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Exact boundary behavior of large solutions to semilinear elliptic equations with a nonlinear gradient term

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Science China Mathematics 2020年第3期63卷 559-574页

作者： Zhijun Zhang School of Mathematics and Information Sciences Yantai University

This paper is concerned with exact boundary behavior of large solutions to semilinear elliptic equations △u=b(x)f(u)(C0+|▽u|q),x∈Ω,where Ω is a bounded domain with a smooth boundary in RN,C0≥0,q E [0,2),b∈Clocα(Ω) is positive in but may be vanishing or appropriately singular on the boundary,f∈C[0,∞),f(0)=0,and f is strictly increasing on [0,∞)(or f∈C(R),f(s)> 0,■s∈R,f is strictly increasing on R).We show unified boundary behavior of such solutions to the problem under a new structure condition on f.

关键词： semilinear elliptic equations a nonlinear gradient term large solutions boundary behavior

Superconvergence and L^(∞)-Error Estimates of the Lowest Order Mixed Methods for Distributed Optimal Control Problems Governed by semilinear elliptic equations

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Numerical Mathematics(Theory,Methods and Applications) 2013年第3期6卷 479-498页

作者： Tianliang Hou School of Mathematical Sciences South China Normal UniversityGuangzhou 510631GuangdongChina

In this paper, we investigate the superconvergence property and the L∞-errorestimates of mixed finite element methods for a semilinear elliptic control problem. Thestate and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant *** derive some superconvergence results for the control variable. Moreover, we derive L^(∞)-error estimates both for the control variable and the state variables. Finally, anumerical example is given to demonstrate the theoretical results.

关键词： semilinear elliptic equations distributed optimal control problems superconvergence L^(∞)-error estimates mixed finite element methods.

A Posteriori Error Estimate of Two Grid Mixed Finite Element Methods for semilinear elliptic equations

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Journal of Applied Mathematics and Physics 2023年第2期11卷 361-376页

作者： Yiming Wen Luoping Chen Jiajia Dai School of Mathematics Southwest Jiaotong University Chengdu China

In this paper, we present the a posteriori error estimate of two-grid mixed finite element methods by averaging techniques for semilinear elliptic equations. We first propose the two-grid algorithms to linearize the mixed method equations. Then, the averaging technique is used to construct the a posteriori error estimates of the two-grid mixed finite element method and theoretical analysis are given for the error estimators. Finally, we give some numerical examples to verify the reliability and efficiency of the a posteriori error estimator.

关键词： Two-Grid Mixed Finite Element Methods Posteriori Error Estimates semilinear elliptic equations Averaging Technique

Multiple Positive Solutions for semilinear elliptic equations Involving Subcritical Nonlinearities in RN

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Journal of Partial Differential equations 2014年第1期27卷 74-94页

作者： KHADEMLOO Somayeh MOHSENHI Rahelh Department of Basic Sciences Babol Noushirvani University of Technology Babol Iran.

In this paper, we study how the shape of the graph of a（z） affects on the number of positive solutions of -△v＋μb（z）v=a（z）vp-1＋λh（z）vq-1,inRN.（0.1） We prove for large enough λ,μ〉 0, there exist at least k＋ 1 positive solutions of the this semilinear elliptic equations where 1 ≤ q 〈 2 〈 p 〈 2＊ = 2N/（N-2） forN ≥ 3.

关键词： Sobolev spaces semilinear elliptic equations critical exponent Nehari manifold Palais-Smale condition.

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求二维半线性椭圆问题混合元解的二重网格法

高等学校计算数学学报 2008年第2期30卷 107-116页

作者：刘伟芮洪兴鲁东大学数学与信息学院山东大学数学与系统科学学院

1引言求解偏微分方程的混合元方法在实际问题(如油藏和地下水的数值模拟)中有广泛的应用.与标准的有限元相比,混合元方法可以同时求解两个变量即压力和流速的近似,而且能保持问题的局部守恒性,从而得到了更好的结果(如Arnold).现已有许... 详细信息

1引言求解偏微分方程的混合元方法在实际问题(如油藏和地下水的数值模拟)中有广泛的应用.与标准的有限元相比,混合元方法可以同时求解两个变量即压力和流速的近似,而且能保持问题的局部守恒性,从而得到了更好的结果(如Arnold).现已有许多文献使用混合元法求解线性的(如),拟线性的(如Milner)和非线性的(如Park)二阶椭圆问题.

