# American Institute of Mathematical Sciences

June  2021, 41(6): 2653-2676. doi: 10.3934/dcds.2020379

## Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case

 1 School of Mathematics, Hefei University of Technology, Hefei 230009, China 2 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, China 3 Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland 4 Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania

* Corresponding author: Vicenţiu D. Răadulescu

Received  July 2020 Revised  September 2020 Published  June 2021 Early access  November 2020

Fund Project: B. Zhang was supported by the National Natural Science Foundation of China (No. 11871199), the Heilongjiang Province Postdoctoral Startup Foundation, PR China (LBH-Q18109), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province. V.D. Răadulescu acknowledges the support through the Project MTM2017-85449-P of the DGISPI (Spain)

In this paper, we study the following coupled nonlocal system
 $\begin{equation*} \begin{cases} (-\Delta)^{s}u-\lambda_{1}u = \mu_{1}|u|^{\alpha}u+\beta|u|^{\frac{\alpha-2}{2}}u|v|^{\frac{\alpha+2}{2}} & \text{in} \ \ \mathbb{R}^{N},\\ (-\Delta)^{s}v-\lambda_{2}v = \mu_{2}|v|^{\alpha}v+\beta|u|^{\frac{\alpha+2}{2}}|v|^{\frac{\alpha-2}{2}}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*}$
 $\int_{\mathbb{R}^{N}}u^{2}dx = b^{2}_{1}\ \text{and} \ \int_{\mathbb{R}^{N}}v^{2}dx = b^{2}_{2},$
where
 $(-\Delta)^{s}$
is the fractional Laplacian,
 $0 , $ \mu_{1},\, \mu_{2}>0 $, $ N>2s $, and $ \frac{4s}{N}<\alpha\leq \frac{2s}{N-2s} $. We are concerned with the attractive case, which corresponds to $ \beta>0 $. In the case of low perturbations of the coupling parameter, by using two-dimensional linking arguments, we show that there exists $ \beta_{1}>0 $such that when $ 0<\beta<\beta_{1} $, then the system has a positive radial solution. Next, in the case of high perturbations of the coupling parameter, we prove that there exists $ \beta_{2}>0 $such that the system has a mountain-pass type solution for all $ \beta>\beta_{2} $. These results correspond to low and high perturbations with respect to the values of the coupling parameter $ \beta $. This paper extends and complements the main results established in [2] for the particular case $ N = 3 $, $ s = 1 $, $ \alpha = 2 $. Citation: Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2653-2676. doi: 10.3934/dcds.2020379 ##### References:  [1] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [2] T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on$\mathbb{R}^{3}$, J. Math. Pures Appl., 106 (2016), 583-614. doi: 10.1016/j.matpur.2016.03.004. [3] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037. doi: 10.1016/j.jfa.2017.01.025. [4] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations, 58 (2019), 24 pp. doi: 10.1007/s00526-018-1476-x. [5] T. Bartsch, X. Zhong and W. Zou, Normalized solutions for a coupled Schrödinger system, Math. Ann., (2020). doi: 10.1007/s00208-020-02000-w. [6] J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303-339. doi: 10.1112/plms/pds072. [7] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] S. Chen, V.D. Rădulescu and X. Tang, Normalized solutions of nonautonomous Kirchhoff equations: Sub- and super-critical cases, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09661-8. [10] S. Cingolani and L. Jeanjean, Stationary waves with prescribed$L^2$-norm for the planar Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568. doi: 10.1137/19M1243907. [11] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in${\mathbb R}$, Acta Math., 210 (2013), 260-318. doi: 10.1007/s11511-013-0095-9. [12] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. [13] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [14] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, vol. 107, Cambridge University Press, 1993. doi: 10.1017/CBO9780511551703. [15] Z. Guo, A. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. [16] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equation, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. [17] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346. [18] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2. [19] G. Molica Bisci, V. Răadulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [20] S. Peng, S. We and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053. [21] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in$\mathbb{R}^N$involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879. [22] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [23] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [24] M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional$p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055. [25] M. Xiang, B. Zhang and V. Răadulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional$p$-Laplacian and critical exponent, Adv. Nonlinear Anal., 9 (2020), 690-709. doi: 10.1515/anona-2020-0021. [26] M. Zhen, J. He and H. