### 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).

Speaker

Mihai Mihăilescu

Research Area
Affiliation

University of Craiova

Date

Tue, 20/11/2012 - 12:00pm to 1:00pm

Venue

RC-4082, Red Centre Building, UNSW