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Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data
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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 |
$ \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}, $ |
$ (-\Delta)^{s} $ |
$ 0<s<1 $ |
$ \mu_{1},\, \mu_{2}>0 $ |
$ N>2s $ |
$ \frac{4s}{N}<\alpha\leq \frac{2s}{N-2s} $ |
$ \beta>0 $ |
$ \beta_{1}>0 $ |
$ 0<\beta<\beta_{1} $ |
$ \beta_{2}>0 $ |
$ \beta>\beta_{2} $ |
$ \beta $ |
$ N = 3 $ |
$ s = 1 $ |
$ \alpha = 2 $ |
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. |
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