Differential Equations and Computational Simulations III J. Graef, R. Shivaji, B. Soni, & J. Zhu (Editors) Electronic Journal of Differential Equations, Conference 01, 1997, pp.109–117. 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)

A BIFURCATION RESULT FOR STURM-LIOUVILLE PROBLEMS WITH A SET-VALUED TERM

Georg Hetzer

Abstract

It is established in this note that −(ku′)′ + g(·, u) ∈ μF (·, u), u′(0) = 0 = u′(1), has a multiple bifurcation point at (0, 0) in the sense that infinitely many continua meet at (0, 0). F is a “set-valued representation” of a function with jump discontinuities along the line segment [0, 1] × {0}. The proof relies on a Sturm-Liouville version of Rabinowitz’s bifurcation theorem and an approximation procedure.

1. Introduction

Of concern is

−(ku′)′(x) + g(x, u(x)) ∈ μF (x, u(x)) x ∈ (0, 1) a.e. (1)

u′(0) = 0, u′(1) = 0 ,

under hypotheses motivated by the situation found for Budyko-North type energy balance climate models (cf. [5, 6, 7, 8, 9] and the references therein). In particular, the set-valued right hand side arises from a jump discontinuity of the albedo at the ice-edge in these models. By filling in such a gap (this is the solution concept we adapt), one arrives at the set-valued problem (1). We are here interested in a considerably simplified version as compared to the situation from climate modeling, e.g. a one-dimensional regular Sturm-Liouville differential operator substitutes for a two-dimensional Laplace-Beltrami operator or a singular Legendre-type operator, and the jump discontinuity is transformed to u = 0 in a way, which resembles only locally the climatological problem. The latter suffices for our purposes, since the global structure has already been investigated in [8] and [3], where the existence of an S-shaped principal solution branch is established. Computer simulations actually suggest that the branch structure of the original problem is very different from what we obtain for (1) in this paper. Here the effect of the discontinuity at zero is a solution branch which consists of infinitely many subbranches all meeting in (0, 0). Two subbranches are distinguished by the number of zeroes of the respective solutions. Thus, one would expect such a “multiple” bifurcation to occur on the relevant segment of the principal solution branch in the climatological setting, too. Unlike in our case, the subbranches would however be bounded (mushrooms in the sense of [13]). An example in Appendix B of [16], a Sturm-Liouville problem of Budyko type with infinitely many solutions, seemingly supports such a conjecture. But, numerical computations do not reveal any bifurcation point on the principal branch, rather the branch structure resembles that for Sellers-type models (continuous setting, cf. [10, 12] and [11] for a survey). Thus, our results are to be understood as an incentive for further investigations in order to clarify this issue.

1991 Subject Classification: 34B15, 34C23, 47H04, 86A10.

Key words and phrases: Differential inclusion, Sturm-Liouville problem, Ra-

binowitz bifurcation.

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

Published November 12, 1998.

Let us illustrate what is going on by means of the following simple example

−u′′ + bu ∈ μ sgn u on [0, π]a.e. (2)
u′(0) = 0 u′(π) = 0 ,

with b > 0 and

sgn(y) = \{ \begin{matrix}\{y/|y|\} & y ∈ ℝ \ \{0\} \\ [−1, 1] & y = 0. \end{matrix}

Assume that one approximates sgn by a sequence of functions h_{j} ∈ C^{∞}(ℝ) satisfying h_{j}(0) = 0, h_{j}′(0) = j, h_{j}|_{(−∞,−1/j]} ≡ −1, h_{j}|_{[1/j,∞)} ≡ 1, h_{j}′(y) > 0 for y ∈ (−1/j, 1/j) and (h_{j})_{j∈ℕ} uniformly convergent on (−∞, −r) ∪ (r, ∞) for r > 0 to y ↦ y/|y|. Then results of Rabinowitz and Crandall and Rabinowitz ensure that μ_{j,n} := (n^{2} + b)/j is a bifurcation point for n ∈ ℤ_{+} of

