In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>...In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>N,V is a spherically symmetric increasing function satisfying V(0)=0,|x|→∞lin V(x)=+∞We discuss the problem in three cases.First,for the case s>2N,e_(s)(ρ)=-∞for anyρ>0.Secondly,for the case N<s<2N,for anyρ>0,we prove that it admits a minimizer which is nonnegative,spherically symmetric and decreasing via the N-Laplacian GagliardoNirenberg inequality.When s=2N,the existence and nonexistence of minimizers of e_(s)(ρ)will also be given.During the arguments,we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.展开更多
AIM:To present an overview of the research on global glaucoma treatment in the last decade in terms of publication year,journals,countries/regions,organizations,references,and keywords,to investigate the current resea...AIM:To present an overview of the research on global glaucoma treatment in the last decade in terms of publication year,journals,countries/regions,organizations,references,and keywords,to investigate the current research international trends and hot topics in this area.METHODS:Bibliometric analysis was conducted on 9128 articles in the Web of Science Core Collection(WoSCC;Clarivate)database.Quantitative and qualitative analysis was employed using VOSviewer(v1.6.18),Pajek(v1.0.0.0),and CiteSpace(v6.1.R2)software.RESULTS:The 9128 papers relating to glaucoma treatment were published from April 2013 to April 2023,of which 7482 articles(82%)were original research articles and 1464(18%)were review articles.The United States(2867)and Johns Hopkins University(166)were the most productive country and institution,respectively,but the University College London had the highest h-index(54).The Journal of Glaucoma was the most productive and Ophthalmology had the highest h-index compared with other journals.The Keywords of interest included treatment surgery,cyclophotocoagulation,minimally invasive glaucoma surgery(MIGS),trabeculectomy,baerveldt,epidemiology,medication adherence,nanoparticle,optical coherence tomography(OCT),gene therapy,and artificial intelligence(AI).Glaucoma surgery appeared as a current research hotspot through the analysis of keywords.CONCLUSION:This study provides insights into the research trends and potential research hotspots in the treatment of glaucoma.This will help researchers to evaluate research policies and to promote international cooperation.展开更多
基金supported by the Xingdian Talents Support Program of Yunnan Province of Youthsthe Yunnan Province Basic Research Project for General Program(202401AT070441)+5 种基金the Yunnan Key Laboratory of Modern Analytical Mathematics and Applications(202302AN360007)supported by the NNSF(12261031)supported by the NNSF(12401145)supported by the NNSF(12371120)the Yunnan Province Basic Research Project for Key Program(202401AS070024)the General Program(202301AT070141)。
文摘In this paper,we investigate the minimization problem e_(s)(p)=_(u∈W_(V)^(1,N))(r^(N)),||u||_(N)^(N)=p>0 inf E(u),where E(u)=1/N∫_(R_(N))|▽_(u)|^(N)dx+1/N∫_(R_(N))V(x)|u|^(N)dx-1/s∫_(R_(N))|u|^(s)dx.Here s>N,V is a spherically symmetric increasing function satisfying V(0)=0,|x|→∞lin V(x)=+∞We discuss the problem in three cases.First,for the case s>2N,e_(s)(ρ)=-∞for anyρ>0.Secondly,for the case N<s<2N,for anyρ>0,we prove that it admits a minimizer which is nonnegative,spherically symmetric and decreasing via the N-Laplacian GagliardoNirenberg inequality.When s=2N,the existence and nonexistence of minimizers of e_(s)(ρ)will also be given.During the arguments,we provide the detailed proof of the N-Laplacian Gagliardo-Nirenberg inequality and N-Laplacian Pohozaev identity.
基金Suppotred by Tianjin Key Medical Discipline Construction Project(No.TJYXZDXK-3-004A-2).
文摘AIM:To present an overview of the research on global glaucoma treatment in the last decade in terms of publication year,journals,countries/regions,organizations,references,and keywords,to investigate the current research international trends and hot topics in this area.METHODS:Bibliometric analysis was conducted on 9128 articles in the Web of Science Core Collection(WoSCC;Clarivate)database.Quantitative and qualitative analysis was employed using VOSviewer(v1.6.18),Pajek(v1.0.0.0),and CiteSpace(v6.1.R2)software.RESULTS:The 9128 papers relating to glaucoma treatment were published from April 2013 to April 2023,of which 7482 articles(82%)were original research articles and 1464(18%)were review articles.The United States(2867)and Johns Hopkins University(166)were the most productive country and institution,respectively,but the University College London had the highest h-index(54).The Journal of Glaucoma was the most productive and Ophthalmology had the highest h-index compared with other journals.The Keywords of interest included treatment surgery,cyclophotocoagulation,minimally invasive glaucoma surgery(MIGS),trabeculectomy,baerveldt,epidemiology,medication adherence,nanoparticle,optical coherence tomography(OCT),gene therapy,and artificial intelligence(AI).Glaucoma surgery appeared as a current research hotspot through the analysis of keywords.CONCLUSION:This study provides insights into the research trends and potential research hotspots in the treatment of glaucoma.This will help researchers to evaluate research policies and to promote international cooperation.
