March  2016, 15(2): 429-444. doi: 10.3934/cpaa.2016.15.429

Schrödinger-Kirchhoff-Poisson type systems

1. 

The Fields Institute for Research in Mathematical Sciences, 222 College Street, 2nd floor, Toronto, Ontario, M5T 3J1, Canada

2. 

Universidade Federal do Pará, Faculdade de Matemática, CEP 66075-110, Belém, Pará, Brazil

Received  March 2015 Revised  October 2015 Published  January 2016

In this article, we are concerned with the boundary value problem \begin{equation} \left\{ \begin{array}{ll} \displaystyle -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} \\ -\Delta \phi= u^{2} &\text{in }\Omega \hbox{} \\ u=\phi=0&\text{on }\partial\Omega, \hbox{} \end{array} \right. \end{equation} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^N$ ($N=1,2$ or $3$), $a>0$, $b\geq0$, and $f:\overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is a continuous function which is globally $3$-superlinear. By using some variants of the mountain pass theorem established in this paper, we show that this problem has at least three solutions: one positive, one negative, and one which changes its sign. Furthermore, in case $f$ is odd with respect to $u$ we obtain an unbounded sequence of sign-changing solutions.
Citation: Cyril Joel Batkam, João R. Santos Júnior. Schrödinger-Kirchhoff-Poisson type systems. Communications on Pure & Applied Analysis, 2016, 15 (2) : 429-444. doi: 10.3934/cpaa.2016.15.429
References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, \emph{Comput. Math. Appl.}, 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1153. doi: 10.1007/s00033-013-0376-3. Google Scholar

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G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type,, \emph{Boundary Value Problems}, (2011). Google Scholar

[5]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, \emph{Contemp. Math.}, 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar

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A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, \emph{J. Math. Anal. Appl.}, 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057. Google Scholar

[7]

T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian,, \emph{Comm. Contemp. Math.}, 234 (2004), 245. doi: 10.1142/S0219199704001306. Google Scholar

[8]

C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems,, arXiv:1501.05942., (). Google Scholar

[9]

C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, arXiv:1501.05733., (). Google Scholar

[10]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Methods Nonlinear Anal.}, 11 (1998), 283. Google Scholar

[11]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems,, \emph{J. Differ. Equ.}, 248 (2010), 521. doi: 10.1016/j.jde.2009.06.017. Google Scholar

[12]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, \emph{Commun. Appl. Anal.}, 7 (2003), 417. Google Scholar

[13]

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: 10.1017/S030821050000353X. Google Scholar

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G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities,, \emph{Arch. Rational Mech. Anal.}, 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[15]

G. M. Figueiredo and J. R. Santos Júnior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method,, \emph{ESAIM Control Optim. Calc. Var.}, 20 (2014), 389. doi: 10.1051/cocv/2013068. Google Scholar

[16]

X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, \emph{J. Differ. Equ.}, (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[17]

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differ. Equ.}, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

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G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar

[19]

J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems,, \textit{Calc. Var. Partial Differential Equations}, 52 (2015), 565. doi: 10.1007/s00526-014-0724-y. Google Scholar

[20]

Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system,, \textit{Annali di Matematica}, (). Google Scholar

[21]

T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, \emph{Applied Mathematics Letters}, 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[22]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, \emph{Nonlinear Anal.}, 63 (2005), 1967. Google Scholar

[23]

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential,, \emph{Nonlinear Anal.}, 75 (2012), 3470. doi: 10.1016/j.na.2012.01.004. Google Scholar

[24]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, \emph{J. Funct. Analysis}, 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar

[25]

D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, \emph{Adv. Nonlinear Stud.}, 8 (2008), 179. Google Scholar

[26]

J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, \emph{J. Differ. Equ.}, 253 (2012), 2314. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[27]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[28]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[29]

F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent,, \emph{Nonlinear Anal.}, 70 (2009), 2150. doi: 10.1016/j.na.2008.02.116. Google Scholar

[30]

W. Zou, On finding sign-changing solutions,, \emph{J. Funct. Anal.}, 234 (2006), 364. doi: 10.1016/j.jfa.2005.09.004. Google Scholar

show all references

References:
[1]

