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  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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  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. BartschX. 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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar
[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  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. BartschX. 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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar
[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119

[2]

Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

[3]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[5]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[8]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[9]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[10]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263

[11]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[12]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[13]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[14]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[15]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[16]

Guangbin CAI, Yang Zhao, Wanzhen Quan, Xiusheng Zhang. Design of LPV fault-tolerant controller for hypersonic vehicle based on state observer. Journal of Industrial & Management Optimization, 2021, 17 (1) : 447-465. doi: 10.3934/jimo.2019120

[17]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261

[18]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[19]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[20]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (30)
  • HTML views (53)
  • Cited by (0)

[Back to Top]