1 Introduction
Equations and systems involving the p-Laplacian operator have been extensively studied in the recent years (see, e.g., [2, 3, 5, 7, 8, 9, 10, 13, 16, 17, 19, 20, 22, 23, 26] and the references therein). In the present paper, we study the critical p-Laplacian system
(1.1)
{
-
Δ
p
u
-
λ
a
p
|
u
|
a
-
2
u
|
v
|
b
=
μ
1
|
u
|
p
∗
-
2
u
+
α
γ
p
∗
|
u
|
α
-
2
u
|
v
|
β
,
x
∈
Ω
,
-
Δ
p
v
-
λ
b
p
|
u
|
a
|
v
|
b
-
2
v
=
μ
2
|
v
|
p
∗
-
2
v
+
β
γ
p
∗
|
u
|
α
|
v
|
β
-
2
v
,
x
∈
Ω
,
u
,
v
in
D
0
1
,
p
(
Ω
)
,
where Δpu:=div(|∇u|p-2∇u) is the p-Laplacian operator defined on
D
1
,
p
(
ℝ
N
)
:=
{
u
∈
L
p
∗
(
ℝ
N
)
:
|
∇
u
|
∈
L
p
(
ℝ
N
)
}
,
endowed with the norm ∥u∥D1,p:=(∫ℝN|∇u|p𝑑x)1p, N≥3, 1<p<N, λ,μ1,μ2≥0, γ≠0, a,b,α,β>1 satisfy a+b=p, α+β=p∗:=NpN-p, the critical Sobolev exponent, Ω is ℝN or a bounded domain in ℝN and D01,p(Ω) is the closure of C0∞(Ω) in D1,p(ℝN). The case for p=2 was thoroughly investigated by Peng, Peng and Wang [21] recently; some uniqueness, synchronization and non-degenerated properties were verified there. Note that we allow the powers in the coupling terms to be unequal. We consider the two cases
Ω is a bounded domain in ℝN, λ>0, μ1,μ2=0, γ=1.
Let
(1.2)
S
:=
inf
u
∈
D
0
1
,
p
(
Ω
)
∖
{
0
}
∫
Ω
|
∇
u
|
p
𝑑
x
(
∫
Ω
|
u
|
p
∗
𝑑
x
)
p
p
∗
be the sharp constant of imbedding for D01,p(Ω)↪Lp∗(Ω) (see, e.g., [1]). Then S is independent of Ω and is attained only when Ω=ℝN. In this case, a minimizer u∈D1,p(ℝN) satisfies the critical p-Laplacian equation
(1.3)
-
Δ
p
u
=
|
u
|
p
∗
-
2
u
,
x
∈
ℝ
N
.
Damascelli, Merchán, Montoro and Sciunzi [14] recently showed that all solutions of (1.3) are radial and radially decreasing about some point in ℝN when 2NN+2≤p<2. Vétois [25] considered a more general equation and extended the result to the case 1<p<2NN+2. Sciunzi [24] extended this result to the case 2<p<N. By exploiting the classification results in [4, 18], we see that, for 1<p<N, all positive solutions of (1.3) are of the form
(1.4)
U
ε
,
y
(
x
)
:=
[
N
(
N
-
p
p
-
1
)
p
-
1
]
N
-
p
p
2
(
ε
1
p
-
1
ε
p
p
-
1
+
|
x
-
y
|
p
p
-
1
)
N
-
p
p
,
ε
>
0
,
y
∈
ℝ
N
,
and
(1.5)
∫
ℝ
N
|
∇
U
ε
,
y
|
p
𝑑
x
=
∫
ℝ
N
|
U
ε
,
y
|
p
∗
𝑑
x
=
S
N
p
.
In case (H1), the energy functional associated with system (1.1) is given by
I
(
u
,
v
)
=
1
p
∫
ℝ
N
(
|
∇
u
|
p
+
|
∇
v
|
p
)
-
1
p
∗
∫
ℝ
N
(
μ
1
|
u
|
p
∗
+
μ
2
|
v
|
p
∗
+
γ
|
u
|
α
|
v
|
β
)
,
(
u
,
v
)
∈
D
,
where D:=D1,p(ℝN)×D1,p(ℝN), endowed with the norm ∥(u,v)∥Dp=∥u∥D1,pp+∥v∥D1,pp. In this case, (1.1) with α=β and p=2 is well studied by Chen and Zou [11, 12]. Define
𝒩
=
{
(
u
,
v
)
∈
D
:
u
≠
0
,
v
≠
0
,
∫
ℝ
N
|
∇
u
|
p
=
∫
ℝ
N
(
μ
1
|
u
|
p
∗
+
α
γ
p
∗
|
u
|
α
|
v
|
β
)
,
∫
ℝ
N
|
∇
v
|
p
=
∫
ℝ
N
(
μ
2
|
v
|
p
∗
+
β
γ
p
∗
|
u
|
α
|
v
|
β
)
}
.
It is easy to see that 𝒩≠∅ and that any nontrivial solution of (1.1) is in 𝒩. By a nontrivial solution we mean a solution (u,v) such that u≠0 and v≠0. A solution is called a least energy solution if its energy is minimal among energies of all nontrivial solutions. A solution (u,v) is positive if u>0 and v>0, and semitrivial if it is of the form (u,0) with u≠0 or (0,v) with v≠0. Set A:=inf(u,v)∈𝒩I(u,v), and note that
A
=
inf
(
u
,
v
)
∈
𝒩
1
N
∫
ℝ
N
(
|
∇
u
|
p
+
|
∇
v
|
p
)
=
inf
(
u
,
v
)
∈
𝒩
1
N
∫
ℝ
N
(
μ
1
|
u
|
p
∗
+
μ
2
|
v
|
p
∗
+
γ
|
u
|
α
|
v
|
β
)
.
Consider the nonlinear system of equations
(1.6)
{
μ
1
k
p
∗
-
p
p
+
α
γ
p
∗
k
α
-
p
p
l
β
p
=
1
,
μ
2
l
p
∗
-
p
p
+
β
γ
p
∗
k
α
p
l
β
-
p
p
=
1
,
k
>
0
,
l
>
0
.
Our main results in this case are the following.
Theorem 1.1.
If (H1) holds and γ<0, then A=1N(μ1-(N-p)/p+μ2-(N-p)/p)SN/p and A is not attained.
Theorem 1.2.
Let (H1) and the following conditions hold:
N2<p<N, α,β>p and
(1.7)
0
<
γ
≤
3
p
2
(
3
-
p
)
2
min
{
μ
1
α
(
α
-
p
β
-
p
)
β
-
p
p
,
μ
2
β
(
β
-
p
α
-
p
)
α
-
p
p
}
;
2NN+2<p<N2, α,β<p and
(1.8)
γ
≥
N
p
2
(
N
-
p
)
2
max
{
μ
1
α
(
p
-
β
p
-
α
)
p
-
β
p
,
μ
2
β
(
p
-
α
p
-
β
)
p
-
α
p
}
.
Then A=1N(k0+l0)SN/p and A is attained by (k0pUε,y,l0pUε,y), where (k0,l0) satisfies (1.6) and
(1.9)
k
0
=
min
{
k
:
(
k
,
l
)
satisfies (1.6)
}
.
Theorem 1.3.
Assume that 2NN+2<p<N2, α,β<p and (H1) holds. If γ>0, then A is attained by some (U,V), where U and V are positive, radially symmetric and decreasing.
Theorem 1.4 (Multiplicity).
Assume that 2NN+2<p<N2, α,β<p and (H1) holds. There exists
γ
1
∈
(
0
,
N
p
2
(
N
-
p
)
2
max
{
μ
1
α
(
2
-
β
2
-
α
)
2
-
β
2
,
μ
2
β
(
2
-
α
2
-
β
)
2
-
α
2
}
]
such that for any γ∈(0,γ1) there exists a solution (k(γ),l(γ)) of (1.6) satisfying
I
(
k
(
γ
)
p
U
ε
,
y
,
l
(
γ
)
p
U
ε
,
y
)
>
A
and (k(γ)pUε,y,l(γ)pUε,y) is a (second) positive solution of (1.1).
