This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.
{"title":"Traveling wave solutions in a higher dimensional lattice delayed cooperation system with nonlocal diffusion","authors":"Kun Li, Yanli He","doi":"10.12775/tmna.2023.011","DOIUrl":"https://doi.org/10.12775/tmna.2023.011","url":null,"abstract":"This paper is concerned with the existence of traveling wave solutions of a higher dimensional lattice delayed cooperation system with nonlocal diffusion. For sufficiently small intraspecific cooperative delays, we construct upper and lower solutions under two different parameters conditions. And then, by using the monotone iterative and Schauder's fixed point theorem, we obtain the existence of traveling wave solutions. The lower bound of the wave speed is in accordance with the properties of linear determined.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135038943","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}
We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.
{"title":"Existence of sign-changing solutions for a third-order boundary value problem with nonlocal conditions of integral type","authors":"Sergey Smirnov","doi":"10.12775/tmna.2022.074","DOIUrl":"https://doi.org/10.12775/tmna.2022.074","url":null,"abstract":"We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966920","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}
In this paper, we study the existence of periodic solutions for a Liénard type $phi$-Laplacian equation $$ (phi(x'))'+f(x)x'+g(x)=p(t). $$ We prove a continuation lemma and use it to prove the existence of periodic solutions for above equation when $g$ or $G$ (the primitive of $g$) satisfies some one-sided or bilateral growth conditions and $F$ (the primitive of $f$) satisfies sublinear condition.
{"title":"On the existence of periodic solutions for Liénard type $phi$-Laplacian equation","authors":"Congmin Yang, Zaihong Wang","doi":"10.12775/tmna.2022.067","DOIUrl":"https://doi.org/10.12775/tmna.2022.067","url":null,"abstract":"In this paper, we study the existence of periodic solutions for a Liénard type $phi$-Laplacian equation $$ (phi(x'))'+f(x)x'+g(x)=p(t). $$ We prove a continuation lemma and use it to prove the existence of periodic solutions for above equation when $g$ or $G$ (the primitive of $g$) satisfies some one-sided or bilateral growth conditions and $F$ (the primitive of $f$) satisfies sublinear condition.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135966113","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}
This paper is concerned with the following supercritical Hénon equation with variable exponent $$ begin{cases} -Delta u=|x|^{alpha}|u|^{2^*_alpha-2+|x|^beta}u&text{in } B, u=0 &text{on } partial B, end{cases} $$% 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.
本文研究了下列变指数超临界hsamnon方程 $$ begin{cases} -Delta u=|x|^{alpha}|u|^{2^*_alpha-2+|x|^beta}u&text{in } B, u=0 &text{on } partial B, end{cases} $$% 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.
{"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 $$ begin{cases} -Delta u=|x|^{alpha}|u|^{2^*_alpha-2+|x|^beta}u&text{in } B, u=0 &text{on } partial B, end{cases} $$% 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":"17 1","pages":"0"},"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}
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.
{"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":"43 1","pages":"0"},"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}
Francisco S. B. Albuquerque, Jonison L. Carvalho, Marcelo F. Furtado, Everaldo S. Medeiros
In this paper we look for ground state solutions of the elliptic system $$ begin{cases} -Delta u+V(x)u+gammaphi K(x)u = Q(x)f(u), &xinmathbb{R}^{2}, Delta phi =K(x) u^{2}, &xinmathbb{R}^{2}, end{cases} $$% where $gamma> 0$ 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.
本文寻找椭圆系统的基态解 $$ begin{cases} -Delta u+V(x)u+gammaphi K(x)u = Q(x)f(u), &xinmathbb{R}^{2}, Delta phi =K(x) u^{2}, &xinmathbb{R}^{2}, end{cases} $$% where $gamma> 0$ 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.
{"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 $$ begin{cases} -Delta u+V(x)u+gammaphi K(x)u = Q(x)f(u), &xinmathbb{R}^{2}, Delta phi =K(x) u^{2}, &xinmathbb{R}^{2}, end{cases} $$% where $gamma> 0$ 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":"36 1","pages":"0"},"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}
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.
Fessler和Gutierrez cite{Fe}, cite{Gu}证明了如果一个非奇异平面映射在$(0,+infty)$中具有没有特征值的雅可比矩阵，则该映射是内射的。我们证明了用与上(下)复半平面分离的任意无界曲线代替$(0,+infty)$成立。另外，我们证明了如果$partial P/partial x + partial Q/partial y$不是满射函数，则雅可比映射$(P,Q)$是内射。
{"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":"73 1","pages":"0"},"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}
In this paper, we are concerned with the parabolic Lichnerowicz equation involving the $Delta_lambda$-Laplacian $$ v_t-Delta_lambda v=v^{-p-2}-v^p,quad v> 0, quad mbox{ in }mathbb R^Ntimesmathbb R, $$ where $p> 0$ 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 $p> 0$. Moreover, in the case $p> 1$, 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 $$ -Delta_lambda u=u^{-p-2}-u^p,quad u> 0,quad mbox{in }mathbb R^N. $$
{"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 $$ v_t-Delta_lambda v=v^{-p-2}-v^p,quad v> 0, quad mbox{ in }mathbb R^Ntimesmathbb R, $$ where $p> 0$ 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 $p> 0$. Moreover, in the case $p> 1$, 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 $$ -Delta_lambda u=u^{-p-2}-u^p,quad u> 0,quad mbox{in }mathbb R^N. $$","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":"22 1","pages":"0"},"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}
Ketty A. De Rezende, Nivaldo G. Grulha Jr., Dahisy V. de S. Lima, Murilo A. J. Zigart
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.
{"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":"43 1","pages":"0"},"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}
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 & text{in }mathbb{R}^{3}, (-Delta )^{t}phi =l(x)u^{2} & 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.
{"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 & text{in }mathbb{R}^{3}, (-Delta )^{t}phi =l(x)u^{2} & 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":"2 1","pages":"0"},"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}
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