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

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

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

Kuan-Hsiang Wang

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.

Keywords: Schrödinger-Poisson system, eigenvalue problem, steep potential well, mountain pass theory, variational methods.

Mathematics Subject Classification: Primary: 35B09, 35B40; Secondary: 35J20, 35J60.

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.

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.
[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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