ [−1, 1]. A simple calculation shows that Lū = \overline{Lu} for every u ∈ X. As a consequence, the eigenfunctions of L are either even or odd; in fact \overline{ϕ}_{j} = (−1)^{j}ϕ_{j}, for j ∈ ℤ_{+}. It follows that if (μ, u) ∈ ℝ_{+} × X_{+} satisfies H(ϵ, μ, u) = rϕ_{j}, for some ϵ ∈ J̄ and r ∈ ℝ, then (μ, ū) satisfies H(ϵ, μ, ū) = (−1)^{j}rϕ_{j}.

Again, let (μ^{∗}, u^{∗}) be one of the bifurcation points of Problem (P) on Σ^{∗}, that is, (μ^{∗}, u^{∗}) = (μ_{j}, u_{j}) for some j ∈ ℕ. Applying Theorem 1.2, with the eigenfunction ϕ_{j} playing the role of the vector y_{0} ∈ Y \R(H_{(μ,u)}(0, μ^{∗}, u^{∗})), we obtain a neighborhood U ⊂ Ω of (μ^{∗}, u^{∗}) and an interval I ⊂ J containing 0 such that for every ϵ ∈ I, the mapping H(ϵ, ·, ·) has a unique singular point (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) ∈ U whose image H(ϵ, μ_{ϵ}^{∗}, u_{ϵ}^{∗}) is a constant multiple of ϕ_{j}. The point (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) is a regular-singular point of H(ϵ, ·, ·), and there is a function ρ ∈ C^{1}(I) such that H(ϵ, μ_{ϵ}^{∗}, u_{ϵ}^{∗}) = ρ(ϵ)ϕ_{j}.

Being a regular-singular point of H(ϵ, ·, ·), the point (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) is a bifurcation point for the equation

H(ϵ, μ, u) = ρ(ϵ)ϕ_{j} ,

and with our earlier observation regarding the symmetries of H and ϕ_{j}, it follows that (μ_{ϵ}^{∗}, \overline{u}_{ϵ}^{∗}) is a bifurcation point for

H(ϵ, μ, u) = (−1)^{j}ρ(ϵ)ϕ_{j} .

Also, since (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) is close to (μ^{∗}, u^{∗}), so is (μ_{ϵ}^{∗}, \overline{u}_{ϵ}^{∗}) (recall that u^{∗} is a constant). That is, the neighborhood U of (μ^{∗}, u^{∗}) provided by Theorem 1.2 can be chosen so that both points, (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) and (μ_{ϵ}^{∗}, \overline{u}_{ϵ}^{∗}), belong to U. But U contains only one singular point of H(ϵ, ·, ·) whose image is a constant multiple of ϕ_{j} . We conclude that \overline{u}_{ϵ}^{∗} = u_{ϵ}^{∗} and (−1)^{j}ρ(ϵ) = ρ(ϵ). If j is odd, we must have ρ(ϵ) = 0, and then (μ_{ϵ}^{∗}, u_{ϵ}^{∗}) is a bifurcation point for Problem (P_{ϵ}). 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.

Theorem 4.1. Let ((μ_{j}, u_{j}))_{j∈ℕ} denote the sequence of bifurcation points on the trivial solution branch Σ^{∗} of Problem (P), as described in Theorem 3.3, and let (ϕ_{j})_{j∈ℤ_{+}} be the sequence of normalized eigenfunctions of the operator L.

(a) For every j ∈ ℕ, if ∫_{−1}^{1} \frac{∂q}{∂ϵ} (0, x)ϕ_{j}(x) dx ≠ 0, then the bifurcation at (μ_{j}, u_{j}) is nonpersistent: None of the problems (P_{ϵ}) with ϵ close to but different from 0 has a bifurcation point near (μ_{j}, u_{j}). In fact, the solution set of (P_{ϵ}), for any such ϵ, is locally, near (μ_{j}, u_{j}), the union of two disjoint, simple C^{1}-arcs.

(b) If the coefficient k and the functions q(ϵ, ·), for ϵ ∈ J̄, are even, then the bifurcations occurring at odd-numbered points (μ_{j}, u_{j}) are persistent: Each of the problems (P_{ϵ}) with ϵ close to 0 has a unique bifurcation point (μ_{j,ϵ}, u_{j,ϵ}) near (μ_{j}, u_{j}). In fact, the solution set of (P_{ϵ}), for any such ϵ, is locally, near (μ_{j}, u_{j}), the union of two simple C^{1}-arcs that intersect transversally at (μ_{j,ϵ}, u_{j,ϵ}).

Example of a trivial solution branch Σ^{∗} with bifurcation points (μ_{j},u_{j}).

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Bettina E. Schmidt Department of Mathematics Auburn University at Montgomery Montgomery, AL 36124-4023, USA Email address: bes@strudel.aum.edu