具有非局部項(xiàng)的非線性橢圓方程的解
發(fā)布時(shí)間：2018-11-09 18:12
【摘要】：本文主要研究了兩類(lèi)帶有非局部項(xiàng)的橢圓方程解的存在性問(wèn)題.首先,我們研究了Kirchhoff類(lèi)型橢圓方程解的存在性,其中包括基態(tài)解、多解、負(fù)能量解和變號(hào)解的存在性.其次,我們對(duì)Schr?dinger-Poisson方程組解的存在性問(wèn)題進(jìn)行了討論.具體的結(jié)論與使用方法如下.本文共分為如下六個(gè)章節(jié):第一章,緒論.主要講述本文所考慮的兩類(lèi)帶有非局部項(xiàng)橢圓方程的物理背景與學(xué)術(shù)意義以及目前國(guó)內(nèi)外的發(fā)展現(xiàn)狀,最后簡(jiǎn)單的介紹本文的主要結(jié)論和本文所使用的符號(hào).第二章,考慮帶有Sobolev臨界指數(shù)的Kirchhoff類(lèi)型的橢圓方程,在(4)N?N?中的有界區(qū)域內(nèi),我們討論了幾種解的存在性問(wèn)題.當(dāng)N?4時(shí),在q?2和q?(2,4)的情況下,我們使用Nehari流形以及Lions的第一集中緊性原理,克服獲得(PS)序列有界性的困難,再結(jié)合Lions的第二集中緊性原理獲得了(PS)序列的局部緊性,最后獲得方程基態(tài)解bu的存在性.我們并且考慮當(dāng)b?0時(shí),Kirchhoff類(lèi)型橢圓方程與Brezis-Nirenberg問(wèn)題的關(guān)系,獲得了當(dāng)b?0時(shí),Kirchhoff類(lèi)型橢圓方程的基態(tài)解bu收斂于Brezis-Nirenberg問(wèn)題的基態(tài)解0u.當(dāng)q?(1,2)時(shí),使用一種截?cái)嗉夹g(shù)截?cái)嗄芰糠汉?再使用Krasnoselskii’s genus(2?指標(biāo))理論,獲得方程有無(wú)窮多負(fù)能量解.當(dāng)N?5時(shí),對(duì)任意的q?(1,2*)和在a,b滿足一些假設(shè)條件情況下,證明了方程存在正解.第三章,討論帶有純冪次非線性項(xiàng)的Kirchhoff類(lèi)型橢圓方程變號(hào)解的存在性.我們使用變分法與不變集的下降流以及2?指標(biāo)理論,獲得了當(dāng)p?(2,6)時(shí),方程存在無(wú)窮多高能量變號(hào)解.我們的這個(gè)工作擴(kuò)展了現(xiàn)有文獻(xiàn)僅在p?(4,6)時(shí)存在變號(hào)解結(jié)論.第四章,討論帶有一般非線性項(xiàng)的Kirchhoff類(lèi)型橢圓方程變號(hào)解的存在性問(wèn)題.通過(guò)對(duì)非線性項(xiàng)的一些較弱假設(shè),運(yùn)用一個(gè)截?cái)嗉夹g(shù),再使用變號(hào)的Nehari流形與集中緊性原理,最后證明方程存在一個(gè)最小能量變號(hào)解和一個(gè)不變號(hào)的基態(tài)解,且這個(gè)變號(hào)解的能量嚴(yán)格大于基態(tài)解能量.第五章,研究另外一種類(lèi)型的帶有非局項(xiàng)的橢圓方程,即帶有Sobolev臨界指數(shù)的Schr?dinger-Poisson方程組.首先由Lax-Milgram定理知Schr?dinger-Poisson方程組可以轉(zhuǎn)化成為一個(gè)包含非局部項(xiàng)的單一方程,再使用變分法,獲得能量泛函.當(dāng)p?1時(shí),把能量泛函約束在Nehari流形上,得到(PS)序列的有界性,再使用集中緊性原理獲得(PS)序列的局部緊性,最后再證明了Nehari流形對(duì)能量泛函的約束是一個(gè)自然約束,因此,獲得原方程組存在一個(gè)基態(tài)解.而當(dāng)p?(0,1)時(shí),運(yùn)用一種截?cái)嗉夹g(shù)截?cái)嗄芰糠汉?再對(duì)獲得的新能量泛函運(yùn)用2?指標(biāo)理論,我們證明了方程存在無(wú)窮多負(fù)能量解.第六章,我們簡(jiǎn)單地把本論文加以總結(jié),同時(shí)希望對(duì)本文中的一些結(jié)論能進(jìn)一步優(yōu)化以及弱化一些假設(shè).
