Construction of Solutions for a Nonlinear Elliptic Problem on Riemannian Manifolds with Boundary

Let (M, g) be a smooth compact n-dimensional Riemannian manifold (n ≥2) with smooth (n − 1)- dimensional boundary δM. We prove that the stable critical points of the mean curvature of the boundary generates H¹(M) solutions for the a singularly perturbed subcritical elliptic problem with Neumann boundary conditions when the perturbation parameter is small