Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

2024/11/29 by Sarbani Pramanik, Pramanik, Sarbani
Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical methods in engineering #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2411.19694

openalex publication_date 2024/11/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal p-Laplacian operator with singular nonlinearity: \beginaligned -Δ_p u + (-Δ_p)^s u = λ(f(u))/(u^β) in Ω\newline u gt;0 in Ω, u =0 in ℝ^N ∖ Ω\endaligned where Ω⊂ ℝ^N is a smooth bounded domain, λ> 0 is a parameter, 0≤ β<1 and f is a non-decreasing C¹-function with f(0)>0. Using the method of sub- and supersolutions, we present a novel multiplicity result and, in specific cases, we also prove a three-solution theorem using Amann's fixed point theorem. Our construction of sub-supersolutions avoids the conventional reliance on ODE techniques and Green's function estimates, thereby making it more adaptable to the nonlinear and nonlocal framework. Additionally, we establish a Hopf-type Strong Comparison Principle for the linear operator with singular nonlinearity, marking the first result of its kind for mixed local-nonlocal operators. This result is crucial in deriving a third solution and holds broader mathematical significance.

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