Critical points of the regular part of the harmonic Green function with Robin boundary condition

In this paper we consider the Green function for the Laplacian in a smooth bounded domain Ω⊂RNΩ⊂RN with Robin boundary condition∂Gλ∂ν+λb(x)Gλ=0,on∂Ω, and its regular part Sλ(x,y)Sλ(x,y), where b>0b>0 is smooth. We show that in general, as λ→∞λ→∞, the Robin function Rλ(x)=Sλ(x,x)Rλ(x)=Sλ(x,x) has at least 3 critical points. Moreover, in the case b≡constb≡const we prove that RλRλ has critical points near non-degenerate critical points of the mean curvature of the boundary, and when b≢constb≢const there are critical points of RλRλ near non-degenerate critical points of b.