*主持人：叶东 教授
*时间：2020年10月16日13：00-14:00
*地点：腾讯会议ID：635 804 429


*讲座内容简介：
In this talk, we consider the semilinear Schr\"odinger equation
$$
\left\{
\begin{array}{ll}
-\triangle u+V(x)u=f(x, u), \ \ \ \ x\in {\R}^{N},\\
u\in H^{1}({\R}^{N}),
\end{array}
\right.
$$
where both $V(x)$ and $f(x, u)$ are periodic in $x$, $0$ belongs to a spectral gap of $-\triangle + V$, and $f(x, u)$ is subcritical and allowed to be super-linear at some $x\in \R^N$ and asymptotically linear at the other $x\in \R^N$.
In the existing literature, it is commonly assumed that $\lim_{|u|\to \infty}\frac{\int_{0}^{u}
f(x, s)\mathrm{d}s}{u^2}=\infty$, uniformly in $x\in \R^N$ to obtain the existence of nontrivial solutions or ground state solutions or infinitely many geometrically distinct solutions. For the first time, we prove the existence of nontrivial solutions or ground state solutions or infinitely many geometrically distinct solutions under the weaker super-quadratic condition $\lim_{|u|\to \infty}\frac{\int_{0}^{u} f(x, s)\mathrm{d}s}{u^2}=\infty$, a.e. $x\in G$ just for some domain $G\subset\R^N$.


*主讲人简介：
唐先华，中南大学教授、博导，研究领域为非线性分析，在《SIAM J. Math. Anal.》《J. Differential Equations》《Calc.Var.Partial Differential Equations》《J. Dynam.Differential Equations》《Nonlinearity》《J. London Math. Soc.》《Discret. Cont. Dyn. Sys.-A》、Proc. Royal Soc. Edinburgh (A)》等刊物上发表了论文300余篇，文章SCI引用3600余次；2014-2018连续5年进入Elesevier发布的中国高被引学者榜单，2018年首次进入科睿唯安（Clarivate Analytics）发布“高被引科学家”名单。先后主持完成5项国家自然科学基金上面项目。2010年获湖南省自然科学一等奖(第1完成人)；2012年获批“享受国务院政府特殊津贴专家”称号。