【作者】 金花；

【导师】 刘文斌；

【作者基本信息】 中国矿业大学 ， 应用数学， 2018， 博士

【摘要】 分数阶微分方程是近年来非常活跃的一个研究领域,其具有深刻的物理背景和丰富的理论内涵,与几何学、泛函分析、量子力学、概率论等分支有着十分紧密的联系.相对局部微分方程问题的研究,非局部问题的处理要变得更加困难.自从Caffarelli和Silvestre引入了分数阶拉普拉斯算子的扩展定义之后,分数阶方程的正则性、极值原理等基本性质才得以建立,从而为各种非线性分析工具的引入打下了基础.本文主要研究具有临界指数的几类分数阶椭圆方程解的存在性、多解性及解的集中性.具体如下:第一部分,在没有单调性条件和(AR)条件下,研究了具有临界指数增长的分数阶Schr?dinger方程基态解的存在性.由于紧性的缺失以及(PS)序列有界性验证带来的困难,本文采用单调性技巧,利用辅助方程构造了原问题有界的(PS)序列,通过有界(PS)序列的分解获得紧性,完成了基态解的存在性证明.第二部分,研究了临界情况下分数阶奇异扰动问题解的存在性和集中性.由于嵌入紧性的缺失以及没有(AR)条件和单调性条件,本文通过截断技巧,利用全空间上Morse迭代得到极限问题基态解集的一致无穷模估计,将临界问题转化为次临界问题,再利用次临界问题解的存在性和集中性,得到临界奇异扰动问题解的存在性和集中性.第三部分,研究了具有临界指数的分数阶Kirchhoff方程解及多解的存在性.由于Kirchhoff项的出现,当维数N>4s时,山路结构不成立且(AR)条件不成立,本文利用扰动的方法得到有界的(PS)序列,进而证明解的存在性及随参数变化的渐近行为;另外,利用截断函数法、集中紧原理和环绕定理,得到一类临界分数阶Kirchhoff方程的多解性.第四部分,研究了具有Hardy-Littlewood-Sobolev临界指数的分数阶Choquard方程基态解的存在性.通过Hardy-Littlewood-Sobolev临界最佳嵌入的达到函数,得到了临界问题最低能量的上界估计.利用逼近的思想,得到了临界问题有界的(PS)序列,通过分解引理和紧性引理得到了临界问题非负径向对称基态解的存在性.更多还原

【Abstract】 The fractional differential equations have profound physical background and rich theoretical connotation.Fractional differential equations are closely related to geom-etry,functional analysis,quantum mechanics,probability theory,and are ones of the most active research fields in the last decade.In the study of fractional differential equations,the nonlocal problems are more difficult than the local problems.In the pi-oneering article of Caffarelli and Silvestre,the basic properties of non-local equations,such as regularity and extremum principle,have been established,which lay the foun-dation for the use of various nonlinear analysis tools.In this thesis,we study the existence,multiplicity and concentration of solutions for several fractional elliptic equations involving critical exponent.The main contents are as follows.In the first part,without the(AR)condition and monotonicity condition on the nonlinearity,we study the existence of ground states of fractional Schr?dinger equation with critical exponent.Because of the loss of compactness and difficulty to get the boundedness of(PS)-sequence,we apply the monotonicity trick and auxiliary equations to obtain a special bounded(PS)-sequence.Then by using of the decomposition of bounded(PS)-sequence,we recover the compactness and then obtain the existence of ground states.In the second part,we study the existence and concentration of the solutions for the singularly perturbed fractional equation with critical exponent.Because of the loss of compactness and without the(AR)condition and monotonicity condition,by the Moser iteration argument in the whole space R~N,we get a priori L~∞-estimate of ground states to the limit problem.Then we reduce the original problem to a subcritical problem by the truncation technique.By use of the existence and concentration of the subcrit-ical problem,we obtain the existence and the concentration of the critical singularly perturbed problem.In the third part,we study the existence and multiplicity of solutions for fractional Kirchhoff equations with critical exponent.Because of the presence of the Kirchhoff term,in high dimension N>4s,mountain pass geometry and the(AR)condition do not hold.We use the perturbation approach to get a bounded(PS)-sequence.Then we obtain the existence of the solutions and the asymptotic behavior of the solutions according to the parameters.In addition,by virtue of the truncation technique,the concentration compactness principle and a linking theorem,we obtain the multiplicity of solutions for a class of fractional Kirchhoff equations.In the final part,we discuss the existence of ground states of the fractional Choquard equation with Hardy-Littlewood-Sobolev critical exponent.In order to get the upper estimate of critical problem,we get the achieved function for the correspond-ing Sobolev imbedding and the corresponding best imbedding constant value.Using the idea of approximation,we get the bounded(PS)-sequence for the critical problem.Then,we present a splitting lemma.Together with the compactness lemma,we obtain the existence of nonnegative ground states for the critical problem.更多还原