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Time discretization of a nonlinear phase field system in general domains
Ground states for asymptotically periodic fractional Kirchhoff equation with critical Sobolev exponent
1. | School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
2. | Department of Mathematics, University of Texas at San Antonio, San Antonio, 78249 Texas, USA |
$ \Big(a+b\int_{\mathbb{R}^3}| (-\Delta)^{\frac{s}{2}} u|^2dx\Big) (-\Delta )^su+V(x) u = K(x)|u|^{2_s^*-2}u+\lambda g(x,u), \; \text{in}\; \mathbb{R}^3, $ |
$ a,b>0 $ |
$ \lambda>0 $ |
$ (-\Delta )^s $ |
$ s\in(\frac{3}{4},1) $ |
$ 2_s^* = \frac{6}{3-2s} $ |
$ V,K $ |
$ g $ |
$ x $ |
References:
[1] |
C. O. Alves and G. M. Figueiredo,
Nonlinear perturbations of a periodic kirchhoff equation in $\mathbb{R}^N$, Nonlinear Anal., 75 (2012), 2750-2759.
doi: 10.1016/j.na.2011.11.017. |
[2] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645.
|
[3] |
A. Arosio and S. Panizzi,
On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.
doi: 10.1090/S0002-9947-96-01532-2. |
[4] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[5] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[6] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana
doi: 10.1007/978-3-319-28739-3. |
[7] |
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. |
[8] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. (4), 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[9] |
S. Chen and X. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[10] |
S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 40, 23.
doi: 10.1007/s00030-018-0531-9. |
[11] |
S. Chen and X. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111.
doi: 10.1016/j.jmaa.2018.12.037. |
[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] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$
doi: 10.1007/978-88-7642-601-8. |
[14] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[15] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior,
Existence and concentration result for the kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[16] |
G. M. Figueiredo and J. R. Santos Júnior, Existence of a least energy nodal solution for a schrödinger-kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506.
doi: 10.1063/1.4921639. |
[17] |
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. |
[18] |
F. Gazzola and M. Lazzarino,
Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50.
|
[19] |
G. Gu, W. Zhang and F. Zhao,
Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55.
doi: 10.1016/j.aml.2018.04.010. |
[20] |
G. Gu, W. Zhang and F. Zhao,
Infinitely many sign-changing solutions for a nonlocal problem, Ann. Mat. Pura Appl., 197 (2018), 1429-14448.
doi: 10.1007/s10231-018-0731-2. |
[21] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[22] |
Y. He, G. Li and S. Peng,
Concentrating bound states for Kirchhoff type problems in $\Bbb R^3$ involving critical Sobolev exponents, Adv. Nonlinear Studies, 14 (2014), 483-510.
doi: 10.1515/ans-2014-0214. |
[23] |
G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883. |
[24] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[25] |
Q. Li, K. Teng and X. Wu, Ground states for fractional schrödinger equations with critical growth, J. Math. Phys., 59 (2018), 033504.
doi: 10.1063/1.5008662. |
[26] |
S. Liang and J. Zhang, Multiplicity of solutions for the noncooperative schrödinger-kirchhoff system involving the fractional p-laplacian in $\mathbb{R}^N$, Z. Angew. Math. Phys., 68 (2017), 63.
doi: 10.1007/s00033-017-0805-9. |
[27] |
J. L. Lions,, On Some Questions in Boundary Value Problems of Mathematical Physics, volume 30 of North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978. |
[28] |
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), no.4, Art. 50, 32.
doi: 10.1007/s00030-017-0473-7. |
[29] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, volume 162 of Encyclopedia of Mathematics and its Applications.
doi: 10.1017/CBO9781316282397. |
[30] |
G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088.
doi: 10.1142/S0219199715500881. |
[31] |
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 Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[32] |
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. |
[33] |
R. Servadei and E. Valdinoci,
Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[34] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[35] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
|
[36] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 36 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[37] |
A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010. |
[38] |
X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110 pp.1-25.
doi: 10.1007/s00526-017-1214-9. |
[39] |
X. Tang and B. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[40] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[41] |
M. Willem, Minimax Theorems, volume 24 of Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[42] |
H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 56 (2015), 091502.
doi: 10.1063/1.4929660. |
[43] |
J. Zhang, Z. Lou, Y. Ji and W. Shao,
Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 462 (2018), 57-83.
doi: 10.1016/j.jmaa.2018.01.060. |
show all references
References:
[1] |
C. O. Alves and G. M. Figueiredo,
Nonlinear perturbations of a periodic kirchhoff equation in $\mathbb{R}^N$, Nonlinear Anal., 75 (2012), 2750-2759.
