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Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces 凸超曲面的非均质完全非线性收缩流
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2024-03-01 DOI: 10.1515/ans-2022-0077
Pengfei Guan, Jiuzhou Huang, Jiawei Liu
{"title":"Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces","authors":"Pengfei Guan, Jiuzhou Huang, Jiawei Liu","doi":"10.1515/ans-2022-0077","DOIUrl":"https://doi.org/10.1515/ans-2022-0077","url":null,"abstract":"We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{n+1}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0077_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula>, and prove the existence and regularity of the flow before extincting to a point in finite time.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the functional ∫Ωf + ∫Ω*g 关于函数∫Ωf + ∫Ω*g
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2024-03-01 DOI: 10.1515/ans-2023-0105
Qiang Guang, Qi-Rui Li, Xu-Jia Wang
{"title":"On the functional ∫Ωf + ∫Ω*g","authors":"Qiang Guang, Qi-Rui Li, Xu-Jia Wang","doi":"10.1515/ans-2023-0105","DOIUrl":"https://doi.org/10.1515/ans-2023-0105","url":null,"abstract":"In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi mathvariant=\"script\">J</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> <m:mi>f</m:mi> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mrow> </m:msub> <m:mi>g</m:mi> </m:math> <jats:tex-math> $mathcal{J}left({Omega},{{Omega}}^{{ast}}right)={int }_{{Omega}}f+{int }_{{{Omega}}^{{ast}}}g$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0105_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula>, where <jats:italic>f</jats:italic>, <jats:italic>g</jats:italic> are given nonnegative functions in a manifold. The duality is a relation <jats:italic>α</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>y</jats:italic>) ≤ 0 ∀ <jats:italic>x</jats:italic> ∈ Ω, <jats:italic>y</jats:italic> ∈ Ω*, for a suitable function <jats:italic>α</jats:italic>. This model covers several geometric and physical applications. In this paper we review two topological methods introduced in the study of the functional, and discuss possible extensions of the methods to related problems.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"54 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Comparison formulas for total mean curvatures of Riemannian hypersurfaces 黎曼超曲面总平均曲率的比较公式
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2024-03-01 DOI: 10.1515/ans-2022-0081
Mohammad Ghomi
{"title":"Comparison formulas for total mean curvatures of Riemannian hypersurfaces","authors":"Mohammad Ghomi","doi":"10.1515/ans-2022-0081","DOIUrl":"https://doi.org/10.1515/ans-2022-0081","url":null,"abstract":"We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"171 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rigidity properties of Colding–Minicozzi entropies 科尔丁-米尼柯齐熵的刚性特性
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2024-03-01 DOI: 10.1515/ans-2022-0082
Jacob Bernstein
{"title":"Rigidity properties of Colding–Minicozzi entropies","authors":"Jacob Bernstein","doi":"10.1515/ans-2022-0082","DOIUrl":"https://doi.org/10.1515/ans-2022-0082","url":null,"abstract":"We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"44 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis 一类具有猎物趋向性/捕食者趋向性的捕食-捕食模型共存状态的全局分岔
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2022-0060
Shanbing Li, Jianhua Wu
{"title":"Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis","authors":"Shanbing Li, Jianhua Wu","doi":"10.1515/ans-2022-0060","DOIUrl":"https://doi.org/10.1515/ans-2022-0060","url":null,"abstract":"Abstract This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42094199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Preface for the special issue in honor of David Jerison 纪念大卫·杰伦的特刊序言
2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2023-0106
Guozhen Lu
{"title":"Preface for the special issue in honor of David Jerison","authors":"Guozhen Lu","doi":"10.1515/ans-2023-0106","DOIUrl":"https://doi.org/10.1515/ans-2023-0106","url":null,"abstract":"","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136008458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity 一类具有超临界非线性的非齐次椭圆方程的非极小解的存在性
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2022-0073
Kazuhiro Ishige, S. Okabe, Tokushi Sato
