In this paper we study the fractional Brezis-Nirenberg type problems in unbounded cylinder-type domains begin{align*} begin{cases} (-Delta)^{s}u-mudfrac{u}{|x|^{2s}}=lambda u+|u|^{2^{ast}_{s}-2}u & text{in } Omega, u=0 & text{in } mathbb{R}^{N}setminus Omega, end{cases} end{align*} where $(-Delta)^{s}$ is the fractional Laplace operator with $sin(0,1)$, $muin[0,Lambda_{N,s})$ with $Lambda_{N,s}$ the best fractional Hardy constant, $lambda> 0$, $N> 2s$ and $2^{ast}_{s}={2N}/({N-2s})$ denotes the fractional critical Sobolev exponent. By applying the fractional Poincaré inequality together with the concentration-compactness principle for fractional Sobolev spaces in unbounded domains, we prove an existence result to the equation.
本文研究了无界圆柱型域中的分数阶Brezis-Nirenberg型问题^{s}u-mudfrac｛u｝｛|x|^｛2s｝｝＝λu+| u | ^｛2^｛ast｝_{s}-2}u&&text｛in｝Omega，u=0&&text{in｝mathbb｛R｝^｛N｝setminusOmega、end｛cases｝end｛align*｝，其中$（-Delta）^｛s｝$是具有$sin（0,1）$的分数拉普拉斯算子，$muin[0]，Lambda_｛N，s｝）$，其中$Lambda_{N，s}$是最佳分式Hardy常数，$Lambda>0$，$N>2s$和$2^｛sast｝_{s｝=｛2N｝/（｛N-2s｝）$$表示分数临界Sobolev指数。
{"title":"Existence results for fractional Brezis-Nirenberg type problems in unbounded domains","authors":"Yansheng Shen, Xumin Wang","doi":"10.12775/tmna.2022.009","DOIUrl":"https://doi.org/10.12775/tmna.2022.009","url":null,"abstract":"In this paper we study the fractional Brezis-Nirenberg type problems in unbounded cylinder-type domains\u0000begin{align*}\u0000begin{cases}\u0000(-Delta)^{s}u-mudfrac{u}{|x|^{2s}}=lambda u+|u|^{2^{ast}_{s}-2}u\u0000 & text{in } Omega,\u0000 u=0 & text{in } mathbb{R}^{N}setminus Omega,\u0000end{cases}\u0000end{align*}\u0000where $(-Delta)^{s}$ is the fractional Laplace operator with $sin(0,1)$,\u0000$muin[0,Lambda_{N,s})$ with $Lambda_{N,s}$ the best fractional Hardy constant, $lambda> 0$, $N> 2s$ and $2^{ast}_{s}={2N}/({N-2s})$\u0000denotes the fractional critical Sobolev exponent. By applying the fractional\u0000Poincaré inequality together with the concentration-compactness principle\u0000for fractional Sobolev spaces in unbounded domains, we prove an existence\u0000result to the equation.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42168030","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 paper, based on a new theoretical framework of time-dependent global attractors (Conti, Pata and Temam cite{CPT13}), we consider the strongly damped wave equations $varepsilon(t)u_{tt}-Delta u_{t}-Delta u+f(u)=g(x)$ and establish the existence of attractors in $mathcal{H}_{t}=H_{0}^{1}(Omega)times L^{2}(Omega)$ and $mathcal{V}_{t}=H_{0}^{1}(Omega)times H_{0}^{1}(Omega)$, respectively.
{"title":"Time-dependent global attractors for the strongly damped wave equations with lower regular forcing term","authors":"Xinyu Mei, T. Sun, Yongqin Xie, Kaixuan Zhu","doi":"10.12775/tmna.2022.022","DOIUrl":"https://doi.org/10.12775/tmna.2022.022","url":null,"abstract":"In this paper, based on a new theoretical framework of\u0000time-dependent global attractors (Conti, Pata and Temam cite{CPT13}),\u0000we consider the strongly damped wave equations $varepsilon(t)u_{tt}-Delta u_{t}-Delta u+f(u)=g(x)$\u0000and establish the existence of attractors\u0000in $mathcal{H}_{t}=H_{0}^{1}(Omega)times L^{2}(Omega)$\u0000and $mathcal{V}_{t}=H_{0}^{1}(Omega)times H_{0}^{1}(Omega)$, respectively.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42920835","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 paper, we study the geometric rigidity of complete Riemannian manifolds admitting local minimizers of energy functionals. More precisely, assuming the existence of a non-trivial local minimizer and under suitable assumptions, a Riemannian manifold under consideration must be a product manifold furnished with a warped metric. Secondly, under similar hypotheses, we deduce a geometrical splitting in the same fashion as in the Cheeger-Gromoll splitting theorem and we also get information about local minimizers.
