Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng¹ ; Chun Lei Tang¹

¹ School of Mathematics and Statistics Southwest University 400715 Chongqing, People’s Republic of China

Annales Polonici Mathematici, Tome 117 (2016) no. 2, pp. 163-179

Résumé

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.

Détail

DOI : 10.4064/ap3783-1-2016

Keywords: consider following kirchhoff type problem involving critical nonlinearity begin equation* begin cases displaystyle bigg bigg int omega nabla bigg bigg delta ast text omega text partial omega end cases end equation* where omega subset mathbb geq smooth bounded domain smooth boundary partial omega geq n under appropriate assumptions existence positive ground state solution via variational method