Abstract.
We deal with positive solutions of Δu = a(x)u p in a bounded smooth domain \(\Omega \subset \mathbb{R}^N\) subject to the boundary condition ∂u/∂v = 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.
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Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.
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García-Melián, J., Sabina De Lis, J.C. & Rossi, J.D. A bifurcation problem governed by the boundary condition I. Nonlinear differ. equ. appl. 14, 499–525 (2007). https://doi.org/10.1007/s00030-007-4064-x
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DOI: https://doi.org/10.1007/s00030-007-4064-x