On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

Abstract

Given that $\left(M,g\right)$ is a smooth compact and symmetric Riemannian $n$-manifold, $n\ge 2$, we prove a multiplicity result for antisymmetric sign changing solutions of the problem $-{\epsilon }^2{\Delta }_{g}u+u={\left|u\right|}^{p-2}u$ in $M$. Here $p>2$ if $n=2$ and if $n\ge 3$.