For the case (H2), we have the following theorem.
Theorem 1.5.
If (H2) holds, p≤N and
0
<
λ
<
p
(
a
a
b
b
)
1
p
λ
1
(
Ω
)
,
where λ1(Ω)>0 is the first Dirichlet eigenvalue of -Δp in Ω, then system (1.1) has a nontrivial nonnegative solution.
2 Proof of Theorem 1.1
Lemma 2.1.
Assume that (H1) holds and -∞<γ<0. If A is attained by a couple (u,v)∈N, then (u,v) is a critical point of I, i.e., (u,v) is a solution of (1.1).
Proof.
Define
𝒩
1
:=
{
(
u
,
v
)
∈
D
:
u
≢
0
,
v
≢
0
,
G
1
(
u
,
v
)
:=
∫
ℝ
N
|
∇
u
|
p
-
∫
ℝ
N
(
μ
1
|
u
|
p
∗
+
α
γ
p
∗
|
u
|
α
|
v
|
β
)
=
0
}
,
𝒩
2
:=
{
(
u
,
v
)
∈
D
:
u
≢
0
,
v
≢
0
,
G
2
(
u
,
v
)
:=
∫
ℝ
N
|
∇
v
|
p
-
∫
ℝ
N
(
μ
2
|
v
|
p
∗
+
β
γ
p
∗
|
u
|
α
|
v
|
β
)
=
0
}
.
Obviously, 𝒩=𝒩1∩𝒩2. Suppose that (u,v)∈𝒩 is a minimizer for I restricted to 𝒩. It follows from the standard minimization theory that there exist two Lagrange multipliers L1,L2∈ℝ such that
I
′
(
u
,
v
)
+
L
1
G
1
′
(
u
,
v
)
+
L
2
G
2
′
(
u
,
v
)
=
0
.
Noticing that
I
′
(
u
,
v
)
(
u
,
0
)
=
G
1
(
u
,
v
)
=
0
,
I
′
(
u
,
v
)
(
0
,
v
)
=
G
2
(
u
,
v
)
=
0
,
G
1
′
(
u
,
v
)
(
u
,
0
)
=
-
(
p
∗
-
p
)
∫
ℝ
N
μ
1
|
u
|
p
∗
+
(
p
-
α
)
∫
ℝ
N
α
γ
p
∗
|
u
|
α
|
v
|
β
,
G
1
′
(
u
,
v
)
(
0
,
v
)
=
-
β
∫
ℝ
N
α
γ
p
∗
|
u
|
α
|
v
|
β
>
0
,
G
2
′
(
u
,
v
)
(
u
,
0
)
=
-
α
∫
ℝ
N
β
γ
p
∗
|
u
|
α
|
v
|
β
>
0
,
G
2
′
(
u
,
v
)
(
0
,
v
)
=
-
(
p
∗
-
p
)
∫
ℝ
N
μ
2
|
v
|
p
∗
+
(
p
-
β
)
∫
ℝ
N
β
γ
p
∗
|
u
|
α
|
v
|
β
,
we get that
{
G
1
′
(
u
,
v
)
(
u
,
0
)
L
1
+
G
2
′
(
u
,
v
)
(
u
,
0
)
L
2
=
0
,
G
1
′
(
u
,
v
)
(
0
,
v
)
L
1
+
G
2
′
(
u
,
v
)
(
0
,
v
)
L
2
=
0
and
G
1
′
(
u
,
v
)
(
u
,
0
)
+
G
1
′
(
u
,
v
)
(
0
,
v
)
=
-
(
p
∗
-
p
)
∫
ℝ
N
|
∇
u
|
p
≤
0
,
G
2
′
(
u
,
v
)
(
u
,
0
)
+
G
2
′
(
u
,
v
)
(
0
,
v
)
=
-
(
p
∗
-
p
)
∫
ℝ
N
|
∇
v
|
p
≤
0
.
We claim that ∫ℝN|∇u|p>0. Indeed, if ∫ℝN|∇u|p=0, then by (1.2) we have
∫
ℝ
N
|
u
|
p
∗
≤
S
-
p
∗
p
(
∫
ℝ
N
|
∇
u
|
p
)
p
∗
p
=
0
.
Thus, a desired contradiction comes out, u≡0 almost everywhere in ℝN. Similarly, ∫ℝN|∇v|p>0. Hence,
|
G
1
′
(
u
,
v
)
(
u
,
0
)
|
=
-
G
1
′
(
u
,
v
)
(
u
,
0
)
>
G
1
′
(
u
,
v
)
(
0
,
v
)
,
|
G
2
′
(
u
,
v
)
(
0
,
v
)
|
=
-
G
2
′
(
u
,
v
)
(
0
,
v
)
>
G
2
′
(
u
,
v
)
(
u
,
0
)
.
Define the matrix
M
:=
(
G
1
′
(
u
,
v
)
(
u
,
0
)
G
2
′
(
u
,
v
)
(
u
,
0
)
G
1
′
(
u
,
v
)
(
0
,
v
)
G
2
′
(
u
,
v
)
(
0
,
v
)
)
.
Then
det
(
M
)
=
|
G
1
′
(
u
,
v
)
(
u
,
0
)
|
⋅
|
G
2
′
(
u
,
v
)
(
0
,
v
)
|
-
G
1
′
(
u
,
v
)
(
0
,
v
)
⋅
G
2
′
(
u
,
v
)
(
u
,
0
)
>
0
,
which means that L1=L2=0. ∎
Proof of Theorem 1.1.
It is standard to see that A>0. By (1.4), we know that ωμi:=μi(p-N)/p2U1,0 satisfies -Δpu=μi|u|p∗-2u in ℝN, where i=1,2. Set e1=(1,0,…,0)∈ℝN and
(
u
R
(
x
)
,
v
R
(
x
)
)
=
(
ω
μ
1
(
x
)
,
ω
μ
2
(
x
+
R
e
1
)
)
,
where R is a positive number. Then vR⇀0 weakly in D1,2(ℝN) and vR⇀0 weakly in Lp∗(ℝN) as R→+∞. Hence,
lim
R
→
+
∞
∫
ℝ
N
u
R
α
v
R
β
𝑑
x
=
lim
R
→
+
∞
∫
ℝ
N
u
R
α
v
R
α
p
∗
-
1
v
R
p
∗
(
β
-
1
)
p
∗
-
1
𝑑
x
≤
lim
R
→
+
∞
(
∫
ℝ
N
u
R
p
∗
-
1
v
R
𝑑
x
)
α
p
∗
-
1
(
∫
ℝ
N
v
R
p
∗
𝑑
x
)
β
-
1
p
∗
-
1
=
0
.
Therefore, for R>0 sufficiently large, the system
{
∫
ℝ
N
|
∇
u
R
|
p
𝑑
x
=
∫
ℝ
N
μ
1
u
R
p
∗
𝑑
x
=
t
R
p
∗
-
p
p
∫
ℝ
N
μ
1
u
R
p
∗
𝑑
x
+
t
R
α
-
p
p
s
R
β
p
∫
ℝ
N
α
γ
p
∗
u
R
α
v
R
β
𝑑
x
,
∫
ℝ
N
|
∇
v
R
|
p
𝑑
x
=
∫
ℝ
N
μ
2
v
R
p
∗
𝑑
x
=
s
R
p
∗
-
p
p
∫
ℝ
N
μ
2
v
R
p
∗
𝑑
x
+
t
R
α
p
s
R
β
-
p
p
∫
ℝ
N
β
γ
p
∗
u
R
α
v
R
β
𝑑
x
has a solution (tR,sR) with
lim
R
→
+
∞
(
|
t
R
-
1
|
+
|
s
R
-
1
|
)
=
0
.
Furthermore, (tRpuR,sRpvR)∈𝒩. Then, by (1.5), we obtain that
A
=
inf
(
u
,
v
)
∈
𝒩
I
(
u
,
v
)
≤
I
(
t
R
p
u
R
,
s
R
p
v
R
)
=
1
N
(
t
R
∫
ℝ
N
|
∇
u
R
|
p
𝑑
x
+
s
R
∫
ℝ
N
|
∇
v
R
|
p
𝑑
x
)
=
1
N
(
t
R
μ
1
-
N
-
p
p
+
s
R
μ
2
-
N
-
p
p
)
S
N
p
,
which implies that A≤1N(μ1-(N-p)/p+μ2-(N-p)/p)SN/p.
