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On the uniqueness of solutions of a semilinear equation in an annulus
An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan |
$ \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*} $ |
$ 4\leq p<6 $ |
$ \mu, \lambda>0 $ |
$ V\in C(\mathbb{R}^3) $ |
$ \overline\Omega: = \{x\in\mathbb{R}^3 : V(x) = 0\} $ |
$ f $ |
$ g $ |
$ -\Delta u+\mu V(x)u = \lambda f(x)u $ |
$ \mathbb{R}^3 $ |
$ \lambda_1(f_{\Omega}) $ |
$ -\Delta u = \lambda f_{\Omega}(x)u $ |
$ \Omega $ |
$ 0<\lambda\leq\lambda_1(f_{\Omega}) $ |
$ \lambda>\lambda_1(f_{\Omega}) $ |
$ \lambda_1(f_{\Omega}) $ |
$ f_{\Omega}: = f|_{\Omega} $ |
$ \mu\to\infty $ |
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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