C. O. Alves, F. J. S. A. Corrêa and T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type,, \emph{Comput. Math. Appl.}, 49 (2005), 85. doi: 10.1016/j.camwa.2005.01.008. Google Scholar

[2]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains,, \emph{Z. Angew. Math. Phys.}, 65 (2014), 1153. doi: 10.1007/s00033-013-0376-3. Google Scholar

[3]

G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem,, \emph{J. Math. Anal. Appl.}, (2011), 248. doi: 10.1016/j.jmaa.2010.07.019. Google Scholar

[4]

G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type,, \emph{Boundary Value Problems}, (2011). Google Scholar

[5]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, \emph{Contemp. Math.}, 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar

[6]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, \emph{J. Math. Anal. Appl.}, 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057. Google Scholar

[7]

T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian,, \emph{Comm. Contemp. Math.}, 234 (2004), 245. doi: 10.1142/S0219199704001306. Google Scholar

[8]

C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems,, arXiv:1501.05942., (). Google Scholar

[9]

C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems,, arXiv:1501.05733., (). Google Scholar

[10]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, \emph{Methods Nonlinear Anal.}, 11 (1998), 283. Google Scholar

[11]

G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems,, \emph{J. Differ. Equ.}, 248 (2010), 521. doi: 10.1016/j.jde.2009.06.017. Google Scholar

[12]

G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations,, \emph{Commun. Appl. Anal.}, 7 (2003), 417. Google Scholar

[13]

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: 10.1017/S030821050000353X. Google Scholar

[14]

G. M. Figueiredo, N. Ikoma and J. R. Santos Júnior, Existence and concentration result for the Kirchhoff type equations with general nonlinearities,, \emph{Arch. Rational Mech. Anal.}, 213 (2014), 931. doi: 10.1007/s00205-014-0747-8. Google Scholar

[15]

G. M. Figueiredo and J. R. Santos Júnior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method,, \emph{ESAIM Control Optim. Calc. Var.}, 20 (2014), 389. doi: 10.1051/cocv/2013068. Google Scholar

[16]

X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbbR^3$,, \emph{J. Differ. Equ.}, (2012), 1813. doi: 10.1016/j.jde.2011.08.035. Google Scholar

[17]

Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differ. Equ.}, 253 (2012), 2285. doi: 10.1016/j.jde.2012.05.017. Google Scholar

[18]

G. Kirchhoff, Mechanik,, Teubner, (1883). Google Scholar

[19]

J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems,, \textit{Calc. Var. Partial Differential Equations}, 52 (2015), 565. doi: 10.1007/s00526-014-0724-y. Google Scholar

[20]

Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system,, \textit{Annali di Matematica}, (). Google Scholar

[21]

T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem,, \emph{Applied Mathematics Letters}, 16 (2003), 243. doi: 10.1016/S0893-9659(03)80038-1. Google Scholar

[22]

T. F. Ma, Remarks on an elliptic equation of Kirchhoff type,, \emph{Nonlinear Anal.}, 63 (2005), 1967. Google Scholar

[23]

J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential,, \emph{Nonlinear Anal.}, 75 (2012), 3470. doi: 10.1016/j.na.2012.01.004. Google Scholar

[24]

D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term,, \emph{J. Funct. Analysis}, 237 (2006), 655. doi: 10.1016/j.jfa.2006.04.005. Google Scholar

[25]

D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains,, \emph{Adv. Nonlinear Stud.}, 8 (2008), 179. Google Scholar

[26]

J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth,, \emph{J. Differ. Equ.}, 253 (2012), 2314. doi: 10.1016/j.jde.2012.05.023. Google Scholar

[27]

M. Willem, Minimax Theorems,, Birkh\, (1996). doi: 10.1007/978-1-4612-4146-1. Google Scholar

[28]

X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Anal. Real World Appl.}, 12 (2011), 1278. doi: 10.1016/j.nonrwa.2010.09.023. Google Scholar

[29]

F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent,, \emph{Nonlinear Anal.}, 70 (2009), 2150. doi: 10.1016/j.na.2008.02.116. Google Scholar

[30]

W. Zou, On finding sign-changing solutions,, \emph{J. Funct. Anal.}, 234 (2006), 364. doi: 10.1016/j.jfa.2005.09.004. Google Scholar

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