含临界位势与临界参数的超线性椭圆型方程

本文考虑一类含临界位势与临界参数的超线性椭圆型方程解的存在性.本文应用Morse理论,考虑非线性项f(x,s)在零点附近以及无穷远处的性质,给出了方程在某个新的Sobolev-Hardy空间中解的存在性.     

A BIHARMONIC EIGENVALUE PROBLEM AND ITS APPLICATION＊

In this article,we focus on the eigenvalue problem of the following linear biharmonic equation in RN:△2u-αu+λg(x)u =0 with u ∈ H2(RN),u ≠ 0,N ≥ 5. (*)Note that there are two parameters α and λ in it,wh...     

基于ADI格式的多尺度图像分解

针对文献[Xu R et al.,IEEE Trans.Biomed.Eng.,2014]的多尺度图像分解模型,该文提出了Alternating Direction Implicit (ADI)格式下的多尺度图像分解算法,并证明了在该模型下ADI格式的收敛性和稳定性,进一步,通过对不同图像的数值实验,验证了该文提出的算法具有更好的纹理提取效果.     

一类相对非线性薛定谔方程解的存在性

利用临界点理论考虑了一类相对非线性薛定谔方程,主要通过变量代换将相对非线性薛定谔方程转化成半线性椭圆型方程.首先考虑位势函数为零时,将经典的场方程结果推广到了相对非线性薛定谔方程;而后利用临界点理论得到了有界位势情形方程非平凡解的存在性,在此情形,改进了文献[12-13]中的超线性条件.     

MULTIPLE SOLUTIONS FOR ASYMPTOTICALLY LINEAR ELLIPTIC EQUATIONS INVOLVING NATURAL GROWTH TERM

In this article, under some conditions on the behaviors of the perturbed function f(x, s) or its primitive F(x,s) =∫so f(x,t)dt near infinity and near zero, a class of asymptotically linear elliptic equations involving natural growth term is studied. By computing the critical group, the existence of three nontrivial solutions is proved.     

含约束临界Kirchhoff型约束变分问题极小解的存在性及其极限行为

非线性Kirchhoff型约束变分问题当非线性项只含一个幂次项且指数为约束临界p=2+8/N时,由现有文献可知该问题不存在极小解.本文考虑了含低阶扰动项和约束临界指数项的Kirchhoff型约束变分问题,利用伸缩技巧、集中紧原理和Pohozaev恒等式,得到了扰动项指数和系数对该变分问题极小解存在性的影响,并证明该极小解是相对应的Kirchhoff方程的基态解.进一步,本文通过精细的能量估计,探讨扰动项指数趋于约束临界指数时极小能量和极小解的极限行为.     

分数阶Kirchhoff型约束变分问题Schwarz对称极小解的存在性

该文主要讨论一般非线性项情形下一类分数阶Kirchhoff型约束变分问题在空间H^s(R^N)上极小解的存在性和Schwarz对称性.利用单调递减重排不等式和伸缩技巧,当非线性项满足合适的条件时,该文细致的讨论了相应约束变分问题在空间H^s(R^N)上极小解的存在性与非线性项的指数和约束f_R^N|u|²dx=c²中参数c的关系.     

一类修正Gross-Pitaevskii方程基态解的存在性

该文利用伸缩变换结合重排不等式等技巧得到了修正Gross-Pitaevskii方程对应极小化问题极小元的存在性与非线性项指数p的依赖关系.当2

0,极小化问题存在极小元.若p=2+4/N且c≤‖φ‖₂或者c>(3/2)^N/4‖φ‖₂(‖φ‖₂的定义见第一节)或p> 2+4/N,问题不存在极小解.而对于p=2+4/N且‖φ‖₂ N/4‖φ‖₂,不知道是否存在极小解.     

EXISTENCE AND STABILITY OF STANDING WAVES FOR A COUPLED NONLINEAR SCHRDINGER SYSTEM

We study the existence and stability of the standing waves of two coupled Schrdinger equations with potentials |x|bi(bi ∈ R, i = 1, 2). Under suitable conditions on the growth of the nonlinear terms, we first establish the existence of standing waves of the Schrdinger system by solving a L2-normalized minimization problem, then prove that the set of all minimizers of this minimization problem is stable. Finally, we obtain the least energy solutions by the Nehari method and prove that the orbit sets of these least energy solutions are unstable, which generalizes the results of [11] where b1= b2= 2.