Let \(\Omega\) be a bounded domain of \({\mathbb{R}}^{2}\) with a smooth boundary such that \(\Omega\) intersects \(\{0\}\times {\mathbb{R}}\) into a segment \(\Gamma\) perpendicular to \(\partial\Omega\) at its endpoints and satisfy a certain nondegeneracy condition involving the curvature of \(\partial\Omega\) at the endpoints of \(\Gamma\), which belong to \(\partial\Omega\). The authors consider the singularly perturbed elliptic problem
\[ \varepsilon^{2}\Delta u-u+u^{p}=0,\;u>0\quad \text{in }\Omega, \qquad \frac{\partial u}{\partial n}=0\quad \text{on }\partial\Omega, \]
where \(\varepsilon>0\) is a small parameter and \(n\) is the unit outward normal field to \(\partial\Omega\), and \(p>1\), and show that if \(\varepsilon\in]0,\varepsilon_{0}[\), and if \(\varepsilon\) remains away from certain critical values, then the above problem has a solution \(u_{\varepsilon}\), and that there exists \(c_{0}>0\) such that \(u_{\varepsilon}(y)\leq\exp [-c_{0}\varepsilon^{-1}{\mathrm{dist}}\,(y,\Gamma_{\varepsilon})]\) if \(y\in\Omega\), where \(\Gamma_{\varepsilon}\) is a certain curve which approaches \(\Gamma\) as \(\varepsilon\) tends to \(0\). The authors also provide some information on the form of \(u_{\varepsilon}\) near \(\Gamma\) as \(\varepsilon\) approaches \(0\).