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

Patrizia Pucci; Mingqi Xiang; Binlin Zhang

doi:10.1515/acv-2016-0049

Published by De Gruyter July 21, 2017

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

Patrizia Pucci , Mingqi Xiang and Binlin Zhang

From the journal Advances in Calculus of Variations

https://doi.org/10.1515/acv-2016-0049

Showing a limited preview of this publication:

Abstract

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+b⁢∥u∥sp⁢(θ-1))⁢[(-Δ)ps⁢u+V⁢(x)⁢|u|p-2⁢u]=λ⁢f⁢(x,u)+(∫ℝN|u|pμ,s*|x-y|μ⁢𝑑y)⁢|u|pμ,s*-2⁢u in ⁢ℝN,

where

∥u∥s=(∬ℝ2⁢N|u⁢(x)-u⁢(y)|p|x-y|N+p⁢s⁢𝑑x⁢𝑑y+∫ℝNV⁢(x)⁢|u|p⁢𝑑x)1p,

a,b∈ℝ0+, with a+b>0, λ>0 is a parameter, s∈(0,1), N>p⁢s, θ∈[1,N/(N-p⁢s)), (-Δ)ps is the fractional p-Laplacian, V:ℝN→ℝ+ is a potential function, 0<μ<N, pμ,s*=(p⁢N-p⁢μ/2)/(N-p⁢s) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f:ℝN×ℝ→ℝ 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.

Keywords: Schrödinger–Choquard–Kirchhoff type; variational methods; critical exponent

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).

Acknowledgements

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

References

[1] C. O. Alves, G. M. Figueiredo and M. Yang, Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field, Asymptot. Anal. 96 (2016), no. 2, 135–159. 10.3233/ASY-151337Search in Google Scholar

[2] C. O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations 257 (2014), no. 11, 4133–4164. 10.1016/j.jde.2014.08.004Search in Google Scholar

[3] D. Applebaum, Lévy processes—From probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), no. 11, 1336–1347. Search in Google Scholar

[4] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure Appl. Math. (New York), John Wiley & Sons, New York, 1984. Search in Google Scholar

[5] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–714. 10.1016/j.na.2015.06.014Search in Google Scholar

[6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ℝN, Comm. Partial Differential Equations 20 (1995), no. 9–10, 1725–1741. 10.1080/03605309508821149Search in Google Scholar

[7] A. Bongers, Existenzaussagen für die Choquard-Gleichung: Ein nichtlineares Eigenwertproblem der Plasma-Physik, Z. Angew. Math. Mech. 60 (1980), no. 7, T240–T242. Search in Google Scholar

[8] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. 10.1007/978-0-387-70914-7Search in Google Scholar

[9] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp. 7, Springer, Heidelberg (2012), 37–52. 10.1007/978-3-642-25361-4_3Search in Google Scholar

[10] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7–9, 1245–1260. 10.1080/03605300600987306Search in Google Scholar

[11] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl. (4) 195 (2016), no. 6, 2099–2129. 10.1007/s10231-016-0555-xSearch in Google Scholar

[12] F. Colasuonno and P. Pucci, Multiplicity of solutions for p⁢(x)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal. 74 (2011), no. 17, 5962–5974. 10.1016/j.na.2011.05.073Search in Google Scholar

[13] P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), no. 2, 247–262. 10.1007/BF02100605Search in Google Scholar

[14] P. d’Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appl. Sci. 25 (2015), no. 8, 1447–1476. 10.1142/S0218202515500384Search in Google Scholar

[15] A. Di Castro, T. Kuusi and G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal. 267 (2014), no. 6, 1807–1836. 10.1016/j.jfa.2014.05.023Search in Google Scholar

[16] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 5, 1279–1299. 10.1016/j.anihpc.2015.04.003Search in Google Scholar

[17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar

[18] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝn, Appunti. Sc. Norm. Super. Pisa (N. S.) 15, Edizioni della Normale, Pisa, 2017. 10.1007/978-88-7642-601-8Search in Google Scholar

