We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in Hn×R ${mathbb{H}}^{n}{times}mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in Hn×R ${mathbb{H}}^{n}{times}mathbb{R}$ : we show that compact connected hypersurfaces of constant r-th mean curvature embedded in Hn×[0,∞) ${mathbb{H}}^{n}{times}left[0,infty right)$ with boundary in the slice Hn×{0} ${mathbb{H}}^{n}{times}left{0right}$ are topological disks under suitable assumptions.
我们对在 H n × R ${mathbb{H}}^{n}{times}mathbb{R}$ 中具有旋转对称性和正常数 r 次平均曲率的超曲面进行了分类。此外，还讨论了在双曲平移下不变的特定恒定高阶平均曲率超曲面。这些不变超曲面中的一些被用作壁垒，以证明 H n × R ${mathbb{H}}^{n}{times}mathbb{R}$ 中的一个 Ros-Rosenberg 型定理：我们证明了嵌入在 H n × [ 0 , ∞ ) ${mathbb{H}}^{n}{times}left[0,infty right)$ 中的边界在切片 H n × { 0 } 中的恒定 r 平均曲率的紧凑连通超曲面。 ${mathbb{H}}^{n}{times}left{0right}$ 在合适的假设条件下是拓扑磁盘。
{"title":"On constant higher order mean curvature hypersurfaces in H n × R ${mathbb{H}}^{n}{times}mathbb{R}$","authors":"Barbara Nelli, Giuseppe Pipoli, Giovanni Russo","doi":"10.1515/ans-2023-0115","DOIUrl":"https://doi.org/10.1515/ans-2023-0115","url":null,"abstract":"We classify hypersurfaces with rotational symmetry and positive constant <jats:italic>r</jats:italic>-th mean curvature in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:math> <jats:tex-math> ${mathbb{H}}^{n}{times}mathbb{R}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0115_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula>. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:math> <jats:tex-math> ${mathbb{H}}^{n}{times}mathbb{R}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0115_ineq_003.png\" /> </jats:alternatives> </jats:inline-formula>: we show that compact connected hypersurfaces of constant <jats:italic>r</jats:italic>-th mean curvature embedded in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:tex-math> ${mathbb{H}}^{n}{times}left[0,infty right)$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0115_ineq_004.png\" /> </jats:alternatives> </jats:inline-formula> with boundary in the slice <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:math> <jats:tex-math> ${mathbb{H}}^{n}{times}left{0right}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0115_ineq_005.png\" /> </jats:alternatives> </jats:inline-formula> are topological disks under suitable assumptions.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"51 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153185","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}
The existence of L2–normalized solutions is studied for the equation −Δu+μu=f(x,u)inRN,∫RNu2dx=m. $-{Delta}u+mu u=fleft(x,uright)quad quad text{in} {mathbf{R}}^{N},quad {int }_{{mathbf{R}}^{N}}{u}^{2} mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L2-subcritical growth and (μ, u) ∈ R × H1(R N) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.
