厦门大学博硕士论文摘要库 - connecting repositories · b ) s 2009 r ... z 9 2...

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F 4 - q ? uBZ'Rl_,0Concentration Phenomenon of Variational Problemh p x

[Or6 ` rg B H H I :$qÆ 2009 4 R$MCÆ 2009 R:!LÆ 2009 RMC Qib) S 2009 4 R厦

门大学博硕士论文摘要库

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厦门大学博硕士论文摘要库

*N;;"%ehJg3yFZ,Y&9/U,p>tg,sFZ,Y9..`('4FQksSHp>tg8p3Ym,p>tU^Q*Ugg1.m'>t,a<(9lYm,p>tgZ%Y&C# eKYm,p>tg)X8;M%IIg,p>tqyI?8U,s,p>t>^1 Iq^yI?8U;<2 1IyqEP$6ZX ” ” ;oOi^k_&oOi^k


rT: 4 ~^$U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2TV 8=M$bt` . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

§0.1 7< . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

§0.2 #a?s_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4TCV [GfwEC<gQ Sobolev DS . . . . . . . . . . . . . . . 6

§1.1 VEen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

§1.2 dV|eO?F| . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7TXV k^WQY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

§2.1 > . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23

§2.2 jZ PLTX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Fy$- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32\2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34


NrT: 5

Contents

Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter 0 Introduction and Main Results . . . . . . . . . . . . . . . 3

§0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

§0.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Chapter 1 Some Lemmas and Local Generalized Sobolev

Inequality and Concentration Theorem . . . . . 6

§1.1 Definitions and Hypotheses . . . . . . . . . . . . . . . . . . . . . 6

§1.2 Notable Theorems and Preliminary Lemmas . . 7

Chapter 2 The Proof of Concentration Theorem . . . . . 23

§2.1 Blow up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

§2.2 Proof of the Existence of an Entire Extremal 26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


rz9 1AY&Qk^+/!t;s':6b+ [6] =s fw/1 'wWsupε−p∗

∫F (u) : ‖∇u‖p ≤ ε, p∂Ω u = 0j p∗ = Np

N−p 65z Sobolev B N > p > 1 u N ≥ 3 ^(0 ≤ F (t) ≤ C|t|

Np

N−p m>|2)NL'-1L'L2>Cf Sobolev 0h0fÆ3C07>CfW r'm|xG H^ ε → 0 ('wWf\U(49/`R>o\U(4r9/ofÆ3!a F X%-u^ F p 0 o7zziM5R>w=\iVCp RN f\U(49/`>U3BoJÆ'wWW 5z Sobolev (


Nrz9 2

CONCENTRATION PHENOMENON OF

VARIATIONAL PROBLEM

ABSTRACT

This dissertation, which expands the result in paper [6], is devoted

to studying the following variational problems

supε−p∗∫

F (u) : ‖∇u‖p ≤ ε, u = 0 on ∂Ω

where p∗ = NpN−p is Sobolev exponent, N > p > 1,N ≥ 3, at the same

time with 0 ≤ F (t) ≤ C|t|Np

N−p It consists of two chapters.

In the first chaper, introducing some famous theorems, we prove

local generalized Sobolev inequality and a generalized concentration-

compactness theorem.

Chaper 2 obtains that as ε → 0, the almost extremals of the above

variational problem concentrate at a single point, and the local be-

haviour of the almost extremals near the concentration point only

depends on F. Moreover, we find that the almost extremal sequence

tend to an extremal for the generalied Sobolev constant on RN pro-

vided that F satifies certain growth conditions at 0 and infinity.

Keywords: Variational problem; Concentration; Critical Sobolev

exponents


&vVeV 3U W 9>N% Aua§0.1 8=xG<Jf hUwWÆ.~ µ ∈ R1

+ u ∈ W 1,p(RN) u 6= 0 .

−div(|Du|p−2Du) = µf(x, u) x ∈ RN (1)V [8] fr' 3.5 p f(x, t) 5>rf\i( (1) 0|Gf µ > 0tXx u > 0 ~ f(x, t) $:5f\i Æ

(f1) f ∈ C1(Rn × R1+)

limt→0f(x,t)tp−1 = 0, limt→+∞

f(x,t)tp−1 = +∞ | x ∈ R

N >k(f2) 0 ≤ f(x, t) ≤ C(1 + |t|q) p − 1 < q < p∗ − 1 (f3) Up θ > p − 1 .

tft(x, t) ≥ θf(x, t) ∀x ∈ RN , t ≥ 0

(f4) lim|x|→+∞ f(x, t) = f(t) | |t| Xz>u|f(x, t) − f(t)| ≤ ε(R)(|t|p−1 + |t|p

∗−1)^ |x| ≥ R (j limR→+∞ ε(R) = 0 TDH f(x, t) f\i^kT~LlG.~f\i%L(1) xfUp! [5] a (1) %f'wW6

supε−p∗∫

F (u) : ‖∇u‖p ≤ ε, u ∈ D1,po (RN) (2)


