CUBO A Mathematical Journal Vol.10, N o ¯ 04, (85–100). December 2008 A Disc-Cutting Theorem and Two-Dimensional Bifurcation of a Reaction-Diffusion System with Inclusions Martin Väth* University of Würzburg, Math. Institut Am Hubland, D-97074 Würzburg, Germany email: vaeth@mathematik.uni-wuerzburg.de ABSTRACT We provide a topological disc-cutting theorem which allows to prove that unilateral inclusions in a reaction-diffusion system of prey-predator type with a two-dimensional bifurcation parameter necessarily have a certain global branch of (global) bifurcation points. RESUMEN Presentamos un teorema “Disc-Cutting” topológico el cual permite probar que in- clusiones unilaterales en un sistema de reación-difusión de tipo predador-presa con parametro de bifurcación 2-dimencional, necessariamente tiene una cierta rama global de puntos de bifucarción (global). *This paper was written in the framework of a Heisenberg fellowship (Az. VA 206/1-2). Financial support by the DFG is gratefully acknowledged. The author wants to thank E. Vogt for valuable comments and suggestions. 86 Martin Väth CUBO 10, 4 (2008) Key words and phrases: Global bifurcation, two-dimensional bifurcation, elliptic equation, in- clusion, Laplace operator. Math. Subj. Class.: 35B32, 35J60, 35K47, 47H04, 47H11. 1 Introduction Although we provide in Section 2 a general topological theorem about the existence of a global branch which is applicable to a large class of bifurcation problems with a parameter from a space of dimension at least 2, our main motivation for the result comes from the following particular problem. Let Ω ⊆ Rn be a bounded domain with a Lipschitz boundary, and let measurable (possibly empty) subsets Ω0 ⊆ Ω and Γ0, Γ ⊆ ∂Ω be fixed with mes(Γ0 ∩ Γ) = 0. We consider the reaction- diffusion system ut = d1∆u + b11u + b12v + f1(d, x, u, v, ∇u, ∇v) = 0 on Ω, vt ∈ d2∆v + b21u + b22v + f2(d, x, u, v, ∇u, ∇v) + { {0} on Ω \ Ω0, m0(d, x, u, v, ∇u, ∇v) on Ω0, (1.1) with the boundary conditions                    u = v = 0 on Γ0, ∂u ∂n = g1(d, x, u, v) on ∂Ω \ Γ0, ∂v ∂n = g2(d, x, u, v) on ∂Ω \ (Γ0 ∪ Γ), ∂v ∂n ∈ g2(d, x, u, v) + m1(d, x, u, v) on Γ. (1.2) Here, d = (d1, d2) ∈ R 2 + is a bifurcation parameter, the nonlinearities fi and gi are small at (u, v) = 0, and mi are nonnegative interval functions specified later. The scalar parameters bij are assumed to satisfy b11 > 0, b12 < 0, b21 > 0, b22 < 0, b11 + b22 < 0, b11b22 − b12b21 > 0, (1.3) which means that system (1.1) is a special system of activator-inhibitor or prey-predator type such that in case d1 = d2 = 0 (i.e. without diffusion) the solution (0, 0) is stable. However, it is known (see e.g. [11] or [2, Appendix] or [1]) that the stability of (1.1)/(1.2) with classical data m0 = m1 = 0 depends on d = (d1, d2). In fact, the domain DS of those d ∈ R 2 + where this system is exponentially stable is the right-hand side of the “envelope” of the sequence of hyperbolas Cn := { (d1, d2) ∈ R 2 + : d2 = b12b21/κ 2 n d1 − b11/κn + b22 κn } . (1.4) CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 87 where κ1 ≤ κ2 ≤ · · · → ∞ denotes the sequence of eigenvalues of −∆, i.e. for which a (weak) nontrivial solution of the problem          −∆u = κnu on Ω, u = 0 on Γ0, ∂u ∂n = 0 on ∂Ω \ Γ0 (1.5) exists. d1 d2 C4 C3 C2 C1 DS Figure 1: Hyperbolas (1.4) determining DS Using degree theory for multivalued maps, it was shown in [6] (for similar results and related systems, see also [2]–[5], [7]–[10]) that in the case of natural unilateral (possibly multivalued) functions m0 and/or m1 there is a destabilizing effect in the sense that even the stationary points admit a global bifurcation along certain paths in DS . In this paper, we will show by a purely topological argument that results of such a type