−u′′ + bu = μh_{j} ∘ u on [0, π]a.e.
u′(0) = 0 u′(π) = 0,

that the branch S_{j,n} emanating from (μ_{j,n}, 0) is unbounded in R_{+} × C([0, π]) and that (μ, w) ∈ S_{j,n} implies that w has exactly n zeroes, which are all simple. Letting j → ∞ yields μ_{j,n} → 0 uniformly for n in any finite subset of Z_{+}, hence any finite number of bifurcation points comes arbitrarily close to 0 for a sufficiently large j ∈ ℕ, thus one expects infinitely many curves of solutions of (2) bifurcating from (0, 0). In fact, these subbranches can be described explicitly as curves μ ↦ (|μ|, u_{μ,j}) (μ ∈ ℝ), where u_{μ,0} ≡ μ/b (no zeroes),

u_{μ,1}(x) := \{ \begin{matrix}\frac{μ}{b}[1 − \frac{cosh(\sqrt{b}x)}{cosh(\sqrt{b}π/2)}] & if x ∈ [0, π/2] \\ −\frac{μ}{b}[1 − \frac{cosh(\sqrt{b}(x−π))}{cosh(\sqrt{b}π/2)}] & if x ∈ (π/2, π] \end{matrix}

(one zero) or

u_{μ,2}(x) := \{ \begin{matrix}\frac{μ}{b}[1 − \frac{cosh(\sqrt{b}x)}{cosh(\sqrt{b}π/4)}] & if x ∈ [0, π/4] \\ −\frac{μ}{b}[1 − \frac{cosh(\sqrt{b}(x−π/2))}{cosh(\sqrt{b}π/4)}] & if x ∈ (π/4, 3π/4] \\ \frac{μ}{b}[1 − \frac{cosh(\sqrt{b}(x−π))}{cosh(\sqrt{b}π/4)}] & if x ∈ (3π/4, π] \end{matrix}

(two zeroes) and so on.

The same structure has been observed in another context by [2; Theorem 2.1.] for a similar reason. The paper deals with a quasilinear Sturm-Liouville problem; the differential operator is the radial part of a p-Laplacian and the right hand side vanishes at zero with a slower rate than |x|^{p−1}, thus the “derivative” is infinite relative to the intrinsic scaling associated with the nonlinear diffusion.

Of course, the use of approximating branches is standard in numerical bifurcation theory, and limiting processes for sets go back at least to the work of Kuratowski. Nowadays, the technical tool for the latter is a so-called Whyburn Lemma, which will be stated in Section 3. It should be mentioned that the approach outlined above has been used in [2].

Our findings are also related to results in [4, 15], where bifurcation from intervals was investigated. In a sense, here the real line is the bifurcation interval, and therefore the (extended) Rabinowitz alternative established in these papers becomes meaningless.

We have not attempted in this note to strive for generality, but shall present the method for the case we are interested in. Obtaining other versions is then a matter of routine. The precise result is stated in the next section, proofs are given in Section 3, which is followed by a concluding remark.

2. The Bifurcation Result

Throughout we assume:

(H1) k ∈ C^{1}([0, 1]), inf k > 0;

(H2) g ∈ C([0, 1] × ℝ), g(x, ·) strictly increasing for x ∈ [0, 1], g_{1}(x) := lim_{y→0} \frac{g(x, y)}{y}

exists uniformly for x ∈ [0, 1];

(H3) f_{+} ∈ C([0, 1]×ℝ_{+}, (0, ∞)), inf f_{+} > 0, f_{−} ∈ C([0, 1]×ℝ_{−}, (−∞, 0)), sup f_{−} < 0.

Let F in (1) be given by

F (x, y) := \{ \begin{matrix}\{f_{+}(x, y)\} & x ∈ [0, 1], y > 0, \\ [f_{−}(x, 0), f_{+}(x, 0)] & x ∈ [0, 1], \\ \{f_{−}(x, y)\} & x ∈ [0, 1], y < 0 \end{matrix}

and set S := {(μ, w) ∈ ℝ × W^{2,∞}([0, 1]), (μ, w) solves (1)}. Throughout S will be considered as subset of the Banach space Y := ℝ × C^{1}([0, 1]) under the norm ∥ · ∥_{Y} : (μ, w) ↦ max{|μ|, ∥w∥_{∞}, ∥w′∥_{∞}}. It is useful to note:

Remark 1. The hypotheses (H1)–(H3) imply that

S ∩ ((−∞, 0] × C^{1}([0, 1])) = (−∞, 0] × {(0, 0)} .

