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 & 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

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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

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T. BartschA. 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

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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

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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

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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

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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

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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

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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

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I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer, 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

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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

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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

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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

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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

show all references

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. BartschA. 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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