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

TL;DR

This paper investigates the existence and multiplicity of positive solutions to a singularly perturbed nonlinear elliptic equation on manifolds, employing Morse theory and topological invariants to estimate solution counts.

Contribution

It introduces a novel approach using Morse theory and the Poincaré polynomial to estimate the number of solutions for perturbed elliptic problems on manifolds.

Findings

Estimates the number of low-energy solutions based on topological invariants.

Shows solutions exist for a residual set of perturbations.

Provides conditions under which solutions are guaranteed.

Abstract

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