-
Previous Article
Combined effects for non-autonomous singular biharmonic problems
- DCDS-S Home
- This Issue
-
Next Article
Hopf bifurcation of a fractional-order octonion-valued neural networks with time delays
Variational analysis for nonlocal Yamabe-type systems
1. | College of Science, Civil Aviation University of China, Tianjin 300300, China |
2. | Dipartimento di Scienze Pure e Applicate (DiSPeA), Universitá degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy |
3. | College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, 266590, China |
$\left\{ \begin{array}{*{35}{l}} \begin{align} & M\left( [u]_{s}^{2}-\mu \int_{{{\mathbb{R}}^{3}}}{V}(x)|u{{|}^{2}}dx \right)\left[ {{(-\Delta )}^{s}}u-\mu V(x)u \right]-\phi |u{{|}^{2_{s,t}^{*}-2}}u \\ & =\lambda h(x)|u{{|}^{p-2}}u+|u{{|}^{2_{s}^{*}-2}}u\quad ~~\text{in}~~~ {{\mathbb{R}}^{\text{3}}} \\ & {{(-\Delta )}^{t}}\phi =|u{{|}^{2_{s,t}^{*}}}~~~ \text{in} ~~{{\mathbb{R}}^{3}}, \\ \end{align} & \ & \ & {} \\\end{array} \right.$ |
$ (-\Delta )^s $ |
$ [u]_{s} $ |
$ u $ |
$ M:\mathbb{R}^+_0\rightarrow \mathbb{R}^+_0 $ |
$ V(x) = {|x|^{-2s}} $ |
$ 2_{s, t}^* = {(3+2t)}/{(3-2s)} $ |
$ s, t\in (0, 1) $ |
$ \lambda, \mu $ |
$ 1<p<2_s^* = {6}/{(3-2s)} $ |
$ h\in L^{{2_s^*}/{(2_s^*-p)}}(\mathbb{R}^3) $ |
$ \lambda $ |
$ \mu $ |
References:
[1] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[2] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[3] |
J. García Azorero and I. Peral Alonso,
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.
doi: 10.1090/S0002-9947-1991-1083144-2. |
[4] |
A. Azzollini and P. d'Avenia,
On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[5] |
A. Azzollini, P. d'Avenia and V. Luisi,
Generalized Schrödinger-Poisson type systems, Commun. Pure Appl. Anal., 12 (2013), 867-879.
doi: 10.3934/cpaa.2013.12.867. |
[6] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods. Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[7] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] |
A. Bongers, Existenzaussagen für die Choquard-Gleichung: Ein nichtlineares eigenwertproblem der plasma-physik, Z. Angew. Math. Mech., 60 (1980), T240–T242. |
[9] |
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[10] |
L. Caffarelli,
Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, Springer, Heidelberg, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[11] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[12] |
Y.-H. Chen and C. G. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[13] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[14] |
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. |
[15] |
P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appli. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[16] |
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. |
[17] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[18] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[19] |
O. Druet and E. Hebey,
Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831-869.
doi: 10.1142/S0219199710004007. |
[20] |
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. |
[21] |
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. |
[22] |
A. Fiscella, G. Molica Bisci and R. Servadei,
Multiplicity results for fractional Laplace problems with critical growth, Manuscripta Math., 155 (2018), 369-388.
doi: 10.1007/s00229-017-0947-2. |
[23] |
G. M. Figueiredo, G. Molica Bisci and R. Servadei,
On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361.
doi: 10.3233/ASY-151316. |
[24] |
F. Gao and M. Yang,
On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.
doi: 10.1007/s11425-016-9067-5. |
[25] |
X. M. He and W. M. Zou, Existence and concetration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[26] |
E. Hebey and P.-D. Thizy, Stationary Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math., 18 (2016), 1550028, 53 pp.
doi: 10.1142/S0219199715500285. |
[27] |
E. Hebey, F. Robert and Y. L. Wen,
Compactness and global estimates for fourth order equation of critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math., 8 (2006), 9-65.
