A uniqueness result for a singular elliptic equation with gradient term. Issue 5 (22nd June 2018)

• Record Type: Journal Article
• Authors: Carmona, José; Leonori, Tommaso
• Is Part Of: Proceedings. Volume 148:Issue 5(2018)
• Page Start: 983
• Page End: 994
• Publication Date: 2018-06-22
• Subjects: nonlinear elliptic equations, -- singular natural growth gradient terms, -- comparison principle; 35J65, -- 35J75, -- 35A02; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0308210518000112
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : We prove the uniqueness of a solution for a problem whose simplest model is with k ≥ 1, 0 f ∈ L ∞ ( Ω ) and Ω is a bounded domain of ℝ N, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λu q with 0 < q < 1 and λ > 0.