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Global existence for a coupled wave system related to the Strauss conjecture
Positive solutions for Kirchhoff-Schrödinger-Poisson systems with general nonlinearity
1. | School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan, 430074, China |
2. | School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, China |
$\left\{ \begin{gathered} - \left( {a + b\int_{{\mathbb{R}^3}} {{{\left| {\nabla u} \right|}^2}{\text{d}}x} } \right)\Delta u + \mu \phi \left( x \right)u =f\left( u \right)\;\;\;&{\text{in}}\;\;{{\mathbb{R}}^3}, \hfill \\ - \Delta \phi =\mu {u^2}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&{\text{in}}\;\;{{\mathbb{R}}^3}, \hfill \\ \end{gathered} \right.$ |
$a>0,b≥q0 $ |
$μ>0 $ |
$f∈ C(\mathbb{R},\mathbb{R}) $ |
$f $ |
$μ $ |
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
C. O. Alves, F. J. S. A. Correa and T. F. Ma,
Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[3] |
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. |
[4] |
A. Azzollini,
Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763.
doi: 10.1016/j.jde.2010.07.007. |
[5] |
A. Azzollini, P. d'Avenia and A. Pomponio,
On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. I. H. Poincaré-AN, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[6] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523.
doi: 10.1007/s10231-010-0160-3. |
[7] |
A. Azzollini,
The elliptic Kirchhoff equation in $\mathbb{R}^{N} $ perturbed by a local nonlinearity, Differential Integral Equations, 25 (2012), 543-554.
doi: 10.1142/S0219199714500394. |
[8] |
C. Batkam and J. R. S. Júnior,
Schrödinger-Kirchhoff-Poisson type systems, Commun. Pure Appl. Anal., 15 (2016), 429-444.
doi: 10.3934/cpaa.2016.15.429. |
[9] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonl. Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[10] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
S. Chen and C. Tang,
Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on $\mathbb{R}^{3} $, Nonlinear Differ. Equ. Appl., 17 (2010), 559-574.
doi: 10.1007/s00030-010-0068-z. |
[12] |
P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data Invent. Math. 108 (1992), 247–262.
doi: 10.1007/BF02100605. |
[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] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[15] |
P. d'Avenia,
Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.
|
[16] |
G. M. Figueiredo, N. Ikoma and J. R. S. Júnior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[17] |
X. He and W. Zou,
Existence and concentration behavior of positive solutions
for a Kirchhoff equation in $\mathbb{R}^{3} $, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[18] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems
in $\mathbb{R}^{3} $ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[19] |
J. Hirata, N. Ikoma and K. Tanaka,
Nonlinear scalar field equations in $R^{N}$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276.
|
[20] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ R^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[21] |
L. Jeanjean and S. Le Coz,
An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.
|
[22] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $R^{N} $, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[23] |
Y. Jiang, Z. Wang and H.-S. Zhou,
Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3} $, Nonlinear Anal., 83 (2013), 50-57.
doi: 10.1016/j.na.2013.01.006. |
[24] |
H. Kikuchi,
Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
doi: 10.1515/ans-2007-0305. |
[25] | |
[26] |
Y. Li, F. Li and J. Shi,
Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[27] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3} $, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[28] |
J. L. Lions,
On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.
doi: 10.1016/S0304-0208(08)70870-3. |
[29] |
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. |
[30] |
A. Salvatore,
Multiple solitary waves for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3} $, Adv. Nonlinear Stud., 6 (2006), 157-169.
doi: 10.1515/ans-2006-0203. |
[31] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[32] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[33] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[34] |
J. Zhang, J. Marcos do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity Commun. Contemp. Math. (2016), 1650028, 16 pp.
doi: 10.1142/S0219199716500280. |
[35] |
G. Zhao, X. Zhu and Y. Li,
Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.
doi: 10.1016/j.amc.2015.01.038. |
show all references
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
C. O. Alves, F. J. S. A. Correa and T. F. Ma,
Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85-93.
doi: 10.1016/j.camwa.2005.01.008. |
[3] |
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. |
[4] |
A. Azzollini,
Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 249 (2010), 1746-1763.