doi: 10.1142/S0219199706002027. |
[28] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[29] |
Y. S. Jiang and H.-S. Zhou.,
Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[30] |
G. Kirchhoff, Vorlesungen Über Mathematische Physik, BG Teubner, 1876. Google Scholar |
[31] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[32] |
F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp.
doi: 10.1142/S0219199714500369. |
[33] |
E. Lieb and M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode island, 2001.
doi: 10.1090/gsm/014. |
[34] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. App. Math., 57 (1976/77), 93-105.
doi: 10.1002/sapm197757293. |
[35] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[36] |
H. D. Liu,
Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212.
doi: 10.1016/j.nonrwa.2016.04.007. |
[37] |
Z. L. Liu, Z.-Q. Wang and J. J. Zhang,
Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.
doi: 10.1007/s10231-015-0489-8. |
[38] |
J. Liu, J.-F. Liao and C.-L. Tang,
Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.
doi: 10.1016/j.jmaa.2015.04.066. |
[39] |
D. F. Lü,
A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48.
doi: 10.1016/j.na.2013.12.022. |
[40] |
V. Maz'ya and T. Shaposhnikova,
On the Bourgain, Brézis, and Mironescu theorem converning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[41] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() |
[42] |
G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp.
doi: 10.1142/S0219199715500881. |
[43] |
S. Pekar, Untersuchung Uber Die Elektronentheorie der Kristalle, Akademie Verlag, 1954. Google Scholar |
[44] |
R. Penrose,
Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[45] |
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. |
[46] |
P. Pucci, M. Q. Xiang and B. L. 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. |
[47] |
P. Pucci, M. Q. Xiang and B. L. 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. |
[48] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional $p$-Laplacian, Adv. Calc. Var., 12 (2019), 253-275.
doi: 10.1515/acv-2016-0049. |
[49] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[50] |
X. Ros-Oton and J. Serra,
Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.
doi: 10.1080/03605302.2014.918144. |
[51] |
Z. F. Shen, F. S. Gao and M. B. Yang,
Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.
doi: 10.1002/mma.3849. |
[52] |
K. M. 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. |
[53] |
D. Wu,
Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity, J. Math. Anal. Appl., 411 (2014), 530-542.
doi: 10.1016/j.jmaa.2013.09.054. |
[54] |
M. Q. Xiang, G. Molica Bisci, G. H. Tian and B. L. Zhang,
Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.
doi: 10.1088/0951-7715/29/2/357. |
[55] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.
doi: 10.1016/j.jde.2015.09.028. |
[56] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 290 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[57] |
J. J. Zhang, J. M. do Ó and M. Squassina,
Fractional Schrödinger-Poisson system with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.
doi: 10.1515/ans-2015-5024. |
[58] |
G. L. Zhao, X. L. Zhu and Y. H. Li,
Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.
doi: 10.1016/j.amc.2015.01.038. |
show all references
References:
[1] |
D. Applebaum,
Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[2] |
G. Autuori, A. Fiscella and P. Pucci,
Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.
doi: 10.1016/j.na.2015.06.014. |
[3] |
J. García Azorero and I. Peral Alonso,
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term, Trans. Amer. Math. Soc., 323 (1991), 877-895.
doi: 10.1090/S0002-9947-1991-1083144-2. |
[4] |
A. Azzollini and P. d'Avenia,
On a system involving a critically growing nonlinearity, J. Math. Anal. Appl., 387 (2012), 433-438.
doi: 10.1016/j.jmaa.2011.09.012. |
[5] |
A. Azzollini, P. d'Avenia and V. Luisi,
Generalized Schrödinger-Poisson type systems, Commun. Pure Appl. Anal., 12 (2013), 867-879.
doi: 10.3934/cpaa.2013.12.867. |
[6] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods. Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[7] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] |
A. Bongers, Existenzaussagen für die Choquard-Gleichung: Ein nichtlineares eigenwertproblem der plasma-physik, Z. Angew. Math. Mech., 60 (1980), T240–T242. |
[9] |
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[10] |
L. Caffarelli,
Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, Springer, Heidelberg, 7 (2012), 37-52.
