一类奇异临界双调和椭圆方程的群不变解

讨论一类含有Hardy-Sobolev临界指数项的奇异双调和椭圆方程,应用Lions集中紧性原理、Palais对称临界原理、Hardy-Rellich型不等式和变分方法,证明了方程在适当条件下群不变解的存在性和多重性.     

二维Newton-Boussinesq方程周期初值问题经典解的整体存在性

研究一类二维Newton-Boussinesq方程的周期初值问题,将方程转化为积分方程,用不动点原理得到解的局部存在性,用能量估计的方法证明经典解的整体存在性．     

具量子效应Zakharov方程弱解的存在性

Zakharov方程具有丰富的物理背景．通过Arzela－Ascoli定理、Faedo－Galerkin方法和紧性原理,得到等离子体模型中具量子效应Zakharov方程弱整体解的存在性．     

一个早期肿瘤生长模型的数学分析

研究了乳腺癌的早期生长模型（DCIS）,它为耦合了抛物方程、椭圆方程的自由边界问题,运用椭圆型方程的变分理论、抛物方程的L^p理论和压缩映照原理,证明了这个问题局部解的存在惟一性,然后用延拓方法得到了整体解的存在惟一性。     

一般形式的一阶椭圆组的非线性Riemann问题

讨论一般形式的一阶线性和拟线性椭圆型方程的非线性Riemann问题,通过把这些问题转化为奇异积分方程,利用压缩原理和广义压缩原理来证明在某些假设条件下所讨论问题的解的存在性．     

电报方程双周期解的极大值原理与强正性估计及应用

本文讨论非线性电报方程u_(tt)-u_(xx)+cu_t=F(t,x,u),(t,x)∈R~2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。     

含临界指数p-Kirchhoff型方程的非平凡解

本文研究了一类含临界指数的p-Kirchhoff型方程.利用变分方法与集中紧性原理,通过证明对应的能量泛函满足局部的(PS)_c条件,得到了这类方程非平凡解的存在性,推广了关于Kirchhoff型方程的相关结果.     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

双调和方程边值问题正解的研究

本文以不动点原理为主要工具 ,证明了一类双调和方程正解的存在性 .     

一类GBBM方程的柯西问题

研究一类非线性拟抛物方程的柯西问题.首先利用与原方程等价的积分方程以及压缩映像原理得到了问题的局部W~(k,p)解的存在性,而后在某些假设下得到了问题W~(k,p)解的整体存在性,最后利用凸性方法得到了解的有限时间内爆破.     

A pre-order principle and set-valued Ekeland variational principle

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results.     

P-distances, q-distances and a generalized Ekeland’s variational principle in uniform spaces

In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle. In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance), which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions of the principle and their improvements.     

Ekeland's variational principle in locally convex spaces and the density of extremal points

In this paper, we prove a general version of Ekeland's variational principle in locally convex spaces, where perturbations contain subadditive functions of topology generating seminorms and nonincreasing functions of the objective function. From this, we obtain a number of special versions of Ekeland's principle, which include all the known extensions of the principle in locally convex spaces. Moreover, we give a general criterion for judging the density of extremal points in the general Ekeland's principle, which extends and improves the related known results.     

H型群上$p$-次Laplace算子的Hopf型引理和强极大值原理

本文首先推广了Capogna,Danielli和Garofalo关于p-次Laplace算子的径向解的一个重要公式,然后通过改进欧氏空间中证明Laplace算子的Hopf引理的方法,证明了H型群上p-次Laplace算子的Hopf型引理,进而证明了一个强极大值原理。     

Generalized Gerstewitz's Functions and Vector Variational Principle for-Efficient Solutions in the Sense of Németh

In this paper, we first generalize Gerstewitz's functions from a single positive vector to a subset of the positive cone. Then, we establish a partial order principle, which is indeed a variant of the pre-order principle [Qiu, J. H.: A pre-order principle and set-valued Ekeland variational principle.J. Math. Anal. Appl., 419, 904–937(2014)]. By using the generalized Gerstewitz's functions and the partial order principle, we obtain a vector EVP for-efficient solutions in the sense of N′emeth, which essentially improves the earlier results by completely removing a usual assumption for boundedness of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.     

On some nonlinear problems induced by an abstract maximal element principle

In this paper, by applying an abstract maximal element principle on quasi-ordered sets established by Lin and Du, we obtain a generalized Brézis–Browder principle, system (vectorial) versions of Ekeland's variational principle and maximal element principle and a vectorial version of Takahashi's nonconvex minimization theorem. Moreover, we investigate the equivalence between scalar versions and vectorial versions of these results.     

Symmetry principles via interactions and symmetry-violations

In this work we would like to continue our ideas concerning the principle mentioned at the end of [Appl. Math. Comput. 113 (2000) 289]. From now on we will call this principle the principle of symmetry of states. We recall that:Principle of symmetry of states. Any system whose states have the same symmetry can take part in “interaction-chain”, until the symmetry of the states will be maximal and the system will be stable. In this case, the system has a minimal number of different states. Each “intermediate system” can be viewed like a homomorphic image of the anterior.We recall the fact that we have used the automata theory for modelling the systems. The “homomorphic image” words mentioned in the statement of our principle refer to automata-homomorphism. We mention also that we have applied this principle concerning radioactivity, making a conjecture, that we have considered each radioactive nucleus like an automaton and for each radioactive chain (viewed as an interaction-chain) applied our principle.Our purpose in this work is to try to find other examples of interactions which are in concordance with our principle (are all the interactions in concordance with this principle ?).     

An invariance principle for discrete-time nonlinear systems

In this paper, we derive an invariance principle generalizing LaSalle's invariance principle for discrete-time nonlinear systems. Next, using the invariance principle, we develop a series of results relating stability, observability, and converse Lyapunov theorems for discrete-time nonlinear systems.     

A revised pre-order principle and set-valued Ekeland variational principles with generalized distances

In my former paper "A pre-order principle and set-valued Ekeland variational principle"(see [J. Math. Anal. Appl., 419, 904–937(2014)]), we established a general pre-order principle.From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles(denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy's EVP in [On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396(2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle,but also can be applied to obtain the above Khanh and Quy's EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.     

A molecular method applied to a non-local PDE in stratified Lie groups

In this article we study a transport-diffusion equation in the framework of the stratified Lie groups. For this equation we will study the existence of the solutions, a maximum principle, a positivity principle and Hölder regularity.