In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results.
{"title":"On the hyperbolic relaxation of the Cahn–Hilliard equation with a mass source","authors":"Dieunel Dor","doi":"10.3233/asy-231844","DOIUrl":"https://doi.org/10.3233/asy-231844","url":null,"abstract":"In this paper, we consider the hyperbolic Cahn–Hilliard equation with a proliferation term, which has applications in biology. First, we study the well-posedness and the regularity of the solutions, which then allow us to study the dissipativity and the high-order dissipativity and finally the existence of the exponential attractor with Dirichlet boundary conditions. Finally, we give numerical simulations that confirm the results.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41988110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.
{"title":"Dynamics and robustness for the 2D Navier–Stokes equations with multi-delays in Lipschitz-like domains","authors":"Keqin Su, Xinguang Yang, A. Miranville, He Yang","doi":"10.3233/asy-231845","DOIUrl":"https://doi.org/10.3233/asy-231845","url":null,"abstract":"This paper is concerned with the dynamics of the two-dimensional Navier–Stokes equations with multi-delays in a Lipschitz-like domain, subject to inhomogeneous Dirichlet boundary conditions. The regularity of global solutions and of pullback attractors, based on tempered universes, is established, extending the results of Yang, Wang, Yan and Miranville (Discrete Contin. Dyn. Syst. 41 (2021) 3343–3366). Furthermore, the robustness of pullback attractors when the delays, considered as small perturbations, disappear is also derived. The key technique in the proofs is the application of a retarded Gronwall inequality and a variable index for the tempered pullback dynamics, allowing to obtain uniform estimates and the compactness of the process.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43612584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.
{"title":"Asymptotic stability of stationary solutions for the Kirchhoff equation","authors":"Min Yu, Weijia Li, Weiping Yan","doi":"10.3233/asy-231841","DOIUrl":"https://doi.org/10.3233/asy-231841","url":null,"abstract":"This paper considers nonlinear Kirchhoff equation with Kelvin–Voigt damping. This model is used to describe the transversal motion of a stretched string. The existence of smooth stationary solutions of nonlinear Kirchhoff equation has been studied widely. In the present contribution, we prove that a class of stationary solutions is asymptotic stable by overcoming the “loss of derivative” phenomenon causing from the Kirchhoff operator. The key point is to find a suitable weighted function when we carry out the energy estimate for the linearized equation.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41877431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we provide a short and robust proof of the Clausius–Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and it can be applied to dilute systems in many other contexts.
{"title":"The Clausius–Mossotti formula","authors":"Mitia Duerinckx, A. Gloria","doi":"10.3233/asy-231840","DOIUrl":"https://doi.org/10.3233/asy-231840","url":null,"abstract":"In this note, we provide a short and robust proof of the Clausius–Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and it can be applied to dilute systems in many other contexts.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48940137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mohammad Akil, Mouhammad Ghader, Z. Hajjej, Mohamad Ali Sammoury
In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem’s well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt–Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams.
{"title":"Well-posedness and polynomial energy decay rate of a transmission problem for Rayleigh beam model with heat conduction","authors":"Mohammad Akil, Mouhammad Ghader, Z. Hajjej, Mohamad Ali Sammoury","doi":"10.3233/asy-231849","DOIUrl":"https://doi.org/10.3233/asy-231849","url":null,"abstract":"In this paper, we investigate the stability of the transmission problem for Rayleigh beam model with heat conduction. First, we reformulate our system into an evolution equation and prove our problem’s well-posedness. Next, we demonstrate the resolvent of the operator is compact in the energy space, then by using the general criteria of Arendt–Batty, we prove that the thermal dissipation is enough to stabilize our model. Finally, a polynomial energy decay rate has been obtained which depends on the mass densities and the moments of inertia of the Rayleigh beams.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44822168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l’Institut Fourier 60 (2010) 2183–2233) and (Differential Integral Equations 27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 1347–1361; SIAM J. Math. Anal. 44 (2012) 1925–1949) for the wave equation with Neumann boundary condition.