In fact, denoting by u^{+} and u^{−} the positive and negative parts of u, respectively, one gets for (μ, u) ∈ S with μ ≤ 0 that

\begin{matrix}∫_{0}^{1}[k(x)u′(x)^{2} + g(x, u(x))u(x)]dx \\ = μ ∫_{0}^{1}[f_{+}(x, u(x))u^{+}(x) + |f_{−}(x, u(x))|u^{−}(x)]dx ≤ 0, \end{matrix}

i.e. u ≡ 0 in view of g(x·) strictly increasing and g(·, 0) ≡ 0.

Our main result is.

Theorem. Let (H1)–(H3) be fulfilled. Then there exist sequences (C_{n}^{±})_{n∈ℤ_{+}} of unbounded, closed, connected subsets of S with (0, 0) ∈ C_{n}^{±} and the property that u has exactly n zeroes, which are all simple, if (μ, u) ∈ C_{n}^{±} \ {(0, 0)}. Moreover, u is positive (negative) on an interval (0, x̃) for some x̃ ∈ (0, 1], if (μ, u) ∈ C_{n}^{+} ((μ, u) ∈ C_{n}^{−}) and u ≢ 0.

Actually, such continua can be obtained as upper limits in the sense of Kuratowski of sequences of solution continua from associated continuous problems. To this end one sets d_{f} := min{inf f_{+}, inf |f_{−}|} and selects an approximation sequence (f_{l}) ∈ C([0, 1] × ℝ, ℝ)^{ℕ} of F satisfying

(A1) f_{l}(x, y) = ly for x ∈ [0, 1] and y ∈ [−\frac{d_{f}}{2l} , \frac{d_{f}}{2l}];

(A2) f_{l}(x, y) × sgn(y) ≥ \frac{d_{f}}{2} for x ∈ [0, 1] and |y| ≥ \frac{d_{f}}{2l}, f_{l} ≤ f_{+} on [0, 1] × [\frac{d_{f}}{2l} , \frac{d_{f}}{l}], f_{l} ≥ f_{−} on [0, 1] × [−\frac{d_{f}}{l} , −\frac{d_{f}}{2l}];

(A3) f_{l}(x, y) = f_{+}(x, y) for x ∈ [0, 1] and y ≥ \frac{d_{f}}{l}; f_{l}(x, y) = f_{−}(x, y) for x ∈ [0, 1] and y ≤ −\frac{d_{f}}{l};

(A4) (f_{l}(x, y))_{l∈ℕ} nondecreasing for (x, y) ∈ [0, 1] × (0, ∞) and nonincreasing for (x, y) ∈ [0, 1] × (−∞, 0).

It is easy to see thanks to (H2) and (A1) that

−(kv′_{l})′(x) + g(x, v_{l}(x)) = μf_{l}(x, v_{l}(x)) x ∈ [0, 1] (3_{l})

v′_{l}(0) = 0 v′_{l}(1) = 0 .

falls into the scope of the Sturm-Liouville version of the celebrated Rabinowitz bifurcation theorem (cf. [14] for a more general, but somewhat different setting).

Indeed, denote the strictly increasing sequence of simple eigenvalues of

−(kψ′)′ + g_{1}ψ = λψ on [0, 1] (4)

ψ′(0) = 0 ψ′(1) = 0

by (λ_{n})_{n∈ℤ_{+}} and set μ_{n,l} := λ_{n}/l. Then (μ_{n,l}, 0) is a bifurcation point of the solution set of (3_{l}) for every n ∈ ℤ_{+}, and for each (n, l) ∈ ℤ_{+} × ℕ there exist two unbounded closed connected subsets C_{n,l}^{±} of the solution set of (3_{l}) with

• C_{n,l}^{+} ∩ C_{n,l}^{−} = {(μ_{n,l}, 0)}. Moreover, (μ_{n,l}, 0) is the only bifurcation point contained in C_{n,l}^{±};

• If (μ, ϑ) ∈ C_{n,l}^{+} and ϑ ≢ 0, then ϑ possesses exactly n simple zeroes (and no multiple zeroes) in (0, 1) and is positive on (0, δ) for some δ > 0;

• C_{n,l}^{−} fulfills the previous statement in case that positive on (0, δ) is replaced by negative on (0, δ).

