Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems

We consider the following singularly perturbed nonlinear elliptic problem where the nonlinearity f is of subcritical growth and Ω is a bounded domain in Rn. Under certain conditions of f, there exists a mountain pass solution u for above problem and the solution u exhibits a spike layer as →0. To see the location of the spike layer for small >0, some additional assumptions, certain nondegeneracy for a limiting problem or monotonicity of f(t)/t, have been assumed. In this paper, we characterize the location of the spike layer without such additional assumptions for n 3.