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March  2020, 28(1): 383-404. doi: 10.3934/era.2020022

## Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system

 1 College of Mathematics and Physics, Fujian Jiangxia University, Fuzhou 350108, China 2 College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Qishan Campus, Fuzhou 350117, China

* Corresponding author: Jianqing Chen

Received  December 2019 Published  March 2020

Fund Project: Lirong Huang is supported by NSF of Fujian (No. 2017J01549); Jianqing Chen is supported by NNSF of China (No. 11871152, 11671085)

This paper is concerned with the following Schrödinger-Poisson system
 $\begin{equation*} (P_\mu): -\Delta u +u + K(x)\phi u = |u|^{p-1}u + \mu h(x)u, \ -\Delta \phi = K(x) u^2, \ x\in\mathbb{R}^3, \end{equation*}$
where
 $p\in (3,5)$
,
 $K(x)$
and
 $h(x)$
are nonnegative functions, and
 $\mu$
is a positive parameter. Let
 $\mu_1 > 0$
be an isolated first eigenvalue of the eigenvalue problem
 $-\Delta u + u = \mu h(x)u$
,
 $u\in H^1(\mathbb{R}^3)$
. As
 $0<\mu\leq\mu_1$
, we prove that
 $(P_{\mu})$
has at least one nonnegative bound state with positive energy. As
 $\mu > \mu_1$
, there is
 $\delta > 0$
such that for any
 $\mu\in (\mu_1, \mu_1 + \delta)$
,
 $(P_\mu)$
has a nonnegative ground state
 $u_{0,\mu}$
with negative energy, and
 $u_{0,\mu^{(n)}}\to 0$
in
 $H^1(\mathbb{R}^3)$
as
 $\mu^{(n)}\downarrow \mu_1$
. Besides,
 $(P_\mu)$
has another nonnegative bound state
 $u_{2,\mu}$
with positive energy, and
 $u_{2,\mu^{(n)}}\to u_{\mu_1}$
in
 $H^1(\mathbb{R}^3)$
as
 $\mu^{(n)}\downarrow \mu_1$
, where
 $u_{\mu_1}$
is a bound state of
 $(P_{\mu_1})$
.
Citation: Lirong Huang, Jianqing Chen. Existence and asymptotic behavior of bound states for a class of nonautonomous Schrödinger-Poisson system. Electronic Research Archive, 2020, 28 (1) : 383-404. doi: 10.3934/era.2020022
##### References:
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A, 134 (2004), no. 5,893–906. doi: 10.1017/S030821050000353X.  Google Scholar [16] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), no. 3,307–322. doi: 10.1515/ans-2004-0305.  Google Scholar [17] T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), no. 1,321–342. doi: 10.1137/S0036141004442793.  Google Scholar [18] P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), no. 2,177–192. doi: 10.1515/ans-2002-0205.  Google Scholar [19] P. D'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal., 74 (2011), no. 16, 5705–5721. doi: 10.1016/j.na.2011.05.057.  Google Scholar [20] L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), no. 8, 2463–2483. doi: 10.1016/j.jde.2013.06.022.  Google Scholar [21] Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), no. 3,582–608. doi: 10.1016/j.jde.2011.05.006.  Google Scholar [22] G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), no. 5, 053505, 19 pp. doi: 10.1063/1.3585657.  Google Scholar [23] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), no. 2,655–674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar [24] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Model. Methods Appl. Sci., 15 (2005), no. 1,141–164. doi: 10.1142/S0218202505003939.  Google Scholar [25] J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), no. 5, 3365–3380. doi: 10.1016/j.jde.2011.12.007.  Google Scholar [26] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), no. 2,263–297. doi: 10.1007/s11587-011-0109-x.  Google Scholar [27] J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differential Equations, 48 (2013), no. 1-2,243–273. doi: 10.1007/s00526-012-0548-6.  Google Scholar [28] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [29] Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), no. 4,809–816. doi: 10.3934/dcds.2007.18.809.  Google Scholar [30] Z. 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show all references