Utilizing a so-called Whyburn Lemma [17] we establish.

Lemma 1. Let (H1) - (H3) and (A1)–(A4) be satisfied and B_{r} denote the ball with center (0, 0) and radius r in Y. Then continua C_{n}^{±} fulfilling the properties stated in the Theorem can be constructed as C_{n}^{±} = ⋃_{j∈ℕ}(lim sup_{l→∞}(C_{n,l}^{±} ∩ B_{j})).

A Bifurcation Result 113 3. Proof of Main Result The proof is given for C_{n}^{+}. Recall Kuratowski’s notion of lower and upper limits

of sequences of sets. Definition 1. Let X be a metric space and (Z_{l})_{l∈N} be a sequence of subsets of X. lim sup_{l→∞} Z_{l} := {x ∈ X : lim inf_{l→∞} dist(x, Z_{l}) = 0} is called the upper limit of the sequence (Z_{l}), whereas

lim inf_{l→∞} Z_{l} := {x ∈ X : lim_{l→∞} dist(x, Z_{l}) = 0}

is called the lower limit of the sequence(Z_{l}). Whyburn’s result tells us that Lemma 2. lim sup_{l→∞} Z_{l} is nonempty, compact and connected provided that X is complete, lim inf_{l→∞} Z_{l} ≠ ∅ and ⋃_{l∈ℕ} Z_{l} is relatively compact.

Fixing n ∈ ℤ_{+} and r ∈ (0, ∞) we are going to apply this lemma to (C_{n,l}^{+}∩B_{r})_{l∈ℕ}. Since φ ↦ −(kφ′)′ + (g_{1} + 1)φ for φ ∈ C^{2}([0, 1]) with φ′(0) = 0 = φ′(1) has a completely continuous inverse from C([0, 1]) into C^{1}([0, 1]), the relative compactness of ∪_{l∈ℕ}(C_{n,l}^{+}∩B_{r}) in Y follows by standard arguments. Moreover, (0, 0) ∈ lim inf_{l→∞}(C_{n,l}^{+}∩ B_{r}) holds because of lim_{l→∞} μ_{n,l} = 0. Thus, C_{n}^{+}(r) := lim sup_{l→∞}(C_{n,l}^{+} ∩ B_{r}) is nonempty, compact and connected in Y. Setting C_{n}^{+} := ⋃_{j∈ℕ} C_{n}^{+}(j) we claim that C_{n}^{+} has the properties as stated in the Theorem. Lemma 3. C_{n}^{+} is unbounded, closed and connected in Y and contains (0, 0). The proof or the above lemma is an easy exercise! Lemma 4. If (μ, u) ∈ C_{n}^{+}, then (μ, u) is a solution of (1) and u ∈ W^{2,∞}([0, 1]). Proof. By definition there exist sequences (κ_{l}) ∈ ℕ^{ℕ} strictly increasing, and ((ν_{κ_{l}}, v_{κ_{l}})) ∈ Y^{ℕ} with (ν_{κ_{l}}, v_{κ_{l}}) ∈ C_{n,κ_{l}}^{+} for l ∈ ℕ and (ν_{κ_{l}}, v_{κ_{l}}) → (μ, u). Since (f_{κ_{l}}(·, v_{κ_{l}}(·)))_{l∈ℕ} is uniformly bounded, we can assume after passing to a subsequence, if necessary, that it converges weakly in L_{2}([0, 1]) to some ϕ. We claim that ϕ(x) ∈ F (x, u(x)) a.e. on (0, 1).

