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positive solutions for a Competition Model with an Inhibitor Involved

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Acta Mathematicae Applicatae Sinica 2008年第4期24卷 589-598页

作者： Bin Chen School of Mathematics Yancheng Teachers University Jiangsu Yancheng 224051 China

In the paper, we study the positive solutions of a diffusive competition model with an inhibitor involved subject to the homogeneous Dirichlet boundary condition. The existence, uniqueness, stability and multiplicity ... 详细信息

关键词： Competition model inhibitor positive solutions bifurcation uniqueness multiplicity

positive solutions of p-th Yamabe type equations on graphs

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Frontiers of Mathematics in China 2018年第6期13卷 1501-1514页

作者： Xiaoxiao ZHANG Aijin LIN ing Jiaotong University Beijing 100044China Department of Mathematics National University of Defense TechnologyChangsha 410073 China

Let G = (V,E) be a finite connected weighted graph, and assume 1 ⩽ α ⩽ p ⩽ q. In this paper, we consider the p-th Yamabe type equation ―∆pu+huq―1 = λfuα―1 on G, where ∆p is the p-th discrete graph Laplacian, h 0 are real functions defined on all vertices of G. Instead of H. Ge’s approach [Proc. Amer. Math. Soc., 2018, 146(5): 2219-2224], we adopt a new approach, and prove that the above equation always has a positive solution u > 0 for some constant λ ∈ ℝ. In particular, when q = p, our result generalizes Ge’s main theorem from the case of α ⩾ p > 1 to the case of 1 ⩽ α ⩽ p, It is interesting that our new approach can also work in the case of α ⩾ p > 1.

关键词： p-th Yamabe type equation graph Laplacian positive solutions

positive solutions OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

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Acta Mathematica Scientia 1996年第3期16卷 321-329页

作者：黄振勋高国柱 FUDAN UNIV DEPT MATHSHANGHAI 200433PEOPLES R CHINA CHINA TEXT UNIV DEPT BASIC SCISHANGHAI 200051PEOPLES R CHINA

This paper is to obtain sufficient conditions under which the neutral functional differential equation d/dx[x(t) + integral(c)(t) x(s)d(s) mu(t, s)] + integral(c)(t) f(t, x(s))d(s) eta(t, s) = 0, t greater than or equal to t(0) greater than or equal to c (1) has a positive solution on [c, +infinity). Some results in [1] are generalized. Then we apply our results to functional differential equations of special form and obtain sufficient conditions for those equations to have a positive solution.

关键词： positive solutions neutral functional differential equations

positive solutions FOR A KIRCHHOFF EQUATION WITH PERTURBED SOURCE TERMS

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Acta Mathematica Scientia 2022年第5期42卷 1817-1830页

作者： Narimane AISSAOUI Wei LONG School of Mathematics and Statistics Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhan 430079China College of Mathematics and Information Science Jiangci Normal UniversityNanchang 330022China

This paper deals with the existence of positive solutions to the following nonlinear Kirchhoff equation with perturbed external source terms:{−(a+b∫_(R^(3))|∇u|^(2)dx)Δu+V(x)u=Q(x)u^(p)+εf(x),u>0,x∈R^(3),u∈H^(1)(R^(3)).Here a,b are positive constants,V(x),Q(x)are positive radial potentials,10 is a small parameter,f(x)is an external source term in L^(2)(R^(3))∩L^(∞)(R^(3)).

关键词： nonlinear Kirchhoff problem Lyapunov-Schmidt reduction positive solutions level set

positive solutions FOR SINGULAR BOUNDARY VALUE PROBLEM OF SECOND ORDER

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Chinese Annals of Mathematics,Series B 2004年第3期25卷 383-392页

作者： LIUJIAQUAN ZENGPING＇AN XIONGMING SchoolofMathematicalScience PekingUniversityBeijing100871China SchoolofMathematlcalScience PekingUniversityBeijing100871China DepartmentofMathematics DaliUniversity（HehuaCampus）Dali671000YunnanChina

Some results of existence of positive solutions for singular boundary value problems{-u″(t) = p(t)f(u(t)), t ∈ (0, 1),u(0) = u(1) = 0are given, where the function p(t) may be singular at t = 0,1.

关键词： Singular boundary value problem positive solutions Variational method

positive solutions for a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-Ⅱ functional response

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Science China Mathematics 2014年第5期57卷 991-1010页

作者： ZHOU Jun KIM Chan-Gyun School of Mathematics and Statistics Southwest University Department of Mathematics College of William and Mary

We consider a Lotka-Volterra prey-predator model with cross-diffusion and Holling type-II functional *** main concern is the existence of positive solutions under the combined effect of cross-diffusion and Holling type-II functional ***,a positive solution corresponds to a coexistence state of the ***,we study the sufficient conditions to ensure the existence of positive solutions by using degree theory and analyze the coexistence region in parameter *** addition,we present the uniqueness of positive solutions in one dimension ***,we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system,and then we characterize the stable/unstable regions of semi-trivial solutions in parameter plane.

关键词： Lotka-Volterra prey-predator model Holling type-II functional response cross-diffusion positive solutions coexistence uniqueness degree theory

positive solutions OF A NONLOCAL AND NONVARIATIONAL ELLIPTIC PROBLEM

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Acta Mathematica Scientia 2021年第5期41卷 1764-1776页

作者： Lingjun LIU Feilin SHI Institute of Applied Mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematics and Statistics Hunan Normal UniversityChangsha 410081China

In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1 详细信息

In this paper,we will study the nonlocal and nonvariational elliptic problem{−(1+a||u||_(q)^(αq))Δu=|u|^(p−1)u+h(x,u,∇_(u))inΩ,u=0 on∂Ω,(0.1)(1)where a>0,α>0,1positive solutionsandΩis a bounded smooth domain in R^(N)(N≥2).Under suitable assumptions about h(x,u,∇u),we obtain\emph{a priori}estimates of positive solutions for the problem(0.1).Furthermore,we establish the existence of positive solutions by making use of these estimates and of the method of continuity.

关键词： positive solutions nonvariational elliptic problem a priori estimates

positive solutions OF BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

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Applied Mathematics and Mechanics(English Edition) 2001年第4期22卷 495-500页

作者： LI Ren-gui(李仁贵) LI Li-shan(刘立山) Department of Mathematics Qufu Normal University Qufu 273165 P R China Department of Mathematics Qufu Normal University Qufu 273165 P R China

New existence results are presented for the singular second-order nonlinear boundary value problems u ' + g(t)f(u) = 0, 0 0)f(u)/u, f(infinity)(-)= lim(u --> infinity)f(u)/u, f(0)(-)= lim(u -->0)f(u)/u, f(infinity)(+) = lim(u --> infinity)f(u)/u, g may be singular at t = 0 and/or t = 1. The proof uses a fixed point theorem in cone theory.

关键词： second-order singular boundary value problems positive solutions cone fixed point

positive solutions to a Singular Third-Order Three-Point Boundary Value Problem

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Journal of Mathematical Research and Exposition 2011年第2期31卷 287-294页

作者： Hong Ping WU College of Mathematics and Information Science Northwest Normal University Gansu 730070 P. R. China

In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly *** we give an example.

关键词： positive solutions singular third-order three-point BVP fixed point theorem