论文标题
部分规律性和liouville定理，用于稳定的各向异性椭圆方程的解决方案
Partial regularity and Liouville theorems for stable solutions of anisotropic elliptic equations
论文作者
论文摘要
We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ Ω\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $ qu：= \ sum_ {i = 1}^{n} \ frac {\ partial} {\ partial x_ {i}}}（f（\ nabla u）f_ {之一f} {\partialξ_{i}} $和$ f：\ mathbb {r}^{n} \ rightArow [0，+\ infty）$是$ c^{2}（\ m马理{r}^{r}^{n}^{n} \ setMinus puthies puthies convex函数。对于有界域$ω$以及上述方程的稳定弱解决方案，我们证明，单数集的Hausdorff尺寸不超过$ n-10 $。在整个空间中，我们在上下文中应用了由舞者 - 法琳娜和crandall-rabinowitz建立的Moser迭代论点，以分别证明liouville定理用于稳定的解决方案，并分别在尺寸$ n <10 $和$ 2 <n <10 $的尺寸上为有限的Morse索引解决方案。我们还提供了一个明确的解决方案，该解决方案在$ n = 2 $中的紧凑型集合外稳定。此外，我们为功率类型的非线性提供类似的liouville定理。
We study the quasilinear elliptic equation \begin{equation*} -Qu=e^u \ \ \text{in} \ \ Ω\subset \mathbb{R}^{N} \end{equation*} where the operator $Q$, known as Finsler-Laplacian (or anisotropic Laplacian), is defined by $$Qu:=\sum_{i=1}^{N}\frac{\partial}{\partial x_{i}}(F(\nabla u)F_{ξ_{i}}(\nabla u)),$$ where $F_{ξ_{i}}=\frac{\partial F}{\partialξ_{i}}$ and $F: \mathbb{R}^{N}\rightarrow[0,+\infty)$ is a convex function of $ C^{2}(\mathbb{R}^{N}\setminus\{0\})$, that satisfies certain assumptions. For bounded domain $Ω$ and for a stable weak solution of the above equation, we prove that the Hausdorff dimension of singular set does not exceed $N-10$. For the entire space, we apply Moser iteration arguments, established by Dancer-Farina and Crandall-Rabinowitz in the context, to prove Liouville theorems for stable solutions and for finite Morse index solutions in dimensions $N<10$ and $2<N<10$, respectively. We also provide an explicit solution that is stable outside a compact set in $N=2$. In addition, we provide similar Liouville theorems for the power-type nonlinearities.