研究了 R N 中方程 - Δ u + μ u = f ( x , u ) 的 L 2 归一化解的存在性 , ∫ R N u 2 d x = m . $-{Delta}u+mu u=fleft(x,uright)quad quad text{in}{mathbf{R}}^{N},quad {int }_{mathbf{R}}^{N}}{u}^{2}这里 m > 0 和 f(x, s) 是给定的，f(x, s) 具有 L 2 次临界增长，且 (μ, u)∈ R × H 1(R N ) 是未知的。本文采用 Hirata 和 Tanaka 的论证（"具有 L 2 约束的非线性标量场方程：山口和对称山口方法"，《非线性研究》，第 19 卷第 2 期，第 263-290 页，2019 年），找到了拉格朗日函数的临界点。为了获得拉格朗日函数的临界点，我们使用了 Palais-Smale-Cerami 条件，而不是 Hirata 和 Tanaka（"带 L 2 约束的非线性标量场方程：山口和对称山口方法"，《非线性研究》，第 19 卷第 2 期，第 263-290 页，2019 年）中的条件 (PSP)。我们还证明了径向对称下的多重性结果。
{"title":"The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients","authors":"Norihisa Ikoma, Mizuki Yamanobe","doi":"10.1515/ans-2022-0056","DOIUrl":"https://doi.org/10.1515/ans-2022-0056","url":null,"abstract":"The existence of <jats:italic>L</jats:italic> <jats:sup>2</jats:sup>–normalized solutions is studied for the equation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>−</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>u</m:mi> <m:mo>+</m:mo> <m:mi>μ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mtext> </m:mtext> <m:mtext> </m:mtext> <m:mtext>in</m:mtext> <m:mspace width=\"0.3333em\" /> <m:msup> <m:mrow> <m:mi mathvariant=\"bold\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:mspace width=\"1em\" /> <m:msub> <m:mrow> <m:mo>∫</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"bold\">R</m:mi> </m:mrow> <m:mrow> <m:mi>N</m:mi> </m:mrow> </m:msup> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mspace width=\"0.17em\" /> <m:mi mathvariant=\"normal\">d</m:mi> <m:mi>x</m:mi> <m:mo>=</m:mo> <m:mi>m</m:mi> <m:mo>.</m:mo> </m:math> <jats:tex-math> $-{Delta}u+mu u=fleft(x,uright)quad quad text{in} {mathbf{R}}^{N},quad {int }_{{mathbf{R}}^{N}}{u}^{2} mathrm{d}x=m.$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0056_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> Here <jats:italic>m</jats:italic> > 0 and <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>s</jats:italic>) are given, <jats:italic>f</jats:italic>(<jats:italic>x</jats:italic>, <jats:italic>s</jats:italic>) has the <jats:italic>L</jats:italic> <jats:sup>2</jats:sup>-subcritical growth and (<jats:italic>μ</jats:italic>, <jats:italic>u</jats:italic>) ∈ R × <jats:italic>H</jats:italic> <jats:sup>1</jats:sup>(R <jats:sup> <jats:italic>N</jats:italic> </jats:sup>) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with <jats:italic>L</jats:italic> <jats:sup>2</jats:sup> constraint: mountain pass and symmetric mountain pass approaches,” <jats:italic>Adv. Nonlinear Stud.</jats:italic>, vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with <jats:italic>L</jats:italic> <jats:sup>2</jats:sup> constraint: mountain pass and symmetric mountain pass approaches,” <jats:italic>Adv. Nonlinear Stud.</jats:italic>, vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"305 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153186","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}
本文研究了以下非线性分数哈特里（或乔夸-佩卡）方程 ( - Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${left(-{Delta}right)}^{s}u+mu u=left({I}_{alpha }{ast}Fleft(uright)right){F}^{prime }left(uright)quad text{in}{mathbb{R}}^{N},$ (*) 其中 μ > 0, s∈ (0, 1), N ≥ 2, α∈ (0, N), I α ∼ 1 | x | N - α ${I}_{alpha }sim frac{1}{vert xvert }^{N-alpha }}$ 是里兹势，F 是一般的次临界非线性。我们的目标是通过假设 F 为奇数或偶数，证明多个（径向对称）解 u∈ H s ( R N ) $uin {H}^{s}left({mathbb{R}}^{N}right)$ 的存在性：我们既考虑了 μ > 0 固定的情况，也考虑了 ∫ R N u 2 = m > 0 ${int }_{mathbb{R}}^{N}}{u}^{2}=m{ >}0$ 规定的情况。这里我们还简化了一些针对 s = 1 的论证（S. Cingolani, M. Gallo, and K. Tanaka, "Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities," Calc.Var.Partial Differ.Equ.，第 61 卷，第 68 期，第 34 页，2022 年）。证明中的一个关键点是研究合适的多维奇数路径，这是由 Berestycki 和 Lions 在局部情况下完成的（H. Berestycki and P.-L. Lions, "Nonlinear scalar field equations II: existence of infinitely many solutions," Arch.Ration.Mech.Anal.4, pp.特别是，在对无约束问题的山口值进行渐近研究（当 μ 变化时）时，需要这些路径的一些特性，然后利用这些特性来描述约束问题的几何形状，并检测出任意 m > 0 的无限多归一化解。
We establish some sharp affine weighted L2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L2 structure of gradient norm, affine invariance and a class of weighted L2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.
