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Gradient Estimates for a Nonlinear Parabolic Equation with Diffusion on Complete Noncompact Manifolds

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Chinese Annals of Mathematics,Series B 2015年第1期36卷 57-66页

作者： Liang ZHAO Zongwei MA Department of Mathematics Nanjing University of Aeronautics and Astronautics College of Mathematics Physics and Information Engineering Jiaxing University

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0<... 详细信息

关键词： Gradient estimates positive solutions Harnack inequality

solutions FOR SUPERLINEAR(n-1,1)CONJUGATE BOUNDARY VALUE PROBLEMS

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Acta Mathematica Scientia 2001年第2期21卷 259-264页

作者：娄本东 Shandong Univ Dept Math Jinan 250100 Peoples R China

The author investigates the existence of positive and nontrivial solutions for superlinear (n - 1, 1) conjugate boundary value problems by means of topological degree theory and cone theory. The main theorems improve ... 详细信息

关键词： Boundary value problems topological degree cone positive solutions non- trivial solutions

Parametric Anisotropic(p,q)-Neumann Problems

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Acta Mathematicae Applicatae Sinica 2023年第4期39卷 926-942页

作者： Zhen-hai LIU Nikolaos S.PAPAGEORGIOU Center for Applied Mathematics of Guangxi Yulin Normal UniversityYulin 537000China Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation Guangxi Minzu UniversityNanning 530006China Department of Mathematics National Technical UniversityZografou Campus15780 AthensGreece

We consider a Neumann problem driven by a(p(z), q(z))-Laplacian(anisotropic problem) plus a parametric potential term with λ > 0 being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ moves on R_(+)=(0,+∞).

关键词： variable exponent spaces superlinear reaction bifurcation-type theorem anisotropic regularity strong comparison positive solutions

Gradient Estimates for p-Laplacian Lichnerowicz Equation on Noncompact Metric Measure Space

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Chinese Annals of Mathematics,Series B 2020年第3期41卷 397-406页

作者： Liang ZHAO Ming SHEN Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 210016China College of Mathematics and Computer Science Fuzhou UniversityFuzhou 350116China

In this paper, the authors obtain the gradient estimates for positive solutions to the weighted p-Laplacian Lichnerowicz equation △p,fu + cuσ= 0 on noncompact smooth metric measure space, where c is a nonnegative constant, and p, σ(1 < p≤2, σ≤p-1) are real constants. Moreover, by the gradient estimate, they can get the corresponding Liouville theorem and Harnack inequality.

关键词： p-Laplacian positive solutions Liouville theorem

The Existence of Semiclassical States for Some P-Laplacian Equation with Critical Exponent

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Acta Mathematicae Applicatae Sinica 2017年第2期33卷 417-434页

作者： Ji-xiu WANG School of Mathematics and Computer Science Hubei University of Arts and Science Xiangyang 441053 China

In this paper, we study the existence of semiclassical states for some p-Laplacian equation. Under given conditions and minimax methods, we show that this problem has at least one positive solution provided that ε≤ε; for any m ∈ N, it has m pairs solutions if ε≤εm, where ε,εm are sufficiently small positive numbers. Moreover, these solutions axe closed to zero in W1,p（RN） as ε→0.

关键词： semiclassical states positive solutions critical exponent

A NONSMOOTH THEORY FOR A LOGARITHMIC ELLIPTIC EQUATION WITH SINGULAR NONLINEARITY

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Acta Mathematica Scientia 2022年第2期42卷 502-510页

作者： Chunyu LEI Jiafeng LIAO Changmu CHU Hongmin SUO School of Sciences GuiZhou Minzu UniversityGuiyang 550025China College of Mathematics Education China West Normal UniversityNanchong 637002China

We consider the logarithmic elliptic equation with singular nonlinearity {Δu+ulogu^(2)+λ/u^(γ)=0,in Ω,u>0,in Ω,u=0,on δΩ,where Ω⊂R^(N)(N≥3)is a bounded domain with a smooth boundary,0positive *** using a variational method and the critical point theory for a nonsmooth functional,we obtain the existence of two positive *** result generalizes and improves upon recent results in the literature.

关键词： Logarithmic elliptic equation singular nonlinearity positive solutions variational method

THE EXISTENCE OF MULTIPLE solutions OF p-LAPLACIAN ELLIPTIC EQUATION

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Acta Mathematica Scientia 2001年第2期21卷 203-212页

作者：谭忠姚正安 Xiamen Univ Dept Math Xiamen 361005 Peoples R China Zhongshan Univ Dept Math Guangzhou 510275 Peoples R China

This paper considers the quasilinear elliptic equation where , and 0 λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, ... 详细信息

This paper considers the quasilinear elliptic equation where , and 0 positive number. Object is to estimate exactly the magnitute of λ* such that (1)λ has at least one positive solution if λ ∈ (0, λ*) and no positive solutions if λ > λ*. Furthermore, (1)λ has at least one positive solution when λ = λ*, and at least two positive solutions when λ ∈ (0, λ*) and . Finally, the authors obtain a multiplicity result with positive energy of (1)λ when 0 < m < p - 1 < q = (Np)/(N-p) - 1.

关键词： Quasilinear elliptic equation super-and subsolution method critical Sobolev exponent positive solutions multiple solutions

SINGULAR DOUBLE PHASE EQUATIONS

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Acta Mathematica Scientia 2023年第3期43卷 1031-1044页

作者：刘振海 Nikolaos S.PAPAGEORGIOU Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal UniversityYulin537000China Guangxi Key Laboratory of Hybrid Computation and IC Design Analysis Guangxi Minzu UniversityNanning530006China Department of Mathematics National Technical UniversityZografou Campus15780AthensGreece

We study a double phase Dirichlet problem with a reaction that has a parametric singular term. Using the Nehari manifold method, we show that for all small values of the parameter, the problem has at least two positiv... 详细信息

关键词： double phase singular term unbalanced growth Musielak-Orlice spaces Nehari manifold positive solutions

Existence, uniqueness and stability of positive steady states to a prey-predator diffusion system

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Science China Mathematics 2009年第5期52卷 1031-1041页

作者： WANG MingXin WANG XuBo Science Research Center Harbin Institute of TechnologyHarbin 150080China Department of Mathematics Southeast UniversityNanjing 210018China Division of Natural Science and Mathematics Macon State CollegeMaconGA 31206USA

In the paper, we study the positive solutions of an elliptic system coming from a preypredator model with modified Leslie-Gower and Holling-Type II schemes. We study the existence, non-existence, bifurcation, uniqueness and stability of positive solutions. In particular, we obtain a continuum of positive solutions connecting a semitrivial solution to the unique positive solution of the limiting system.

关键词： prey-predator model positive solutions bifurcation existence and uniqueness stability 35J55 35B25 92B05