【作者】 何毅；

【导师】 李工宝；

【作者基本信息】 华中师范大学 ， 基础数学， 2015， 博士

【摘要】 本文主要研究几类含有奇异摄动的Kirchhoff型方程,拟线性Schrodinger方程以及Schrodinger-Poisson方程的解的存在性,集中性以及多解性.本文共分为五章：在第一章中,我们先概述本文所研究的问题的背景以及国内外的研究现状,并简要介绍本文所做的主要工作以及相关的预备知识和一些记号.在第二章与第三章中,我们研究了两类含奇异摄动的Kirchhoff型方程.在第二章中,我们研究了下述含有临界Sobolev指标的Kirchhoff型方程的解的存在性、集中性以及多解性,其中ε是一个足够小的正数,α,b是正常数,f∈C1(R+,R)且次临界增长(f(u)～up-1(4<p<6)),V:R3→R是一个局部Holder连续函数,通过运用文献[M. del Pino, P. L. Felmer, Calc. Var. Partial Differential Equations 4 (1996) 121-137]中提出的惩罚函数方法,我们首先证明当ε>0足够小时,上述方程具有一个指数衰减的弱解u。,并且,当ε→0时,uε集中于位势V的局部极小值点.根据极小极大定理和Ljusternik-Schnirelmann理论,我们通过研究位势V(x)的局部极小值点所构成的集合的拓扑结构得到了上述方程的一个多解性的结果.需要指出的是,为了克服含临界指标的非线性项u5的出现所引起的伸缩失紧,我们需要将泛函的能量降低至某一临界能量之下.在文献[J. Wang, L. Tian, J. Xu, F. Zhang, J. Differential Equations 253 (2012) 2314-2351]中, J.Wang et al.给出的临界能量是其中S是D1,2(R3)→L6(R3)的最佳Sobolev嵌入常数,但是c*不是最优的.在本章中,我们给出了这一类Kirchhoff型方程的确切的临界能量值：我们的结果是文献[X. He, W. Zou, J. Differential Equations 2 (2012)1813-1834]关于全空间上的Kirchhoff型方程次临界问题的存在性、集中性与多解性的结果的部分推广.在第三章中,我们研究下述含有临界Sobolev指标的Kirchhoff型方程的解的存在性与集中性.其中ε是一个足够小的正数,α,b是正常数,λ>0,2<p≤4,我们构造出一族解uε∈H1(R3),并且使得当ε→0时,u3将集中于位势V的局部极小值点.尽管,在文献[Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014),441-468]中,Y. He, G. Li和S. Peng研究过下述含有临界Sobolev指标的Kirchhoff型方程其中f(u)～up-1(4< p< 6),并且满足Ambrosetti-Rabinowitz条件(以下简称(AR)条件),从而保证了Palais-Smale序列(以下简称(PS)序列)的有界性.但是,在本章中,g(u):=λ|u|p-2u+|u|4u(2<p≤4)不满足(AR)条件((?)μ>4,0<μ∫0ug(s)ds≤g(u)u),在证明正解的存在性的过程中,寻找有界的(PS)序列变成了一个主要的困难.另外,g(s)/s3(s>0)不具备严格增性也导致了Nehari流形无法引入.我们的结果是文献[Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014),441-468]的关于f(u)～|u|p-2u(4<p<6)的存在性与集中性结果的推广.在第四章中,我们研究下列含临界Sobolev指标的Schrodinger-Poisson方程:的解的存在性与集中性.其中ε是一个足够小的正数,λ>0,3<p≤4,我们构造出一族解uε∈H1(R3),并且使得当ε→0时,uε将集中于位势V的局部极小值点.尽管,对于含次临界增长的Schrodinger-Poiss(?)n方程非线性项.f(u)～|u|p-2u(4<p<6)满足(AR)条件,容易获取有界的(PS)序列的情形已经被广泛地研究了,但是本章考虑的带临界项的情形更为复杂.由于9(u)：=λ|u|p-2u+|u|4u(3<p≤4)不满足(AR)条件((?)μ>4,0<μ∫0ug(s)ds≤g(u)u),在证明正解的存在性的过程中,寻找有界的(PS)序列变成了一个主要的困难.另外,g(s)/s3(s>0)不具备严格增性也导致了Nehari流形无法引入.我们的结果是新的.在第五章中,我们研究下列含临界Sobolev指标的拟线性Schrodinger方程的解的存在性、集中性以及多解性.其中ε是一个足够小的正数,N≥3,2*=2N/(n-2),4<q<2·2*,min V>0,inf W>0,在某种适当的条件下,我们将证明上述方程的解的存在性与集中现象,再利用极小极大定理和Ljusternik-Schnirelmann理论,我们通过研究位势V的全局极小值点所构成的集合与位势W的全局极大值点所构成的集合的拓扑结构来得到上述方程的多解性的结果.我们的结果是文献[X.He,A.Qian,W.Zou,Nonlinearity 26(2013), 3137-3168]的结果的部分推广.文献[X.He,A.Qian,W.Zou,Nonlinearity 26(2013),313-3168]考虑了下述带奇异摄动的拟线性Schrodinger方程其中h(u)次临界增长,V(x)满足著名工作文献[P.Rabinowitz,Z.Angew.Math. Phys.43(1992)270-291]的位势条件：更多还原