关键词： mixed finite element method semilinear elliptic equations two-grid method.

On S-Shaped Bifurcation Curves for a Class of Perturbed semilinear equations

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Chinese Annals of Mathematics,Series B 2008年第6期29卷 641-662页

作者： Benlong XU Zhongliang WANG Department of Mathematics Shanghai Normal University Shanghai 200234 China

By making use of bifurcation analysis and continuation method, the authors discuss the exact number of positive solutions for a class of perturbed equations. The nonlinearities concerned are the so-called convex-concave functions and their behaviors may be asymptotic sublinear or asymptotic linear. Moreover, precise global bifurcation diagrams are obtained.

关键词： semilinear elliptic equations Exact multiplicity of solutions S-Shaped bifurcation curve

INFINITELY MANY SOLUTIONS FOR AN elliptic PROBLEM INVOLVING CRITICAL NONLINEARITY

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Acta Mathematica Scientia 2010年第6期30卷 2017-2032页

作者：曹道民严树森 Institute of Applied Mathematics AMSS Chinese Academy of Sciences Department of Mathematics The University of New England

We study the following elliptic problem：{-div（a（x）Du）=Q（x）｜u｜2-2u＋λu x∈Ω,u=0 onδΩ Under certain assumptions on a and Q, we obtain existence of infinitely many solutions by variational method.

关键词： semilinear elliptic equations infinitely many solutions variational method

MULTIPLICITY OF POSITIVE SOLUTIONS FOR A CLASS OF CONCAVE-CONVEX elliptic equations WITH CRITICAL GROWTH

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Acta Mathematica Scientia 2018年第2期38卷 497-518页

作者：廖家锋蒲洋唐春雷 School of Mathematics and Statistics Southwest University Chongqing 400715 China School of Mathematics and Information China West Normal University Nanchong 637002 China

In this article, the following concave and convex nonlinearities elliptic equations involving critical growth is considered,{-△u=g（x）｜u｜2＊-2u＋λf（x）｜u｜q-2u,x∈Ω u=0,x∈δΩ where Ω RN（N ≥ 3） is an open bounded domain with smooth boundary, 1 〈 q 〈 2, λ 〉 0. 2＊= 2N/N-2 is the critical Sobolev exponent, f ∈L2＊/2N/N-2 is nonzero and nonnegative, and g E （Ω） is a positive function with k local maximum points. By the Nehari method and variational method, k ＋ 1 positive solutions are obtained. Our results complement and optimize the previous work by Lin [MR2870946, Nonlinear Anal. 75（2012） 2660-26711.

关键词： semilinear elliptic equations critical growth positive solutions Nehari method variational method

THE EXISTENCE AND GLOBAL OPTIMAL ASYMPTOTIC BEHAVIOUR OF LARGE SOLUTIONS FOR A semilinear elliptic PROBLEM

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Acta Mathematica Scientia 2008年第3期28卷 595-603页

作者：张志军 School of Mathematics and Informational Science Yantai University

By Karamata regular variation theory and constructing comparison functions, the author shows the existence and global optimal asymptotic behaviour of solutions for a semilinear elliptic problem Δu = k（x）g（u）, u 〉 0, x ∈ Ω, u｜δΩ =＋∞, where Ω is a bounded domain with smooth boundary in R^N; g ∈ C^1[0, ∞）, g（0） = g＇（0） = 0, and there exists p 〉 1, such that lim g（sξ）/g（s）=ξ^p, ↓Aξ 〉 0, and k ∈ Cloc^α（Ω） is non-negative non-trivial in D which may be singular on the boundary.

关键词： semilinear elliptic equations explosive subsolutions explosive supersolutions existence the global optimal asymptotic behaviour