Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

Yi  He; Lu  Lu; Wei Shuai

doi:10.3934/cpaa.2016.15.103

Previous Article
Riemann problem for the relativistic generalized Chaplygin Euler equations
CPAA Home
This Issue
Next Article
Multiple nontrivial solutions to a $p$-Kirchhoff equation

January 2016, 15(1): 103-125. doi: 10.3934/cpaa.2016.15.103

Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

Yi He ^1, , Lu Lu ^2, and Wei Shuai ^3,

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

Received April 2015 Revised August 2015 Published December 2015

Abstract
Related Papers

We are concerned with standing waves for the following Schrödinger-Poisson equation with critical nonlinearity: \begin{eqnarray} && - {\varepsilon ^2}\Delta u + V(x)u + \psi (x)u = \lambda W(x){\left| u \right|^{p - 2}}u + {\left| u \right|^4}u\;\;{\text{ in }}\mathbb{R}^3, \\ && - {\varepsilon ^2}\Delta \psi = {u^2}\;\;{\text{ in }}\mathbb{R}^3, u>0, u \in {H^1}(\mathbb{R}^3), \end{eqnarray} where $\varepsilon $ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$, $V$ and $W$ are two potentials. Under proper assumptions, we prove that for $\varepsilon > 0$ sufficiently small, the above problem has a positive ground-state solution ${u_\varepsilon }$ by using a monotonicity trick and a new version of global compactness lemma. Moreover, we use another global compactness method due to [C. Gui, Commun. Partial Differential Equations 21 (1996) 787-820] to show that ${u_\varepsilon }$ concentrates around a set which is related to the set where the potential $V(x)$ attains its global minima or the set where the potential $W(x)$ attains its global maxima as $\varepsilon \to 0$. As far as we know, the existence and concentration behavior of the positive solutions to the Schrödinger-Poisson equation with critical nonlinearity $g(u): = \lambda W(x)|u{|^{p - 2}}u + |u{|^4}u$ $(3

Keywords: critical growth., Schrödinger-Poisson equation, Existence, concentration.

Mathematics Subject Classification: Primary: 35J20, 35J60, 35J9.

Citation: Yi He, Lu Lu, Wei Shuai. Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents. Communications on Pure & Applied Analysis, 2016, 15 (1) : 103-125. doi: 10.3934/cpaa.2016.15.103

References:

[1]	A. Ambrosetti, On Schrödinger-Poisson systems,, \emph{Milan J. Math.}, 76 (2008), 257. doi: 10.1007/s00032-008-0094-z. Google Scholar
[2]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 11 (1998), 283. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 . Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	D. Mugnai, The Schrödinger-Poisson system with positive potential,, \emph{Commun. Partial Differential Equations}, 36 (2011), 1099. doi: 10.1080/03605302.2011.558551. Google Scholar
[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. Google Scholar
[36]	P. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	G. Vaira, Ground states for Schrödinger-Poisson type systems,, \emph{Ricerche mat.}, 60 (2011), 263. doi: 10.1007/s11587-011-0109-x. Google Scholar
[40]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Commun. Math. Phys.}, 153 (1993), 229. Google Scholar
[41]	M. Willem, Minimax theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical points theory and applications,, \emph{J. Funct. Anal.}, 14 (1973), 349. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Topol. Methods Nonlinear Anal.}, 11 (1998), 283. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	I. Ekeland, On the variational principle,, \emph{J. Math. Anal. Appl.}, 47 (1974), 324. doi: 10.1016/0022-247X(74)90025-0. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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 . Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[34]	D. Mugnai, The Schrödinger-Poisson system with positive potential,, \emph{Commun. Partial Differential Equations}, 36 (2011), 1099. doi: 10.1080/03605302.2011.558551. Google Scholar
[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. Google Scholar
[36]	P. Rabinowitz, On a class of nonlinear Schrödinger equations,, \emph{Z. Angew. Math. Phys.}, 43 (1992), 270. doi: 10.1007/BF00946631. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	G. Vaira, Ground states for Schrödinger-Poisson type systems,, \emph{Ricerche mat.}, 60 (2011), 263. doi: 10.1007/s11587-011-0109-x. Google Scholar
[40]	X. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Commun. Math. Phys.}, 153 (1993), 229. Google Scholar
[41]	M. Willem, Minimax theorems,, Progress in Nonlinear Differential Equations and their Applications, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Xu Zhang, Shiwang Ma, Qilin Xie. Bound state solutions of Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 605-625. doi: 10.3934/dcds.2017025
[2]	Lun Guo, Wentao Huang, Huifang Jia. Ground state solutions for the fractional Schrödinger-Poisson systems involving critical growth in $ \mathbb{R} ^{3} $. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1663-1693. doi: 10.3934/cpaa.2019079
[3]	Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867
[4]	Claudianor O. Alves, Minbo Yang. Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5881-5910. doi: 10.3934/dcds.2016058
[5]	Sitong Chen, Xianhua Tang. Existence of ground state solutions for the planar axially symmetric Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4685-4702. doi: 10.3934/dcdsb.2018329
[6]	Vincenzo Ambrosio. Concentration phenomena for critical fractional Schrödinger systems. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2085-2123. doi: 10.3934/cpaa.2018099
[7]	Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076
[8]	Marius Ghergu, Gurpreet Singh. On a class of mixed Choquard-Schrödinger-Poisson systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 297-309. doi: 10.3934/dcdss.2019021
[9]	Pierre-Damien Thizy. Schrödinger-Poisson systems in $4$-dimensional closed manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2257-2284. doi: 10.3934/dcds.2016.36.2257
[10]	Chunhua Wang, Jing Yang. Positive solutions for a nonlinear Schrödinger-Poisson system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5461-5504. doi: 10.3934/dcds.2018241
[11]	Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang. Standing waves for Schrödinger-Poisson system with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 6103-6129. doi: 10.3934/dcds.2019266
[12]	Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119
[13]	César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535
[14]	Yong-Yong Li, Yan-Fang Xue, Chun-Lei Tang. Ground state solutions for asymptotically periodic modified Schr$ \ddot{\mbox{o}} $dinger-Poisson system involving critical exponent. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2299-2324. doi: 10.3934/cpaa.2019104
[15]	Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[16]	Claudianor Oliveira Alves, M. A.S. Souto. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Communications on Pure & Applied Analysis, 2002, 1 (3) : 417-431. doi: 10.3934/cpaa.2002.1.417
[17]	Daniele Cassani, João Marcos do Ó, Abbas Moameni. Existence and concentration of solitary waves for a class of quasilinear Schrödinger equations. Communications on Pure & Applied Analysis, 2010, 9 (2) : 281-306. doi: 10.3934/cpaa.2010.9.281
[18]	Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457
[19]	Mingzheng Sun, Jiabao Su, Leiga Zhao. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 427-440. doi: 10.3934/dcds.2015.35.427
[20]	Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411

2018 Impact Factor: 0.925

American Institute of Mathematical Sciences

Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Concentrating ground-state solutions for a class of Schödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors