这篇论文分为五个部分. 第一章绪论，简要介绍了论文的选题背景、研究现状以及本文主要研究内容.并且给出了论文中所涉及的相关概念、定理等预备知识. 第二章，讨论了一类带周期位势的Schrodinger--Kirchhoff方程正解的存在性问题.利用山路引理、问题的平移不变性以及Lions集中紧致原理，该章证明了问题至少存在一个正基态解. 第三章，讨论了带势阱的Schrodinger--Kirchhoff方程正解的存在性问题.利用山路引理并通过比较原泛函与极限泛函的能量，该章证明了此问题至少存在一个正基态解. 第四章，研究了一类4次超线性的Schrodinger--Kirchhoff方程.此...This dissertation consists of five parts. In chapter 1, we briefly introduce the research background, present status and the overview of this work. Moreover, we give some preliminaries involved in this dissertation. In chapter 2, we discuss a class of Schrodinger--Kirchhoff equations with periodic potentials. By using the mountain pass lemma, the translation invariance of this problem an...学位：理学博士院系专业：数学科学学院_基础数学学号：1902013015415

吴越

Xiamen University Institutional Repository

Æ?èµ10384ÆÒµ19020130154154©aÒ ?UDCÆ ¬ Æ   Ø ©Aa Schro¨dinger–Kirchhoff.ý§)35ïÄStudies on the existence of solutions for someSchro¨dinger–Kirchhoff-type elliptic equationsÇ 6¶µ 4çÅ Ç;  ¶ ¡µ Ä : ê ÆØ©JFÏµ 2016c 4 Ø©Fmµ 2016c 5 Æ ÇFÏµ 2016c 6 F¬ÌRµµ  <µ2016c 5厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库fÆÆ Ø©M5(²<¥Æ Ø©´<3e,Õá¤ïÄ¤J"<3Ø©¥ëÙ¦<½8N®²uLïÄ¤J§þ3©¥±·ª²(I²§¿ÎÜ{Æ5Ú5fÆïÄ)Æâ¹Ä5£Á1¤6",	§TÆ Ø©I[g,ÆÄ7 (8]ÏÒ:11171204)4ïg,ÆÄ7–#ÑcÆÄ7 (8]ÏÒ:2014J06002)]Ï"(²<£\¶¤µc  F厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库fÆÆ Ø©Í¦^(²<Ó¿fÆâ5¥u<¬ÚIÆ ^~61¢{65½3Ú¦^dÆ Ø©§¿Ì+Ü½Ù½ÅxÆ Ø©£)Ú>f¤§#NÆ Ø©?\fÆãÖ,9Ùêâ¥!/"<Ó¿fÆòÆ Ø©\\IÆ¬!a¬Æ Ø©ïü êâ¥?1u¢§òÆ Ø©IKÚÁ®?Ñ§æ^K<! <½öÙ§ªÜnEÆ Ø©"  Æ Ø©áuµ  £ ¤1.²fÆ¬"Ø½Æ Ø©§u  c    F)§) ·^þãÇ"  £ ¤2.Ø§·^þãÇ"  £3±þA)ÒS/K0½WþASN"Æ Ø©A´®²fÆ¬"½LÆ Ø©§²fÆ¬"½Æ Ø©þúmÆ Ø©"d(²9ØW§%@úmÆ Ø©§þ·^þãÇ"¤(²<£\¶¤µc  F厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库Á ùØ©©ÊÜ©.1ÙXØ,{0Ø©ÀKµ!ïÄyG±9©ÌïÄSN.¿ÑØ©¥¤9'Vg!½ný£.1Ù,?Øa±Ï ³ Schro¨dinger–Kirchhoff§)35¯K.|^ì´Ún!¯K Z3–²£ØC5±9 Lions8¥;n,TÙy²¯K3Ä).1nÙ,?Ø³² Schro¨dinger–Kirchhoff§)35¯K.|^ì´Ún¿ÏL'¼4¼Uþ,TÙy²d¯K3Ä).1oÙ,ïÄa 4g5 Schro¨dinger–Kirchhoff§. d? ³´CÒ,Ïd Schro¨dingerfäkkKm. TÙÄky² Palais-Smale^¤á,?|^MorsenØÑ²)35(J. Ù,ïÄa5g¼ê up  Schro¨dinger–Kirchhoff§. TÙE#6/,ÏLéUþ¼3T6/þå4,y² 1 < p < 5,¯K3»é¡).'cµSchro¨dinger–Kirchhoff.§;C©{; 4g5· I ·厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库ABSTRACTThis dissertation consists of five parts.In chapter 1, we briefly introduce the research background, present status and theoverview of this work. Moreover, we give some preliminaries involved in this dissertation.In chapter 2, we discuss a class of Schro¨dinger–Kirchhoff equations with periodic po-tentials. By using the mountain pass lemma, the Z3-translation invariance of this problemand the Lions’ concentration-compactness principle, the existence of positive ground stateis proved.Chapter 3 is devoted to a class of Schro¨dinger–Kirchhoff equations with potential well.We obtain a positive ground state solution by using the mountain pass lemma and comparingthe energy between functionals.In chapter 4, we consider a class of 4-superlinear Schro¨dinger–Kirchhoff equations.The potentials are indefinite of sign so that the Schro¨dinger operator possesses a finite di-mensional negative space. By proving the Palais-Smale condition hold and making use ofthe Morse theory, we get nontrivial