{"title":"A note on local minimizers of energy on complete manifolds","authors":"M. Batista, José I. Santos","doi":"10.12775/tmna.2022.013","DOIUrl":"https://doi.org/10.12775/tmna.2022.013","url":null,"abstract":"In this paper, we study the geometric rigidity of complete Riemannian manifolds admitting local minimizers of energy functionals.\u0000More precisely, assuming the existence of a non-trivial local minimizer and under suitable assumptions, a Riemannian manifold under consideration must be\u0000a product manifold furnished with a warped metric.\u0000Secondly, under similar hypotheses, we deduce a geometrical splitting in\u0000the same fashion as in the Cheeger-Gromoll splitting theorem and we also get information about local minimizers.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41321676","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 work, we address the existence of solutions for a biharmonic elliptic equation with homogeneous Navier boundary condition. The problem is asymmetric and has linear behavior on $-infty$ and superlinear on $+infty$. To obtain the results we apply topological methods.
{"title":"On a semilinear fourth order elliptic problem with asymmetric nonlinearity","authors":"Fabiana Ferreira, E. Medeiros, Wallisom Rosa","doi":"10.12775/tmna.2022.028","DOIUrl":"https://doi.org/10.12775/tmna.2022.028","url":null,"abstract":"In this work, we address the existence of solutions for a biharmonic elliptic equation \u0000with homogeneous Navier boundary condition. The problem is asymmetric and has linear behavior on $-infty$ and superlinear on $+infty$. \u0000To obtain the results we apply topological methods.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48889974","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 are concerned with periodic solutions of the fractional Laplace equation begin{equation*} {(-partial_{xx})^s}u(x)+F'(u(x))=0 quad mbox{in }mathbb{R}, end{equation*} where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at $+1$ and $-1$. We show that the value of least positive period is $2{pi}times({1}/{-F''(0)})^{{1}/({2s})}$. The axial symmetry of odd periodic solutions is obtained by moving plane method. We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution of the same equation as periods $Trightarrow+infty$.
{"title":"Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit","authors":"Zhenping Feng, Zhuoran Du","doi":"10.12775/tmna.2022.016","DOIUrl":"https://doi.org/10.12775/tmna.2022.016","url":null,"abstract":"We are concerned with periodic solutions of the fractional Laplace equation\u0000begin{equation*}\u0000{(-partial_{xx})^s}u(x)+F'(u(x))=0 quad mbox{in }mathbb{R},\u0000end{equation*}\u0000where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at\u0000$+1$ and $-1$. We show that the value of least positive period is\u0000$2{pi}times({1}/{-F''(0)})^{{1}/({2s})}$.\u0000 The axial symmetry of odd periodic solutions is obtained by moving plane method.\u0000We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution\u0000 of the same equation as periods $Trightarrow+infty$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44242890","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 paper, we establish the existence of solutions to a class of elliptic systems. The nonlinearities include exponential growth terms and convection terms. The exponential growth term means it could be critical growth at $infty$. The Trudinger-Moser inequality is used to deal with it. The convection term means it involves the gradient of unknown function. The strong convergence of sequences is employed to overcome the difficulties caused by convection terms. The variational methods are invalid and the Galerkin method and an approximation scheme are applied to obtain four different solutions. Our results supplements those from cite{Araujo2018}.
{"title":"Some existence results for elliptic systems with exponential nonlinearities and convection terms in dimension two","authors":"W. Liu","doi":"10.12775/tmna.2022.025","DOIUrl":"https://doi.org/10.12775/tmna.2022.025","url":null,"abstract":"In this paper, we establish the existence of solutions to a class of elliptic systems.\u0000The nonlinearities include exponential growth terms and convection terms.\u0000 The exponential growth term means it could be critical growth at $infty$.\u0000The Trudinger-Moser inequality is used to deal with it. The convection term means\u0000 it involves the gradient of unknown function.\u0000The strong convergence of sequences is employed to overcome the difficulties caused by convection terms.\u0000The variational methods are invalid and the Galerkin method and an approximation scheme are applied to obtain four different solutions.\u0000Our results supplements those from cite{Araujo2018}.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49360756","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 investigate the boundedness and regularity of solutions to degenerate elliptic equations with variable exponents that are subject to the Dirichlet boundary condition. By employing the De Giorgi iteration, we obtain a-priori bounds and the Hölder continuity for solutions. As an application, we obtain the existence of infinitely many small solutions for a class of degenerate elliptic equations involving variable exponents.