actually imply the existence of certain global branches of bifurcation points (i.e. not only the bifurcation branch is global but actually even the set of bifurcation points itself). Such phenomena can naturally arise only because our bifurcation parameter d is not only from a one-dimensional space. We point out that although we concentrate only on the system (1.1)/(1.2), the same results (and proofs) hold for all systems for which corresponding results along paths are available. In particular, this is the case when we consider instead of (1.2) the Signorini type boundary conditions (see e.g. [9], [10])                    u = v = 0 on Γ0, ∂u ∂n = 0 on ∂Ω \ Γ0, ∂v ∂n = 0 on ∂Ω \ (Γ0 ∪ Γ), ∂v ∂n ≥ 0, v ≥ 0, ∂v ∂n · v = 0 on Γ. 88 Martin Väth CUBO 10, 4 (2008) 2 The Disc-Cutting Theorem Our main topological tool is based on a generalization of the Whyburn lemma yielding global branches [12] and on the following result. Theorem 2.1 (Boundaries connect squaresides). Let X be a topological space into which a square with (compact) “square-sides” A1, A2, A3, A4 and “square-interior” Q is homeomorphically embed- ded. Let V ⊆ X be open such that A1 ⊆ V and V ∩ A3 = ∅. Then there is a connected subset C ⊆ Q ∩ ∂V such that C ∩ Ai 6= ∅ for i = 2, 4. The afore mentioned generalization of the Whyburn lemma is the following (this version can be proved using only the [countable] axiom of dependent choices): Theorem 2.2. Let X be a regular space, A ⊆ X compact, and S ⊆ X. Then for each open set U ⊇ A for which U ∩ S is compact and metrizable the following statements are equivalent: 1. For each open set Ω ⊇ A with Ω ⊆ U there is some x ∈ ∂Ω in S. 2. There is a connected set C ⊆ (S ∩ U ) \ A such that C ∩ S intersects A and ∂U . For the proof of Theorem 2.2 we refer to [12]. Let us now use this result and some winding number theory to prove Theorem 2.1. Proof of Theorem 2.1. We show first that it suffices to show the claim for the case X = R2 and Q = (0, 1)×(0, 1) with A1 := [0, 1]×{0}, A2 := {1}×[0, 1], A3 := [0, 1]×{1}, and A4 := {0}×[0, 1]. To see that then the general case holds also, assume that we have a homeomorphism f of Q onto a subset X0 of a general space X. Note that X0 is the union of the sets ˜Q := f (Q) and ˜Ai := f (Ai). Note that we do not assume that X is Hausdorff, so X0 might not be closed, although it is compact. However, if V ⊆ X is open with ˜A1 ⊆ V and V ∩ ˜A3 = ∅, then V0 := f −1 (V ∩ X0) is open in Q, and so there is some open V1 ⊆ R 2 with V1 ∩ Q = V0. Since V 0 ⊆ f −1 (V ∩ X0) is disjoint from the compact set f −1( ˜A3) = A3, and since R 2 is regular, we thus find a closed neighborhood of A3 which is disjoint from V0. Eliminating this neighborhood from V1 if necessary, we may thus assume without loss of generality that V 1 ∩ A3 = ∅. By the special case of the Theorem, we thus find a connected set C ⊆ Q ∩ ∂V1 with C ∩ Ai 6= ∅ for i = 2, 4. Then ˜C := f (C) is connected, and its closure contains f (C) and thus intersects ˜Ai = f (Ai) for i = 2, 4. Moreover, ˜C is contained in f (Q ∩ ∂V1). Note that, since Q is open, V 1 ∩ Q = (V1 ∩ Q) ∩ Q = V 0 ∩ Q, and so ˜C ⊆ f (V 0 ∩ Q) ⊆ ˜Q ∩ V . Moreover, ˜C is disjoint from V , since ˜C = f (C) is contained in X0 and disjoint from V ∩ X0 = f (V0) in view of C ∩ V0 = ∅. Hence, we have indeed found a connected set ˜C ⊆ Q ∩ V \ V = Q ∩ ∂V whose closure intersects ˜Ai for i = 2, 4. This proves the general case of the claim. We prove now the claim in the special case X := R2, Q := (0, 1) × (0, 1) and Ai as above. Thus, let V ⊆ R2 be open with A1 ⊆ V and V ∩ A3 = ∅. Without loss of generality, we may CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 89 assume in addition that V is contained in Q0 := (−1, 2) × (−1, 1). Indeed, otherwise, we could replace V by the intersection