For any (u,v)∈𝒩,
∫
ℝ
N
|
∇
u
|
p
𝑑
x
≤
μ
1
∫
ℝ
N
|
u
|
p
∗
𝑑
x
≤
μ
1
S
-
p
∗
p
(
∫
ℝ
N
|
∇
u
|
p
𝑑
x
)
p
∗
p
.
Therefore, ∫ℝN|∇u|p𝑑x≥μ1-(N-p)/pSN/p. Similarly,
∫
ℝ
N
|
∇
v
|
p
𝑑
x
≥
μ
2
-
N
-
p
p
S
N
p
.
Then A≥1N(μ1-(N-p)/p+μ2-(N-p)/p)SN/p. Hence,
(2.1)
A
=
1
N
(
μ
1
-
N
-
p
p
+
μ
2
-
N
-
p
p
)
S
N
p
.
Suppose by contradiction that A is attained by some (u,v)∈𝒩. Then (|u|,|v|)∈𝒩 and I(|u|,|v|)=A. By Lemma 2.1, we see that (|u|,|v|) is a nontrivial solution of (1.1). By the strong maximum principle, we may assume that u>0, v>0, and so ∫ℝNuαvβ𝑑x>0. Then
∫
ℝ
N
|
∇
u
|
p
𝑑
x
<
μ
1
∫
ℝ
N
|
u
|
p
∗
𝑑
x
≤
μ
1
S
-
p
∗
p
(
∫
ℝ
N
|
∇
u
|
p
𝑑
x
)
p
∗
p
,
which yields that
∫
ℝ
N
|
∇
u
|
p
𝑑
x
>
μ
1
-
N
-
p
p
S
N
p
.
Similarly,
∫
ℝ
N
|
∇
v
|
p
𝑑
x
>
μ
2
-
N
-
p
p
S
N
p
.
Therefore,
A
=
I
(
u
,
v
)
=
1
N
∫
ℝ
N
(
|
∇
u
|
p
+
|
∇
v
|
p
)
𝑑
x
>
1
N
(
μ
1
-
N
-
p
p
+
μ
2
-
N
-
p
p
)
S
N
p
,
which contradicts (2.1). ∎
3 Proof of Theorem 1.2
Proposition 3.1.
Assume that c,d∈R satisfy
(3.1)
{
μ
1
c
p
∗
-
p
p
+
α
γ
p
∗
c
α
-
p
p
d
β
p
≥
1
,
μ
2
d
p
∗
-
p
p
+
β
γ
p
∗
c
α
p
d
β
-
p
p
≥
1
,
c
>
0
,
d
>
0
.
If N2<p<N, α,β>p and (1.7) holds, then c+d≥k+l, where k,l∈R satisfy (1.6).
Proof.
Let y=c+d, x=cd, y0=k+l and x0=kl. By (3.1) and (1.6), we have that
y
p
∗
-
p
p
≥
(
x
+
1
)
p
∗
-
p
p
μ
1
x
p
∗
-
p
p
+
α
γ
p
∗
x
α
-
p
p
:=
f
1
(
x
)
,
y
0
p
∗
-
p
p
=
f
1
(
x
0
)
,
y
p
∗
-
p
p
≥
(
x
+
1
)
p
∗
-
p
p
μ
2
+
β
γ
p
∗
x
α
p
:=
f
2
(
x
)
,
y
0
p
∗
-
p
p
=
f
2
(
x
0
)
.
Thus,
f
1
′
(
x
)
=
α
γ
(
x
+
1
)
p
∗
-
2
p
p
x
α
-
2
p
p
p
p
∗
(
μ
1
x
p
∗
-
p
p
+
α
γ
p
∗
x
α
-
p
p
)
2
[
-
p
∗
(
p
∗
-
p
)
μ
1
α
γ
x
β
p
+
β
x
-
(
α
-
p
)
]
,
f
2
′
(
x
)
=
β
γ
(
x
+
1
)
p
∗
-
2
p
p
p
p
∗
(
μ
2
+
β
γ
p
∗
x
α
p
)
2
[
(
β
-
p
)
x
α
p
-
α
x
α
-
p
p
+
p
∗
(
p
∗
-
p
)
μ
2
β
γ
]
.
Let x1=(pαγp∗(p∗-p)μ1)p/(β-p), x2=α-pβ-p and
g
1
(
x
)
=
-
p
∗
(
p
∗
-
p
)
μ
1
α
γ
x
β
p
+
β
x
-
(
α
-
p
)
,
g
2
(
x
)
=
(
β
-
p
)
x
α
p
-
α
x
α
-
p
p
+
p
∗
(
p
∗
-
p
)
μ
2
β
γ
.
It follows from (1.7) that
max
x
∈
(
0
,
+
∞
)
g
1
(
x
)
=
g
1
(
x
1
)
=
(
β
-
p
)
(
p
α
γ
p
∗
(
p
∗
-
p
)
μ
1
)
p
β
-
p
-
(
α
-
p
)
≤
0
,
min
x
∈
(
0
,
+
∞
)
g
2
(
x
)
=
g
2
(
x
2
)
=
-
p
(
α
-
p
β
-
p
)
α
-
p
p
+
p
∗
(
p
∗
-
p
)
μ
2
β
γ
≥
0
.
That is, f1(x) is strictly decreasing in (0,+∞) and f2(x) is strictly increasing in (0,+∞). Hence,
y
p
∗
-
p
p
≥
max
{
f
1
(
x
)
,
f
2
(
x
)
}
≥
min
x
∈
(
0
,
+
∞
)
(
max
{
f
1
(
x
)
,
f
2
(
x
)
}
)
=
min
{
f
1
=
f
2
}
(
max
{
f
1
(
x
)
,
f
2
(
x
)
}
)
=
y
0
p
∗
-
p
p
,
where {f1=f2}:={x∈(0,+∞):f1(x)=f2(x)}. ∎
Remark 3.1.
From the proof of Proposition 3.1 it is easy to see that system (1.6), under the assumption of Proposition 3.1, has only one real solution (k,l)=(k0,l0), where (k0,l0) is defined as in (1.9).
Define the functions
(3.2)
{
F
1
(
k
,
l
)
:=
μ
1
k
p
∗
-
p
p
+
α
γ
p
∗
k
α
-
p
p
l
β
p
-
1
,
k
>
0
,
l
≥
0
,
F
2
(
k
,
l
)
:=
μ
2
l
p
∗
-
p
p
+
β
γ
p
∗
k
α
p
l
β
-
p
p
-
1
,
k
≥
0
,
l
>
0
,
l
(
k
)
:=
(
p
∗
α
γ
)
p
β
k
p
-
α
β
(
1
-
μ
1
k
p
∗
-
p
p
)
p
β
,
0
<
k
≤
μ
1
-
p
p
∗
-
p
,
k
(
l
)
:=
(
p
∗
β
γ
)
p
α
l
p
-
β
α
(
1
-
μ
2
l
p
∗
-
p
p
)
p
α
,
0
<
l
≤
μ
2
-
p
p
∗
-
p
.
Then F1(k,l(k))≡0 and F2(k(l),l)≡0.
Lemma 3.2.
Assume that 2NN+2<p<N2, α,β<p and γ>0. Then
(3.3)
F
1
(
k
,
l
)
=
0
,
F
2
(
k
,
l
)
=
0
,
k
,
l
>
0
,
has a solution (k0,l0) such that
(3.4)
F
2
(
k
,
l
(
k
)
)
<
0
for all
k
∈
(
0
,
k
0
)
,
that is, (k0,l0) satisfies (1.9). Similarly, (3.3) has a solution (k1,l1) such that
(3.5)
F
1
(
k
(
l
)
,
l
)
<
0
for all
l
∈
(
0
,
l
1
)
,
that is,
(
k
1
,
l
1
)
satisfies (1.6) and
l
1
=
min
{
l
:
(
k
,
l
)
is a solution of (1.6)
}
.
