【摘要】：本文主要研究幾類非線性橢圓方程變號(hào)解的存在性,涉及到的方程包括含有臨界指標(biāo)的擬線性Schrodinger方程,含有分?jǐn)?shù)次Laplacian的Kirchhoff型方程以及含有分?jǐn)?shù)次Laplacian的非線性Choquard方程.本文共分四章:在第一章中,我們概述本文所研究問(wèn)題的背景及國(guó)內(nèi)外研究現(xiàn)狀,并簡(jiǎn)要介紹本文的主要工作及相關(guān)預(yù)備知識(shí)和一些記號(hào).在第二章中,我們研究JRN中含臨界指標(biāo)的擬線性Schrodinger方程—div(g2(u)%絬)+ g(u)g'(u)|%絬|2 + V(x)u = K(u),x ∈ RN,節(jié)點(diǎn)解的存在性,其中N≥ 3,g:R → R+是可微的偶函數(shù).并且對(duì)任意的s ≥ 0,我們有g(shù)'(s)≥ 0.此外,我們還假定K:R →R是一個(gè)連續(xù)函數(shù),位勢(shì)函數(shù)V:R → R是一個(gè)正的徑向?qū)ΨQ函數(shù).我們發(fā)現(xiàn)了上述擬線性Schrodinger方程的臨界指標(biāo).進(jìn)一步地,對(duì)任意正整數(shù)k ≥ 0,我們證明了該方程存在一個(gè)恰好變號(hào)kk次的節(jié)點(diǎn)解.在第三章中,我們研究含有分?jǐn)?shù)次Laplacian算子的Kirchhoff型問(wèn)題極小能量變號(hào)解的存在性及其漸近行為.其中s∈(0,1),N2s,a和b是兩個(gè)正常數(shù),位勢(shì)函數(shù)V(x)∈ C(RN,R)是非負(fù)有正下界函數(shù).利用約束變分的方法以及數(shù)量形變引理,我們證明了在適當(dāng)?shù)奈粍?shì)條件下,上述問(wèn)題有一個(gè)極小能量變號(hào)解ub.進(jìn)一步地,我們證明了該變號(hào)解的能量嚴(yán)格大于兩倍的基態(tài)能量.作為這一章的一個(gè)附帶結(jié)果,我們給出了當(dāng)b(?)0時(shí),ub的一個(gè)收斂性質(zhì).在第四章中,我們研究含有分?jǐn)?shù)次Laplacian算子的非線性Choquard方程(-△)su+V(x)u=(Iα*|u|p)|u|p-2u,x∈RN,≥其中 s ∈(0,1),N2s,0αN,p∈(N+α/N,N+α/N-2s),位勢(shì)V ∈C(RN,R)是正函數(shù)并且滿足適當(dāng)?shù)奈粍?shì)條件.Iα是如下定義在每一個(gè)點(diǎn)x∈\{0}的Riesz位勢(shì)(?),其中(?)利用約束變分方法,我們證明了上述方程有一個(gè)非負(fù)的基態(tài)解.利用噴泉定理,我們證明了該方程有無(wú)窮多解.進(jìn)一步地,當(dāng)p∈(2,N+α/N-2s),時(shí),我們利用約束變分方法和數(shù)量形變引理證明了上述方程極小能量變號(hào)解的存在性.
[Abstract]:In this paper, we study the existence of sign variation solutions for several kinds of nonlinear elliptic equations, including quasilinear Schrodinger equations with critical indices, Kirchhoff type equations with fractional Laplacian and nonlinear Choquard equations with fractional Laplacian. This paper is divided into four chapters: in the first chapter, we summarize the background of this paper and the current situation of research at home and abroad, and briefly introduce the main work of this paper and related preparatory knowledge and some symbols. In the second chapter, we study the existence of solutions of quasilinear Schrodinger equation -div (g2 (u)%) g (u)'(u)% V (x) u = K (u) x 鈭，

本文編號(hào)：2188206