Multi-peak solutions for some singular perturbation problems

We consider the problem {epsilon 2 Delta u - 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 epsilon approaches zero, at a maximum of the function d(x) = d(.,partial derivative Omega), the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain Omega presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity.