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 with Robin boundary condition

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 bconst there are critical points of R_λ near non-degenerate critical points of b.