American Institute of Mathematical Sciences

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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

Maoding Zhen 1, , Binlin Zhang 2, and  Vicenţiu D. Rădulescu 3,4,,

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*} $
satisfying the additional conditions
$ \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<s<1 $
,
$ \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 $
.
Keywords: fractional Laplacian, normalized solution, ground state solution.
Mathematics Subject Classification: Primary: 35J50, 35B33, 35R11, 58E05.
Citation: Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[17]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[17]

J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Mathematics Studies, 30 (1978), 284-346.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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