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Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in \({\mathbb {R}}^N\)

  • Patrizia Pucci
  • Mingqi Xiang
  • Binlin Zhang
Article

Abstract

In this paper we investigate the existence of multiple solutions for the nonhomogeneous fractional p-Laplacian equations of Schrödinger–Kirchhoff type
$$\begin{aligned} M\left( \iint _{R^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right) (-\varDelta )^s_pu+V(x)|u|^{p-2}u=f(x,u)+g(x) \end{aligned}$$
in \({\mathbb {R}}^N\), where \((-\varDelta )^s_p\) is the fractional p-Laplacian operator, with \(0<s<1<p<\infty \) and \(ps<N\), the nonlinearity \(f:{\mathbb {R}}^N\times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is a Carathéodory function and satisfies the Ambrosetti–Rabinowitz condition, \(V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^+\) is a potential function and \(g:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is a perturbation term. We first establish Batsch–Wang type compact embedding theorem for the fractional Sobolev spaces. Then multiplicity results are obtained by using the Ekeland variational principle and the Mountain Pass theorem.

Mathematics Subject Classification

35R11 35A15 35J60 47G20 

Notes

Acknowledgments

P. Pucci is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica “G. Severi” (INdAM) and was partially supported by the MIUR Project Aspetti variazionali e perturbativi nei problemi differenziali nonlineari, and finally the manuscript was realized within the auspices of the INDAM-GNAMPA Project 2015 titled Modelli ed equazioni non-locali di tipo frazionario (Prot_2015_000368). M. Xiang was support by the Fundamental Research Funds for the Central Universities (No. 3122015L014). B. Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667) and Doctoral Research Foundation of Heilongjiang Institute of Technology (No. 2013BJ15).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaPerugiaItaly
  2. 2.College of ScienceCivil Aviation University of ChinaTianjinPeople’s Republic of China
  3. 3.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China
  4. 4.Department of MathematicsHeilongjiang Institute of TechnologyHarbinPeople’s Republic of China

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