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Existence results for Schrödinger–Choquard–Kirchhoff equations involving the fractional p-Laplacian

  • Patrizia Pucci ORCID logo EMAIL logo , Mingqi Xiang and Binlin Zhang


The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case

(a+busp(θ-1))[(-Δ)psu+V(x)|u|p-2u]=λf(x,u)+(N|u|pμ,s*|x-y|μ𝑑y)|u|pμ,s*-2uin N,



a,b0+, with a+b>0, λ>0 is a parameter, s(0,1), N>ps, θ[1,N/(N-ps)), (-Δ)ps is the fractional p-Laplacian, V:N+ is a potential function, 0<μ<N, pμ,s*=(pN-pμ/2)/(N-ps) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f:N× is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when f satisfies superlinear growth conditions and λ is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when f is sublinear at infinity and λ is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter a can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.

MSC 2010: 35R11; 35A15; 47G20

Communicated by Giuseppe Mingione

Funding statement: Patrizia 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 (INdAM). The manuscript was realized within the auspices of the INdAM–GNAMPA Project Equazioni Differenziali non lineari (Prot_2017_0000265). Patrizia Pucci was partly supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009). Mingqi Xiang was supported by National Nature Science Foundation of China (No. 11601515) and Fundamental Research Funds for the Central Universities (No. 3122017080). Binlin Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306) and Research Foundation of Heilongjiang Educational Committee (No. 12541667).


We greatly thank the referee for his/her thorough review and highly appreciate the references he/she requested to add.


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Received: 2016-10-25
Revised: 2017-04-11
Accepted: 2017-06-12
Published Online: 2017-07-21
Published in Print: 2019-07-01

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