Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

2023/05/20 by Biagio Ricceri, Ricceri, Biagio · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2305.12180

openalex publication_date 2023/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω⊂ \bf Rⁿ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let q∈ ]0,1[, α∈ L^∞(Ω), with α>0, and k∈ \bf N. Then, the problem \cases -tan(∫_Ω|∇ u(x)|²dx)Δu= α(x)u^q amp; in Ω\cr amp; \cr ugt;0 amp; in Ω\cr amp; \cr u=0 amp; on ∂ Ω \cr amp; \cr (k-1)πlt;∫_Ω|∇ u(x)|²dxlt;(k-1)π+π\over 2 \cr has a unique weak solution u which is the unique global minimum in H¹₀(Ω) of the functional u→ 1\over 2tan (∫_Ω|∇ u(x)|²dx)∫_Ω|∇ u(x)|²dx-1\over q+1∫_Ωα(x)|u⁺(x)|^q+1dx , where u⁺=max\0,u\.

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