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Predicted Critical State Based on Invariance of the Lyapunov Exponent in Dual Spaces
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作者 刘通 夏旭 《Chinese Physics Letters》 SCIE EI CAS CSCD 2024年第1期68-76,共9页
Critical states in disordered systems,fascinating and subtle eigenstates,have attracted a lot of research interests.However,the nature of critical states is difficult to describe quantitatively,and in general,it canno... Critical states in disordered systems,fascinating and subtle eigenstates,have attracted a lot of research interests.However,the nature of critical states is difficult to describe quantitatively,and in general,it cannot predict a system that hosts the critical state.We propose an explicit criterion whereby the Lyapunov exponent of the critical state should be 0 simultaneously in dual spaces,namely the Lyapunov exponent remains invariant under the Fourier transform.With this criterion,we can exactly predict a one-dimensional quasiperiodic model which is not of self-duality,but hosts a large number of critical states.Then,we perform numerical verification of the theoretical prediction and display the self-similarity of the critical state.Due to computational complexity,calculations are not performed for higher dimensional models.However,since the description of extended and localized states by the Lyapunov exponent is universal and dimensionless,utilizing the Lyapunov exponent of dual spaces to describe critical states should also be universal.Finally,we conjecture that some kind of connection exists between the invariance of the Lyapunov exponent and conformal invariance,which can promote the research of critical phenomena. 展开更多
关键词 STATE exponent critical
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TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS
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作者 Khalid BOUABID Rachid ECHARGHAOUI Mohssine EL MANSOUR 《Acta Mathematica Scientia》 SCIE CSCD 2023年第5期2061-2074,共14页
In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with ... In this paper,by an approximating argument,we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents■whereΩis a smooth bounded domain in RN with 0∈?Ωand all the principle curvatures of?Ωat 0 are negative,a∈C1(Ω,R*+),μ>0,0<s<2,1<q<2 and N>2(q+1)/(q-1).By2*:=2N/(N-2)and 2*(s):(2(N-s))/(N-2)we denote the critical Sobolev exponent and Hardy-Sobolev exponent,respectively. 展开更多
关键词 Laplacien critical Sobolev-Hardy exponent critical Sobolev exponent infinitely many solutions Pohozaev identity
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Solutions for Schrodinger-Poisson system involving nonlocal term and critical exponent
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作者 MO Xiu-ming MAO An-min WANG Xiang-xiang 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2023年第3期357-372,共16页
In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev crit... In this paper,we consider the following Kirchhoff-Schrodinger-Poisson system:{−(a+b∫_(R^(3))|∇u|^(2))△u+u+ϕu=μQ(x)|u|^(q-2)u+K(x)|u|^(4)u,in R^(3),−△ϕ=u^(2) the nonlinear growth of|u|^(4)u reaches the Sobolev critical exponent.By combining the variational method with the concentration-compactness principle of Lions,we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions.The nonlinear term includes the nonlinearity f(u)~|u|^(q-2)u for the well-studied case q∈[4,6),and the less-studied case q∈(2,3),we adopt two different strategies to handle these cases.Our result improves and extends some related works in the literature. 展开更多
关键词 variational methods critical exponent concentration-compactness principle
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CALCULATION OF CRITICAL EXPONENT β(L) OF MAGNETIC THIN FILMS BY USING ISING MODEL
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作者 霍炳海 区镜添 《Transactions of Tianjin University》 EI CAS 1998年第2期51-54,共4页
The variational cumulant expansion developed in recent years has been extended to treat the Ising model in statistical physics.In this paper,a detailed calculation of the critical temperature T c (L) and criti... The variational cumulant expansion developed in recent years has been extended to treat the Ising model in statistical physics.In this paper,a detailed calculation of the critical temperature T c (L) and critical exponent β(L) for the magnetic film of L layers are presented by means of the variational cumulant expansion.For L >1,the results of our theoretical calculations are in approximate coincidence with the experimental ones made before,and for the special case of L =1 (2 D),the results of the calculation are identical to the data from other reports. 展开更多
