报告题目1：On the ground states of the Schrödinger–Poisson–Slater equation

报告人：刘增副教授，苏州科技大学

报告时间：2023年12月23日（周六）9:00-10:00

报告地点：20-306

报告摘要：In this talk, I will report our recent results on the Schrödinger–Poisson–Slater equation

where , ,  and  is the Newtonian potential.  For , we prove the nonexistence of nontrivial solutions, existence of positive nonradial (and radial) ground state solutions. Furthermore, we show that a symmetry breaking occurs for the ground state solutions. For  and , we obtain a characterization of the limit profile of the positive ground states for by using direct variational analysis based on the comparison of the ground state energy levels. It is a joint work with Vitaly Moroz, Yisheng huang and Yuejuan Tang.

报告人简介：刘增，苏州科技大学副教授、硕士生导师。2014年于苏州大学获博士学位。主要研究领域为非线性分析和非线性椭圆型偏微分方程。2018年9月-2019年3月到英国斯旺西大学访问Vitaly Moroz教授。主持完成国家自然科学基金青年基金、数学天元基金各1项，参与国家自然科学基金面上项目等各类项目5项。主要成果发表在Calc. Var. PDEs，Potential Analysis，Nonlinear Anal.，Topol. Methods Nonlinear Anal., Adv. Nonlinear Stud.，Pro. Amer. Math. Soc.等国际重要学术刊物上。

报告题目2：Non-existence and multiplicity of positive solutions for Choquard type equations with combined nonlinearities

报告人：马世旺教授，南开大学

报告时间：2023年12月25日（周一）10:00-11:00

报告地点：20-306

报告摘要：We study the non-existence and multiplicity of positive  solutions of the nonlinear  Choquard type equation $$-\Delta u+ \varepsilon u=(I_\alpha \ast |u|^{p})|u|^{p-2}u+ |u|^{q-2}u, \quad {\rm in} \ \mathbb R^N, \eqno(P_\varepsilon)$$ where $N\ge 3$ is an integer, $p\in [\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}]$, $q\in (2,\frac{2N}{N-2}]$, $I_\alpha$ is the Riesz potential of order $\alpha\in (0,N)$ and  $\varepsilon>0$ is  a  parameter.  We fix one of  $p,q$ as a critical exponent (in the sense of Hardy-Littlewood-Sobolev and Sobolev inequalities ) and  view the others in $p,q,\varepsilon, \alpha$ as parameters, we find regions in the $(p,q,\alpha, \varepsilon)$-parameter space, such that the corresponding equation has no positive ground state or admits multiple positive solutions. This is a counterpart of the Brezis-Nirenberg Conjecture (Brezis-Nirenberg, CPAM, 1983) for nonlocal elliptic equation in the whole space. Particularly, some  threshold results for the existence of ground states and some conditions which insure  two positive solutions are obtained.  These results are quite different in nature from the corresponding local equation with combined powers nonlinearity and reveal the  special influence of the nonlocal term. To the best of our knowledge, the only two papers concerning the multiplicity of positive solutions of elliptic equations with critical growth nonlinearity  are  given by Atkinson, Peletier (1986) for elliptic equation on a ball and Juncheng Wei, Yuanze Wu (2022) for elliptic equation with a combined powers nonlinearity in the whole space. The ODE technique is main ingredient in the proofs of the above mentioned papers, however, ODE technique does not work any more in our model equation due to the presence of the nonlocal term.

报告人简介：马世旺，1997年于湖南大学获得理学博士学位，先后工作于上海交通大学、南开大学，现为南开大学数学学院教授、博士生导师。主持完成多项国家自然科学基金项目，目前为国家自然科学基金委员会、教育部学位与研究生教育发展中心通讯评审专家，中国生物数学会理事。马世旺教授的研究领域为非线性分析、微分方程与动力系统，在诸如发展方程行波解、微分方程多解性等问题研究中取得一些重要成果，目前已在包括 JDE、JDDE、DCDS 等期刊发表学术论文90余篇。这些研究成果被同行广泛引用，单篇论文最高引用次数超过100次。

邀请人：非线性分析与PDE团队