# American Institute of Mathematical Sciences

March 2018, 17(2): 605-626. doi: 10.3934/cpaa.2018033

## Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity

 1 School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China 2 School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, China

Received  May 2017 Revised  July 2017 Published  March 2018

In the present paper the following Kirchhoff-Schrödinger-Poisson system is studied:
 $\left\{ \begin{gathered} - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}{\text{d}}x} } \right)\Delta u + \mu \phi \left( x \right)u =f\left( u \right)\;\;\;&{\text{in}}\;\;{{\mathbb{R}}^3}, \hfill \\ - \Delta \phi =\mu {u^2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&{\text{in}}\;\;{{\mathbb{R}}^3}, \hfill \\ \end{gathered} \right.$
where
 $a>0,b≥q0$
are constants and
 $μ>0$
is a parameter,
 $f∈ C(\mathbb{R},\mathbb{R})$
. Without assuming the Ambrosetti-Rabinowitz type condition and monotonicity condition on
 $f$
, we establish the existence of positive radial solutions for the above system by using variational methods combining a monotonicity approach with a delicate cut-off technique. We also study the asymptotic behavior of solutions with respect to the parameter
 $μ$
. In addition, we obtain the existence of multiple solutions for the nonhomogeneous case corresponding to the above problem. Our results improve and generalize some known results in the literature.
Citation: Dengfeng Lü. Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (2) : 605-626. doi: 10.3934/cpaa.2018033
##### References:
 [1] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. [2] C. O. Alves, F. J. S. A. Correa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. [3] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [4] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763. doi: 10.1016/j.jde.2010.07.007. [5] A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. I. H. Poincaré-AN, 27 (2010), 779-791. doi: 10.1016/j.anihpc.2009.11.012. [6] A. Azzollini, P. d'Avenia and A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3. [7] A. Azzollini, The elliptic Kirchhoff equation in $\mathbb{R}^{N}$ perturbed by a local nonlinearity, Differential Integral Equations, 25 (2012), 543-554. doi: 10.1142/S0219199714500394. [8] C. Batkam and J. R. S. Júnior, Schrödinger-Kirchhoff-Poisson type systems, Commun. Pure Appl. Anal., 15 (2016), 429-444. doi: 10.3934/cpaa.2016.15.429. [9] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonl. Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [11] S. Chen and C. Tang, Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on $\mathbb{R}^{3}$, Nonlinear Differ. Equ. Appl., 17 (2010), 559-574. doi: 10.1007/s00030-010-0068-z. [12] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data Invent. Math. 108 (1992), 247–262. doi: 10.1007/BF02100605. [13] 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: 10.1017/S030821050000353X. [14] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. doi: 10.1515/ans-2004-0305. [15] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [16] G. M. Figueiredo, N. Ikoma and J. R. S. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [17] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [18] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. [19] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $R^{N}$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [20] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [21] L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. [22] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^{N}$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1. [23] Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3}$, Nonlinear Anal., 83 (2013), 50-57. doi: 10.1016/j.na.2013.01.006. [24] H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437. doi: 10.1515/ans-2007-0305. [25] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [26] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [27] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3}$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [28] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. doi: 10.1016/S0304-0208(08)70870-3. [29] 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. [30] A. Salvatore, Multiple solitary waves for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3}$, Adv. Nonlinear Stud., 6 (2006), 157-169. doi: 10.1515/ans-2006-0203. [31] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. [32] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. [33] J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401. doi: 10.1016/j.na.2012.07.008. [34] J. Zhang, J. Marcos do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity Commun. Contemp. Math. (2016), 1650028, 16 pp. doi: 10.1142/S0219199716500280. [35] G. Zhao, X. Zhu and Y. Li, Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581. doi: 10.1016/j.amc.2015.01.038.

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##### References:
 [1] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X. [2] C. O. Alves, F. J. S. A. Correa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008. [3] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108. doi: 10.1016/j.jmaa.2008.03.057. [4] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763. doi: 10.1016/j.jde.2010.07.007. [5] A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. I. H. Poincaré-AN, 27 (2010), 779-791. doi: 10.1016/j.anihpc.2009.11.012. [6] A. Azzollini, P. d'Avenia and A. Pomponio, Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523. doi: 10.1007/s10231-010-0160-3. [7] A. Azzollini, The elliptic Kirchhoff equation in $\mathbb{R}^{N}$ perturbed by a local nonlinearity, Differential Integral Equations, 25 (2012), 543-554. doi: 10.1142/S0219199714500394. [8] C. Batkam and J. R. S. Júnior, Schrödinger-Kirchhoff-Poisson type systems, Commun. Pure Appl. Anal., 15 (2016), 429-444. doi: 10.3934/cpaa.2016.15.429. [9] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonl. Anal., 11 (1998), 283-293. doi: 10.12775/TMNA.1998.019. [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555. [11] S. Chen and C. Tang, Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on $\mathbb{R}^{3}$, Nonlinear Differ. Equ. Appl., 17 (2010), 559-574. doi: 10.1007/s00030-010-0068-z. [12] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data Invent. Math. 108 (1992), 247–262. doi: 10.1007/BF02100605. [13] 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: 10.1017/S030821050000353X. [14] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322. doi: 10.1515/ans-2004-0305. [15] P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192. [16] G. M. Figueiredo, N. Ikoma and J. R. S. Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979. doi: 10.1007/s00205-014-0747-8. [17] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differential Equations, 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035. [18] Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106. doi: 10.1007/s00526-015-0894-2. [19] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $R^{N}$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276. [20] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $R^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [21] L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840. [22] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $R^{N}$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408. doi: 10.1090/S0002-9939-02-06821-1. [23] Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3}$, Nonlinear Anal., 83 (2013), 50-57. doi: 10.1016/j.na.2013.01.006. [24] H. Kikuchi, Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437. doi: 10.1515/ans-2007-0305. [25] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [26] Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017. [27] G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3}$, J. Differential Equations, 257 (2014), 566-600. doi: 10.1016/j.jde.2014.04.011. [28] J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. doi: 10.1016/S0304-0208(08)70870-3. [29] 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. [30] A. Salvatore, Multiple solitary waves for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3}$, Adv. Nonlinear Stud., 6 (2006), 157-169. doi: 10.1515/ans-2006-0203. [31] W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. [32] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023. [33] J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401. doi: 10.1016/j.na.2012.07.008. [34] J. Zhang, J. Marcos do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity Commun. Contemp. Math. (2016), 1650028, 16 pp. doi: 10.1142/S0219199716500280. [35] G. Zhao, X. Zhu and Y. Li, Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581. doi: 10.1016/j.amc.2015.01.038.
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