bài toán biên cho một vài lớp phương trình có chứa toán tử elliptic suy biến mạnh

R
N
N
C
k
(Ω) k Ω
L
p
(Ω) p
Ω

x

x
=

∂
∂x
1
, ,
∂
∂x
N
1

x


y

y
=

∂
∂y
1
, ,
∂
∂y
N
2

1 y

z

z
=

∂
∂z
1
, ,
∂
∂z
N
3


z
∆
x
x : ∆
x
=
N
1

i=1
∂
2
∂x
2
i
∆
y
y : ∆
y
=
N
2

j=1
∂
2
∂y
2
j

∆
z
z : ∆
z
=
N
3

l=1
∂
2
∂z
2
l
(., .) L
2
(Ω)
P
α,β
P
α,β
u = ∆
x
u + ∆
y
u + |x|
2α
|y|
2β
∆

z
u α, β ≥ 0 α + β > 0,
|x|
2α
=

N
1

i=1
x
2
i

α
, |y|
2β
=

N
2

j=1
y
2
j

β
,
dx = dx

1
dx
2
dx
N
1
, dy = dy
1
dy
2
dy
N
2
, dz = dz
1
dz
2
dz
N
3
C(X, Y ) X Y.
C
1
(X, Y ) X Y.
P
α,β
u = ∆
x
u + ∆

y
u + |x|
2α
|y|
2β
∆
z
u, α, β ≥ 0, α + β > 0.
P
α,β
S(t)
G
k
u = ∆
x
u + |x|
2k
∆
y
u (x, y) ∈ Ω ⊂ R
N
1
+N
2
, N
1
, N
2
≥ 1, k ∈ Z
+

,
• k = 0 G
0
Ω.
• k > 0 G
k
Ω ⊂ R
N
1
+N
2
x = 0
G
k
u
Ω Ω
G
k
∆u =
∂
2
u
∂x
2
1
+
∂
2
u
∂x

2
2
+ +
∂
2
u
∂x
2
n
.





∆u + f(u) = 0 Ω,
u = 0 ∂Ω,
(1)
Ω R
n
(n ≥ 2),
f(u) = λu + |u|
p−1
u.
• n = 2 1 < p < ∞,
• n ≥ 3 λ = 0 p ≥
n + 2
n − 2
Ω
• n ≥ 3 λ = 0 1 < p <

n + 2
n − 2
,
n ≥ 3 p
0
=
n + 2
n − 2
p
0
+ 1 =
2n
n − 2
p
0





−∆u = λu + u
n+2
n−2
Ω,
u = 0 ∂Ω,
(2)
Ω R
n
, n ≥ 3
• n ≥ 4 0 < λ < λ

1
λ
1
• n = 3 0 < λ
∗
< λ < λ
1
Ω
λ
∗
=
1
4
λ
1





−L
k
u + f(u) = 0 Ω,
u = 0 ∂Ω,
(3)
Ω R
2
L
k
u =

∂
2
u
∂x
2
+ x
2k
∂
2
u
∂y
2
(k ≥ 1)
f(u) = u|u|
γ−1
• γ ≥
4
k
Ω L
k
• 0 < γ <
4
k
4 + k
k
L
k






−P
k
u + f(u) = 0 Ω,
u = 0 ∂Ω,
Ω R
2
(k ≥ 1) f (u) = u|u|
γ−1
P
k
u =
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+ x
2k
∂
2
u
∂z

2
.
γ ≥
4
k + 1
Ω P
k





−G
k
u + f(u) = 0 Ω,
u = 0 ∂Ω,
G
k
u = ∆
x
u + |x|
2k
∆
y
u, k ≥ 1,
Ω R
N
1
+N
2

x ∈ R
N
1
y ∈ R
N
2
∂Ω
f(u) = u|u|
γ−1
γ >
4
N
1
+ N
2
(k + 1) − 2
Ω
G
k
2
∗
k
=
2N(k)
N(k) −2
N(k) = N
1
+(k+1)N
2
N

1
+ N
2
(k + 1) + 2
N
1
+ N
2
(k + 1) − 2
.













u
t
− G
k
u + f(u) + g(x, y) = 0 (x, y) ∈ Ω, t > 0,
u(x, t) = 0 (x, y) ∈ ∂Ω, t > 0,
u(x, 0) = u

0
(x) (x, y) ∈ Ω,
(4)
Ω R
N
1
+N
2
∂Ω u
0
∈ S
1
0
(Ω)
g(x, y) ∈ L
2
(Ω) f : R → R
|f(u) − f(v)| ≤ C
0
|u − v|(1 + |u|
ρ
+ |v|
ρ
)
2
N(k) −2
< ρ <
4
N(k) −2
,