solutions for this type of problems.The last chapter is concerned with a class of Schro¨dinger–Kirchhoff equations withpure power nonlinearities up. In this chapter, we construct a new manifold and then mini-mize the energy functional on this manifold. We get the existence of positive radial solutionsfor this problem whenever 1 < p < 5.Key Words: Schro¨dinger–Kirchhoff-type equations; variational methods; 4-superlinear· III ·厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库8 ¹Á  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IABSTRACT.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III1Ù X Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 ïÄµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 ©Vã . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 ý£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Ù ±Ï ³ Schro¨dinger–Kirchhoff.¯K . . . . . . . . . . . . . . . . . 132.1 Úó9(Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 ì´AÛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 (Øy² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181nÙ ³² Schro¨dinger–Kirchhoff.¯K . . . . . . . . . . . . . . . . . . . . . . 213.1 Úó9(Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 (Øy² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221oÙ Ø½ ³ Schro¨dinger–Kirchhoff.¯K . . . . . . . . . . . . . . . . . 274.1 Úó9(Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Palais-Smale^. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 .+9½n 4.1y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 ½n 4.2y² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361ÊÙ g5 Schro¨dinger–Kirchhoff¯K . . . . . . . . . . . . . . 395.1 Úó9(Ø . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 (Øy² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40ë©z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47ôÖÆ¬Æ ÏmïÄ¤J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57· V ·厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库CONTENTSAbstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IAbstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IIIChapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Overview of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter 2 The Schro¨dinger–Kirchhoff-type problems with periodicpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The mountain pass geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Proof of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Chapter 3 The Schro¨dinger–Kirchhoff-type problems with potentialwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Proof of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Chapter 4 The Schro¨dinger–Kirchhoff-type problems withindefinite potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 The Palais-Smale condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Critical groups and proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Chapter 5 The Schro¨dinger–Kirchhoff-type problems with purepower nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Proof of main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Academic achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57· VII ·厦门大学博硕士论文摘要库厦门大学博硕士论文摘要库1Ù X Ø1.1 ïÄµC©n´g,.