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Multiple nontrivial solutions to a $p$-Kirchhoff equation
Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents
1. | School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074, China |
2. | School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China |
3. | Department of Mathematics, Huazhong Normal University, Wuhan, 430079 |
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems,, \emph{Milan J. Math.}, 76 (2008), 257.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.
|
[3] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation,, \emph{Commun. Contemp. Math.}, 10 (2008), 1.
doi: 10.1142/S021919970800282X. |
[4] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 779.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations,, \emph{J. Math. Anal. Appl.}, 345 (2008), 90.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 11 (1998), 283.
|
[7] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations,, \emph{Rev. Math. Phys.}, 14 (2002), 409.
doi: 10.1142/S0129055X02001168. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, \emph{Proc. Amer. Math. Soc.}, 88 (1983), 486.
doi: 10.2307/2044999. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent,, \emph{Commun. Pure Appl. Math.}, 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[10] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non autonomous Schrödinger-Poisson systems,, \emph{J. Differential Equations}, 248 (2010), 521.
doi: 10.1016/j.jde.2009.06.017. |
[12] |
S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 10 (1997), 1.
|
[13] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem,, \emph{Ann. Inst. Henri Poincar\'e.}, 15 (1998), 73.
doi: 10.1016/S0294-1449(99)80021-3. |
[14] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893.
doi: http://dx.doi.org/10.1017/S030821050000353X. |
[15] |
T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation,, \emph{SIAM J. Math. Anal.}, 37 (2005), 321.
doi: 10.1137/S0036141004442793. |
[16] |
T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem,, \emph{Calc. Var. Partial Differential Equations}, 25 (2006), 105.
doi: 10.1007/s00526-005-0342-9. |
[17] |
Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, \emph{Manuscripta Math.}, 140 (2013), 51.
doi: 10.1007/s00229-011-0530-1. |
[18] |
I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.
doi: 10.1016/0022-247X(74)90025-0. |
[19] |
C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[20] |
X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations,, \emph{Z. Angew. Math. Phys.}, 5 (2011), 869.
doi: 1007/s00033-011-0120-9. |
[21] |
X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth,, \emph{J. Math. Phys.}, 53 (2012).
doi: http://dx.doi.org/10.1063/1.3683156. |
[22] |
J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: mountain pass and symmetric mountain pass approaches,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 253.
|
[23] |
Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains,, \emph{Sci. China Math.}, 57 (2014), 1927.
doi: 10.1007/s11425-014-4830-2. |
[24] |
Y. He and G. Li, Standing waves for a class of Schrödinger-Poisson equations in $\mathbbR^3$ involving critical Sobolev exponents,, to appear in \emph{Annales Academi\ae Scientiarum Fennic\ae, (). Google Scholar |
[25] |
E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 25.
doi: 10.1007/s00526-012-0509-0. |
[26] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edingburgh Sect. A}, 129 (1999), 787.
doi: http://dx.doi.org/10.1017/S0308210500013147 . |
[27] |
Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well,, \emph{J. Differential Equations}, 251 (2011), 582.
doi: 10.1016/j.jde.2011.05.006. |
[28] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations,, \emph{Ann. Acad. Sci. Fenn. A I Math.}, 15 (1990), 27.
doi: 10.5186/aasfm.1990.1521. |
[29] |
Z. Liu, S. Guo and Y. Fang, Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth,, \emph{J. Math. Phys.}, 56 (2015).
doi: http://dx.doi.org/10.1063/1.4919543. |
[30] |
G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$,, \emph{Commun. Partial Differential Equations}, (1989), 1291.
doi: 10.1080/03605308908820654. |
[31] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.
doi: 10.2307/2007032. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non. Lin\'eaire}, 2 (1984), 223.
|
[33] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I,, \emph{Rev. Mat. H. Iberoamericano 1}, 2 (1985), 145.
doi: 10.4171/RMI/6. |
[34] |
D. Mugnai, The Schrödinger-Poisson system with positive potential,, \emph{Commun. Partial Differential Equations}, 36 (2011), 1099.
doi: 10.1080/03605302.2011.558551. |
[35] |
P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.
doi: 10.1512/iumj.1986.35.35036. |
[36] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270.
doi: 10.1007/BF00946631. |
[37] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, \emph{J. Funct. Anal.}, 237 (2006), 655.
doi: 10.1016/j.jfa.2006.04.005. |
[38] |
D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases,, \emph{Arch. Rational Mech. Anal.}, 198 (2010), 349.
doi: 10.1007/s00205-010-0299-5. |
[39] |
G. Vaira, Ground states for Schrödinger-Poisson type systems,, \emph{Ricerche mat.}, 60 (2011), 263.
doi: 10.1007/s11587-011-0109-x. |
[40] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Commun. Math. Phys.}, 153 (1993), 229.
|
[41] |
M. Willem, Minimax theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[42] |
J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbbR^3$,, \emph{Calc. Var. Partial Differential Equations}, 48 (2013), 243.
doi: 10.1007/s00526-012-0548-6. |
[43] |
J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth,, \emph{J. Math. Phys.} \textbf{55} (2014), 55 (2014).
doi: http://dx.doi.org/10.1063/1.4868617. |
[44] |
L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations,, \emph{J. Math. Anal. Appl.}, 346 (2008), 155.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems,, \emph{Milan J. Math.}, 76 (2008), 257.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349.
|
[3] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation,, \emph{Commun. Contemp. Math.}, 10 (2008), 1.
doi: 10.1142/S021919970800282X. |
[4] |
A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 27 (2010), 779.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations,, \emph{J. Math. Anal. Appl.}, 345 (2008), 90.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 11 (1998), 283.
