Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
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Alessio Fiscella
Abstract
This paper deals with the existence of nontrivial solutions for critical possibly degenerate Kirchhoff fractional (p, q) systems. For clarity, the results are first presented in the scalar case, and then extended into the vectorial framework. The main features and novelty of the paper are the (p, q) growth of the fractional operator, the double lack of compactness as well as the fact that the systems can be degenerate. As far as we know the results are new even in the scalar case and when the Kirchhoff model considered is non–degenerate.
Acknowledgements
A. Fiscella realized the manuscript within the auspices {of the FAPESP Project titled Operators with non standard growth (2019/23917-3), of the FAPESP Thematic Project titled Systems and partial differential equations (2019/02512-5) and of the CNPq Project titled Variational methods for singular fractional problems (3787749185990982).
A. Fiscella and P. Pucci are members of the Gruppo Nazionale per ľAnalisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and are partly supported by the INdAM – GNAMPA Project Equazioni alle derivate parziali: problemi e modelli (Prot_U-UFMBAZ-2020-000761). P. Pucci was partly supported by the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis.
References
[1] E. Abreu, H.S. Medeiros, Local behaviour and existence of solutions of the fractional (p, q)–Laplacian. Submitted paper, https://arxiv.org/abs/1812.01466.Suche in Google Scholar
[2] V. Ambrosio, Concentration phenomena for critical fractional Schrödinger systems. Commun. Pure Appl. Anal. 17, No 5 (2018), 2085–2123.10.3934/cpaa.2018099Suche in Google Scholar
[3] V. Ambrosio, Fractional p&q Laplacian problems in ℝN with critical growth. To appear in: Z. Anal. Anwend., https://arxiv.org/abs/1801.10449.Suche in Google Scholar
[4] V. Ambrosio, T. Isernia, On a fractional p&q Laplacian problem with critical Sobolev–Hardy exponents, Mediterr. J. Math. 15, No 6 (2018) Art. 219, pages 17.10.1007/s00009-018-1259-9Suche in Google Scholar
[5] V. Ambrosio, R. Servadei, Supercritical fractional Kirchhoff type problems. Fract. Calc. Appl. Anal. 22, No 5 (2019), 1351–1377; 10.1515/fca-2019-0071; https://www.degruyter.com/view/journals/fca/22/5/fca.22.issue-5.xml.Suche in Google Scholar
[6] G. Autuori, A. Fiscella, P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125 (2015), 699–714.10.1016/j.na.2015.06.014Suche in Google Scholar
[7] G. Autuori, P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations. NoDEA Nonlinear Diff. Equations Appl. 20, No 3 (2013), 977–1009.10.1007/s00030-012-0193-ySuche in Google Scholar
[8] G. Autuori, P. Pucci, Elliptic problems involving the fractional Laplacian in ℝN. J. Diff. Equations255, No 8 (2013), 2340–2362.10.1016/j.jde.2013.06.016Suche in Google Scholar
[9] M. Bhakta, D. Mukherjee, Multiplicity results for (p, q) fractional elliptic equations involving critical nonlinearities. Adv. Diff. Equations24, No 3-4 (2019), 185–228.Suche in Google Scholar
[10] L. Brasco, E. Parini, M. Squassina, Stability of variational eigenvalues for the fractional p–Laplacian. Discrete Contin. Dyn. Syst. 36, No 4 (2016), 1813–1845.10.3934/dcds.2016.36.1813Suche in Google Scholar
[11] H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functional. Proc. Amer. Math. Soc. 88, No 3 (1983), 486–490.10.1007/978-3-642-55925-9_42Suche in Google Scholar
[12] L. Caffarelli, J.M. Roquejoffre, O. Savin, Nonlocal minimal surfaces. Comm. Pure Appl. Math. 63, No 9 (2010), 1111–1144.10.1002/cpa.20331Suche in Google Scholar
