Existence and L‐estimates for elliptic equations involving convolution. Issue 5 (24th June 2020) - Marino, Greta; Motreanu, Dumitru
Abstract : In this article, with a fixed p ∈ (1, + ∞ ) and a bounded domain Ω ⊂ ℝ N, N ≥2, whose boundary ∂ Ω fulfills the Lipschitz regularity, we study the following boundary value problem − div 𝒜 ( x, u, ∇ u ) + a | u | p − 2 u = ℬ ( x, ρ ∗ E ( u ), ∇ ( ρ ∗ E ( u ) ) ) in Ω, 𝒜 ( x, u, ∇ u ) · ν = 𝒞 ( x, u ) on ∂ Ω, ( P ) where 𝒜 : Ω × ℝ × ℝ N → ℝ N, ℬ : Ω × ℝ × ℝ N → ℝ, 𝒞 : ∂ Ω × ℝ → ℝ are Carathéodory functions, a > 0 is a constant, E : W 1, p ( Ω ) → W 1, p ( ℝ N ) is an extension operator related to Ω, and ρ is an integrable function on ℝ N . This is a novel problem that involves the nonlocal operator assigning to u the convolution ρ ∗ E ( u ) of ρ with E ( u ). Under verifiable conditions, we prove the existence of a (weak) solution to problem (P) by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to (P) is bounded.