On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds

TL;DR

This paper investigates the existence and multiplicity of sign-changing solutions to a nonlinear elliptic PDE on symmetric Riemannian manifolds, highlighting the influence of symmetry on solution count.

Contribution

It provides new multiplicity results for antisymmetric sign-changing solutions on symmetric Riemannian manifolds, expanding understanding of solution structures in nonlinear elliptic problems.

Findings

Established lower bounds on the number of sign-changing solutions.

Demonstrated the role of symmetry in solution multiplicity.

Extended previous results to a broader class of manifolds.

Abstract

We consider the problem -{\epsilon}^2\Delta_gu+u = |u|^{p-2}u in M, where (M,g) is a symmetric Riemannian manifold. We give a multiplicity result for antisymmetric changing sign solutions.