一类拟线性方程组的可解性

E m w+ = 0
−∆u = f x, T u, T v, D(T u), D(T v) ,
"Z u.
−∆u ≥ f x, u, T v, D(u), D(T v) .
−∆w ≤ f x, T u, T v, D(T u), D(T v) − f x, u, T v, D(u), D(T v) .
4 F @N > A @N y ^e u b v 0l
< 0 | Oyv- |
- W # j'D L m *3 F K~0& bu
(1 )
sg
Sattinger
V1 =<
⎧ ⎪⎨ −∆u = f (x, u, v, Du, Dv),
x ∈ Ω,
@Ω
L 2 2 C= 5 ⎪⎩
−∆v = g(x, u = v = 0,
u + u + 2u − |u − u| u + u − 2u − |u − u|
T u(x) =
+
,
4
4
v + v + 2v − |v − v| v + v − 2v − |v − v|
T v(x) =
+
% L F @ % L * R 4
4
l h [5] A.6 T u, T v ∈ W 1,q0(Ω).
3 xbnl
@ E# T {
P( 8 1 0X
W %3 F
⎧
⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩
−∆u −∆v
= =
λ431λφ21(φx()x+) +2λ9λ11221uu++vλ4+1 vλ+1φ√|D43uλ|11232
, φ(x)
|Dv|
1 2
,
@ L 2 2 C=5 u = v = 0,
Ω RN , N > 2
(9)
i = 1, 2, 3, 4.
(5)–(9), $ (4) r"
u 2,p + v 2,p ≤ C1ε
<ε
=
1 2C1
,
$NZ.
C
u 2,p +
v 2,p
+. 2
u 2,p + v 2,p ≤ C,
M U " V 37 91 F 0 (4) F # 10.2) &
;;
2 E /}
Y :58 u 1,q0 + v 1,q0 ≤ C. = Y. m (u, v) ∈ W 1,q0(Ω) 2,
}m (1) Q& ~ u, u, v, v 0z
⎧ ⎪⎨ u(x),
u ≤ u,
T u(x)
=
⎪⎩
u(x), u(x),
⎧
⎪⎨ v(x),
u ≤ u ≤ u, u ≤ u, v ≤ v,
)w
T v(x)
=
⎪⎩
v(x), v(x),
v ≤ v ≤ v, v ≤ v.
T u(x) ≤ max |u|, |u| = M, a.e. in Ω,
Leabharlann Baidu6.4
(2) (3)
w1, w2 ∈ W 2,p(Ω) ∩ W01,p(Ω),
2 ( R 3 # -?8 0g A $ :H8 S
S(0, u, v) = (0, 0).
Sobolev
W 2,p(Ω) → →
F R E T W1,q0(Ω)
S !g
= =.P ( u, v ∈ W 1,q0(Ω),
t ∈ [0, 1],
⎪⎩
L2v u=
= g(x, v = 0,
u,
v),
x ∈ Ω, x ∈ ∂Ω, A!!
@ L 2 2 5 Ω RN (N ≥ 1)
Ye ∂Ω 0 =
0 #N F @N #J 6 R f},gm:
Ω×R (1 )
×
R0→
R
F"CPRN80~^FeC
Lk,~k)=h1, 2,
"PF@Cy^
L 0 W - L 2 $ [4] Y -F }mN * D L] }m (1 ) 0H
[Z, W ] = y(x) ∈ Lp(Ω) : Z(x) ≤ y(x) ≤ W (x), a.e. in Ω .
`k # = I s8 1
(H1)–(H4)
#W 4~_ [W2,p(Ω)]2.
G\}m (1) & =KN ~
1 F @ =t H (H2)
[αp, p∗)
p∗
=
Np N −p
.
4 <^We8m C X N q0 ∈ [αp, p∗), T : W 1,q0(Ω) → W 1,q0(Ω) ∩ L∞(Ω)
u αip ≤ C, v αip ≤ C, Du αip ≤ C, Du αip ≤ C,
Gs8 K= 2 S Dv αip ≤ C, u 2,αip ≤ C u 2,p, v 2,αip ≤ C v 2,p,
u 2,p ≥ 1, v 2,p ≥ 1 (
u 2,p, v 2,p
).