关键词 variational cumulant expansion critical temperature critical exponent
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MULTIPLICITY OF POSITIVE SOLUTIONS FOR SINGULAR ELLIPTIC SYSTEMS WITH CRITICAL SOBOLEV-HARDY AND CONCAVE EXPONENTS 被引量:9
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作者 Tsing-San Hsu Huei-Lin Li 《Acta Mathematica Scientia》 SCIE CSCD 2011年第3期791-804,共14页
In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions ... In this paper,we consider a singular elliptic system with both concave non-linearities and critical Sobolev-Hardy growth terms in bounded domains.By means of variational methods,the multiplicity of positive solutions to this problem is obtained. 展开更多
关键词 elliptic system critical Sobolev-Hardy exponent concave exponents Nehari manifold
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SOLUTIONS FOR A NONHOMOGENEOUS ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENT IN R^N 被引量:10
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作者 王征平 周焕松 《Acta Mathematica Scientia》 SCIE CSCD 2006年第3期525-536,共12页
For the following elliptic problem {-△u-μu/|x|^2=|u|^2^*(s)-2u/|x|^s+h(x), on R^N u∈D^1,2(R^N), N≥3, 0≤μ〈μ^-=(N-2)^2/4, 0≤s〈2, where 2^*(s)=2(N-s)/N-2 is the critical Sobolev-Hardy expon... For the following elliptic problem {-△u-μu/|x|^2=|u|^2^*(s)-2u/|x|^s+h(x), on R^N u∈D^1,2(R^N), N≥3, 0≤μ〈μ^-=(N-2)^2/4, 0≤s〈2, where 2^*(s)=2(N-s)/N-2 is the critical Sobolev-Hardy exponent, h(x) ∈ (D^1,2(R^N))^*, the dual space of (D^1,2(R^N)), with h(x)≥(≠)0. By Ekeland's variational principle, subsuper solutions and a Mountain Pass theorem, the authors prove that the above problem has at least two distinct solutions if ||h||*〈CN,sAs^N-s/4-2s(1-μ/μ)^1/2, CN,s=4-2s/N-2(N-2/N+2-2s)^N+2-2s/4-2s and As = inf u∈D^1,2(R^N)/{0}∫R^N(|△↓u|^2-μu^2/|x|^2)dx/(∫R^N|u|^2^*(s)/|x|^sdx)^2/2^*(s). 展开更多
关键词 critical Sobolev-Hardy exponent elliptic equation Mountain Pass theorem subsuper solutions NONHOMOGENEOUS
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NONTRIVIAL SOLUTION FOR A CLASS OF SEMILINEAR BIHARMONIC EQUATION INVOLVING CRITICAL EXPONENTS 被引量:9
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作者 姚仰新 王荣鑫 沈尧天 《Acta Mathematica Scientia》 SCIE CSCD 2007年第3期509-514,共6页
In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequal... In this article, the authors prove the existence and the nonexistence of nontrivial solutions for a semilinear biharmonic equation involving critical exponent by virtue of Mountain Pass Lemma and Sobolev-Hardy inequality. 展开更多
关键词 Biharmonic equation critical exponent singular term nontrivial solution Sobolev-Hardy inequality
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SOLUTIONS FOR THE QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS 被引量:6
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作者 康东升 《Acta Mathematica Scientia》 SCIE CSCD 2010年第5期1529-1540,共12页
In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions... In this article, we study the quasilinear elliptic problem involving critical Hardy Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign-changing solutions to the problem. 展开更多
关键词 quasilinear problem critical exponent solution variational method
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EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS 被引量:6
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作者 李圆晓 高文杰 《Acta Mathematica Scientia》 SCIE CSCD 2013年第1期107-121,共15页
The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means o... The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions. 展开更多
关键词 singular elliptic system concave-convex nonlinearities positive solution Nehari manifold critical Sobolev-Hardy exponent
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MULTIPLE SOLUTIONS FOR THE p&q-LAPLACIAN PROBLEM WITH CRITICAL EXPONENT 被引量:6
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作者 李工宝 张国 《Acta Mathematica Scientia》 SCIE CSCD 2009年第4期903-918,共16页
