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Ground state solution for a class of supercritical Hénon equation with variable exponent 一类变指数超临界hsamnon方程的基态解
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.065
Xiaojing Feng
{"title":"Ground state solution for a class of supercritical Hénon equation with variable exponent","authors":"Xiaojing Feng","doi":"10.12775/tmna.2022.065","DOIUrl":"https://doi.org/10.12775/tmna.2022.065","url":null,"abstract":"This paper is concerned with the following supercritical Hénon equation with variable exponent $$ begin{cases} -Delta u=|x|^{alpha}|u|^{2^*_alpha-2+|x|^beta}u&amp;text{in } B, u=0 &amp;text{on } partial B, end{cases} $$% where $Bsubsetmathbb{R}^N$ $(Ngeq 3)$ is the unit ball, $alpha!> !0$, $ 0!< !beta!< !min{(N!+!alpha)/2,N!-!2}$ and $2^*_alpha=({2N+2alpha})/({N-2})$. We obtain the existence of positive ground state solution by applying the mountain pass theorem, concentration-compactness principle and approximation techniques.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966913","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}
引用次数: 0
A class of singular $k_i$-Hessian systems 一类奇异k_i -Hessian系统
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.072
Meiqiang Feng
{"title":"A class of singular $k_i$-Hessian systems","authors":"Meiqiang Feng","doi":"10.12775/tmna.2022.072","DOIUrl":"https://doi.org/10.12775/tmna.2022.072","url":null,"abstract":"Our main objective of this article is to investigate a class of singular $k_i$-Hessian systems. Among others, we obtain new theorems on the existence and multiplicity of positive radial solutions. Several nonexistence theorems are also derived.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966919","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}
引用次数: 0
A planar Schrödinger-Poisson system with vanishing potentials and exponential critical growth 具有消失势和指数临界增长的平面Schrödinger-Poisson系统
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.058
Francisco S. B. Albuquerque, Jonison L. Carvalho, Marcelo F. Furtado, Everaldo S. Medeiros
{"title":"A planar Schrödinger-Poisson system with vanishing potentials and exponential critical growth","authors":"Francisco S. B. Albuquerque, Jonison L. Carvalho, Marcelo F. Furtado, Everaldo S. Medeiros","doi":"10.12775/tmna.2022.058","DOIUrl":"https://doi.org/10.12775/tmna.2022.058","url":null,"abstract":"In this paper we look for ground state solutions of the elliptic system $$ begin{cases} -Delta u+V(x)u+gammaphi K(x)u = Q(x)f(u), &amp;xinmathbb{R}^{2}, Delta phi =K(x) u^{2}, &amp;xinmathbb{R}^{2}, end{cases} $$% where $gamma&gt; 0$ and the continuous potentials $V$, $K$, $Q$ satisfy some mild growth conditions and the nonlinearity $f$ has exponential critical growth. The key point of our approach is a new version of the Trudinger-Moser inequality for weighted Sobolev space.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966914","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}
引用次数: 0
Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space 特征值空间中具有断开曲线的非奇异平面映射的注入性
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.073
Marco Sabatini
{"title":"Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space","authors":"Marco Sabatini","doi":"10.12775/tmna.2022.073","DOIUrl":"https://doi.org/10.12775/tmna.2022.073","url":null,"abstract":"Fessler and Gutierrez cite{Fe}, cite{Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+infty)$, then it is injective. We prove that the same holds replacing $(0,+infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $partial P/partial x + partial Q/partial y$ is not a surjective function.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966747","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}
引用次数: 0
A note on positive solutions of Lichnerowicz equations involving the $Delta_lambda$-Laplacian 涉及$Delta_lambda$ -拉普拉斯式的Lichnerowicz方程正解的注记
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.076
Anh Tuan Duong, Thi Quynh Nguyen
