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Critical Kirchhoff-type equation with singular potential 具有奇异势的临界kirchhoff型方程
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-06-23 DOI: 10.12775/tmna.2022.051
Yujian Su, Senli Liu
{"title":"Critical Kirchhoff-type equation with singular potential","authors":"Yujian Su, Senli Liu","doi":"10.12775/tmna.2022.051","DOIUrl":"https://doi.org/10.12775/tmna.2022.051","url":null,"abstract":"In this paper, we deal with the following Kirchhoff-type equation:\u0000begin{equation*}\u0000-bigg(1\u0000+int_{mathbb{R}^{3}}|nabla u|^{2}dxbigg)\u0000Delta u\u0000+frac{A}{|x|^{alpha}}u\u0000=f(u),quad xinmathbb{R}^{3},\u0000end{equation*}\u0000where $A> 0$ is a real parameter and $alphain(0,1)cup ({4}/{3},2)$.\u0000Remark that $f(u)=|u|^{2_{alpha}^{*}-2}u +lambda|u|^{q-2}u\u0000+|u|^{4}u$,\u0000where $lambda> 0$, $qin(2_{alpha}^{*},6)$,\u0000$2_{alpha}^{*}=2+{4alpha}/({4-alpha})$\u0000is the embedding bottom index, and $6$ is the embedding top index and Sobolev critical exponent.\u0000We point out that the nonlinearity $f$ is the almost ``optimal'' choice.\u0000First, for $alphain({4}/{3},2)$, applying the generalized version of Lions-type\u0000 theorem and the Nehari manifold, we show the existence of nonnegative\u0000Nehari-type ground sate solution for above equation. Second, for $alphain(0,1)$,\u0000 using the generalized version of Lions-type theorem and the Pohov{z}aev\u0000 manifold, we establish the existence of nonnegative Pohov{z}aev-type ground\u0000state solution for above equation. Based on our new generalized version\u0000of Lions-type theorem, our works extend the results in Li-Su [Z. Angew. Math. Phys. {bf 66} (2015)].","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42518484","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}
引用次数: 0
Existence of nontrivial solutions to Schrödinger systems with linear and nonlinear couplings via Morse theory 基于Morse理论的线性和非线性耦合Schrödinger系统非平凡解的存在性
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-06-23 DOI: 10.12775/tmna.2022.032
Zhitao Zhang, Meng Yu, Xiaotian Zheng
{"title":"Existence of nontrivial solutions to Schrödinger systems with linear and nonlinear couplings via Morse theory","authors":"Zhitao Zhang, Meng Yu, Xiaotian Zheng","doi":"10.12775/tmna.2022.032","DOIUrl":"https://doi.org/10.12775/tmna.2022.032","url":null,"abstract":"In this paper, we use Morse theory to study existence of nontrivial solutions to the following Schrödinger system with linear and nonlinear couplings which arises from Bose-Einstein condensates:\u0000$$\u0000begin{cases}\u0000-Delta u+lambda_{1} u+kappa v=mu_{1} u^{3}+beta uv^{2}\u0000& text{in } Omega,\u0000-Delta v+lambda_{2} v+kappa u=mu_{2} v^{3}+beta vu^{2}\u0000& text{in } Omega,\u0000u=v=0 & text{on } partialOmega,\u0000end{cases}\u0000$$\u0000where $Omega$ is a bounded smooth domain in $mathbb{R}^{N}$($N=2,3$),\u0000$lambda_{1},lambda_{2},mu_{1},mu_{2} in mathbb{R} setminus { 0 }$,\u0000$beta, kappa in mathbb{R}$.\u0000 In two cases of\u0000$kappa=0$ and $kappaneq 0$, by transferring an eigenvalue problem into an algebraic problem, we compute the Morse index and critical groups of the trivial\u0000 solution. Furthermore, even when the trivial solution is degenerate,\u0000we show a local linking structure of energy functional at zero within a suitable\u0000 parameter range and then get critical groups of the trivial solution.\u0000As an application, we use Morse theory to get an existence theorem on existence\u0000of nontrivial solutions under some conditions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42699179","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}
引用次数: 0
Multiple solutions to Bahri-Coron problem involving fractional $p$-Laplacian in some domain with nontrivial topology 非平凡拓扑域上涉及分数阶拉普拉斯算子的Bahri-Coron问题的多重解
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-06-23 DOI: 10.12775/tmna.2022.033
Uttam Kumar, Sweta Tiwari