[19] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal. 94 (2014), 156–170. 10.1016/j.na.2013.08.011Search in Google Scholar

[20] F. Gao and M. Yang, On the Brézis–Nirenberg type critical problem for nonlinear Choquard equation, preprint (2016), https://arxiv.org/abs/1604.00826v3. 10.1007/s11425-016-9067-5Search in Google Scholar

[21] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy–Littlewood–Sobolev critical exponents, J. Math. Anal. Appl. 448 (2017), no. 2, 1006–1041. 10.1016/j.jmaa.2016.11.015Search in Google Scholar

[22] T. Kuusi, G. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys. 337 (2015), no. 3, 1317–1368. 10.1007/s00220-015-2356-2Search in Google Scholar

[23] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57 (1976/77), no. 2, 93–105. 10.1007/978-3-642-55925-9_37Search in Google Scholar

[24] E. Lieb and M. Loss, Analysis, 2nd ed., Grad. Stud. Math. 14, American Mathematical Society, Providence, 2001. Search in Google Scholar

[25] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), no. 6, 1063–1072. 10.1016/0362-546X(80)90016-4Search in Google Scholar

[26] D. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal. 99 (2014), 35–48. 10.1016/j.na.2013.12.022Search in Google Scholar

[27] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), no. 2, 153–184. 10.1016/j.jfa.2013.04.007Search in Google Scholar

[28] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, Article ID 1550005. 10.1142/S0219199715500054Search in Google Scholar

[29] X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional p-Laplacian, Nonlinearity 29 (2016), no. 2, 357–374. 10.1088/0951-7715/29/2/357Search in Google Scholar

[30] G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia Math. Appl. 162, Cambridge University Press, Cambridge, 2016. 10.1017/CBO9781316282397Search in Google Scholar

[31] T. Mukherjee and K. Sreenadh, Existence and multiplicity results for Brezis–Nirenberg type fractional Choquard equation, preprint (2016), https://arxiv.org/abs/1605.06805. Search in Google Scholar

[32] S. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie-Verlag, Berlin, 1954. 10.1515/9783112649305Search in Google Scholar

[33] R. Penrose, Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 356 (1998), no. 1743, 1927–1939. 10.1098/rsta.1998.0256Search in Google Scholar

[34] P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., to appear. 10.5565/PUBLMAT6211801Search in Google Scholar

[35] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ℝN involving nonlocal operators, Rev. Mat. Iberoam. 32 (2016), no. 1, 1–22. 10.4171/RMI/879Search in Google Scholar

[36] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian in ℝN, Calc. Var. Partial Differential Equations 54 (2015), no. 3, 2785–2806. 10.1007/s00526-015-0883-5Search in Google Scholar

[37] P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal. 5 (2016), no. 1, 27–55. 10.1515/anona-2015-0102Search in Google Scholar

[38] D. Wu, Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity, J. Math. Anal. Appl. 411 (2014), no. 2, 530–542. 10.1016/j.jmaa.2013.09.054Search in Google Scholar

[39] M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional p-Laplacian, J. Math. Anal. Appl. 424 (2015), no. 2, 1021–1041. 10.1016/j.jmaa.2014.11.055Search in Google Scholar

[40] M. Xiang, B. Zhang and M. Ferrara, Multiplicity results for the non-homogeneous fractional p-Kirchhoff equations with concave-convex nonlinearities, Proc. A. 471 (2015), no. 2177, Article ID 20150034. 10.1098/rspa.2015.0034Search in Google Scholar

[41] M. Xiang, B. Zhang and X. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via fountain theorem, Nonlinear Anal. 120 (2015), 299–313. 10.1016/j.na.2015.03.015Search in Google Scholar

[42] H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in ℝN, J. Math. Anal. Appl. 431 (2015), no. 2, 935–954. 10.1016/j.jmaa.2015.06.012Search in Google Scholar

Received: 2016-10-25

Revised: 2017-04-11

Accepted: 2017-06-12

Published Online: 2017-07-21

Published in Print: 2019-07-01

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

Abstract

Acknowledgements

References

Journal and Issue

Articles in the same Issue