F (u) ≥ −
µ.u
2
2
− C
1
,
f(u)u ≥ −µu
2
− C
2
,
C
0
, C
1
, C
2
≥ 0 µ < λ
1
, λ
1
−G
k
Ω
X
1
2
f : R → R
C

1
|u|
p
− C
0
≤ f(u)u ≤ C
2
|u|
p
+ C
0
, p > 2,
f

(u) ≥ −C
3
, u ∈ R,
C
0
, C
1
, C
2
, C
3
G
k
•
•
•

g(x, y, z, t) = λt + |t|
γ
t λ ≤ 0
γ ≥
4
N
α,β
− 2
u ∈ H
2
(Ω)
g(x, y, z, t)
g(x, y, z, t) (I)
1
−
(I)
6
L
p
(Ω)
D
−γ
γ >
2
∗
α,β
(2 − p)
2p(2
∗
α,β

− 2)
f f : u(x, y, z) → f(x, y, z, u(x, y, z))
D
1
2
→ D
−γ
0
γ
0
=
ρ
2.(2
∗
α,β
− 2)
u ∈ C([0, T ], D
1
2
)
u ∈ C([0, T ], D
1
2
)
f
S(t) D
1
2
S(t)
S

1
0
(Ω)
f,
S(t)
L
2
(Ω)
P (x, D) =

|α|≤m
a
α
(x)D,
x = (x
1
, x
2
, , x
n
) ∈ Ω ⊂ R
n
α = (α
1
, , α
n
) ∈ N
n
|α| = α
1

+ α
2
+ + α
n
, D = (D
1
, , D
n
), D
j
= −i
∂
∂x
j
,
D
α
= D
α
1
D
α
2
D
α
n
= (−i)
|α|
∂
|α|

∂
α
1
x
1
∂
α
n
x
n
a
α
(x) ∈ C(Ω)
ξ
α
= ξ
α
1
1
ξ
α
2
2
ξ
α
n
n
, ξ = (ξ
1
, ξ

2
, , ξ
n
) ∈ R
n
.
P (α, ξ) =

|α|≤m
a
α
(x)ξ
α
P (x, D)
P
m
(x, D) =

|α|=m
a
α
(x)ξ
α
P (x, D)
P (x, D) x ∈ Ω
∀ξ ∈ R
n
: P
m
(x, ξ) = 0 ξ = 0. (5)

P (x, D) Ω x ∈ Ω
G
k
k > 0 Ω
x = 0 G
k
P
α,β
u = ∆
x
u + ∆
y
u + |x|
2α
|y|
2β
∆
z
u
x = (x
1
, x
2
, , x
N
1
) ∈ R
N
1
y = (y

1
, y
2
, , y
N
2
) ∈ R
N
2
z = (z
1
, z
2
, , z
N
3
) ∈ R
N
3
, α + β > 0, α ≥ 0, β ≥ 0,
P
α,β
(x, y, z, ξ) = −(ξ
2
1
+ ξ
2
2
+ + ξ
2

N
1
+ ξ
2
N
1
+1
+ + ξ
2
N
1
+N
2
+(ξ
2
N
1
+N
2
+1
+ + ξ
2
N
1
+N
2
+N
3
)|x|
2α

|y|
2β
).
M
0
(x
0
, y
0
) x
0
= 0, y
0
= 0
P
α,β
(x, y, z, ξ) = 0 ξ
i
= 0 i = 1, , N
1
+ N
2
+ N
3
P
α,β
x
0
= 0 y
0

= 0 ξ = (0, , 0, 1) = 0
P
α,β
(x, y, z, ξ) = 0 Ω
M
0
. P
α,β
P
α,β
Ω ⊂ R
N
1
+N
2
+N
3
x = 0 y = 0 P
α,β
u ∈ H
A D(A)(D(A) = H)
(Au, v) = (u, Av) u, v ∈ D(A).
A
(Au, u) ≥ C||u||
2
inf
u∈D(A)
(Au, u)
||u||
2

> 0
u ∈ D(A) \ {0}.
u, v ∈ D(A),
[u, v]
A
= (Au, v)
H
D(A) |u|
A
=

[u, u]
A
D(A).
H
A
D(A)
|u|
A
H
A
⊂ H.
inf
a∈H
A
\{0}
|a|
2
A
||u||

2
= λ
1
> 0.
inf
a∈H
A
\{0}
|a|
2
A
||u||
2
u
1
, u
1
λ
1
Au
1
= λ
1
u
1
.
H
(1)
A
= {u ∈ H

A
: [u, u
1
]
A
= (Au, u
1
)
H
= 0, }
inf
a∈H
(1)
A
\{0}
|a|
2
A
||u||
2
= λ
2
≥ λ
1
.
inf
a∈H
(1)
A
\{0}

|a|
2
A
||u||
2
u
2
, u
2
λ
2
Au
2
= λ
2
u
2
.
0 < λ
1
≤ λ
2
≤ λ
3
≤ λ
n
≤ u
1
, u
2