Proof.
We only prove the existence of (k0,l0). It follows from F1(k,l)=0, k,l>0, that
l
=
l
(
k
)
for all
k
∈
(
0
,
μ
1
-
p
p
∗
-
p
)
.
Substituting this into F2(k,l)=0, we have
(3.6)
μ
2
(
p
∗
α
γ
)
α
β
(
1
-
μ
1
k
p
∗
-
p
p
)
α
β
+
β
γ
p
∗
k
(
p
∗
-
p
)
α
p
β
-
(
p
∗
α
γ
)
p
-
β
β
k
-
(
p
∗
-
p
)
(
p
-
α
)
p
β
(
1
-
μ
1
k
p
∗
-
p
p
)
p
-
β
β
=
0
.
By setting
f
(
k
)
:=
μ
2
(
p
∗
α
γ
)
α
β
(
1
-
μ
1
k
p
∗
-
p
p
)
α
β
+
β
γ
p
∗
k
(
p
∗
-
p
)
α
p
β
-
(
p
∗
α
γ
)
p
-
β
β
k
-
(
p
∗
-
p
)
(
p
-
α
)
p
β
(
1
-
μ
1
k
p
∗
-
p
p
)
p
-
β
β
,
the existence of a solution of (3.6) in (0,μ1-p/(p∗-p)) is equivalent to f(k)=0 possessing a solution in (0,μ1-p/(p∗-p)). Since α,β<p, we get that
lim
k
→
0
+
f
(
k
)
=
-
∞
,
f
(
μ
1
-
p
p
∗
-
p
)
=
β
γ
p
∗
μ
1
-
α
β
>
0
,
which implies that there exists k0∈(0,μ1-p/(p∗-p)) such that f(k0)=0 and f(k)<0 for k∈(0,k0). Let l0=l(k0). Then (k0,l0) is a solution of (3.3) and (3.4) holds. ∎
Remark 3.2.
From 2NN+2<p<N2 and α,β<p we get that 2<p∗<2p. It can be seen from N2<p<N and α,β>p that 2<2p<p∗.
Lemma 3.3.
Assume that 2NN+2<p<N2, α,β<p and (1.8) holds. Let (k0,l0) be the same as in Lemma 3.2. Then
(
k
0
+
l
0
)
p
∗
-
p
p
max
{
μ
1
,
μ
2
}
<
1
and
(3.7)
F
2
(
k
,
l
(
k
)
)
<
0
for all
k
∈
(
0
,
k
0
)
,
F
1
(
k
(
k
)
,
l
)
<
0
for all
l
∈
(
0
,
l
0
)
.
Proof.
Recalling (3.2), we obtain that
l
′
(
k
)
=
(
p
∗
α
γ
)
p
β
p
β
(
k
p
-
α
p
-
μ
1
k
β
p
)
p
-
β
β
(
p
-
α
p
k
-
α
p
-
μ
1
β
p
k
β
-
p
p
)
=
(
p
∗
μ
1
α
γ
)
p
β
k
p
-
p
∗
β
(
μ
1
-
1
-
k
p
∗
-
p
p
)
p
-
β
β
(
p
-
α
μ
1
β
-
k
p
∗
-
p
p
)
,
l
′
(
(
p
-
α
μ
1
β
)
p
p
∗
-
p
)
=
l
′
(
μ
1
-
p
p
∗
-
p
)
=
0
,
l
′
(
k
)
>
0
for
k
∈
(
0
,
(
p
-
α
μ
1
β
)
p
p
∗
-
p
)
,
l
′
(
k
)
<
0
for
k
∈
(
(
p
-
α
μ
1
β
)
p
p
∗
-
p
,
μ
1
-
p
p
∗
-
p
)
.
From
l
′′
(
k
¯
)
=
p
-
β
β
(
p
∗
μ
1
α
γ
)
p
β
k
¯
p
-
2
β
-
α
β
(
μ
1
-
1
-
k
¯
p
∗
-
p
p
)
p
-
2
β
β
[
(
p
-
α
μ
1
β
-
k
¯
p
∗
-
p
p
)
2
-
(
μ
1
-
1
-
k
¯
p
∗
-
p
p
)
(
α
(
p
-
α
)
μ
1
β
(
p
-
β
)
-
k
¯
p
∗
-
p
p
)
]
=
0
and k¯∈((p-αμ1β)p/(p∗-p),μ1-p/(p∗-p)), we have k¯=(p(p-α)(2p-p∗)μ1β)p/(p∗-p). Then, by (1.8), we get that
min
k
∈
(
0
,
μ
1
-
p
/
(
p
∗
-
p
)
]
l
′
(
k
)
=
min
k
∈
(
(
p
-
α
μ
1
β
)
p
/
(
p
∗
-
p
)
,
μ
1
-
p
/
(
p
∗
-
p
)
]
l
′
(
k
)
=
l
′
(
k
¯
)
=
-
(
p
∗
(
p
∗
-
p
)
μ
1
p
α
γ
)
p
β
(
p
-
β
p
-
α
)
p
-
β
β
≥
-
1
.
Therefore, l′(k)>-1 for k∈(0,μ1-p/(p∗-p)] with
k
≠
(
p
(
p
-
α
)
(
2
p
-
p
∗
)
μ
1
β
)
p
p
∗
-
p
,
which implies that l(k)+k is strictly increasing on [0,μ1-p/(p∗-p)]. Noticing that k0<μ1-p/(p∗-p), we have
μ
1
-
p
p
∗
-
p
=
l
(
μ
1
-
p
p
∗
-
p
)
+
μ
1
-
p
p
∗
-
p
>
l
(
k
0
)
+
k
0
=
l
0
+
k
0
,
that is, μ1(k0+l0)(p∗-p)/p<1. Similarly, μ2(k0+l0)(p∗-p)/p<1. To prove (3.7), by Lemma 3.2 it suffices to show that (k0,l0)=(k1,l1). It follows from (3.4) and (3.5) that k1≥k0 and l0≥l1. Suppose by contradiction that k1>k0. Then l(k1)+k1>l(k0)+k0. Hence, l1+k(l1)=l(k1)+k1>l(k0)+k0=l0+k(l0). Following the arguments as in the beginning of the current proof, we have that l+k(l) is strictly increasing for l∈[0,μ2-p/(p∗-p)]. Therefore, l1>l0, which contradicts l0≥l1. Then k1=k0, and similarly l0=l1. ∎
Remark 3.3.
For any γ>0, condition (1.8) always holds for dimension N large enough.
Proposition 3.4.
Assume that 2NN+2<p<N2, α,β<p and (1.8) holds. Then
(3.8)
{
k
+
l
≤
k
0
+
l
0
,
F
1
(
k
,
l
)
≥
0
,
F
2
(
k
,
l
)
≥
0
,
k
,
l
≥
0
,
(
k
,
l
)
≠
(
0
,
0
)
,
has a unique solution (k,l)=(k0,l0).
Proof.
Obviously, (k0,l0) satisfies (3.8). Suppose that (k~,l~) is any solution of (3.8) and, without loss of generality, assume that k~>0. We claim that l~>0. In fact, if l~=0, then k~≤k0+l0 and F1(k~,0)=μ1k~(p∗-p)/p-1≥0. Thus,
1
≤
μ
1
k
~
p
∗
-
p
p
≤
μ
1
(
k
0
+
l
0
)
p
∗
-
p
p
,
a contradiction to Lemma 3.3.
Suppose by contradiction that k~<k0. It can be seen that k(l) is strictly increasing on (0,(p-βμ2α)p/(p∗-p)] and strictly decreasing on
[
(
2
-
β
μ
2
α
)
p
p
∗
-
p
,
μ
2
-
p
p
∗
-
p
]
and
k
(
0
)
=
k
(
μ
2
-
p
p
∗
-
p
)
=
0
.
Since 0<k~<k0=k(l0), there exist 0<l1<l2<μ2-p/(p∗-p) such that k(l1)=k(l2)=k~ and
(3.9)
F
2
(
k
~
,
l
)
<
0
⇔
k
~
<
k
(
l
)
⇔
l
1
<
l
<
l
2
.
