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*} $
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 $
.
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. BarriosE. ColoradoA. 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. BartschL. 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. BartschX. Zhong and W. Zou, Normalized solutions for a coupled Schrödinger system, Math. Ann., (2020).  doi: 10.1007/s00208-020-02000-w.

[6]

J. BellazziniL. 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. FrankE. 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. GuoA. 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. MelletS. 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 BisciV. 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. PengS. 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. XiangB. 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. XiangB. 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. ZhenJ. He and H. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253.  doi: 10.3934/cpaa.2019013.

[27]

M. ZhenJ. HeH. 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. ZhenJ. HeH. 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. BarriosE. ColoradoA. 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. BartschL. 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. BartschX. Zhong and W. Zou, Normalized solutions for a coupled Schrödinger system, Math. Ann., (2020).  doi: 10.1007/s00208-020-02000-w.

[6]

J. BellazziniL. 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. FrankE. 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. GuoA. 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. MelletS. 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 BisciV. 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. PengS. 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. XiangB. 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. XiangB. 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. ZhenJ. He and H. Xu, Critical system involving fractional Laplacian, Commun. Pure Appl. Anal., 18 (2019), 237-253.  doi: 10.3934/cpaa.2019013.

[27]

M. ZhenJ. HeH. 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. ZhenJ. HeH. 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.

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