, , u
n
,
H
A
H
inf
a∈H
(i)
A
\{0}
|a|
2
A
||u||
2
H
(i)
A
i = 1, 2 ,
λ
1
, λ
2
, , λ
i
, i = 1, 2 ,
λ
i
,

λ
i
, i = 1, 2 , lim
m→∞
λ
m
= +∞
X Y U(x)
x f : U(x) ⊂ X → Y
x T ∈ L(X, Y )
f(x + h) − f(x) − T h = o(h), h > 0,
h 0 T
f x f

(x) = T
df(x; h) = f

(x)h
α ≥ 0 f ∈ L
p
(R
n
) 1 ≤ p < ∞ g
α
(f)
g
α
(f) = G
α
∗ f G

α
(x) = (1 + 4π
2
|x|
2
)
−
2
α
α > 0 g
0
(f) = f
|G
α
|
1
= 1 |g
α
(f)|
p
≤ |f|
p
.
L
p
α
(R
n
) L
p

α
(R
n
) = g
α
(L
p
(R
n
)) α ≥ 0
1 ≤ p < ∞
X
S(t) : X → X t ≥ 0
S(0) = I
S(t + s) = S(t)S(s) = S(s)S(t) t, s ≥ 0
S(t)u
0
(t, u
0
) ∈ [0, +∞) × X.
{S(t)}
t≥0
X
S(t)
Φ ∈ C
0
(X, R) Φ(S(t)u) ≤ Φ(u)
t ≥ 0, u ∈ X, Φ(S(t)u) = Φ(u) t ≥ 0, u
S(t)u = u t ≥ 0. Φ
S(t)

S(t)
X A ⊂ X
S(t)
A
A S(t)A = A t ≥ 0
A B ⊂ X
dist(S(t)B, A) → 0 t → +∞
dist(S(t)B, A) = sup
a∈S(t)B
inf
b∈A
d(a, b).
X S(t)
t > 0, S(t)
S(t) = S
(1)
(t) + S
(2)
(t)
S
(1)
(t) S
(2)
(t)
B ⊂ X
sup
y∈B
||S
(1)
(t)||

X
→ 0, t → +∞.
t
0
∪
t≥t
0
S
(2)
(t)B X.
S(t), t ≥ 0
S(t)
E Σ(E)
E E \{0} 0 A ∈ Σ(E)
n γ(A) = n n
Φ ∈ C(A, R
n
\ {0})
n γ(A) = +∞ γ(φ) = 0.
X C([0, T ]; X)
u : [0, T ] → X
u
C([0,T ];X)
= max
0≤t≤T
||u(t)||
X
.
X
x ∈ X γ

+
(x) = {S(t)x : t ≥ 0}.
B ⊂ X B
γ
+
(B) = ∪
t≥0
S(t)B = ∪
z∈B
γ
+
(z).
L
p
((a, b); X)
u : (a, b) → X
||u||
p
L
p
((a,b);X)
=
b

a
||u||
p
X
dt < +∞.
ϕ(t)

(0, T )
ϕ(t) ≤ c
0
t
−γ
0
+ c
1
t

0
(t − s)
−γ
1
ϕ(s)ds, t ∈ (0, T ),
c
0
, c
1
≥ 0 0 ≤ γ
0
γ
1
< 1
K = K(γ
1
, c
1
, T )
ϕ(t) ≤

c
0
1 − γ
0
t
−γ
0
K(γ
1
, c
1
, T ), t ∈ (0, T ).
X
0
, X, X
1
X
0
→ X → X
1
, X X
1
X
0
X X
0
, X
1
1 < α
0

, α
1
< ∞,
E = {u ∈ L
α
0
(0, T ; X
0
),
du
dt
∈ L
α
1
(0, T ; X
1
)},
||u||
E
= ||u||
L
α
0
(0,T ;X
0
)
+




du
dt



L
α
1
(0,T ;X
1
)
.
E → L
α
0
(0, T ; X)
P
α,β
u − C(x, y, z)u + g(x, y, z, u) = 0 Ω,
u = 0 ∂Ω,
Ω R
N
1
+N
2
+N
3
∂Ω
x = (x
1

, x
2
, , x
N
1
) ∈ R
N
1
y = (y
1
, y
2
, , y
N
2
) ∈ R
N
2
z = (z
1
, z
2
, , z
N
3
) ∈ R
N
3
C(x, y, z) ≥ 0 C(x, y, z) ∈ C
0,σ