It follows from F1(k~,l~)≥0 and F2(k~,l~)≥0 that l~≥l(k~) and l~≤l1 or l~≥l2. By (3.7), we see F2(k~,l(k~))<0. By (3.9), we get that l1<l(k~)<l2. Therefore, l~≥l2.
On the other hand, set l3:=k0+l0-k~. Then l3>l0 and, moreover,
k
(
l
3
)
+
k
0
+
l
0
-
k
~
=
k
(
l
3
)
+
l
3
>
k
(
l
0
)
+
l
0
=
k
0
+
l
0
,
that is, k(l3)>k~. By (3.9), we have l1<l3<l2. Since k~+l~≤k0+l0, we obtain that l~≤k0+l0-k~=l3<l2. This contradicts l~≥l2. ∎
Proof of Theorem 1.2.
Recalling (1.4) and (1.6), we see that (k0pUε,y,l0pUε,y)∈𝒩 is a nontrivial solution of (1.1), and
(3.10)
A
≤
I
(
k
0
p
U
ε
,
y
,
l
0
p
U
ε
,
y
)
=
1
N
(
k
0
+
l
0
)
S
N
p
.
Let {(un,vn)}⊂𝒩 be a minimizing sequence for A, i.e., I(un,vn)→A as n→∞. Define
c
n
=
(
∫
ℝ
N
|
u
n
|
p
∗
𝑑
x
)
p
p
∗
and
d
n
=
(
∫
ℝ
N
|
v
n
|
p
∗
𝑑
x
)
p
p
∗
.
Then
S
c
n
≤
∫
ℝ
N
|
∇
u
n
|
p
𝑑
x
=
∫
ℝ
N
(
μ
1
|
u
n
|
p
∗
+
α
γ
p
∗
|
u
n
|
α
|
v
n
|
β
)
𝑑
x
(3.11)
≤
μ
1
c
n
p
∗
p
+
α
γ
p
∗
c
n
α
p
d
n
β
p
,
S
d
n
≤
∫
ℝ
N
|
∇
v
n
|
p
𝑑
x
=
∫
ℝ
N
(
μ
2
|
v
n
|
p
∗
+
β
γ
p
∗
|
u
n
|
α
|
v
n
|
β
)
𝑑
x
(3.12)
≤
μ
2
d
n
p
∗
p
+
β
γ
p
∗
c
n
α
p
d
n
β
p
.
Dividing both sides of these inequalities by Scn and Sdn, respectively, and denoting
c
~
n
=
c
n
S
p
p
∗
-
p
,
d
~
n
=
d
n
S
p
p
∗
-
p
,
we deduce that
μ
1
c
~
n
p
∗
-
p
p
+
α
γ
p
∗
c
~
n
α
-
p
p
d
~
n
β
p
≥
1
,
μ
2
d
~
n
p
∗
-
p
p
+
β
γ
p
∗
c
~
n
α
p
d
~
n
β
-
p
p
≥
1
,
that is, F1(c~n,d~n)≥0 and F2(c~n,d~n)≥0. Then, for N2<p<N and α,β>p, Proposition 3.1 and Remark 3.1 ensure that c~n+d~n≥k+l=k0+l0, whereas for 2NN+2<p<N2 and α,β<p Proposition 3.4 guarantees that c~n+d~n≥k0+l0. Therefore,
(3.13)
c
n
+
d
n
≥
(
k
0
+
l
0
)
S
p
p
∗
-
p
=
(
k
0
+
l
0
)
S
N
-
p
p
.
Noticing that I(un,vn)=1N∫ℝN(|∇un|p+|∇vn|p), by (3.10)–(3.12) we have
S
(
c
n
+
d
n
)
≤
N
I
(
u
n
,
v
n
)
=
N
A
+
o
(
1
)
≤
(
k
0
+
l
0
)
S
N
p
+
o
(
1
)
.
Combining this with (3.13), we get that cn+dn→(k0+l0)S(N-p)/p as n→∞. Thus,
A
=
lim
n
→
∞
I
(
u
n
,
v
n
)
≥
lim
n
→
∞
1
N
S
(
c
n
+
d
n
)
=
1
N
(
k
0
+
l
0
)
S
N
p
.
Hence,
A
=
1
N
(
k
0
+
l
0
)
S
N
p
=
I
(
k
0
p
U
ε
,
y
,
l
0
p
U
ε
,
y
)
.
∎
4 Proofs of Theorems 1.3 and 1.4
For (H1) holding and γ>0, define
A
′
:=
inf
(
u
,
v
)
∈
𝒩
′
I
(
u
,
v
)
,
where
𝒩
′
:=
{
(
u
,
v
)
∈
D
∖
{
(
0
,
0
)
}
:
∫
ℝ
N
(
|
∇
u
|
p
+
|
∇
v
|
p
)
=
∫
ℝ
N
(
μ
1
|
u
|
p
∗
+
μ
2
|
v
|
p
∗
+
γ
|
u
|
α
|
v
|
β
)
}
.
It follows from 𝒩⊂𝒩′ that A′≤A. By the Sobolev inequality, we see that A′>0. Consider
{
-
Δ
p
u
=
μ
1
|
u
|
p
∗
-
2
u
+
α
γ
p
∗
|
u
|
α
-
2
u
|
v
|
β
,
x
∈
B
(
0
,
R
)
,
-
Δ
p
v
=
μ
2
|
v
|
p
∗
-
2
v
+
β
γ
p
∗
|
u
|
α
|
v
|
β
-
2
v
,
x
∈
B
(
0
,
R
)
,
u
,
v
∈
H
0
1
(
B
(
0
,
R
)
)
,
where B(0,R):={x∈ℝN:|x|<R}. Define
(4.1)
𝒩
′
(
R
)
:=
{
(
u
,
v
)
∈
H
(
0
,
R
)
∖
{
(
0
,
0
)
}
:
∫
B
(
0
,
R
)
(
|
∇
u
|
p
+
|
∇
v
|
p
)
=
∫
B
(
0
,
R
)
(
μ
1
|
u
|
p
∗
+
μ
2
|
v
|
p
∗
+
γ
|
u
|
α
|
v
|
β
)
}
and
A
′
(
R
)
:=
inf
(
u
,
v
)
∈
𝒩
′
(
R
)
I
(
u
,
v
)
,
where H(0,R):=H01(B(0,R))×H01(B(0,R)). For ε∈[0,min{α,β}-1), consider
(4.2)
{
-
Δ
p
u
=
μ
1
|
u
|
p
∗
-
2
-
2
ε
u
+
(
α
-
ε
)
γ
p
∗
-
2
ε
|
u
|
α
-
2
-
ε
u
|
v
|
β
-
ε
,
x
∈
B
(
0
,
1
)
,
-
Δ
p
v
=
μ
2
|
v
|
p
∗
-
2
-
2
ε
v
+
(
β
-
ε
)
γ
p
∗
-
2
ε
|
u
|
α
-
ε
|
v
|
β
-
2
-
ε
v
,
x
∈
B
(
0
,
1
)
,
u
,
v
∈
H
0
1
(
B
(
0
,
1
)
)
.
Define
(4.3)
{
I
ε
(
u
,
v
)
:=
1
p
∫
B
(
0
,
1
)
(
|
∇
u
|
p
+
|
∇
v
|
p
)
-
1
p
∗
-
2
ε
∫
B
(
0
,
1
)
(
μ
1
|
u
|
p
∗
-
2
ε
+
μ
2
|
v
|
p
∗
-
2
ε
+
γ
|
u
|
α
-
ε
|
v
|
β
-
ε
)
,
𝒩
ε
′
:=
{
(
u
,
v
)
∈
H
(
0
,
1
)
∖
{
(
0
,
0
)
}
:
G
ε
(
u
,
v
)
:=
∫
B
(
0
,
1
)
(
|
∇
u
|
p
+
|
∇
v
|
p
)
-
∫
B
(
0
,
1
)
(
μ
1
|
u
|
p
∗
-
2
ε
+
μ
2
|
v
|
p
∗
-
2
ε
+
γ
|
u
|
α
-
ε
|
v
|
β
-
ε
)
=
0
}
and
A
ε
:=
inf
(
u
,
v
)
∈
𝒩
ε
′
I
ε
(
u
,
v
)
.
