Classification of isolated singularities for equations involving the Finsler-Laplace operator

Abstract

In this talk, we consider anisotropic elliptic operators such as

where $H$ and $H^{\circ }$ are polar Finsler norms on ${ℝ}^N$ ($N\ge 3$) and $-\infty <\lambda \le {\left(N-2\right)}^2∕4$. When $H\left(x\right)=\left|x\right|$, where $\left|x\right|$ denotes the euclidian norm on ${ℝ}^N$, operator $L_{\lambda ,H}u$ becomes the classical Hardy-Sobolev operator$-\Delta u-\frac{\lambda }{|x{|}^2}u$. We completely classify the behavior near the origin for all positive weak solutions of $L_{\lambda ,H}u=0$ in $\left\{x\in {ℝ}^N:0<H^{\circ }\left(x\right)<1\right\}$. We establish that either $u∕{\Phi }_{\lambda }^{+}\to {\gamma }^{+}\in \left(0,\infty \right)$ or $u∕{\Phi }_{\lambda }^{-}\to {\gamma }^{-}\in \left(0,\infty \right)$, as $\left|x\right|\to 0$, where ${\Phi }_{\lambda }^{\pm }$ denote the fundamental solutions of $L_{\lambda ,H}u=0$. This is a joint work with F. Cīrstea.