江苏省应用数学（中国矿业大学）中心系列学术报告
报告题目：On positive normalized solutions to a mass mixed coupled Schrödinger system with Sobolev critical exponent
报告人：钟学秀
报告时间：2024年7月1日（周一）下午15:30-16:30
报告地点：数学学院A302
主持人：范海宁
报告摘要：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 $\nu\rightarrow 0^+$.
报告人简介：钟学秀，2015年博士毕业于清华大学，师从邹文明教授。2015-2017年于台湾大学博士后；2017-2019年于中山大学专职科研人员；2019年至今为华南师范大学副研究员，华南数学应用与交叉研究中心青年拔尖引进人才，最新ESI高被引学者。研究方向为运用非线性分析、变分法等方法来研究几何分析学、数学物理中椭圆型偏微分方程和方程组以及某些不等式问题。主持国家青年基金和面上基金各一项。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等国际重要刊物上发表多篇学术论文。在非线性泛函分析和椭圆偏微分方程领域的Li-Lin 公开问题，Sirakov 公开问题，Bartsch-Jeanjean-Soave公开问题等方面获得了重要进展。