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A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents
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. | College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao, 266590, China |
$ \mathbb{R}^N $ |
$ \begin{equation*} \begin{cases} \left(a_{1}+b_{1}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u +\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u,\\ \left(a_{2}+b_{2}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx\right)(-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+ \frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\\ \end{cases} \end{equation*} $ |
$ (-\Delta)^{s} $ |
$ 0<s<1 $ |
$ N>2s, $ |
$ 2_{s}^{\ast} = 2N/(N-2s) $ |
$ \mu_{1},\mu_{2},\gamma, k>0 $ |
$ \alpha+\beta = 2_{s}^{\ast},\ 1<p<2_{s}^{\ast}-1 $ |
$ a_{i},b_{i}\geq 0, $ |
$ a_{i}+b_{i}>0,\ \ i = 1,2 $ |
$ \begin{equation*} \begin{cases} (-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u+\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u, \ \ x\in \mathbb{R}^N, \\ (-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+\frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\ \ x\in \mathbb{R}^N,\\ \lambda_{1}^{s}-a_{1}-b_{1}\lambda_{1}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx = 0, \ \ \lambda_{1}\in \mathbb{R}^+,\\ \lambda_{2}^{s}-a_{2}-b_{2}\lambda_{2}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx = 0, \ \ \lambda_{2}\in \mathbb{R}^+. \end{cases} \end{equation*} $ |
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Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal., 6 (2017), 85-93.
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G. M. Bisci,
Sequence of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 241-253.
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H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
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Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[8] |
Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.
doi: 10.1007/s00526-014-0717-x. |
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A. Cotsiolis and N. K. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
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Y. Ding, F. Gao and M. Yang,
Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.
doi: 10.1088/1361-6544/aba88d. |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
L. Du and M. Yang,
Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847-5866.
doi: 10.3934/dcds.2019219. |
[14] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[15] |
A. Fiscella and P. Pucci,
$p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.
doi: 10.1016/j.nonrwa.2016.11.004. |
[16] |
F. Gao, E. Silva, M. Yang and J. Zhou,
Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.
doi: 10.1017/prm.2018.131. |
[17] |
Z. Guo, S. 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. |
[18] |
X. He and W. Zou,
Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.
doi: 10.1007/s11425-017-9399-6. |
[19] |
P. Han,
The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257.
|
[20] |
Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 32 pp.
doi: 10.1007/s00030-017-0473-7. |
[21] |
D. Lü and S. Peng,
Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differential Equation, 263 (2017), 8947-8978.
doi: 10.1016/j.jde.2017.08.062. |
[22] |
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() ![]() |
[23] |
A. Mellet, S. Mischler and C. Mouhotg,
Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.
doi: 10.1007/s00205-010-0354-2. |
[24] |
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differetial Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[25] |
P. Pucci, M. Xiang and B. Zhang,
Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.
doi: 10.1515/anona-2015-0102. |
[26] |
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. |
[27] |
R. Servadei and E. Valdinoci,
The Brézis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[28] |
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. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
K. Wu and F. Zhou,
Nodal solutions for a Kirchhoff type problem in $\mathbb{R}^N$, Appl. Math. Lett., 88 (2019), 58-63.
doi: 10.1016/j.aml.2018.08.008. |
[31] |
M. Xiang, B. Zhang and V. Rădulescu,
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. |
[32] |
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. |
[33] |
M. Zhen, B. Zhang and V. Rădulescu,
Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653-2676.
doi: 10.3934/dcds.2020379. |
[34] |
F. Zhou and M. Yang, Solutions for a Kirchhoff type problem with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 494 (2021), 124638, 7pp.
doi: 10.1016/j. jmaa. 2020.124638. |
show all references
References:
[1] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[2] |
V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity, Commun. Contemp. Math., 20 (2018), 1750054, 17pp.
doi: 10.1142/S0219199717500547. |
[3] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure. Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[4] |
S. Baraket and G. Molica Bisci,
Multiplicity results for elliptic Kirchhoff-type problems, Adv. Nonlinear Anal., 6 (2017), 85-93.