{"title":"Multiple solutions to Bahri-Coron problem involving fractional $p$-Laplacian in some domain with nontrivial topology","authors":"Uttam Kumar, Sweta Tiwari","doi":"10.12775/tmna.2022.033","DOIUrl":"https://doi.org/10.12775/tmna.2022.033","url":null,"abstract":"In this article, we establish the existence of positive and multiple\u0000 sign-changing solutions to the fractional $p$-Laplacian equation with purely critical nonlinearity\u0000 begin{equation}\u0000label{Ppomegas-a}tag{P$_{p,Omega}^{s}$}\u0000begin{cases}\u0000 (-Delta)_{p}^s u =|u|^{p_s^*-2} u& text{in }Omega, \u0000 u =0 & text{on }Omega^{c},\u0000 end{cases}\u0000end{equation}\u0000in a bounded domain $Omegasubset mathbb{R}^{N}$ for $sin (0,1)$,\u0000$pin (1,infty)$, and the fractional critical Sobolev exponent\u0000$p^{*}_{s}={Np}/({N-sp})$ under some symmetry assumptions.\u0000We study Struwe's type global compactness results for the Palais-Smale sequence\u0000in the presence of symmetries.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45702993","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}
引用次数: 0
Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions 无渐近条件的Kirchhoff型问题正解的多重性
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-06-23 DOI: 10.12775/tmna.2022.031
X. Qian
{"title":"Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions","authors":"X. Qian","doi":"10.12775/tmna.2022.031","DOIUrl":"https://doi.org/10.12775/tmna.2022.031","url":null,"abstract":"In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\u0000[\u0000begin{cases}\u0000-bigg({varepsilon}^2a+{varepsilon}bint_{mathbb{R}^3} |n u|^2dxbigg)Delta u+u=Q(x)|u|^{p-2}u, & xinmathbb{R}^3,\u0000uin H^1big(mathbb{R}^3big), quad u> 0, & xinmathbb{R}^3,\u0000end{cases}\u0000]\u0000where $varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$\u0000 is a nonnegative continuous potential and does not satisfy any asymptotic condition.\u0000 Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48374354","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}
引用次数: 0
On the relative category in the brake orbits problem 关于制动轨道问题中的相对范畴
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-03-04 DOI: 10.12775/tmna.2022.057
Dario Corona, R. Giambò, F. Giannoni, P. Piccione
{"title":"On the relative category in the brake orbits problem","authors":"Dario Corona, R. Giambò, F. Giannoni, P. Piccione","doi":"10.12775/tmna.2022.057","DOIUrl":"https://doi.org/10.12775/tmna.2022.057","url":null,"abstract":"In this paper %dedicated to the memory of Edward Fadell and Sufian Husseini\u0000we show how the notion of the Lusternik-Schnirelmann relative category can be used\u0000to study a multiplicity problem for brake orbits in a potential well\u0000which is homeomorphic to the $N$-dimensional unit disk.\u0000The estimate of the relative category of the set of chords with endpoints on the\u0000$(N-1)$-unit sphere was shown to the third author by\u0000Fadell and Husseini while he was visiting the University of Wisconsin at Madison.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46213723","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}
引用次数: 0
Reeb graphs of circle-valued functions: A survey and basic facts 圆值函数的Reeb图：综述与基本事实
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-03-04 DOI: 10.12775/tmna.2022.023
Irina Gelbukh
{"title":"Reeb graphs of circle-valued functions: A survey and basic facts","authors":"Irina Gelbukh","doi":"10.12775/tmna.2022.023","DOIUrl":"https://doi.org/10.12775/tmna.2022.023","url":null,"abstract":"The Reeb graph of a circle-valued function is a topological space obtained by contracting connected components of level sets (preimages of points) to points.\u0000For some smooth functions, the Reeb graph has the structure of a finite graph.\u0000This notion finds numerous applications in the theory of dynamical systems, as well as in the topological classification of circle-valued functions and the study of their homotopy properties.\u0000However, important theoretical facts on the topological properties of the Reeb graphs of circle-valued functions are scattered across numerous papers on different topics, according to the specific needs of the corresponding application.\u0000In this paper, we systematize the existing results on the Reeb graphs of circle-valued functions and generalize some of them to wider classes of functions or spaces.\u0000We also show how some results can be carried out from real-valued functions. Finally, we adapt some facts from the theory of foliations to the Reeb graphs of circle-valued functions.\u0000In particular, we analyze the cycle rank of the Reeb graph and address the problem of realization of a finite graph as a Reeb graph.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44244132","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}
