论文标题
带有振荡和奇异数据的准线性riccati类型方程
Quasilinear Riccati type equations with oscillatory and singular data
论文作者
论文摘要
我们表征了quasilinear riccati类型方程的解决方案的存在\ begin {eqnarray*} \ left \ {\ oken {array} {rcl} {rcl} - {\ rm div} \ \ rm div} \，\ nathcal \ nathcal {a} \ text {in} 〜Ω，\\ u＆=＆0 \ quad \ text {on}〜\ partialω，\ end end {array} \ right。 \ end {eqnarray*}带有分配或量度基准$σ$。这里$ {\ rm div} \，\ Mathcal {a}（x，x，\ nabla u）$是一种以$ p $ -laplacian（$ p> 1 $）为模型的准省利操作员，而$ phaplacian（$ p> 1 $），而$ω$是一个有界的域，其边界足够平坦（在RefifeNberg的含义上）。对于分销数据，我们假设$ p> 1 $和$ q> p $。对于测量数据，我们假设它们在$ω$，$ p> \ frac {3n-2} {2n-1} $中得到紧凑的支持，而$ q $在$ p-1 $ p-1 <q <1 $中。在这种情况下，我们还假设$ \ Mathcal {a} $和$ \partialΩ$上的更多规律性条件。
We characterize the existence of solutions to the quasilinear Riccati type equation \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}\,\mathcal{A}(x, \nabla u)&=& |\nabla u|^q + σ\quad \text{in} ~Ω, \\ u&=&0 \quad \text{on}~ \partial Ω, \end{array}\right. \end{eqnarray*} with a distributional or measure datum $σ$. Here ${\rm div}\,\mathcal{A}(x, \nabla u)$ is a quasilinear elliptic operator modeled after the $p$-Laplacian ($p>1$), and $Ω$ is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that $p>1$ and $q>p$. For measure data, we assume that they are compactly supported in $Ω$, $p>\frac{3n-2}{2n-1}$, and $q$ is in the sub-linear range $p-1<q<1$. We also assume more regularity conditions on $\mathcal{A}$ and on $\partialΩ$ in this case.
