Asymptotic profile of $2$-nodal solutions to a semilinear elliptic problem on a Riemannian manifold

Abstract

We describe the asymptotic profile of sign changing solutions with small energy to the singularly perturbed nonlinear elliptic equation $-\varepsilon ^{2}\Delta_{g}u+u=|u|^{p-2}u$ on an $n$-dimensional compact Riemannian manifold $M$ without boundary, $p\in(2,2^{\ast}),$ as $\varepsilon \rightarrow0.$