【作者】 陈莉萍；

【导师】 孙建平；

【作者基本信息】 兰州理工大学 ， 应用数学， 2023， 硕士

【摘要】 本硕士论文主要研究两类非局部问题解的存在性,共分为四章:第一章,首先对Kirchhoff方程和分数阶Choquard方程的背景和国内外研究现状做了叙述,接着介绍了本文的主要结果.第二章,介绍了本文中所用到的一些记号、定义及相关引理.第三章,研究如下一类具有变号权的Kirchhoff方程(?)的非平凡解,其中a,b>0,3<p<5,V(x)是一个连续的变号权并在无穷远处小于零.通过山路定理和Sobolev不等式证明了上述方程至少存在一个非平凡解.第四章,研究如下带有势函数的质量超临界分数阶Choquard方程(?)的正规化解,其中 N ≥ 3,s ∈(0,1),α ∈(N-2s,N),p ∈(1+α+2s/N,N+α/N-2s),μ 是拉格朗日乘子,势函数V∈C1(RN)且在无穷远处消失.当V满足某些条件假设时,利用分裂引理和广义的极小极大方法证明了上述方程正规化解的存在性.更多还原

【Abstract】 This master’s thesis mainly studies the existence of two kinds of nonlocal problem solutions,which are divided into four chapters:In Chapter 1,firstly,the background and research status of Kirchhoff equation and fractional Choquard equation are described,and then the main results of this paper are introduced.In Chapter 2,we briefly introduce some of notations,definitions and related lemmas used in this article.In Chapter 3,we research nontrivial solution of the following Kirchhoff equations with sign-changing weight(?) where a,b>0,3<p<5,V(x)is a continuous and sign-changing function and is less than zero at infinity.The existence of at least one nontrivial solution to the above equation is proved by using the mountain pass theorem and Sobolev inequality.In Chapter 4,we study normalized solution of the following mass supercritical fractional Choquard equation with potential function (?) where N>3,s∈(0,1),α∈(N-2s,N),p ∈(1+α+2s/N,N+α/N-2s),μ is a Lagrange multiplie,the potential function V ∈C1(RN)disappears at infinity.When V satisfies some conditional assumptions,the use of split lemma and generalized the minimax method of normal existence of the above equation.更多还原