{"title":"A-priori bound and Hölder continuity of solutions to degenerate elliptic equations with variable exponents","authors":"Ky Ho, L. Nhan, L. Truong","doi":"10.12775/tmna.2022.021","DOIUrl":"https://doi.org/10.12775/tmna.2022.021","url":null,"abstract":"We investigate the boundedness and regularity of solutions to degenerate elliptic equations with variable exponents that are subject to the Dirichlet boundary condition. \u0000By employing the De Giorgi iteration, we obtain a-priori bounds and the Hölder continuity for solutions. \u0000As an application, we obtain the existence of infinitely many small solutions for a class of degenerate elliptic equations involving variable exponents.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46074872","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}
It is known that the concept of affine-periodicity encompasses classic notions of symmetries as the classic periodicity, anti-periodicity and rotating symmetries (in particular, quasi-periodicity). The aim of this paper is to establish the basis of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible $ntimes n$ matrix $Q$, with entries in $mathbb C$, we establish conditions for the existence of a $(Q,T)$-affine-periodic solution within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ. We apply our main results to measure differential equations with Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.
{"title":"Affine-periodic solutions for generalized ODEs and other equations","authors":"M. Federson, R. Grau, Carolina Mesquita","doi":"10.12775/tmna.2022.027","DOIUrl":"https://doi.org/10.12775/tmna.2022.027","url":null,"abstract":"It is known that the concept of affine-periodicity encompasses classic notions\u0000of symmetries as the classic periodicity, anti-periodicity and rotating symmetries\u0000(in particular, quasi-periodicity). The aim of this paper is to establish the basis\u0000 of affine-periodic solutions of generalized ODEs. Thus, for a given real number $T> 0$ and an invertible $ntimes n$ matrix $Q$, with entries in $mathbb C$,\u0000we establish conditions for the existence of a $(Q,T)$-affine-periodic solution\u0000within the framework of nonautonomous generalized ODEs, whose integral form displays the nonabsolute Kurzweil integral, which encompasses many types\u0000of integrals, such as the Riemann, the Lebesgue integral, among others. The main tools employed here are the fixed point theorems of Banach and of Krasnosel'skiĭ.\u0000We apply our main results to measure differential equations with\u0000Henstock-Kurzweil-Stiejtes righthand sides as well as to impulsive differential equations and dynamic equations on time scales which are particular cases of the former.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43136090","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}
Background and objectives: The Alberta Stroke Program CT Score (ASPECTS) is a widely used rating system for assessing infarct extent and location. We aimed to investigate the prognostic value of ASPECTS subregions' involvement in the long-term functional outcomes of acute ischemic stroke (AIS).
Materials and methods: Consecutive patients with AIS and anterior circulation large-vessel stenosis and occlusion between January 2019 and December 2020 were included. The ASPECTS score and subregion involvement for each patient was assessed using posttreatment magnetic resonance diffusion-weighted imaging. Univariate and multivariable regression analyses were conducted to identify subregions related to 3-month poor functional outcome (modified Rankin Scale scores, 3-6) in the reperfusion and medical therapy cohorts, respectively. In addition, prognostic efficiency between the region-based ASPECTS and ASPECTS score methods were compared using receiver operating characteristic curves and DeLong's test.
Results: A total of 365 patients (median age, 64 years; 70% men) were included, of whom 169 had poor outcomes. In the reperfusion therapy cohort, multivariable regression analyses revealed that the involvement of the left M4 cortical region in left-hemisphere stroke (adjusted odds ratio [aOR] 5.39, 95% confidence interval [CI] 1.53-19.02) and the involvement of the right M3 cortical region in right-hemisphere stroke (aOR 4.21, 95% CI 1.05-16.78) were independently associated with poor functional outcomes. In the medical therapy cohort, left-hemisphere stroke with left M5 cortical region (aOR 2.87, 95% CI 1.08-7.59) and caudate nucleus (aOR 3.14, 95% CI 1.00-9.85) involved and right-hemisphere stroke with right M3 cortical region (aOR 4.15, 95% CI 1.29-8.18) and internal capsule (aOR 3.94, 95% CI 1.22-12.78) affected were related to the increased risks of poststroke disability. In addition, region-based ASPECTS significantly improved the prognostic efficiency compared with the conventional ASPECTS score method.
