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Annali di Matematica Pura ed Applicata (1923 -)

, Volume 195, Issue 6, pp 2099–2129 | Cite as

Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations

  • Maicol Caponi
  • Patrizia Pucci
Article

Abstract

The paper deals with existence, multiplicity and asymptotic behavior of entire solutions for a series of stationary Kirchhoff fractional p-Laplacian equations. The existence presents several difficulties due to the intrinsic lack of compactness arising from different reasons, and the suitable strategies adopted to overcome the technical hurdles depend on the specific problem under consideration. The results of the paper extend in several directions recent theorems. Furthermore, the main assumptions required in the paper weaken the hypotheses used in the recent literature on stationary Kirchhoff fractional problems. Some equations treated in the paper cover the so-called degenerate case that is the case in which the Kirchhoff function M is zero at zero. In other words, from a physical point of view, when the base tension of the string modeled by the equation is zero, it is a very realistic case. Last but not least no monotonicity assumption is required on M, and also this aspect makes the models more believable in several physical applications.

Keywords

Stationary Kirchhoff problems Non-local p-Laplacian operators  Hardy coefficients Critical exponents 

Mathematics Subject Classification

35R11 35J60 35J20 35B09 

Notes

Acknowledgments

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 Project Modelli ed equazioni nonlocali di tipo frazionario (Prot_2015_000368). The second author was partly supported by the Italian MIUR project Variational and perturbative aspects of nonlinear differential problems (201274FYK7).

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Scuola Internazionale Superiore di Studi Avanzati – SISSATriesteItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly

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