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

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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

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

Received September 2013 Revised January 2014 Published July 2014

Abstract
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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:

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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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