Abstract

In this paper we consider the Green function for the Laplacian in a smooth bounded domain Ω ⊂ R N with Robin boundary conditionfrac(∂ G λ, ∂ ν) + λ b (x) G λ = 0, on ∂ Ω, and its regular part S λ (x, y), where b > 0 is smooth. We show that in general, as λ → ∞, the Robin function R λ (x) = S λ (x, x) has at least 3 critical points. Moreover, in the case b ≡ const we prove that R λ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b ≢ const there are critical points of R λ near non-degenerate critical points of b. © 2008 Elsevier Inc. All rights reserved.