# American Institute of Mathematical Sciences

April  2021, 20(4): 1497-1519. doi: 10.3934/cpaa.2021030

## An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well

 Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Received  October 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The author is supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 108-2115-M-390-007-MY2,108-2811-M-390-500)

In this paper, we study an eigenvalue problem for Schrödinger-Poisson system with indefinite nonlinearity and potential well as follows:
 $\begin{equation*} \begin{cases} -\Delta u+\mu V(x)u+K(x)\phi u = \lambda f(x)u+g(x)|u|^{p-2}u\ \ &\mbox{in}\ \ \mathbb{R}^3, \\ -\Delta \phi = K(x)u^2\ \ &\mbox{in}\ \ \mathbb{R}^3, \end{cases} \end{equation*}$
where
 $4\leq p<6$
, the parameters
 $\mu, \lambda>0$
,
 $V\in C(\mathbb{R}^3)$
is a potential well with the bottom
 $\overline\Omega: = \{x\in\mathbb{R}^3 : V(x) = 0\}$
, and the functions
 $f$
and
 $g$
are allowed to be sign-changing. By establishing an approximate estimate between the positive principal eigenvalue of
 $-\Delta u+\mu V(x)u = \lambda f(x)u$
in
 $\mathbb{R}^3$
and the positive principal eigenvalue
 $\lambda_1(f_{\Omega})$
of
 $-\Delta u = \lambda f_{\Omega}(x)u$
in
 $\Omega$
, we prove that at least a positive solution exists in
 $0<\lambda\leq\lambda_1(f_{\Omega})$
while at least two positive solutions exist in
 $\lambda>\lambda_1(f_{\Omega})$
and near
 $\lambda_1(f_{\Omega})$
, where
 $f_{\Omega}: = f|_{\Omega}$
. The results are obtained via variational method and steep potential. Furthermore, we also study the concentration of solutions as
 $\mu\to\infty$
and the decay rate of solutions at infinity.
Citation: Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1497-1519. doi: 10.3934/cpaa.2021030
##### References:
 [1] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differ. Equ., 1 (1993), 439-475.  doi: 10.1007/BF01206962. [2] 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. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X. [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] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494. [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for superlinear elliptic problems on $\mathbb{R}^3$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [7] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019. [8] H. Brézis and E. H. Lieb, A relation between point convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999. [9] G. Cerami and G. Vaira, Positive solution for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [10] J. Chabrowski and D. G. Costa, On a class of Schrödinger-Type equations with indefinite weight functions, Commun. Partial Differ. Equ., 33 (2008), 1368-1394. doi: 10.1080/03605300601088880. [11] J. Chen, Multiple positive solutions of a class of nonautonomous Schrödinger-Poisson systems, Nonlinear Anal., 21 (2015), 13-26. doi: 10.1016/j.nonrwa.2014.06.002. [12] D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 13 (2001), 159-189. doi: 10.1007/PL00009927. [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] M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrodinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502. doi: 10.1063/1.4941036. [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3. [16] L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differ. Equ., 255 (2013), 2463-2483. doi: 10.1016/j.jde.2013.06.022. [17] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [18] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 109-145. [19] 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. [20] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/978-3-662-02624-3. [21] Z. Shen and Z. Han, Multiple solutions for a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Math. Anal. Appl., 426 (2015), 839-854. doi: 10.1016/j.jmaa.2015.01.071. [22] J. Sun, T. F. Wu and Y. Wu, Existence of nontrivial solution for Schrödinger-Poisson systems with indefinite steep potential well, Z. Angew. Math. Phys., 68 (2017), 1-22. doi: 10.1007/s00033-017-0817-5. [23] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche Math., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. [24] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [25] T. F. Wu, On a class of nonlocal nonlinear Schrödinger equations with potential well, Adv. Nonlinear Anal., 9 (2020), 665-689. doi: 10.1515/anona-2020-0020. [26] Y. Ye and C. Tang, Existence and multiplicity of solutions for Schrödinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differ. Equ., 53 (2015), 383-411. doi: 10.1007/s00526-014-0753-6. [27] F. Zhang and M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 85-106.  doi: 10.1016/j.jde.2020.07.013. [28] L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1-23. doi: 10.1016/j.jde.2013.03.005.

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##### References:
 [1] S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differ. Equ., 1 (1993), 439-475.  doi: 10.1007/BF01206962. [2] 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. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X. [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] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569.  doi: 10.1142/S0219199701000494. [6] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for superlinear elliptic problems on $\mathbb{R}^3$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741. doi: 10.1080/03605309508821149. [7] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.  doi: 10.12775/TMNA.1998.019. [8] H. Brézis and E. H. Lieb, A relation between point convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999. [9] G. Cerami and G. Vaira, Positive solution for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.  doi: 10.1016/j.jde.2009.06.017. [10] J. Chabrowski and D. G. Costa, On a class of Schrödinger-Type equations with indefinite weight functions, Commun. Partial Differ. Equ., 33 (2008), 1368-1394. doi: 10.1080/03605300601088880. [11] J. Chen, Multiple positive solutions of a class of nonautonomous Schrödinger-Poisson systems, Nonlinear Anal., 21 (2015), 13-26. doi: 10.1016/j.nonrwa.2014.06.002. [12] D. G. Costa and H. Tehrani, Existence of positive solutions for a class of indefinite elliptic problems in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 13 (2001), 159-189. doi: 10.1007/PL00009927. [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] M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrodinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502. doi: 10.1063/1.4941036. [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3. [16] L. Huang, E. M. Rocha and J. Chen, Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differ. Equ., 255 (2013), 2463-2483. doi: 10.1016/j.jde.2013.06.022. [17] Y. Jiang and H. Zhou, Schrödinger-Poisson system with steep potential well, J. Differ. Equ., 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. [18] P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, part I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1 (1984), 109-145. [19] 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. [20] M. Struwe, Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/978-3-662-02624-3. [21] Z. Shen and Z. Han, Multiple solutions for a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Math. Anal. Appl., 426 (2015), 839-854. doi: 10.1016/j.jmaa.2015.01.071. [22] J. Sun, T. F. Wu and Y. Wu, Existence of nontrivial solution for Schrödinger-Poisson systems with indefinite steep potential well, Z. Angew. Math. Phys., 68 (2017), 1-22. doi: 10.1007/s00033-017-0817-5. [23] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche Math., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x. [24] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [25] T. F. Wu, On a class of nonlocal nonlinear Schrödinger equations with potential well, Adv. Nonlinear Anal., 9 (2020), 665-689. doi: 10.1515/anona-2020-0020. [26] Y. Ye and C. Tang, Existence and multiplicity of solutions for Schrödinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differ. Equ., 53 (2015), 383-411. doi: 10.1007/s00526-014-0753-6. [27] F. Zhang and M. Du, Existence and asymptotic behavior of positive solutions for Kirchhoff type problems with steep potential well, J. Differ. Equ., 269 (2020), 85-106.  doi: 10.1016/j.jde.2020.07.013. [28] L. Zhao, H. Liu and F. Zhao, Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differ. Equ., 255 (2013), 1-23. doi: 10.1016/j.jde.2013.03.005.
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