Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.5.691
Haiyang He
In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation [-Delta_{mathbb{B}^{N}}u=int_{mathbb{B}^N}dfrac{|u(y)|^{p}}{|2sinhfrac{rho(T_y(x))}{2}|^mu} dV_y cdot |u|^{p-2}u +lambda u] on the hyperbolic space (mathbb{B}^N), where (Delta_{mathbb{B}^{N}}) denotes the Laplace-Beltrami operator on (mathbb{B}^N), [sinhfrac{rho(T_y(x))}{2}=dfrac{|T_y(x)|}{sqrt{1-|T_y(x)|^2}}=dfrac{|x-y|}{sqrt{(1-|x|^2)(1-|y|^2)}},] (lambda) is a real parameter, (0lt mult N), (1lt pleq 2_mu^*), (Ngeq 3) and (2_mu^*:=frac{2N-mu}{N-2}) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
{"title":"Nonlinear Choquard equations on hyperbolic space","authors":"Haiyang He","doi":"10.7494/opmath.2022.42.5.691","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.691","url":null,"abstract":"In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation [-Delta_{mathbb{B}^{N}}u=int_{mathbb{B}^N}dfrac{|u(y)|^{p}}{|2sinhfrac{rho(T_y(x))}{2}|^mu} dV_y cdot |u|^{p-2}u +lambda u] on the hyperbolic space (mathbb{B}^N), where (Delta_{mathbb{B}^{N}}) denotes the Laplace-Beltrami operator on (mathbb{B}^N), [sinhfrac{rho(T_y(x))}{2}=dfrac{|T_y(x)|}{sqrt{1-|T_y(x)|^2}}=dfrac{|x-y|}{sqrt{(1-|x|^2)(1-|y|^2)}},] (lambda) is a real parameter, (0lt mult N), (1lt pleq 2_mu^*), (Ngeq 3) and (2_mu^*:=frac{2N-mu}{N-2}) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.5.727
A. Orpel
We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation [Delta u(x)+f(x,u(x))+g(x)xcdot nabla u(x)=0,] for (xin Omega_{R}={ x in mathbb{R}^n, |x|gt R }), (ngt 2). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when (f(x,cdot)) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient (g) without radial symmetry.
研究了一类外域对流扩散方程正平稳解的存在性和多重性。这个问题导致如下的椭圆方程[Delta u(x)+f(x,u(x))+g(x)xcdot nabla u(x)=0,]对于(xin Omega_{R}={ x in mathbb{R}^n, |x|gt R }), (ngt 2)。本文的目的是证明我们的问题在无穷邻域中具有有限能量的极小解的不可减数列。我们还证明了这些序列中的每一个都会产生问题的另一个解。还考虑了(f(x,cdot))在原点为负的情况，即所谓的半正负问题。我们的结果是基于一定的迭代模式，其中我们采用了由Noussair和Swanson开发的下解和上解方法。该方法允许我们考虑包含泛函系数(g)的对流项的超线性问题，而不需要径向对称。
{"title":"Positive stationary solutions of convection-diffusion equations for superlinear sources","authors":"A. Orpel","doi":"10.7494/opmath.2022.42.5.727","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.727","url":null,"abstract":"We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation [Delta u(x)+f(x,u(x))+g(x)xcdot nabla u(x)=0,] for (xin Omega_{R}={ x in mathbb{R}^n, |x|gt R }), (ngt 2). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when (f(x,cdot)) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient (g) without radial symmetry.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.2.257
Nikolaos S. Papageorgiou
We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.
{"title":"Double phase problems: a survey of some recent results","authors":"Nikolaos S. Papageorgiou","doi":"10.7494/opmath.2022.42.2.257","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.257","url":null,"abstract":"We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.5.659
J. Džurina
This paper is concerned with oscillatory behavior of linear functional differential equations of the type [y^{(n)}(t)=p(t)y(tau(t))] with mixed deviating arguments which means that its both delayed and advanced parts are unbounded subset of ((0,infty)). Our attention is oriented to the Euler type of equation, i.e. when (p(t)sim a/t^n.)
{"title":"Properties of even order linear functional differential equations with deviating arguments of mixed type","authors":"J. Džurina","doi":"10.7494/opmath.2022.42.5.659","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.659","url":null,"abstract":"This paper is concerned with oscillatory behavior of linear functional differential equations of the type [y^{(n)}(t)=p(t)y(tau(t))] with mixed deviating arguments which means that its both delayed and advanced parts are unbounded subset of ((0,infty)). Our attention is oriented to the Euler type of equation, i.e. when (p(t)sim a/t^n.)","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.5.709
M. Jenaliyev, M. Ramazanov, M. Yergaliyev
The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.
