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

Abstract:

In this talk, we consider certain anisotropic elliptic operators such as

\[

  L_{\lambda,H}u:= -{\rm div}(H(\nabla u)(\nabla H)(\nabla u))-\frac{\lambda}{H^\circ(x)^2}u,\;\;\;{\rm on}\;\{x\in\mathbb{R}^N:\;0<H^\circ(x)<1\}\,,

\]where \(H\) and \(H^\circ\) are polar Finsler norms on \(\mathbb{R}^N\) (\(N \geq 3\)) and \(\lambda \leq \frac{1}{4}(N-2)^2\). When  \(H(x) = |x|\), the Euclidian norm on \(\mathbb{R}^N\), the 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 \(\{x\in\mathbb{R}^N:\;0<H^\circ(x)<1\}\), and establish that either \(u/\Phi_\lambda^+\rightarrow\gamma^+\in(0,\infty)\) or \(u/\Phi_\lambda^-\rightarrow\gamma^-\in(0,\infty)\), as \(|x|\rightarrow 0\), where \(\Phi_\lambda^\pm\) denote the fundamental solutions of \(L_{\lambda,H}u = 0\).

This is a joint work with Florica C. Cîrstea (University of Sydney).