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Perturbation methods for nonlocal Kirchhoff–type problems

  • Luigi D’Onofrio EMAIL logo , Alessio Fiscella and Giovanni Molica Bisci
Published/Copyright: August 8, 2017
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 20 Issue 4

Abstract

This paper deals with the existence of infinitely many solutions for a class of Dirichlet elliptic problems driven by a bi–nonlocal operator u ↦ M(∥u∥2)(−Δ)su, where M models a Kirchhoff–type coefficient while (−Δ)s denotes the fractional Laplace operator. More precisely, by adapting to our bi–nonlocal framework the variational and topological tools introduced in [16], we establish the existence of infinitely many solutions. The main feature and difficulty of our problems is due to the possible degenerate nature of the Kirchhoff term M.

MSC 2010: Primary 35J60; 35R11; Secondary 35J20; 35S15
Key Words and Phrases: Kirchhoff–type problems; existence of solutions; fractional Sobolev spaces; variational methods

Dedicated to Tania Restuccia on the occasion of her 70th birthday with all our affection and esteem

Acknowledgements

The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Projects Problemi variazionali su varietà Riemanniane e gruppi di Carnot (Prot_2016_000421) and Teoria e modelli per problemi non locali.

A. Fiscella was supported by Coordenação de Aperfeiçonamento de pessoal de nível superior (CAPES) through the fellowship 33003017003P5–PNPD20131750–UNICAMP/MATEMÁTICA.

L. D’Onofrio was supported by Sostegno alla Ricerca Individuale Universitá degli Studi di Napoli “Parthenope”.

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Received: 2017-3-14
Revised: 2017-5-5
Published Online: 2017-8-8
Published in Print: 2017-8-28

© 2017 Diogenes Co., Sofia

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  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
  5. Perturbation methods for nonlocal Kirchhoff–type problems
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  27. Corrigendum: Fractional integral on martingale hardy spaces with variable exponents
https://doi.org/10.1515/fca-2017-0044
Keywords for this article
Kirchhoff–type problems; existence of solutions; fractional Sobolev spaces; variational methods

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 20–4–2017)
  4. Research Paper
  5. Perturbation methods for nonlocal Kirchhoff–type problems
  6. Research Paper
  7. On infinite order differential operators in fractional viscoelasticity
  8. Research Paper
  9. Fundamental solution of the multi-dimensional time fractional telegraph equation
  10. Research Paper
  11. Robustness and convergence of fractional systems and their applications to adaptive schemes
  12. Research Paper
  13. Stability analysis of linear distributed order fractional systems with distributed delays
  14. Research Paper
  15. Fractional sobolev spaces and functions of bounded variation of one variable
  16. Research Paper
  17. Approximate controllability for fractional differential equations of sobolev type via properties on resolvent operators
  18. Research Paper
  19. On moduli of smoothness and averaged differences of fractional order
  20. Research Paper
  21. A piecewise memory principle for fractional derivatives
  22. Research Paper
  23. A computational approach for the solution of a class of variable-order fractional integro-differential equations with weakly singular kernels
  24. Shopt Paper
  25. Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method
  26. Corrigendum
  27. Corrigendum: Fractional integral on martingale hardy spaces with variable exponents
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