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Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity
Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold
1. | College of Science, Hohai University, Nanjing, 210098, China, China |
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Differential Equations}, 254 (2013), 1977.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Eur. Math. Soc.}, 7 (2005), 117.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential and Integral Equations}, 18 (2005), 1321.
|
[4] |
M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners,, 1$^{nd}$ edition, (2011).
doi: 10.1007/978-0-85729-227-8. |
[5] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[6] |
J. Byeon and Z. Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potential,, \emph{J. Eur. Math. Soc.}, 8 (2006), 217.
doi: 10.4171/JEMS/48. |
[7] |
C. S. Chen, L. Chen and Z. H. Xiu, Existence of nontrivial solutions for singular quasilinear elliptic equations on $\mathbbR^N$,, \emph{Computers and Mathematics with Applications}, 6 (2013), 1909.
doi: 10.1016/j.camwa.2013.04.017. |
[8] |
C. S. Chen and Q. Zhu, Existence of positive solutions to $p-$Kirchhoff-type problem without compactness conditions,, \emph{Applied Mathematics Letters}, 28 (2014), 82.
doi: 10.1016/j.aml.2013.10.005. |
[9] |
S. J. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbbR^N$,, \emph{Nonlinear Analysis: Real World Applications}, 14 (2013), 1477.
doi: 10.1016/j.nonrwa.2012.10.010. |
[10] |
C. S. Chen, H. X. Song and Z. H. Xiu, Multiple solutions for $ p-$Kirchhoff equations in $\mathbbR^N$,, \emph{Nonlinear Analysis}, 86 (2013), 146.
doi: 10.1016/j.na.2013.03.017. |
[11] |
W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, \emph{Arch. Rational Mech. Anal.}, 91 (1986), 283.
doi: 10.1007/BF00282336. |
[12] |
L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics Vol.19, (1998).
|
[13] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285.
doi: 10.1016/j.jde.2012.05.017. |
[14] |
Y. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eire}, 23 (2006), 829.
doi: 10.1016/j.anihpc.2006.01.003. |
[15] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.
|
[16] |
W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, \emph{J. Appl. Math. Comput.}, 39 (2012), 473.
doi: 10.1007/s12190-012-0536-1. |
[17] |
J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, \emph{Nonlinear Analysis}, 75 (2012), 3470.
doi: 10.1016/j.na.2012.01.004. |
[18] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 1278.
doi: 10.1016/j.nonrwa.2010.09.023. |
[19] |
L. Wang, On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials,, \emph{Nonlinear Analysis}, 83 (2013), 58.
doi: 10.1016/j.na.2012.12.012. |
show all references
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Differential Equations}, 254 (2013), 1977.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potential vanishing at infinity,, \emph{J. Eur. Math. Soc.}, 7 (2005), 117.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential and Integral Equations}, 18 (2005), 1321.
|
[4] |
M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners,, 1$^{nd}$ edition, (2011).
doi: 10.1007/978-0-85729-227-8. |
[5] |
H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313.
doi: 10.1007/BF00250555. |
[6] |
J. Byeon and Z. Q. Wang, Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potential,, \emph{J. Eur. Math. Soc.}, 8 (2006), 217.
doi: 10.4171/JEMS/48. |
[7] |
C. S. Chen, L. Chen and Z. H. Xiu, Existence of nontrivial solutions for singular quasilinear elliptic equations on $\mathbbR^N$,, \emph{Computers and Mathematics with Applications}, 6 (2013), 1909.
doi: 10.1016/j.camwa.2013.04.017. |
[8] |
C. S. Chen and Q. Zhu, Existence of positive solutions to $p-$Kirchhoff-type problem without compactness conditions,, \emph{Applied Mathematics Letters}, 28 (2014), 82.
doi: 10.1016/j.aml.2013.10.005. |
[9] |
S. J. Chen and L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on $\mathbbR^N$,, \emph{Nonlinear Analysis: Real World Applications}, 14 (2013), 1477.
doi: 10.1016/j.nonrwa.2012.10.010. |
[10] |
C. S. Chen, H. X. Song and Z. H. Xiu, Multiple solutions for $ p-$Kirchhoff equations in $\mathbbR^N$,, \emph{Nonlinear Analysis}, 86 (2013), 146.
doi: 10.1016/j.na.2013.03.017. |
[11] |
W. Y. Ding and W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation,, \emph{Arch. Rational Mech. Anal.}, 91 (1986), 283.
doi: 10.1007/BF00282336. |
[12] |
L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics Vol.19, (1998).
|
[13] |
Y. H. Li, F. Y. Li and J. P. Shi, Existence of positive solution to Kirchhoff type problems without compactness conditions,, \emph{J. Differential Equations}, 253 (2012), 2285.
doi: 10.1016/j.jde.2012.05.017. |
[14] |
Y. Li, Z. Q. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eire}, 23 (2006), 829.
doi: 10.1016/j.anihpc.2006.01.003. |
[15] |
P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.
|
[16] |
W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations,, \emph{J. Appl. Math. Comput.}, 39 (2012), 473.
doi: 10.1007/s12190-012-0536-1. |
[17] |
J. J. Nie and X. Wu, Existence and multiplicity of non-trivial solutions for Schrödinger-Kirchhoff-type equations with radial potential,, \emph{Nonlinear Analysis}, 75 (2012), 3470.
doi: 10.1016/j.na.2012.01.004. |
[18] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $\mathbbR^N$,, \emph{Nonlinear Analysis: Real World Applications}, 12 (2011), 1278.
doi: 10.1016/j.nonrwa.2010.09.023. |
[19] |
L. Wang, On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials,, \emph{Nonlinear Analysis}, 83 (2013), 58.
doi: 10.1016/j.na.2012.12.012. |
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