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.  Google Scholar

[2]

C. O. AlvesF. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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A. AzzolliniP. 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.  Google Scholar

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A. AzzolliniP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.   Google Scholar

[16]

G. M. FigueiredoN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. HirataN. 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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

Y. JiangZ. 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.  Google Scholar

[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.  Google Scholar

[25]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[26]

Y. LiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[32]

J. WangL. TianJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

G. ZhaoX. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

C. O. AlvesF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. AzzolliniP. 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.  Google Scholar

[6]

A. AzzolliniP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. d'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.   Google Scholar

[16]

G. M. FigueiredoN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. HirataN. 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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

Y. JiangZ. 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.  Google Scholar

[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.  Google Scholar

[25]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[26]

Y. LiF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.   Google Scholar

[32]

J. WangL. TianJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

G. ZhaoX. 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.  Google Scholar

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