### Electronic Research Archive

2020, Issue 1: 383-404. doi: 10.3934/era.2020022
Special Issues

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

• Primary: 35J20, 35J70

• 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[J]. Electronic Research Archive, 2020, 28(1): 383-404. doi: 10.3934/era.2020022

### Related Papers:

• 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 $(P_{\mu_1})$.

 [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 [2] On Schrödinger-Poisson systems. Milan J. Math. (2008) 76: 257-274. [3] Dual variational methods in critical point theory and applications. J. Funct. Anal. (1973) 14: 349-381. [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 [5] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984. [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 [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 [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 [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 [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 [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 [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 [13] G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), no. 2-3,417–423. [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [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 [28] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1 [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 [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 [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
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