Global Structure of the Solutions of Non-Linear Real Analytic Eigenvalue Problems

Abstract

Proc. London Math. Soc. (3) 27 (1973) 747-765 748 E. N. DANCER equations than those in [1]. If E is a Banach space with norm || ||, EJ^a) will be used to denote {x e E: \\x -a\\ < a}. However, we shall write E a instead of E^O). Finally, E a denotes {x e E: || x || ^ a}. Finite-dimensional real analytic germsBefore obtaining our results in infinite-dimensional spaces, we need to prove some results about finite-dimensional real analytic germs. The definitions and results from the theory of real analytic germs that we shall need can be found in Narasimhan's book [4]. The main part of the following lemma is part of Federer's version ([5], Theorem 3.4.8, part 10) of the representation theorem for real analytic germs. The last part follows from Narasimhan's Proposition 3.3 in [4].Proof. Let I denote the ideal of real analytic functions vanishing on S o . Since S o is irreducible, / is prime. By examining Narasimhan's proof of NON-LINEAR REAL ANALYTIC EIGENVALUE PROBLEMS 749 Proposition 3.2 in [4], we see that it suffices to show that, if L is a subspace of R n x R containing {0} x R and / : L -$• R is a non-trivial irreducible real analytic function vanishing on S o , then there exists a one-dimensional subspace T of R w x {0} such that / does not vanish on S o nT. Here we are identifying / with its natural extension (by way of the orthogonal projection) to R n x R. Suppose, by way of contradiction, that this is false. Then, by the Weierstrass preparation theorem (Narasimhan,[4], p. 15), f = Agr, where g is real analytic in a neighbourhood of zero. Since / is irreducible, g(0,0) # 0 and so A vanishes on S Q , that is, S 0^R n x{0}. This gives a contradiction and so we have the required result. LEMMA 4. Suppose that S is a subset of R n x R such that S o is a real analytic germ with dim R S o = 1. Then there exist y, 8 > 0 and non-negative integers T +1 , T_ x such that, for each A with

Some global results for nonlinear fourth order eigenvalue problems

“…Rabinowitz [27] has shown that there exists a connected set of nontrivial solutions of = G(λ ) bifurcating from (µ 0) which either is unbounded in R × E or must also bifurcate from another point ( µ 0) where µ = µ is a characteristic value of L. Therefore, bifurcation from characteristic values of odd multiplicity is a global rather than a local phenomenon. He also obtains stronger results for bifurcation from a characteristic value of multiplicity 1 (these results are strengthened in Dancer [13]) and gives applications of these results to some second order linearizable Sturm-Liouville problems. Note that the eigenfunction of corresponding linear Sturm-Liouville problems is characterized by the fact that it has only simple nodal zeros and the number of zeros of the eigenfunction is equal to the serial number of the corresponding eigenvalue increased by 1.…”

“…Note that if (λ ˆ ) is a solution of (1), (2) We decomposeD = N N + 1 into two subcontinuaD ν , ν = + or −, = N N + 1 which meet (λ 0 ) as in [13], which is more convenient and more natural than the one in [27]. The proof of Theorem 2 in [13] shows…”

Steady Water Waves

“…According to standard local bifurcation theory a branch Γ 1 = {(u 1 s , λ 1 s ) : s ∈ (−1, 1)} of smallamplitude solutions to (2.6) emerges from (u 0 , λ 0 ) = (0, 1); in fact a symmetric, subcritical pitchfork bifurcation takes place. The following global bifurcation result asserts that the portion of Γ 1 to the right of the primary bifurcation point extends to a global solution branch; the result is proved by an application of real-analytic global bifurcation theory (Dancer [25,27]). …”

Solutions of Multiparameter Systems of Elliptic Differential Equations

“…It is known that a change of the Leray-Schauder degree implies the existence of global bifurcation point, which completes the proof. [2,3,9,17,19,29] for discussion and examples.…”