doi: 10.1515/anona-2015-0168. |
[5] |
G. M. Bisci,
Sequence of weak solutions for fractional equations, Math. Res. Lett., 21 (2014), 241-253.
doi: 10.4310/MRL.2014.v21.n2.a3. |
[6] |
H. Brézis and E. Lieb,
A relation between pointwise convergence of functions and functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[7] |
Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8. |
[8] |
Z. Chen and W. Zou,
Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.
doi: 10.1007/s00526-014-0717-x. |
[9] |
A. Cotsiolis and N. K. Tavoularis,
Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[10] |
P. D'Ancona and S. Spagnolo,
Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent Math, 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[11] |
Y. Ding, F. Gao and M. Yang,
Semiclassical states for Choquard type equations with critical growth: Critical frequency case, Nonlinearity, 33 (2020), 6695-6728.
doi: 10.1088/1361-6544/aba88d. |
[12] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[13] |
L. Du and M. Yang,
Uniqueness and nondegeneracy of solutions for a critical nonlocal equation, Discrete Contin. Dyn. Syst., 39 (2019), 5847-5866.
doi: 10.3934/dcds.2019219. |
[14] |
A. Fiscella and E. Valdinoci,
A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.
doi: 10.1016/j.na.2013.08.011. |
[15] |
A. Fiscella and P. Pucci,
$p$-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.
doi: 10.1016/j.nonrwa.2016.11.004. |
[16] |
F. Gao, E. Silva, M. Yang and J. Zhou,
Existence of solutions for critical Choquard equations via the concentration-compactness method, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 921-954.
doi: 10.1017/prm.2018.131. |
[17] |
Z. Guo, S. 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. |
[18] |
X. He and W. Zou,
Ground state solutions for a class of fractional Kirchhoff equations with critical growth, Sci. China Math., 62 (2019), 853-890.
doi: 10.1007/s11425-017-9399-6. |
[19] |
P. Han,
The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257.
|
[20] |
Z. Liu, M. Squassina and J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, NoDEA Nonlinear Differential Equations Appl., 24 (2017), 32 pp.
doi: 10.1007/s00030-017-0473-7. |
[21] |
D. Lü and S. Peng,
Existence and asymptotic behavior of vector solutions for coupled nonlinear Kirchhoff-type system, J. Differential Equation, 263 (2017), 8947-8978.
doi: 10.1016/j.jde.2017.08.062. |
[22] |
G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, vol. 162, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() ![]() |
[23] |
A. Mellet, S. Mischler and C. Mouhotg,
Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.
doi: 10.1007/s00205-010-0354-2. |
[24] |
P. Pucci, M. Xiang and B. Zhang,
Multiple solutions for nonhomogeneous Schrödinger–Kirchhoff type equations involving the fractional $p$-Laplacian in ${\mathbb {R}}^N$, Calc. Var. Partial Differetial Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[25] |
P. Pucci, M. Xiang and B. Zhang,
Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.
doi: 10.1515/anona-2015-0102. |
[26] |
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. |
[27] |
R. Servadei and E. Valdinoci,
The Brézis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[28] |
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. |
[29] |
M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[30] |
K. Wu and F. Zhou,
Nodal solutions for a Kirchhoff type problem in $\mathbb{R}^N$, Appl. Math. Lett., 88 (2019), 58-63.
doi: 10.1016/j.aml.2018.08.008. |
[31] |
M. Xiang, B. Zhang and V. Rădulescu,
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. |
[32] |
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. |
[33] |
M. Zhen, B. Zhang and V. Rădulescu,
Normalized solutions for nonlinear coupled fractional systems: low and high perturbations in the attractive case, Discrete Contin. Dyn. Syst., 41 (2021), 2653-2676.
doi: 10.3934/dcds.2020379. |
[34] |
F. Zhou and M. Yang, Solutions for a Kirchhoff type problem with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 494 (2021), 124638, 7pp.
doi: 10.1016/j. jmaa. 2020.124638. |
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