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The paper is concerned with the local and global bifurcation structure of positiveThe system arises in nonlinear optics and in the Hartree-Fock theory for a double condensate. Local and global bifurcations in terms of the nonlinear coupling parameter β of the system are investigated by using spectral analysis and by establishing a new Liouville type theorem for nonlinear elliptic systems which provides a-priori bounds of solution branches. If the domain is radial, possibly unbounded, then we also control the nodal structure of a certain weighted difference of the components of the solutions along the bifurcating branches. Mathematics Subject Classification (2000)35B05 · 35B32 · 35J50 · 35J55 · 58C40 · 58E07 Dedicated to Paul Rabinowitz on the occasion of his 70th birthday.

SynopsisWe study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.

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

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We study the set of solutions of the nonlinear elliptic systemin a smooth bounded domain Ω ⊂ R N , N 3, with coupling parameter β ∈ R. This system arises in the study of Bose-Einstein double condensates. We show that the value β = − √ μ 1 μ 2 is critical for the existence of a priori bounds for solutions of (P). More precisely, we show that for β > − √ μ 1 μ 2 , solutions of (P) are a priori bounded. In contrast, when λ 1 = λ 2 , μ 1 = μ 2 , (P) admits an unbounded sequence of solutions if β − √ μ 1 μ 2 .

In this paper, we obtain a version of the sliding plane method of Gidas, Ni and Nirenberg which applies to domains with no smoothness condition on the boundary. The method obtains results on the symmetry of positive solutions of boundary value problems for nonlinear elliptic equations. We also show how our techniques apply to some problems on half spaces.In this note, we show how to apply the method of moving planes [6] Gidas and Spruck [8].In Section 1, we prove our main result on bounded domains and discuss generalisations and in Section 2 we prove some technical lemmas needed in Section 1. In Section 3, we consider the problem on half spaces.