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Ground State and Multiple Solutions for Kirchhoff Type Equations with Critical Exponent

Published:2017-09-08
Printed: Jun 2018
• Dongdong Qin,
School of Mathematics and Statistics, Central South University, Changsha, Hunan, P.R.C.
• Yubo He,
School of Mathematics and Statistics, Central South University, Changsha, Hunan, P.R.C.
• Xianhua Tang,
School of Mathematics and Statistics, Central South University, Changsha, Hunan, P.R.C.
 Format: LaTeX MathJax PDF

Abstract

In this paper, we consider the following critical Kirchhoff type equation: \begin{align*} \left\{ \begin{array}{lll} - \left(a+b\int_{\Omega}|\nabla u|^2 \right)\Delta u=Q(x)|u|^4u + \lambda |u|^{q-1}u,~~\mbox{in}~~\Omega, \\ u=0,\quad \text{on}\quad \partial \Omega, \end{array} \right. \end{align*} By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\lt q\lt 5$. The relation between the number of maxima of $Q$ and the number of positive solutions for the problem is also investigated.
 Keywords: Kirchhoff type equation, variational methods, critical exponent, Nehari manifold, ground state
 MSC Classifications: 35J20 - Variational methods for second-order elliptic equations 35J60 - Nonlinear elliptic equations 35J25 - Boundary value problems for second-order elliptic equations

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