Bifurcation from infinity for an asymptotically linear problem on the half-line Fran¸ cois Genoud 1 Department of Mathematics, Heriot-Watt University Edinburgh, EH14 4AS, Scotland Abstract This paper is concerned with asymptotic bifurcation for a semilinear equation on the half-line. For an asymptotically linear nonlinearity, the existence of a continuum of solutions ‘bifurcating from infinity’ is obtained using a topological degree. Under additional monotonicity conditions, the continuum is shown to be a continuous curve. Applications to nonlinear planar waveguides are mentioned. Keywords: asymptotic bifurcation, topological degree, unbounded domain, saturable nonlinearity, nonlinear waveguides 1. Introduction We study asymptotic bifurcation for the nonlinear eigenvalue problem    u ′′ (x)+ f (x, u(x) 2 )u(x)= λu(x) for x> 0, u(x) > 0 for x  0, u ′ (0) = lim x→∞ u(x)=0, (N) where the nonlinearity is supposed to be asymptotically linear in the sense that there is f ∞ ∈ C(R + ) such that f (x, s) → f ∞ (x) as s →∞ for all x  0. This problem arises in the modelling of nonlinear planar waveguides. The asymptotic linearity assumption describes saturation of the refractive index as the intensity of the beam becomes large and allows one to go beyond the Kerr approximation (a power-law nonlinearity) usually made in the waveguide lit- erature. The bifurcation result presented here yields existence of high power (large |u| L 2 ) symmetric TE travelling waves for self-focusing planar waveguides, as described in [2, 11]. Email address: F.Genoud@hw.ac.uk (Fran¸ cois Genoud ) 1 The author acknowledges funding from the Swiss NSF, grant PBELP2-125494, and from the UK EPSRC, grant EP/H030514/1. Preprint submitted to Elsevier April 18, 2011