Proceedings of the Royal Society of Edinburgh, 137A, 225–252, 2007 Bifurcation and stability of equilibria with asymptotically linear boundary conditions at infinity Jos´ e M. Arrieta, Rosa Pardo and Anibal Rodr´ ıguez-Bernal Departamento de Matem´atica Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain (arrieta@mat.ucm.es; rpardo@mat.ucm.es; arober@mat.ucm.es) (MS received 18 March 2005; accepted 8 March 2006) We consider an elliptic equation with a nonlinear boundary condition which is asymptotically linear at infinity and which depends on a parameter. As the parameter crosses some critical values, there appear certain resonances in the equation producing solutions that bifurcate from infinity. We study the bifurcation branches, characterize when they are sub- or supercritical and analyse the stability type of the solutions. Furthermore, we apply these results and techniques to obtain Landesman–Lazer-type conditions guaranteeing the existence of solutions in the resonant case and to obtain an anti-maximum principle. 1. Introduction Over the last decade a lot of attention has been paid to problems with nonlinear boundary conditions. Hence, nowadays, the underlying mechanisms for dissipative- ness or blow-up of solutions is fairly well understood (see, for example, [3,5,7,18,19]). Therefore, it is natural to analyse the dynamics and bifurcations induced by the nonlinear boundary conditions, and compare their effects in the case of an interior reaction term, which has been more widely studied. For example, in [6] the existence of patterns for such problems, i.e. a stable non-trivial equilibrium, was considered (see also the references therein for some previous and related results). In this work we consider the evolutionary equation of parabolic type, u t − ∆u + u =0 in Ω, t > 0, ∂u ∂n = λu + g(λ, x, u) on ∂Ω, t > 0, u(0,x)= u 0 (x) in Ω, ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (1.1) in a bounded and sufficiently smooth domain Ω ⊂ R N with N  2 and analyse the behaviour and stability properties of the equilibrium solutions. These equilibria are solutions of the following elliptic problem with nonlinear boundary conditions: −∆u + u =0 in Ω, ∂u ∂n = λu + g(λ, x, u) on ∂Ω. ⎫ ⎬ ⎭ (1.2) 225 c  2007 The Royal Society of Edinburgh