我们在上半空间建立了一些尖锐的仿射加权 L 2 Sobolev 不等式，其中涉及边界上具有退化性的发散算子。此外，对于某些指数情况，我们还描述了极值函数和最佳常数的特征。我们的方法仅依赖于梯度规范的 L 2 结构、仿射不变性和上半空间的一类加权 L 2 Sobolev 不等式。这是一种简单的方法，不依赖于欧几里得空间的几何结构，如凸几何上的布伦-闵科夫斯基理论。
{"title":"Sharp affine weighted L 2 Sobolev inequalities on the upper half space","authors":"Jingbo Dou, Yunyun Hu, Caihui Yue","doi":"10.1515/ans-2023-0117","DOIUrl":"https://doi.org/10.1515/ans-2023-0117","url":null,"abstract":"We establish some sharp affine weighted <jats:italic>L</jats:italic> <jats:sup>2</jats:sup> Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the <jats:italic>L</jats:italic> <jats:sup>2</jats:sup> structure of gradient norm, affine invariance and a class of weighted <jats:italic>L</jats:italic> <jats:sup>2</jats:sup> Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"8 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153182","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}
We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation −Δu=f(u),u∈D1,2(RN), $-{Delta}u=fleft(uright), uin {D}^{1,2}left({mathrm{R}}^{N}right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c0 if N = 5, 6 and smaller than 10c0 if N ≥ 7, where c0 is the ground state energy.
我们给出了非线性标量场方程- Δ u = f ( u ) , u ∈ D 1,2 ( R N ) 的符号变化解的最小能量上限、 $-{{Delta}u=fleft(uright), uin {D}^{1,2}left({mathrm{R}}^{N}right),$ 其中 N ≥ 5，非线性 f 在无穷远处是次临界的，在原点附近是超临界的。更确切地说，我们证明了非径向符号变化解的存在，当 N = 5, 6 时，其能量小于 12c 0；当 N ≥ 7 时，其能量小于 10c 0，其中 c 0 为基态能量。
{"title":"An upper bound for the least energy of a sign-changing solution to a zero mass problem","authors":"Mónica Clapp, Liliane Maia, Benedetta Pellacci","doi":"10.1515/ans-2022-0065","DOIUrl":"https://doi.org/10.1515/ans-2022-0065","url":null,"abstract":"We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mo>−</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mi>u</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width=\"0.17em\" /> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>D</m:mi> <m:mn>1,2</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:msup> <m:mi mathvariant=\"normal\">R</m:mi> <m:mi>N</m:mi> </m:msup> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> <jats:tex-math> $-{Delta}u=fleft(uright), uin {D}^{1,2}left({mathrm{R}}^{N}right),$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2022-0065_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula> where <jats:italic>N</jats:italic> ≥ 5 and the nonlinearity <jats:italic>f</jats:italic> is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12<jats:italic>c</jats:italic> <jats:sub>0</jats:sub> if <jats:italic>N</jats:italic> = 5, 6 and smaller than 10<jats:italic>c</jats:italic> <jats:sub>0</jats:sub> if <jats:italic>N</jats:italic> ≥ 7, where <jats:italic>c</jats:italic> <jats:sub>0</jats:sub> is the ground state energy.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"143 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153187","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}
In this paper, we give an alternative approach to the C2,α,β estimate for complex Monge-Ampère equations with cone singularities along simple normal crossing divisors.
