【作者】 周静；

【导师】 彭双阶；

【作者基本信息】 华中师范大学 ， 基础数学， 2017， 博士

【摘要】 本文主要研究几类非线性椭圆型方程解的存在性.全文共分四章:在第一章中,我们主要阐述本文所讨论问题的背景及研究现状,并简要介绍本文的主要工作.在第二章中,我们研究下述带有对称位势函数的χ(2)二次谐波SHG(Second Harmonic Generation)系统:同步正解的存在性,其中2 ≤N<6,μ>0且≥ 7.我们建立了该系统的非退化性.有了这个系统的非退化性,我们利用Liapunov-Schmidt约化构造出该系统的无穷多个非径向对称的同步正解.在第三章中,我们考虑在≤N<6)中的带有非对称位势的χ(2)二次谐波系统:其中位势函数’P(x),Q(x)是满足某适当退化性的连续函数,而且不需要任何对称性质,ε为一正常数,μ和γ都是参数.我们对具有非对称位势函数的问题提出了新的结论,使用方法有别于前一章.主要利用Liapunov-Schmidt约化方法.目前我们有两个主要的困难.首先,我们要证明极大值点不会跑到无穷远处,这点可以由对于位势函数的慢衰减性假设可以保证.其次,当波峰靠近位形空间的边界时,我们要注意到其能量的差.这个关键的估计将在一个引理中给出.在引理中我们给出了从第m步到第(m + 1)步所产生的累积误差是可控的.在第四章中,我们主要考虑分数阶的带有Hardy位势的非局部方程其中μ≥0满足>(?)>2N/N-6s;a为一正常数,且(-Δ)s表示的是在Ω上的带有零Dirichlet边界(?)Ω条件的分数阶拉普拉斯算子.我们利用逼近法得到了该问题的无穷多解的存在性.更多还原

【Abstract】 In this thesis,we mainly deal with the problems on the existence of solutions for a χ(2)SHG system.There are three chapters in this thesis.In Chapter One,we summarize the background of the related problems and state the main results of the present thesis.In Chapter Two,we consider about the solutions for the following χ(2)SHG system where 2≤N<6,μ>0 and μ>γ.We establish the non-degeneracy of this problem.With the knowledge of the non-degeneracy of this system,we construct many non-radial symmetric synchronized positive solutions,by utilizing Liapunov-Schmidt reduction.The Chapter Three is concerned with the following χ(2)SHG(Second Harmonic Generation)system in RN(2≤N<6),where the potentials P(x),Q(x)are continuous functions satisfying suitable decay assumptions,but without any symmetry properties,∈ is a positive constant,μ andβ are some parameters.We mainly use the Liapunov-Schmidt reduction method.There are two main difficulties.Firstly,we need to show that the maximum points will not go to infinity.This is guaranteed by the slow decay assumption.Secondly,we have to detect the difference in the energy when the spikes move to the boundary of the configuration space.A crucial estimate will be given in a Lemma,in which we prove that the accumulated error can be controlled from step m to step(m + 1).In Chapter four,by an approximating argument,we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth provided N>6s,μ>0,0<s<1,2s*=2N/N-2s,a>0 is a constant and Ω is an open bounded domain in RN which corntains the origin.更多还原