{"title":"On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations","authors":"M. Jenaliyev, M. Ramazanov, M. Yergaliyev","doi":"10.7494/opmath.2022.42.5.709","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.709","url":null,"abstract":"The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.2.337
Gaili Zhu, C. Duan, Jianjun Zhang, Huixing Zhang
{"title":"Ground states of coupled critical Choquard equations with weighted potentials","authors":"Gaili Zhu, C. Duan, Jianjun Zhang, Huixing Zhang","doi":"10.7494/opmath.2022.42.2.337","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.337","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342357","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.5.673
Y. Hata, H. Matsunaga
This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays [x'(t)=-ax(t)-bx(t-tau)-cx(t-2tau),quad tgeq 0,] where (a), (b), and (c) are real numbers and (taugt 0). We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when (tau) increases only if (c-alt 0) and (sqrt{-8c(c-a)}lt |b| lt a+c). The explicit stability dependence on the changing (tau) is also described.
{"title":"Stability switches in a linear differential equation with two delays","authors":"Y. Hata, H. Matsunaga","doi":"10.7494/opmath.2022.42.5.673","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.5.673","url":null,"abstract":"This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays [x'(t)=-ax(t)-bx(t-tau)-cx(t-2tau),quad tgeq 0,] where (a), (b), and (c) are real numbers and (taugt 0). We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when (tau) increases only if (c-alt 0) and (sqrt{-8c(c-a)}lt |b| lt a+c). The explicit stability dependence on the changing (tau) is also described.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.3.439
Faisal Susanto, K. Wijaya, Slamin, Andrea Semani�ov�-Fe�ov��kov�
A vertex (k)-labeling (phi:V(G)rightarrow{1,2,dots,k}) on a simple graph (G) is said to be a distance irregular vertex (k)-labeling of (G) if the weights of all vertices of (G) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in (G). The least integer (k) for which (G) has a distance irregular vertex (k)-labeling is called the distance irregularity strength of (G) and denoted by (mathrm{dis}(G)). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.
{"title":"Distance irregularity strength of graphs with pendant vertices","authors":"Faisal Susanto, K. Wijaya, Slamin, Andrea Semani�ov�-Fe�ov��kov�","doi":"10.7494/opmath.2022.42.3.439","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.3.439","url":null,"abstract":"A vertex (k)-labeling (phi:V(G)rightarrow{1,2,dots,k}) on a simple graph (G) is said to be a distance irregular vertex (k)-labeling of (G) if the weights of all vertices of (G) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in (G). The least integer (k) for which (G) has a distance irregular vertex (k)-labeling is called the distance irregularity strength of (G) and denoted by (mathrm{dis}(G)). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.4.527
Z. Awanis, A. Salman
We introduce a strong (k)-rainbow index of graphs as modification of well-known (k)-rainbow index of graphs. A tree in an edge-colored connected graph (G), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let (k) be an integer with (2leq kleq n). The strong (k)-rainbow index of (G), denoted by (srx_k(G)), is the minimum number of colors needed in an edge-coloring of (G) so that every (k) vertices of (G) is connected by a rainbow tree with minimum size. We focus on (k=3). We determine the strong (3)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong (3)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong (3)-rainbow index of amalgamation of some graphs.
{"title":"The strong 3-rainbow index of some certain graphs and its amalgamation","authors":"Z. Awanis, A. Salman","doi":"10.7494/opmath.2022.42.4.527","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.4.527","url":null,"abstract":"We introduce a strong (k)-rainbow index of graphs as modification of well-known (k)-rainbow index of graphs. A tree in an edge-colored connected graph (G), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let (k) be an integer with (2leq kleq n). The strong (k)-rainbow index of (G), denoted by (srx_k(G)), is the minimum number of colors needed in an edge-coloring of (G) so that every (k) vertices of (G) is connected by a rainbow tree with minimum size. We focus on (k=3). We determine the strong (3)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong (3)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong (3)-rainbow index of amalgamation of some graphs.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.7494/opmath.2022.42.2.157
Lifeng Guo, Y. Sun, Guannan Shi
In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by [begin{cases}mathcal{L}_{K}u(x)+ulog|u|+|u|^{q-2}u=0, & xinOmega, u=0, & xinmathbb{R}^{n}setminusOmega,end{cases}] where (2lt qlt 2^{*}_s), (L_{K}) is a non-local operator, (Omega) is an open bounded set of (mathbb{R}^{n}) with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.
{"title":"Ground states for fractional nonlocal equations with logarithmic nonlinearity","authors":"Lifeng Guo, Y. Sun, Guannan Shi","doi":"10.7494/opmath.2022.42.2.157","DOIUrl":"https://doi.org/10.7494/opmath.2022.42.2.157","url":null,"abstract":"In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by [begin{cases}mathcal{L}_{K}u(x)+ulog|u|+|u|^{q-2}u=0, & xinOmega, u=0, & xinmathbb{R}^{n}setminusOmega,end{cases}] where (2lt qlt 2^{*}_s), (L_{K}) is a non-local operator, (Omega) is an open bounded set of (mathbb{R}^{n}) with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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