Multiplicity and asymptotic profile of 2-nodal solutions to a semilinear elliptic problem on a Riemannian manifold

TL;DR

This paper investigates the number and asymptotic behavior of sign-changing solutions with two nodal domains for a singularly perturbed nonlinear elliptic equation on a Riemannian manifold, linking solutions to topological invariants.

Contribution

It provides a lower bound on the number of such solutions based on the cup-length of the configuration space and describes their asymptotic profiles as perturbation parameter approaches zero.

Findings

Established a lower bound for sign-changing solutions.

Described the asymptotic profile of solutions as epsilon approaches zero.

Provided new estimates for the cup-length of the configuration space.

Abstract

We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -{\epsilon}^{2}{\Delta}_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold M, p in (2,2n/(n-2)), in terms of the cup-length of the configuration space of M. We give a precise description of the asymptotic profile of these solutions as {\epsilon} goes to 0. We also provide new estimates for the cup-length of the configuration space of M.