Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold

Let (M,g)(M,g) be a compact, connected, orientable smooth NN-dimensional Riemannian manifold without boundary. Let p∈(2,2∗)p∈(2,2∗), where 2∗=2N/(N−2)2∗=2N/(N−2). We consider the multiple existence of solutions of the problem equation(P)−ε2Δgu+u=|u|p−2uon M, where ΔgΔg stands for the Laplacian in MM.