江蘇省應用數學（中國礦業大學）中心系列學術報告
報告題目：On positive normalized solutions to a mass mixed coupled Schrödinger system with Sobolev critical exponent
報告人：鐘學秀
報告時間：2024年7月1日（周一）下午15:30-16:30
報告地點：伟德bvA302
主持人：範海甯
報告摘要：I will report a recent work (Joint with Qing Guo, Qihan He and Wei Shuai) concerning positive normalized solutions to a coupled Schr\odinger system
subject to the normalization constraint $$ \int_{\mathbb{R}^N}|u|^2\mathrm{d}x=a, \int_{\mathbb{R}^N}|v|^2\mathrm{d}x=b.$$ Here, $\mu_1,\mu_2, \nu>0$ are given parameters, and $a,b>0$ denote the masses. We are particularly interested in the mass mixed with a Sobolev critical coupled case where $2<p, q<2+\frac{4}{N}, \alpha>1, \beta>1$, and $\alpha+\beta=2^*:=\frac{2N}{N-2}$. For sufficiently small $\nu>0$, we demonstrate that the above system admits two positive solutions, one of which acts as a local minimizer, and the other as a mountain pass solution. This result resolves Soave's open problem [{\it J. Funct. Anal.}, 2020, Remark 1.1] within the context of the system case. Notably, our existence result holds true for all dimensions $N\geq 3$. Our results also significantly extending the result of Gou and Jeanjean[{\it Nonlinearity}, 2018, Theorem 1.1] to the Sobolev critical coupled case and by removing the constraint ``either $p,q\leq \alpha+\beta-\frac{2}{N}$ or $|p-q|\leq \frac{2}{N}$ for $N\geq 5$. Additionally, we also establish a sequence of properties for the local minimizer, including local uniqueness, continuity with respect to the small parameter $\nu$, and the asymptotic behavior as $\nu\rightarrow 0^+$.
報告人簡介：鐘學秀，2015年博士畢業于清華大學，師從鄒文明教授。2015-2017年于台灣大學博士後；2017-2019年于中山大學專職科研人員；2019年至今為華南師範大學副研究員，華南數學應用與交叉研究中心青年拔尖引進人才，最新ESI高被引學者。研究方向為運用非線性分析、變分法等方法來研究幾何分析學、數學物理中橢圓型偏微分方程和方程組以及某些不等式問題。主持國家青年基金和面上基金各一項。目前已在J.Differential Geom.， J. Math. Pures Appl., Math. Ann.， Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)，Calc. Var. PDE，J. Differential Equations等國際重要刊物上發表多篇學術論文。在非線性泛函分析和橢圓偏微分方程領域的Li-Lin 公開問題，Sirakov 公開問題，Bartsch-Jeanjean-Soave公開問題等方面獲得了重要進展。