{"title":"A note on positive solutions of Lichnerowicz equations involving the $Delta_lambda$-Laplacian","authors":"Anh Tuan Duong, Thi Quynh Nguyen","doi":"10.12775/tmna.2022.076","DOIUrl":"https://doi.org/10.12775/tmna.2022.076","url":null,"abstract":"In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $Delta_lambda$-Laplacian $$ v_t-Delta_lambda v=v^{-p-2}-v^p,quad v&gt; 0, quad mbox{ in }mathbb R^Ntimesmathbb R, $$ where $p&gt; 0$ and $Delta_lambda$ is a sub-elliptic operator of the form $$ Delta_lambda=sum_{i=1}^Npartial_{x_i}big(lambda_i^2partial_{x_i}big). $$ Under some general assumptions of $lambda_i$ introduced by A.E. Kogoj and E. Lanconelli in Nonlinear Anal. {bf 75} (2012), no. 12, 4637-4649, we shall prove a uniform lower bound of positive solutions of the equation provided that $p&gt; 0$. Moreover, in the case $p&gt; 1$, we shall show that the equation has only the trivial solution $v=1$. As a consequence, when $v$ is independent of the time variable, we obtain the similar results for the elliptic Lichnerowicz equation involving the $Delta_lambda$-Laplacian $$ -Delta_lambda u=u^{-p-2}-u^p,quad u&gt; 0,quad mbox{in }mathbb R^N. $$","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966110","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}
引用次数: 0
Conley index theory for Gutierrez-Sotomayor flows on singular 3-manifolds 奇异3-流形上Gutierrez-Sotomayor流的Conley指标理论
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.070
Ketty A. De Rezende, Nivaldo G. Grulha Jr., Dahisy V. de S. Lima, Murilo A. J. Zigart
{"title":"Conley index theory for Gutierrez-Sotomayor flows on singular 3-manifolds","authors":"Ketty A. De Rezende, Nivaldo G. Grulha Jr., Dahisy V. de S. Lima, Murilo A. J. Zigart","doi":"10.12775/tmna.2022.070","DOIUrl":"https://doi.org/10.12775/tmna.2022.070","url":null,"abstract":"This paper is a continuation of the investigation done in dimension two, this time for the Gutierrez-Sotomayor vector fields on singular $3$-manifolds. The singularities of Gutierrez-Sotomayor flows (GS flows, for short) in this setting are the 3-dimensional counterparts of cones, cross-caps, double and triple crossing points. First, we prove the existence of a Lyapunov function in a neighborhood of a given singularity of a GS flow, i.e. a GS singularity. In these neighbourhoods, index pairs are defined and allow a direct computation of the Conley indices for the different types of GS singularities. The Conley indices are used to prove local necessary conditions on the number of connected boundary components of an isolating block for a GS singularity as well as their Euler characteristic. Lyapunov semi-graphs are introduced as a tool to record this topological and dynamical information. Lastly, we construct isolating blocks so as to prove the sufficiency of the connectivity bounds on the boundaries of isolating blocks given by the Lyapunov semi-graphs.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966745","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}
引用次数: 0
Three positive solutions for the indefinite fractional Schrödinger-Poisson systems 不定分数阶Schrödinger-Poisson系统的三个正解
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.046
Guofeng Che, Tsung-fang Wu
{"title":"Three positive solutions for the indefinite fractional Schrödinger-Poisson systems","authors":"Guofeng Che, Tsung-fang Wu","doi":"10.12775/tmna.2022.046","DOIUrl":"https://doi.org/10.12775/tmna.2022.046","url":null,"abstract":"In this paper, we are concerned with the following fractionalSchrödinger-Poisson systems with concave-convex nonlinearities: begin{equation*} begin{cases} (-Delta )^{s}u+u+mu l(x)phi u=f(x)|u|^{p-2}u+g(x)|u|^{q-2}u &amp; text{in }mathbb{R}^{3}, (-Delta )^{t}phi =l(x)u^{2} &amp; text{in }mathbb{R}^{3},% end{cases} end{equation*} where ${1}/{2}< tleq s< 1$, $1< q< 2< p< min {4,2_{s}^{ast }}$, $2_{s}^{ast }={6}/({3-2s})$, and $mu > 0$ is a parameter, $fin Cbig(mathbb{R}^{3}big)$ is sign-changing in $mathbb{R}^{3}$ and $gin L^{p/(p-q)}big(mathbb{R}^{3}big)$. Under some suitable assumptions on $l(x)$, $f(x)$ and $g(x)$, we explore that the energy functional corresponding to the system is coercive and bounded below on $H^{alpha }big(mathbb{R}^{3}big)$ which gets a positive solution. Furthermore, we constructed some new estimation techniques, and obtained other two positive solutions. Recent results from the literature are generally improved and extended.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966748","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}
