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Positive solution for the Kirchhoff-type equations involving general subcritical growth
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Schrödinger-Kirchhoff-Poisson type systems
1. | The Fields Institute for Research in Mathematical Sciences, 222 College Street, 2nd floor, Toronto, Ontario, M5T 3J1, Canada |
2. | Universidade Federal do Pará, Faculdade de Matemática, CEP 66075-110, Belém, Pará, Brazil |
References:
[1] |
C. O. Alves, F. J. S. A. Corrêa 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. |
[2] |
C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[3] |
G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[4] |
G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, Boundary Value Problems, (2011) doi:10.1155/2011/891430. |
[5] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[6] |
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. |
[7] |
T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Comm. Contemp. Math., 234 (2004), 245-258.
doi: 10.1142/S0219199704001306. |
[8] |
C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems, arXiv:1501.05942. |
[9] |
C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv:1501.05733. |
[10] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Methods Nonlinear Anal., 11 (1998), 283-293. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[12] |
G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. |
[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] |
G. M. Figueiredo, N. Ikoma and J. R. Santos 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. |
[15] |
G. M. Figueiredo and J. R. Santos Júnior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415.
doi: 10.1051/cocv/2013068. |
[16] |
X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[17] |
Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[18] | |
[19] |
J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[20] |
Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Annali di Matematica, in press. |
[21] |
T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Applied Mathematics Letters, 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1. |
[22] |
T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. |
[23] |
J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[24] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[25] |
D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. |
[26] |
J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[27] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[28] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[29] |
F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[30] |
W. Zou, On finding sign-changing solutions, J. Funct. Anal., 234 (2006), 364-419.
doi: 10.1016/j.jfa.2005.09.004. |
show all references
References:
[1] |
C. O. Alves, F. J. S. A. Corrêa 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. |
[2] |
C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.
doi: 10.1007/s00033-013-0376-3. |
[3] |
G. Anelo, A uniqueness result for a nonlocal equation of Kirchhoff equation type and some related open problem, J. Math. Anal. Appl., 373 (2011), 248-251.
doi: 10.1016/j.jmaa.2010.07.019. |
[4] |
G. Anelo, On a perturbed Dirichlet problem for a nonlocal differential equation of Kirchhoff type, Boundary Value Problems, (2011) doi:10.1155/2011/891430. |
[5] |
A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[6] |
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. |
[7] |
T. Bartsch and Z. Liu, Multiple sign changing solutions of a quasilinear elliptic eigenvalue problem involving the p-Laplacian, Comm. Contemp. Math., 234 (2004), 245-258.
doi: 10.1142/S0219199704001306. |
[8] |
C. J. Batkam, High energy sign-changing solutions to Scrhödinger-Poisson type systems, arXiv:1501.05942. |
[9] |
C. J. Batkam, Multiple sign-changing solutions to a class of Kirchhoff type problems, arXiv:1501.05733. |
[10] |
V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Methods Nonlinear Anal., 11 (1998), 283-293. |
[11] |
G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differ. Equ., 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[12] |
G. M. Coclite, A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423. |
[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] |
G. M. Figueiredo, N. Ikoma and J. R. Santos 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. |
[15] |
G. M. Figueiredo and J. R. Santos Júnior, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415.
doi: 10.1051/cocv/2013068. |
[16] |
X. He and W. Zou, Existence and concentration of positive solutions for a Kirchhoff equation in $\mathbb{R}^3$, J. Differ. Equ., 252 (2012), 1813-1834.
doi: 10.1016/j.jde.2011.08.035. |
[17] |
Y. Li, F. Li and J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equ., 253 (2012), 2285-2294.
doi: 10.1016/j.jde.2012.05.017. |
[18] | |
[19] |
J. Liu, X. Liu and Z.-Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differential Equations, 52 (2015), 565-586.
doi: 10.1007/s00526-014-0724-y. |
[20] |
Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Annali di Matematica, in press. |
[21] |
T. F. Ma and J. E. M. Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Applied Mathematics Letters, 16 (2003), 243-248.
doi: 10.1016/S0893-9659(03)80038-1. |
[22] |
T. F. Ma, Remarks on an elliptic equation of Kirchhoff type, Nonlinear Anal., 63 (2005), 1967-1977. |
[23] |
J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff equations with radial potential, Nonlinear Anal., 75 (2012), 3470-3479.
doi: 10.1016/j.na.2012.01.004. |
[24] |
D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Analysis, 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[25] |
D. Ruiz and G. Siciliano, A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2008), 179-190. |
[26] |
J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differ. Equ., 253 (2012), 2314-2351.
doi: 10.1016/j.jde.2012.05.023. |
[27] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[28] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbb{R}^N2$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287.
doi: 10.1016/j.nonrwa.2010.09.023. |
[29] |
F. Zhao and L. Zhao, Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
[30] |
W. Zou, On finding sign-changing solutions, J. Funct. Anal., 234 (2006), 364-419.
doi: 10.1016/j.jfa.2005.09.004. |
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