# American Institute of Mathematical Sciences

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. 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. 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