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Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent

  • Zhisu Liu
  • Shangjiang Guo
Article

Abstract

In this paper, we consider the following semilinear Kirchhoff type equation
$$\left\{\begin{array}{ll}-\left(\epsilon^2a + \epsilon b \int \limits_{\mathbb{R}^3}|\nabla u|^2 \right) \triangle {u}+V(x)u = \mu K(x)|u|^{p-1}u + Q(x)|u|^4u, \,\, \mathrm{in}\, \mathbb{R}^3, \\ u \in H^1(\mathbb{R}^3), \,\, u > 0,\end{array}\right.$$
where \({\epsilon > 0}\) is a small parameter, \({p \in [3,5)}\), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u|4 u reaches the Sobolev critical exponent since 2* = 6 for three spatial dimensions. We prove the existence of a positive ground state solution \({u_\epsilon}\) with exponential decay at infinity for μ > 0 and \({\epsilon}\) sufficiently small under some suitable conditions on the nonnegative functions V, K and Q. Moreover, \({u_\epsilon}\) concentrates around a global minimum point of V as \({\epsilon \rightarrow 0^+}\). The methods used here are based on the concentration-compactness principle of Lions.

Mathematics Subject Classification (2010)

35J60 35J65 53C35 

Keywords

Kirchhoff type equation Critical Sobolev exponent Concentration-compactness principle Variational method 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan University, ChangshaHunanPeople’s Republic of China

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