{"title":"Concentration of solutions for ( N , q )-Laplacian equatio","authors":"L. Wang, Jun Wang, Binlin Zhang","doi":"10.14232/ejqtde.2023.1.14","DOIUrl":null,"url":null,"abstract":"<jats:p>In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity: <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign=\"left left\" rowspacing=\".2em\" columnspacing=\"1em\" displaystyle=\"false\"> <mml:mtr> <mml:mtd> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=\"normal\">Δ<!-- Δ --></mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mi>x</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>N</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>q</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <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:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mi>u</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </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:mi>N</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> </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:mi>N</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:mo fence=\"true\" stretchy=\"true\" symmetric=\"true\" /> </mml:mrow> </mml:math> where <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:math> is a positive continuous function and has a local minimum, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> is a small parameter, <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>2</mml:mn> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>N</mml:mi> <mml:mo><</mml:mo> <mml:mi>q</mml:mi> <mml:mo><</mml:mo> <mml:mo>+</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:math>, <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:math> is <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> with subcritical growth. When <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: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:math> satisfy some appropriate assumptions, we construct the solution <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>ε<!-- ε --></mml:mi> </mml:mrow> </mml:msub> </mml:math> that concentrates around any given isolated local minimum of <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:math> by applying the penalization method for the above equation.</jats:p>","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Qualitative Theory of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.14232/ejqtde.2023.1.14","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: 0
Abstract
In this article, we consider the concentration of positive solutions for the following equation with Trudinger–Moser nonlinearity: {−ΔNu−Δqu+V(εx)(|u|N−2u+|u|q−2u)=f(u),x∈RN,u∈W1,N(RN)∩W1,q(RN),x∈RN, where V is a positive continuous function and has a local minimum, ε>0 is a small parameter, 2≤N<q<+∞, f is C1 with subcritical growth. When V and f satisfy some appropriate assumptions, we construct the solution uε that concentrates around any given isolated local minimum of V by applying the penalization method for the above equation.
在这篇文章中，我们考虑到采用Trudinger等式的积极解决方案的浓度——莫泽非线性:{ − Δ N u − Δ q u + V ( ε x ) ( | u | N − 2 u + | u| q − 2 u ) = f ( u ) , x ∈ R N , u ∈ W 1 ,N ( R N ) ∩ W 1 , q ( R N ) , x ∈ R N ,V是一个积极挑战功能和在哪里有a local最低,ε> 0是a small参数即可,2≤N q +∞,f是C和subcritical增长1。当V和f满足一些appropriate assumptions solution u》,我们构造 周围concentrates任何最低给孤立的local的εV by applying equation头顶的penalization方法》一书。
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.
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