Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation −ε^2Δu+u = u^p in Ω ⊆ RN, where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N \geq 3 and for k ∈ {1, . . . , N − 2}. We impose Neumann boundary conditions, assuming 1 < p < (N −k +2)/(N −k − 2) and ε→0+. This result settles in full generality a phenomenon previously considered only in the particular case N = 3 and k = 1.