题目：Solutions of Semilinear Equations on the Integer Lattice Graphs

报告人：陈虎元，上海数学与交叉学科研究

时间：10月13日，14:30-15:20

地点：新楼308

摘要：In this talk, I will present results on positive solutions to semilinear elliptic equations on lattice graphs, focusing on two specific models.

The first part concerns Kazdan–Warner-type equations $$-\Delta u=\epsilon e^{ku} +\beta \delta_0$$ on the two-dimensional integer lattice graph $\mathbb{Z}^2$. We establish the existence of a continuous family of finite-energy solutions.

In the second part, we study the existence and nonexistence of positive solutions to the Lane–Emden equation $$ -\Delta u = Q u^p $$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$, as well as in half-space and quadrant domains, with zero Dirichlet boundary conditions imposed in the latter two cases.