V0 := V ∩ {(x1, x2) ∈ (−1/2, 3/2) × (−1/2, 1) : x2 < 1 − dist(x1, [0, 1])} and note that A1 ⊆ V0, V 0 ∩ A3 = ∅, and Q ∩ ∂V0 = Q ∩ ∂V . Put S := Q ∩ ∂V and assume by contradiction that there is no connected subset C ⊆ S ∩ Q with C ∩ Ai 6= ∅ (i = 2, 4). Applying Theorem 2.2 in the space Q with A := A2 and U := Q \ A4, we find some open (in Q) set Ω ⊆ Q with A2 ⊆ Ω and Ω ∩ A4 = ∅ such that the boundary B0 of Ω with respect to Q contains no point from ∂V . By the compactness of these sets, we thus find some closed neighborhood M ⊆ R2 of ∂V which is disjoint from B0. Since ∂V is a compact subset of Q0 and R 2 is regular, we may assume in addition that M ⊆ Q0. Covering the compact set V with sufficiently small open balls, we find an open set G ⊆ V ∪ M containing V such that the boundary of G consists of finitely many piecewise smooth closed curves. Fix some a ∈ A1. Since a ∈ V ⊆ G, the argument principle of complex analysis (or, in other words, the well-known connection between the degree of the identity function with the winding number of the boundary) implies that at least one of these curves must have nonzero winding number with respect to a. We think of such a closed curve as a continuous map γ : S1 → Q0 (where S 1 denotes the unit circle). Note that this curve lies completely in ∂G ⊆ M . In particular, γ(S1) is disjoint from B0 and thus contained in the union of the three sets Ω2 := Ω ∩ Q, Ω4 := (Q \ Ω) ∩ Q = Q \ Ω, R := Q0 \ (Q ∪ {a}). Since Ω is open in Q, it follows that Ω2 is open in Q and thus open in R 2 . Analogously, also Ω4 is open in R 2 . With the notation γ = (γ1, γ2), we define now a homotopy h : [0, 1] × S 1 → R2 \ {a} by h(t, s) :=      ((1 − t)γ1(s), γ2(s)) if s ∈ γ −1 (Ω4), ((1 − t)γ1(s) + t, γ2(s)) if s ∈ γ −1 (Ω2), (γ1(s), γ2(s)) otherwise. This map is indeed continuous by the glueing lemma, because γ can cross the boundary of Ωi only at Ai (i = 2, 4). We thus have shown that γ is homotopic (in R 2 \ {a}) to a curve which assumes only values in R. Since R is obviously simply connected, γ is actually homotopic to a constant (in R 2 \ {a}). Hence, the homotopy invariance of the winding number shows that γ actually has winding number 0 around a. This is the required contradiction. Using Theorems 2.1 and 2.2, we can now prove the following disc-cutting theorem: Theorem 2.3 (Disc-Cutting). Let X be a topological space into which a (compact) disc with “disc- interior” Q is homeomorphically embedded. Let the “disc-boundary” be the union of four nonempty disjoint connected sets A1, A2, A3, A4, enumerated in order along the boundary. Assume also that A2 and A4 both contain at least two points. 90 Martin Väth CUBO 10, 4 (2008) Let S ⊆ Q be closed in Q such that each compact smooth (via the embedding) injective path P in Q ∪ A2 ∪ A4 with P ∩ Ai 6= ∅ (i = 2, 4) contains some point from S. Then there is a connected subset C ⊆ S such that C ∩ Ai 6= ∅ for i = 1, 3. Remark 2.1. One could also replace “smooth path” by “polygonal path” in the statement of Theorem 2.3 with the obvious modification in the following proof. Proof. We show first that it suffices to show the claim for the case X = R2 and the unit disc Q. Indeed, if f : Q → X is a homeomorphism onto a subset of a general space X, let ˜Q := f (Q) and ˜Ai := f (Ai). Let S ⊆ ˜Q be as in the claim; in particular, S is closed in ˜Q. Then S0 := f −1 (S) is closed in Q. The hypothesis on S means that each smooth path connecting A2 with A4 in Q meets S0. The special case of the result thus implies that there is a connected subset C0 ⊆ S0 such that C0 ∩ Ai 6= ∅ for i = 1, 3. Then C := f (C0) ⊆ S is connected, and C ∩ ˜Ai ⊇ f (C0) ∩ f (Ai) 6= ∅. Hence, the statement holds also in the general case. Thus, to prove the theorem, we may assume without loss of