In this paper,we investigate the Sobolev spaces W^(1,p)(V)and W_(0)^(1,p)(V)on a locally finite graph G=(V,E),which are fundamental tools when we apply the variational methods to partial differential equations on grap...In this paper,we investigate the Sobolev spaces W^(1,p)(V)and W_(0)^(1,p)(V)on a locally finite graph G=(V,E),which are fundamental tools when we apply the variational methods to partial differential equations on graphs.As a key contribution of this note,we show that in general,W^(1,p)(V)≠W_(0)^(1,p)(V)on locally finite graphs,which is different from the situation on Euclidean space RN.展开更多
Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up p...Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.展开更多
We study a system of equations arising in the Chern-Simons model on finite graphs.Using the iteration scheme and the upper and lower solutions method,we get existence of solutions in the non-critical case.The critical...We study a system of equations arising in the Chern-Simons model on finite graphs.Using the iteration scheme and the upper and lower solutions method,we get existence of solutions in the non-critical case.The critical case is dealt with by priori estimates.Our results generalize those of Huang et al.Journal of Functional Analysis 281(10)(2021)Paper No.109218).展开更多
In this paper,we study the Kato's inequality on locally finite graphs.We also study the application of Kato's inequality to Ginzburg-Landau equations on such graphs.Interesting properties of elliptic and parab...In this paper,we study the Kato's inequality on locally finite graphs.We also study the application of Kato's inequality to Ginzburg-Landau equations on such graphs.Interesting properties of elliptic and parabolic equations on the graphs and a Liouville type theorem are also derived.展开更多
In this paper,we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs.Our method is based on calculus of variations,which was built on connect...In this paper,we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs.Our method is based on calculus of variations,which was built on connected finite graphs by Grigor'yan,Lin and Yang.展开更多
This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the disc...This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.展开更多
A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast dominat...A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.展开更多
In this paper,we study the p-Laplacian Choquard equation-Δ_(p)u+V(x)|u|^(p-2)u=(∑_(y∈N^(n)_(y≠x))|u(y)|^(q)/d(x,y)^(n-a))|u|^(q-2)u on a finite lattice graph Nn with n∈N_(+),where p>1,q>1 and 0≤α≤n are s...In this paper,we study the p-Laplacian Choquard equation-Δ_(p)u+V(x)|u|^(p-2)u=(∑_(y∈N^(n)_(y≠x))|u(y)|^(q)/d(x,y)^(n-a))|u|^(q-2)u on a finite lattice graph Nn with n∈N_(+),where p>1,q>1 and 0≤α≤n are some constants,V(x)is a positive function on Nn.Using the Nehari method,we prove that if 1<p<q<+∞,then the above equation admits a ground state solution.Previously,the p-Laplacian Choquard equation on finite lattice graph has not been studied,and our result contains the critical cases α=0 and α=n,which further improves the study of Choquard equations on lattice graphs.展开更多
In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topologic...In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given.Finally,examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.展开更多
We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|^(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, t...We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|^(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|^(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.展开更多
In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to...In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.展开更多
The original version of the article was published in[1].Unfortunately,the original version of this article contains a mistake:in Theorem 6.2 appears thatβ(n,△)=(n-△+5)/4 but the correct statement isβ(n,△)=(n-△+4...The original version of the article was published in[1].Unfortunately,the original version of this article contains a mistake:in Theorem 6.2 appears thatβ(n,△)=(n-△+5)/4 but the correct statement isβ(n,△)=(n-△+4)/4.In this erratum we correct the theorem and give the correct proof.展开更多
Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certa...Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.展开更多
Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In ...Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359-409 （2015）]. In many topics, such as eigenvalue estimate and heat kernel estimate （not including the Liouville type theorems）, replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.展开更多
By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main re...By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.展开更多
We consider a kind of nonlinear systems on a locally finite graph G=(V,E).We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solutionwhich depends on the parameterλwith som...We consider a kind of nonlinear systems on a locally finite graph G=(V,E).We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solutionwhich depends on the parameterλwith some suitable assumptions on the potentials.Moreover,we pay attention to the concentration behavior of these solutions and prove that asλ→∞,these solutions converge to a ground state solution of a corresponding Dirichlet problem.Finally,we also provide some numerical experiments to illustrate our results.展开更多
基金supported by the National Key R and D Program of China(2020YFA0713100)the National Natural Science Foundation of China(12271039)the Open Project Program(K202303)of Key Laboratory of Mathematics and Complex Systems,Beijing Normal University。
文摘In this paper,we investigate the Sobolev spaces W^(1,p)(V)and W_(0)^(1,p)(V)on a locally finite graph G=(V,E),which are fundamental tools when we apply the variational methods to partial differential equations on graphs.As a key contribution of this note,we show that in general,W^(1,p)(V)≠W_(0)^(1,p)(V)on locally finite graphs,which is different from the situation on Euclidean space RN.
