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Science China Mathematics 2015年第11期58卷 2301-2316页

作者： GUO ZongMing ZHOU Feng Department of Mathematics Henan Normal University Center for PDEs and Department of Mathematics East China Normal University

A monotonicity formula for stable solutions to a class of weighted semilinear elliptic equations with ＂negative exponent＂ is established. It is well known that such a monotonieity formula plays an essential role in the study of finite Morse index solutions of equations with ＂positive exponent＂. Unlike the positive exponent case, we will see that both the monotonicity formula and the sub-harmonicity play crucial roles in classifying positive finite Morse index solutions to the equations with negative exponent and obtaining sharp results for their asymptotic behaviors.

关键词： sub-harmonicity monotonicity formula singular nonlinearity finite Morse index solutions

monotonicity formulas of EF-Critical Maps with Potential

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Journal of Mathematical Research with Applications 2015年第6期35卷 681-691页

作者： Yingbo HAN College of Mathematics and Information Science Xinyang Normal University

In this paper, we introduce the notion of EF-critical map with potential with respect to the functional EF,H(u). By using the stress-energy tensor, we obtain some monotonicity formulas and vanishing results for these ... 详细信息

关键词： EF-critical map with potential stress-energy tensor monotonicity formula

New monotonicity formulae for Semi-linear Ellipti and Parabolic Systems

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Chinese Annals of Mathematics,Series B 2010年第3期31卷 411-432页

作者： Li MA Xianfa SONG Lin ZHAO Department of Mathematical Sciences Tsinghua University Beijing 100084 China. Department of Mathematics Tianjin University Tianjin 300072 China.

The authors establish a general monotonicity formula for the following elliptic system △ui＋fi（x,ui,…,um）=0 in Ω,where Ω belong to belong to R^n is a regular domain, （fi（x, u1,... ,um）） = △↓F（x,→↑u）, F（x,→↑u ） is a given smooth function of x ∈ R^n and →↑u = （u1,…,um） ∈ R^m. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic systemδtui-△ui-fi（x,ui,…,um）=0 in（ti,t2）×R^n,where t1 〈 t2 are two constants, （fi（x,→↑u ） is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonieity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.

关键词： Elliptic systems Parabolic system monotonicity formula Ginzburg-Landau model

Finite Morse Index Solutions of a Nonlinear Schr?dinger Equation

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Acta Mathematica Sinica,English Series 2023年第3期39卷 513-522页

作者： Phuong LE Faculty of Economic Mathematics University of Economics and LawHo Chi Minh CityVietnam Vietnam National University Ho Chi Minh CityVietnam

We prove Liouville type theorems for stable and finite Morse index H_(loc)^(1)∩L_(loc)^(∞)solutions of the nonlinear Schrodinger equation -Δu+λ|x|^(a)u=|x|b|^(u)|^(p-1)u in R^(N),where N≥2,λ>0,a,b>-2 and p>1,Our analysis reveals that all stable solutions of the equation must be zero for all p>1,Furthermore,finite Morse index solutions must be zero if N≥3 an p≥(N+2+2b)/(N-2).The main tools we use are integral estimates,a Pohozaev type identity and a monotonicity formula.

关键词： Schrodinger equation Liouville type theorems stable solutions finite Morse index solutions monotonicity formula

Partial energies monotonicity and holomorphicity of Hermitian pluriharmonic maps

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Science China Mathematics 2013年第5期56卷 1019-1032页

作者： YANG GuiLin HAN YingBo DONG YuXin School of Mathematical Sciences Fudan University Key Laboratory of Mathematics for Nonlinear Sciences Ministry of EducationFudan University College of Mathematics and Information Science Xinyang Normal University

In this paper, we introduce the notion of Hermitian pluriharmonic maps from Hermitian manifold into Kiihler manifold. Assuming the domain manifolds possess some special exhaustion functions and the vecotor field V = JMδJM satisfies some decay conditions, we use stress-energy tensors to establish some monotonicity formulas of partial energies of Hermitian pluriharmonic maps. These monotonicity inequalities enable us to derive some holomorphicity for these Hermitian pluriharmonic maps.

关键词： stress energy tensor monotonicity formula Hermitian pluriharmonic map holomorphic map

Gradient Flow of the L_(β)-Functional

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Communications in Mathematical Research 2021年第1期37卷 113-140页

作者： Xiaoli Han Jiayu Li Jun Sun Department of Mathematical Sciences TsinghuaUniversityBeijing 100084China School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026 AMSS CASBeijing 100190China School of Mathematics and Statistics Wuhan UniversityWuhan 430072China Hubei Key Laboratory of Computational Science Wuhan UniversityWuhan 430072China

In this paper,we start to study the gradient flow of the functional L_(β) introduced by Han-Li-Sun in[8].As a first step,we show that if the initial surface is symplectic in a Kähler surface,then the symplectic property is preserved along the gradient *** we show that the singularity of the flow is characterized by the maximal norm of the second fundamental *** β=1,we derive a monotonicity formula for the *** applications,we show that the l-tangent cone of the flow consists of the finite flat planes.

关键词： β-symplectic critical surfaces gradient flow monotonicity formula tangent cone