Lemma 4.1.
Assume that 2NN+2<p<N2, α,β<p. For ε∈(0,min{α,β}-1), there holds
A
ε
<
min
{
inf
(
u
,
0
)
∈
𝒩
ε
′
I
ε
(
u
,
0
)
,
inf
(
0
,
v
)
∈
𝒩
ε
′
I
ε
(
0
,
v
)
}
.
Proof.
From min{α,β}≤p∗2 it is easy to see that 2<p∗-2ε<p∗. Then we may assume that ui is a least energy solution of
-
Δ
p
u
=
μ
i
|
u
|
p
∗
-
2
-
2
ε
u
,
u
∈
H
0
1
(
B
(
0
,
1
)
)
,
i
=
1
,
2
.
Therefore,
I
ε
(
u
1
,
0
)
=
a
1
:=
inf
(
u
,
0
)
∈
𝒩
ε
′
I
ε
(
u
,
0
)
,
I
ε
(
0
,
u
2
)
=
a
2
:=
inf
(
0
,
v
)
∈
𝒩
ε
′
I
ε
(
0
,
v
)
.
We claim that, for any s∈ℝ, there exists a unique t(s)>0 such that (t(s)pu1, t(s)psu2)∈𝒩′ε. In fact,
t
(
s
)
p
∗
-
p
-
2
ε
p
=
∫
B
(
0
,
1
)
(
|
∇
u
1
|
p
+
|
s
|
p
|
∇
u
2
|
p
)
∫
B
(
0
,
1
)
(
μ
1
|
u
1
|
p
∗
-
2
ε
+
μ
2
|
s
u
2
|
p
∗
-
2
ε
+
γ
|
u
1
|
α
-
ε
|
s
u
2
|
β
-
ε
)
=
q
a
1
+
q
a
2
|
s
|
p
q
a
1
+
q
a
2
|
s
|
p
∗
-
2
ε
+
|
s
|
β
-
ε
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
,
where q:=p(p∗-2ε)p∗-p-2ε=p(Np-2ε+2εp)p2-2εN+2εp→N as ε→0. Noticing that t(0)=1, we have
lim
s
→
0
t
′
(
s
)
|
s
|
β
-
ε
-
2
s
=
-
(
β
-
ε
)
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
(
p
∗
-
2
ε
)
a
1
,
that is,
t
′
(
s
)
=
-
(
β
-
ε
)
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
(
p
∗
-
2
ε
)
a
1
|
s
|
β
-
ε
-
2
s
(
1
+
o
(
1
)
)
as
s
→
0
.
Then
t
(
s
)
=
1
-
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
(
p
∗
-
2
ε
)
a
1
|
s
|
β
-
ε
(
1
+
o
(
1
)
)
as
s
→
0
,
and so,
t
(
s
)
p
∗
-
2
ε
p
=
1
-
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
p
a
1
|
s
|
β
-
ε
(
1
+
o
(
1
)
)
as
s
→
0
.
Since 1p-1q=1p∗-2ε, we have
A
ε
≤
I
ε
(
t
(
s
)
p
u
1
,
t
(
s
)
p
s
u
2
)
=
(
1
p
-
1
p
∗
-
2
ε
)
(
q
a
1
+
q
a
2
|
s
|
p
∗
-
2
ε
+
|
s
|
β
-
ε
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
)
t
p
∗
-
2
ε
p
=
a
1
-
(
1
p
-
1
q
)
|
s
|
β
-
ε
∫
B
(
0
,
1
)
γ
|
u
1
|
α
-
ε
|
u
2
|
β
-
ε
+
o
(
|
s
|
β
-
ε
)
<
a
1
=
inf
(
u
,
0
)
∈
𝒩
ε
′
I
ε
(
u
,
0
)
as
|
s
|
is small enough
.
Similarly, Aε<inf(0,v)∈𝒩ε′Iε(0,v). ∎
Noticing the definition of ωμi in the proof of Theorem 1.1, similarly to Lemma 4.1, we obtain that
A
′
<
min
{
inf
(
u
,
0
)
∈
𝒩
′
I
(
u
,
0
)
,
inf
(
0
,
v
)
∈
𝒩
′
I
(
0
,
v
)
}
=
min
{
I
(
ω
μ
1
,
0
)
,
I
(
0
,
ω
μ
2
)
}
(4.4)
=
min
{
1
N
μ
1
-
N
-
p
p
S
N
p
,
1
N
μ
2
-
N
-
p
p
S
N
p
}
.
Proposition 4.2.
For any ε∈(0,min{α,β}-1), system (4.2) has a classical positive least energy solution (uε,vε) and uε, vε are radially symmetric decreasing.
Proof.
It is standard to see that Aε>0. For (u,v)∈𝒩ε′ with u≥0 and v≥0, we denote by (u∗,v∗) its Schwartz symmetrization. By the properties of the Schwartz symmetrization and γ>0, we get that
∫
B
(
0
,
1
)
(
|
∇
u
∗
|
p
+
|
∇
v
∗
|
p
)
≤
∫
B
(
0
,
1
)
(
μ
1
|
u
∗
|
p
∗
-
2
ε
+
μ
2
|
v
∗
|
p
∗
-
2
ε
+
γ
|
u
∗
|
α
-
ε
|
v
∗
|
β
-
ε
)
.
Obviously, there exists t∗∈(0,1] such that (t∗pu∗,t∗pv∗)∈𝒩ε′. Therefore,
I
ε
(
t
∗
p
u
∗
,
t
∗
p
v
∗
)
=
(
1
p
-
1
p
∗
-
2
ε
)
t
∗
∫
B
(
0
,
1
)
(
|
∇
u
∗
|
p
+
|
∇
v
∗
|
p
)
≤
p
∗
-
2
ε
-
p
p
(
p
∗
-
2
ε
)
∫
B
(
0
,
1
)
(
|
∇
u
|
p
+
|
∇
v
|
p
)
(4.5)
=
I
ε
(
u
,
v
)
.
Therefore, we may choose a minimizing sequence (un,vn)∈𝒩ε′ of Aε such that (un,vn)=(un∗,vn∗) and Iε(un,vn)→Aε as n→∞. By (4.5), we see that un, vn are uniformly bounded in H01(B(0,1)). Passing to a subsequence, we may assume that un⇀uε, vn⇀vε weakly in H01(B(0,1)). Since H01(B(0,1))↪Lp∗-2ε(B(0,1)) is compact, we deduce that
∫
B
(
0
,
1
)
(
μ
1
|
u
ε
|
p
∗
-
2
ε
+
μ
2
|
v
ε
|
p
∗
-
2
ε
+
γ
|
u
ε
|
α
-
ε
|
v
ε
|
β
-
ε
)
=
lim
n
→
∞
∫
B
(
0
,
1
)
(
μ
1
|
u
n
|
p
∗
-
2
ε
+
μ
2
|
v
n
|
p
∗
-
2
ε
+
γ
|
u
n
|
α
-
ε
|
v
n
|
β
-
ε
)
=
p
(
p
∗
-
2
ε
)
p
∗
-
2
ε
-
p
lim
n
→
∞
I
ε
(
u
n
,
v
n
)
=
p
(
p
∗
-
2
ε
)
p
∗
-
2
ε
-
p
A
ε
>
0
,
which implies that (uε,vε)≠(0,0). Moreover, uε≥0, vε≥0 are radially symmetric. Noticing that
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
≤
lim
n
→
∞
∫
B
(
0
,
1
)
(
|
∇
u
n
|
p
+
|
∇
v
n
|
p
)
,
we get that
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
≤
∫
B
(
0
,
1
)
(
μ
1
|
u
ε
|
p
∗
-
2
ε
+
μ
2
|
v
ε
|
p
∗
-
2
ε
+
γ
|
u
ε
|
α
-
ε
|
v
ε
|
β
-
ε
)
.