[Abstract]:In this paper, we study the existence of solutions for two classes of elliptic equations with nonlocal terms. First, we study the existence of solutions for Kirchhoff type elliptic equations, including ground state solutions, multiple solutions, negative energy solutions and sign changing solutions. Secondly, we discuss the existence of solutions for Schr?dinger-Poisson equations. The specific conclusions and usage methods are as follows. This article is divided into the following six chapters: the first chapter, introduction. This paper mainly describes the physical background and academic significance of the two kinds of elliptic equations with nonlocal terms considered in this paper, as well as the present development situation at home and abroad. Finally, the main conclusions and symbols used in this paper are briefly introduced. In chapter 2, we consider the elliptic equation of Kirchhoff type with Sobolev critical exponent. In this paper, we discuss the existence of some solutions in the bounded domain of. When N4, we use the Nehari manifold and the first concentrated compactness principle of Lions to overcome the difficulty of obtaining the boundedness of (PS) sequences in the case of Q2 and Q2 (2? 4). The local compactness of (PS) sequences is obtained by combining the second set compactness principle of Lions. Finally, the existence of the ground state solution bu of the equation is obtained. We also consider the relation between the Kirchhoff type elliptic equation and the Brezis-Nirenberg problem when b0, and we obtain that the ground state solution of the Kirchhoff type elliptic equation converges to the ground state solution of the Brezis-Nirenberg problem when b0. When Q2, a truncation technique is used to truncate the energy functional, and then Krasnoselskii's genus (2? Index) theory, the equation has infinitely many negative energy solutions. The existence of positive solutions to the equation is proved when N5 satisfies some hypotheses for any Q? (1 ~ 2 *) and a b. In chapter 3, we discuss the existence of sign change solutions for Kirchhoff type elliptic equations with pure power nonlinear terms. We use the variational method and the descending flow of invariant sets and 2? Based on the index theory, it is obtained that the equation has infinitely high energy sign solutions when p _ (2 ~ (2) ~ (6). Our work extends the existing literature only when p? (4 ~ (6) has a sign solution. In chapter 4, we discuss the existence of sign variation solutions for Kirchhoff type elliptic equations with general nonlinear terms. By using a truncation technique and the Nehari manifold of sign variation and the principle of centralization compactness, it is proved that the equation has a minimum energy sign solution and a ground state solution of invariant sign. Moreover, the energy of the signed solution is strictly larger than that of the ground state solution. In chapter 5, we study another type of elliptic equations with nonlocal terms, that is, Schr?dinger-Poisson equations with Sobolev critical exponents. First, the Lax-Milgram theorem shows that the Schr?dinger-Poisson equations can be transformed into a single equation containing nonlocal terms, and then the energy functional is obtained by using the variational method. When p? 1, the energy functional is confined to the Nehari manifold, the boundedness of the (PS) sequence is obtained, and the local compactness of the (PS) sequence is obtained by using the centralization compactness principle. Finally, it is proved that the constraint of the Nehari manifold on the energy functional is a natural constraint. Therefore, the existence of a ground state solution for the original equations is obtained. When p? (0 ~ 1), a truncation technique is used to truncate the energy functional, and then 2? In the index theory, we prove that there are infinite negative energy solutions to the equation. In the sixth chapter, we summarize this paper briefly, and hope that some conclusions in this paper can be further optimized and some hypotheses can be weakened.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O175.25
本文編號(hào)：2321180