doi: 10.1007/978-3-642-25361-4_3. |
[11] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[12] |
Y.-H. Chen and C. G. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[13] |
F. Colasuonno and P. Pucci,
Multiplicity of solutions for $p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.
doi: 10.1016/j.na.2011.05.073. |
[14] |
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. |
[15] |
P. d'Avenia, G. Siciliano and M. Squassina,
On fractional Choquard equations, Math. Models Methods Appli. Sci., 25 (2015), 1447-1476.
doi: 10.1142/S0218202515500384. |
[16] |
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. |
[17] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[18] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[19] |
O. Druet and E. Hebey,
Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces, Commun. Contemp. Math., 12 (2010), 831-869.
doi: 10.1142/S0219199710004007. |
[20] |
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. |
[21] |
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. |
[22] |
A. Fiscella, G. Molica Bisci and R. Servadei,
Multiplicity results for fractional Laplace problems with critical growth, Manuscripta Math., 155 (2018), 369-388.
doi: 10.1007/s00229-017-0947-2. |
[23] |
G. M. Figueiredo, G. Molica Bisci and R. Servadei,
On a fractional Kirchhoff-type equation via Krasnoselskii's genus, Asymptot. Anal., 94 (2015), 347-361.
doi: 10.3233/ASY-151316. |
[24] |
F. Gao and M. Yang,
On the Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.
doi: 10.1007/s11425-016-9067-5. |
[25] |
X. M. He and W. M. Zou, Existence and concetration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[26] |
E. Hebey and P.-D. Thizy, Stationary Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math., 18 (2016), 1550028, 53 pp.
doi: 10.1142/S0219199715500285. |
[27] |
E. Hebey, F. Robert and Y. L. Wen,
Compactness and global estimates for fourth order equation of critical Sobolev growth arising from conformal geometry, Commun. Contemp. Math., 8 (2006), 9-65.
doi: 10.1142/S0219199706002027. |
[28] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[29] |
Y. S. Jiang and H.-S. Zhou.,
Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[30] |
G. Kirchhoff, Vorlesungen Über Mathematische Physik, BG Teubner, 1876. Google Scholar |
[31] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E (3), 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[32] |
F. Y. Li, Y. H. Li and J. P. Shi, Existence of positive solutions to Schrödinger-Poisson type systems with critical exponent, Commun. Contemp. Math., 16 (2014), 1450036, 28 pp.
doi: 10.1142/S0219199714500369. |
[33] |
E. Lieb and M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14, AMS, Providence, Rhode island, 2001.
doi: 10.1090/gsm/014. |
[34] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. App. Math., 57 (1976/77), 93-105.
doi: 10.1002/sapm197757293. |
[35] |
P.-L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[36] |
H. D. Liu,
Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal. Real World Appl., 32 (2016), 198-212.
doi: 10.1016/j.nonrwa.2016.04.007. |
[37] |
Z. L. Liu, Z.-Q. Wang and J. J. Zhang,
Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.
doi: 10.1007/s10231-015-0489-8. |
[38] |
J. Liu, J.-F. Liao and C.-L. Tang,
Positive solutions for Kirchhoff-type equations with critical exponent in $\mathbb{R}^N$, J. Math. Anal. Appl., 429 (2015), 1153-1172.
doi: 10.1016/j.jmaa.2015.04.066. |
[39] |
D. F. Lü,
A note on Kirchhoff-type equations with Hartree-type nonlinearities, Nonlinear Anal., 99 (2014), 35-48.
doi: 10.1016/j.na.2013.12.022. |
[40] |
V. Maz'ya and T. Shaposhnikova,
On the Bourgain, Brézis, and Mironescu theorem converning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[41] |
G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016.
doi: 10.1017/CBO9781316282397.![]() ![]() |
[42] |
G. Molica Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp.
doi: 10.1142/S0219199715500881. |
[43] |
S. Pekar, Untersuchung Uber Die Elektronentheorie der Kristalle, Akademie Verlag, 1954. Google Scholar |
[44] |
R. Penrose,
Quantum computation, entanglement and state reduction, Philos. Trans. Roy. Soc., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[45] |
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. |
[46] |
P. Pucci, M. Q. Xiang and B. L. 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. |
[47] |
P. Pucci, M. Q. Xiang and B. L. 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. |
[48] |
P. Pucci, M. Q. Xiang and B. L. Zhang,
Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional $p$-Laplacian, Adv. Calc. Var., 12 (2019), 253-275.
