Nondegenerate solutions for constrained semilinear elliptic problems on Riemannian manifolds

Abstract

Let $$M^n$$ be a connected compact smooth manifold, where $$n \ge 2$$ . In this article, we prove that nondegeneracy of nonconstant solutions for a class of singularly perturbed semilinear elliptic problems on M is generic with respect to the pair $$(\epsilon ,g)$$ , where $$\epsilon >0$$ and g is a metric of class $$C^k$$ , $$k\ge 1$$ . As applications, we show that under certain growth conditions, such result generalizes to nondegeneracy of any solution for the Allen-Cahn or nonlinear Schrödinger equations.