Let x_{0} ∈ (0, 1) with u(x_{0}) > 0. Then there exist ρ > 0 and δ ∈ (0, min{x_{0}, 1 − x_{0}}) with u(x) > ρ for all x ∈ (x_{0} − δ, x_{0} + δ), hence there is an l_{0} ∈ ℕ with v_{κ_{l}}(x) > \frac{ρ}{2} for all l ≥ l_{0} and x ∈ (x_{0} − δ, x_{0} + δ). Choose l_{1} > l_{0} with \frac{d_{f}}{κ_{l_{1}}} < \frac{ρ}{2}. Then f_{κ_{l}}(x, v_{κ_{l}}(x)) = f_{+}(x, v_{κ_{l}}(x)) for all l ≥ l_{1} and all x ∈ (x_{0} − δ, x_{0} + δ), which yields ϕ(x) = f_{+}(x, u(x)) for x ∈ (x_{0} − δ, x_{0} + δ) a.e.. Likewise, the case u(x_{0}) < 0 is treated.

Next, let Ξ := {x ∈ (0, 1): ϕ(x) > f_{+}(x, 0)}. Suppose that meas(Ξ) > 0. Then ε := ∫_{Ξ}[ϕ(x) − f_{+}(x, 0)]dx > 0, and one finds η ∈ (0, ∞) with meas(Ξ)|f_{+}(x, y) − f_{+}(x, 0)| ≤ \frac{ε}{2} for x ∈ [0, 1] and y ∈ [0, η]. Choosing l_{2} ∈ ℕ with ∥v_{κ_{l}} − u∥_{∞} < η for l ≥ l_{2} and observing that f_{κ_{l}}(·, y) < 0 if y < 0, one obtains for l ≥ l_{2}:

∫_{Ξ}[ϕ(x) − f_{κ_{l}}(x, v_{κ_{l}}(x))]dx = ∫_{Ξ}[ϕ(x) − f_{+}(x, 0)]dx
+ ∫_{Ξ}[f_{+}(x, 0) − f_{κ_{l}}(x, v_{κ_{l}}(x))]dx
≥ε + ∫_{Ξ}[f_{+}(x, 0) − f_{κ_{l}}(x, v_{κ_{l}}^{+}(x))]dx
≥\frac{ε}{2},

which contradicts f_{κ_{l}}(·, v_{κ_{l}}(·)) ⇀^{L_{2}} ϕ. Thus, meas(Ξ) = 0.

Likewise, one derives ϕ(x) ≥ f_{−}(x, 0) for almost every x ∈ [0, 1] with u(x) = 0, hence ϕ(x) ∈ F (x, u(x)) holds for x ∈ [0, 1] a.e..

Now, let A be the closed linear operator in L_{2}([0, 1]) defined by dom(A) := {φ ∈ W^{2,2}([0, 1]) : φ′(0) = 0 = φ′(1)} and Aφ := −(kφ′)′. Clearly, ν_{κ_{l}}f_{κ_{l}}(·, v_{κ_{l}}(·)) − g(·, v_{κ_{l}}(·) ⇀^{L_{2}} μϕ − g(·, u(·)), hence v_{κ_{l}} → u and the fact that A is weakly closed yield Au = μϕ − g(·, u(·)), i.e. Au + g(·, u(·)) ∈ F (·, u(·)) a.e.. Finally, note that u ∈ W^{2,∞}([0, 1]) thanks to the uniform boundedness of u and assumptions (H2) and (H3).

Let Z_{u} := {x ∈ [0, 1] : u(x) = 0} denote the set of zeroes of u for u ∈ C([0, 1]) and SZ_{u} := {x ∈ Z_{u} : u′(x) ≠ 0} the set of simple zeroes of u for u ∈ C^{1}([0, 1]).

Lemma 5. Let (μ, u) ∈ S, u ≢ 0 and Z_{u} \ SZ_{u} ≠ ∅. Then SZ_{u} is an infinite set.

