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AN IMPROVEMENT ON THE CONCENTRATION-COMPACTNESS PRINCIPLE
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作者 邱兴 洪毅 沈尧天 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2001年第1期60-67,共8页
In this paper we first improve the concentration- compactness lemma by proving that the vanishing case is a special case of dichotomy, then we apply this improved concentration- compactness lemma to a typical restrct... In this paper we first improve the concentration- compactness lemma by proving that the vanishing case is a special case of dichotomy, then we apply this improved concentration- compactness lemma to a typical restrcted minimization problem, and get some new results. 展开更多
关键词 concentration-compactness lemma minimization problem G-invariant function
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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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Infinitely many solutions of p-Laplacian equations with limit subcritical growth
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作者 耿堤 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2007年第10期1373-1382,共10页
We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functiona... We discussed a class of p-Laplacian boundary problems on a bounded smooth domain, the nonlinearity is odd symmetric and limit subcritical growing at infinite. A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified. We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical. 展开更多
关键词 p-Laplacian operators limit subcritical growth concentration-compactness principle Palais-Smale condition asymptotic minimax principle
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EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES
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作者 Nemat NYAMORADI Abdolrahman RAZANI 《Acta Mathematica Scientia》 SCIE CSCD 2021年第4期1321-1332,共12页
In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△... In this paper,we consider the following new Kirchhoff-type equations involving the fractional p-Laplacian and Hardy-Littlewood-Sobolev critical nonlinearity:(A+B∫∫_(R^(2N))|u(x)-u(y)|^(p)/|x-y|^(N+ps)dxdy)^(p-1)(-△)_(p)^(s)u+λV(x)|u|^(p-2)u=(∫_(R^(N))|U|^(P_(μ,S)^(*))/|x-y|^(μ)dy)|u|^(P_(μ,S)^(*))^(-2)u,x∈R^(N),where(-△)_(p)^(s)is the fractional p-Laplacian with 0<s<1<p,0<μ<N,N>ps,a,b>0,λ>0 is a parameter,V:R^(N)→R^(+)is a potential function,θ∈[1,2_(μ,s)^(*))and P_(μ,S)^(*)=pN-pμ/2/N-ps is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.We get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus theory.To the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case. 展开更多
关键词 Hardy-Littlewood-Sobolev inequality concentration-compactness principle variational method Fractional p-Laplacian operators multiple solutions
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Concentration phenomena in the semilinear parabolic equation 被引量:1
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作者 谭忠 《Science China Mathematics》 SCIE 2001年第1期40-47,共8页
We prove the existence of the global, but unbounded solution of the semilinear heat equations with critical Sobolev exponent, and that under some assumptions, the global unbounded classical solution concentrates on or... We prove the existence of the global, but unbounded solution of the semilinear heat equations with critical Sobolev exponent, and that under some assumptions, the global unbounded classical solution concentrates on origin as t→∞. 展开更多
关键词 semilinear parabolic equation critical Sobolev exponent EXISTENCE ASYMPTOTIC concentration-compactness principle
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Existence and Concentration of Ground States of Coupled Nonlinear Schr■dinger Equations with Bounded Potentials 被引量:1
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作者 Gongming WEI 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 2008年第3期247-264,共18页
A 2-coupled nonlinear Schrbdinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for suf... A 2-coupled nonlinear Schrbdinger equations with bounded varying potentials and strongly attractive interactions is considered. When the attractive interaction is strong enough, the existence of a ground state for sufficiently small Planck constant is proved. As the Planck constant approaches zero, it is proved that one of the components concentrates at a minimum point of the ground state energy function which is defined in Section 4. 展开更多
关键词 CONCENTRATION Nehari's manifold Critical point theory concentration-compactness principle
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Scattering for Focusing Combined Power-type NLS 被引量:1
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作者 Jian XIE 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2014年第5期805-826,共22页
We consider the scattering of Cauchy problem for the focusing combined power-type Schroodinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condit... We consider the scattering of Cauchy problem for the focusing combined power-type Schroodinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi. 展开更多
关键词 Nonlinear Schroodinger equation SCATTERING concentration-compactness method
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Existence of Solutions to Nonlinear Schr?dinger Equations Involving N-Laplacian and Potentials Vanishing at Infinity
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作者 Mao Chun ZHU Jun WANG Xiao Yong QIAN 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2020年第10期1151-1170,共20页
We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential crit... We study the existence of solutions for the following class of nonlinear Schr?dinger equations-ΔN u+V(x)u=K(x)f(u)in R^N where V and K are bounded and decaying potentials and the nonlinearity f(s)has exponential critical growth.The approaches used here are based on a version of the Trudinger–Moser inequality and a minimax theorem. 展开更多
