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Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations

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Abstract

The existence of one non-trivial solution for a nonlinear problem on compact d-dimensional (\({d \geq 3}\)) Riemannian manifolds without boundary, is established. More precisely, a recent critical point result for differentiable functionals is exploited, in order to prove the existence of a determined open interval of positive eigenvalues for which the considered problem admits at least one non-trivial weak solution. Moreover, as a consequence of our approach, a multiplicity result is presented, requiring the validity of the Ambrosetti–Rabinowitz hypothesis. Successively, the Cerami compactness condition is studied in order to obtain a similar multiplicity theorem in superlinear cases. Finally, applications to Emden-Fowler type equations are presented.

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Authors and Affiliations

  1. Mathematics Section, Department of Science for Engineering and Architecture, Engineering Faculty, University of Messina, 98166, Messina, Italy

    Gabriele Bonanno

  2. Dipartimento MECMAT, University of Reggio Calabria, Via Graziella, Feo di Vito, 89124, Reggio Calabria, Italy

    Giovanni Molica Bisci

  3. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 014700, Bucharest, Romania

    Vicenţiu D. Rădulescu

  4. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Vicenţiu D. Rădulescu

Authors
  1. Gabriele Bonanno
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  2. Giovanni Molica Bisci
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  3. Vicenţiu D. Rădulescu
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Correspondence to Vicenţiu D. Rădulescu.

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Bonanno, G., Bisci, G.M. & Rădulescu, V.D. Nonlinear elliptic problems on Riemannian manifolds and applications to Emden–Fowler type equations. manuscripta math. 142, 157–185 (2013). https://doi.org/10.1007/s00229-012-0596-4

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  • DOI: https://doi.org/10.1007/s00229-012-0596-4

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