generality that X = R2 and that Q is the unit disc. Assume by contradiction that a set C as in the claim does not exist. We apply Theorem 2.2 in the space Q with A := A1, U := Q\A3, and S instead of S. Observing that A ⊆ U , because A2 and A4 are nondegenerate, we find some open in Q set Ω ⊇ A1 with Ω ∩ A3 = ∅ such that the boundary B0 of Ω with respect to Q contains no element of S. Note that B0 is a closed subset of Q and thus compact. Note also that B0 is disjoint from S and from Ai (i = 1, 3). We thus find an open neighborhood M ⊆ R2 of B0 which is disjoint from S ∪ A1 ∪ A3. Moreover, if we let Âi (i = 1, 3) be compact “intervals” of the circle boundary which contain the corresponding “intervals” Ai (i = 1, 3) in their interior (with respect to the circle boundary) but still satisfy Â1 ⊆ Ω and Â3 ⊆ Q \ Ω, and if we let Âi (i = 2, 4) denote closure of the corresponding remaining intervals (contained in Ai), we can apply Theorem 2.1 with V := Ω and the four “square-sides” Âi. We thus find a connected set C ⊆ B0 such that there are points ai ∈ C ∩ Âi for i = 2, 4, and so ai ∈ C ∩ Ai (i = 2, 4). Since C ⊆ B0 is connected, if follows that a2 and a4 belong to the same connected component of B0. Since M ⊆ R 2 is an open neighborhood of B0, we may thus connect a2 and a4 by a smooth injective path in M . Since M is disjoint from Ai (i = 1, 3), we thus find a compact smooth injective path P in M ∩ (Q ∪ A2 ∪ A4) with P ∩ Ai 6= ∅ (i = 2, 4). Since M ∩ S = ∅, this path cannot contain a point from S, contradicting the hypothesis. 3 The Reaction-Diffusion System with Inclusions 3.1 Detailed Hypotheses We will consider the weak formulation of the stationary problem corresponding to (1.1)/(1.2), i.e. we will consider the weak formulation of d1∆u + b11u + b12v + f1(d, u, v, ∇u, ∇v) = 0 d2∆v + b21u + b22v + f2(d, u, v, ∇u, ∇v) ∈ −m0(d, u, v, ∇u, ∇v) in Ω (3.1) CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 91 with boundary conditions            u = v = 0 on Γ0, ∂u ∂n = g1(d, u, v) on ∂Ω \ Γ0, ∂v ∂n ∈ g2(d, u, v) + m1(d, x, u, v) on ∂Ω \ Γ0, (3.2) where we will assume that the (possibly multivalued) functions mi have the form m0(d, x, u, v, w, z) := [c0(d)m0(x, u, v, w, z), c0(d)m0(x, u, v, w, z)] and m1(d, x, u, v) := [c1(d)m1(x, u, v), c1(d)m1(x, u, v)], and where we assumed for the simplicity of notation that m 0 , m0, m1 and m1 vanish for x /∈ Ω0 or x /∈ Γ, respectively, where Ω0 ⊆ Ω and Γ ⊆ ∂Ω \ Γ0 are measurable. In order to require nontrivial situations, we will assume that mesΩ0 > 0 or mesΓ > 0 (or both). (3.3) For our considerations it will be crucial that mesΓ0 > 0 (3.4) so that we can equip the space H of all functions from W 1,2(Ω, R2) vanishing on Γ0 with the scalar product 〈U, V 〉 := ∫ Ω 〈∇U (x), ∇V (x)〉 dx, which under hypothesis (3.4) generates the inherited topology, see e.g. [13, Theorem 4.8.1]. We assume (1.3), and by DS ⊆ R 2 + , we denote the (open) domain of stability mentioned in the introduction. Note that all points of R 2 + ∩ ∂DS belong to some of the hyperbolas Cn defined by (1.4). We will assume that all of the above functions are at least defined for d ∈ DS ∪ {d ∗} where the point d∗ ∈ Cn ∩ ∂DS will be specified later on. For i = 0, 1, we fix exponents pi, qi, and q ∗ i according to the following restrictions. { pi ∈ [1, ∞), 1 ≤ q ∗ i < qi < ∞ arbitrary if n ≤ 2, p0 := n n−2 , p1 := n−1 n−2 , ∞ > q0 > q ∗ 0 := 2n n+2 , ∞ > q1 > q ∗ 1 := 2n−2 n if n > 2. Moreover, we assume the following hypothesis. 1. ci, ci are continuous on DS ∪ {d ∗} and without zeros on DS . 