On the Role of Mean Curvature in Some Singularly Perturbed Neumann Problems

We construct solutions exhibiting a single spike-layer shape around some point of the boundary as $\var \to 0$ for the problem \left\{ \begin{array}{l} \var^2 \tri u - u + u^p = 0 \quad \mbox{in} \ \Omega, \\ u > 0 \quad \mbox{in} \ \Omega, \\ \frac{\partial u}{\partial \nu} = 0 \quad \mbox{on} \ \partial \Omega, \end{array} \right.\label {1.1} where $ \Omega $ is a bounded domain with smooth boundary in $R^N$, $p > 1 $, and $p< {N+2\over N-2}$ if $N\ge 3$. Our main result states that given a topologically nontrivial critical point of the mean curvature function of $\partial \Omega$, for instance, a possibly degenerate local maximum, local minimum, or saddle point, there is a solution with a single local maximum, which is located at the boundary and approaches this point as $\var\to 0$ while vanishing asymptotically elsewhere.