doi: 10.1515/acv-2016-0049. |
[49] |
P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65. Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 1986.
doi: 10.1090/cbms/065. |
[50] |
X. Ros-Oton and J. Serra,
Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.
doi: 10.1080/03605302.2014.918144. |
[51] |
Z. F. Shen, F. S. Gao and M. B. Yang,
Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci., 39 (2016), 4082-4098.
doi: 10.1002/mma.3849. |
[52] |
K. M. 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. |
[53] |
D. Wu,
Existence and stability of standing waves for nonlinear fractional Schrödinger equations with Hartree type nonlinearity, J. Math. Anal. Appl., 411 (2014), 530-542.
doi: 10.1016/j.jmaa.2013.09.054. |
[54] |
M. Q. Xiang, G. Molica Bisci, G. H. Tian and B. L. Zhang,
Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.
doi: 10.1088/0951-7715/29/2/357. |
[55] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Existence of solutions for perturbed fractional $p$-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413.
doi: 10.1016/j.jde.2015.09.028. |
[56] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 290 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[57] |
J. J. Zhang, J. M. do Ó and M. Squassina,
Fractional Schrödinger-Poisson system with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.
doi: 10.1515/ans-2015-5024. |
[58] |
G. L. Zhao, X. L. Zhu and Y. H. Li,
Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.
doi: 10.1016/j.amc.2015.01.038. |
[1] |
Shaodong Wang. Infinitely many blowing-up solutions for Yamabe-type problems on manifolds with boundary. Communications on Pure & Applied Analysis, 2018, 17 (1) : 209-230. doi: 10.3934/cpaa.2018013 |
[2] |
L. Brandolini, M. Rigoli and A. G. Setti. On the existence of positive solutions of Yamabe-type equations on the Heisenberg group. Electronic Research Announcements, 1996, 2: 101-107. |
[3] |
Guozhen Lu and Juncheng Wei. On positive entire solutions to the Yamabe-type problem on the Heisenberg and stratified groups. Electronic Research Announcements, 1997, 3: 83-89. |
[4] |
Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008 |
[5] |
Qiang Lin, Xueteng Tian, Runzhang Xu, Meina Zhang. Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020160 |
[6] |
Andrea Tellini. Imperfect bifurcations via topological methods in superlinear indefinite problems. Conference Publications, 2015, 2015 (special) : 1050-1059. doi: 10.3934/proc.2015.1050 |
[7] |
Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857 |
[8] |
Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171 |
[9] |
Yan Hu. Layer solutions for an Allen-Cahn type system driven by the fractional Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 947-964. doi: 10.3934/cpaa.2016.15.947 |
[10] |
Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $ p $–Laplacian type problems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020151 |
[11] |
Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013 |
[12] |
Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 |
[13] |
Qi S. Zhang. Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem. Electronic Research Announcements, 1997, 3: 45-51. |
[14] |
Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113 |
[15] |
Omid S. Fard, Javad Soolaki, Delfim F. M. Torres. A necessary condition of Pontryagin type for fuzzy fractional optimal control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 59-76. doi: 10.3934/dcdss.2018004 |
[16] |
Wenxiong Chen, Shijie Qi. Direct methods on fractional equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1269-1310. doi: 10.3934/dcds.2019055 |
[17] |
Andrea Malchiodi. Topological methods for an elliptic equation with exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 277-294. doi: 10.3934/dcds.2008.21.277 |
[18] |
Zalman Balanov, Carlos García-Azpeitia, Wieslaw Krawcewicz. On variational and topological methods in nonlinear difference equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2813-2844. doi: 10.3934/cpaa.2018133 |
[19] |
Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295 |
[20] |
Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483 |
2018 Impact Factor: 0.545
Tools
Metrics
Other articles
by authors
[Back to Top]