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.
Highlights
Introduction and main resultsWe address in this note the question of existence for the quasilinear Riccati type equation (1.1)−div A(x, ∇u) = |∇u|q + σ in Ω, u = 0 on ∂Ω, where the datum σ is generally a signed distribution given on a bounded domain Ω ⊂ Rn, n ≥ 2.In (1.1) the nonlinearity A : Rn × Rn → Rn is a Caratheodory vector valued function, i.e., A(x, ξ) is measurable in x for every ξ and continuous in ξ for a.e. x
See [4, 5, 11, 12] in which the case q = p is considered. Note that in this special case, the Riccati type equation −div A(x, ∇u) = |∇u|p + σ is strongly related to the Schrodinger type equation −div A(x, ∇u) = σ|u|p−2u
Main results: The first main result of this paper is to treat (1.1) with oscillatory data in the framework of the natural space
Summary
Η|2 for any (ξ, η) ∈ Rn × Rn \ (0, 0) and a.e. x ∈ Rn. The special case A(x, ξ) = |ξ|p−2ξ gives rise to the standard p-Laplacian ∆pu = div (|∇u|p−2∇u). The special case A(x, ξ) = |ξ|p−2ξ gives rise to the standard p-Laplacian ∆pu = div (|∇u|p−2∇u) Note that these conditions imply that A(x, 0) = 0 for a.e. x ∈ Rn, and. Λ−1|ξ|p−2|λ|2 for every (λ, ξ) ∈ Rn × Rn \ {(0, 0)} and a.e. x ∈ Rn. More regularity conditions will be imposed later on the nonlinearity A(x, ξ) in the x-variable and on the boundary ∂Ω of Ω. If σ is a signed measure these necessary conditions can be quantified as (1.4).
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