Bubbling solutions for supercritical problems on manifolds

Abstract

Let (M,g) be an n-dimensional compact Riemannian manifold without boundary and Γ be a non-degenerate closed geodesic of (M,g). We prove that the supercritical problem−Δgu+hu=un+1n−3±ϵ,u>0, in (M,g) has a solution that concentrates along Γ as ϵ goes to zero, provided the function h and the sectional curvatures along Γ satisfy a suitable condition. A connection with the solution of a class of periodic Ordinary Differential Equations with singularity of attractive or repulsive type is established.