Existence, non-existence and degeneracy of limit solutions to p-Laplace problems involving Hardy potentials as p→1⁺. The case of a critical drift

2024/07/18 by Juan Carlos Ortiz Chata, Chata, Juan Carlos Ortiz, Francesco Petitta +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2407.13411

openalex publication_date 2024/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we analyze the asymptotic behaviour as p→ 1⁺ of solutions u_p to \ -Δ_pu · amp;= · amp;λ|∇ u|^p-2∇ u⋅(x)/(|x|²)+ f · amp; in Ω,
u_p · amp;= · amp;0 · amp; on ∂Ω,. where Ω is a bounded open subset of ℝ^N with Lipschitz boundary containing the origin, λ∈ℝ, and f is a nonnegative datum in L^N,∞(Ω). As a consequence, under suitable smallness assumptions on f and λ, we show sharp existence results of bounded solutions to the Dirichlet problems \begincases - Δ₁ u = λ(D u)/(|D u|)⋅ (x)/(|x|²)+f amp; in Ω, u=0 amp; on ∂ Ω, \endcases where Δ₁u=\hboxdiv ((Du)/(|Du|)) is the 1-Laplacian operator. The case of a generic drift term in L^N,∞(Ω) is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.

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