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013. [27] M. Zhen, J. He, H. Xu and M. Yang, Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent, Bound. Value Probl., 96 (2018), 25 pp. doi: 10.1186/s13661-018-1016-9. [28] M. Zhen, J. He, H. Xu and M. Yang, Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent, Discrete Contin. Dyn. Syst., 39 (2019), 6523-6539. doi: 10.3934/dcds.2019283. show all references ##### References:  [1] B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023. [2] T. Bartsch, L. Jeanjean and N. Soave, Normalized solutions for a system of coupled cubic Schrödinger equations on$\mathbb{R}^{3}$, J. Math. Pures Appl., 106 (2016), 583-614. doi: 10.1016/j.matpur.2016.03.004. [3] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037. doi: 10.1016/j.jfa.2017.01.025. [4] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differential Equations, 58 (2019), 24 pp. doi: 10.1007/s00526-018-1476-x. [5] T. Bartsch, X. Zhong and W. Zou, Normalized solutions for a coupled Schrödinger system, Math. Ann., (2020). doi: 10.1007/s00208-020-02000-w. [6] J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations, Proc. Lond. Math. Soc., 107 (2013), 303-339. doi: 10.1112/plms/pds072. [7] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001. [8] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. [9] S. Chen, V.D. Rădulescu and X. Tang, Normalized solutions of nonautonomous Kirchhoff equations: Sub- and super-critical cases, Appl. Math. Optim., (2020). doi: 10.1007/s00245-020-09661-8. [10] S. Cingolani and L. Jeanjean, Stationary waves with prescribed$L^2$-norm for the planar Schrödinger-Poisson system, SIAM J. Math. Anal., 51 (2019), 3533-3568. doi: 10.1137/19M1243907. [11] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in${\mathbb R}$, Acta Math., 210 (2013), 260-318. doi: 10.1007/s11511-013-0095-9. [12] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. [13] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [14] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge Tracts in Mathematics, vol. 107, Cambridge University Press, 1993. doi: 10.1017/CBO9780511551703. [15] Z. Guo, A. Luo and W. Zou, On critical systems involving fractional Laplacian, J. Math. Anal. Appl., 446 (2017), 681-706. doi: 10.1016/j.jmaa.2016.08.069. [16] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equation, Nonlinear Anal., 28 (1997), 1633-1659. doi: 10.1016/S0362-546X(96)00021-1. [17] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346. [18] A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2. [19] G. Molica Bisci, V. Răadulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [20] S. Peng, S. We and Q. Wang, Multiple positive solutions for linearly coupled nonlinear elliptic systems with critical exponent, J. Differential Equations, 263 (2017), 709-731. doi: 10.1016/j.jde.2017.02.053. [21] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in$\mathbb{R}^N$involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22. doi: 10.4171/RMI/879. [22] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. [23] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [24] M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional$p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041. doi: 10.1016/j.jmaa.2014.11.055. [25] M. Xiang, B. Zhang and V. Răadulescu, Superlinear Schrödinger-Kirchhoff type problems involving the fractional$p$-Laplacian and critical exponent, Adv. Nonlinear Anal., 9 (2020), 690-709. doi: 10.1515/anona-2020-0021. [26] M. Zhen, J. He and H. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253. doi: 10.3934/cpaa.2019013. [27] M. Zhen, J. He, H. Xu and M. Yang, Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent, Bound. Value Probl., 96 (2018), 25 pp. doi: 10.1186/s13661-018-1016-9. [28] M. Zhen, J. He, H. Xu and M. Yang, Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent, Discrete Contin. Dyn. Syst., 39 (2019), 6523-6539. doi: 10.3934/dcds.2019283.  [1] Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126 [2] Yu Su. Ground state solution of critical Schrödinger equation with singular potential. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3347-3371. doi: 10.3934/cpaa.2021108 [3] Xi Wang, Zuhan Liu, Ling Zhou. 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