Proof. Note that u ≢ 0, hence μ > 0 by Remark 1. Let ẑ ∈ Z_{u} \ SZ_{u} be a boundary point of Z_{u}. Consider the case ẑ > 0 and assume that there is an ε ∈ (0, ẑ) with (ẑ − ε, ẑ) ∩ Z_{u} = ∅. Then either Z_{u} ∩ [0, ẑ) = ∅, and one sets \underline{z} = 0 or there exists a \underline{z} ∈ [0, ẑ − ε] with u(\underline{z}) = 0 and (\underline{z}, ẑ) ∩ Z_{u} = ∅. Let σ := sgn(u|_{(\underline{z},ẑ)}), v := σu and ȳ:= inf{|y| : y ∈ ℝ, |g(x, y)| ≥ d_{f}}, where d_{f} has the same meaning as in Section 2. It follows from (H2) and (H3) that ȳ > 0. Setting

ς := \{ \begin{matrix}+ & if σ = 1 \\ − & if σ = −1 \end{matrix}

and ρ := ∥g(·, u)∥_{∞}/ȳ, one obtains

μσf_{ς}(x, σv(x)) − σg(x, σv(x)) + ρv(x) ≥ 0 for x ∈ [\underline{z}, ẑ] a.e..

Hence v satisfies −(kv′)′ + ρv ≥ 0 on (\underline{z}, ẑ) and either v′(\underline{z}) = 0 = v(ẑ) or v(\underline{z}) = 0 = v(ẑ). Moreover, the fact that v(x) > 0 for x ∈ (\underline{z}, ẑ) allows us to assume that v ∈ C^{2}((\underline{z}, ẑ)), thus the strong maximum principle yields v′(ẑ) < 0, a contradiction. Consequently, ẑ is a accumulation point of (0, ẑ) ∩ Z_{u}, and one finds a strictly increasing sequence (z_{j})_{j∈ℕ} such that z_{j} → ẑ, (z_{2j−1}, z_{2j}) ∩ Z_{u} = ∅ and u(z_{j}) = 0. The same argument as above shows u′(z_{j}) ≠ 0, and SZ_{u} is an infinite set. The other cases are treated likewise.

Remark 2. Let (μ, u) ∈ C_{n}^{±}. By construction, one finds a sequence (v_{j})_{j∈ℕ} with

Z_{v_{j}} = SZ_{v_{j}} for j ∈ ℕ, SZ_{v_{j}} contains n elements and v_{j} →^{C^{1}} u, hence u cannot have more than n simple zeroes, consequently Lemma 5 implies u ≡ 0 in case that Z_{u} \ SZ_{u} ≠ ∅.

The following lemma excludes this latter possibility if μ > 0. In order to state it, we introduce the following notation. The principal eigenvalue of

−∥k∥_{∞}ψ′′ + ∥g_{1}∥_{∞}ψ = νψ on (0, \frac{1}{n + 1}) (5)

ψ(0) = 0 ψ(\frac{1}{n + 1}) = 0

will be denoted by ν^{⋆}(n).

Lemma 6. Let (H1)–(H3) and (A1)–(A4) be fulfilled, n ∈ ℤ_{+}, \underline{μ} ∈ (0, ∞) and l_{0} ∈ ℕ with \underline{μ}l_{0} > ν^{⋆}(n). Then there exists a δ > 0 such that ∥u∥_{∞} ≥ δ for all (μ, u) ∈ C_{n,l}^{+} with μ ≥ \underline{μ} and l ≥ l_{0}.

Proof. (cf. [2] for a similar reasoning) Let ε ∈ (0, \frac{1}{2}(\underline{μ}l_{0} − ν^{⋆}(n))) and choose δ ∈ (0, \frac{d_{f}}{2l_{0}} )] with |g(x, y) − g_{1}(x)y| ≤ ε|y| for all x ∈ [0, 1] and |y| ≤ δ. Let l ∈ ℕ with l ≥ l_{0} and (μ, u) ∈ C_{n,l}^{+} with μ ≥ \underline{μ}. Let us fix ideas by considering the case, where one can find \underline{x}, x̄ ∈ (0, 1) with x̄ − \underline{x} ≥ \frac{1}{n+1}, u(\underline{x}) = 0 = u(x̄) and u(x) > 0 for x ∈ (\underline{x}, x̄).

