Existence of solutions of a class of nonlinear singular equations in Lorentz spaces

Abstract

We consider the following nonlinear elliptic Dirichlet problem involving a Leray-Lions type differential operator − div(𝜓(𝑥, 𝑢(𝑥), ∇𝑢(𝑥))) + 𝑎(𝑥)𝑢(𝑥) = 𝑓(𝑥), in Ω, 𝑢 ∈ 𝑊1,𝑝 0 (Ω), where Ω ⊂ ℝ𝑁 is a bounded domain with smooth boundary, 2 ≤ 𝑝<𝑁, 𝑎 ∈ 𝐿∞ loc(Ω; ℝ+ 0 ) and 𝑓 ∈ 𝐿𝑞,𝑞1 (Ω) is a function in a Lorentz space. We show the existence of a solution 𝑢 ∈ 𝑊1,𝑝 0 (Ω) ∩ 𝐿𝑟,𝑠(Ω) and an a priori estimate for the solution with respect to the Lorentz space norm of 𝑓 ∈ 𝐿𝑞,𝑞1 (Ω), for suitable values 𝑝, 𝑞, 𝑞1, 𝑟 and 𝑠.