Partial regularity for weak solutions of anisotropic Lane-Emden equation

Abstract:

We study positive weak solutions of the quasilinear Lane-Emden equation \begin{equation*} -Qu=u^{\alpha } \quad \text {in}\quad \Omega \subset \mathbb {R}^{n}, \end{equation*} where $\alpha \geq \frac {n+2}{n-2}$, for $n\ge 3$, is supercritical and the operator $Q$, known as Finsler-Laplacian or anisotropic Laplacian, is defined by \begin{equation*} Qu≔\sum _{i=1}^{n}\frac {\partial }{\partial x_{i}}(F(\nabla u)F_{\xi _{i}}(\nabla u)). \end{equation*} Here, $F_{\xi _{i}}=\frac {\partial F}{\partial \xi _{i}}$ and $F: \mathbb {R}^{n}\rightarrow [0,+\infty )$ is a convex function of $C^{2}(\mathbb {R}^{n}\setminus \{0\})$, that satisfies positive homogeneity of first order and other certain assumptions. We prove that the Hausdorff dimension of singular set of $u$ is less than $n-2\frac {\alpha +1}{\alpha -1}$.