Dv αip ≤ C,
T v(x) ≤ max |v|, |v| = m, a.e. in Ω,
$?=E' u, v ∈ W 1,q0(Ω), T u, T v ∈ L∞(Ω).
3= %l :H- F $ u, v ∈ W1,q0(Ω),
W 2,p(Ω)
W 1,q0 (Ω) u, u, v, v ∈ W 1,q0 (Ω).
(4)
@ $ 4 3hi(x, Tu, Tv) p ≤ C,
C
b u, u, v, v p.
i = 1, 2.
(5)
⎧ ⎪⎨
Du αip αi ,
u ≥ u,
|D(T u)|αi p =
D(T u) αip αi = ⎪⎩ ⎧ ⎪⎨
Du αip αi , Du αip αi , Dv αip αi ,
(2)
w1 = 0,
x ∈ ∂Ω.
−∆w2 = tg x, T u, T v, D(T u), D(T v) , x ∈ Ω,
(3)
w2 = 0,
x ∈ ∂Ω.
0 F ' nw E (H1)–(H4) f, g ∈ Lp(Ω),
u, v ∈ W 1,q0 (Ω).
91 L F # ) 8 b J & u ~ [6]
(4) −∆u − f (x, u, v, Du, Dv) ≤ 0 ≤ −∆u − f (x, u, v, Du, Dv), v ∈ [v, v],
= 8( −∆v − g(x, u, v, Du, Dv) ≤ 0 ≤ −∆v − g(x, u, v, Du, Dv), u ∈ [u, u]. Z(x), W (x) ∈ Lp(Ω),
T u, T v ∈ W 1,q0 (Ω) ∩ L∞(Ω).
g S:
[0, 1] × W 1,q0 (Ω) 2 → W 1,q0 (Ω) 2,
8 b J S(t, u, v) = (w1, w2),
w1 w2
%
422
+.b
26
}m0~
−∆w1 = tf x, T u, T v, D(T u), D(T v) , x ∈ Ω,
Ye 0
x ∈ Ω,
x ∈ Ω, x ∈ ∂Ω,
(13)
φ(x)
=
ϕ(x) sup |ϕ| + sup |Dϕ|
≤
1.
Ω
Ω
+ . 424
-L B ) R 80kλ1
> 0, ϕ(x)
^e
>
#
0
J
b g −∆
LΩ 0-Dirichlet
# BG26
nw03 k b
−∆φ(x)
=
−∆ϕ(x) sup |ϕ| + sup |Dϕ|
u,
v,
Du,
Dv),
RN (N ≥ 2)
Ye ∂Ω 0
x ∈ Ω, x ∈ ∂Ω,
=
(1)
T R8nw
^e ( (H1) f, g : Ω × R × R × RN × RN → R Caratheodory
f (x, s, t, ξ, η) ≤ h1(x, s, t) + k1|ξ|α1 + k2|η|α2 ,
! T b ^e ( h1(x, s, t) h2(x, s, t)
Caratheodory
'=E (H2)
r > 0,
sup
hi(·, s, t) ∈ Lp(Ω),
2N N +1
< p < N,
421
|s|≤r, |t|≤r
2
}wfd h (H3) max {α1, α2, α3, α4} = α ≤ 1,
A
L 4 % w+ = 0. B E Ω u ≤ u. q
$ . h m T u = u. b u ≤ u
(12)
Ω |∇w+|2 dx = 0,
v ≤ v ≤ v.
