两类p-Kirchhoff型方程的正解

发布时间：2018-04-08 19:26

本文选题：p-Kirchhoff型方程　切入点：变分方法　出处：《太原理工大学》2017年硕士论文

【摘要】：本文利用变分方法研究了两类P-Kirchhoff型方程正解的存在性.首先,研究了一类带有临界指数的P-Kirchhoff型方程正解的存在性.其次,研究了一类带参数的p-Kirchhoff型方程解的存在性,不存在性和指数衰减性.主要理论依据是山路引理、集中紧性原理、隐函数定理、最优化理论和比较原理.在第三章,研究了下列带有临界指数的p-Kirchhoff型方程定理1.1如果下列情形之一满足,在第四章,研究了下列带参数的p-Kirchhoff型方程主要结果如下:定理1.2如果(V_1),(V_2),(f_1)和(f_2)满足,则(ⅰ)存在b*0,当b ∈(0,b*)时,方程(P_b)至少有一个正解,且在无穷远处呈指数衰减,即存在常数C,R,τ0,使得|x|R时,u(x)≤Ce-τ|x|.(ⅱ)当2Pp*时,存在b0,使得b ≥b时,方程(P_b)不存在非平凡解.全文结构如下:第一章介绍了近年来Kirchhoff型方程的研究进展以及本文的主要结果.第二章主要给出了本文所用到的变分方法的相关知识.第三章首先给出了带临界指数的p-Kirchhoff型方程的变分结构,其次利用变分方法和集中紧性原理证明了该方程正解的存在性.第四章首先给出了带参数的P-Kirchhoff型方程的变分结构,其次利用变分方法和比较原理证明了当参数b充分小时方程至少存在一个正解且在无穷远处呈指数衰减;当2pp*参数b充分大时方程不存在非平凡解。
[Abstract]:In this paper, the existence of positive solutions for two classes of P-Kirchhoff type equations is studied by means of variational method.Firstly, the existence of positive solutions for a class of P-Kirchhoff type equations with critical exponents is studied.Secondly, the existence, non-existence and exponential decay of solutions for a class of p-Kirchhoff type equations with parameters are studied.The main theoretical basis is mountain pass Lemma, centralization compactness principle, implicit function theorem, optimization theory and comparison principle.The structure of the paper is as follows: in chapter 1, the research progress of Kirchhoff type equations and the main results of this paper are introduced.The second chapter gives the knowledge of the variational method used in this paper.In chapter 3, the variational structure of the p-Kirchhoff type equation with critical exponents is first given, and the existence of positive solutions of the equation is proved by means of the variational method and the principle of centralization compactness.In chapter 4, the variational structure of the P-Kirchhoff type equation with parameters is first given. Secondly, the variational method and comparison principle are used to prove that the equation has at least one positive solution and exponential decay at infinity when the parameter b is sufficiently small.When the 2 pp* parameter b is sufficiently large, there is no nontrivial solution for the equation.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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