Then there exists tε∈(0,1] such that (tεpuε,tεpvε)∈𝒩ε′, and therefore
A
ε
≤
I
ε
(
t
ε
p
u
ε
,
t
ε
p
v
ε
)
=
(
1
p
-
1
p
∗
-
2
ε
)
t
ε
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
≤
lim
n
→
∞
p
∗
-
2
ε
-
p
p
(
p
∗
-
2
ε
)
∫
B
(
0
,
1
)
(
|
∇
u
n
|
p
+
|
∇
v
n
|
p
)
=
lim
n
→
∞
I
ε
(
u
n
,
v
n
)
=
A
ε
,
which yields that tε=1, (uε,vε)∈𝒩ε′, I(uε,vε)=Aε and
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
=
lim
n
→
∞
∫
B
(
0
,
1
)
(
|
∇
u
n
|
p
+
|
∇
v
n
|
p
)
.
That is, un→uε, vn→vε strongly in H01(B(0,1)). It follows from the standard minimization theory that there exists a Lagrange multiplier L∈ℝ satisfying
I
ε
′
(
u
ε
,
v
ε
)
+
L
G
ε
′
(
u
ε
,
v
ε
)
=
0
.
Since Iε′(uε,vε)(uε,vε)=Gε(uε,vε)=0 and
G
ε
′
(
u
ε
,
v
ε
)
(
u
ε
,
v
ε
)
=
-
(
p
∗
-
2
ε
-
p
)
∫
B
(
0
,
1
)
(
μ
1
|
u
ε
|
p
∗
-
2
ε
+
μ
2
|
v
ε
|
p
∗
-
2
ε
+
γ
|
u
ε
|
α
-
ε
|
v
ε
|
β
-
ε
)
<
0
,
we get that L=0, and so Iε′(uε,vε)=0. By Aε=I(uε,vε) and Lemma 4.1, we have uε≢0 and vε≢0. Since uε,vε≥0 are radially symmetric decreasing, by the regularity theory and the maximum principle, we obtain that (uε,vε) is a classical positive least energy solution of (4.2). ∎
Proof of Theorem 1.3.
We claim that
(4.6)
A
′
(
R
)
≡
A
′
for all
R
>
0
.
Indeed, assume R1<R2. Since 𝒩′(R1)⊂𝒩′(R2), we get that A′(R2)≤A′(R1). On the other hand, for every (u,v)∈𝒩′(R2), define
(
u
1
(
x
)
,
v
1
(
x
)
)
:=
(
(
R
2
R
1
)
N
-
p
p
u
(
R
2
R
1
x
)
,
(
R
2
R
1
)
N
-
p
p
v
(
R
2
R
1
x
)
)
.
Then it is easy to see that (u1,v1)∈𝒩′(R1). Thus, we have
A
′
(
R
1
)
≤
I
(
u
1
,
v
1
)
=
I
(
u
,
v
)
for all
(
u
,
v
)
∈
𝒩
′
(
R
2
)
,
which means that A′(R1)≤A′(R2). Hence, A′(R1)=A′(R2). Obviously, A′≤A′(R). Let (un,vn)∈𝒩′ be a minimizing sequence of A′. We assume that un,vn∈H01(B(0,Rn)) for some Rn>0. Therefore, (un,vn)∈𝒩′(Rn) and
A
′
=
lim
n
→
∞
I
(
u
n
,
v
n
)
≥
lim
n
→
∞
A
′
(
R
n
)
=
A
′
(
R
)
,
which completes the proof of the claim.
By recalling (4.1) and (4.3), for every (u,v)∈𝒩′(1), there exists tε>0 with tε→1 as ε→0 such that (tεpu,tεpv)∈𝒩ε′. Then
lim sup
ε
→
0
A
ε
≤
lim sup
ε
→
0
I
ε
(
t
ε
p
u
,
t
ε
p
v
)
=
I
(
u
,
v
)
for all
(
u
,
v
)
∈
𝒩
′
(
1
)
.
It follows from (4.6) that
(4.7)
lim sup
ε
→
0
A
ε
≤
A
′
(
1
)
=
A
′
.
According to Proposition 4.2, we may let (uε,vε) be a positive least energy solution of (4.2), which is radially symmetric decreasing. By (4.3) and the Sobolev inequality, we have
(4.8)
A
ε
=
p
∗
-
2
ε
-
2
2
(
p
∗
-
2
ε
)
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
≥
C
>
0
for all
ε
∈
(
0
,
min
{
α
,
β
}
-
1
2
]
,
where C is independent of ε. Then it follows from (4.7) that uε, vε are uniformly bounded in H01(B(0,1)). We may assume that uε⇀u0, vε⇀v0, up to a subsequence, weakly in H01(B(0,1)). Hence, (u0,v0) is a solution of
{
-
Δ
p
u
=
μ
1
|
u
|
p
∗
-
2
u
+
α
γ
p
∗
|
u
|
α
-
2
u
|
v
|
β
,
x
∈
B
(
0
,
1
)
,
-
Δ
p
v
=
μ
2
|
v
|
p
∗
-
2
v
+
β
γ
p
∗
|
u
|
α
|
v
|
β
-
2
v
,
x
∈
B
(
0
,
1
)
,
u
,
v
∈
H
0
1
(
B
(
0
,
1
)
)
.
Suppose by contradiction that ∥uε∥∞+∥vε∥∞ is uniformly bounded. Then, by the dominated convergent theorem, we get that
lim
ε
→
0
∫
B
(
0
,
1
)
u
ε
p
∗
-
2
ε
=
∫
B
(
0
,
1
)
u
0
p
∗
,
lim
ε
→
0
∫
B
(
0
,
1
)
v
ε
p
∗
-
2
ε
=
∫
B
(
0
,
1
)
v
0
p
∗
,
lim
ε
→
0
∫
B
(
0
,
1
)
u
ε
α
-
ε
v
ε
β
-
ε
=
∫
B
(
0
,
1
)
u
0
α
v
0
β
.
Combining these with Iε′(uε,vε)=I′(u0,v0), similarly to the proof of Proposition 4.2, we see that uε→u0, vε→v0 strongly in H01(B(0,1)). It follows from (4.8) that (u0,v0)≠(0,0) and, moreover, u0≥0, v0≥0. Without loss of generality, we may assume that u0≢0. By the strong maximum principle, we obtain that u0>0 in B(0,1). By the Pohozaev identity, we have a contradiction
0
<
∫
∂
B
(
0
,
1
)
(
|
∇
u
0
|
p
+
|
∇
v
0
|
p
)
(
x
⋅
ν
)
𝑑
σ
=
0
,
where ν is the outward unit normal vector on ∂B(0,1). Hence, ∥uε∥∞+∥vε∥∞→∞ as ε→0. Let
K
ε
:=
max
{
u
ε
(
0
)
,
v
ε
(
0
)
}
.
Since uε(0)=maxB(0,1)uε(x) and vε(0)=maxB(0,1)vε(x), we see that Kε→+∞ as ε→0. Setting
U
ε
(
x
)
:=
K
ε
-
1
u
ε
(
K
ε
-
a
ε
x
)
,
V
ε
(
x
)
:=
K
ε
-
1
v
ε
(
K
ε
-
a
ε
x
)
,
a
ε
:=
p
∗
-
p
-
p
ε
p
,
we have
(4.9)
max
{
U
ε
(
0
)
,
V
ε
(
0
)
}
=
max
{
max
x
∈
B
(
0
,
K
ε
a
ε
)
U
ε
(
x
)
,
max
x
∈
B
(
0
,
K
ε
a
ε
)
V
ε
(
x
)
}
=
1
,
and (Uε,Vε) is a solution of
{
-
Δ
p
U
ε
=
μ
1
U
ε
p
∗
-
2
ε
-
1
+
(
α
-
ε
)
γ
p
∗
-
2
ε
U
ε
α
-
1
-
ε
V
ε
β
-
ε
,
x
∈
B
(
0
,
K
ε
a
ε
)
,
-
Δ
p
V
ε
=
μ
2
V
ε
p
∗
-
2
ε
-
1
+
(
β
-
ε
)
γ
p
∗
-
2
ε
U
ε
α
-
ε
V
ε
β
-
1
-
ε
,
x
∈
B
(
0
,
K
ε
a
ε
)
.
