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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 & Continuous Dynamical Systems - S, 2020, 13 (7) : 2069-2094. doi: 10.3934/dcdss.2020159
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show all references

References:
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D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[21]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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[30]

G. Kirchhoff, Vorlesungen Über Mathematische Physik, BG Teubner, 1876. Google Scholar

[31]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[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.  Google Scholar

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E. Lieb and M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode island, 2001. doi: 10.1090/gsm/014.  Google Scholar

[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.  Google Scholar

[35]

P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[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.  Google Scholar

[37]

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