The paper deals with positive solutions to singularly perturbed semilinear elliptic problems of the following form:
Here is a bounded domain with smooth boundary and is smooth; a typical choice is . The purpose of the paper is to investigate the existence of single interior spike solutions. A family of solutions is called single peaked if: the energy is bounded for any ; has only one local maximum point , , in and as . The limiting point is called a boundary spike if , respectively an interior spike if . In both cases the problem of existence and location of spikes has stimulated quite a few papers in recent years. The paper provides a unified approach to this problem under rather general assumptions on the function (both for Dirichlet and for Neumann homogeneous boundary conditions); in particular, necessary and sufficient conditions for the existence of interior spike solutions are given. A variety of methods is used to this purpose, including in particular weak convergence of measures and Lyapunov-Schmidt reduction.