This paper deals with the Neumann problem
where is a smooth bounded domain, when and when . Associated with (1) is the functional
Solutions of (1) are called single boundary peaked if , with , , where is the positive radial solution of in , as , .
The author states the theorem: if is a family of single boundary peaked solutions of (1), then, as , has only one local maximum point and . Moreover, the tangential derivative of the mean curvature of at tends to zero. A converse of this theorem is also investigated. The particular case of the least-energy solutions of (1) was studied previously by W-M. Ni and I. Takagi [Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042), and Duke Math. J. 70, No. 2, 247-281 (1993; Zbl 0796.35056)].
The proof is based on a decomposition of of the form , where is the solution of in and on , and on fine estimates for and the error term .