论文标题
具有一阶项和措施的分数椭圆方程的解决方案的解决方案的条件
A capacity-based condition for existence of solutions to fractional elliptic equations with first-order terms and measures
论文作者
论文摘要
在此手稿中，我们呼吁潜在理论为存在分布解决方案提供足够条件，以通过非线性一阶术语和数据$ω$：$$ \ left \ weak {array} {array} {rcll} {rcll} {rcll}（ - }\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in } \mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$在$ Q $和$ω$上的合适假设下。粗略地说，EXIS \ tesce的条件指出，如果该度量数据由Riesz分数容量局部控制，则有一个全局的方程解决方案。我们还表明，如果存在一个积极的解决方案，那么对于相关的Riesz容量，$ω$必然将是绝对连续的，这给出了这项工作的主要结果的部分倒数。最后，在不同的功能空间中也给出了$ω$的$ U $的估计。
In this manuscript, we appeal to Potential Theory to provide a sufficient condition for existence of distributional solutions to fractional elliptic problems with non-linear first-order terms and measure data $ω$: $$ \left\{ \begin{array}{rcll} (-Δ)^su&=&|\nabla u|^q + ω\quad \text{in }\mathbb{R}^n,\, \,\,s \in (1/2, 1)\\u & > &0 \quad \text{in } \mathbb{R}^{n}\\\lim_{|x|\to \infty}u(x) & =& 0, \end{array} \right. $$under suitable assumptions on $q$ and $ω$. Roughly speaking, the condition for exis\-tence states that if the measure data is locally controlled by the Riesz fractional capacity, then there is a global solution for the equation. We also show that if a positive solution exists, necessarily the measure $ω$ will be absolutely continuous with respect to the associated Riesz capacity, which gives a partial reciprocal of the main result of this work. Finally, estimates of $u$ in terms of $ω$ are also given in different function spaces.
