On the necessity of the connectedness condition of Ω in : “Nontrivial solutions for the Laplace equation with a nonlinear Goldstein–Wentzell boundary condition”

Abstract

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 Ω 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 Ω combined with a partition (Γ₀, Γ₁) 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 Ω is connected, this phenomenon does not show up and all assertions in the paper are correct.

Keywords

Laplace equation / Laplace–Beltrami operator / existence and multiplicity for nontrivial solutions / Wentzell boundary conditions / Ventcel boundary conditions / Mountain Pass Theorem

References

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