【作者】 周末；

【导师】 何进春；

【作者基本信息】 华中科技大学 ， 基础数学， 2020， 硕士

【摘要】 本文讨论了以下广义拟线性Schr(?)dinger方程径向解的存在性:-div(g2(u)▽u)+g(u)g’(u)|▽u|2+V(|x|)u=h(|x|,u),x∈RN,(A)其中g(t)=(1+2t2s)1/2,s∈[0,1],N≥2。当函数V,h满足某些条件时,利用变分法和对称山路引理得出了方程有无穷多个径向对称解并且其能量趋于无穷。另外还给出了对于一般的函数g(t),当其满足一定的条件时方程仍然有相同的结论。本篇文章的结构编排如下:第一部分简要介绍了问题的研究背景和主要结论,即定理1.1和定理1.2。第二部分首先介绍了本篇文章需要涉及到的基本知识点和结论,这些结论很容易在任何一本变分法理论的书籍中找到。其次是介绍了证明定理成立涉及到的变量代换的方法,改方法用于间接证明解的存在性。第三部分首先给出了证明定理1.1需要涉及到的若干引理的证明细节,利用这些引理即可证明方程(A)对应的泛函满足对称山路定理的所有条件,这样即可得出定理1.1是成立的。如法炮制,也能够得出定理1.2的结论。第四部分是对结论和问题的展望。更多还原

【Abstract】 This paper discusses the existence of radial solutions for the generalized quasilinear schrodinger equation:-div(g2(u)▽u)+g(u)g’(u)|▽u|2+V(|x|)u=h(|x|,u),x∈RN,(A)where g(t)=(1+2t2s)1/2,s E[0,1],N≥2.When the functions V,h,satisfy some conditions,the variational method and symmetric mountain pass lemma are used to obtain that the equation has infinite radial symmetric solutions and its energy tends to infinity.In addition,for the general function g(t),when it satisfies certain conditions,the equation still has the same conclusion.The structure of this article is as follows:The first part briefly introduces the research background and main conclusions of the problem,namely theorem 1.1 and theorem 1.2.The second part first introduces the basic knowledge and conclusions which need to be involved in this paper.These conclusions can be easily found in any book of variational theory.Secondly,it introduces the method of variable substitution,which is used to prove the existence of solution indirectly.In the third part,we first give the proof details of some lemmas which need to be involved in the proof of theorem 1.1.By using these lemmas,we can prove that the functional corresponding to equation(A)satisfies all the conditions of the symmetric mountain path theorem,and then we can conclude that the theorem 1.1 is tenable.If we do this,we can also draw the conclusion that theorem 1.2.The fourth part is the prospect of the conclusion and the problem.更多还原