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 − ε 2 Δ g u + u = u p − 1 on 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 − ε 2 div h ( c ( x ) ∇ h u ) + a ( x ) u = b ( x ) u p − 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 .