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

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

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

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.

[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, ().

[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, ().

[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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