# American Institute of Mathematical Sciences

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 $$\left\{ \begin{array}{ll} \displaystyle -\left(a+b\int_{\Omega}|\nabla u|^{2}\right)\Delta u +\phi u= f(x, u) &\text{in }\Omega \hbox{} \\ -\Delta \phi= u^{2} &\text{in }\Omega \hbox{} \\ u=\phi=0&\text{on }\partial\Omega, \hbox{} \end{array} \right.$$ 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:
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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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