Singularly perturbed Neumann problems with potentials

Such a problem was intensively studied in several works. For example, Ni & Takagi, in [11, 12], show that, for ε sufficiently small, there exists a solution uε of (2) which concentrates in a point Qε ∈ ∂Ω andH(Qε) → max∂ΩH , here H denotes the mean curvature of ∂Ω. Moreover in [10], using the LiapunovSchmidt reduction, Li constructs solutions with single peak and multi-peaks on ∂Ω located near any stable critical points of H . Since the publication of [11, 12], there have been many works on spike-layer solutions of (2), see for example [5, 6, 7, 8, 9, 14] and references therein.