Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent
Volume 117 / 2016
Annales Polonici Mathematici 117 (2016), 163-179
MSC: Primary 35A15; Secondary 35B09.
DOI: 10.4064/ap3783-1-2016
Published online: 16 May 2016
Abstract
We consider the following Kirchhoff type problem involving a critical nonlinearity: \begin{equation*} \begin{cases} \displaystyle-\bigg[a+b\bigg(\int_{\Omega}|\nabla u|^{2}\,dx\bigg)^{m}\bigg]\Delta u= f(x,u)+|u|^{2^{\ast}-2}u &\text{in $\Omega$,} \\ u=0 &\text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^{N}$ $(N\geq3)$ is a smooth bounded domain with smooth boundary $\partial\Omega$, $a \gt 0$, $b\geq 0,$ and $0 \lt m \lt {2}/({N-2}) $. Under appropriate assumptions on $f$, we show the existence of a positive ground state solution via the variational method.
