【摘要】：本文我們研究橢圓型方程基態(tài)解和集中性等若干問(wèn)題。我們著重研究 Neumann 邊值問(wèn)題、分?jǐn)?shù)階 Schrodinger 方程、Schrodinger-Poisson 系統(tǒng)和Kirchhoff方程。主要內(nèi)容安排如下:第一章:我們回顧一些記號(hào)和約定,并給出在后面章節(jié)中會(huì)用到的一些有用的初步結(jié)果。第二章:我們研究次線性Neumann問(wèn)題。與之前關(guān)于Dirichlet問(wèn)題的工作相比,由于相應(yīng)的變分泛函沒(méi)有下界,因此一些困難隨之產(chǎn)生。我們證明該Neumann問(wèn)題有無(wú)窮多小負(fù)能量解,這補(bǔ)充了 Parini和Weth在[93]中關(guān)于最小能量解的近期工作。第三章:我們研究非線性分?jǐn)?shù)階Schrodinger方程L2-標(biāo)準(zhǔn)化解的存在性、不存在性和質(zhì)量集中。與之前關(guān)于Schrodinger方程的工作相比,由于分?jǐn)?shù)階Laplacian的非局部性,因此我們遇到一些新的挑戰(zhàn)。我們先證明了分?jǐn)?shù)階Gagliardo-Nirenberg-Sobolev不等式的最佳嵌入常數(shù)可以由精確的形式表達(dá)出來(lái),這提高了 [57, 58]的工作。做到了以上這些,我們?nèi)缓蠼⒃摲匠蘈2-標(biāo)準(zhǔn)化解的存在性和不存在性。最后通過(guò)利用一些精巧的能量估計(jì),在某類(lèi)勢(shì)阱下我們給出質(zhì)量臨界情形時(shí)的L2-標(biāo)準(zhǔn)化解集中行為的詳細(xì)分析。第四章:我們研究Schrodinger-Poisson系統(tǒng)。由于它的物理相關(guān)性,因此三維Schrodinger-Poisson系統(tǒng)已被廣泛研究和充分了解。相反地,本文關(guān)注的二維Schrodinger-Poisson系統(tǒng)的信息要少很多。Cingolani和Weth在[36]中已觀察到Schrodinger-Poisson系統(tǒng)的變分結(jié)構(gòu)會(huì)在二維情形時(shí)出現(xiàn)本質(zhì)差異,這導(dǎo)致其解集會(huì)有更豐富的結(jié)構(gòu)。然而[36]的變分方法僅限于p≥4的情形,這排除了一些物理有關(guān)的指標(biāo)。本章我們將去掉這個(gè)令人不愉快的限制,并在2 p 4的情形下使用一個(gè)不同的變分方法探索更復(fù)雜的底層泛函幾何。第五章:我們研究一類(lèi)Kirchhoff方程。在適當(dāng)?shù)募僭O(shè)下,通過(guò)變分方法我們證明該方程基態(tài)解的存在性。此外我們還調(diào)查基態(tài)解的集中現(xiàn)象。
[Abstract]:In this paper, we study some problems such as the ground state solution and the centralization of the elliptic equation. We focus on the Neumann boundary value problem, fractional Schrodinger equation, Schrodinger-Poisson system and Kirchhoff equation. The main contents are as follows: chapter 1: we review some notation and conventions and give some useful preliminary results that will be used in later chapters. Chapter 2: we study sublinear Neumann problem. Compared with the previous work on the Dirichlet problem, some difficulties arise because the corresponding variational functional has no lower bound. We prove that the Neumann problem has infinitely small negative energy solutions, which complements the recent work of Parini and Weth on the minimum energy solution in [93]. Chapter 3: we study the existence, nonexistence and mass concentration of L _ 2-standard solutions for nonlinear fractional Schrodinger equations. Compared with previous work on Schrodinger equations, we meet some new challenges due to the nonlocality of fractional Laplacian. We first prove that the best embedding constant of fractional order Gagliardo-Nirenberg-Sobolev inequality can be expressed in exact form, which improves the work of [57, 58]. Then we establish the existence and nonexistence of the L 2-standard solution of the equation. Finally, by using some subtle energy estimates, we give a detailed analysis of the concentration behavior of the L2-standard solution in the critical case of mass under a certain kind of potential well. Chapter 4: we study Schrodinger-Poisson system. Because of its physical correlation, 3D Schrodinger-Poisson system has been widely studied and fully understood. On the contrary, the information of two-dimensional Schrodinger-Poisson systems is much less concerned in this paper. In [36], Cingolani and Weth have observed that the variational structures of Schrodinger-Poisson systems are essentially different in two-dimensional cases, which leads to a richer structure of their solution sets. However, the variational method of [36] is limited to the case of p 鈮，

本文編號(hào)：2344720