(Ω)
g(x, y, z, 0) = 0 g(x, y, z, t) = 0 α, β ≥ 0 α + β > 0 0 < σ ≤ 1
G(x, y, z, t) =

t
0
g(x, y, z, s)ds,
ν = (ν
x
, ν
y
, ν
z
) (ν
x
1
, ν
x
2
, , ν
x
N
1
, ν
y
1
, ν
y
2
, , ν

y
N
2
, ν
z
1
, ν
z
2
, , ν
z
N
3
)
∂Ω
N = N
1
+ N
2
+ N
3
, N
α,β
= N
1
+ N
2
+ N
3
(α + β + 1),

(a, b, c) = (a
1
, , a
N
1
, b
1
, , b
N
2
, c
1
, , c
N
3
) ∈ R
N
,
P
α,β
u = ∆
x
u + ∆
y
u + |x|
2α
|y|
2β
∆
z

u.
Ω P
α,β
{(0, 0, 0)}
(x, ν
x
) + (y, ν
y
) + (α + β + 1)(z, ν
z
) > 0
(x, y, z) ∈ ∂Ω.
C(x, y, z) = 0, g(x, y, z, t) = g(t) u(x, y, z)
H
2
(Ω)
u(x, y, z)

Ω

N
α,β
G(u) −
N
α,β
− 2
2
g(u)u

dxdydz

+

Ω
|x|
2(α−1)
|y|
2(β−1)
|
z
u|
2
(α(x, a)|y|
2
+ β(y, b)|x|
2
)dxdydz
=
1
2

∂Ω
ν
α,β
ν
α,β
(a,b,c)

∂u
∂ν


2
ds,
ν
α,β
= |ν
x
|
2
+ |ν
y
|
2
+ |x|
2α
|y|
2β
|ν
z
|
2
,
ν
α,β
(a,b,c)
= (x − a, ν
x
) + (y − b, ν
y
) + (α + β + 1)(z − c, ν
z

).

Ω
G(u)dxdydz = −

Ω
x
i
g(u)
∂u
∂x
i
dxdydz,
N
1

Ω
G(u)dxdydz = −

Ω

(x − a), 
x
u

g(u)dxdydz.
α
1
, β
1

α
1

Ω
G(u)dxdydz = −
α
1
N
1

Ω

(x − a), 
x
u

g(u)dxdydz,
β
1

Ω
G(u)dxdydz = −
β
1
N
2

Ω

(y − b), 

y
u

g(u)dxdydz,

Ω
G(u)dxdydz = −
1
N
3

Ω

(z −c), 
z
u

g(u)dxdydz.
(α
1
+β
1
+1)

Ω
G(u)dxdydz = −

Ω

α

1
((x − a), 
x
u)
N
1
+
β
1
((y − b), 
y
u)
N
2
+
((z −c), 
z
u)
N
3

g(u)dxdydz,
α
1
, β
1
g(u) = −(∆
x
u + ∆
y

u + |x|
2α
|y|
2β
∆
z
).
(α
1
+ β
1
+ 1)

Ω
G(u)dxdydz =

Ω

α
1
((x − a), 
x
u)
N
1
+
β
1
((y − b), 
y

u)
N
2
+
((z −c), 
z
u)
N
3

× (∆
x
u + ∆
y
u + |x|
2α
|y|
2β
∆
z
)dxdydz.
(α
1
+ β
1
+ 1)

Ω
G(u)dxdydz =
9


i=1
I
i
,
I
1
=

Ω
α
1

(x − a), 
x
u

N
1
∆
x
udxdydz,
I
2
=

Ω
α
1


(x − a), 
x
u

N
1
∆
y
udxdydz,
I
3
=

Ω
α
1
N
1

(x − a), 
x
u

|x|
2α
|y|
2β
∆
z
udxdydz,

I
4
=

Ω
β
1
N
2

(y − b), 
y
u

∆
x
udxdydz,
I
5
=

Ω
β
1
N
2

(y − b), 
y
u


∆
y
udxdydz,
I
6
=

Ω
β
1
N
2

(y − b), 
y
u

|x|
2α
|y|
2β
∆
z
udxdydz,
I
7
=

Ω


(z −c), 
z
u

N
3
∆
x
udxdydz,
I
8
=

Ω

(z −c), 
z
u

N
3
∆
y
udxdydz,
I
9
=

Ω


(z −c), 
z
u

N
3
|x|
2α
|y|
2β
∆
z
udxdydz.
I
i
i = 1, 9
I
1
.
I
1
=
α
1
N
1

Ω


(x
1
− a
1
)
∂u
∂x
1
+ (x
2
− a
2
)
∂u
∂x
2
+ + (x
N
1
− a
N
1
)
∂u
∂x
N
1

×


∂
2
u
∂x
2
1
+ +
∂
2
u
∂x
2
N
1

dxdydz.