2. For each d ∈ DS ∪ {d ∗} the following holds: The functions fi(d, · , u, v, w, z) and gi(d, · , u, v) are measurable, and fi(d, x, · , · , · , · ) and gi(d, x, · , · ) are continuous for almost all x. More- over, fi and gi satisfy the growth estimates |fi(d, x, u, v, w, z)| ≤ ad(x) + bd · ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q0 , 92 Martin Väth CUBO 10, 4 (2008) and |gi(d, x, u, v, w, z)| ≤ ãd(x) +˜bd · ((|u| + |v|) p1 ) 2/q1 , where the quantities ‖ad‖Lq0 (Ω) , ‖ãd‖Lq1 (∂Ω\Γ0) , bd, and ˜bd are locally bounded with respect to d. 3. For each d0 ∈ DS ∪ {d ∗} there are estimates of the form |fi(d, x, u, v, w, z) − fi(d0, x, u, v, w, z)| ≤ cd0 (d) ( ad0,d(x) + ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q ∗ 0 ) and |gi(d, x, u, v) − gi(d0, x, u, v)| ≤ c̃d0 (d) ( ãd0,d(x) + (|u| + |v|) 2p1/q ∗ 1 ) where ‖ad0,d‖L q ∗ 0 (Ω) , ‖ãd0,d‖L q ∗ 1 (∂Ω\Γ0) ≤ 1 and cd0 (d), c̃d0 (d) → 0 as d → d0. 4. fi and gi become uniformly small at (u, v) = 0 in the sense that for each sufficiently small ball B in DS (and thus for each nonempty compact subset B ⊆ DS) the following holds: sup w,z∈Rn sup d∈B |fi(d, x, u, v, w, z)| ≤ cB max { (|u| + |v|)2p0/q0 , |u| + |v| } lim (u,v,w,z)→0 sup d∈B fi(d, x, u, v, w, z) |u| + |v| + ‖w‖ + ‖z‖ = 0 sup w,z∈Rn sup d∈B |gi(d, x, u, v)| ≤ cB max { (|u| + |v|)2p1/q1 , |u| + |v| } lim (u,v,w,z)→0 sup d∈B gi(d, x, u, v) |u| + |v| = 0 5. The functions m 0 ( · , u, v, w, z) and m0( · , u, v, w, z) are measurable, m0(x, · , · , · , · ) is lower semicontinuous, m0(x, · , · , · , · ) is upper semicontinuous, and the corresponding superposi- tion operators M 0 (u, v, w, z)(x) := m 0 (x, u(x), v(x), w(x), z(x)) and M 0(u, v, w, z)(x) := m0(x, u(x), v(x), w(x), z(x)) send continuous (and thus measurable) functions into measurable functions. Moreover, we require for some a0 ∈ Lq0 (Ω) and b0 < ∞ the growth estimates max {|m 0 (x, u, v, w, z)| , |m0(x, u, v, w, z)|} ≤ a0(x) + b0 · ((|u| + |v|) p0 + ‖w‖ + ‖z‖) 2/q0 . CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 93 6. The functions m 1 ( · , u, v) and m1( · , u, v) are measurable, m1(x, · , · ) is lower semicontinuous, m1(x, · , · ) is upper semicontinuous, and the corresponding superposition operators M 1(u, v)(x) := m1(x, u(x), v(x)) and M 1(u, v)(x) := m1(x, u(x), v(x)) send continuous (and thus measurable) functions into measurable functions. Moreover, we require the following growth estimates for some a1 ∈ Lq1 (Γ) and b1 < ∞: max {|m 1 (x, u, v)| , |m1(d, x, u, v)|} ≤ a1(x) + b1 · (|u| + |v|) 2p1/q1 . 7. The following unilateral conditions hold: 0 = c 0 (d)m 0 (x, u, v, w, z) = c0(d)m0(x, u, v, w, z) if v > 0, 0 = c 0 (d)m 0 (x, u, 0, w, z) ≤ c0(d)m0(x, u, 0, w, z) 0 ≤ c0(d)m0(x, u, v, w, z) ≤ c0(d)m0(x, u, v, w, z) if v < 0, 0 = c 1 (d)m 1 (x, u, v) = c1(d)m1(x, u, v) if v > 0, 0 = c1(d)m1(x, u, 0) ≤ c1(d)m1(x, u, 0) 0 ≤ c1(d)m1(x, u, v) ≤ c1(d)m1(x, u, v) if v < 0. lim (u,v,w,z)→0 v<0 |m 0 (x, u, v, w, z)| v = −∞ for almost all x ∈ Ω0, lim (u,v)→0 v<0 |m 1 (x, u, v)| v = −∞ for almost all x ∈ Γ. 3.2 Definition of weak solutions We consider the cone K := {U = (u1, u2) ∈ H : u2|Ω0 ≥ 0 and u2|Γ ≥ 0} and define operators A(d), G(d, · ), M (d, · ) : H → H by 〈A(d)U, V 〉 := ∫ Ω 〈( d −1 1 b11 d −1 1 b12 d −1 2 b21 d −1 2 b22 ) U (x), V (x) 〉 dx, 〈G(d, U ), V 〉 := ∫ Ω 〈( d −1 1 f1(d, U (x), ∇U (x)) d −1 2 f2(d, U (x), ∇U (x)) ) , V (x) 〉 dx + ∫ ∂Ω\Γ0 〈( g1(d, U (x)) g2(d, U (x)) ) , V (x) 〉 dx, 94 Martin Väth CUBO 10, 4 (2008) and M (d, U ) := ⋂ V ∈K { Z ∈ H : 〈Z, V 〉 ∈ ∫ Ω0 〈( 0 · d−1 1 d −1 2 m0(d, x, U (x), ∇U (x)) ) , V (x) 〉 dx + ∫ Γ 〈( 0 m1(d, x, U (x)) ) , V (x) 〉 dx } := ⋂ V =(ṽ,v)∈K { Z = ( 0 z ) ∈ H : ∫ Ω0 d −1 2 c 0 (d)m 0 (x, U (x), ∇U (x))v(x) dx + ∫ Γ c 1 (d)m 1 (x, U (x))v(x) dx ≤ 〈Z, V 〉 ≤ ∫ Ω0 d −1 2 c0(d)m0(x, U (x), ∇U (x))v(x) dx + ∫ Γ c1(d)m1(x, U (x))v(x) dx } , respectively. We define weak solutions of problem (3.1)/(3.2) as solutions of the inclusion U − A(d)U − G(d, U ) ∈ M (d, U ). Our hypotheses imply in particular (see e.g. [6]): Proposition 3.1. F (d, U ) := A(d)U − G(d, U ) − M (d, U ) is an upper semicontinuous map with nonempty compact values. Moreover, F is compact in the sense that if D0 ⊆ DS ∪ {d ∗} is compact and B ⊆ H is bounded then F (D0 × B) is precompact. 