[13] M. Caponi, P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p–Laplacian equations. Ann. Mat. Pura Appl. 195, No 6 (2016), 2099–2129.10.1007/s10231-016-0555-xSuche in Google Scholar
[14] C. Chen, Multiple solutions for a class of quasilinear Schrödinger equations in ℝN. J. Math. Phys., No 7 56 (2015), # 071507.10.1063/1.4927254Suche in Google Scholar
[15] W. Chen, M. Squassina, Critical nonlocal systems with concave–convex powers. Adv. Nonlinear Stud. 16, No 4 (2016), 821–842.10.1515/ans-2015-5055Suche in Google Scholar
[16] S. Dipierro, M. Medina, E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of ℝn. Lecture Notes, Scuola Normale Superiore Pisa, Vol. 15, viii+158 pages (2017).10.1007/978-88-7642-601-8Suche in Google Scholar
[17] L. D’Onofrio, A. Fiscella, G. Molica Bisci, Perturbation methods for nonlocal Kirchhoff–type problems. Fract. Calc. Appl. Anal. 20, No 4 (2017), 829–853; 10.1515/fca-2017-0044; https://www.degruyter.com/view/journals/fca/20/4/fca.20.issue-4.xml.Suche in Google Scholar
[18] H. Fan, Multiple positive solutions for a fractional elliptic system with critical nonlinearities. Bound. Value Probl. 2016 (2016), # 196, 18 pages.10.1186/s13661-016-0703-7Suche in Google Scholar
[19] L.F.O. Faria, O.H. Miyagaki, Critical Brezis-Nirenberg problem for nonlocal systems. Topol. Methods Nonlinear Anal. 50, No 1 (2017), 333–355.10.12775/TMNA.2017.017Suche in Google Scholar
[20] L.F.O. Faria, O.H. Miyagaki, F.R. Pereira, M. Squassina, C. Zhang, The Brezis–Nirenberg problem for nonlocal systems. Adv. Nonlinear Anal. 5, No 1 (2016), 85–103.10.1515/anona-2015-0114Suche in Google Scholar
[21] G.M. Figueiredo, Existence of positive solutions for a class of p&q elliptic problems with critical growth on ℝN. J. Math. Anal. Appl. 378, No 2 (2011), 507–518.10.1016/j.jmaa.2011.02.017Suche in Google Scholar
[22] A. Fiscella, P. Pucci, Kirchhoff–Hardy fractional problems with lack of compactness. Adv. Nonlinear Stud. 17, No 3 (2017), 429–456.10.1515/ans-2017-6021Suche in Google Scholar
[23] A. Fiscella, P. Pucci, p–fractional Kirchhoff equations involving critical nonlinearities. Nonlinear Anal. Real World Appl. 35 (2017), 350–378.10.1016/j.nonrwa.2016.11.004Suche in Google Scholar
[24] A. Fiscella, P. Pucci, (p, q) systems with critical terms in ℝN. Nonlinear Anal. 177 Part B (2018), In: Special Issue Nonlinear PDEs and Geometric Function Theory, in Honor of Carlo Sbordone on his 70th birthday, 454–479.10.1016/j.na.2018.03.012Suche in Google Scholar
[25] A. Fiscella, P. Pucci, S. Saldi, Existence of entire solutions for Schrödinger–Hardy systems involving two fractional operators. Nonlinear Anal. 158 (2017), 109–131.10.1016/j.na.2017.04.005Suche in Google Scholar
[26] A. Fiscella, P. Pucci, B. Zhang, p–fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities. Adv. Nonlinear Anal. 8, No 1 (2019), 1111–1131.10.1515/anona-2018-0033Suche in Google Scholar
[27] A. Fiscella, E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94 (2014), 156–170.10.1016/j.na.2013.08.011Suche in Google Scholar
[28] Z. Guo, S. Luo, W. Zou, On critical systems involving fractional Laplacian. J. Math. Anal. Appl. 446, No 1 (2017), 681–706.10.1016/j.jmaa.2016.08.069Suche in Google Scholar
[29] X. He, M. Squassina, W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities. Commun. Pure Appl. Anal. 15, No 4 (2016), 1285–1308.10.3934/cpaa.2016.15.1285Suche in Google Scholar
[30] L. Li, S. Tersian, Fractional problems with critical nonlinearities by a sublinear perturbation. Fract. Calc. Appl. Anal. 23, No 2 (2020), 484–503; 10.1515/fca-2020-0023; https://www.degruyter.com/view/journals/fca/23/2/fca.23.issue-2.xml.Suche in Google Scholar
