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The Existence of Positive Ground State Solutions for the Nonlinear Kirchhoff Type Equations in $R^3$
Prof. Li Gongbao(Central China Normal University)
2018-01-01 12:13  华东师范大学

ABSTRACT OF THE TALK
In this talk, we study the following nonlinear problem of the Kirchhoff type with pure power nonlinearities:
$\left\{ \begin{array}{*{35}{l}}
{} & -\left( a+b\int_{{{R}^{3}}}{{{\left| Du \right|}^{2}}} \right)\Delta u+V\left( x \right)={{\left| u \right|}^{p-1}}u,x\in {{R}^{3}}, \\
{} & u\in {{H}^{1}}\left( {{R}^{3}} \right),u>0,x\in {{R}^{3}}, \\
\end{array} \right.\left( 1 \right)$
where $a$,$b>0$ are constants, $2$$<$$p$$<$$5$ and $V:{{R}^{3}}\to R$. Under certain assumptions on$V$, we prove that $\left( 1 \right)$ has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma. Our main results especially solve problem $\left( 1 \right)$ in the case where$p\in (2,3]$, which has been an open problem for the Kirchhoff equations.
BIOGRAPHY
Gongbao Li is professor of Mathematics at Central China Normal University in Wuhan.