Abstract We characterize the existence of solutions to the quasilinear Riccati-type equation { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) = | ∇ ⁡ u | q + σ in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with a distributional or measure datum σ. Here div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ( p > 1 {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 {p>1} and q > p {q>p} . For measure data, we assume that they are compactly supported in Ω, p > 3 ⁢ n - 2 2 ⁢ n - 1 {p>\frac{3n-2}{2n-1}} , and q is in the sub-linear range p - 1 < q < 1 {p-1

中文翻译：

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具有振荡和奇异数据的拟线性 Riccati 型方程

摘要 我们刻画了拟线性 Riccati 型方程 { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) = | 解的存在性。∇ ⁡ u | q + σ in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle= |\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{对齐}\right。具有分布或测量数据 σ。这里 div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) {\operatorname{div}\mathcal{A}(x,\nabla u)} 是一个拟线性椭圆算子，它模仿 p-Laplacian ( p > 1 {p>1 } )，Ω 是一个边界足够平坦的有界域（在 Reifenberg 的意义上）。对于分布数据，我们假设 p > 1 {p>1} 和 q > p {q>p} 。对于测量数据，