[31] S. Liang, J. Zhang, Multiplicity of solutions for the noncooperative Schrödinger–Kirchhoff system involving the fractional p–Laplacian in ℝN. Z. Angew. Math. Phys. 68, No 3 (2017), Art. 63, 18 pages.10.1007/s00033-017-0805-9Suche in Google Scholar
[32] V. Maz’ya, T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, No 2 (2002), 230–238.10.1006/jfan.2002.3955Suche in Google Scholar
[33] X. Mingqi, V. Rădulescu, B. Zhang, Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM Control Optim. Calc. Var. 24, No 3 (2018), 1249–1273.10.1051/cocv/2017036Suche in Google Scholar
[34] P.K. Mishra, K. Sreenadh, Fractional p–Kirchhoff system with sign changing nonlinearities. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. (RACSAM)111, No 1 (2017), 281–296.10.1007/s13398-016-0294-2Suche in Google Scholar
[35] O.H. Miyagaki, F.R. Pereira, Existence results for non–local elliptic systems with nonlinearities interacting with the spectrum. Adv. Diff. Equations23, No 7-8 (2018), 555–580.Suche in Google Scholar
[36] G. Molica Bisci, V. Rădulescu, R. Servadei, Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and its Applications 162, Cambridge University Press, Cambridge (2016).10.1017/CBO9781316282397Suche in Google Scholar
[37] K. Perera, M. Squassina, Y. Yang, Bifurcation and multiplicity results for critical fractional p–Laplacian problems. Math. Nachr. 289, No 2-3 (2016), 332–342.10.1002/mana.201400259Suche in Google Scholar
[38] P. Pucci, S. Saldi, Critical stationary Kirchhoff equations in ℝN involving nonlocal operators. Rev. Mat. Iberoam. 32, No 1 (2016), 1–22.10.4171/RMI/879Suche in Google Scholar
[39] P. Pucci, M. Xiang, B. Zhang, Multiple solutions for nonhomogenous Schrödinger-Kirchhoff type equations involving the fractional p–Laplacian in ℝN. Calc. Var. Partial Diff. Equations54, No 3 (2015), 2785–2806.10.1007/s00526-015-0883-5Suche in Google Scholar
[40] P. Pucci, M. Xiang, B. Zhang, Existence and multiplicity of entire solutions for fractional p–Kirchhoff equations. Adv. Nonlinear Anal. 5, No 1 (2016), 27–55.10.1515/anona-2015-0102Suche in Google Scholar
[41] M. Xiang, B. Zhang, V. Rădulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p–Laplacian. Nonlinearity29, No 10 (2016), 3186–3205.10.1088/0951-7715/29/10/3186Suche in Google Scholar
© 2020 Diogenes Co., Sofia
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
Artikel in diesem Heft
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–Volume 23–3–2020)
- Survey Paper
- Why fractional derivatives with nonsingular kernels should not be used
- Fractional-order susceptible-infected model: Definition and applications to the study of COVID-19 main protease
- Generalized fractional Poisson process and related stochastic dynamics
- Research Paper
- Determination of the fractional order in semilinear subdiffusion equations
- Degenerate Kirchhoff (p, q)–Fractional systems with critical nonlinearities
- Solution of linear fractional order systems with variable coefficients
- “Fuzzy” calculus: The link between quantum mechanics and discrete fractional operators
- The green function for a class of Caputo fractional differential equations with a convection term
- Inverse problem for a multi-term fractional differential equation
- Maximum principles for a class of generalized time-fractional diffusion equations
- Multiple positive solutions for a nonlocal PDE with critical Sobolev-Hardy and singular nonlinearities via perturbation method
- Variational approximation for fractional Sturm–Liouville problem
- The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field
- Construction of fixed point operators for nonlinear difference equations of non integer order with impulses
- An averaging principle for stochastic differential equations of fractional order 0 < α < 1
- Weak solvability of the variable-order subdiffusion equation
