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concentration phenomena in the semilinear parabolic equation

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Science China Mathematics 2001年第1期44卷 40-47页

作者：谭忠 1. Department of Mathematics Xiamen University 361005 Xiamen China

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... 详细信息

关键词： semilinear parabolic equation critical Sobolev exponent existence asymptotic concentration-compactness principle

EXISTENCE TO FRACTIONAL CRITICAL EQUATION WITH HARDY-LITTLEWOOD-SOBOLEV NONLINEARITIES

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Acta Mathematica Scientia 2021年第4期41卷 1321-1332页

作者： Nemat NYAMORADI Abdolrahman RAZANI Department of Mathematics Razi UniversityKermanshahIran Department of Pure Mathematics Faculty of ScienceImam Khomeini International University34149-16818QazvinIran

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 0ps,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 *** get the existence of infinitely many solutions for the above problem by using the concentration compactness principle and Krasnoselskii’s genus *** the best of our knowledge,our result is new even in Choquard-Kirchhoff-type equations involving the p-Laplacian case.

关键词： Hardy-Littlewood-Sobolev inequality concentration-compactness principle variational method Fractional p-Laplacian operators multiple solutions

Solutions for Schrodinger-Poisson system involving nonlocal term and critical exponent

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Applied Mathematics(A Journal of Chinese Universities) 2023年第3期38卷 357-372页

作者： MO Xiu-ming MAO An-min WANG Xiang-xiang School of Biomedicine Beijing City UniversityBeijing 100094China School of Mathematical Sciences Qufu Normal UniversityQufu 273165China.

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 *** 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 *** 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 *** result improves and extends some related works in the literature.

关键词： variational methods critical exponent concentration-compactness principle

Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

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Acta Mathematica Sinica,English Series 2020年第8期36卷 873-888页

作者： Xu Min WANG School of Mathematical Sciences Beijing Normal UniversityBeijing 100875P.R.China

In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥*** precisely,we show that for any given α>0 and 0 详细信息

In this paper,we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in Rn,where n≥*** precisely,we show that for any given α>0 and 0concentration-compactness principle,we establish the existence of *** results extend the known results when t=0 to the singular version for 0

关键词： Singular supercritical Trudinger-Moser inequality radial lemma concentration-compactness principle extremal functions

A Note on Solutions for Asymptotically Linear Elliptic Systems

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Acta Mathematicae Applicatae Sinica 2008年第3期24卷 511-522页

作者： Lei-ga Zhao Department of Mathematics Beijing University of Chemical Technology 100029 Beijing China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China Department of Mathematics Yunnan Normal University Kunming 650092 China

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.

关键词： Elliptic system ground state solution variational methods concentration-compactness principle

Infinitely many solutions of p-Laplacian equations with limit subcritical growth

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Applied Mathematics and Mechanics(English Edition) 2007年第10期28卷 1373-1382页

作者：耿堤 School of Mathematical Sciences South China Normal UniversityGuangzhou 510631P.R.China

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.

关键词： p-Laplacian operators limit subcritical growth concentration-compactness principle Palais-Smale condition asymptotic minimax principle

Bifurcation and multiplicity of positive solutions for nonhomogeneous fractional Schrödinger equations with critical growth

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Science China Mathematics 2020年第8期63卷 1571-1612页

作者： Xiaoming He Wenming Zou College of Science Minzu University of ChinaBeijing 100081China Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China

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 *** show that there exists 0λ*,a unique solution(λ*,uλ*)ifλ=λ*,which shows that(λ*,uλ*)is a turning point in Hs(RN)for the *** proofs are based on the variational methods and the principle of concentration-compactness.

关键词： fractional Schrödinger equation bifurcation and multiplicity concentration-compactness principle critical Sobolev exponent

EXISTENCE OF POSITIVE SOLUTION FOR A BIHARMONIC EQUATION

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Annals of Differential Equations 2005年第1期21卷 59-64页

作者： XieChaodong YaoYangxin YangJun CollegeofEconomicandManagement GuizhouUniversityforEthnicMinoriticsGuiyang550025 Dept.ofAppl.Math. SouthChinaUniversityofTechnologyGuangzhou510640

In this paper we consider the biharmonic equation (?) in a smooth bounded domain Ω (?) RN with boundary condition (?) where N ≥5, 1 0 and 2= (2N)/(N-4). We prove the existence of λ such that for 0 < λ < λ the abo... 详细信息

关键词： biharmonic equation critical point concentration-compactness principle