论文标题
在没有均匀性的某些均方根方程的淋巴结解决方案中
On the nodal set of solutions to some sublinear equations without homogeneity
论文作者
论文摘要
我们研究了对不稳定的Alt-Phillips类型问题的结构集结构\ [-ΔU=λ_+（U^+）^{p-1} {p-1}-λ_-（u^ - ）^{q-1} \]其中$ 1 \ le p <q <q <q <q <q <q <q <q <q <q <q <q <q <q <q <2 $，$λ_+>λ_+> 0 $，$ c $，$λ__-该方程的特征在于右手方的均方根不均匀特征，这使得难以从自由边界问题（例如单调性公式和爆炸论点）中以标准方式适应经典工具。我们的主要结果是：接近节点集的解决方案的局部行为；可接受的消失命令的完整分类，以及对当地最小化器的奇异套件的Hausdorff尺寸的估计；退化（不是局部最小）溶液的存在。
We investigate the structure of the nodal set of solutions to an unstable Alt-Phillips type problem \[ -Δu = λ_+(u^+)^{p-1}-λ_-(u^-)^{q-1} \] where $1 \le p<q<2$, $λ_+ >0$, $λ_- \ge 0$. The equation is characterized by the sublinear inhomogeneous character of the right hand-side, which makes difficult to adapt in a standard way classical tools from free-boundary problems, such as monotonicity formulas and blow-up arguments. Our main results are: the local behavior of solutions close to the nodal set; the complete classification of the admissible vanishing orders, and estimates on the Hausdorff dimension of the singular set, for local minimizers; the existence of degenerate (not locally minimal) solutions.