{"title":"Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity","authors":"Kazuhiro Ishige, S. Okabe, Tokushi Sato","doi":"10.1515/ans-2022-0073","DOIUrl":"https://doi.org/10.1515/ans-2022-0073","url":null,"abstract":"Abstract In our previous paper [K. Ishige, S. Okabe, and T. Sato, A supercritical scalar field equation with a forcing term, J. Math. Pures Appl. 128 (2019), pp. 183–212], we proved the existence of a threshold κ ∗ > 0 {kappa }^{ast }gt 0 such that the elliptic problem for an inhomogeneous elliptic equation − Δ u + u = u p + κ μ -Delta u+u={u}^{p}+kappa mu in R N {{bf{R}}}^{N} possesses a positive minimal solution decaying at the space infinity if and only if 0 < κ ≤ κ ∗ 0lt kappa le {kappa }^{ast } . Here, N ≥ 2 Nge 2 , μ mu is a nontrivial nonnegative Radon measure in R N {{bf{R}}}^{N} with a compact support, and p > 1 pgt 1 is in the Joseph-Lundgren subcritical case. In this article, we prove the existence of nonminimal positive solutions to the elliptic problem. Our arguments are also applicable to inhomogeneous semilinear elliptic equations with exponential nonlinearity.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47802723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation p-k-Hessian方程无穷多个边界爆破解的存在性
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2022-0074
M. Feng, Xuemei Zhang
{"title":"The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation","authors":"M. Feng, Xuemei Zhang","doi":"10.1515/ans-2022-0074","DOIUrl":"https://doi.org/10.1515/ans-2022-0074","url":null,"abstract":"Abstract The primary objective of this article is to analyze the existence of infinitely many radial p p - k k -convex solutions to the boundary blow-up p p - k k -Hessian problem σ k ( λ ( D i ( ∣ D u ∣ p − 2 D j u ) ) ) = H ( ∣ x ∣ ) f ( u ) in Ω , u = + ∞ on ∂ Ω . {sigma }_{k}left(lambda left({D}_{i}left({| Du| }^{p-2}{D}_{j}u)))=Hleft(| x| )fleft(u)hspace{0.33em}hspace{0.1em}text{in}hspace{0.1em}hspace{0.33em}Omega ,hspace{0.33em}u=+infty hspace{0.33em}hspace{0.1em}text{on}hspace{0.1em}hspace{0.33em}partial Omega . Here, k ∈ { 1 , 2 , … , N } kin left{1,2,ldots ,Nright} , σ k ( λ ) {sigma }_{k}left(lambda ) is the k k -Hessian operator, and Ω Omega is a ball in R N ( N ≥ 2 ) {{mathbb{R}}}^{N}hspace{0.33em}left(Nge 2) . Our methods are mainly based on the sub- and super-solutions method.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45752357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem Schrödinger-Poisson问题多峰解的非退化性
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2022-0079
Lin Chen, Hui Ding, Benniao Li, Jianghua Ye
{"title":"Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem","authors":"Lin Chen, Hui Ding, Benniao Li, Jianghua Ye","doi":"10.1515/ans-2022-0079","DOIUrl":"https://doi.org/10.1515/ans-2022-0079","url":null,"abstract":"Abstract In this article, we consider the following Schrödinger-Poisson problem: − ε 2 Δ u + V ( y ) u + Φ ( y ) u = ∣ u ∣ p − 1 u , y ∈ R 3 , − Δ Φ ( y ) = u 2 , y ∈ R 3 , left{begin{array}{ll}-{varepsilon }^{2}Delta u+V(y)u+Phi (y)u={| u| }^{p-1}u,& yin {{mathbb{R}}}^{3}, -Delta Phi (y)={u}^{2},& yin {{mathbb{R}}}^{3},end{array}right. where ε > 0 varepsilon gt 0 is a small parameter, 1 < p < 5 1lt plt 5 , and V ( y ) V(y) is a potential function. We construct multi-peak solution concentrating at the critical points of V ( y ) V(y) through the Lyapunov-Schmidt reduction method. Moreover, by using blow-up analysis and local Pohozaev identities, we prove that the multi-peak solution we construct is non-degenerate. To our knowledge, it seems be the first non-degeneracy result on the Schödinger-Poisson system.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46698882","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Asymptotic properties of critical points for subcritical Trudinger-Moser functional 次临界Trudinger-Moser泛函临界点的渐近性质
IF 1.8 2区 数学
Advanced Nonlinear Studies Pub Date : 2023-01-01 DOI: 10.1515/ans-2022-0042
Masato Hashizume
{"title":"Asymptotic properties of critical points for subcritical Trudinger-Moser functional","authors":"Masato Hashizume","doi":"10.1515/ans-2022-0042","DOIUrl":"https://doi.org/10.1515/ans-2022-0042","url":null,"abstract":"Abstract On a smooth bounded domain we study the Trudinger-Moser functional E α ( u ) ≔ ∫ Ω ( e α u 2 − 1 ) d x , u ∈ H 1 ( Ω ) {E}_{alpha }left(u):= mathop{int }limits_{Omega }({e}^{alpha {u}^{2}}-1){rm{d}}x,hspace{1.0em}uin {H}^{1}left(Omega ) for α ∈ ( 0 , 2 π ) alpha in left(0,2pi ) and its restriction E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} , where Σ λ ≔ u ∈ H 1 ( Ω ) ∣ ∫ Ω ( ∣ ∇ u ∣ 2 + λ u 2 ) d x = 1 {Sigma }_{lambda }:= left{uin {H}^{1}left(Omega )| {int }_{Omega }(| nabla u{| }^{2}+lambda {u}^{2}){rm{d}}x=1right} for λ > 0 lambda gt 0 . By applying the asymptotic analysis and the variational method, we obtain asymptotic behavior of critical points of E α ∣ Σ λ {E}_{alpha }{| }_{{Sigma }_{lambda }} both as λ → 0 lambda to 0 and as λ → + ∞ lambda to +infty . In particular, we prove that when α alpha is sufficiently small, maximizers for sup u ∈ Σ λ E α ( u ) {sup }_{uin {Sigma }_{lambda }}{E}_{alpha }left(u) tend to 0 in C ( Ω ¯ ) Cleft(overline{Omega }) as λ → + ∞ lambda to +infty .","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44892795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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