在本文中，我们给出了另一种方法，即沿简单法线交叉除数具有圆锥奇点的复杂蒙日-安培方程的 C 2,α,β 估计。
{"title":"A C 2,α,β estimate for complex Monge–Ampère type equations with conic sigularities","authors":"Liding Huang, Gang Tian, Jiaxiang Wang","doi":"10.1515/ans-2023-0113","DOIUrl":"https://doi.org/10.1515/ans-2023-0113","url":null,"abstract":"In this paper, we give an alternative approach to the <jats:italic>C</jats:italic> <jats:sup>2,<jats:italic>α</jats:italic>,<jats:italic>β</jats:italic> </jats:sup> estimate for complex Monge-Ampère equations with cone singularities along simple normal crossing divisors.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"72 6 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140129560","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}
Given <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>I</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> <jats:tex-math> $I,Bin mathbb{N}cup left{0right}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0118_ineq_001.png" /> </jats:alternatives> </jats:inline-formula>, we investigate the existence and geometry of complete finitely branched minimal surfaces <jats:italic>M</jats:italic> 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:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0118_ineq_002.png" /> </jats:alternatives> </jats:inline-formula> with Morse index at most <jats:italic>I</jats:italic> and total branching order at most <jats:italic>B</jats:italic>. Previous works of Fischer-Colbrie (“On complete minimal surfaces with finite Morse index in 3-manifolds,” <jats:italic>Invent. Math.</jats:italic>, vol. 82, pp. 121–132, 1985) and Ros (“One-sided complete stable minimal surfaces,” <jats:italic>J. Differ. Geom.</jats:italic>, vol. 74, pp. 69–92, 2006) explain that such surfaces are precisely the complete minimal surfaces 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:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2023-0118_ineq_003.png" /> </jats:alternatives> </jats:inline-formula> of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an <jats:italic>M</jats:italic> with estimates that are given in terms of <jats:italic>I</jats:italic> and <jats:italic>B</jats:italic>. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for <jats:italic>m</jats:italic>-dimensional submanifolds Σ of an <jats:italic>n</jats:italic>-dimensional Riemannian manifold <jats:italic>X</jats:italic>, where these area estimates depend on the geometry of <jats:italic>X</jats:italic> and upper bounds on the lengths of the mean curvature vectors of Σ. We also describe a family of complete, finitely branched minimal surfaces in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"
给定 I , B ∈ N∪ { 0 } $I,Bin mathbb{N}cup left{0right}$, 我们研究了 R 3 ${mathbb{R}}^{3}$ 中具有最多 I 的莫尔斯指数和最多 B 的总分支序的完整有限分支极小曲面 M 的存在性和几何性质。Math., vol. 82, pp.Geom., vol. 74, pp.为了得到这些特殊曲面的一些主要结果，我们得到了 n 维黎曼流形 X 的 m 维子流形 Σ 的一般内在单调性面积公式，其中这些面积估计值取决于 X 的几何形状和 Σ 的平均曲率向量长度的上限。我们还描述了 R 3 ${mathbb{R}}^{3}$ 中一系列稳定且不可定向的完整有限分支极小曲面；这些例子概括了经典的 Henneberg 极小曲面。
{"title":"Geometry of branched minimal surfaces of finite index","authors":"William H. Meeks, Joaquín Pérez","doi":"10.1515/ans-2023-0118","DOIUrl":"https://doi.org/10.1515/ans-2023-0118","url":null,"abstract":"Given <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" overflow=\"scroll\"> <m:mi>I</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:math> <jats:tex-math> $I,Bin mathbb{N}cup left{0right}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0118_ineq_001.png\" /> </jats:alternatives> </jats:inline-formula>, we investigate the existence and geometry of complete finitely branched minimal surfaces <jats:italic>M</jats:italic> 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:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0118_ineq_002.png\" /> </jats:alternatives> </jats:inline-formula> with Morse index at most <jats:italic>I</jats:italic> and total branching order at most <jats:italic>B</jats:italic>. Previous works of Fischer-Colbrie (“On complete minimal surfaces with finite Morse index in 3-manifolds,” <jats:italic>Invent. Math.</jats:italic>, vol. 82, pp. 121–132, 1985) and Ros (“One-sided complete stable minimal surfaces,” <jats:italic>J. Differ. Geom.</jats:italic>, vol. 74, pp. 69–92, 2006) explain that such surfaces are precisely the complete minimal surfaces 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:mn>3</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math> ${mathbb{R}}^{3}$ </jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_ans-2023-0118_ineq_003.png\" /> </jats:alternatives> </jats:inline-formula> of finite total curvature and finite total branching order. Among other things, we derive scale-invariant weak chord-arc type results for such an <jats:italic>M</jats:italic> with estimates that are given in terms of <jats:italic>I</jats:italic> and <jats:italic>B</jats:italic>. In order to obtain some of our main results for these special surfaces, we obtain general intrinsic monotonicity of area formulas for <jats:italic>m</jats:italic>-dimensional submanifolds Σ of an <jats:italic>n</jats:italic>-dimensional Riemannian manifold <jats:italic>X</jats:italic>, where these area estimates depend on the geometry of <jats:italic>X</jats:italic> and upper bounds on the lengths of the mean curvature vectors of Σ. We also describe a family of complete, finitely branched minimal surfaces in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" ","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"58 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140114990","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}
In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.
