Schrödinger-Kirchhoff-Poisson type systems

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

Schrödinger-Kirchhoff-Poisson type systems

Cyril Joel Batkam ^1, and João R. Santos Júnior ^2,

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

Abstract
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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.

Keywords: Schrödinger-Kirchhoff-Poisson system, signed solution, sign-changing solution., mountain pass theorem, variational methods.

Mathematics Subject Classification: Primary: 35J20, 35J47; Secondary: 35B33, 35J25, 35J50, 35J6.

Citation: Cyril Joel Batkam, João R. Santos Júnior. Schrödinger-Kirchhoff-Poisson type systems. Communications on Pure and 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, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008.
[2]	C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3.
[3]	G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251. doi: 10.1016/j.jmaa.2010.07.019.
[4]	G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, Boundary Value Problems, (2011) doi:10.1155/2011/891430.
[5]	A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X.
[6]	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.
[7]	T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Comm. Contemp. Math., 234 (2004), 245-258. doi: 10.1142/S0219199704001306.
[8]	C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems, arXiv:1501.05942.
[9]	C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv:1501.05733.
[10]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Methods Nonlinear Anal., 11 (1998), 283-293.
[11]	G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017.
[12]	G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
[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]	G. M. Figueiredo, N. Ikoma and J. R. Santos 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.
[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, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415. doi: 10.1051/cocv/2013068.
[16]	X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035.
[17]	Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017.
[18]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[19]	J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586. doi: 10.1007/s00526-014-0724-y.
[20]	Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Annali di Matematica, in press.
[21]	T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Applied Mathematics Letters, 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1.
[22]	T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.
[23]	J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004.
[24]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005.
[25]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190.
[26]	J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023.
[27]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[28]	X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.
[29]	F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.
[30]	W. Zou, On finding sign-changing solutions, J. Funct. Anal., 234 (2006), 364-419. doi: 10.1016/j.jfa.2005.09.004.

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, Comput. Math. Appl., 49 (2005), 85-93. doi: 10.1016/j.camwa.2005.01.008.
[2]	C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3.
[3]	G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251. doi: 10.1016/j.jmaa.2010.07.019.
[4]	G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, Boundary Value Problems, (2011) doi:10.1155/2011/891430.
[5]	A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Contemp. Math., 10 (2008), 391-404. doi: 10.1142/S021919970800282X.
[6]	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.
[7]	T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Comm. Contemp. Math., 234 (2004), 245-258. doi: 10.1142/S0219199704001306.
[8]	C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems, arXiv:1501.05942.
[9]	C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv:1501.05733.
[10]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Methods Nonlinear Anal., 11 (1998), 283-293.
[11]	G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543. doi: 10.1016/j.jde.2009.06.017.
[12]	G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
[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]	G. M. Figueiredo, N. Ikoma and J. R. Santos 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.
[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, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415. doi: 10.1051/cocv/2013068.
[16]	X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813-1834. doi: 10.1016/j.jde.2011.08.035.
[17]	Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017.
[18]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[19]	J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586. doi: 10.1007/s00526-014-0724-y.
[20]	Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Annali di Matematica, in press.
[21]	T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Applied Mathematics Letters, 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1.
[22]	T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977.
[23]	J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479. doi: 10.1016/j.na.2012.01.004.
[24]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674. doi: 10.1016/j.jfa.2006.04.005.
[25]	D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190.
[26]	J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351. doi: 10.1016/j.jde.2012.05.023.
[27]	M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
[28]	X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.
[29]	F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164. doi: 10.1016/j.na.2008.02.116.
[30]	W. Zou, On finding sign-changing solutions, J. Funct. Anal., 234 (2006), 364-419. doi: 10.1016/j.jfa.2005.09.004.

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