[Abstract]:In this paper, we study the existence of solutions for two classes of elliptic equations with nonlocal terms. First, we study the existence of solutions for Kirchhoff type elliptic equations, including ground state solutions, multiple solutions, negative energy solutions and sign changing solutions. Secondly, we discuss the existence of solutions for Schr?dinger-Poisson equations. The specific conclusions and usage methods are as follows. This article is divided into the following six chapters: the first chapter, introduction. This paper mainly describes the physical background and academic significance of the two kinds of elliptic equations with nonlocal terms considered in this paper, as well as the present development situation at home and abroad. Finally, the main conclusions and symbols used in this paper are briefly introduced. In chapter 2, we consider the elliptic equation of Kirchhoff type with Sobolev critical exponent. In this paper, we discuss the existence of some solutions in the bounded domain of. When N4, we use the Nehari manifold and the first concentrated compactness principle of Lions to overcome the difficulty of obtaining the boundedness of (PS) sequences in the case of Q2 and Q2 (2? 4). The local compactness of (PS) sequences is obtained by combining the second set compactness principle of Lions. Finally, the existence of the ground state solution bu of the equation is obtained. We also consider the relation between the Kirchhoff type elliptic equation and the Brezis-Nirenberg problem when b0, and we obtain that the ground state solution of the Kirchhoff type elliptic equation converges to the ground state solution of the Brezis-Nirenberg problem when b0. When Q2, a truncation technique is used to truncate the energy functional, and then Krasnoselskii's genus (2? Index) theory, the equation has infinitely many negative energy solutions. The existence of positive solutions to the equation is proved when N5 satisfies some hypotheses for any Q? (1 ~ 2 *) and a b. In chapter 3, we discuss the existence of sign change solutions for Kirchhoff type elliptic equations with pure power nonlinear terms. We use the variational method and the descending flow of invariant sets and 2? Based on the index theory, it is obtained that the equation has infinitely high energy sign solutions when p _ (2 ~ (2) ~ (6). Our work extends the existing literature only when p? (4 ~ (6) has a sign solution. In chapter 4, we discuss the existence of sign variation solutions for Kirchhoff type elliptic equations with general nonlinear terms. By using a truncation technique and the Nehari manifold of sign variation and the principle of centralization compactness, it is proved that the equation has a minimum energy sign solution and a ground state solution of invariant sign. Moreover, the energy of the signed solution is strictly larger than that of the ground state solution. In chapter 5, we study another type of elliptic equations with nonlocal terms, that is, Schr?dinger-Poisson equations with Sobolev critical exponents. First, the Lax-Milgram theorem shows that the Schr?dinger-Poisson equations can be transformed into a single equation containing nonlocal terms, and then the energy functional is obtained by using the variational method. When p? 1, the energy functional is confined to the Nehari manifold, the boundedness of the (PS) sequence is obtained, and the local compactness of the (PS) sequence is obtained by using the centralization compactness principle. Finally, it is proved that the constraint of the Nehari manifold on the energy functional is a natural constraint. Therefore, the existence of a ground state solution for the original equations is obtained. When p? (0 ~ 1), a truncation technique is used to truncate the energy functional, and then 2? In the index theory, we prove that there are infinite negative energy solutions to the equation. In the sixth chapter, we summarize this paper briefly, and hope that some conclusions in this paper can be further optimized and some hypotheses can be weakened.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O175.25
【參考文獻(xiàn)】
相關(guān)期刊論文 前1條
1 ;Multiplicity of Solutions for a Class of Kirchhoff Type Problems[J];Acta Mathematicae Applicatae Sinica(English Series);2010年03期
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