&vVeV 4j λ > 0 F (x, t) =∫ t

0 f(x, τ)dτ |>f F (x, t) P.L.Lionsp [1] OTW g'H26f\iF (x, t) = −c(x)|t|p + F0(x, t) (3)j 0 < m ≤ c(x) ≤ M F0(x, t) ≥ 0 uUp q > p . F0(x,t)

|t|q pt > 0 (f~p t < 0 (w6z=6H*C6f\i (3) f\in# (f1) ∼ (f4) \i~pe=s xGb F ~5(F ) |R;B C 0 ≤ F (t) ≤ C|t|p

∗ ^( F (t) 6.)-up L1 GI F 6= 0.

(F+) max(F+0 , F+

∞) < SF/S∗ F+0 , F+

∞ rI ÆF+

0 := lim supt→0

F (t)

|t|p∗, F−

0 := lim inft→0

F (t)

|t|p∗,

F+∞ := lim sup

|t|→∞

F (t)

|t|p∗, F−

∞ := lim inf|t|→∞

F (t)

|t|p∗.^ F+

0 = F−0 7 F+

∞ = F−∞ xG,^ F0 := F+

0 = F−0 7 F∞ := F+

∞ =

F−∞ xGF(f F 6D0.)f

§0.2 $bt`sf':1^ ε → 0 4'wW (2) fGx uε fk!:T3B F 5 (F ) \i^ F 6)BfDU3B uε 5 (1)Cj f = F ′ VC=s': `r' 7 [Ær' 7.(W r') b (Ω) 7 (F ) + uε 6 SFε (Ω) f>\U


&vVeV 53B(4< 6Eε−p∗

∫

Ω

F (uε) = SFε (Ω)(1 + o(ε)) ^ε → 0.

1. / F+0 = F−

0 -upTfGI Ω 6= RN [ capRN (RN/Ω) > 0 VC(4 uε 9/pRo x0 ∈ Ω < 6E

|∇uε|p

εp∗ δx0

pM(Ω) ε−p∗F (uε)dx ∗ SF δx0

pM(Ω) / F+0 < SF/S∗ P Ω 6XTfVCb F+

0 = F−0 6D#:f

2. / (F+) \i+VCUp>o x0 . xε → x0 u3Bωε(y) = uε(xε + ε

nn−py)| SF f>U3B[p D1,p

o (RN) ωε → ω, ‖∇ω‖p = 1 u∫

RN F (ω) = SF TDHr' 7 ':X0w/Æm>\U3B(4 (uε) W Ω fR>ompW ofR>UfkYU3B(4|>YU3B~3B 6 (2) ε = 1 f>x厦门大学博硕士论文摘要库

&vVeV 6TCV [GfwEC<g Sobolev DS§1.1. VEenpv xGrI D1,p

o (Ω) = C∞o (Ω)

‖∇u‖p

i.e. D1,po (Ω)rIl C∞

o (Ω)%B ‖∇v‖p = (∫

Ω |∇v|p)1p f"VC Sobolev 0h0 (h [7]) xGF 405+

1. p D1,po (Ω) u = C v u = 0

2. / Ω XzVC D1,po (Ω) = W 1,p

o (Ω) r0rI D1,po (Ω) f9/ Æ

uε u pD1,po (Ω) h

∇uε ∇u pLp(Ω) VCV Sobolev rr' Up24. uε u pLp∗(Ω) uε → u pLq

loc(Ω) ,j 1 ≤ q < p,

uε → u pΩ a.e.^(xGX (Ω) (F ) 7 (F+) bÆ(Ω) Ω 6 R

N f> N ≥ 3.

(F ) |R;B C 0 ≤ F (t) ≤ C|t|p∗ ^( F (t) 6.)-u


&vVeV 7p L1 GI F 6= 0.

(F+) max(F+0 , F+

∞) < SF/S∗ F+0 , F+

∞ rI ÆF+

0 := lim supt→0

F (t)

|t|p∗, F−

0 := lim inft→0

F (t)

|t|p∗,

F+∞ := lim sup

|t|→∞

F (t)

|t|p∗, F−

∞ := lim inf|t|→∞

F (t)

|t|p∗.^ F+

0 = F−0 7 F+

∞ = F−∞ xG,^ F0 := F+

0 = F−0 7 F∞ := F+

∞ =

F−∞6>ps xG^ Ω l Ω f"M(Ω) l Ω f Borel 8wfW;+_M(Ω) 6 C(Ω)∗ ffIfW;KQM(Ω) f9/rI Æ

µε∗ µh |Gf3B φ ∈ C(Ω) X

∫

Ω

φdµε →

∫

Ω

φdµ

§1.2. dV|eO?F|vxG':JHfj \%:f0h0Poincare 0h07 Sobolev0h0DYL\%:fL'r'7L'pD>fL K;T`1. Poincare 0h0b Q 6[X)BP Lipschitz $zfXTf. -u θ ∈