关键词 Potentials vanishing at infinity concentration-compactness Principles Mountain-pass theorem exponential critical growth N-Laplacian bound solution
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A Note on Solutions for Asymptotically Linear Elliptic Systems
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作者 Lei-ga Zhao 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2008年第3期511-522,共12页
In this paper, we are concerned with the elliptic system of{ -△u+V(x)u=g(x,v), x∈R^N, -△v+V(x)v=f(x,u), x∈R^N, where V(x) is a continuous potential well, f, g are continuous and asymptotical... In this paper, we are concerned with the elliptic system of{ -△u+V(x)u=g(x,v), x∈R^N, -△v+V(x)v=f(x,u), x∈R^N, where V(x) is a continuous potential well, f, g are continuous and asymptotically linear as t→∞. The existence of a positive solution and ground state solution are established via variational methods. 展开更多
关键词 Elliptic system ground state solution variational methods concentration-compactness principle
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HOMOCLINIC ORBITS FOR LAGRANGIAN SYSTEMS
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作者 Wu SHAOPING Departmentof Mathematics, Zhejiang University, Hangzhou, 310027, China. 《Chinese Annals of Mathematics,Series B》 SCIE CSCD 1996年第2期245-256,共12页
The existence of at least two homoclinic orbits for Lagrangian system (LS) is proved, wherethe Lagrangian L(t,x,y) =1/2∑aij(x)yiyj-V(t, x), in which the potential V(t,x) is globallysurperquadratic in x and T-periodic... The existence of at least two homoclinic orbits for Lagrangian system (LS) is proved, wherethe Lagrangian L(t,x,y) =1/2∑aij(x)yiyj-V(t, x), in which the potential V(t,x) is globallysurperquadratic in x and T-periodic in t. The Concentration-Compactness Lemma and Mini-max argument are used to prove the existences. 展开更多
关键词 Lagrangian systerm Superquadratic growth concentration-compactness Minimax argument
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Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals
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作者 Xu Min WANG 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2020年第8期873-888,共16页
In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalitie... In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥2.More precisely,we show that for any given α>0 and 0<t<n,then the following two inequalities hold for ∀u∈W^1,n0,r(B),∫Bsup∣▽u∣^ndx≤1∫Bexp((αn,t+∣x∣^α∣)u∣^n/n-1)/∣x∣^tdx<∞ and ∫Bsup∣▽u∣^ndx≤1∫Bexp(αn,t+∣u∣^n/n-1+∣x∣^α)/∣x∣^tdx<∞.We also consider the problem of the sharpness of the constantαn,t.Furthermore,by employing the method of estimating the lower bound and using the concentration-compactness principle,we establish the existence of extremals.These results extend the known results when t=0 to the singular version for 0<t<n. 展开更多
关键词 Singular supercritical Trudinger-Moser inequality radial lemma concentration-compactness principle extremal functions
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Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth
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作者 Xiaoming He Wenming Zou 《Science China Mathematics》 SCIE CSCD 2020年第8期1571-1612,共42页
In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4... In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth{(−∆)su + u = u^2∗s−1 + λ(f(x, u) + h(x)), x ∈ R^N ,u ∈ Hs(R^N ), u(x) > 0, x ∈ RN ,where s∈(0,1),N>4 s,andλ>0 is a parameter,2s*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0<λ*<+∞such that the problem has exactly two positive solutions ifλ∈(0,λ*),no positive solutions forλ>λ*,a unique solution(λ*,uλ*)ifλ=λ*,which shows that(λ*,uλ*)is a turning point in Hs(RN)for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness. 展开更多
关键词 fractional Schrödinger equation bifurcation and multiplicity concentration-compactness principle critical Sobolev exponent
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Quantization of the Blow-Up Value for the Liouville Equation with Exponential Neumann Boundary Condition
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作者 Tao Zhang Changliang Zhou Chunqin Zhou 《Communications in Mathematics and Statistics》 SCIE 2018年第1期29-48,共20页
In this paper,we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition.In particular,we will obtain a sharp mass quantization result for the solution sequ... In this paper,we analyze the asymptotic behavior of solution sequences of the Liouville-type equation with Neumann boundary condition.In particular,we will obtain a sharp mass quantization result for the solution sequences at a blow-up point. 展开更多
关键词 Neumann problem concentration-compactness phenomena Blow-up behaviors Mass quantization
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EXISTENCE OF POSITIVE SOLUTION FOR A BIHARMONIC EQUATION
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作者 XieChaodong YaoYangxin YangJun 《Annals of Differential Equations》 2005年第1期59-64,共6页
In this paper we consider the biharmonic equation (?) in a smooth bounded domain Ω (?) RN with boundary condition (?) where N ≥5, 1 < q < 2, λ>0 and 2= (2N)/(N-4). We prove the existence of λ such that fo... In this paper we consider the biharmonic equation (?) in a smooth bounded domain Ω (?) RN with boundary condition (?) where N ≥5, 1 < q < 2, λ>0 and 2= (2N)/(N-4). We prove the existence of λ such that for 0 < λ < λ the above problem has a positive solution. 展开更多
关键词 biharmonic equation critical point concentration-compactness principle
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