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Positive Solutions for a Class of Fractional p-Laplacian Equation with Critical Sobolev Exponent and Decaying Potentials

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Acta Mathematicae Applicatae Sinica 2022年第2期38卷 463-483页

作者： Na LI Xiao-ming HE College of Science Minzu University of ChinaBeijing 100081China

In this paper,we study the existence of positive solution for the p-Laplacian equations with frac-tional critical nonlinearity{-Δ)_(p)^(s)u+V(x)|u|^(p-2)u=K(x)f(u)+P(x)|u|p_(s)^(*)-^(2)u,x∈R^(N),u∈Ds,p(RN),where s∈(0,1),p_(s)^(*)=Np/N-sp,N>sp,p>1 and V(x),K(x)are positive continuous functions which vanish at infinity,f is a function with a subcritical growth,and P(x)is bounded,nonnegative continuous *** using variational method in the weighted spaces,we prove the above problem has at least one positive solution.

关键词： Fractional p-Laplacian Variational methods mountain pass theorem Critical growth Vanishing potential

Existence of Solutions for a Quasilinear Schr?dinger Equation with Potential Vanishing

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Acta Mathematicae Applicatae Sinica 2023年第3期39卷 696-706页

作者： Yan-fang XUE Jian-xin HAN Xin-cai ZHU School of Mathematics and Statistics Xinyang Normal UniversityHenan 464000China

We study the following quasilinear Schrodinger equation-△u+V(x)u-△(u^(2))u=K(x)g(u),x∈R^(3),where the nonlinearity g(u)is asymptotically cubic at infinity,the potential V(x)may vanish at *** appropriate assumptions... 详细信息

关键词： quasilinear SchrOdinger equation vanishing potential asymptotically cubic mountain pass theorem

Positive Ground State Solutions for Schrodinger-Poisson System with General Nonlinearity and Critical Exponent

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Journal of Partial Differential Equations 2023年第1期36卷 68-81页

作者： CHEN Qingfang LIAO Jiafeng School of Mathematics and Information China West Normal UniversityNanchong 637009China Nanchong NO.1 Middle School Nanchong 637000China College of Mathematics Education China West Normal UniversityNanchong 637009China

In this paper,we consider the following Schrodinger-Poisson system{-Δu+ηΦu=f(x,μ)+μ^(5),x∈Ω,-ΔФ=μ^(2),x∈Ω,μ=Φ=0,x∈■Ω,whereΩis a smooth bounded domain in R^(3),η=±1 and the continuous function f sat... 详细信息

关键词： Schrodinger-Poisson system Sobolev critical exponent positive ground state solu-tion mountain pass theorem

EXISTENCE OF SOLITARY WAVES TO A GENERALIZED KADOMTSEV-PETVIASHVILI EQUATION

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Acta Mathematica Scientia 2012年第3期32卷 1149-1156页

作者：梁占平苏加宝 School of Mathematical Sciences Shanxi University School of Mathematical Sciences Capital Normal University

In this article, we study the existence of nontrivial solitary waves of a gener- alized Kadomtsev-Petviashvili equation via variational methods.

关键词： Generalized Kadomtsev-Petviashvili equation solitary waves mountain pass theorem

HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS

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Acta Mathematica Scientia 2010年第1期30卷 312-318页

作者：万莉莉唐春雷 Department of Mathematics Southwest University

The existence of homoclinic orbits is obtained by the variational approach for a class of second order Hamiltonian systems q（t）＋ ↓△V（t, q（t）） = 0, where V（t, x） = -K（t, x）＋ W（t, x）, K（t, x） is neit... 详细信息

关键词： homoclinic orbits second order Hamiltonian systems mountain pass theorem

BIHARMONIC EQUATIONS WITH ASYMPTOTICALLY LINEAR NONLINEARITIES

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Acta Mathematica Scientia 2007年第3期27卷 549-560页

作者：刘玥王征平 Wuhan Institute of Physics and Mathematics Chinese Academy of sciences

This article considers the equation △2u = f（x,u）with boundary conditions either u｜aΩ = au/an｜aΩ = 0 or u｜aΩ = △u｜aΩ = 0, where f（x, t） is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in R^N, N 〉 4. By a variant version of mountain pass theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f（x, t）.

关键词： Biharmonic mountain pass theorem asymptotically linear

A Variational ODE and its Application to an Elliptic Problem

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Acta Mathematicae Applicatae Sinica 2007年第4期23卷 685-696页

作者： Huan-song Zhou Hong-bo Zhu Wuhan Institute of Physics and Mathematics Chinese Academy of Sciences P.O. Box 71010 Wuhan 430071 China

In this paper, we consider the following ODE problem { （-u＂（τ）＋（N-1）（N-3）/4τ^2 ）u（τ）＋λu（τ）=f（τ,τ（1-N）/2 u）u（τ）,τ〉0, u∈H, N≥3. （P）,where f ∈ C（（0,＋∞） ×R,R）, f（τ,s） goes t... 详细信息

In this paper, we consider the following ODE problem { （-u＂（τ）＋（N-1）（N-3）/4τ^2 ）u（τ）＋λu（τ）=f（τ,τ（1-N）/2 u）u（τ）,τ〉0, u∈H, N≥3. （P）,where f ∈ C（（0,＋∞） ×R,R）, f（τ,s） goes to p（τ） and q（τ） uniformly in τ 〉 0 as s→ 0 and s→＋∞ respectively, 0≤ p（τ） ≤ q（τ） ∈L^∞（0,∞）. Moreover, for τ 〉 0, f（τ, s） is nondecreasing in s≥ 0. Some existence and non-existence of positive solutions to problem （P） are proved without assuming that p（τ） = 0 and q（τ） has a limit at infinity. Based on these results, we get the existence of positive solutions for an elliptic problem.

关键词： Elliptic equation asymptotically linear mountain pass theorem

Existence and Multiplicity Results for a Degenerate Elliptic Equation

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Acta Mathematica Sinica,English Series 2006年第3期22卷 665-670页

作者： Wei DONG Jian Tao CHEN Hebei University of Engineering

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.

关键词： degenerate elliptic equation mountain pass theorem maximal positive solution

Multiple Solutions for a Fourth-order Asymptotically Linear Elliptic Problem

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Acta Mathematica Sinica,English Series 2006年第4期22卷 1121-1126页

作者： Ai Xia QIAN Shu Jie LI Department of Mathematics Qufu Normal University Qufu 273165 P. R. China Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080 P. R. China

Under simple conditions, we prove the existence of three solutions for a fourth-order asymptotically linear elliptic boundary value problem. For the resonance case at infinity, we do not need to assume any more condit... 详细信息

关键词： Fourth-order elliptic boundary value problem mountain pass theorem Morse theory