Assume that u(x) ≤ δ for x ∈ (\underline{x}, x̄). Noting that (A1) and the first part of (A2) guarantee that |f_{l}(x, y)| ≥ l_{0}|y| for all x ∈ [0, 1], |y| ≤ \frac{d_{f}}{2l_{0}} and l ≥ l_{0}, one obtains:

−(ku′)′(x) + g_{1}(x)u(x) = μf_{l}(x, u(x) − (g(x, u(x)) − g_{1}(x)u(x))

≥ \underline{μ}l_{0}u(x) − εu(x)

> \frac{1}{2}(\underline{μ}l_{0} + ν^{⋆}(n))u(x)

> ν^{⋆}(n)u(x)

for x ∈ (\underline{x}, x̄). The principal eigenvalue ν_{0} of

−(kψ′)′ + g_{1}(·)ψ = νψ on (\underline{x}, x̄) (6)

ψ(\underline{x}) = 0 ψ(x̄) = 0 .

satisfies ν_{0} ≤ ν^{⋆}(n) in view of x̄ − \underline{x} ≥ \frac{1}{n+1}, k ≤ ∥k∥_{∞} and g_{1} ≤ ∥g_{1}∥_{∞}, hence

\begin{matrix}ν_{0} ∫ x̄ ; \underline{x} y(x)u(x) dx = & ∫ x̄ ; \underline{x}[−(ky′)′(x) + g_{1}(x)y(x)]u(x)] dx \\ = & ∫ x̄ ; \underline{x}[−(ku′)′(x) + g_{1}(x)u(x)]y(x)] dx \\ > & ν_{0} ∫ x̄ ; \underline{x} u(x)y(x) dx \end{matrix}

for any nonnegative principal eigenfunction y of (6) which is a contradiction, and ∥u∥_{∞} > δ is established in that case.

The case u(x) < 0 on (\underline{x}, x̄) can be handled by dealing with −u as above. If there is no such interval (\underline{x}, x̄) of length at least 1/(n + 1), one finds either an x̄ ≥ 1/(n + 1) such that u has no zero in (0, x̄) or an \underline{x} ≤ 1 − \frac{1}{n+1} such that u has no zero in (\underline{x}, 1). Noting that the principal eigenvalue of (6) becomes smaller, if one replaces ψ(\underline{x}) = 0 by ψ′(0) = 0 or ψ(x̄) = 0 by ψ′(1) = 0, one can use the same reasoning for these two cases, too.

Now, we can finish the proof of the theorem. It follows from Lemma 4 that C_{n}^{+} ⊆ S. Moreover, Lemma 6 shows that (μ, 0) ∈ C_{n}^{+} implies μ = 0. Remark 2 therefore implies that given (μ, u) ∈ C_{n}^{+} \ {(0, 0)}, u has only finitely many zeroes which are all simple. The construction of C_{n}^{+} and the fact that the zero number is constant in a C^{1}-neighborhood of a C^{1}-function with finitely many simple zeroes imply therefore that u has exactly n simple zeroes.

4. Concluding Remark

We have focused in this paper on a class of Sturm-Liouville problems with some special features that were suggested by Budyko-North type energy balance climate models. Of course, one can also deal by this technique with other boundary conditions and nonlinearities and can allow nontrivial solutions for negative parameter values. Furthermore, it is possible to consider radial solvability of second order elliptic boundary value problems including degenerate cases similar to [2]; roughly speaking, the approach works whenever one has sufficient regularity of the solutions and nodal properties as in the Sturm-Liouville case. One challenge ahead is to come up with substitutes for this nodal properties (e.g. Morse indices etc.) in order to address general second order elliptic boundary value problems. The other central issue is to analyze the more complex situation arising from the climate model.

Acknowledgment. It is a pleasure to thank a referee for suggesting a simplification in the proof of Lemma 6 and for pointing out several misprints.

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Georg Hetzer Department of Mathematics, Auburn University, Auburn, AL 36849-5310, USA E-mail address: hetzege@mail.auburn.edu