本文主要研究内容
作者姜瑞廷（2019）在《四阶微分方程解的研究》一文中研究指出：本文分五章.第一章,引言部分.主要介绍四阶微分方程的一些背景,国内外的研究现状,本文用到的重要引理及本文得到的主要结果.第二章,我们将讨论如下非周期四阶微分方程U(4)+wu"+a(x)u=f(x)|u|q-2u+g(x)|u|p-2u,x∈ R,其中1<q<2<p<+∞,a(x),f(x)和g(x)是连续函数且满足如下条件:(H1)a ∈ C(R,R),且存在正常数a1,使得ω≤2(?),且当|x|→+∞时,有0<a1<a(x)→+∞;(H3)g ∈C(服)∩L∞(R),且对于几乎处处的x∈ R,有g(x)>0.为了得到方程解的存在性,我们在空间X:={u ∈H2(R)| ∫R[u"(x)2-wu’(x)2+a(x)u(x)2]dx<+∞中考虑泛函J:X→R:J(u)=1/2∫R[u"(x)2-wu’(x)2+a(x)u(x)2]dx-1/q∫Rf(x)|u|qdx q-1/p∫ g(x)|u|pdx=1/2‖u‖2-1/qf(x)|u|qdx-1/p∫R g(x)|u|pdx的临界点.因为J在X上不是下方有界的,所以我们引入Nehari流形N={u∈ X:{J’(u),u)=0},其中<·,·>是X与X’之间的对偶.而Nehari流形N与形如Nu:t→J(fu)(t>0)的函数密切相关.对于u ∈ X,令因为对于u ∈ X{0}及t>0,有则tu∈N当且仅当Nu’(t)=0,即Nu(t)的临界点与Nehari流形上的点.特别地,u∈N当且仅当Nu’(1)=0.定义N+={u ∈ N:<(1)>0},首先,我们证得J在N上是下方有界的,强制的,N0=(?),接着,证明了 J在N+和N-上的极小值点均为J的临界点,进而得到解的多重性.本章,我们证明了如下定理.定理2.1.3假设(H1)-(H3)成立,若|f|q*|g|∞(2-q)/(p-2)∈(0,σ),则问题(2.1.1)有两个非平凡解,且一个解的能量是负的,其中,σ=(p-2)(2-q)(2—q)/(p—2)(Sp/(p-g))(p-q)/(p-2).定理2.1.5假设(H1)-(H3)成立,若|f|q*|g|∞(2-q)/(0,σ*)∈(0,σ*),则问题(2.1.1)有两个非平凡解,且一个解的能量是正的,另一个解的能量是负的,其中,0<σ*:=q/2σ<σ.进一步,负能量解是一个基态解.第三章,我们考虑四阶非线性系统:借助Green函数,将系统的正解转化为算子的不动点,运用Guo-Krasnosel’skii不动点定理,得到了如下结论:定理3.3.2 假设(H1)和(H2)成立,σ∈(0,1/2),f0,f∞σ,g0,g∞σ ∈(0,∞),α1,α2 ∈[0,1],α3,α4 ∈(0,1),a∈[0,1],∈(0,1),L1<L2及L3<L4.则对于任意的λ ∈(L1,L2)及μ ∈(L3,L4),问题(3.1.1)存在正解(u(t),v(t)∈[0,1].定理3.3.10 假设(H1)和(H2)成立,a ∈(0,1/2),f0σ，f∞,g0σ,g∞ ∈(0,∞),α1,α2[0,1],α3,α4 ∈(0,1):a ∈[0,1],b ∈(0,1),L1<L2RL3<L4.则对于任意的λ∈(/1,Lλ 及μ(L3,L4),问题(3.1.1)有正解(u(2),v(t),t ∈[0,1].第四章,我们考虑如下方程其中,N1,β ∈ R,△pu=div(|▽u|p-2▽u),pP>2.在本章中,我们假设 Vλ(x)=λa(x)-b(x),且a(x)和b(x)满足以下条件:(a1)a ∈ C(RN),且对于任意的x∈RN a(x)≥ 0成立.存在a0>0使得当N≥4时,集合{a<a0}:={x RN|a(x)<a0}有有限的正测度;当N ≤3时,其中,|·|表示Lebesgue测度,S∞是嵌入H2(RN)(N≤ 3)→L∞(RN)的最佳Sobolev常数,A0将在引理中给出;(a2)Ω=int{x∈RN:a(x)=0}是一个具有光滑边界的非空集合且Ω={x∈RN:a(x)=0};(b)b(x)是RN上的可测函数且存在0<b0<γ使得0 ≤ b(x)≤b0/|x|4对于所有的x∈RN 均成立,其中,γ=N2(N-4)2/16是Hardy-Sobolev常数.同时,我们假设f∈C(R ×R,R).令F(x,u)=∫0uf(x,.s)ds,且F(x,u)=F(.x,u)+α(x)|u|s,其中,1<s<2,F,α满足下列条件:(f1)α(x)∈ α(x)≥ 0且|α|2/2-s<П*:= min{γ-b0/γ(?)2s/2,П0,П},其中(?)2,П0,П1在注4.1.2给出;(f2)F(x,u)∈ C1(RN× R,R),F(x,0)三 0,存在r>p及两个连续有界函数p,q:RN→ R使得q>0在Ω上成立且对于x∈一致成立;(f3)存在d0满足0≤0<(P-2(γ-b0)/2pγ(?)2使得F(x,u)-1/pf(x,u)u≤d0|u|2,x∈RN,u∈ R,其中,f(x,u)=F(x,u).我们在空间Xλ={u∈ H2(RN)| ∫RN(|Δu|2+λa(x)u2)dx<+∞)}中考虑能量泛函Iλ(u)—O/[|Δu|2λa(x)u2]dx-1/2∫RNb(x)u2dx+β/p∫RN |▽u|pdx—∫RNF(x,u)dx=1/2‖u‖λ2-1/2∫RNb(x)u2dx+β/p∫RN |▽u|pdx-∫RNF(x,u)dx,u Xλ.