# 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  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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] J. 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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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche Math., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x.  Google Scholar [24] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] G. Vaira, Ground states for Schrödinger-Poisson type systems, Ricerche Math., 60 (2011), 263-297. doi: 10.1007/s11587-011-0109-x.  Google Scholar [24] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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