【学术报告】2018年6月14日上午罗鹏副教授来我院举办学术讲座

2018-06-06 08:50:38    浏览次数:

\begin{equation*}

-\varepsilon^2\Delta u+ V(x)u=|u|^{p-2}u,~u\in H^1(\mathbb{R}^N),

\end{equation*}

where $\varepsilon>0$ is a small parameter, $N\geq 1$, $2<p<2^*$.

For a class of $V(x)$ which attains the minimum at a closed hypersurface $\Gamma$, we give the location of the concentrated point to the positive ground state solution and establish the local uniqueness of the positive solution with concentration under certain conditions on $V(x)$, which implies the uniqueness of positive ground state solution. Also some partial symmetry can be obtained by the uniqueness. Here our main tools are the local Pohozaev identities and blow-up analysis. And the crucial key is to analysis the precise algebra relation of the concentrated point caused by the degeneracy and inhomogeneity of $V(x)$ at this point.