In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:{-△pu-△qu=│u│^p*-2u+μ│u│^r-2u in Ω u... In this paper, we study the existence of multiple solutions for the following nonlinear elliptic problem of p&q-Laplacian type involving the critical Sobolev exponent:{-△pu-△qu=│u│^p*-2u+μ│u│^r-2u in Ω u│δΩ=0,where Ω belong to R^N is a bounded domain,N〉p,p^*=Np/N-p is the critical Sobolev exponent and μ 〉0. We prove that if 1 〈 r 〈 q 〈 p 〈 N, then there is a μ0 〉 0, such that for any μ∈ (0, μ0), the above mentioned problem possesses infinitely many weak solutions. Our result generalizes a similar result in [8] for p-Laplacian type problem. 展开更多
关键词 p&q-Laplacian multiplicity of solutions critical exponent
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MULTIPLE POSITIVE SOLUTIONS FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT 被引量:4
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作者 范海宁 刘晓春 《Acta Mathematica Scientia》 SCIE CSCD 2014年第4期1111-1126,共16页
In this paper, we study the multiplicity results of positive solutions for a class of quasi-linear elliptic equations involving critical Sobolev exponent. With the help of Nehari manifold and a mini-max principle, we ... In this paper, we study the multiplicity results of positive solutions for a class of quasi-linear elliptic equations involving critical Sobolev exponent. With the help of Nehari manifold and a mini-max principle, we prove that problem admits at least two or three positive solutions under different conditions. 展开更多
关键词 Nehari manifold critical Sobolev exponent quasi-linear problem mini-max principle multiple positive solutions
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CRITICAL EXPONENTS OF EVOLUTIONARY p-LAPLACIAN WITH INTERIOR AND BOUNDARY SOURCES 被引量:3
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作者 尹景学 金春花 杨莹 《Acta Mathematica Scientia》 SCIE CSCD 2011年第3期778-790,共13页
This paper is concerned with the evolutionary p-Laplacian with interior and boundary sources.The critical exponents for the nonlinear sources are determined.
关键词 critical exponent P-LAPLACIAN global existence BLOW-UP
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PERTURBATION METHODS IN SEMILINEAR ELLIPTIC PROBLEMS INVOLVING CRITICAL HARDY-SOBOLEV EXPONENT 被引量:4
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作者 蓝永艺 唐春雷 《Acta Mathematica Scientia》 SCIE CSCD 2014年第3期703-712,共10页
In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation me... In this article, we deal with a class of semilinear elliptic equations which are perturbations of the problems with the critical Hardy-Sobolev exponent. Some existence results are given via an abstract perturbation method in critical point theory. 展开更多
关键词 critical Hardy-Sobolev exponent semilinear elliptic equation perturbation methods positive radial solution
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NEUMANN PROBLEMS OF A CLASS OF ELLIPTIC EQUATIONS WITH DOUBLY CRITICAL SOBOLEV EXPONENTS 被引量:3
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作者 韩丕功 《Acta Mathematica Scientia》 SCIE CSCD 2004年第4期633-638,共6页
This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes... This paper deals with the Neumann problem for a class of semilinear elliptic equations -△u + u =|u|2*-2u+ μ|u|q-2u in Ω, au/ar= |u|(?)*-2u on aΩ, where 2 = 2N/N-2, s=2(N-1)/N-2, 1 <q<2,N(?)3,μ>γ denotes the unit outward normal to boundary aΩ. By vaxiational method and dual fountain theorem, the existence of infinitely many solutions with negative energy is proved. 展开更多
关键词 Neumann problem semilinear elliptic equation (PS)·c condition critical Sobolev exponent
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p-LAPLACE EQUATIONS WITH MULTIPLE CRITICAL EXPONENTS AND SINGULAR CYLINDRICAL POTENTIAL 被引量:2
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作者 孙小妹 《Acta Mathematica Scientia》 SCIE CSCD 2013年第4期1099-1112,共14页
In this paper, we deal with the following problem:By variational method, we prove the existenceof a nontrivial weak solution whenand the existence of a cylindricalweak solution when
关键词 p-Laplace equation cylindrical potential critical exponents
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EXISTENCE OF NODAL SOLUTION FOR SEMI-LINEAR ELLIPTIC EQUATIONS WITH CRITICAL SOBOLEV EXPONENT ON SINGULAR MANIFOLD 被引量:2
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作者 刘晓春 梅媛 《Acta Mathematica Scientia》 SCIE CSCD 2013年第2期543-555,共13页
In this article, we prove that semi-linear elliptic equations with critical cone Sobolev exponents possess a nodal solution.
