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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, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[3] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., 10 (2008), 1-14.
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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[7] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[10] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.
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, Topol. Methods Nonlinear Anal., 10 (1997), 1-13. |
[13] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. Henri Poincaré., 15 (1998), 73-111.
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, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
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, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[16] |
T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations, 25 (2006), 105-137.
doi: 10.1007/s00526-005-0342-9. |
[17] |
Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.
doi: 10.1007/s00229-011-0530-1. |
[18] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[19] |
C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820.
doi: 10.1080/03605309608821208. |
[20] |
X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889.
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, J. Math. Phys., 53 (2012), 19pp.
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, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. |
[23] |
Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952.
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, ().
|
[25] |
E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54.
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 $\mathbbR^N$, Proc. Roy. Soc. Edingburgh Sect. A, 129 (1999), 787-809.
doi: http://dx.doi.org/10.1017/S0308210500013147 . |
[27] |
Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[28] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36.
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, J. Math. Phys., 56 (2015), 22pp.
doi: http://dx.doi.org/10.1063/1.4919543. |
[30] |
G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314.
doi: 10.1080/03605308908820654. |
[31] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 2 (1984) 223-283. |
[33] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I, Rev. Mat. H. Iberoamericano 1, 2 (1985), 145-201.
doi: 10.4171/RMI/6. |
[34] |
D. Mugnai, The Schrödinger-Poisson system with positive potential, Commun. Partial Differential Equations, 36 (2011), 1099-1117.
doi: 10.1080/03605302.2011.558551. |
[35] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[36] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[37] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[38] |
D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases, Arch. Rational Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[39] |
G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[40] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. |
[41] |
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 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 $\mathbbR^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.
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, J. Math. Phys. 55 (2014), 14pp.
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, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications, J. Funct. Anal., 14 (1973), 349-381. |
[3] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., 10 (2008), 1-14.
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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[5] |
A. Azzollini and A. Pomponio, Groud state solutions for nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[6] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. |
[7] |
V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[8] |
H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.2307/2044999. |
[9] |
H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic problems involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[10] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations, I existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.
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, Topol. Methods Nonlinear Anal., 10 (1997), 1-13. |
[13] |
D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. Henri Poincaré., 15 (1998), 73-111.
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, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893-906.
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, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[16] |
T. D'Aprile and J. Wei, Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem, Calc. Var. Partial Differential Equations, 25 (2006), 105-137.
doi: 10.1007/s00526-005-0342-9. |
[17] |
Y. Ding and X. Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities, Manuscripta Math., 140 (2013), 51-82.
doi: 10.1007/s00229-011-0530-1. |
[18] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[19] |
C. Gui, Existence of multi-bumb solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21 (1996), 787-820.
doi: 10.1080/03605309608821208. |
[20] |
X. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 5 (2011), 869-889.
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, J. Math. Phys., 53 (2012), 19pp.
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, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. |
[23] |
Y. He and G. Li, The existence and concentration of weak solutions to a class of $p$-Laplacian type problems in unbounded domains, Sci. China Math., 57 (2014), 1927-1952.
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, ().
|
[25] |
E. Hebey and J. Wei, Schrödinger-Poisson systems in the 3-sphere, Calc. Var. Partial Differential Equations, 47 (2013), 25-54.
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 $\mathbbR^N$, Proc. Roy. Soc. Edingburgh Sect. A, 129 (1999), 787-809.
doi: http://dx.doi.org/10.1017/S0308210500013147 . |
[27] |
Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.
doi: 10.1016/j.jde.2011.05.006. |
[28] |
G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. A I Math., 15 (1990), 27-36.
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, J. Math. Phys., 56 (2015), 22pp.
doi: http://dx.doi.org/10.1063/1.4919543. |
[30] |
G. Li and S. Yan, Eigenvalue problems for quasilinear elliptic equations on $\mathbbR^N$, Commun. Partial Differential Equations, 14 (1989), 1291-1314.
doi: 10.1080/03605308908820654. |
[31] |
E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[32] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non. Linéaire, 2 (1984) 223-283. |
[33] |
P. L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part I, Rev. Mat. H. Iberoamericano 1, 2 (1985), 145-201.
doi: 10.4171/RMI/6. |
[34] |
D. Mugnai, The Schrödinger-Poisson system with positive potential, Commun. Partial Differential Equations, 36 (2011), 1099-1117.
doi: 10.1080/03605302.2011.558551. |
[35] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[36] |
P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[37] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[38] |
D. Ruiz, On the Schrödinger-Poisson-Slater System: behavior of minimizers, radial and nonradial cases, Arch. Rational Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[39] |
G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[40] |
X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244. |
[41] |
M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 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 $\mathbbR^3$, Calc. Var. Partial Differential Equations, 48 (2013), 243-273.
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, J. Math. Phys. 55 (2014), 14pp.
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, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
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