H. Hajaiej, Rohit Kumar, Tuhina Mukherjee, Linjie Song
This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in R 2 , − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in R 2 , u , v ⩾ 0 in R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α, β > 1, α + β ⩽ min { 2 s 1 , 2 s 2 }, and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2. The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h. In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s, α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h.
本文主要研究以下局部和非局部型系统解的存在性和不存在性 - ∂ x x u + ( - Δ ) y s 1 u + u - u 2 s 1 - 1 = κ α h ( x , y ) u α - 1 v β in R 2 、- ∂ x x v + ( - Δ ) y s 2 v + v - v 2 s 2 - 1 = κ β h ( x , y ) u α v β - 1 in R 2 , u , v ⩾ 0 in R 2 、其中 s 1 , s 2∈ ( 0 , 1 ) , α, β > 1, α + β ⩽ min { 2 s 1 , 2 s 2 }, 2 s i = 2 ( 1 + s i ) 1 - s i , i = 1 , 2。本文将证明，如果 κ 足够大且 h 满足 (H)，则在亚临界情况下存在基态解。此外，如果 κ 变得非常小，那么我们的系统就没有解。对临界情况，即 s 1 = s 2 = s, α + β = 2 s 的研究更为复杂，只有在大κ 和径向 h 满足 (H1) 时才存在解。最后，我们建立了一个 Pohozaev 特性，它使我们能够在一些关于 h 的平滑假设下证明不存在结果。
{"title":"Existence and non-existence results to a mixed anisotropic Schrödinger system in a plane","authors":"H. Hajaiej, Rohit Kumar, Tuhina Mukherjee, Linjie Song","doi":"10.3233/asy-241922","DOIUrl":"https://doi.org/10.3233/asy-241922","url":null,"abstract":"This article focuses on the existence and non-existence of solutions for the following system of local and nonlocal type − ∂ x x u + ( − Δ ) y s 1 u + u − u 2 s 1 − 1 = κ α h ( x , y ) u α − 1 v β in R 2 , − ∂ x x v + ( − Δ ) y s 2 v + v − v 2 s 2 − 1 = κ β h ( x , y ) u α v β − 1 in R 2 , u , v ⩾ 0 in R 2 , where s 1 , s 2 ∈ ( 0 , 1 ) , α, β > 1, α + β ⩽ min { 2 s 1 , 2 s 2 }, and 2 s i = 2 ( 1 + s i ) 1 − s i , i = 1 , 2. The existence of a ground state solution entirely depends on the behaviour of the parameter κ > 0 and on the function h. In this article, we prove that a ground state solution exists in the subcritical case if κ is large enough and h satisfies (H). Further, if κ becomes very small, then there is no solution to our system. The study of the critical case, i.e., s 1 = s 2 = s, α + β = 2 s , is more complex, and the solution exists only for large κ and radial h satisfying (H1). Finally, we establish a Pohozaev identity which enables us to prove the non-existence results under some smooth assumptions on h.","PeriodicalId":505560,"journal":{"name":"Asymptotic Analysis","volume":"32 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141658673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.
在本文中，我们将继续研究（Yang 和 Estrada 在 Asymptot.Anal.95(1-2) (2015) 1-19）中关于厚分布渐近展开的研究。我们利用在（Yang and Estrada in Asymptot.Anal.95(1-2) (2015) 1-19).此外，我们还为允许点奇异性的情况推导了一个新的 "拉普拉斯公式"。
{"title":"Asymptotic expansion of thick distributions II","authors":"Jiajia Ding, Ricardo Estrada, Yunyun Yang","doi":"10.3233/asy-241924","DOIUrl":"https://doi.org/10.3233/asy-241924","url":null,"abstract":"In this article we continue our research in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19), about the asymptotic expansion of thick distributions. We compute more examples of asymptotic expansion of integral transforms using the techniques developed in (Yang and Estrada in Asymptot. Anal. 95(1–2) (2015) 1–19). Besides, we derive a new “Laplace Formula” for the situation in which a point singularity is allowed.","PeriodicalId":505560,"journal":{"name":"Asymptotic Analysis","volume":"15 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141656170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .
