Nodal-set boundary contact for the two-peak solution

Characterize, in terms of the Green and Robin functions, whether and where the nodal set of the positive–negative two-peak solution to the four-dimensional Brezis–Nirenberg problem meets the boundary, and determine whether the balance condition τ_Ω(ξ_1^*)=τ_Ω(ξ_2^*) is necessary for such contact.

Background

Theorem 1 establishes that the closure of the nodal set meets the boundary when the boundary is connected and the Robin-function values at the two limiting concentration points are equal. The proof uses convergence away from the concentration points to the difference of two Green functions and a boundary normal-derivative argument.

The theorem does not establish whether equality of the two Robin values is necessary, nor does it describe the boundary contact points in general. These questions are explicitly left unresolved.