本文旨在确定半空间中弱适定双曲型边值问题的分类。这类带有导数损失的问题在文献中是相当经典的，例如在(Arch)。合理的机械。肛门。101(1988)261-292)或(In analysis mathacimmatique et Applications (1988) 319-356 Gauthier-Villars)。已知根据均匀Kreiss-Lopatinskii条件退化所处的频率空间边界面积的种类，能量估计可以包含不同的损失。在(Annales de l’institut傅里叶60(2010)2183-2233)和(微分积分方程27(2014)531-562)中，使用几何光学展开研究了退化的三个第一可能区域。在本文中，我们使用相同的工具来处理最后一种剩余的情况，即掠射区域的简并。与第一个案例相比，我们会看到给出展开中阶项振幅的方程，从而初始化整个展开的结构，不再是输运方程而是由傅里叶乘数给出的。为了恢复第一个振幅，这个乘法器需要反转。作为一个应用，我们讨论了离散连续的现有估计。直流发电机系统。,爵士。B 23 (2018) 1347-1361;SIAM J. Math。与诺伊曼边界条件的波动方程。44(2012)1925-1949)。
{"title":"Geometric optics expansion for weakly well-posed hyperbolic boundary value problem: The glancing degeneracy","authors":"Antoine Benoit, R. Loyer","doi":"10.3233/asy-231838","DOIUrl":"https://doi.org/10.3233/asy-231838","url":null,"abstract":"This article aims to finalize the classification of weakly well-posed hyperbolic boundary value problems in the half-space. Such problems with loss of derivatives are rather classical in the literature and appear for example in (Arch. Rational Mech. Anal. 101 (1988) 261–292) or (In Analyse Mathématique et Applications (1988) 319–356 Gauthier-Villars). It is known that depending on the kind of the area of the boundary of the frequency space on which the uniform Kreiss–Lopatinskii condition degenerates then the energy estimate can include different losses. The three first possible areas of degeneracy have been studied in (Annales de l’Institut Fourier 60 (2010) 2183–2233) and (Differential Integral Equations 27 (2014) 531–562) by the use of geometric optics expansions. In this article we use the same kind of tools in order to deal with the last remaining case, namely a degeneracy in the glancing area. In comparison to the first cases studied we will see that the equation giving the amplitude of the leading order term in the expansion, and thus initializing the whole construction of the expansion, is not a transport equation anymore but it is given by some Fourier multiplier. This multiplier needs to be invert in order to recover the first amplitude. As an application we discuss the existing estimates of (Discrete Contin. Dyn. Syst., Ser. B 23 (2018) 1347–1361; SIAM J. Math. Anal. 44 (2012) 1925–1949) for the wave equation with Neumann boundary condition.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42874583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0, ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.
对于标量平流扩散方程y t ε + M (x, t) y x ε−ε y x x ε = 0， (x, t)∈(0,1)× (0, t)的解，在Dirichlet边界条件下，对参数ε > 0进行了渐近分析。当ε值较小时，解y ε在x = 1邻域(假设M > 0)有一个尺寸为O (ε)的边界层，在点(0,0)开始的特征邻域有一个尺寸为O (ε 1 / 2)的内层。假设这些层在有限时间后相互作用，并使用匹配渐近展开的方法，我们构造了一个显式近似P ε满足‖y ε−P ε‖L∞(0,T;l2 (0,1)) = 0 (ε 1 / 2)。我们强调关于作者最近考虑的M常数情况的额外困难。
{"title":"Internal layer intersecting the boundary of a domain in a singular advection–diffusion equation","authors":"Y. Amirat, A. Münch","doi":"10.3233/asy-231836","DOIUrl":"https://doi.org/10.3233/asy-231836","url":null,"abstract":"We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M ( x , t ) y x ε − ε y x x ε = 0, ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ), supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O ( ε ) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O ( ε 1 / 2 ) in the neighborhood of the characteristic starting from the point ( 0 , 0 ). Assuming that these layers interact each other after a finite time T > 0 and using the method of matched asymptotic expansions, we construct an explicit approximation P ε satisfying ‖ y ε − P ε ‖ L ∞ ( 0 , T ; L 2 ( 0 , 1 ) ) = O ( ε 1 / 2 ). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41877961","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology − κ Δ u = λ u + f ( x , u ) − v , − Δ v = u − v , in a bounded domain Ω ⊂ R N . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.