引用次数: 0
Compactness in normed spaces: a unified approach through semi-norms 赋范空间中的紧性:通过半规范的统一方法
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.064
Jacek Gulgowski, Piotr Kasprzak, Piotr Maćkowiak
{"title":"Compactness in normed spaces: a unified approach through semi-norms","authors":"Jacek Gulgowski, Piotr Kasprzak, Piotr Maćkowiak","doi":"10.12775/tmna.2022.064","DOIUrl":"https://doi.org/10.12775/tmna.2022.064","url":null,"abstract":"In this paper we prove two new abstract compactness criteria in normed spaces. To this end we first introduce the notion of an equinormed set using a suitable family of semi-norms on the given normed space satisfying some natural conditions. Those conditions, roughly speaking, state that the norm can be approximated (on the equinormed sets even uniformly) by the elements of this family. As we are given some freedom of choice of the underlying semi-normed structure that is used to define equinormed sets, our approach opens a new perspective for building compactness criteria in specific normed spaces. As an example we show that natural selections of families of semi-norms in spaces $C(X,R)$ and $l^p$ for $pin[1,+infty)$ lead to the well-known compactness criteria (including the Arzel`a-Ascoli theorem). In the second part of the paper, applying the abstract theorems, we construct a simple compactness criterion in the space of functions of bounded Schramm variation.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135959832","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}
引用次数: 2
Liouville type theorems for Kirchhoff sub-elliptic equations involving $Delta_lambda$-operators 涉及$Delta_lambda$ -算子的Kirchhoff次椭圆方程的Liouville型定理
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.071
Thi Thu Huong Nguyen, Dao Trong Quyet, Thi Hien Anh Vu
{"title":"Liouville type theorems for Kirchhoff sub-elliptic equations involving $Delta_lambda$-operators","authors":"Thi Thu Huong Nguyen, Dao Trong Quyet, Thi Hien Anh Vu","doi":"10.12775/tmna.2022.071","DOIUrl":"https://doi.org/10.12775/tmna.2022.071","url":null,"abstract":"In this paper, we study the Kirchhoff elliptic equations of the form $$ -M(|nabla_lambda u|^2)Delta_lambda u=w(x)f(u) quad mbox{in }mathbb R^{N}, $$ where $M$ is a smooth monotone function, $w$ is a weight function and $f(u)$ is of the form $u^p, e^u$ or $-u^{-p}$. The operator $Delta_lambda$ is strongly degenerate and given by $$ Delta_lambda=sum_{j=1}^N frac{partial}{partial x_j}bigg(lambda_j^2(x)frac{partial }{partial x_j}bigg). $$ We shall prove some classifications of stable solutions to the equation above under general assumptions on $M$ and $lambda_j$, $j=1,ldots,N$.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966916","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}
引用次数: 0
A Fredholm alternative for elliptic equations with interior and boundary nonlinear reactions 具有内部和边界非线性反应的椭圆方程的Fredholm替代
4区 数学
Topological Methods in Nonlinear Analysis Pub Date : 2023-09-23 DOI: 10.12775/tmna.2022.054
Daniel Maroncelli, Mauricio A. Rivas
{"title":"A Fredholm alternative for elliptic equations with interior and boundary nonlinear reactions","authors":"Daniel Maroncelli, Mauricio A. Rivas","doi":"10.12775/tmna.2022.054","DOIUrl":"https://doi.org/10.12775/tmna.2022.054","url":null,"abstract":"In this paper we study the existence of solutions to the following generalized nonlinear two-parameter problem begin{equation*} a(u, v) = lambda b(u, v) + mu m(u, v) + varepsilon F(u, v), end{equation*} for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$. This problem is a natural abstraction of nonlinear problems that occur for a large class of differential operators, various elliptic pde's with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. Lastly, the Steklov-Robin Fredholm equation is used to exemplify the abstract results.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966917","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}
引用次数: 0
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