c  Birkh¨auser Verlag, Basel, 2007 NoDEA Nonlinear differ. equ. appl. 14 (2007) 499—525 1021–9722/07/060499–27 DOI 10.1007/s00030-007-4064-x A bifurcation problem governed by the boundary condition I ∗ Jorge GARC ´ IA-MELI ´ AN, and Jos´ e C. Sabina DE LIS Dpto. de An´ alisis Matem´atico, Universidad de La Laguna C/ Astrof´ ısico Francisco S´ anchez s/n 38271 - La Laguna, Spain e-mail: jjgarmel@ull.es, josabina@ull.es Julio D. ROSSI Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires 1428, Buenos Aires, Argentina e-mail: jrossi@dm.uba.ar Abstract. We deal with positive solutions of Δu = a(x)u p in a bounded smooth domain Ω ⊂ R N subject to the boundary condition ∂u/∂ν = λu, λ a parameter, p> 1. We prove that this problem has a unique positive solution if and only if 0 <λ<σ1 where, roughly speaking, σ1 is finite if and only if |∂Ω ∩{a =0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ1. 2000 Mathematics Subject Classification: 35J60, 35B32, 35J25 Key words: Elliptic problems, bifurcation, eigenvalues 1 Introduction It is the main concern of the present work the study of the following semilinear boundary value problem:    ∆u = a(x)u p x ∈ Ω ∂u ∂ν = λu x ∈ ∂ Ω , (1.1) * Supported by DGES and FEDER under grant BFM2001-3894 (J. Garc´ ıa-Meli´anand J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.