In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infin...In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.展开更多
In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o...In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.展开更多
The existence of high energy periodic solutions for the second-order Hamiltonian system -ü(t)+A(t)u(t)=▽F(t,u(t)) with convex and concave nonlinearities is studied, where F(t, u) = F1(t,u)+F2(t,...The existence of high energy periodic solutions for the second-order Hamiltonian system -ü(t)+A(t)u(t)=▽F(t,u(t)) with convex and concave nonlinearities is studied, where F(t, u) = F1(t,u)+F2(t,u). Under the condition that F is an even functional, infinitely many solutions for it are obtained by the variant fountain theorem. The result is a complement for some known ones in the critical point theory.展开更多
Recently we have suggested that the state of a single particle should be jointly defined by two wave functions.In this letter we use this suggestion to discuss the motion of a single particle in an infinitely deep squ...Recently we have suggested that the state of a single particle should be jointly defined by two wave functions.In this letter we use this suggestion to discuss the motion of a single particle in an infinitely deep square potential well;the results obiained in the classical limit correctly describe the motion of the single particle with clarity.展开更多
By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions o...By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions of stress fields of the interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are obtained. They indicate that the stress concentration occurs at the dislocation source and the tip of the crack, and the value of the stress increases with the number of the dislocations increasing. These results are the development of interaction among the finitely many defects of quasicrystals, which possesses an important reference value for studying the interaction problems of infinitely many defects in fracture mechanics of quasicrystal.展开更多
Abstract This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The...Abstract This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.展开更多
We study the following elliptic problem:{-div(a(x)Du)=Q(x)|u|2-2u+λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.
We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability...We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.展开更多
We study a Schrodinger system with the sum of linear and nonlinear couplings.Applying index theory,we obtain infinitely many solutions for the system with periodic potent ials.Moreover,by using the concentration compa...We study a Schrodinger system with the sum of linear and nonlinear couplings.Applying index theory,we obtain infinitely many solutions for the system with periodic potent ials.Moreover,by using the concentration compactness met hod,we prove the exis tence and nonexistence of ground state solutions for the system with close-to-periodic potentials.展开更多
In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).
In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not b...In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.展开更多
This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical ex...This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical exponent for the Sobolev imbedding, λ > 0 and f(x, u) satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p = 2 or f(x,u) = |u|q-2u, where 1 < q < p, are generalized.展开更多
A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This appr...A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.展开更多
This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an infinitely long cylinder. We study the effects of the strength of the source on quenching. Cr...This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an infinitely long cylinder. We study the effects of the strength of the source on quenching. Criteria for global existence of the solution and for quenching are investigated.展开更多
An instantaneous velocity where a moment of the clock only corresponds to an arbitrary distance or position in space cannot be implied in Axiom 1, but it indicates that there is only one dimensional existence, space o...An instantaneous velocity where a moment of the clock only corresponds to an arbitrary distance or position in space cannot be implied in Axiom 1, but it indicates that there is only one dimensional existence, space or time, where a certain moment only corresponds to itself specifically, not to any other time or any given length of space. Further, a definition of velocity that consists of two dimensions representing the relationship between space and time is not valid and there is only one-dimensional space or time that is independent of each other in Axiom 1. As a result, the principle of relativity and the principle of the constant velocity of light are replaced by the principle of an inertial system and the principle of universal invariant velocity in Axiom 1. Unlike two dimensions whose magnitude is determined by the ratio, the magnitude of a single dimension is determined by the unit values of one dimension, which indicates that an infinitely great velocity is meaningless. Further, if the two inertial systems are infinite versus finite in Axiom 3, then this extension of the infinitely great velocity can be defined as inextensible.展开更多
In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> ...In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>展开更多
A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lo...A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.展开更多
We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in t...We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.展开更多
We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by 1.We consider the algorithms t...We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by 1.We consider the algorithms that use finitely many function values as information(called standard information).On the one hand,we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights.On the other hand,by using the Smolyak al-gorithm with the above interpolatory quadratures,we proved that the weighted integral problem is of exponential convergence in the worst case setting.展开更多
In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of c...In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.展开更多
文摘In this paper, we intend to consider a kind of nonlinear Klein-Gordon equation coupled with Born-Infeld theory. By using critical point theory and the method of Nehari manifold, we obtain two existing results of infinitely many high-energy radial solutions and a ground-state solution for this kind of system, which improve and generalize some related results in the literature.
文摘In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.
文摘The existence of high energy periodic solutions for the second-order Hamiltonian system -ü(t)+A(t)u(t)=▽F(t,u(t)) with convex and concave nonlinearities is studied, where F(t, u) = F1(t,u)+F2(t,u). Under the condition that F is an even functional, infinitely many solutions for it are obtained by the variant fountain theorem. The result is a complement for some known ones in the critical point theory.
基金supported by Institute of Theoretical Physics,Academia Sinica.
文摘Recently we have suggested that the state of a single particle should be jointly defined by two wave functions.In this letter we use this suggestion to discuss the motion of a single particle in an infinitely deep square potential well;the results obiained in the classical limit correctly describe the motion of the single particle with clarity.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11462020,11262017,and 11262012)the Key Project of Inner Mongolia Normal University,China(Grant No.2014ZD03)
文摘By means of analytic function theory, the problems of interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are studied. The analytic solutions of stress fields of the interaction between infinitely many parallel dislocations and a semi-infinite crack in one-dimensional hexagonal quasicrystal are obtained. They indicate that the stress concentration occurs at the dislocation source and the tip of the crack, and the value of the stress increases with the number of the dislocations increasing. These results are the development of interaction among the finitely many defects of quasicrystals, which possesses an important reference value for studying the interaction problems of infinitely many defects in fracture mechanics of quasicrystal.