【Abstract】 In this paper, we mainly study the existence, concentration and multiplicity of nontrivial solutions to several classes of singularly perturbed Kirchhoff type equa-tions, Schrodinger-Poisson equations and quasilinear Schrodinger equations.The thesis consists of six chapters:In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In Chapter Two and Chapter Three, we study two classes of singularly per-turbed Kirchhoff type equations.In Chapter Two, we study the existence, concentration and multiplicity of weak solutions to the Kirchhoff type equation with critical Sobolev growth where ε is a small positive parameter and a, b> 0 are constants,f∈C1(R+,R) is subcritical, V:R3→R is a locally Holder continuous function. We first prove that for ε>0 sufficiently small, the above problem has a weak solution uε with exponential decay at infinity by using penalization method due to [M. del Pino, P. L. Felmer, Calc. Var. Partial Differential Equations 4 (1996) 121-137]. Moreover, uε concentrates around a local minimum point of V as ε→0. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple solutions by employing the topological construction of the set where the potential V(x) attains its local minima.We point out that to overcome the obstacle due to the appearance of the critical nonlinearity u5, we need to pull the energy level down below a certain critical level. In [J. Wang, L. Tian, J. Xu, F. Zhang, J. Differential Equations 253 (2012) 2314-2351], J. Wang el al. give a critical level where S is the best Sobolev constant of the imbedding D1,2(R3)→L6(R3). But the energy level c* is not the best. In this chapter, we prove that the precise threshold value for the Kirchhoff type equation is: Our result can be viewed as a partial extension of [X. He, W. Zou, J. Differential Equations 2 (2012) 1813-1834] concerning Kirchhoff type equations with subcritical nonlinearities.In Chapter Three, we are concerned with the following Kirchhoff type equation with critical nonlinearity: where ε is a small positive parameter, a, b>0, λ>0,2<p≤4. Under certain assumptions on the potential V, we construct a family of positive solutions uε∈ H1(R3) which concentrates around a local minimum of V as ε→0.Critical growth Kirchhoff type problem has been studied in [Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014), 441-468], where the assumption for f(u) is that f(u)～|u|p-2u with 4<p<6 and satisfies the Ambrosetti-Rabinowitz condition ((AR)condition in short) which forces the boundedness of any Palais-Smale sequence ((PS) condition in short) of the corresponding energy functional of the equation. As g(u):=λ|u|p-2u+|u|4u with 2<p≤4 does not satisfy the (AR) condition ((?)μ>4,0<μ∫0ug(s)ds≤g(u)u), the boundedness of (PS) sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function g(s)/s3 is not increasing for s>0 prevents us from using the Nehari manifold directly as usual. Our result extends the main result in [Y. He, G. Li, S. Peng, Adv. Nonlinear Stud.14 (2014),441-468] concerning the existence and concentration of positive solutions to the case where f(u)～|u|p-2u with 4<p<6.In Chapter Four, we are concerned with the following Schrodinger-Poisson e-quation with critical nonlinearity: where ε> 0 is a small positive parameter, λ>0,3<p≤4. Under certain assumptions on the potential V, we construct a family of positive solutions uε∈ H1(R3) which concentrates around a local minimum of V as ε→0.Subcritical growth Schrodinger-Poisson equation has been studied extensively, where the assumption for f(u) is that f(u)～|u|p-2u with 4<p<6 and satisfies the (AR) condition which forces the boundedness of any (PS) sequence of the corresponding energy functional of the equation. The more difficult critical case is studied in this paper. As g(u):=λ|u|p-2u+|u|4u with 3<p≤4 does not satisfy the (AR) condition ((?)μ>4,0<μ∫0ug(s)ds≤g(u)u), the boundedness of (PS) sequence becomes a major difficulty in proving the existence of a positive solution. Also, the fact that the function s3/g(s) is not increasing for s>0 prevents us from using the Nehari manifold directly as usual. The main result we obtained in this paper is new.In Chapter Five, we study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrodinger equation with critical Sobolev growthwhere ε is a small positive parameter, N≥3,2*=N-2/2N,4<q<2·2*, min V> 0 and inf W>0. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials V(x) attains its minima and W(x) attains its maxima.Our result can be seen as a partial extension of [X. He, A. Qian, W. Zou, Non-linearity 26 (2013),3137-3168] concerning the following type singularly perturbed quasilinear Schrodinger equationwhere h(u) is of subcritical growth and the potential V(x) satisfies the following condition due to the celebrated work [P. Rabinowitz, Z. Angew. Math. Phys.43 (1992) 270-291]:更多还原