结合Ekeland变分原理和山路引理,我们得到了以下结论:定理4.1.3假设N ≥ 1,2<p<4,β∈ R且(a1),(a2),(b)成立.另外,假设f满足(f1)-(f3).则存在A*>0,使得当λ:A*时,问题(4.1.1)至少有两个非平凡解.当b(x)=0时,上述问题退化为自然地,我们可得以下结论:推论4.1.5假设(a1)和(a2)成立,N≥ 1,∈ R,2<p<4.另外,假设f满足(f2)及(f1’)α(x)∈L2/2-s(EN)且|α|2/2-s<min{1/(?)2s/2,П,П1’},其中,D,A0,e2p/(4-p)是正常数.(乃)存在d0’满足0≤d0’≤(p-2)/2p(?)2 使得F(x,u)-1/pf(x,u)u≤d0’|u|2,x ∈RN,u ∈R.则存在A**>0使得当λ≥ Λ**时,该问题至少有两个非平凡解.第五章,考虑以下ε4Δ2u+V(x)u=λf(x)|u|q-2u+g(x)|u|r-2u,x ∈RN,其中,ε,λ>0是参数,N≥ 5,1<q<2<r<2*=2N/N-4,假设函数f,g和V满足以下条件:(F)f≥ 0,≠0,f∈q(RN)∩ C(RN)(q*= r/(r-q))且|f|q*>0,fmax=maxf(x)=1;(G1)g是定义在RN上的连续的正函数;(G2)存在k个点a1,a2,.ak∈RN使得g(ai)=max g(x)=1,1 ≤i ≤k,且0<g∞=lim|x|→+∞ g(x)<1;(V1)V∈C(RN,R)满足V∞:=liminf V(x)≥ V0:=inf V(x)>0(V2)V(ai)=V0,i=1,2,...,k.对于ε>0,定义空间Hε{u∈H:∫RNV(εχ)|u|2dx<+∞},其范数为:(?)由条件(V1)知,嵌入H→H是连续的.如果我们做变量替换x→εx则上述问题转化为△2u+V(εx)u=λf(εx)|u|q-2u+g(εx)|u|r-2u,x ∈ RN.(0.0.1)考虑 Euler-Lagrange 泛函(?)则Iε是C1的,但在Hε上不是下方有界的.类似于第二章,我们考虑Nehari流形Nε={u∈Hε{0}:
Abstract
ben wen fen wu zhang .di yi zhang ,yin yan bu fen .zhu yao jie shao si jie wei fen fang cheng de yi xie bei jing ,guo nei wai de yan jiu xian zhuang ,ben wen yong dao de chong yao yin li ji ben wen de dao de zhu yao jie guo .di er zhang ,wo men jiang tao lun ru xia fei zhou ji si jie wei fen fang cheng U(4)+wu"+a(x)u=f(x)|u|q-2u+g(x)|u|p-2u,x∈ R,ji zhong 1<q<2<p<+∞,a(x),f(x)he g(x)shi lian xu han shu ju man zu ru xia tiao jian :(H1)a ∈ C(R,R),ju cun zai zheng chang shu a1,shi de ω≤2(?),ju dang |x|→+∞shi ,you 0<a1<a(x)→+∞;(H3)g ∈C(fu )∩L∞(R),ju dui yu ji hu chu chu de x∈ R,you g(x)>0.wei le de dao fang cheng jie de cun zai xing ,wo men zai kong jian X:={u ∈H2(R)| ∫R[u"(x)2-wu’(x)2+a(x)u(x)2]dx<+∞zhong kao lv fan han J:X→R:J(u)=1/2∫R[u"(x)2-wu’(x)2+a(x)u(x)2]dx-1/q∫Rf(x)|u|qdx q-1/p∫ g(x)|u|pdx=1/2‖u‖2-1/qf(x)|u|qdx-1/p∫R g(x)|u|pdxde lin jie dian .yin wei Jzai Xshang bu shi xia fang you jie de ,suo yi wo men yin ru Nehariliu xing N={u∈ X:{J’(u),u)=0},ji zhong <·,·>shi Xyu X’zhi jian de dui ou .er Nehariliu xing Nyu xing ru Nu:t→J(fu)(t>0)de han shu mi qie xiang guan .dui yu u ∈ X,ling yin wei dui yu u ∈ X{0}ji t>0,you ze tu∈Ndang ju jin dang Nu’(t)=0,ji Nu(t)de lin jie dian yu Nehariliu xing shang de dian .te bie de ,u∈Ndang ju jin dang Nu’(1)=0.ding yi N+={u ∈ N:<(1)>0},shou xian ,wo men zheng de Jzai Nshang shi xia fang you jie de ,jiang