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    Positive solutions for quasilinear Schrödinger equations with critical growth and potential vanishing at infinity
November  2014, 13(6): 2289-2303. doi: 10.3934/cpaa.2014.13.2289

Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold

Caisheng Chen 1, and  Qing Yuan 1,

1. 

College of Science, Hohai University, Nanjing, 210098, China, China

Received  September 2013 Revised  January 2014 Published  July 2014

In this paper, we study the existence of positive solution to $p-$Kirchhoff type problem \begin{eqnarray} &(a+\mu(\int_{\mathbb{R}^N}(|\nabla u|^p+V(x)|u|^p)dx)^{\tau})(-\Delta_pu+V(x)|u|^{p-2}u)=|u|^{m-2}u\\ &+\lambda |u|^{q-2}u, \; {\rm in}\; \mathbb{R}^N \\ &u(x)>0, \;\;{\rm in}\;\; \mathbb{R}^N,\;\; u\in W^{1,p}(\mathbb{R}^N), \end{eqnarray} where $a, \mu>0, \tau\ge 0, \lambda\in \mathbb{R} $ and $1 < p < N, p < q < m < p^*=\frac{pN}{N-p}$. The potential $V(x)\in C(\mathbb{R}^N)$ and $0 < \inf_{x\in\mathbb{R}^N}V(x) < \sup_{x\in\mathbb{R}^N}V(x) < \infty$. The existence of solution will be obtained by the Nehari manifold and variational method.
Keywords: variational method., $p-$Kirchhoff elliptic equation, the Nehari manifold.
Mathematics Subject Classification: 35J50, 35J75, 35J9.
Citation: Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289
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.  Google Scholar

[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.  Google Scholar

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A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential and Integral Equations}, 18 (2005), 1321.   Google Scholar

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M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners,, 1$^{nd}$ edition, (2011).  doi: 10.1007/978-0-85729-227-8.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics Vol.19, (1998).   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, \emph{Differential and Integral Equations}, 18 (2005), 1321.   Google Scholar

[4]

M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners,, 1$^{nd}$ edition, (2011).  doi: 10.1007/978-0-85729-227-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics Vol.19, (1998).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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