# American Institute of Mathematical Sciences

May  2022, 5(2): 113-128. doi: 10.3934/mfc.2021036

## Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

 1 School of Mathematics and Computer Application Technology, Jining university, Shandong 273155, China 2 School of Mathematical Sciences, Qufu Normal University, Shandong 273165, China

* Corresponding author: Caixia Chen

Received  July 2021 Published  May 2022 Early access  December 2021

Fund Project: Supported by the Shandong Province Science Foundation ZR2021MA096 and ZR2020MA005

In this paper, we consider the following Schrödinger-Poisson equation
 \left\{\begin{aligned} &-\triangle u + u + \phi u = u^{5}+\lambda g(u), &\hbox{in}\ \ \Omega, \\\ & -\triangle \phi = u^{2}, & \hbox{in}\ \ \Omega, \\\ & u, \phi = 0, & \hbox{on}\ \ \partial\Omega.\end{aligned}\right.
where
 $\Omega$
is a bounded smooth domain in
 $\mathbb{R}^{3}$
,
 $\lambda>0$
and the nonlinear growth of
 $u^{5}$
reaches the Sobolev critical exponent in three spatial dimensions. With the aid of variational methods and the concentration compactness principle, we prove the problem admits at least two positive solutions and one positive ground state solution.
Citation: Caixia Chen, Aixia Qian. Multiple positive solutions for the Schrödinger-Poisson equation with critical growth. Mathematical Foundations of Computing, 2022, 5 (2) : 113-128. doi: 10.3934/mfc.2021036
##### References:
 [1] C. O. Alves and M. A. 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. [2] C. O. Alves, M. A. Souto and S. H. M. Soares, Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl, 377 (2011), 584-592.  doi: 10.1016/j.jmaa.2010.11.031. [3] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z. [4] 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. [5] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168. [6] I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [8] S. Chen and C. Tang, High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050. [9] H. Guo, Nonexistence of least energy nodal solutions for Schrödinger-Poisson equation, Appl. Math. Lett., 68 (2017), 135-142.  doi: 10.1016/j.aml.2016.12.016. [10] L. Huang, E. M. Rocha and J. Chen, On the Schrödinger-Poisson system with a general indefinite nonlinear, Nonlinear Anal., 28 (2016), 1-19.  doi: 10.1016/j.nonrwa.2015.09.001. [11] C. Y. Lei, G. S. Liu and L. T. Guo, Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 343-355.  doi: 10.1016/j.nonrwa.2016.01.018. [12] H. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal., 32 (2016), 198-212.  doi: 10.1016/j.nonrwa.2016.04.007. [13] Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066. [14] A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [15] 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. [16] J. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.  doi: 10.1016/j.jmaa.2012.01.057. [17] J. Sun and T. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115.  doi: 10.1016/j.na.2014.11.009. [18] M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

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##### References:
 [1] C. O. Alves and M. A. 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. [2] C. O. Alves, M. A. Souto and S. H. M. Soares, Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl, 377 (2011), 584-592.  doi: 10.1016/j.jmaa.2010.11.031. [3] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z. [4] 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. [5] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.  doi: 10.1142/S0129055X02001168. [6] I. Catto and P. L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110. [7] G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [8] S. Chen and C. Tang, High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050. [9] H. Guo, Nonexistence of least energy nodal solutions for Schrödinger-Poisson equation, Appl. Math. Lett., 68 (2017), 135-142.  doi: 10.1016/j.aml.2016.12.016. [10] L. Huang, E. M. Rocha and J. Chen, On the Schrödinger-Poisson system with a general indefinite nonlinear, Nonlinear Anal., 28 (2016), 1-19.  doi: 10.1016/j.nonrwa.2015.09.001. [11] C. Y. Lei, G. S. Liu and L. T. Guo, Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 343-355.  doi: 10.1016/j.nonrwa.2016.01.018. [12] H. Liu, Positive solutions of an asymptotically periodic Schrödinger-Poisson system with critical exponent, Nonlinear Anal., 32 (2016), 198-212.  doi: 10.1016/j.nonrwa.2016.04.007. [13] Z. Liu and S. Guo, On ground state solutions for the Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 412 (2014), 435-448.  doi: 10.1016/j.jmaa.2013.10.066. [14] A. Mao, L. Yang, A. Qian and S. Luan, Existence and concentration of solutions of Schrödinger-Poisson system, Appl. Math. Lett, 68 (2017), 8-12.  doi: 10.1016/j.aml.2016.12.014. [15] 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. [16] J. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 390 (2012), 514-522.  doi: 10.1016/j.jmaa.2012.01.057. [17] J. Sun and T. Wu, Multiplicity and concentration of homoclinic solutions for some second order Hamiltonian systems, Nonlinear Anal., 114 (2015), 105-115.  doi: 10.1016/j.na.2014.11.009. [18] M. Willem, Minimax Theorems, Birthäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.
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