# American Institute of Mathematical Sciences

## Variational analysis for nonlocal Yamabe-type systems

 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 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. Citation: Mingqi Xiang, Giovanni Molica Bisci, Binlin Zhang. Variational analysis for nonlocal Yamabe-type systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020159 ##### References:  [1] 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 [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] A. Bongers, Existenzaussagen für die Choquard-Gleichung: Ein nichtlineares eigenwertproblem der plasma-physik, Z. Angew. Math. Mech., 60 (1980), T240–T242. Google Scholar [9] H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar [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. Google Scholar [11] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional$p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x. Google Scholar [12] Y.-H. Chen and C. G. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842. doi: 10.1088/0951-7715/29/6/1827. Google Scholar [13] F. Colasuonno and P. Pucci, Multiplicity of solutions for$p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974. doi: 10.1016/j.na.2011.05.073. Google Scholar [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [15] P. d'Avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Models Methods Appli. Sci., 25 (2015), 1447-1476. doi: 10.1142/S0218202515500384. Google Scholar [16] P. D'Ancona and S. 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Fiscella and P. Pucci,$p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. doi: 10.1016/j.nonrwa.2016.11.004. Google Scholar [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. 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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 [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. 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. 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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. Google Scholar [11] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional$p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129. doi: 10.1007/s10231-016-0555-x. Google Scholar [12] Y.-H. Chen and C. G. Liu, Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842. doi: 10.1088/0951-7715/29/6/1827. Google Scholar [13] F. Colasuonno and P. Pucci, Multiplicity of solutions for$p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974. doi: 10.1016/j.na.2011.05.073. Google Scholar [14] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. 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Math., 12 (2010), 831-869. doi: 10.1142/S0219199710004007. Google Scholar [20] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170. doi: 10.1016/j.na.2013.08.011. Google Scholar [21] A. Fiscella and P. Pucci,$p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378. doi: 10.1016/j.nonrwa.2016.11.004. Google Scholar [22] A. Fiscella, G. Molica Bisci and R. Servadei, Multiplicity results for fractional Laplace problems with critical growth, Manuscripta Math., 155 (2018), 369-388. doi: 10.1007/s00229-017-0947-2. Google Scholar [23] G. M. Figueiredo, G. Molica Bisci and R. Servadei, On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361. doi: 10.3233/ASY-151316. Google Scholar [24] F. Gao and M. Yang, On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242. doi: 10.1007/s11425-016-9067-5. Google Scholar [25] X. M. He and W. M. Zou, Existence and concetration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp. doi: 10.1063/1.3683156. Google Scholar [26] E. Hebey and P.-D. Thizy, Stationary Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math., 18 (2016), 1550028, 53 pp. doi: 10.1142/S0219199715500285. Google Scholar [27] E. Hebey, F. Robert and Y. L. Wen, Compactness and global estimates for fourth order equation of critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math., 8 (2006), 9-65. doi: 10.1142/S0219199706002027. Google Scholar [28] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on$\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. Google Scholar [29] Y. S. Jiang and H.-S. Zhou., Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. Google Scholar [30] G. Kirchhoff, Vorlesungen Über Mathematische Physik, BG Teubner, 1876. Google Scholar [31] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [32] F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp. doi: 10.1142/S0219199714500369. Google Scholar [33] E. Lieb and M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode island, 2001. doi: 10.1090/gsm/014. Google Scholar [34] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. App. Math., 57 (1976/77), 93-105. doi: 10.1002/sapm197757293. Google Scholar [35] P.-L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [36] H. D. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212. doi: 10.1016/j.nonrwa.2016.04.007. Google Scholar [37] Z. L. Liu, Z.-Q. Wang and J. J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794. doi: 10.1007/s10231-015-0489-8. Google Scholar [38] J. Liu, J.-F. Liao and C.-L. Tang, Positive solutions for Kirchhoff-type equations with critical exponent in$\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172. doi: 10.1016/j.jmaa.2015.04.066. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [43] S. Pekar, Untersuchung Uber Die Elektronentheorie der Kristalle, Akademie Verlag, 1954. Google Scholar [44] R. Penrose, Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc., 356 (1998), 1927-1939. doi: 10.1098/rsta.1998.0256. Google Scholar [45] P. Pucci and S. 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