doi: 10.1016/j.na.2011.11.017. |
[2] |
V. Ambrosio and T. Isernia,
Concentration phenomena for a fractional schrödinger-kirchhoff type equation, Math. Meth. Appl. Sci., 41 (2018), 15-645.
|
[3] |
A. Arosio and S. Panizzi,
On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.
doi: 10.1090/S0002-9947-96-01532-2. |
[4] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[5] |
V. Benci and G. Cerami,
The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93.
doi: 10.1007/BF00375686. |
[6] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, volume 20 of Lecture Notes of the Unione Matematica Italiana
doi: 10.1007/978-3-319-28739-3. |
[7] |
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. |
[8] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl. (4), 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[9] |
S. Chen and X. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin. Dyn. Syst., 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[10] |
S. Chen, X. Tang and F. Liao, Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions, NoDEA Nonlinear Differential Equations Appl., 25 (2018), Art. 40, 23.
doi: 10.1007/s00030-018-0531-9. |
[11] |
S. Chen and X. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111.
doi: 10.1016/j.jmaa.2018.12.037. |
[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] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^n$
doi: 10.1007/978-88-7642-601-8. |
[14] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[15] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior,
Existence and concentration result for the kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[16] |
G. M. Figueiredo and J. R. Santos Júnior, Existence of a least energy nodal solution for a schrödinger-kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506.
doi: 10.1063/1.4921639. |
[17] |
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. |
[18] |
F. Gazzola and M. Lazzarino,
Existence results for general critical growth semilinear elliptic equations, Commun. Appl. Anal., 4 (2000), 39-50.
|
[19] |
G. Gu, W. Zhang and F. Zhao,
Infinitely many positive solutions for a nonlocal problem, Appl. Math. Lett., 84 (2018), 49-55.
doi: 10.1016/j.aml.2018.04.010. |
[20] |
G. Gu, W. Zhang and F. Zhao,
Infinitely many sign-changing solutions for a nonlocal problem, Ann. Mat. Pura Appl., 197 (2018), 1429-14448.
doi: 10.1007/s10231-018-0731-2. |
[21] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[22] |
Y. He, G. Li and S. Peng,
Concentrating bound states for Kirchhoff type problems in $\Bbb R^3$ involving critical Sobolev exponents, Adv. Nonlinear Studies, 14 (2014), 483-510.
doi: 10.1515/ans-2014-0214. |
[23] |
G. Kirchhoff,, Vorlesungen über Mechanik, Birkhäuser Basel, 1883. |
[24] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[25] |
Q. Li, K. Teng and X. Wu, Ground states for fractional schrödinger equations with critical growth, J. Math. Phys., 59 (2018), 033504.
doi: 10.1063/1.5008662. |
[26] |
S. Liang and J. Zhang, Multiplicity of solutions for the noncooperative schrödinger-kirchhoff system involving the fractional p-laplacian in $\mathbb{R}^N$, Z. Angew. Math. Phys., 68 (2017), 63.
doi: 10.1007/s00033-017-0805-9. |
[27] |
J. L. Lions,, On Some Questions in Boundary Value Problems of Mathematical Physics, volume 30 of North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1978. |
[28] |
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), no.4, Art. 50, 32.
doi: 10.1007/s00030-017-0473-7. |
[29] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, volume 162 of Encyclopedia of Mathematics and its Applications.
doi: 10.1017/CBO9781316282397. |
[30] |
G. Molica Bisci and L. Vilasi, On a fractional degenerate kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088.
doi: 10.1142/S0219199715500881. |
[31] |
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 Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5. |
[32] |
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. |
[33] |
R. Servadei and E. Valdinoci,
Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[34] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
doi: 10.3934/dcds.2013.33.2105. |
[35] |
R. Servadei and E. Valdinoci,
Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154.
|
[36] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 36 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[37] |
A. Szulkin and T. Weth,, The methods of Nehari manifold, Handbook of Nonconvex Analysis ans Applications. International Press, Boston, 2010. |
[38] |
X. Tang and S. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110 pp.1-25.
doi: 10.1007/s00526-017-1214-9. |
[39] |
X. Tang and B. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[40] |
K. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[41] |
M. Willem, Minimax Theorems, volume 24 of Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[42] |
H. Zhang, J. Xu and F. Zhang, Existence and multiplicity of solutions for superlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 56 (2015), 091502.
doi: 10.1063/1.4929660. |
[43] |
J. Zhang, Z. Lou, Y. Ji and W. Shao,
Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 462 (2018), 57-83.
doi: 10.1016/j.jmaa.2018.01.060. |
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