关键词 Cone Sobolev space critical exponent nodal solution
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LOCATION OF THE BLOW UP POINT FOR POSITIVE SOLUTIONS OF A BIHARMONIC EQUATION INVOLVING NEARLY CRITICAL EXPONENT 被引量:1
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作者 耿堤 《Acta Mathematica Scientia》 SCIE CSCD 2005年第2期283-295,共13页
In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point ... In this paper a semilinear biharmonic problem involving nearly critical growth with Navier boundary condition is considered on an any bounded smooth domain. It is proved that positive solutions concentrate on a point in the domain, which is also a critical point of the Robin’s function corresponding to the Green’s function of biharmonic operator with the same boundary condition. Similar conclusion has been obtained in [6] under the condition that the domain is strictly convex. 展开更多
关键词 Biharmonic operator Navier boundary conditions asymptotic behavior critical exponents Green's function
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INFINITELY MANY SOLUTIONS FOR A SINGULAR ELLIPTIC EQUATION INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS IN R^N 被引量:1
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作者 贺小明 邹文明 《Acta Mathematica Scientia》 SCIE CSCD 2010年第3期830-840,共11页
In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ|x|^2^-μ=α|x|^s^-|μ|^2*(s)-2u+βα(x)|u|^... In this article, we study the existence of multiple solutions for the singular semilinear elliptic equation involving critical Sobolev-Hardy exponents -△μ-μ|x|^2^-μ=α|x|^s^-|μ|^2*(s)-2u+βα(x)|u|^r-2u,x∈R^n. By means of the concentration-compactness principle and minimax methods, we obtain infinitely many solutions which tend to zero for suitable positive parameters α,β. 展开更多
关键词 Singular elliptic equation Multiple solutions critical Sobolev-Hardy exponent Minimax method
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MULTIPLICITY OF POSITIVE SOLUTIONS FOR A NONLOCAL ELLIPTIC PROBLEM INVOLVING CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX NONLINEARITIES 被引量:2
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作者 Jinguo ZHANG Tsing-San HSU 《Acta Mathematica Scientia》 SCIE CSCD 2020年第3期679-699,共21页
In this article,we study the following critical problem involving the fractional Laplacian:{(−Δ)^α/2u−γu/|x|^α=λ|u|^q−2/|x|^s+|u|^2^∗α^(t)−2u/|x|^t in Ω,u=0 in R^N∖Ω,whereΩ⊂R^N(N>α)is a bounded smooth dom... In this article,we study the following critical problem involving the fractional Laplacian:{(−Δ)^α/2u−γu/|x|^α=λ|u|^q−2/|x|^s+|u|^2^∗α^(t)−2u/|x|^t in Ω,u=0 in R^N∖Ω,whereΩ⊂R^N(N>α)is a bounded smooth domain containing the origin,α∈(0,2),0≤s,t<α,1≤q<2,λ>0,2α^*(t)=2(N-t)/N-αis the fractional critical Sobolev-Hardy exponent,0≤γ<γH,and γH is the sharp constant of the Sobolev-Hardy inequality.We deal with the existence of multiple solutions for the above problem by means of variational methods and analytic techniques. 展开更多
关键词 Fractional Laplacian Hardy potential multiple positive solutions critical Sobolev-Hardy exponent
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EXISTENCE OF SOLUTIONS FOR THE FRACTIONAL(p,q)-LAPLACIAN PROBLEMS INVOLVING A CRITICAL SOBOLEV EXPONENT 被引量:1
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作者 Fanfan CHEN Yang YANG 《Acta Mathematica Scientia》 SCIE CSCD 2020年第6期1666-1678,共13页
In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N i... In this article,we study the following fractional(p,q)-Laplacian equations involving the critical Sobolev exponent:(Pμ,λ){(−Δ)s 1 p u+(−Δ)s 2 q u=μ|u|q−2 u+λ|u|p−2 u+|u|p∗s 1−2 u,u=0,inΩ,in R N∖Ω,whereΩ⊂R N is a smooth and bounded domain,λ,μ>0,0<s 2<s 1<1,1<q<p<Ns 1.We establish the existence of a non-negative nontrivial weak solution to(Pμ,λ)by using the Mountain Pass Theorem.The lack of compactness associated with problems involving critical Sobolev exponents is overcome by working with certain asymptotic estimates for minimizers. 展开更多
关键词 fractional(p q)-Laplacian non-negative solutions critical Sobolev exponents
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