基金supported by the National Science Foundation of China(11671401)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(17XNH106)
文摘Let G =（V, E） be a locally finite connected weighted graph, and ？ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ？u ＋ f（u） on G. The blow-up phenomenons for ut = ？u ＋ f（u） are discussed in terms of two cases：（i） an initial condition is given;（ii） a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.
基金partially supported by the National Key Research and Development Program of China 2020YFA0713100by the National Natural Science Foundation of China(Grant No.11721101).
文摘We study a system of equations arising in the Chern-Simons model on finite graphs.Using the iteration scheme and the upper and lower solutions method,we get existence of solutions in the non-critical case.The critical case is dealt with by priori estimates.Our results generalize those of Huang et al.Journal of Functional Analysis 281(10)(2021)Paper No.109218).
基金supported by National Natural Science Foundation of China (Grant No.10631020)Doctoral Program Foundation of the Ministry of Education of China (Grant No. 20090002110019)
文摘In this paper,we study the Kato's inequality on locally finite graphs.We also study the application of Kato's inequality to Ginzburg-Landau equations on such graphs.Interesting properties of elliptic and parabolic equations on the graphs and a Liouville type theorem are also derived.
基金partially supported by the National Science Foundation of China(Grant No.11401575 and 11721101).
文摘In this paper,we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs.Our method is based on calculus of variations,which was built on connected finite graphs by Grigor'yan,Lin and Yang.
文摘This paper is mainly concerned with the following nonlinear p-Laplacian equation-△pu(x)+(λa(x)+1)|u|^(p-2)(x)u(x)=f(x,u(x)),in V on a locally finite graph G=(V,E)with more general nonlinear term,whereΔp is the discrete pLaplacian on graphs,p≥2.Under some suitable conditions on f and a(x),we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλvia the method of Nehari manifold,for anyλ>1.In addition,asλ→+∞,we prove that the solution uλconverge to a solution of the following Dirichlet problem{-△pu(x)+|u|^(p-2)(x)u(x)=f(x,u(x)),inΩ,u(x)=0,onδΩwhereΩ={x∈V:a(x)=0}is the potential well and δΩ denotes the the boundary ofΩ.
文摘A dominating set of a graph G is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of G is the order of a minimum dominating set of G. The (t, r) broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength t that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least r from its set of broadcasting neighbors. In this paper, we extend the study of (t, r) broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed (t, r) broadcast domination numbers of all orientations of a graph G. In particular, we prove that in the cases r = 1 and (t, r) = (2, 2), for every integer value in this interval, there exists an orientation of G which has directed (t, r) broadcast domination number equal to that value. We also investigate directed (t, r) broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.
基金Supported by the National Natural Science Foundation of China(Grant No.12471088)。
文摘In this paper,we study the p-Laplacian Choquard equation-Δ_(p)u+V(x)|u|^(p-2)u=(∑_(y∈N^(n)_(y≠x))|u(y)|^(q)/d(x,y)^(n-a))|u|^(q-2)u on a finite lattice graph Nn with n∈N_(+),where p>1,q>1 and 0≤α≤n are some constants,V(x)is a positive function on Nn.Using the Nehari method,we prove that if 1<p<q<+∞,then the above equation admits a ground state solution.Previously,the p-Laplacian Choquard equation on finite lattice graph has not been studied,and our result contains the critical cases α=0 and α=n,which further improves the study of Choquard equations on lattice graphs.
