{"title":"一类活化剂-抑制剂体系解的存在性","authors":"G. Figueiredo, M. Montenegro","doi":"10.1017/S0017089522000131","DOIUrl":null,"url":null,"abstract":"Abstract We prove the existence of a solution for a class of activator–inhibitor system of type \n$- \\Delta u +u = f(u) -v$\n , \n$-\\Delta v+ v=u$\n in \n$\\mathbb{R}^{N}$\n . The function f is a general nonlinearity which can grow polynomially in dimension \n$N\\geq 3$\n or exponentiallly if \n$N=2$\n . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.","PeriodicalId":50417,"journal":{"name":"Glasgow Mathematical Journal","volume":"65 1","pages":"98 - 113"},"PeriodicalIF":0.5000,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence of solution for a class of activator–inhibitor systems\",\"authors\":\"G. Figueiredo, M. Montenegro\",\"doi\":\"10.1017/S0017089522000131\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We prove the existence of a solution for a class of activator–inhibitor system of type \\n$- \\\\Delta u +u = f(u) -v$\\n , \\n$-\\\\Delta v+ v=u$\\n in \\n$\\\\mathbb{R}^{N}$\\n . The function f is a general nonlinearity which can grow polynomially in dimension \\n$N\\\\geq 3$\\n or exponentiallly if \\n$N=2$\\n . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.\",\"PeriodicalId\":50417,\"journal\":{\"name\":\"Glasgow Mathematical Journal\",\"volume\":\"65 1\",\"pages\":\"98 - 113\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-04-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Glasgow Mathematical Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1017/S0017089522000131\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Glasgow Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/S0017089522000131","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Existence of solution for a class of activator–inhibitor systems
Abstract We prove the existence of a solution for a class of activator–inhibitor system of type
$- \Delta u +u = f(u) -v$
,
$-\Delta v+ v=u$
in
$\mathbb{R}^{N}$
. The function f is a general nonlinearity which can grow polynomially in dimension
$N\geq 3$
or exponentiallly if
$N=2$
. We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.
期刊介绍:
Glasgow Mathematical Journal publishes original research papers in any branch of pure and applied mathematics. An international journal, its policy is to feature a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics.
The journal has a web-based submission system for articles. For details of how to to upload your paper see GMJ - Online Submission Guidelines or go directly to the submission site.