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Existence and Concentration of sign-changing solutions of Quasilinear Choquard Equation

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Journal of Applied Mathematics and Physics 2023年第4期11卷 1124-1151页

作者： Die Wang Yuqi Wang Shaoxiong Chen Department of Mathematics Yunnan Normal University Kunming China

In this paper, we study the following quasilinear equation of choquard type: where A(x,t) is given real functions on RN × R and with N ≥ 3, 1 p N, max{N-2p,1} α N, , and ε > 0 is a small parameter, Iα is the Riesz potential. We establish for small ε the existence of a sequence of sign-changing solutions concentrating near a given local minimum point of the bounded potential function V by using the method of invariant sets of descending flow, perturbation method and truncation technique. .

关键词： Quasilinear Choquard Equation The Method of Invariant Sets of Descending Flow Truncation sign-changing solutions

sign-changing solutions FOR p-BIHARMONIC EQUATIONS WITH HARDY POTENTIAL IN R^N

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Acta Mathematica Scientia 2017年第3期37卷 593-606页

作者：杨瑞瑞张薇刘祥清 Department of Mathematics Yunnan Normal University School of Mathematics and Statistics Yunnan University

In this article, by using the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions of p-biharmonic equations with Hardy potential in RN.

关键词： sign-changing solutions p-biharmonic equations Hardy potential

MULTIPLE sign-changing solutions FOR A CLASS OF SCHRODINGER EQUATIONS WITH SATURABLE NONLINEARITY

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Acta Mathematica Scientia 2021年第2期41卷 493-504页

作者：刘忠原 School of Mathematics and Statistics Henan UniversityKaifeng 475004China

In this paper,we construct sign-changing radial solutions for a class of Schrodinger equations with saturable nonlinearity which arises from several models in mathematical physics.More precisely,for any given nonnegative integer k,by using a minimization argument,we first obtain a sign-changing minimizer with k nodes of a constrained minimization problem,and show,by a deformation lemma and Miranda's theorem,that the minimizer is the desired solution.

关键词： sign-changing solutions saturable nonlinearity Nehari manifold variational methods

sign-changing solutions FOR SCHRDINGER EQUATIONS WITH VANISHING AND sign-changing POTENTIALS

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Acta Mathematica Scientia 2014年第3期34卷 691-702页

作者：吴元泽黄毅生刘增 College of Sciences China University of Mining and Technology Department of Mathematics Soochow University

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation -△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞, 2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].

关键词： Variational methods bounded state solutions sign-changing solutions

sign-changing solutions FOR THE STATIONARY KIRCHHOFF PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN IN R^N

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Acta Mathematica Scientia 2018年第6期38卷 1712-1730页

作者：程琨高琦 Department of Information Engineering Jingdezhen Ceramic Institute Department of Mathematics School of Science Wuhan University of Technology

In this paper, we study the existence of least energy sign-changing solutions for aKirchhoff-type problem involving the fractional Laplacian operator. By using the constraintvariation method and quantitative deformation lemma, we obtain a least energy nodal solu-tion ub for the given problem. Moreover, we show that the energy of ub is strictly larger thantwice the ground state energy. We also give a convergence property of ub as b O, where bis regarded as a positive parameter.

关键词： Kirchhoff equation fractional Laplaciau sign-changing solutions

sign-changing and Multiple solutions Theorems for Semilinear Elliptic Boundary Value Problems with Jumping Nonlinearities

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Acta Mathematica Sinica,English Series 2000年第1期16卷 113-122页

作者： Shujie Li Zhitao Zhang Institute of Mathematics,Academia Sinica,Beijing 100080,P.R.China Institute of Mathematics Academia Sinica Beijing 100080 P.R. China

In this paper,we use the ordinary differential equation theory of Banach spaces and minimax theory,and in particular,the relative mountain pass lemma to study semilinear elliptic boundary value problems with jumping nonlinearities at zero or infinity,and get new multiple solutions and sign- changing solutions theorems,at last we get up to six nontrivial solutions.

关键词： sign-changing solutions Dirichlet problems Multiple solutions Jumping nonlinearities

INFINITELY MANY sign-changing solutions FOR THE BRZIS-NIRENBERG PROBLEM INVOLVING HARDY POTENTIAL

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Acta Mathematica Scientia 2016年第2期36卷 527-536页

作者：张靖马世旺 Mathematics Science College Inner Mongolia Normal University Hohhot 010022 China School of Mathematical Sciences and LPMC Nankai University Tianjin 300071 China

In this article, we give a new proof on the existence of infinitely many sign- changing solutions for the following Brezis-Nirenberg problem with critical exponent and a Hardy potential -△u-μ（u/｜x｜^2）=λu＋｜u｜^2＊-2u inΩ, u=0 on eΩ,where Ω is a smooth open bounded domain of R^N which contains the origin, 2＊=2N/n-2 is the critical Sobolev exponent. More precisely, under the assumptions that N ≥ 7, μ ∈ [0, μ- 4）, and μ=（N-2）^2/4, we show that the problem admits infinitely many sign-changing solutions for each fixed λ 〉 0. Our proof is based on a combination of invariant sets method and Lj usternik-Schnirelman theory.

关键词： Critical exponent sign-changing solutions minimax method hardy potential

Quasilinear Schrodinger-Poisson equations involving a nonlocal term and an integral constraint

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Science China Mathematics 2022年第11期65卷 2297-2324页

作者： Xiaojing Dong Anmin Mao School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China Academy of Mathematics and Systems Science Chinese Academy of SciencesBeijing 100190China School of Mathematical Sciences Qufu Normal UniversityQufu 273165China

In this paper,we consider a class of quasilinear Schrodinger-Poisson problems of the form∫-(a+b∫_(R)^(N)|■μ|^(2)dx)■μ+V(x)u+Фu-1/2u■(u^(2))-λ|u|^(p-2)u=0 in R^(N),-ΔФ=u^(2),u(x)→0,|x|→∞in R^(N),∫_(R)^(N)|u|^(p)dx=1,where a>0,b≥0,N≥3,λappears as a Lagrangian multiplier,and 4sign-changing solutions.Our results extend and improve some recent work.

关键词： quasilinear problem nonlocal term sign-changing solutions genus theory

Nodal solutions of the Brezis-Nirenberg Problem in Dimension 6

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Analysis in Theory and Applications 2022年第1期38卷 1-25页

作者： Angela Pistoia Giusi Vaira 无 Dipartimento di Scienze di Base e Applicate per Vlngegneria Sapienza Universita di RomaVia Scarpa 1600161 RomaItaly Dipartimento di Matematica Universita degli studi di Bari“Aldo Moro”via Edoardo Orabona 470125 BariItaly

We show that the classical Brezis-Nirenberg problem-△μ=μ|μ|+λμinΩ,μ=0 on■Ω,when O is a bounded domain in R^(6)has a sign-changing solution which blows-up at a point in n asλapproaches a suitable valueλ_(0)>0.

关键词： sign-changing solutions blow-up phenomenon Ljapunov-Schmidt reduction Transversality theorem