{"title":"The time horizon for stochastic homogenization of the one-dimensional wave equation","authors":"M. Schäffner, B. Schweizer","doi":"10.3233/asy-241923","DOIUrl":"https://doi.org/10.3233/asy-241923","url":null,"abstract":"The wave equation with stochastic rapidly oscillating coefficients can be classically homogenized on bounded time intervals; solutions converge in the homogenization limit to solutions of a wave equation with constant coefficients. This is no longer true on large time scales: Even in the periodic case with periodicity ε, classical homogenization fails for times of the order ε − 2 . We consider the one-dimensional wave equation with random rapidly oscillation coefficients on scale ε and are interested in the critical time scale ε − β from where on classical homogenization fails. In the general setting, we derive upper and lower bounds for β in terms of the growth rate of correctors. In the specific setting of i.i.d. coefficients with matched impedance, we show that the critical time scale is ε − 1 .","PeriodicalId":505560,"journal":{"name":"Asymptotic Analysis","volume":"84 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141664453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish the existence of global-in-time weak solutions for the Landau–Lifschitz–Gilbert equation with magnetostriction in the case of mixed boundary conditions. From this model, we derive by asymptotic method a two-dimensional model for thin ferromagnetic plates taking into account magnetostrictive effects.
{"title":"Thin ferromagnetic plates with magnetostriction","authors":"Mouna Kassan, Gilles Carbou, Mustapha Jazar","doi":"10.3233/asy-241899","DOIUrl":"https://doi.org/10.3233/asy-241899","url":null,"abstract":"In this paper, we establish the existence of global-in-time weak solutions for the Landau–Lifschitz–Gilbert equation with magnetostriction in the case of mixed boundary conditions. From this model, we derive by asymptotic method a two-dimensional model for thin ferromagnetic plates taking into account magnetostrictive effects.","PeriodicalId":505560,"journal":{"name":"Asymptotic Analysis","volume":"568 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140246775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞, if N = 2, λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.
我们研究双曲空间中以下一类非线性椭圆问题正解的存在与不存在 - Δ B N u - λ u = a ( x ) u p - 1 + ε u 2 ∗ - 1 in B N , u ∈ H 1 ( B N ) , 其中 B N 表示双曲空间, 2 < p < 2 ∗ ：= 2 N N - 2 , 若 N ⩾ 3 ; 2 < p < + ∞ , 若 N = 2, λ < ( N - 1 ) 2 4 , 且 0 < a∈ L ∞ ( B N ).我们首先证明 a ( x ) ≡ 1 时存在正径向对称的基态解。接下来，我们证明对于 a ( x ) ⩾ 1，存在一个ε很小的基态解。为了证明这一点，我们采用了 "度量的保角变化"，它允许我们把原方程转化为 R N 中一个球上的奇异方程。然后，通过使用炸毁论证对能级进行仔细分析，我们证明了基态解的存在。最后，我们考虑 a ( x ) ⩽ 1 的情况，首先证明不存在基态解，然后证明在 ε 较小的情况下存在边界解（高能量解）。我们以 Bahri-Li 的精神运用变分论证来证明双曲空间中高能束缚态解的存在。
{"title":"On a class of elliptic equations with critical perturbations in the hyperbolic space","authors":"D. Ganguly, Diksha Gupta, K. Sreenadh","doi":"10.3233/asy-241895","DOIUrl":"https://doi.org/10.3233/asy-241895","url":null,"abstract":"We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space − Δ B N u − λ u = a ( x ) u p − 1 + ε u 2 ∗ − 1 in B N , u ∈ H 1 ( B N ) , where B N denotes the hyperbolic space, 2 < p < 2 ∗ : = 2 N N − 2 , if N ⩾ 3 ; 2 < p < + ∞, if N = 2, λ < ( N − 1 ) 2 4 , and 0 < a ∈ L ∞ ( B N ). We first prove the existence of a positive radially symmetric ground-state solution for a ( x ) ≡ 1. Next, we prove that for a ( x ) ⩾ 1, there exists a ground-state solution for ε small. For proof, we employ “conformal change of metric” which allows us to transform the original equation into a singular equation in a ball in R N . Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case a ( x ) ⩽ 1 is considered where we first show that there is no ground-state solution, and prove the existence of a bound-state solution (high energy solution) for ε small. We employ variational arguments in the spirit of Bahri–Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.","PeriodicalId":505560,"journal":{"name":"Asymptotic Analysis","volume":"9 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139962955","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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