{"title":"Bifurcation from infinity and multiplicity results for an elliptic system from biology","authors":"Chunqiu Li, Zhen Peng","doi":"10.3233/asy-231839","DOIUrl":"https://doi.org/10.3233/asy-231839","url":null,"abstract":"This article is concerned with the bifurcation from infinity of the following elliptic system arising from biology − κ Δ u = λ u + f ( x , u ) − v , − Δ v = u − v , in a bounded domain Ω ⊂ R N . We regard this problem as a stationary problem of some reaction-diffusion system. By using a method of a pure dynamical nature, we will establish some multiplicity results on bifurcations from infinity for this system under an appropriate Landesman-Lazer type condition.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45299308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.
我们考虑以下非线性奇异椭圆问题(1)−div (M (x) |∇u | p−2∇u) + b | u | r−2 u = a u p−1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on∂Ω，其中1 < p < N;Ω R N是一个有界正则域，包含原点和0 < γ < 1, a小于或等于0,b小于或等于0,0≤f∈L m (Ω)和1 < m < N p。本文的主要目的是研究Dirichlet问题中一些低阶项的存在性和正则化效果，尽管在Dirichlet问题的右侧存在哈代势和奇异项。
{"title":"Singular elliptic problem involving a Hardy potential and lower order term","authors":"A. Sbai, Y. El hadfi, Mounim El Ouardy","doi":"10.3233/asy-231832","DOIUrl":"https://doi.org/10.3233/asy-231832","url":null,"abstract":"We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42479028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let s ∈ ( 0 , 1 ), N > 2 s and D s , 2 ( R N ) : = { u ∈ L 2 s ∗ ( R N ) : ‖ u ‖ D s , 2 ( R N ) : = ( C N , s 2 ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 ( R N ) → R of the type J ( u ) : = 1 2 ‖ u ‖ D s , 2 ( R N ) 2 − ∫ R N b ( x ) G ( u ) d x , where G ( t ) : = ∫ 0 t g ( τ ) d τ, g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 ( R N ) : u ∈ C ( R N ) with sup x ∈ R N ( 1 + | x | N − 2 s ) | u ( x ) | < ∞ } must be a local minimizer of J in the D s , 2 ( R N )-topology.
让s∈(0,1),N > 2 s和D, N (R): R ={美国洛杉矶∈2∗(N):‖美国‖D, 2 (N): R = C (N, s 2∬R | u (x)−y y u (x) | 2 |−| N + 2 s D×D y) 1 2 <∞,哪里的s∗:s = N N−2是《fractional连接exponent C和N, s是一个积极、康斯坦。我们认为functionals j.r.: D s, 2处(N)→R J型》(u): = 1‖D‖美国,2 (R N) 2−∫R N b (x) G (u) dx,哪里G (t): t =∫0 G(ττ)D, G: a R→R是挑战功能subcritical增长at无限,和b: R N→R是a suitable)功能。我们证明那a local minimizer J在topology》之子空间V s: R ={美国∈D, 2 (N): u R∈C (N)和汤x∈R N (1 + s | x | N−2)| u (x) | <∞的一定是a local minimizer j.r.》D s, 2处(N) -topology。
{"title":"D s , 2 ( R N ) versus C ( R N ) local minimizers","authors":"V. Ambrosio","doi":"10.3233/asy-231833","DOIUrl":"https://doi.org/10.3233/asy-231833","url":null,"abstract":"Let s ∈ ( 0 , 1 ), N > 2 s and D s , 2 ( R N ) : = { u ∈ L 2 s ∗ ( R N ) : ‖ u ‖ D s , 2 ( R N ) : = ( C N , s 2 ∬ R 2 N | u ( x ) − u ( y ) | 2 | x − y | N + 2 s d x d y ) 1 2 < ∞ } , where 2 s ∗ : = 2 N N − 2 s is the fractional critical exponent and C N , s is a positive constant. We consider functionals J : D s , 2 ( R N ) → R of the type J ( u ) : = 1 2 ‖ u ‖ D s , 2 ( R N ) 2 − ∫ R N b ( x ) G ( u ) d x , where G ( t ) : = ∫ 0 t g ( τ ) d τ, g : R → R is a continuous function with subcritical growth at infinity, and b : R N → R is a suitable weight function. We prove that a local minimizer of J in the topology of the subspace V s : = { u ∈ D s , 2 ( R N ) : u ∈ C ( R N ) with sup x ∈ R N ( 1 + | x | N − 2 s ) | u ( x ) | < ∞ } must be a local minimizer of J in the D s , 2 ( R N )-topology.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44010650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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