几类椭圆型方程组解的存在性

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摘要

本文主要研究了几类非线性椭圆型方程组解的存在性问题，运用的工具主要

有不动点定理，直接变分法，山路引理，Pohozaev等式等.

作者首先利用不动点定理证明了拟线性椭圆型方程组径向正解的存在性问题

和三个径向正解的存在性问题. 其次利用 Pohozaev 等式研究了拟线性椭圆型方程

组临界和超临界指标情况下解的不存在性问题；接着利用 Lagrange 乘子定理证明

了拟线性椭圆型方程组次临界情况下解的存在性；然后利用山路引理证明了次临

界和临界情况下解的存在性问题. 最后利用 Lagrange 乘子定理证明了一类

(p,q)-Laplace 非线性椭圆型方程组极小能量解的存在性问题.

本文主要分为四章：在第一章，主要介绍了本文的一些主要历史背景和本文

的一些主要结果.

在第二章，作者通过锥上不动点定理研究了非线性椭圆方程组正径向解的存

在性及三个正径向解的存在性问题. 本章主要是把微分方程转化为积分方程，这

样就可以把求微分方程的弱解问题转化为求积分方程的不动点问题，从而利用锥

上不动点定理解决了原问题.

在第三章，作者通过 Pohozaev 等式，证明了一类非线性椭圆型方程组临界和

超临界情况下解的不存在性问题.然后又利用 Lagrange 乘子定理证明了这个方程

组次临界情况下解的存在性，接着利用山路引理证明了这个方程组次临界情况下

解的存在性以及临界情况下解的存在性.

在第四章，主要利用直接变分原理，通过 Nehari 流形的方法证明了

(p,q)-Laplace 方程组的耦合项相互分离时极小能量解的存在性问题.

关键词：正径向解全连续算子 Pohozaev 等式 Lagrange 乘子山路引

理 Nehari 流形极小能量解

ABSTRACT

In this thesis, we studied the existence of solutions for some kinds of elliptic

systems. We also studied the nonexistence of solutions for a kind of nonlinear elliptic

systems with overcritical and subcritical states. Some important tools are applied in this

paper, which are fixed point theorem, variational method, Mountain pass Lemma, and

Pohozaev identity.

In this paper, we study the existence of radially positive solutions for a kind of

nonlinear elliptic systems with nonhomogeneous conditions and the existence of three

radially positive solutions for this kind of nonlinear elliptic systems. Secondly, we

mainly use Pohozaev identity to prove nonexistence of solutions for a kind of nonlinear

elliptic systems with overcritical and subcritical states. Via Lagrange multiplier theorem,

we studied existence of solution for this systems with subcritical states. Using Mountain

pass Lemma, we studied the existence of solution for this systems with critical and

subcritical states. Finally, we studied the existence of nontrivially least energy solutions

for a class of nonlinear (p, q)-Laplace elliptic systems.

This paper consists of four parts. In chapter one, we talked about the main

backgrounds and results of the paper.

In chapter two, using fixed point theorem, we study the existence of radially

positive solutions for a kind of nonlinear elliptic systems with nonhomogeneous

conditions and the existence of three radially positive solutions for this kind of

nonlinear elliptic systems. Transferring the differential equations to the integral

equations, we can obtain that the fixed points of the integral equations are equal to the

solutions of the nonlinear elliptic systems. Using the fixed point theorem, we can obtain

the fixed points of the integral equations, which are the solutions of the differential

equations.

In chapter three, firstly, we use the Pohozaev identity to prove nonexistence of

solutions for a kind of nonlinear elliptic systems with overcritical and subcritical states.

Secondly, via Lagrange multiplier theorem, we studied the existence of solutions for the

kind of nonlinear elliptic systems with subcritical states. Finally, using Mountain pass

lemma, we studied the existence of solutions for this systems with critical and

subcritical states.

In chapter four, using directly variational method, we prove the existence of

nontrivially least energy solution for a class of nonlinear (p, q)-Laplace elliptic systems

with the coupling term separated.

Key Words: Radial solution, Completely continuous operator, Lagr-

ange multiplier, Pohozaev identity, Mountain pass lemma, Nehari me-

thod, Least energy solution

中文摘要

ABSTRACT

第一章绪论............................................................................................................1

§1.1 有关历史背景............................................................................................1

§1.2 本文工作....................................................................................................3

第二章非线性椭圆方程组的三个正径向解的存在性..........................................6

§2.1 引言..........................................................................................................6

§2.2 预备知识....................................................................................................7

§2.3 主要结果的证明.........................................................................................9

第三章非线性椭圆方程组的三个正径向解的存在性........................................15

§3.1 引言........................................................................................................15

§3.2 解的不存在性..........................................................................................16

§3.3 次临界情况下解的存在性.......................................................................17

第四章非线性椭圆方程组的三个正径向解的存在性........................................21

§2.1 引言........................................................................................................21

§2.2 预备知识..................................................................................................22

§2.3 主要结果的证明.......................................................................................26

参考文献..................................................................................................................29

在读期间公开发表的论文和承担科研项目及取得成果......................................33

致谢........................................................................................................................34

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摘要：

摘要本文主要研究了几类非线性椭圆型方程组解的存在性问题，运用的工具主要有不动点定理，直接变分法，山路引理，Pohozaev等式等.作者首先利用不动点定理证明了拟线性椭圆型方程组径向正解的存在性问题和三个径向正解的存在性问题.其次利用Pohozaev等式研究了拟线性椭圆型方程组临界和超临界指标情况下解的不存在性问题；接着利用Lagrange乘子定理证明了拟线性椭圆型方程组次临界情况下解的存在性；然后利用山路引理证明了次临界和临界情况下解的存在性问题.最后利用Lagrange乘子定理证明了一类(p,q)-Laplace非线性椭圆型方程组极小能量解的存在性问题.本文主要分为四章：在第一章，主要介绍了...

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