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

In this paper we analyze the asymptotic behaviour as p→ 1⁺ of solutions uₚ to \ arrayrclr -Δₚu&=&λ|∇ u|ᵖ⁻²∇ u/|x|²+ f& in Ω,\\ uₚ&=&0 & on ∂Ω, array. where Ω is a bounded open subset of RN with Lipschitz boundary containing the origin, λ, and f is a nonnegative datum in LN,∞(Ω). As a consequence, under suitable smallness assumptions on f and λ, we show sharp existence results of bounded solutions to the Dirichlet problems cases - Δ₁ u = u/|D u|· x/|x|²+f & in\, Ω, u=0 & on\ ∂ Ω, cases where Δ₁u=div\,(Du/|Du|) is the $1$-Laplacian operator. The case of a generic drift term in LN,∞(Ω) is also considered. Explicits examples are given in order to show the optimality of the main assumptions on the data.