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Multiple solutions for Schrödinger equations on Riemannian manifolds via \(\nabla \)-theorems

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Abstract

We consider a smooth, complete and non-compact Riemannian manifold \((\mathcal {M},g)\) of dimension \(d \ge 3\), and we look for solutions to the semilinear elliptic equation

$$\begin{aligned} -\varDelta _g w + V(\sigma ) w = \alpha (\sigma ) f(w) + \lambda w \quad \hbox {in }\mathcal {M}. \end{aligned}$$

The potential \(V :\mathcal {M} \rightarrow \mathbb {R}\) is a continuous function which is coercive in a suitable sense, while the nonlinearity f has a subcritical growth in the sense of Sobolev embeddings. By means of \(\nabla \)-theorems introduced by Marino and Saccon, we prove that at least three non-trivial solutions exist as soon as the parameter \(\lambda \) is sufficiently close to an eigenvalue of the operator \(-\varDelta _g\).

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Acknowledgements

The authors would like to thank the anonymous referee for her/his valuable comments and remarks.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, Milan, Italy

    Luigi Appolloni & Simone Secchi

  2. Dipartimento di Scienze Pure e Applicate, Università di Urbino Carlo Bo, Urbino, Italy

    Giovanni Molica Bisci

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  1. Luigi Appolloni
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  2. Giovanni Molica Bisci
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Correspondence to Giovanni Molica Bisci.

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The first and third authors are supported by GNAMPA, project “Equazioni alle derivate parziali: problemi e modelli.”.

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Appolloni, L., Molica Bisci, G. & Secchi, S. Multiple solutions for Schrödinger equations on Riemannian manifolds via \(\nabla \)-theorems. Ann Glob Anal Geom 63, 11 (2023). https://doi.org/10.1007/s10455-023-09885-1

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