On the role of distance function in some singular perturbation problems

We consider the problem {epsilon 2 Delta u - + f(u) = 0 in Omega u > 0 in Omega, u = 0 on partial derivative Omega where Omega is a smooth domain in R-N, not necessarily bounded, epsilon > 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as epsi lon approaches zero, at a maximum of the function d(x, partial derivative O mega), the distance to the boundary. We obtain single-peaked solutions associated to any topologically nontrivia l critical point of the distance function such as for instance a local, pos sibly degenerate, saddle point. The construction relies on a variational lo calization argument to control a certain minmax value for an associated mod ified energy functional as well as on a precise asymptotic estimate for thi s energy level.