Remarks on some quasilinear equations with gradient terms and measure data

2012/11/28 by Marie-Françoise Bidaut-Véron, Bidaut-Véron, Marie-Françoise, Marta Garcia-Huidobro +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1211.6542

To appear in Contemporary Mathematics

arxiv created 2013/02/13 · arxiv updated 2013/02/14

Abstract

Let Ω⊂ ℝ^N be a smooth bounded domain, H a Caratheodory function defined in Ω× \mathbbR× R^N, and μ a bounded Radon measure in Ω. We study the problem% -Δ_pu+H(x,u,∇ u)=μ inΩ, u=0 on∂ Ω, where Δ_p is the p-Laplacian (p>1), and we emphasize the case H(x,u,∇ u)=± ‖ ∇ u‖ ^q (q>0). We obtain an existence result under subcritical growth assumptions on H, we give necessary conditions of existence in terms of capacity properties, and we prove removability results of eventual singularities. In the supercritical case, when μ\geqq 0 and H is an absorption term, i.e. % H\geqq 0, we give two sufficient conditions for existence of a nonnegative solution.

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