基金supported by National Natural Science Foundation of China under grant 51378384Key Project of Natural Science Foundation of Tianjin Municipality under Grant 12JCZDJC29000
文摘Abstract This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.
基金supported by Key Project (10631030) of NSFCKnowledge Innovation Funds of CAS in Chinasupported by ARC in Australia
文摘We study the following elliptic problem:{-div(a(x)Du)=Q(x)|u|2-2u+λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.
基金National Natural Science Foundation of China(Grant Nos.11672257,11632008,11772306,and 11972173)the Natural Science Foundation of Jiangsu Province of China(Grant No.BK20161314)+1 种基金the 5th 333 High-level Personnel Training Project of Jiangsu Province of China(Grant No.BRA2018324)the Excellent Scientific and Technological Innovation Team of Jiangsu University.
文摘We study a novel class of two-dimensional maps with infinitely many coexisting attractors.Firstly,the mathematical model of these maps is formulated by introducing a sinusoidal function.The existence and the stability of the fixed points in the model are studied indicating that they are infinitely many and all unstable.In particular,a computer searching program is employed to explore the chaotic attractors in these maps,and a simple map is exemplified to show their complex dynamics.Interestingly,this map contains infinitely many coexisting attractors which has been rarely reported in the literature.Further studies on these coexisting attractors are carried out by investigating their time histories,phase trajectories,basins of attraction,Lyapunov exponents spectrum,and Lyapunov(Kaplan–Yorke)dimension.Bifurcation analysis reveals that the map has periodic and chaotic solutions,and more importantly,exhibits extreme multi-stability.
文摘We study a Schrodinger system with the sum of linear and nonlinear couplings.Applying index theory,we obtain infinitely many solutions for the system with periodic potent ials.Moreover,by using the concentration compactness met hod,we prove the exis tence and nonexistence of ground state solutions for the system with close-to-periodic potentials.
文摘In this paper, it is proved that the following boundary value problem [GRAPHICS] admits infinitely many solution for 0 < lambda < lambda-1, n greater-than-or-equal-to 5 and for ball regions OMEGA = B(R)(0).
文摘In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.
基金Supported by NSFC(10171032) NSF of Guangdong Proviance (011606)
文摘This paper is concerned with the following nonlinear Dirichlet problem:where △pu = div(| ▽u|p- 2 ▽u) is the p-Laplacian of u, Ω is a bounded domain in Rn (n > 3), 1 < p < n, p = -pn/n-p is the critical exponent for the Sobolev imbedding, λ > 0 and f(x, u) satisfies some conditions. It reaches the conclusion that this problem has infinitely many solutions. Some results as p = 2 or f(x,u) = |u|q-2u, where 1 < q < p, are generalized.
文摘A fixed mesh variational formulation is used to establish existence and uniqueness of the solution of ordinary differential equations with (in finitely many) state-dependent in pulses on the right-hand side. This approach gives a natural numerical scheme to approximate the solution.The convergence of the approximation is proved and its asymptatic order obtained.
文摘This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an infinitely long cylinder. We study the effects of the strength of the source on quenching. Criteria for global existence of the solution and for quenching are investigated.
文摘An instantaneous velocity where a moment of the clock only corresponds to an arbitrary distance or position in space cannot be implied in Axiom 1, but it indicates that there is only one dimensional existence, space or time, where a certain moment only corresponds to itself specifically, not to any other time or any given length of space. Further, a definition of velocity that consists of two dimensions representing the relationship between space and time is not valid and there is only one-dimensional space or time that is independent of each other in Axiom 1. As a result, the principle of relativity and the principle of the constant velocity of light are replaced by the principle of an inertial system and the principle of universal invariant velocity in Axiom 1. Unlike two dimensions whose magnitude is determined by the ratio, the magnitude of a single dimension is determined by the unit values of one dimension, which indicates that an infinitely great velocity is meaningless. Further, if the two inertial systems are infinite versus finite in Axiom 3, then this extension of the infinitely great velocity can be defined as inextensible.
文摘In this paper, we consider the following fourth-order equation of Kirchhoff type<br /> <p> <img src="Edit_bcc9844d-7cbc-494d-90c4-d75364de5658.bmp" alt="" /> </p> <p> where <i>a</i>, <i>b</i> > 0 are constants, 3 < <i>p</i> < 5, <i>V</i> ∈ <i>C</i> (R<sup>3</sup>, R);Δ<sup>2</sup>: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on <i>V</i> (<i>x</i>). We make some assumptions on the potential <i>V</i> (<i>x</i>) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature. </p>
基金浙江省自然科学基金,浙江省宁波市博士基金,the State Key Laboratory of Oil/Gas Reservoir Geology and Exploitation,Scientific Research Fund of Education Department of Zhejiang Province under
文摘A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.
基金supported in part by grants from the National Natural Science Foundation of China and the NBRP of Chinasupported in part by grants from the National Natural Science Foundation of Chinathe PSC-CUNY and the Hundred Talents Program from the Chinese Academy of Sciences.
文摘We study a conformal measure for an infinitely renormalizable quadratic polynomial. We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds. The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11471043,11671271)by the Beijing Natural Science Foundation(Grant No.1172004)。
文摘We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by 1.We consider the algorithms that use finitely many function values as information(called standard information).On the one hand,we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights.On the other hand,by using the Smolyak al-gorithm with the above interpolatory quadratures,we proved that the weighted integral problem is of exponential convergence in the worst case setting.
文摘In this paper we consider the existence of infinitely many positive solutions for second order nonlinear eigenvalue problem with singular coefficient function. By the use of Krasnosel'skii fixed point theorem of cone expansion-compression type we give several sufficient conditions.