|
[7] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations,, \emph{Rev. Math. Phys.}, 14 (2002), 409.
doi: 10.1142/S0129055X02001168. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, \emph{Proc. Amer. Math. Soc.}, 88 (1983), 486.
doi: 10.2307/2044999. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent,, \emph{Commun. Pure Appl. Math.}, 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[10] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state,, \emph{Arch. Ration. Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non autonomous Schrödinger-Poisson systems,, \emph{J. Differential Equations}, 248 (2010), 521.
doi: 10.1016/j.jde.2009.06.017. |
[12] |
S. Cingolani and N. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations,, \emph{Topol. Methods Nonlinear Anal.}, 10 (1997), 1.
|
[13] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem,, \emph{Ann. Inst. Henri Poincar\'e.}, 15 (1998), 73.
doi: 10.1016/S0294-1449(99)80021-3. |
[14] |
T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 134 (2004), 893.
doi: http://dx.doi.org/10.1017/S030821050000353X. |
[15] |
T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation,, \emph{SIAM J. Math. Anal.}, 37 (2005), 321.
doi: 10.1137/S0036141004442793. |
[16] |
T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem,, \emph{Calc. Var. Partial Differential Equations}, 25 (2006), 105.
doi: 10.1007/s00526-005-0342-9. |
[17] |
Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, \emph{Manuscripta Math.}, 140 (2013), 51.
doi: 10.1007/s00229-011-0530-1. |
[18] |
I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324.
doi: 10.1016/0022-247X(74)90025-0. |
[19] |
C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[20] |
X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations,, \emph{Z. Angew. Math. Phys.}, 5 (2011), 869.
doi: 1007/s00033-011-0120-9. |
[21] |
X. He and W. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth,, \emph{J. Math. Phys.}, 53 (2012).
doi: http://dx.doi.org/10.1063/1.3683156. |
[22] |
J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbbR^N$: mountain pass and symmetric mountain pass approaches,, \emph{Topol. Methods Nonlinear Anal.}, 35 (2010), 253.
|
[23] |
Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains,, \emph{Sci. China Math.}, 57 (2014), 1927.
doi: 10.1007/s11425-014-4830-2. |
[24] |
Y. He and G. Li, Standing waves for a class of Schrödinger-Poisson equations in $\mathbbR^3$ involving critical Sobolev exponents,, to appear in \emph{Annales Academi\ae Scientiarum Fennic\ae, (). Google Scholar |
[25] |
E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere,, \emph{Calc. Var. Partial Differential Equations}, 47 (2013), 25.
doi: 10.1007/s00526-012-0509-0. |
[26] |
L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landsman-Lazer-type problem set on $\mathbbR^N$,, \emph{Proc. Roy. Soc. Edingburgh Sect. A}, 129 (1999), 787.
doi: http://dx.doi.org/10.1017/S0308210500013147 . |
[27] |
Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well,, \emph{J. Differential Equations}, 251 (2011), 582.
doi: 10.1016/j.jde.2011.05.006. |
[28] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations,, \emph{Ann. Acad. Sci. Fenn. A I Math.}, 15 (1990), 27.
doi: 10.5186/aasfm.1990.1521. |
[29] |
Z. Liu, S. Guo and Y. Fang, Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth,, \emph{J. Math. Phys.}, 56 (2015).
doi: http://dx.doi.org/10.1063/1.4919543. |
[30] |
G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$,, \emph{Commun. Partial Differential Equations}, (1989), 1291.
doi: 10.1080/03605308908820654. |
[31] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.
doi: 10.2307/2007032. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non. Lin\'eaire}, 2 (1984), 223.
|
[33] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I,, \emph{Rev. Mat. H. Iberoamericano 1}, 2 (1985), 145.
doi: 10.4171/RMI/6. |
[34] |
D. Mugnai, The Schrödinger-Poisson system with positive potential,, \emph{Commun. Partial Differential Equations}, 36 (2011), 1099.
doi: 10.1080/03605302.2011.558551. |
[35] |
P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. Math. J.}, 35 (1986), 681.
doi: 10.1512/iumj.1986.35.35036. |
[36] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270.
doi: 10.1007/BF00946631. |
[37] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, \emph{J. Funct. Anal.}, 237 (2006), 655.
doi: 10.1016/j.jfa.2006.04.005. |
[38] |
D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases,, \emph{Arch. Rational Mech. Anal.}, 198 (2010), 349.
doi: 10.1007/s00205-010-0299-5. |
[39] |
G. Vaira, Ground states for Schrödinger-Poisson type systems,, \emph{Ricerche mat.}, 60 (2011), 263.
doi: 10.1007/s11587-011-0109-x. |
[40] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Commun. Math. Phys.}, 153 (1993), 229.
|
[41] |
M. Willem, Minimax theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[42] |
J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbbR^3$,, \emph{Calc. Var. Partial Differential Equations}, 48 (2013), 243.
doi: 10.1007/s00526-012-0548-6. |
[43] |
J. Zhang, The existence and concentration of positive solutions for a nonlinear Schrödinger-Poisson system with critical growth,, \emph{J. Math. Phys.} \textbf{55} (2014), 55 (2014).
doi: http://dx.doi.org/10.1063/1.4868617. |
[44] |
L. Zhao and F. Zhao, On the existence of solutions for the Schrödinger-Poisson equations,, \emph{J. Math. Anal. Appl.}, 346 (2008), 155.
doi: 10.1016/j.jmaa.2008.04.053. |
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