New results on the existence of ground state solutions for generalized quasilinear Schrödinger equations coupled with the Chern–Simons gauge theory

IF 0.8 4区数学 Q1 MATHEMATICS

Electronic Journal of Qualitative Theory of Differential Equations Pub Date : 2021-01-01 DOI:10.14232/ejqtde.2021.1.73

Yingying Xiao, Chuanxi Zhu

{"title":"New results on the existence of ground state solutions for generalized quasilinear Schrödinger equations coupled with the Chern–Simons gauge theory","authors":"Yingying Xiao, Chuanxi Zhu","doi":"10.14232/ejqtde.2021.1.73","DOIUrl":null,"url":null,"abstract":"In this paper, we study the following quasilinear Schrödinger equation <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mtable columnalign=\"right left\" rowspacing=\"3pt\" columnspacing=\"0em\" displaystyle=\"true\"> <mml:mtr> <mml:mtd> <mml:mo>−</mml:mo> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mi>u</mml:mi> </mml:mtd> <mml:mtd> <mml:mi /> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>−</mml:mo> <mml:mi>κ</mml:mi> <mml:mi>u</mml:mi> <mml:mi mathvariant=\"normal\">Δ</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>μ</mml:mi> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>κ</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mi>u</mml:mi> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd /> <mml:mtd> <mml:mi /> <mml:mo>+</mml:mo> <mml:mi>μ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>+</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:mfrac> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:mi>s</mml:mi> </mml:mfrac> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>κ</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>s</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mtext>d</mml:mtext> <mml:mi>s</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mspace width=\"1em\" /> <mml:mtext>in</mml:mtext> <mml:mtext> </mml:mtext> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>κ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> and <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>f</mml:mi> <mml:mo>∈</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:math> By using a constraint minimization of Pohožaev–Nehari type and analytic techniques, we obtain the existence of ground state solutions.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Qualitative Theory of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2021.1.73","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 2

Abstract

In this paper, we study the following quasilinear Schrödinger equation − Δ u + V ( x ) u − κ u Δ ( u 2 ) + μ h 2 ( | x | ) | x | 2 ( 1 + κ u 2 ) u + μ ( ∫ | x | + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s ) u = f ( u ) in R 2 , κ > 0 V ∈ C 1 ( R 2 , R ) and f ∈ C ( R , R ) By using a constraint minimization of Pohožaev–Nehari type and analytic techniques, we obtain the existence of ground state solutions.

查看原文本刊更多论文

与chen - simons规范理论耦合的广义拟线性Schrödinger方程基态解存在性的新结果

在这篇论文中，我们学习《薛定谔的跟踪quasilinear equation −Δu + V ( x ) u u−κΔ ( u 2 ) + μ h 2 ( | x | )| x | 2 ( 1 + κ u 2 ) u + μ ( ∫ | x |+ ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s )u = f ( u ) 在 R 2 , κ> 0∈V C 1 ( R 2 , R)和f∈C ( R , 最新的R):用a水量minimizationžaev——Nehari类型和分析techniques,我们得到地面state university)之存在解决方案。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Electronic Journal of Qualitative Theory of Differential Equations 数学-数学

CiteScore

1.40

自引率

9.10%

发文量

审稿时长

3 months

期刊介绍： The Electronic Journal of Qualitative Theory of Differential Equations (EJQTDE) is a completely open access journal dedicated to bringing you high quality papers on the qualitative theory of differential equations. Papers appearing in EJQTDE are available in PDF format that can be previewed, or downloaded to your computer. The EJQTDE is covered by the Mathematical Reviews, Zentralblatt and Scopus. It is also selected for coverage in Thomson Reuters products and custom information services, which means that its content is indexed in Science Citation Index, Current Contents and Journal Citation Reports. Our journal has an impact factor of 1.827, and the International Standard Serial Number HU ISSN 1417-3875. All topics related to the qualitative theory (stability, periodicity, boundedness, etc.) of differential equations (ODE''s, PDE''s, integral equations, functional differential equations, etc.) and their applications will be considered for publication. Research articles are refereed under the same standards as those used by any journal covered by the Mathematical Reviews or the Zentralblatt (blind peer review). Long papers and proceedings of conferences are accepted as monographs at the discretion of the editors.