From periodic ODE’s to supercritical PDE’s

Let (M,g)(M,g) be an mm-dimensional compact Riemannian manifold without boundary. Assume κ∈C2(M)κ∈C2(M) is such that −Δg+κ−Δg+κ is coercive. We prove the existence of solutions to the supercritical problems −Δgu+κu=up,u>0in(M,g)and−Δgu+κu=λeuin(M,g) which concentrate along an (m−1)(m−1)-dimensional submanifold of MM as p→∞p→∞ and λ→0λ→0, respectively, under suitable symmetry assumptions on the manifold MM. A connection with the solutions of a class of periodic ODE’s is established.