引用次数: 0
Maps of degree one, LS category and higher topological complexities 一阶映射、LS范畴和更高拓扑复杂性
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-03-04 DOI: 10.12775/tmna.2021.051
Yuli B. Rudyak, Soumen Sarkar
{"title":"Maps of degree one, LS category and higher topological complexities","authors":"Yuli B. Rudyak, Soumen Sarkar","doi":"10.12775/tmna.2021.051","DOIUrl":"https://doi.org/10.12775/tmna.2021.051","url":null,"abstract":"In this paper, we study the relation between the Lusternik-Schnirelmann category\u0000and the topological complexity of two closed oriented manifolds connected by a degree one map.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49353195","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}
引用次数: 0
Irrationally elliptic closed characteristics on symmetric compact convex hypersurfaces in R^8 R^8中对称紧致凸超曲面的非有理椭圆闭特性
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-03-04 DOI: 10.12775/tmna.2021.057
Wen Wang
{"title":"Irrationally elliptic closed characteristics on symmetric compact convex hypersurfaces in R^8","authors":"Wen Wang","doi":"10.12775/tmna.2021.057","DOIUrl":"https://doi.org/10.12775/tmna.2021.057","url":null,"abstract":"Let $Sigma$ be a $C^3$ compact symmetric convex hypersurface in\u0000$mathbb{R}^{8}$. We prove that when $Sigma$ carries exactly four\u0000geometrically distinct closed characteristics, then there are at least two\u0000irrationally elliptic closed characteristics on $Sigma$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47487568","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}
引用次数: 0
Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system 仿射控制系统解的Randers-Kropina度量的多个连通测地线
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-02-26 DOI: 10.12775/tmna.2022.066
E. Caponio, M. Javaloyes, A. Masiello
{"title":"Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system","authors":"E. Caponio, M. Javaloyes, A. Masiello","doi":"10.12775/tmna.2022.066","DOIUrl":"https://doi.org/10.12775/tmna.2022.066","url":null,"abstract":"We consider a geodesic problem in a manifold endowed with\u0000a Randers-Kropina metric. This is a type of a singular Finsler metric arising both\u0000in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo's navigation problem with a wind represented\u0000by a vector field having norm not greater than one.\u0000By using Lusternik-Schnirelman theory, we prove existence of infinitely many\u0000geodesics between two given points when the manifold is not contractible.\u0000Due to the type of non-holonomic constraints that the velocity vectors must satisfy,\u0000this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with \u0000a totally non-integrable distribution.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43067152","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}
引用次数: 1
Global multiplicity for parametric anisotropic Neumann (p,q)-equations 参数各向异性Neumann（p，q）-方程的全局多重性
IF 0.7 4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-02-26 DOI: 10.12775/TMNA.2022.010
Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš
{"title":"Global multiplicity for parametric anisotropic Neumann (p,q)-equations","authors":"Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš","doi":"10.12775/TMNA.2022.010","DOIUrl":"https://doi.org/10.12775/TMNA.2022.010","url":null,"abstract":"We consider a Neumann boundary value problem driven by the anisotropic\u0000 $(p,q)$-Laplacian plus a parametric potential term. \u0000The reaction is ``superlinear\". We prove a global (with respect to the parameter) multiplicity result for positive solutions. \u0000Also, we show the existence of a minimal positive solution and finally, we produce\u0000 a nodal solution.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44218099","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}
引用次数: 0
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