Interior spikes of a singularly perturbed Neumann problem with potentials

where Ø is a smooth bounded domain of R with external normal ν, N ≥ 3, 1 < p < (N + 2)/(N − 2), J : R → R and V : R → R are C functions. In [5], the first author, extending the classical results by Ni and Takagi, in [3, 4], proved that there exist solutions of (1) that concentrate at maximum and minimum points of a suitable auxiliary function defined on the boundary ∂Ø and depending only on J and V . Here we study the existence of solutions which concentrate in the interior of Ø and we will show that the concentration occurs at maximum and minimum points of the same auxiliary function introduced in [5], but now defined in Ø. We assume that the reader has familiarity with [5]. When J ≡ 1 and V ≡ 1, interior spikes have been found by Wei (see [6]) showing that concentration occurs at local maxima of the distance function dist(·, ∂Ø). On J and V we will do the following assumptions: