Topological and Variational Methods for Partial Differential Equations

In recent years, there has been a major impact of topological and variational methods on the study of nonlinear elliptic and parabolic partial differential equations. In particular, surprising results for classical open problems have been obtained with new techniques far beyond the classical approaches. The purpose of the meeting was to provide a forum for these developments and to bring together experts with complementary background

Direct Method of Scaling Spheres for the Laplacian and Fractional Laplacian Equations with Hardy-Henon Type Nonlinearity

In this paper, we focus on the partial differential equation \begin{equation*} (-\Delta)^\frac{\alpha}{2} u(x)=f(x,u(x))\;\;\;\;\text{ in }\mathbb{R}^n, \end{equation*} where $0<\alpha\leq 2$. By the direct method of scaling spheres investigated by Dai and Qin (\cite{dai2023liouville}, \textit{International Mathematics Research Notices, 2023}), we derive a Liouville-type theorem. This mildly extends the previous researches on Liouville-type theorem for the semi-linear equation $(-\Delta)^\frac{\alpha}{2} u(x)=f(u(x))$ where the nonlinearity $f$ depends solely on the solution $u(x)$, and covers the Liouville-type theorem for Hardy-H\'enon equations $(-\Delta)^\frac{\alpha}{2} u(x)=|x|^au^p(x)$