Existence results for mixed local and nonlocal elliptic equations involving singularity and nonregular data

2024/10/06 by Sekhar Ghosh, Bhowmick, Souvik, Ghosh, Sekhar · 2 citations
Computer Science · Mathematics · #35J20 #35J75 #35M12 #35R06 #35R11 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2410.04441

openalex publication_date 2024/10/06 · openalex created_date 2024/11/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove the existence of weak, veryweak and duality solutions to a class of elliptic problems involving singularity and measure data which is given by: -Δu+(-Δ)^s u = (f(x))/(u^γ) +μ in Ω with the zero Dirichlet boundary data u=0 in \mathbb R^N ∖ Ω. The existence of weak solutions is obtained by approximating a sequence of problems for 0<γ≤1 and γ>1. We employ Schauder's fixed point theorem and embeddings of Marcinkiewicz spaces. The novelty of our work is that we prove the existence of a duality solution and its equivalence with weak solutions to the problem Lu=μ. Moreover, we prove a veryweak maximum principle and a Kato-type inequality for the mixed local-nonlocal operator L=-Δ+(-Δ)^s, which are crucial tools to guarantee the existence of veryweak solutions to the problem. Using a Kato-type inequality, maximum principle together with sub-super solution method, we prove the existence of veryweak solution for 0<γ<1. Our work extends the studies due to Oliva and Petitta [ESAIM Control Optim. Calc. Var., 22(1):289--308, 2016.] and Petitta [Adv. Nonlinear Stud., 16(1):115--124, 2016.] for the mixed local-nonlocal operator.

Cited by

• Regularity and existence for a mixed local-nonlocal parabolic equation with variable singularities and measure data
• Existence Theory for a class of semilinear mixed local and nonlocal equations involving variable singularities and singular measures

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