doi: 10.1016/j.jde.2010.07.007. |
[5] |
A. Azzollini, P. d'Avenia and A. Pomponio,
On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. I. H. Poincaré-AN, 27 (2010), 779-791.
doi: 10.1016/j.anihpc.2009.11.012. |
[6] |
A. Azzollini, P. d'Avenia and A. Pomponio,
Multiple critical points for a class of nonlinear functionals, Ann. Mat. Pura Appl., 190 (2011), 507-523.
doi: 10.1007/s10231-010-0160-3. |
[7] |
A. Azzollini,
The elliptic Kirchhoff equation in $\mathbb{R}^{N} $ perturbed by a local nonlinearity, Differential Integral Equations, 25 (2012), 543-554.
doi: 10.1142/S0219199714500394. |
[8] |
C. Batkam and J. R. S. Júnior,
Schrödinger-Kirchhoff-Poisson type systems, Commun. Pure Appl. Anal., 15 (2016), 429-444.
doi: 10.3934/cpaa.2016.15.429. |
[9] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonl. Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[10] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[11] |
S. Chen and C. Tang,
Multiple solutions for nonhomogeneous Schrödinger-Maxwell and Klein-Gordon-Maxwell equations on $\mathbb{R}^{3} $, Nonlinear Differ. Equ. Appl., 17 (2010), 559-574.
doi: 10.1007/s00030-010-0068-z. |
[12] |
P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data Invent. Math. 108 (1992), 247–262.
doi: 10.1007/BF02100605. |
[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] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
doi: 10.1515/ans-2004-0305. |
[15] |
P. d'Avenia,
Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2 (2002), 177-192.
|
[16] |
G. M. Figueiredo, N. Ikoma and J. R. S. Júnior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Rational Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[17] |
X. He and W. Zou,
Existence and concentration behavior of positive solutions
for a Kirchhoff equation in $\mathbb{R}^{3} $, J. Differential Equations, 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[18] |
Y. He and G. Li,
Standing waves for a class of Kirchhoff type problems
in $\mathbb{R}^{3} $ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.
doi: 10.1007/s00526-015-0894-2. |
[19] |
J. Hirata, N. Ikoma and K. Tanaka,
Nonlinear scalar field equations in $R^{N}$: mountain pass and symmetric mountain pass approaches, Topol. Methods Nonlinear Anal., 35 (2010), 253-276.
|
[20] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $ R^{N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[21] |
L. Jeanjean and S. Le Coz,
An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11 (2006), 813-840.
|
[22] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $R^{N} $, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[23] |
Y. Jiang, Z. Wang and H.-S. Zhou,
Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3} $, Nonlinear Anal., 83 (2013), 50-57.
doi: 10.1016/j.na.2013.01.006. |
[24] |
H. Kikuchi,
Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7 (2007), 403-437.
doi: 10.1515/ans-2007-0305. |
[25] | |
[26] |
Y. Li, F. Li and J. Shi,
Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[27] |
G. Li and H. Ye,
Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3} $, J. Differential Equations, 257 (2014), 566-600.
doi: 10.1016/j.jde.2014.04.011. |
[28] |
J. L. Lions,
On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346.
doi: 10.1016/S0304-0208(08)70870-3. |
[29] |
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. |
[30] |
A. Salvatore,
Multiple solitary waves for a nonhomogeneous Schrödinger-Maxwell system in $\mathbb{R}^{3} $, Adv. Nonlinear Stud., 6 (2006), 157-169.
doi: 10.1515/ans-2006-0203. |
[31] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162.
|
[32] |
J. Wang, L. Tian, J. Xu and F. Zhang,
Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[33] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[34] |
J. Zhang, J. Marcos do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity Commun. Contemp. Math. (2016), 1650028, 16 pp.
doi: 10.1142/S0219199716500280. |
[35] |
G. Zhao, X. Zhu and Y. Li,
Existence of infinitely many solutions to a class of Kirchhoff-Schrödinger-Poisson system, Appl. Math. Comput., 256 (2015), 572-581.
doi: 10.1016/j.amc.2015.01.038. |
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