w+ = 0, x ∈ Ω,
) = # T u = u b T v = v. )H (10) Z .- 0~ (u, v) ∈ W 2,p(Ω) 2 (1)
S(t, u, v) =
1 % C= 0:H8 bnw l~0 (u,v) u,v
u 1,q0 + v 1,q0 ≤ C. (H1)
8 .-
u 1,q0 ≤ C u 2,p ≤ C
91 (F v 1,q0 ≤ C v 2,p ≤ C Tu b Tv 08
h1(x, T u, T v) p + k1 |D(T u)|α1 p + k2 |D(T v)|α2 p , h2(x, T u, T v) p + k3 |D(T u)|α3 p + k4 |D(T v)t|α4 p),
7 2 26 2 3 7
2003 4 7
,/c
ACTA MATHEMATICAE APPLICATAE SINICA
Vol. 26 No. 3 July, 2003
yjmrta]~_igt ∗
K(
;; o`' d cI 230026)
| v: . V " Y D+. " " V /% 7z pMU }E#BE` }/t Leray-Schauder 9v477 / ,# Q!2# OEW J}
ce^ Q M }
f 947 Leray-Schauder
{u 1
X - n6 S
0 - W ) * Z? @ W 1U
D
X @ > @ # k( > [1–3]
W G0 }m
Si ALF0
⎧
BP ' f D D :i F*" 0d> +|a ? }9m$0e PT6 =Z ZF|w *F0
⎪⎨ L1u = f (x, u, v), x ∈ Ω,
(H4)
αi
≥
1 p
αi = 0, i = 1, 2, 3, 4.
`z # (u, u), (v, v) }m (1) 0 =KN
~
TI(
(1) u, u, v, v ∈ W 2,p(Ω) ∩ L∞(Ω) in Ω;
(2) u ≤ u, v ≤ v in Ω;
(3) u ≤ 0 ≤ u, v ≤ 0 ≤ v on ∂Ω;
Leray-Schauder S(1, u, v) = (u, v),
L 423
(v [8] 8
⎧ ⎪⎨ −∆u = f x, T u, T v, D(T u), D(T v) ,
x ∈ Ω,
⎪⎩
−∆v = g x, T u, T v, D(T u), D(T v) u = v = 0,
,
x ∈ Ω, x ∈ ∂Ω.
v ≤ v ≤ v, v ≤ v, i = 2, 4.
(7)
Du αip ≤ k1ε u 2,αip + k2(ε) u αip, u ≤ u ≤ u,
(8)
)w b F Dv αip ≤ k3ε v 2,αip + k4(ε) v αip,
v ≤ v ≤ v.
u ≤ u ≤ u, v ≤ v ≤ v, αi ≤ 1, i = 1, 2, 3, 4 u, u, v, v ∈ W 1,q0 (Ω),
(11)
(11) " N w+, Ω NjJ.
∇w · ∇w+ dx ≤ f x, T u, T v, D(T u), D(T v)
Ω
Ω
− f x, u, T v, D(u), D(T v) w+ dx.
(12)
= R A N A = x ∈ Ω : u(x) ≤ u(x) , B = x ∈ Ω : u(x) > u(x) , Ω = A ∪ B.
S 8 8 S PO g(x, s, t, ξ, η) ≤ h2(x, s, t) + k3|ξ|α3 + k4|η|α4,
∗
{q200@1 5
4
i
F], 5 17
2002
_ a (10071080 )
6gT1x8 xF]@,
MiN
_ (99J10142 )
/
M U " V 37
;;
2 E /}
@ e αi, ki ∈ R+, i = 1, 2, 3, 4.
u ≤ u ≤ u, u ≤ u, i = 1, 3. v ≥ v,
(6)
|D(T v)|αi p =
D(T v) αip αi = ⎪⎩
Dv αip αi , Dv αip αi ,
*~)Vp b L0 b (6) (7)
Du αip
Dv αip.
S L8 .- [7]
4.14 (Ehrling-Nirenberg-Gagliardo)
(10)
1 F E T b * - ( (u, v) ∈ W 1,q0(Ω) 2 f, g ∈ Lp(Ω) (u, v) ∈ W 2,p(Ω) 2.
u, v
u ≤ u ≤ u, v ≤ v ≤ v.
E u ≤ u. = ( H R w = u − u, )w
8 w ∈ W 2,p(Ω),
w+(x) = max 0, w(x) ,
=
λ1ϕ(x) sup |ϕ| + sup |Dϕ|
=
λ1φ(x),
&E T + 2 < p < N V
Ω