基金Lin Wang is supported by the National Natural Science Foundation of China(No.11801336,11771118)the Science and Technology Innovation Project of Shanxi Higher Education(No.2019L0475)the Applied Basic Research Program of Shanxi Province(No:201901D211417).
文摘In this paper,the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced.Moreover,formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given.Finally,examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.
基金supported by the Funding of Beijing Philosophy and Social Science(Grant No.15JGC153)the Ministry of Education Project of Humanities and Social Sciences(Grant No.16YJCZH148)+1 种基金supported by the Fundamental Research Funds for the Central Universitiessupported by the Ministry of Education Project of Key Research Institute of Humanities and Social Sciences at Universities(Grant No.16JJD790060)
文摘We consider the nonlinear Schr¨odinger equation-?u +(λa(x) + 1)u = |u|^(p-1) u on a locally finite graph G =(V, E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution uλ. Moreover, as λ→∞, the solution uλconverges to a solution of the Dirichlet problem-?u + u = |u|^(p-1) u which is defined on the potential well ?. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.
基金Supported in part by two grants from Ministerio de Economía y Competitividad,Spain:MTM2013-46374-P and MTM2015-69323-REDT
文摘In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs （in the Gromov sense）. To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H（n, δ0） be the set of graphs G with n vertices and minimum degree 50, and J（n, Δ） be the set of graphs G with n vertices and maximum degree A. We study the four following extremal problems on graphs： a（n,δ0） = min{δ（G） ｜ G ∈H（n, δ0）}, b（n, δ0） =- max{δ（G）｜ e ∈H（n, δ0）}, α（n, Δ） = min{δ（G） [ G ∈ J（n, Δ）} and β（n,Δ） = max{δ（G） ] G∈Π（n,Δ）}. In particular, we obtain bounds for b（n, δ0） and we compute the precise value of a（n, δ0）, α（n, Δ） and w（n, Δ） for all values of n, r0 and A, respectively.
基金Supported by two grants from Ministerio de Economía y Competitividad,Spain(Grant Nos.MTM2013-46374-P and MTM2015-69323-REDT)
文摘The original version of the article was published in[1].Unfortunately,the original version of this article contains a mistake:in Theorem 6.2 appears thatβ(n,△)=(n-△+5)/4 but the correct statement isβ(n,△)=(n-△+4)/4.In this erratum we correct the theorem and give the correct proof.
文摘Let be a function on locally finite connect graph G=(V,E)andΩbe a bounded subset of V.We consider the nonlinear Dirichlet boundary condition problem{-△u=f(u),inΩ,u=0,onδΩ.Let f:R→R be a function satisfying certain assumptions.Then under the functional framework we use the three-solution theorem and the variational method to prove that the above equation has at least three solutions,of which one is trivial and the others are strictly positive.
基金supported by National Natural Science Foundation of China(Grant No.11271011)supported by National Natural Science Foundation of China(Grant Nos.11171347 and 11471014)
文摘Continuing our previous work （arXiv：1509.07981vl）, we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li-Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359-409 （2015）]. In many topics, such as eigenvalue estimate and heat kernel estimate （not including the Liouville type theorems）, replacing the Li-Yau inequality by the global gradient estimate, we can get similar results.
文摘By using the perpetual cutoff method,we prove two discrete versions of gradient estimates for bounded Laplacian on locally finite graphs with exception sets under the condition of CDE′(K,N).This generalizes a main result of F.Münch who considers the case of CD(K,∞)curvature.Hence,we answer a question raised by Münch.For that purpose,we characterize some basic properties of radical form of the perpetual cutoff semigroup and give a weak commutation relation between bounded LaplacianΔand perpetual cutoff semigroup P w t in our setting.
文摘We consider a kind of nonlinear systems on a locally finite graph G=(V,E).We prove via the mountain pass theorem that this kind of systems has a nontrivial ground state solutionwhich depends on the parameterλwith some suitable assumptions on the potentials.Moreover,we pay attention to the concentration behavior of these solutions and prove that asλ→∞,these solutions converge to a ground state solution of a corresponding Dirichlet problem.Finally,we also provide some numerical experiments to illustrate our results.
