On the necessity of the connectedness condition of $ \Omega $ in : "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition"

This note corrects an inaccuracy in the paper "Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition" by the author, which appeared in Commun. Anal. Mech. 15 (2023), no. 4,811–830. In particular, we point out that the bounded open set must be connected for some of the main results in the paper to hold. To motivate this claim, we give an example of a disconnected open set combined with a partition of its boundary, for which the potential–well depth associated to the problem vanishes. When this occurs, the framework developed in the paper breaks down. We also show that, conversely, when is connected, this phenomenon does not show up and all assertions in the paper are correct.