Abstract

We construct solutions exhibiting a single spike-layer shape around some point of the boundary as ? ? 0 for the problem (0.1) { ?2 ?u - u + up = 0 in ?, u > 0 in ?, ?u/?v = 0 on ??, where ? is a bounded domain with smooth boundary in RN, p > 1, and p < N+2/N-1 if N ? 3. Our main result states that given a topologically nontrivial critical point of the mean curvature function of ??, 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 ? ? 0 while vanishing asymptotically elsewhere.