{"title":"Existence of solutions for 4p-order PDES","authors":"F. Moradi, N. Moradi, M. Addam, S. E. Habib","doi":"10.2478/mjpaa-2022-0013","DOIUrl":null,"url":null,"abstract":"Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ2pu=λm(x)u in Ω,u=Δu=…Δ2p−1u=0 on ∂Ω. \\left\\{ {\\matrix{ {{\\Delta ^{2p}}u = \\lambda m\\left( x \\right)u\\,\\,\\,in\\,\\,\\Omega ,} \\cr {u = \\Delta u = \\ldots {\\Delta ^{2p - 1}}u = 0\\,\\,\\,\\,on\\,\\,\\partial \\Omega .} \\cr } } \\right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.","PeriodicalId":36270,"journal":{"name":"Moroccan Journal of Pure and Applied Analysis","volume":"8 1","pages":"179 - 190"},"PeriodicalIF":0.0000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moroccan Journal of Pure and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/mjpaa-2022-0013","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ2pu=λm(x)u in Ω,u=Δu=…Δ2p−1u=0 on ∂Ω. \left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\Omega ,} \cr {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 0\,\,\,\,on\,\,\partial \Omega .} \cr } } \right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.