3.3 Local and Global Bifurcation Points Note that (d, 0) ∈ DS × H is always a solution of (3.1)/(3.2). We call a pair (d, U ) ∈ DS × H a nontrivial solution if U = (u, v) 6= 0, and if (d, u, v) is a weak solution of (3.1)/(3.2). The local bifurcation points (in DS ) are the elements of the set Blocal := {d ∈ DS : Each neighborhood of (d, 0) ∈ DS × H contains a nontrivial solution} . We call a point d ∈ DS a global bifurcation point (with respect to a point d ∗ ∈ Cn ∩ ∂DS) if there is a connected set C ⊆ DS × (H \ {0}) consisting only of nontrivial solutions such that (d, 0) ∈ C and such that C is a global branch in the sense that at least one of the following holds: 1. C is unbounded. 2. C reaches d∗, i.e. C contains some point (d∗, U ) which is a weak solution of (3.1)/(3.2). Note that in the second case, we do not exclude U = 0, i.e. C might return to the trivial branch at the hyperbola point d∗ ∈ Cn. We denote the set of global bifurcation points (with respect to d∗) by Bglobal(d ∗ ). CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 95 Proposition 3.2. Each global bifurcation point is a local bifurcation point. Moreover, Blocal is closed in DS. In particular, Bglobal(d∗) ∩ DS ⊆ Blocal. In our considerations an important role will be played by the vertical asymptote of the right- most hyperbola {(d1, d2) ∈ DS : d1 = b11/κ1} (3.5) and the corresponding part to the right of this asymptote, i.e. H := {(d1, d2) ∈ DS : d1 > b11/κ1} . (3.6) The following has been shown in [6]: Proposition 3.3. H ∩ Blocal = ∅. We also need another terminology. We say that a point d ∈ ∂DS is n-interior if d ∈ Cn and if there is some eigenfunction e of −∆ for the eigenvalue κn, i.e. e = u is a weak solution of (1.5), such that, for some constant ε > 0, e ≥ ε > 0 almost everywhere on Ω0 and e ≥ ε > 0 almost everywhere on Γ. (3.7) Recall in this connection that we require (3.3) For the case that Γ is a smooth manifold with boundary and Ω0 = ∅, we replace (3.7) by the milder requirement e(x) > 0 for almost all x ∈ Γ. (3.8) We say that d ∈ ∂DS is (n, m)-interior if d ∈ Cn ∩ Cm and if there is a function e which is a linear combination of eigenfunctions to the eigenvalues κn and κm such that (3.7) or (3.8) holds, respectively. If d ∈ Cn ∩ Cm ∩ ∂DS and d is n-interior or m-interior then d is also (n, m)-interior. However, d might be (n, m)-interior without being n-interior or m-interior. Using the main results from [6], we will prove now: Lemma 3.1. Let d ∈ ∂DS be n-interior or (n, m)-interior. Then there is an open neighborhood U0 ⊆ R 2 of d such that U0 ∩ DS ∩ Blocal = ∅. Moreover, if the hypotheses are satisfied with d ∗ = d, then each continuous compact path γ in DS ∪ {d ∗} connecting d∗ = d with some point from (3.6) contains some point from Bglobal(d ∗ ) ⊆ DS. Lemma 3.1 would follow rather straightforwardly from the results of [6] if we would allow that the connected set C in the definition of global bifurcation points is contained in (DS ∪{d ∗}) × (H \ {0}). However, it might happen that C \({d∗}×H) fails to be connected. Therefore, we need some additional arguments. We use the following result which is actually a consequence of Theorem 2.2 (and can also be proved using only the [countable] axiom of dependent choices, see [12]): 96 Martin Väth CUBO 10, 4 (2008) Theorem 3.1. Let X be a regular space, A ⊆ X compact, and S ⊆ X be closed. Suppose that S is locally compact, metrizable and σ-compact. Then for each open set U ⊇ A the following statements are equivalent: 1. There is a connected set C ⊆ S which intersects A and is either noncompact or intersects ∂U . 