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Infinitely Many Solutions for a Class of Quasi-linear Elliptic Problem

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Acta Mathematicae Applicatae Sinica 2024年第3期40卷 728-743页

作者： Xiao-yao JIA Zhen-luo LOU School of Economics and Management Zhongyuan University of TechnologyZhengzhou 450007China Mathematics and Statistics School Henan University of Science and TechnologyLuoyang 471023China

In this paper,we study the following quasi-linear elliptic equation:■where Ω?R^(N) is a bounded domain,λ>0 is a parameter.The function ψ(|t|)t is the subcritical term,and φ(|t|)t is the critical Orlicz-Sobolev growth term with respect to φ.Under appropriate conditions on φ,ψ and φ,we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation,for λ∈(0,λ_(0)),where λ_(0)> 0 is a fixed constant.

关键词： quasi-linear elliptic equations variational methods Orlicz-Sobolev spaces

MULTIPLICITY OF NORMALIZED SOLUTIONS FOR THE FRACTIONAL SCHR?DINGER-POISSON SYSTEM WITH DOUBLY CRITICAL GROWTH

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Acta Mathematica Scientia 2024年第3期44卷 997-1019页

作者：孟禹希贺小明 School of Mathematics and Statistics Beijing Institute of TechnologyBeijing100081China College of Science Minzu University of ChinaBeijing100081China

In this paper,we are concerned with solutions to the fractional Schrodinger-Poisson system■ with prescribed mass ∫_(R^(3))|u|^(2)dx=a^(2),where a> 0 is a prescribed number,μ> 0 is a paremeter,s ∈(0,1),2 详细信息

关键词： fractional Schrodinger-Poisson system normalized solutions variational methods L^(2)-subcritical L^(2)-supercritical

MULTIPLE POSITIVE SOLUTIONS TO A CLASS OF MODIFIED NONLINEAR SCHR?DINGER EQUATION IN A HIGH DIMENSION

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Acta Mathematica Scientia 2023年第4期43卷 1819-1840页

作者：钟延生李永青 School of Mathematics and Statistics&FJKLMAA &Center for Applied Mathematics of Fujian Province(FJNU)Fujian Normal UniversityFuzhou 350117China

The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4 详细信息

The existence and multiplicity of positive solutions for equation(1.1)with the new critical exponent 4

关键词： variational methods nonlinear Schrodinger equation sign-changing

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

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Acta Mathematica Scientia 2022年第5期42卷 2063-2077页

作者：皮慧荣曾勇 School of Mathematics and Information Guangxi UniversityNanning 530004China

This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation−(a+b∫_(R^(3))|∇u|^(2))Δu+V(x)u=f(u)in R^(3),with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions.Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity,especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

关键词： Kirchhoff type equation general nonlinearity variational methods Pohozaev identity

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 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.

关键词： variational methods critical exponent concentration-compactness principle

Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well

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Science China Mathematics 2022年第8期65卷 1577-1598页

作者： Claudianor O.Alves Chao Ji Unidade Acadêmica de Matemática Universidade Federal de Campina GrandeCampina Grande 58429-900Brazil Department of Mathematics East China University of Science and TechnologyShanghai 200237China

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation:{−Δu+λV(x)u=ulogu^(2)inRN,u∈H^(1)(R^(N)),where N≥1,⋋>0 is a parameter and the nonnegative continuous function V:ℝ^(N)→ℝhas potential wellΩ:=int V^(−1)(0)which possesses k disjoint bounded componentsΩ=∪^(k)_(j)=1Ω_(j).Using the variational methods,we prove that if the parameter⋋>0 is large enough,then the equation has at least 2^(k)−1 multi-bump positive solutions.

关键词： variational methods logarithmic Schrödinger equation multi-bump solutions deepening potential well

EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS

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Acta Mathematica Scientia 2023年第1期43卷 303-323页

作者：罗缘圆高冬梅王俊 Institute of Applied System Analysis Jiangsu UniversityZhenjiang212013China

We study the Choquard equation-Δu+V(x)u-b(x)∫R3|u(y)|2/|x-y|dyu,x∈R3,where V(x)=V1(x),b(x)=b1(x)for x1>0 and V(x)=V2(x),b(x)=b2(x)for x1<0,and V1,V2,b1and b2are periodic in each coordinate direction.Under some suitable assumptions,we prove the existence of a ground state solution of the equation.Additionally,we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.

关键词： Choquard equation ground state solution critical points variational methods

Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

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Frontiers of Mathematics in China 2022年第5期17卷 813-828页

作者： Chongqing WEI Anran LI School of Mathematical Sciences Shanxi UniversityTaiyuan 030006China

In this paper,a class of Kirchhoff type equations in R^(N)(N≥3)with zero mass and a critical term is studied.Under some appropriate conditions,the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem.The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem.Compared to the usual Kirchhoff-type problems,we only require the nonlinearity to satisfy the classical superquadratic condition(Ambrosetti-Rabinowitz condition).

关键词： Kirchhoff type equations with a critical term variational methods Symmetric Mountain Pass theorem Second Concentration Compactness lemma

THE MULTIPLICITY AND CONCENTRATION OF POSITIVE SOLUTIONS FOR THE KIRCHHOFF-CHOQUARD EQUATION WITH MAGNETIC FIELDS

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Acta Mathematica Scientia 2022年第4期42卷 1453-1484页

作者：王莉程琨汪继秀 College of Science East China Jiaotong UniversityNanchang 330013China Department of Information Engineering Jingdezhen Ceramic InstituteJingdezhen333403China School of Mathematics and Statistics Hubei University of Arts and ScienceXiangyang441053China

In this paper,we study the multiplicity and concentration of positive solutions for the following fractional Kirchhoff-Choquard equation with magnetic fields:(aε^(2s)+bε^(4 s-3)[u]_(ε)^(2),A/ε)(-Δ)_(A/ε)^(s)u+V(x)u=ε^(-α)(Iα*F(|u|^(2)))f(|u|^(2))u in R^(3).Hereε>0 is a small parameter,a,b>0 are constants,s E(0,1),(-Δ)As is the fractional magnetic Laplacian,A:R^(3)→R^(3) is a smooth magnetic potential,Iα=Γ(3-α/2)/2απ3/2Γ(α/2)·1/|x|^(α) is the Riesz potential,the potential V is a positive continuous function having a local minimum,and f:R→R is a C^(1) subcritical nonlinearity.Under some proper assumptions regarding V and f,we show the multiplicity and concentration of positive solutions with the topology of the set M:={x∈R^(3):V(x)=inf V}by applying the penalization method and LjusternikSchnirelmann theory for the above equation.

关键词： Fractional Kirchhoff-Choquard problem penalization method Ljusternik-Schnirelmann theory variational methods