【摘要】：本文概述作者承擔(dān)的國(guó)家自然科學(xué)基金項(xiàng)目所獲得的部分成果,特別是從源頭出發(fā)系統(tǒng)地培育強(qiáng)不定問(wèn)題變分方法的特色方向,并開(kāi)啟一些應(yīng)用問(wèn)題的研究,包括(1)建立強(qiáng)不定問(wèn)題的變分框架的基本方法;(2)建立局部凸拓?fù)渚�性空間的形變理論,相應(yīng)得到處理強(qiáng)不定問(wèn)題的臨界點(diǎn)定理;(3)首次研究非自治穩(wěn)態(tài)Dirac系統(tǒng)解的存在性,特別是突破強(qiáng)不定困難獲得其半經(jīng)典解的存在性、集中現(xiàn)象和指數(shù)衰減性;(4)首次得到非線性(非自治、無(wú)界Hamilton型)反應(yīng)-擴(kuò)散系統(tǒng)整體解的存在性和多重性,特別是奇異擾動(dòng)下其基態(tài)解的存在性、集中現(xiàn)象和衰減性;(5)深入研究Hamilton系統(tǒng)的同宿軌和Schr銉dinger方程的全局解;(6)其他初始性工作,如自旋流形上的Dirac方程的分歧現(xiàn)象.
[Abstract]:This paper summarizes some of the achievements of the National Natural Science Foundation project undertaken by the author, especially the characteristic direction of systematically cultivating the variational methods for strongly indefinite problems from the source, and opens the research on some application problems. The main contents are as follows: (1) the basic method of establishing a variational framework for strongly indefinite problems; (2) the deformation theory of locally convex topological linear space is established, and the critical point theorem for dealing with strongly indefinite problems is obtained. (3) the existence of solutions for nonautonomous steady-state Dirac systems is studied for the first time, especially the existence, concentration and exponential decay of semi-classical solutions are obtained by breaking through the strong indeterminacy. (4) for the first time, we obtain the existence and multiplicity of global solutions for nonlinear (nonautonomous, unbounded Hamilton type) reaction-diffusion systems, especially the existence, concentration and attenuation of ground state solutions under singular perturbations; (5) the global solutions of homoclinic orbits and Schr's dinger equations for Hamilton systems are studied in depth, and (6) other initiality work, such as bifurcation of Dirac equations on spheroidal manifolds.
【作者單位】： 中國(guó)科學(xué)院數(shù)學(xué)與系統(tǒng)科學(xué)研究院;中國(guó)科學(xué)院大學(xué)數(shù)學(xué)科學(xué)學(xué)院;
【基金】：國(guó)家自然科學(xué)基金(批準(zhǔn)號(hào):10831005,11331010,10421001和10640420049)資助項(xiàng)目
【分類號(hào)】：O175;O177

【相似文獻(xiàn)】

相關(guān)期刊論文 前5條

1 石鐘慈,許學(xué)軍;非對(duì)稱不定問(wèn)題的非協(xié)調(diào)多重網(wǎng)格法[J];計(jì)算數(shù)學(xué);1999年04期

2 王國(guó)棟;A~＊E＿0A與AE＿0A~＊的有定及不定問(wèn)題[J];云南大學(xué)學(xué)報(bào)(自然科學(xué)版);1994年04期

3 梁恒,白峰杉;對(duì)稱不定問(wèn)題的不精確Newton法[J];計(jì)算數(shù)學(xué);2002年03期

4 王文璞;彭振峗;;一類矩陣不定問(wèn)題有解的條件[J];計(jì)算技術(shù)與自動(dòng)化;2010年03期

5 ;[J];;年期

相關(guān)碩士學(xué)位論文 前1條

1 谷龍江;兩類強(qiáng)不定問(wèn)題的無(wú)窮多小能量解[D];蘭州大學(xué);2014年

，

本文編號(hào)：2365465