Conclusion: The involvement of specific ASPECTS subregions depending on the affected hemisphere was associated with worse functional outcomes 3 months after stroke, and the critical subregion distribution varied by clinical management. Therefore, region-based ASPECTS could provide additional value in guiding individual decision making and neurological recovery in patients with AIS.
{"title":"Impact of the Alberta Stroke Program CT Score subregions on long-term functional outcomes in acute ischemic stroke: Results from two multicenter studies in China.","authors":"Xinrui Wang, Caohui Duan, Jinhao Lyu, Dongshan Han, Kun Cheng, Zhihua Meng, Xiaoyan Wu, Wen Chen, Guohua Wang, Qingliang Niu, Xin Li, Yitong Bian, Dan Han, Weiting Guo, Shuai Yang, Ximing Wang, Tijiang Zhang, Junying Bi, Feiyun Wu, Shuang Xia, Dan Tong, Kai Duan, Zhi Li, Rongpin Wang, Jinan Wang, Xin Lou","doi":"10.2478/jtim-2022-0057","DOIUrl":"10.2478/jtim-2022-0057","url":null,"abstract":"<p><strong>Background and objectives: </strong>The Alberta Stroke Program CT Score (ASPECTS) is a widely used rating system for assessing infarct extent and location. We aimed to investigate the prognostic value of ASPECTS subregions' involvement in the long-term functional outcomes of acute ischemic stroke (AIS).</p><p><strong>Materials and methods: </strong>Consecutive patients with AIS and anterior circulation large-vessel stenosis and occlusion between January 2019 and December 2020 were included. The ASPECTS score and subregion involvement for each patient was assessed using posttreatment magnetic resonance diffusion-weighted imaging. Univariate and multivariable regression analyses were conducted to identify subregions related to 3-month poor functional outcome (modified Rankin Scale scores, 3-6) in the reperfusion and medical therapy cohorts, respectively. In addition, prognostic efficiency between the region-based ASPECTS and ASPECTS score methods were compared using receiver operating characteristic curves and DeLong's test.</p><p><strong>Results: </strong>A total of 365 patients (median age, 64 years; 70% men) were included, of whom 169 had poor outcomes. In the reperfusion therapy cohort, multivariable regression analyses revealed that the involvement of the left M4 cortical region in left-hemisphere stroke (adjusted odds ratio [aOR] 5.39, 95% confidence interval [CI] 1.53-19.02) and the involvement of the right M3 cortical region in right-hemisphere stroke (aOR 4.21, 95% CI 1.05-16.78) were independently associated with poor functional outcomes. In the medical therapy cohort, left-hemisphere stroke with left M5 cortical region (aOR 2.87, 95% CI 1.08-7.59) and caudate nucleus (aOR 3.14, 95% CI 1.00-9.85) involved and right-hemisphere stroke with right M3 cortical region (aOR 4.15, 95% CI 1.29-8.18) and internal capsule (aOR 3.94, 95% CI 1.22-12.78) affected were related to the increased risks of poststroke disability. In addition, region-based ASPECTS significantly improved the prognostic efficiency compared with the conventional ASPECTS score method.</p><p><strong>Conclusion: </strong>The involvement of specific ASPECTS subregions depending on the affected hemisphere was associated with worse functional outcomes 3 months after stroke, and the critical subregion distribution varied by clinical management. Therefore, region-based ASPECTS could provide additional value in guiding individual decision making and neurological recovery in patients with AIS.</p>","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"24 1","pages":"197-208"},"PeriodicalIF":4.9,"publicationDate":"2022-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11107184/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88765297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A word on Professor Kazimierz Goebel (1940-2022)","authors":"Stanisław Prus","doi":"10.12775/tmna.2022.019","DOIUrl":"https://doi.org/10.12775/tmna.2022.019","url":null,"abstract":"<jats:p />","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44483707","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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