##### References:
 [1] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), no. 4,439–475. doi: 10.1007/BF01206962.  Google Scholar [2] A. Ambrosetti, On Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.  doi: 10.1007/s00032-008-0094-z.  Google Scholar [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [4] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), no. 3,391–404. doi: 10.1142/S021919970800282X.  Google Scholar [5] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.  Google Scholar [6] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), no. 7, 1746–1765. doi: 10.1016/j.jde.2010.07.007.  Google Scholar [7] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), no. 1, 90–108. doi: 10.1016/j.jmaa.2008.03.057.  Google Scholar [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), no. 2,283–293. doi: 10.12775/TMNA.1998.019.  Google Scholar [9] V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), no. 4,409–420. doi: 10.1142/S0129055X02001168.  Google Scholar [10] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), no. 3,486–490. doi: 10.1090/S0002-9939-1983-0699419-3.  Google Scholar [11] G. Cerami and G. Vaira, Positive solution for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 248 (2010), no. 3,521–543. doi: 10.1016/j.jde.2009.06.017.  Google Scholar [12] J. Chen, Z. Wang and X. Zhang, Standing waves for nonlinear Schrödinger-Poisson equation with high frequency, Topol. Methods Nonlinear Anal., 45 (2015), no. 2,601–614. doi: 10.12775/TMNA.2015.028.  Google Scholar [13] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), no. 2-3,417–423.  Google Scholar [14] D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $R^N$, Calc. Var. Partial Differential Equations, 13 (2001), no. 2,159–189. doi: 10.1007/PL00009927.  Google Scholar [15] 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), no. 5,893–906. doi: 10.1017/S030821050000353X.  Google Scholar [16] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), no. 3,307–322. doi: 10.1515/ans-2004-0305.  Google Scholar [17] T. D'Aprile and J. Wei, On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), no. 1,321–342. doi: 10.1137/S0036141004442793.  Google Scholar [18] P. D'Avenia, Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), no. 2,177–192. doi: 10.1515/ans-2002-0205.  Google Scholar [19] P. D'Avenia, A. Pomponio and G. Vaira, Infinitely many positive solutions for a Schrödinger-Poisson system, Nonlinear Anal., 74 (2011), no. 16, 5705–5721. doi: 10.1016/j.na.2011.05.057.  Google Scholar [20] L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), no. 8, 2463–2483. doi: 10.1016/j.jde.2013.06.022.  Google Scholar [21] Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), no. 3,582–608. doi: 10.1016/j.jde.2011.05.006.  Google Scholar [22] G. Li, S. Peng and C. Wang, Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 52 (2011), no. 5, 053505, 19 pp. doi: 10.1063/1.3585657.  Google Scholar [23] D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), no. 2,655–674. doi: 10.1016/j.jfa.2006.04.005.  Google Scholar [24] D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Model. Methods Appl. Sci., 15 (2005), no. 1,141–164. doi: 10.1142/S0218202505003939.  Google Scholar [25] J. Sun, H. Chen and J. J. Nieto, On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), no. 5, 3365–3380. doi: 10.1016/j.jde.2011.12.007.  Google Scholar [26] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), no. 2,263–297. doi: 10.1007/s11587-011-0109-x.  Google Scholar [27] J. Wang, L. Tian, J. Xu and F. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differential Equations, 48 (2013), no. 1-2,243–273. doi: 10.1007/s00526-012-0548-6.  Google Scholar [28] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [29] Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), no. 4,809–816. doi: 10.3934/dcds.2007.18.809.  Google Scholar [30] Z. Wang and H.-S. Zhou, Positive solutions for nonlinear Schrödinger equations with deepening potential well, J. Eur. Math. Soc., 11 (2009), no. 3,545–573. doi: 10.4171/JEMS/160.  Google Scholar [31] L. Zhao and F. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), no. 6, 2150–2164. doi: 10.1016/j.na.2008.02.116.  Google Scholar
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