zhi de ,N0=(?),jie zhao ,zheng ming le Jzai N+he N-shang de ji xiao zhi dian jun wei Jde lin jie dian ,jin er de dao jie de duo chong xing .ben zhang ,wo men zheng ming le ru xia ding li .ding li 2.1.3jia she (H1)-(H3)cheng li ,re |f|q*|g|∞(2-q)/(p-2)∈(0,σ),ze wen ti (2.1.1)you liang ge fei ping fan jie ,ju yi ge jie de neng liang shi fu de ,ji zhong ,σ=(p-2)(2-q)(2—q)/(p—2)(Sp/(p-g))(p-q)/(p-2).ding li 2.1.5jia she (H1)-(H3)cheng li ,re |f|q*|g|∞(2-q)/(0,σ*)∈(0,σ*),ze wen ti (2.1.1)you liang ge fei ping fan jie ,ju yi ge jie de neng liang shi zheng de ,ling yi ge jie de neng liang shi fu de ,ji zhong ,0<σ*:=q/2σ<σ.jin yi bu ,fu neng liang jie shi yi ge ji tai jie .di san zhang ,wo men kao lv si jie fei xian xing ji tong :jie zhu Greenhan shu ,jiang ji tong de zheng jie zhuai hua wei suan zi de bu dong dian ,yun yong Guo-Krasnosel’skiibu dong dian ding li ,de dao le ru xia jie lun :ding li 3.3.2 jia she (H1)he (H2)cheng li ,σ∈(0,1/2),f0,f∞σ,g0,g∞σ ∈(0,∞),α1,α2 ∈[0,1],α3,α4 ∈(0,1),a∈[0,1],∈(0,1),L1<L2ji L3<L4.ze dui yu ren yi de λ ∈(L1,L2)ji μ ∈(L3,L4),wen ti (3.1.1)cun zai zheng jie (u(t),v(t)∈[0,1].ding li 3.3.10 jia she (H1)he (H2)cheng li ,a ∈(0,1/2),f0σ，f∞,g0σ,g∞ ∈(0,∞),α1,α2[0,1],α3,α4 ∈(0,1):a ∈[0,1],b ∈(0,1),L1<L2RL3<L4.ze dui yu ren yi de λ∈(/1,Lλ ji μ(L3,L4),wen ti (3.1.1)you zheng jie (u(2),v(t),t ∈[0,1].di si zhang ,wo men kao lv ru xia fang cheng ji zhong ,N1,β ∈ R,△pu=div(|▽u|p-2▽u),pP>2.zai ben zhang zhong ,wo men jia she Vλ(x)=λa(x)-b(x),ju a(x)he b(x)man zu yi xia tiao jian :(a1)a ∈ C(RN),ju dui yu ren yi de x∈RN a(x)≥ 0cheng li .cun zai a0>0shi de dang N≥4shi ,ji ge {a<a0}:={x RN|a(x)<a0}you you xian de zheng ce du ;dang N ≤3shi ,ji zhong ,|·|biao shi Lebesguece du ,S∞shi qian ru H2(RN)(N≤ 3)→L∞(RN)de zui jia Sobolevchang shu ,A0jiang zai yin li zhong gei chu ;(a2)Ω=int{x∈RN:a(x)=0}shi yi ge ju you guang hua bian jie de fei kong ji ge ju Ω={x∈RN:a(x)=0};(b)b(x)shi RNshang de ke ce han shu ju cun zai 0<b0<γshi de 0 ≤ b(x)≤b0/|x|4dui yu suo you de x∈RN jun cheng li ,ji zhong ,γ=N2(N-4)2/16shi Hardy-Sobolevchang shu .tong shi ,wo men jia she f∈C(R ×R,R).ling F(x,u)=∫0uf(x,.s)ds,ju F(x,u)=F(.x,u)+α(x)|u|s,ji zhong ,1<s<2,F,αman zu xia lie tiao jian :(f1)α(x)∈ α(x)≥ 0ju |α|2/2-s<П*:= min{γ-b0/γ(?)2s/2,П0,П},ji zhong (?)2,П0,П1zai zhu 4.1.2gei chu ;(f2)F(x,u)∈ C1(RN× R,R),F(x,0)san 0,cun zai r>pji liang ge lian xu you jie han shu p,q:RN→ Rshi de q>0zai Ωshang cheng li ju dui yu x∈yi zhi cheng li ;(f3)cun zai d0man zu 0≤0<(P-2(γ-b0)/2pγ(?)2shi de F(x,u)-1/pf(x,u)u≤d0|u|2,x∈RN,u∈ R,ji zhong ,f(x,u)=F(x,u).wo men zai kong jian Xλ={u∈ H2(RN)| ∫RN(|Δu|2+λa(x)u2)dx<+∞)}zhong kao lv neng liang fan han Iλ(u)—O/[|Δu|2λa(x)u2]dx-1/2∫RNb(x)u2dx+β/p∫RN |▽u|pdx—∫RNF(x,u)dx=1/2‖u‖λ2-1/2∫RNb(x)u2dx+β/p∫RN |▽u|pdx-∫RNF(x,u)dx,u Xλ.jie ge Ekelandbian fen yuan li he shan lu yin li ,wo men de dao le yi xia jie lun :ding li 4.1.3jia she N ≥ 1,2<p<4,β∈ Rju (a1),(a2),(b)cheng li .ling wai ,jia she fman zu (f1)-(f3).ze cun zai A*>0,shi de dang λ:A*shi ,wen ti (4.1.1)zhi shao you liang ge fei ping fan jie .dang b(x)=0shi ,shang shu wen ti tui hua wei zi ran de ,wo men ke de yi xia jie lun :tui lun 4.1.5jia she (a1)he (a2)cheng li ,N≥ 1,∈ R,2<p<4.ling wai ,jia she fman zu (f2)ji (f1’)α(x)∈L2/2-s(EN)ju |α|2/2-s<min{1/(?)2s/2,П,П1’},ji zhong ,D,A0,e2p/(4-p)shi zheng chang shu .(nai )cun zai d0’man zu 0≤d0’≤(p-2)/2p(?)2 shi de F(x,u)-1/pf(x,u)u≤d0’|u|2,x ∈RN,u ∈R.ze cun zai A**>0shi de dang λ≥ Λ**shi ,gai wen ti zhi shao you liang ge fei ping fan jie .di wu zhang ,kao lv yi xia ε4Δ2u+V(x)u=λf(x)|u|q-2u+g(x)|u|r-2u,x ∈RN,ji zhong ,ε,λ>0shi can shu ,N≥ 5,1<q<2<r<2*=2N/N-4,jia she han shu f,ghe Vman zu yi xia tiao jian :(F)f≥ 0,≠0,f∈q(RN)∩ C(RN)(q*= r/(r-q))ju |f|q*>0,fmax=maxf(x)=1;(G1)gshi ding yi zai RNshang de lian xu de zheng han shu ;(G2)cun zai kge dian a1,a2,.ak∈RNshi de g(ai)=max g(x)=1,1 ≤i ≤k,ju 0<g∞=lim|x|→+∞ g(x)<1;(V1)V∈C(RN,R)man zu V∞:=liminf V(x)≥ V0:=inf V(x)>0(V2)V(ai)=V0,i=1,2,...,k.dui yu ε>0,ding yi kong jian Hε{u∈H:∫RNV(εχ)|u|2dx<+∞},ji fan shu wei :(?)you tiao jian (V1)zhi ,qian ru H→Hshi lian xu de .ru guo wo men zuo bian liang ti huan x→εxze shang shu wen ti zhuai hua wei △2u+V(εx)u=λf(εx)|u|q-2u+g(εx)|u|r-2u,x ∈ RN.(0.0.1)kao lv Euler-Lagrange fan han (?)ze Iεshi C1de ,dan zai Hεshang bu shi xia fang you jie de .lei shi yu di er zhang ,wo men kao lv Nehariliu xing Nε={u∈Hε{0}:
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论文作者分别是来自山西大学的姜瑞廷，发表于刊物山西大学2019-11-12论文，是一篇关于四阶微分方程论文,凹凸项论文,存在性论文,多重性论文,流形论文,山西大学2019-11-12论文的文章。本文可供学术参考使用，各位学者可以免费参考阅读下载，文章观点不代表本站观点，资料来自山西大学2019-11-12论文网站，若本站收录的文献无意侵犯了您的著作版权，请联系我们删除。