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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

 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.
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
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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, J. Differential Equations, 254 (2013), 1977-1991. 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, J. Eur. Math. Soc., 7 (2005), 117-144. doi: 10.4171/JEMS/24.  Google Scholar [3] A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential and Integral Equations, 18 (2005), 1321-1332.  Google Scholar [4] M. Badiale and E. Serra, Semilinear Elliptic Equations for Beginners, 1$^{nd}$ edition, Springer-Verlag, London, 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, Arch. Rational Mech. Anal., 82 (1983), 313-345. 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, J. Eur. Math. Soc., 8 (2006), 217-228. 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 $\mathbbR^N$, Computers and Mathematics with Applications, 6 (2013), 1909-1919. 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, Applied Mathematics Letters, 28 (2014), 82-87. 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 $\mathbbR^N$, Nonlinear Analysis: Real World Applications, 14 (2013), 1477-1486. 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 $\mathbbR^N$, Nonlinear Analysis, 86 (2013), 146-156. 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, Arch. Rational Mech. Anal., 91 (1986), 283-308. doi: 10.1007/BF00282336.  Google Scholar [12] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics Vol.19, $1^{nd}$ edition, American Mathematical Society, Providence, 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, J. Differential Equations, 253 (2012), 2285-2294. 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, Ann. Inst. H. Poincaré Anal. Non Linéire, 23 (2006), 829-837. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  Google Scholar [16] W. Liu and X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. 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, Nonlinear Analysis, 75 (2012), 3470-3479. 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 $\mathbbR^N$, Nonlinear Analysis: Real World Applications, 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023.  Google Scholar [19] L. Wang, On a quasilinear Schrödinger-Kirchhoff-type equation with radial potentials, Nonlinear Analysis, 83 (2013), 58-68. doi: 10.1016/j.na.2012.12.012.  Google Scholar
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