W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) versus C 0 1 ( Ω ) × C 0 1 ( Ω ) local minimizers

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-05-09 DOI:10.3233/asy-241911

João Pablo P. Da Silva

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.

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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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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