Solutions to a singularly perturbed supercritical elliptic equation on a Riemannian manifold concentrating at a submanifold

Given a smooth Riemannian manifold (M,g) we investigate the existence of positive solutions to the equation−ε2Δgu+u=up−1on M which concentrate at some submanifold of M as ε→0, for supercritical nonlinearities. We obtain a positive answer for some manifolds, which include warped products. Using one of the projections of the warped product or some harmonic morphism, we reduce this problem to a problem of the form−ε2divh(c(x)∇hu)+a(x)u=b(x)up−1, with the same exponent p, on a Riemannian manifold (M,h) of smaller dimension, so that p turns out to be subcritical for this last problem. Then, applying Lyapunov-Schmidt reduction, we establish existence of a solution to the last problem which concentrates at a point as ε→0.