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Nondegenerate solutions for constrained semilinear elliptic problems on Riemannian manifolds

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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.

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Acknowledgements

The author thanks Paolo Piccione, Gaetano Siciliano and the anonymous referee respectively for suggesting the topic, discussing drafts of such paper and valuable suggestions. During the development of this article, the author was partially supported by Capes grant 88887.614697/2021-00.

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Correspondence to Gustavo de Paula Ramos.

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de Paula Ramos, G. Nondegenerate solutions for constrained semilinear elliptic problems on Riemannian manifolds. Nonlinear Differ. Equ. Appl. 28, 64 (2021). https://doi.org/10.1007/s00030-021-00726-3

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