变分方法在几类Kirchhoff型椭圆方程中的应用

发布时间：2018-05-21 05:24

本文选题：Kirchhoff型椭圆方程 + 次临界增长项　；参考：《西南大学》2016年博士论文

【摘要】：本学位论文主要是利用变分方法和一些分析技巧研究了全空间中三类Kirchhoff型椭圆方程正解的存在性或多重性.在第1章中,我们主要介绍了Kirchhoff型椭圆方程具有的物理背景、研究现状、一些工具性定理和引理以及论文结构安排.在第2章中,我们考虑下列具有零质量和临界项的Kirchhoff型椭圆方程一些恰当的条件时,我们得出方程正解的存在性、多重性以及不存在性.在第3章中,我们考虑下列具有一般次临界项的Kirchhoff型椭圆方程其中a0,b0,λ≥0并且g满足一般的次临界增长条件.当λ很小时,我们得出方程具有一个正解.这推广了Li等人[Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations 253 (2012), no.7,2285-2294.]的结果.当λ很大时,我们得出方程没有非平凡解.在第4章中,我们考虑下列具有次临界项或临界项的Kirchhoff型椭圆方程其中a0,b0,4q≤6, g满足次临界增长条件并且V,K,g关于x是渐近周期的函数.首先,我们得出方程,其中4q6,正基态解的存在性.然后,让指数q→6-,我们得出方程,其中q=6,正基态解的存在性.
[Abstract]:In this thesis, the existence or multiplicity of the positive solutions of three types of Kirchhoff type elliptic equations in the whole space is studied by using the variational method and some analytical techniques. In the first chapter, we mainly introduce the physical background of the Kirchhoff elliptic equation, the present research status, some theorems and lemmas and the arrangement of the structure of the paper. In the second chapter, when we consider the following appropriate conditions for the Kirchhoff elliptic equation with zero mass and critical term, we obtain the existence, multiplicity and nonexistence of the positive solution of the equation. In the third chapter, we consider the following Kirchhoff elliptic equations with general subcritical terms, including A0, B0, lambda > 0 and G satisfying the general order The critical growth condition. When lambda is very small, we conclude that the equation has a positive solution. This extends Li et al. [Existence of a positive solution to Kirchhoff type problems without compactness. In the fourth chapter, we consider the following elliptic equation of Kirchhoff type with subcritical term or critical term in which A0, b0,4q < 6, G satisfies the subcritical growth condition and V, K, g about X is a function of asymptotic period. First, we get the equation, in which 4q6, the existence of the positive basic state solution. Then, the exponential Q to 6-, we get the equation, where q=6, positive The existence of the ground state solution.
【学位授予单位】：西南大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.25

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