Abstract view
Ground state and multiple solutions for Kirchhoff type equations with critical exponent


Yubo He,
School of Mathematics and Statistics, Central South University, Changsha, Hunan, P.R.C.
Dongdong Qin,
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.
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^{q1}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.