主标题：微分方程理论、计算与应用讨论班（5）

系列讲座：“Introduction for the Theory of Nonlinear Integral Equations”-I&II

报告题目：The Existence Results for a Class of Integral Equations

报告人：湖南农业大学理学院 许建开（副教授）

报告时间：12月10日下午14:00-15:30

报告地点：数计学院4号楼229会议室

摘要: In this topic, we consider the following integral equations

u(x)=\\int_{\\mathbb{R}^n}\\frac{v^{q}(y)}{|x-y|^{\\lambda}}dy,

v(x)=\\int_{\\mathbb{R}^n}\\frac{u^{p}(y)}{|x-y|^{\\lambda}}dy,

for $p<0 $, $q<0$ and $ \\lambda\\in (-\\infty, 0)$. we establish the sharp criteria for the existence of positive solution to the system above and show that if $(u,v)$ is a pair of non-negative lebesgue measurable solution of the integral system, then $$\\frac{1}{p+1}+\\frac{1}{q+1}="\\frac{\\lambda}{n}$$

which is distinct from the originated Hardy-Littlewood-Sobolev type integral system. Moreover, we also get the explicit form of positive solution for the integral system in the case: $p=q$.

报告题目：Radial Symmetry and Asymptotic Behaviors of Positive Solutions for Certain Nonlinear Integral Equations

报告人：湖南农业大学理学院 许建开（副教授）

报告时间：12月10日下午16:00-17:30

报告地点：数计学院4号楼229会议室

摘要:  In this topic, we investigate some properties of positive solutions for a nonlinear integral system. Using the method of moving planes in an integral form, which was recently introduced by Chen, Li, and Ou in [Comm.Partial Diff. Equations 30 (2005), 59-65; Comm. Pure Appl. Math. 54 (2006), 330-343], we show that all positive solutions of the integral system in certain functional spaces are radially symmetric with respect to the origin and decreasing in critical space. Furthermore, with the help of regularity lifting lemma, we obtain the optimal integrable intervals and the asymptotic estimates for such solutions around the origin and near infinity.