2. There is a connected set C ⊆ (S∩U )\A such that C ∩S intersects A and is either noncompact or intersects ∂U . Proof of Lemma 3.1. Only the last claim is not immediately contained in some of the results from [6]. To see this last claim, we apply the main result from [6] first to show that there is a connected set C0 ⊆ (DS ∪ {d ∗}) × (H \ {0}) such that C0 intersects (γ ∩ DS ) × {0} and such that either C0 is unbounded or C0 intersects also {d ∗} × H. Moreover, we will arrange it that, in the space X := R2 × H, C0 has the additional property that closures of bounded subsets of S := C0 are compact and consist only of (weak) solutions and satisfies C0 ∩ (R 2 × {0}) = (γ \ (U0 ∩ DS )) × {0} . (3.9) Indeed, assume that γ = σ([a, b]) with some continuous σ : [a, b] → DS ∪ {d ∗} satisfying σ(a) = d∗ and σ(b) ∈ H. We extend σ to a continuous σ : [a, ∞) → DS ∪ {d ∗} with σ(s) ∈ H for all s ≥ b such that both components of σ(s) tend to ∞ as s → ∞. For all sufficiently small s0 ∈ (a, b) we have σ([a, s0]) ⊆ U0, and by the main result from [6], we find some connected set C1 ⊆ [a, ∞) × (H \ {0}) such that C0 := {(σ(s), u) : (s, u) ∈ C1} consists only of (nontrivial) weak solutions of (3.1)/(3.2) and such that C1 contains some point from [s0, b] × {0} and such that either C1 is unbounded or C1 contains some point from {a} × H or from ([a, s0) ∪ (b, ∞)) × {0} The set C0 has all required properties. Indeed, since C0 consists only of nontrivial solutions, the closure of σ([a, ∞)) is contained in γ ∪ (U0 ∩ DS ) ∪ H, and no point of (U0 ∩ DS) ∪ H is a local bifurcation point, we obtain (3.9). The set C0 is connected, because it is the image of the connected set C1 under the continuous map T (s, u) := (σ(s), u). The set C0 contains T (C1) and thus intersects T ([s0, b] × {0}) ⊆ (γ ∩ DS ) × {0} and is either unbounded (by our choice of the extension of σ) or intersects {d∗} × H or (U0 ∪ H) × {0}. In the latter case, C0 actually intersects {d∗} × {0} by (3.9). To see these remaining properties, recall that with F from Proposition 3.1 the weak solutions of (3.1)/(3.2) are the elements of {(d, u) ∈ (DS ∪ {d ∗}) × H : u ∈ F (d, u)} . CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 97 Since C0 is contained in this set, F is upper semicontinuous, and the closure of σ([a, ∞)) is contained in DS ∪ {d ∗}, also S = C0 is contained in this set. The compactness of F described in Proposition 3.1 and our choice of the extension of σ implies that closed bounded subsets of S are compact. Hence, S = C0 has all required properties. In particular, S is locally compact and σ-compact. We apply Theorem 3.1 with A := (γ \ U0) × {0} and U := X \ ({d ∗} × H). The connected set C0 witnesses that the first statement of Theorem 3.1 is satisfied: Note that this set indeed intersects A in view of (3.9), because C0 intersects (γ ∩ DS ) × {0}. Hence, also the second statement of Theorem 3.1 holds which means that there is a connected set C ⊆ (S ∩ U ) \ A such that the set C contains some point (d0, 0) with d0 ∈ γ \ U0 ⊆ γ ∩ DS and such that either C is noncompact (and thus unbounded) or intersects ∂U = {d∗} × H. Thus, d0 ∈ Bglobal(d ∗ ). 4 The main result Theorem 4.1. Let D0 := DS \ H where H is from (3.6). Let d∗ ∈ ∂DS be m-interior or (n, m)-interior (n ≤ m) and such that the hypotheses de- scribed at the beginning of Section 3 are satisfied with this d∗. Then there is a connected set B ⊆ Bglobal(d∗) ∩ D0 ⊆ Blocal such that B intersects the d1-axis or some hyperbola Ck “strictly under” d∗. More precisely, we have k ≥ n, and the case Ck = Cm is only possible if d ∗ is an intersection point of two different hyperbolas. In all cases, the intersection B ∩ Ck does not contain d ∗ (i.e. is strictly under d∗). Moreover, this branch B satisfies in addition the following: 1. If Cn is the right-most hyperbola (i.e. if Cn = C1) then B is unbounded. 