In this paper,

${\Omega}$ is a smooth domain in

${\mathbb{R}}^{n}$, which satisfies the following condition: there exists an open bounded subset

${\Lambda}$ of its boundary

$\partial {\Omega}$ with a smooth boundary

$\partial {\Lambda}$ and closed subsets

$B,{B}_{0}$ of

${\Lambda}$, such that

$B$ is connected and

${B}_{0}\subset B$. The authors consider the problem (1):

${\epsilon}^{2}{\Delta}u-u+f\left(u\right)=0$ and

$u>0$ in

${\Omega}$ with the boundary condition

$\frac{\partial u}{\partial r}=0$ on

$\partial {\Omega}$. Here

$f$ is a suitable function

$\mathbb{R}\to \mathbb{R}$; the example

$f\left(t\right)={t}^{p}$,

$p>1$ and

$p<\frac{n+2}{n-2}$ if

$n\ge 3$, is allowed. Let

$H\left(P\right)$ be the mean curvature function at

$P\in \partial {\Omega}$. Let

$c$ be a topologically nontrivial critical point for

$H$ in

${\Lambda}$, characterized through a max-min scheme. The main result states the existence of a boundary-spike family of solutions of (1), with maxima

${P}_{\epsilon}\in {\Lambda}$, so that

$H\left({P}_{\epsilon}\right)\to c$. As in

*M. del Pino* and

*P. L. Felmer* [J. Funct. Anal. 149, 245-265 (1997;

Zbl 0887.35058)], the proof is based on the introduction of a modified version of the energy-functional and ideas developed by W.-M. Ni, I. Takagi and J. Wei in recent papers on related problems.