Mountain pass solutions for singularly perturbed nonlinear Dirichlet problems

We consider the following singularly perturbed nonlinear elliptic problemɛ2Δu-u+f(u)=0,u>0inΩ,u=0on∂Ωwhere 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 uɛ for above problem and the solution uɛ exhibits a spike layer as ɛ→0. To see the location of the spike layer for small ɛ>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.