{"title":"W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers","authors":"João Pablo P. Da Silva","doi":"10.3233/asy-241911","DOIUrl":null,"url":null,"abstract":"In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω, for some C > 0, ∀ t , s ∈ R, p < σ i ⩽ p ∗ : = N p / ( N − p ), i = 1 , 2, and 1 < p < N. We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asymptotic Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3233/asy-241911","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω, for some C > 0, ∀ t , s ∈ R, p < σ i ⩽ p ∗ : = N p / ( N − p ), i = 1 , 2, and 1 < p < N. We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ). An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.
W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) 对 C 0 1 ( Ω ) × C 0 1 ( Ω ) 的局部最小值
在这项工作中,我们考虑一个函数 I :W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R 的形式 I ( u , v ) = 1 p ∫ Ω ( |∇ u | p + |∇ v | p ) d x - ∫ Ω H ( x 、u ( x ) , v ( x ) ) d x 其中 Ω ⊂ R N 是一个光滑有界域,max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | }⩽ C ( 1 + | s | σ 1 - 1 + | t | σ 2 - 1 ) a.e. x∈ Ω, 对于某个 C > 0, ∀ t , s∈ R, p < σ i ⩽ p∗ : = N p / ( N - p ), i = 1 , 2, 且 1 < p < N。我们证明 C 0 1 ( Ω ) × C 0 1 ( Ω ) 拓扑中的局部最小值就是 W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) 拓扑中的局部最小值。这一结果的一个重要应用与一类凹凸型非线性系统的解的多重性问题有关。
期刊介绍:
The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand. Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.