2. Otherwise (i.e. if Cn 6= C1) the set B is unbounded, or B intersects some hyperbola Ck “strictly over” d∗ (i.e. k ≤ n, and the case Ck = Cn is only possible if d ∗ is an intersection point of two different hyperbolas; B ∩ Ck does not contain d ∗). Moreover, for any k for which there is some k-interior point we have B ∩Ck = ∅, and for any pair (k, ℓ) for which the intersection point is (k, ℓ)-interior point this intersection point is not contained in B. Figure 2 illustrates qualitatively the four possibilities of branches B of bifurcation points if there is some n-interior point with Cn 6= C1; one of these possibilities must (qualitatively) occur according to Theorem 4.1. Similarly, Figure 3 illustrates the two possibilities of branches B if there is some 1-interior point. 98 Martin Väth CUBO 10, 4 (2008) In particular, if there are n-interior points for every n, then the last statement of Theorem 4.1 implies that only one possibility can occur: There must be a branch B which is unbounded and such that B intersects the d1-axis (possibly at (0, 0)). We point out that (contrary to what the figures might suggest) the theorem does not state that the branch B is pathwise connected, i.e. it might look “weird” (but it is connected in the topological sense). d1 d2 C4 C3 C2 C1 D0 L H Figure 2: The four qualitative different possible branches B of bifurcation points if there is some 2-interior point (one of these must occur) d1 d2 C4 C3 C2 C1 D0 L H Figure 3: The two qualitative different possible branches B of bifurcation points if there is some 1-interior point (one of these two must occur) Proof. The last statement of Theorem 4.1 is automatically satisfied by the first claim of Lemma 3.1, since Bglobal(d ∗) ⊆ Blocal must be disjoint from any k-interior or (k, ℓ)-interior point. CUBO 10, 4 (2008) A Disc-Cutting Theorem and Two-Dimensional Bifurcation ... 99 Using this fact with d∗, we find some open neighborhood U0 ⊆ R 2 which is disjoint from Bglobal(d ∗) ⊆ Blocal. Let L0 ⊆ H be some line which is parallel but strictly to the right of the line (3.5). Let Q, H0 ⊆ DS be that parts to the left and right of this line L0, respectively. Lemma 3.1 implies Blocal ∩ DS ⊆ Q. Using the one-point compactification X of Q, we consider Q as the disc-interior of some homeomorphically embedded disc, whose boundary corresponds to the union of the d1-axis, the line L0, the point ∞, and the “envelope” E = R 2 + ∩ ∂DS of all of the hyperbolas Cn. Let A2 be that part of the boundary which corresponds to L0 (without the two “boundary points” at ∞ and at the d1-axis), and let A4 correspond to U0 ∩ E. Let A1 and A3 denote the ramining (compact) connected subsets of the boundary of the disc Q. Now we can apply the disc-cutting theorem with S = Q ∩ Bglobal(d∗). In fact, each continuous compact path in Q connecting A2 with A4 must intersect S by Lemma 3.1. Hence, the disc-cutting Theorem 2.3 implies the existence of a connected set B ⊆ S with B ∩ Ai 6= ∅ for i = 1, 3. Since B cannot intersect L0, the property B ∩ A3 means that either B is unbounded or that B intersects some point of some Ck “strictly above” d ∗ . The property B ∩ A1 means that B intersect some point of some Ck “strictly below” d ∗ or the d1-axis. Received: March 2008. Revised: March 2008. References [1] Drábek, P. and Kučera, M., Reaction-diffusion systems: Destabilizing effect of unilateral conditions, Nonlinear Anal., 12 (1988), no. 11, 1172–1192. [2] Eisner, J., Reaction-diffusion systems: Destabilizing effect of conditions given by inclusions, Math. Bohem., 125 (2000), no. 4, 385–420. 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