Variational analysis for nonlocal Yamabe-type systems

Mingqi Xiang; Giovanni Molica Bisci; Binlin Zhang

doi:10.3934/dcdss.2020159

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July 2020, 13(7): 2069-2094. doi: 10.3934/dcdss.2020159

Variational analysis for nonlocal Yamabe-type systems

Mingqi Xiang ^1, , Giovanni Molica Bisci ^2,, and Binlin Zhang ^3,

1.	College of Science, Civil Aviation University of China, Tianjin 300300, China
2.	Dipartimento di Scienze Pure e Applicate (DiSPeA), Universitá degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy
3.	College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China

^* Corresponding author: Giovanni Molica Bisci

Dedicated to Professor Patrizia Pucci with deep esteem and admiration

Received March 2018 Revised July 2018 Published November 2019

Fund Project: M. Xiang was supported by the National Natural Science Foundation of China (No. 11601515) and the Tianjin Youth Talent Special Support Program. G. Molica Bisci is 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 Projects Problemi variazionali su varietà Riemanniane e gruppi di Carnot (Prot 2016 000421) and Teoria e modelli per problemi non locali. B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199)

The paper is concerned with existence, multiplicity and asymptotic behavior of (weak) solutions for nonlocal systems involving critical nonlinearities. More precisely, we consider

$\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$

where

$ (-\Delta )^s $

is the fractional Lapalcian,

$ [u]_{s} $

is the Gagliardo seminorm of

$ u $

$ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $

is a continuous function satisfying certain assumptions,

$ V(x) = {|x|^{-2s}} $

is the Hardy potential function,

$ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $

$ s, t\in (0, 1) $

$ \lambda, \mu $

are two positive parameters,

$ 1<p<2_s^* = {6}/{(3-2s)} $

and

$ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $

. By using topological methods and the Krasnoleskii's genus theory, we obtain the existence, multiplicity and asymptotic behaviour of solutions for above problem under suitable positive parameters

$ \lambda $

and

$ \mu $

. Moreover, we also consider the existence of nonnegative radial solutions and non-radial sign-changing solutions. The main novelties are that our results involve the possibly degenerate Kirchhoff function and the upper critical exponent in the sense of Hardy–Littlehood–Sobolev inequality. We emphasize that some of the results contained in the paper are also valid for nonlocal Schrödinger–Maxwell systems on Cartan–Hadamard manifolds.

Keywords: Fractional Laplacian, Yamabe-type problems, topological methods, genus theory.

Mathematics Subject Classification: Primary: 35R11, 35A15; Secondary: 35J60.

Citation: Mingqi Xiang, Giovanni Molica Bisci, Binlin Zhang. Variational analysis for nonlocal Yamabe-type systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 2069-2094. doi: 10.3934/dcdss.2020159

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[47]	P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
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show all references

References:

[1]	D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
[2]	G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714. doi: 10.1016/j.na.2015.06.014.
[3]	J. García Azorero and I. Peral Alonso, Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895. doi: 10.1090/S0002-9947-1991-1083144-2.
[4]	A. Azzollini and P. d'Avenia, On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438. doi: 10.1016/j.jmaa.2011.09.012.
[5]	A. Azzollini, P. d'Avenia and V. Luisi, Generalized Schrödinger-Poisson type systems, Commun. Pure Appl. Anal., 12 (2013), 867-879. doi: 10.3934/cpaa.2013.12.867.
[6]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods. Nonlinear Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019.
[7]	V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420. doi: 10.1142/S0129055X02001168.
[8]	A. Bongers, Existenzaussagen für die Choquard-Gleichung: Ein nichtlineares eigenwertproblem der plasma-physik, Z. Angew. Math. Mech., 60 (1980), T240–T242.
[9]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
[10]	L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, Springer, Heidelberg, 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3.
[11]	M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x.
[12]	Y.-H. Chen and C. G. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842. doi: 10.1088/0951-7715/29/6/1827.
[13]	F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974. doi: 10.1016/j.na.2011.05.073.
[14]	E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.
[15]	P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appli. Sci., 25 (2015), 1447-1476. doi: 10.1142/S0218202515500384.
[16]	P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. doi: 10.1007/BF02100605.
[17]	T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906. doi: 10.1017/S030821050000353X.
[18]	T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. doi: 10.1515/ans-2004-0305.
[19]	O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831-869. doi: 10.1142/S0219199710004007.
[20]	A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011.
[21]	A. Fiscella and P. Pucci, $p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. doi: 10.1016/j.nonrwa.2016.11.004.
[22]	A. Fiscella, G. Molica Bisci and R. Servadei, Multiplicity results for fractional Laplace problems with critical growth, Manuscripta Math., 155 (2018), 369-388. doi: 10.1007/s00229-017-0947-2.
[23]	G. M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361. doi: 10.3233/ASY-151316.
[24]	F. Gao and M. Yang, On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5.
[25]	X. M. He and W. M. Zou, Existence and concetration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156.
[26]	E. Hebey and P.-D. Thizy, Stationary Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math., 18 (2016), 1550028, 53 pp. doi: 10.1142/S0219199715500285.
[27]	E. Hebey, F. Robert and Y. L. Wen, Compactness and global estimates for fourth order equation of critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math., 8 (2006), 9-65. doi: 10.1142/S0219199706002027.
[28]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.
[29]	Y. S. Jiang and H.-S. Zhou., Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006.
[30]	G. Kirchhoff, Vorlesungen Über Mathematische Physik, BG Teubner, 1876.
[31]	N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.
[32]	F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp. doi: 10.1142/S0219199714500369.
[33]	E. Lieb and M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode island, 2001. doi: 10.1090/gsm/014.
[34]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. App. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293.
[35]	P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4.
[36]	H. D. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212. doi: 10.1016/j.nonrwa.2016.04.007.
[37]	Z. L. Liu, Z.-Q. Wang and J. J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794. doi: 10.1007/s10231-015-0489-8.
[38]	J. Liu, J.-F. Liao and C.-L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172. doi: 10.1016/j.jmaa.2015.04.066.
[39]	D. F. Lü, A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48. doi: 10.1016/j.na.2013.12.022.
[40]	V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brézis, and Mironescu theorem converning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238. doi: 10.1006/jfan.2002.3955.
[41]	G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.
[42]	G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp. doi: 10.1142/S0219199715500881.
[43]	S. Pekar, Untersuchung Uber Die Elektronentheorie der Kristalle, Akademie Verlag, 1954.
[44]	R. Penrose, Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256.
[45]	P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879.
[46]	P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55. doi: 10.1515/anona-2015-0102.
[47]	P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806. doi: 10.1007/s00526-015-0883-5.
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