{"title":"Segregated solutions for nonlinear Schrödinger systems with a large number of components","authors":"Haixia Chen, Angela Pistoia","doi":"10.1515/ans-2022-0076","DOIUrl":"https://doi.org/10.1515/ans-2022-0076","url":null,"abstract":"In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.","PeriodicalId":7191,"journal":{"name":"Advanced Nonlinear Studies","volume":"22 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140117566","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}
我们研究涉及上半空间 R + n = x ∈ R n ∣ x 1 > 0 ${mathbb{R}}_{+}^{n}=left{xin {mathbb{R}}^{n}mid {x}_{1}{ >}0right}$ : ( - Δ ) s u ( x ) = f ( u ( x ) ) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x ) = 0 , x ∉ R + n 。 begin{cases}quad hfill & {left(-{Delta}right)}^{s}uleft(xright)=fleft(uleft(xright)right),qquad xin {mathbb{R}}_{+}^{n},hfill quad hfill &；qquad uleft(xright){ >}0,qquad xin {mathbb{R}}_{+}^{n},hfill quad hfill & qquad uleft(xright)=0,qquad xnotin {mathbb{R}}_{+}^{n}.hfill end{cases}. .我们证明正解在 x 1 方向上是单调递增的，假设 u(x) 的增长速度不超过 |x| γ，且 γ ∈ (0, 2s)为 |x| 大。首先，我们建立了狭长区域的最大原则。然后，我们应用分数拉普拉卡方移动平面的直接方法来推导单调性。作为单调性结果的一个应用，我们用它来证明 f(u) = u p , p∈ 1 , n - 1 + 2 s n - 1 - 2 s $pin left(1,frac{n-1+2s}{n-1-2s}right)$ 的有界正解在 R + n ${mathbb{R}}_{+}^{n}$ 中不存在。
本文关注的是半空间 R + 2 ${mathbb{R}}_{+}^{2}$ 上具有指数非线性的 Hénon-Hardy 型系统： ( - Δ ) α 2 u ( x ) = | x | a u p 1 ( x ) e q 1 v ( x ) , x ∈ R + 2 , ( - Δ ) v ( x ) = | x | b u p 2 ( x ) e q 2 v ( x ) , x ∈ R + 2 、 $begin{cases}{left(-{Delta}right)}^{frac{alpha }{2}}uleft(xright)=vert x{vert }^{a}{u}^{{p}_{1}}left(xright){e}^{{q}_{1}vleft(xright)}, xin {mathbb{R}}_{+}^{2},quad hfill left(-{Delta}right)vleft(xright)=vert x{vert }^{b}{u}^{p}_{2}}left(xright){e}^{q}_{2}vleft(xright)}、xin {mathbb{R}}_{+}^{2},quad hfill end{cases}$ with Dirichlet boundary conditions, where 0 <；α < 2 和 p 1, p 2, q 1, q 2 > 0。首先，我们在假设 p 1 ≥ - 2 a α - 1 ${p}_{1}ge -frac{2a}{alpha }-1$ 的条件下导出了与上述系统相对应的积分表示公式。然后，我们通过缩放球方法证明上述系统解的柳维尔定理。
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