¥^ÊHn,§òg,.¥éõ¯K=z?Ø,¼3A½^e.:3¯K.~X©AÛ¥±¯K!ÿ/¯K±94­¡¯KÑ±w¤C©¯K.qX3Ôn!ó§!²L!+nNõÆkX´LA^.4:´aAÏ/ª.:. ;C©nØØ%SNB´Ïé¼4.·,3km¥,¼ê´ëYrB±íÑ¼ê4.´du"y;5,ù(Ø3Ã¡m¿Ø¤á. òí2Ã¡m,±ò¼3;8þ, ,3¢SA^¥ù^Lr. 19­V, Poincare´(1887)Ú Hilbert(1898)óy² Dirichletn, dd)4zS{. 20­VÐ,¿|êÆ[ TonelliÚ?¼eëYVg. ù,¼´rfeëY,Ã¡mäkf;5,B±y4zS34.y¼feëY5,~I\þà5^.ù«{8UÑ´ïÄ¼4¯K­Ãã. R. S. PalaisÚ S. Smaleu 1964c3¦ó [1]¥ò6/Ú¼ê(Üå5Ä,JÑ Palais-Smale^.T^´V\3¼êgþ«;5^,m;5±9¼êgr5à5. T^(Ü EkelandC©n [2]Ñ435½n,äk`5.Euler-Lagrange§)¹e¿Ø´ÙéA¼4:,´éA¼.:. d,;C©{w,ØU?nda¯K,·IØÓnØ.duÌ´ÿÀóä,¤±ùÜ©.:nØ~¡C©{. §Ì?Ö´ïÄ¼.:35õ­5,±9.:NCÿÀ5,ÌSNk44n, MorsenØ, Ljusternik-SchnirelmannnØ±9InØ.ùnØÌgÄ:´/CÚn,=e¼ê3üØÓY²8mvk.:,@o3½^e,Ù¥Y²87,±/CÂ ,Y²8. Ïd,üY²8ÿÀ5ØÓ,§m7,3.:. /CÚng@J­Vnc A. P. MorseÚ L. Ljusternik, L. Schnirelmanng±ØÓÝïáå5nØ,y3©O¡MorsenØÚ Ljusternik-SchnirelmannnØ.ùünØÓA:´ÏLïÄY²8ÿÀ55ïÄ.:35,Ïd§êÿÀ,©ÿÀ±9AÛÆmk;éX.8c, R. S. PalaisÚ S. Smaleqòùü· 1 ·厦门大学博硕士论文摘要库1.1ïÄµnØí2Ã¡6/þ,¦±2A^.44nØ´d G.D. Birkhoffu 1917cÄkÚ\. TnØâ»5?Ð´ AmbrosettiÚ Rabinowitzu 1973c3ó [3]¥JÑì´Ún±9d ZyÑX#44½n. ì´Ún9Ùí2´.:nØ­|¤Ü©,Rabinowitz3 [4]¥o(ì´Ún9Ù«CN,¿òÙA^u©§ïÄ¥. Willem3 [5]¥?Ø«½n9Ù3©§¥A^. C 30c5§k'.:35Úõ­5¡ó,M,ZyÑÃXQ:½n,7½n,ÛÜ7½n,½näk2A^¿Â(J (X [6–12]). ù½n±?nQÃþ.qÃe.¼C©¯K,3ý>¯K,uÄ±Ï¯K±9 HamiltonXÚ±Ï;¯KïÄ¥,ék¿Â#(J (X [13–28]9Ù¤ë©z).©ÌïÄ.:nØ3¤¢ Schro¨dinger–Kirchhoff.§−(a+ b∫R3|∇u|2 dx)∆u+ V (x)u = f(x, u), x ∈ R3, (SK)¥A^,Ù¥ a > 0, b ≥ 0~ê. T§ 1883c Kirchhoff3©z [29]¥JÑÔn.k'. äN5`, Kirchhoff3ïÄîÄÚåuUC¯K,QÑ±e§:ρ∂2u∂t2−(P0h+E2L∫ L0∣∣∣∣∂u∂x∣∣∣∣2 dx)∂2u∂x2= 0, (1.1)Ù¥ ρ, h, P , L ´~ê, u = u(x, t) L«u . T§,«¿Âþ`´D’Alembert ÅÄ§ò. T§âÑA:´§¹kÛÜP0h+ E2L∫ L0∣∣∂u∂x∣∣2 dx, T6u²þ 12L∫ L0∣∣∂u∂x∣∣2 dx, ¤±T§Ø2´Å:½Â,Ïd¡ÛÜ¯K. 1978c J.L. Lions3¦ó [30]¥é§utt −(a+ b∫Ω|∇u|2 dx)∆u = f(x, u), (1.2)Ú?Ä¼©Ûµe ,§ (1.1)<þ'5ÚïÄ.d	,ù«ÛÜ§3Ù¦+,X)XÚ,k9,Ù¥ u^5£ã6ug²þÄL§. ~Xë©z [31, 32]¥, uQ^5£ã<Ý.'uaqu (1.1)§õ0,ë [33]9Ù¤ë©z.e¡§ {− (a+ b ∫Ω |∇u|2 dx)∆u = f(x, u), x ∈ Ω,u = 0, x ∈ ∂Ω(1.3)· 2 ·厦门大学博硕士论文摘要库   Degree papers are in the “Xiamen University Electronic Theses and Dissertations Database”.  Fulltexts are available in the following ways: 1. If your library is a CALIS member libraries, please log on http://etd.calis.edu.cn/ and submit requests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to etd@xmu.edu.cn for delivery details. 厦门大学博硕士论文摘要库

Studies on the existence of solutions for some Schr&ouml;dinger--Kirchhoff-type elliptic equations

http://dspace.xmu.edu.cn/bitstream/handle/2288/133916/%e5%87%a0%e7%b1%bb%20Schr%26ouml%3bdinger--Kirchhoff%20%e5%9e%8b%e6%a4%ad%e5%9c%86%e6%96%b9%e7%a8%8b%e8%a7%a3%e7%9a%84%e5%ad%98%e5%9c%a8%e6%80%a7%e7%a0%94%e7%a9%b6.pdf?sequence=1&isAllowed=y

Studies on the existence of solutions for some Schrödinger--Kirchhoff-type elliptic equations

Abstract

Similar works

Full text

Available Versions

Xiamen University Institutional Repository

Studies on the existence of solutions for some Schr&ouml;dinger--Kirchhoff-type elliptic equations

Abstract

Similar works

Full text

Available Versions

Xiamen University Institutional Repository

Studies on the existence of solutions for some Schrödinger--Kirchhoff-type elliptic equations