(0, 1) VCUp>;B C(Q, θ) . |Gf ω ∈ W 1,p0 (Q) u |ω 6=


&vVeV 8

0| ≤ θ|Q|, J0h0+‖ω‖p∗ ≤ C(Q, θ)‖∇ω‖p| Poincare 0h0 ( h [10]) xGXl>f0Æ / u ∈ W 1,p(Ω) VC4002t+

∫

Ω

|u|p∗

≤ C( ∫

Ω

|∇u|p +

∫

∂Ω

|u|p)

∫

Ω

|u|p∗

≤ C( ∫

Ω

|∇u|p +

∫

Γ

|u|p), m(Γ) > 0.

2. Sobolev 0h0|"d D1,po (Ω) f3B v 5

∫

Ω

|v|p∗

≤ S∗(

∫

Ω

|∇v|pdx)p∗

pj S∗ 6 Sobolev ;Ba Ω z%-u^ Ω = RN (U S∗ W` x6 [7].

3.>Cf Sobolev 0h0l2L>Cf Sobolev 0h0xG$rI>Cf Sobolev ;BxG^SF

ε (Ω) := ε−p∗ sup

∫

Ω

F (u) : ‖∇u‖p ≤ ε, u ∈ D1,po (Ω)7

SF := SF1 (Rn).VC SF 6?l>Cf Sobolev ;B^(xGE uε | SF

ε (Ω)


&vVeV 96\U(4^u~^ uε a SFε (Ω) frI-u5

ε−p∗∫

Ω

F (uε) = SFε (Ω)(1 + o(1)) ^ε → 0.^e=s':S='wWf\U(4~0S='wWftUfUpwWpxGL>C Sobolev0h0pxG> ∫

Ω F (u)f!>1>L'%L>C Sobolev f0h0:Tb (F ) xG 0 < SF ≤ cS∗ < ∞. T3BfJ!xGrIÆuε := u(x/s).VC

∫

sΩ

F (us) = sn

∫

Ω

F (u),∫

sΩ

|us|p = sn−p

∫

Ω

|u|p. /xG s = ‖∇u‖− p

n−p

p , VC ‖∇us‖p = 1 -u∫

sΩ

F (us) = ‖∇u‖−p∗

p

∫

Ω

F (u)r%xGOT\L'%LÆ3>Cf Sobolev 0h0L' 1, |Gf δ > 0 Up>;B K(δ) > 0 . |Gf 0 <

r < R 5 r/R ≤ K(δ) VCUp>u3B φrR(x) ∈ W 1,∞(RN) -u

φrR(x) =

1 ^x ∈ Br

0,

0 ^x ∈ RN/BR

0 .


&vVeV 10. j5∫

BRx

|∇(φrRu)|p ≤

∫

BRx

|∇u|p + δ

∫

RN

|∇u|pLÆφr

R(x) =

1 ^x ∈ Br0,

log(|x|/R)log(r/R)

r ≤ |x| ≤ R

0 ^x ∈ RN/BR

0 .VCF ∫

BR0

|∇φrR|

N =ωN−1

(log Rr )N−1

,

∫

BR0

|∇φrR|

p =1

N − p

ωN−1(RN−p − rN−p)

(log Rr )N−1

(p < N),LDxG limR→∞

∫BR

0|∇φr

R|p = ∞(p < N)6limR→∞

∫BR

0|∇φr

R|N =

0 gQxGX(

∫

BRx

|∇(φrRu)|p)

1p ≤ (

∫

BRx

|φrR∇u|p)

1p + (

∫

BRx

|∇φrRu|p)

1p

≤ (

∫

BRx

|∇u|p)1p + (

∫

BRx

|∇φrR|

N)1N (

∫

BRx

|u|p∗

)1p∗

= (

∫

BRx

|∇u|p)1p + (

ωn−1

(log Rr )N−1

)1N (

∫

BRx

|u|p∗

)1

p∗

≤ (

∫

BRx

|∇u|p)1p + (

ωn−1(S∗)

P∗

n

(log Rr )n−1

)1N (

∫

RN

|∇u|p)1pLD∫

BRx|∇(φr

Ru)|p ≤ ((∫

BRx|∇u|p)

1p +CR(

∫RN |∇u|p)

1p )p j CR =

(ωn−1(S∗)

P∗

n

log(Rr)n−1 )

1N . \ [9]

|a + b|p ≤ (1 + ε)|a|p + ((1 + ε)1

p−1 − 1)−p

p∗ (1 + ε)|b|p ∀ε > 0


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 submitrequests online, or consult the interlibrary loan department in your library. 2. For users of non-CALIS member libraries, please mail to [email protected] for delivery details.


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