Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theory

Abstract

Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with low energy for (ε, h) belonging to a residual subset of (0, ε0) \times Bρ, for (ε0, ρ) small enough.