Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

Abstract

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<q<1} . We also assume more regularity conditions on 𝒜 {\mathcal{A}} and on ∂ ⁡ Ω ⁢ Ω {\partial\Omega\Omega} in this case.