Since
∫
ℝ
N
|
∇
U
ε
(
x
)
|
p
𝑑
x
=
K
ε
a
ε
(
N
-
p
)
-
p
∫
ℝ
N
|
∇
u
ε
(
y
)
|
p
𝑑
y
=
K
ε
-
(
N
-
p
)
ε
∫
ℝ
N
|
∇
u
ε
(
x
)
|
p
𝑑
x
≤
∫
ℝ
N
|
∇
u
ε
(
x
)
|
p
𝑑
x
,
we see that {(Uε,Vε)}n≥1 is bounded in D. By elliptic estimates, we get that, up to a subsequence,
(
U
ε
,
V
ε
)
→
(
U
,
V
)
∈
D
uniformly in every compact subset of ℝN as ε→0, and (U,V) is a solution of (1.1), that is, I′(U,V)=0. Moreover, U≥0, V≥0 are radially symmetric decreasing. By (4.9), we have (U,V)≠(0,0), and so (U,V)∈𝒩′. Thus,
A
′
≤
I
(
U
,
V
)
=
(
1
p
-
1
p
∗
)
∫
ℝ
N
(
|
∇
U
|
p
+
|
∇
V
|
p
)
𝑑
x
≤
lim inf
ε
→
0
(
1
p
-
1
p
∗
)
∫
B
(
0
,
K
ε
a
ε
)
(
|
∇
U
ε
|
p
+
|
∇
V
ε
|
p
)
𝑑
x
=
lim inf
ε
→
0
(
1
p
-
1
p
∗
-
2
ε
)
∫
B
(
0
,
K
ε
a
ε
)
(
|
∇
U
ε
|
p
+
|
∇
V
ε
|
p
)
𝑑
x
≤
lim inf
ε
→
0
(
1
p
-
1
p
∗
-
2
ε
)
∫
B
(
0
,
1
)
(
|
∇
u
ε
|
p
+
|
∇
v
ε
|
p
)
𝑑
x
=
lim inf
ε
→
0
A
ε
.
It follows from (4.7) that A′≤I(U,V)≤lim infε→0Aε≤A′, which means that I(U,V)=A′. By (4.4), we get that U≢0 and V≢0. The strong maximum principle guarantees that U>0 and V>0. Since (U,V)∈𝒩, we have I(U,V)≥A≥A′. Therefore,
(4.10)
I
(
U
,
V
)
=
A
=
A
′
,
that is, (U,V) is a positive least energy solution of (1.1) with (H1) holding, which is radially symmetric decreasing. This completes the proof. ∎
Remark 4.1.
If (H1) and (C2) hold, then it can be seen from Theorems 1.2 and 1.3 that (k0pUε,y,l0pUε,y) is a positive least energy solution of (1.1), where (k0,l0) is defined by (1.9) and Uε,y is defined by (1.4).
Proof of Theorem 1.4.
To prove the existence of (k(γ),l(γ)) for γ>0 small, recalling (3.2), we denote Fi(k,l,γ) by Fi(k,l), i=1,2, in this proof. Let k(0)=μ1-p/(p∗-p) and l(0)=μ2-p/(p∗-p). Then
F
1
(
k
(
0
)
,
l
(
0
)
,
0
)
=
F
2
(
k
(
0
)
,
l
(
0
)
,
0
)
=
0
.
Obviously, we have
∂
k
F
1
(
k
(
0
)
,
l
(
0
)
,
0
)
=
p
∗
-
p
p
μ
1
k
p
∗
-
2
p
p
>
0
,
∂
l
F
1
(
k
(
0
)
,
l
(
0
)
,
0
)
=
∂
k
F
2
(
k
(
0
)
,
l
(
0
)
,
0
)
=
0
,
∂
l
F
2
(
k
(
0
)
,
l
(
0
)
,
0
)
=
p
∗
-
p
p
μ
2
l
p
∗
-
2
p
p
>
0
,
which implies that
det
(
∂
k
F
1
(
k
(
0
)
,
l
(
0
)
,
0
)
∂
l
F
1
(
k
(
0
)
,
l
(
0
)
,
0
)
∂
k
F
2
(
k
(
0
)
,
l
(
0
)
,
0
)
∂
l
F
2
(
k
(
0
)
,
l
(
0
)
,
0
)
)
>
0
.
By the implicit function theorem, we see that k(γ), l(γ) are well defined and of class C1 in (-γ2,γ2) for some γ2>0, and F1(k(γ),l(γ),γ)=F2(k(γ),l(γ),γ)=0. Then (k(γ)pUε,y,l(γ)pUε,y) is a positive solution of (1.1). Noticing that
lim
γ
→
0
(
k
(
γ
)
+
l
(
γ
)
)
=
k
(
0
)
+
l
(
0
)
=
μ
1
-
N
-
p
p
+
μ
2
-
N
-
p
p
,
we obtain that there exists γ1∈(0,γ2] such that
k
(
γ
)
+
l
(
γ
)
>
min
{
μ
1
-
N
-
p
p
,
μ
2
-
N
-
p
p
}
for all
γ
∈
(
0
,
γ
1
)
.
It follows from (4.4) and (4.10) that
I
(
k
(
γ
)
p
U
ε
,
y
,
l
(
γ
)
p
U
ε
,
y
)
=
1
N
(
k
(
γ
)
+
l
(
γ
)
)
S
N
p
>
min
{
1
N
μ
1
-
N
-
p
p
S
N
p
,
1
N
μ
2
-
N
-
p
p
S
N
p
}
>
A
′
=
A
=
I
(
U
,
V
)
,
that is, when (H1) is satisfied, (k(γ)pUε,y,l(γ)pUε,y) is a different positive solution of (1.1) with respect to (U,V). ∎
5 Proof of Theorem 1.5
In this section, we consider the case (H2).
Proposition 5.1.
Let q,r>1 satisfy q+r≤p∗, and set
S
q
,
r
(
Ω
)
=
inf
u
,
v
∈
W
0
1
,
p
(
Ω
)
u
,
v
≠
0
∫
Ω
(
|
∇
u
|
p
+
|
∇
v
|
p
)
𝑑
x
(
∫
Ω
|
u
|
q
|
v
|
r
𝑑
x
)
p
q
+
r
,
S
q
+
r
(
Ω
)
=
inf
u
∈
W
0
1
,
p
(
Ω
)
u
≠
0
∫
Ω
|
∇
u
|
p
𝑑
x
(
∫
Ω
|
u
|
q
+
r
𝑑
x
)
p
q
+
r
.
Then
(5.1)
S
q
,
r
(
Ω
)
=
q
+
r
(
q
q
r
r
)
1
q
+
r
S
q
+
r
(
Ω
)
.
Moreover, if u0 is a minimizer for Sq+r(Ω), then (q1/pu0,r1/pu0) is a minimizer for Sq,r(Ω).
Proof.
For u≠0 in W01,p(Ω) and t>0, taking v=t-1/pu in the first quotient gives
S
q
,
r
(
Ω
)
≤
[
t
r
q
+
r
+
t
-
q
q
+
r
]
∫
Ω
|
∇
u
|
p
𝑑
x
(
∫
Ω
|
u
|
q
+
r
𝑑
x
)
p
q
+
r
,
and minimizing the right-hand side over u and t shows that Sq,r(Ω) is less than or equal to the right-hand side of (5.1). For u,v≠0 in W01,p(Ω), let w=t1/pv, where
t
q
+
r
p
=
∫
Ω
|
u
|
q
+
r
𝑑
x
∫
Ω
|
v
|
q
+
r
𝑑
x
.
Then ∫Ω|u|q+r𝑑x=∫Ω|w|q+r𝑑x, and hence
∫
Ω
|
u
|
q
|
w
|
r
𝑑
x
≤
∫
Ω
|
u
|
q
+
r
𝑑
x
=
∫
Ω
|
w
|
q
+
r
𝑑
x
by the Hölder inequality, so
∫
Ω
(
|
∇
u
|
p
+
|
∇
v
|
p
)
𝑑
x
(
∫
Ω
|
u
|
q
|
v
|
r
𝑑
x
)
p
q
+
r
=
∫
Ω
(
t
r
q
+
r
|
∇
u
|
p
+
t
-
q
q
+
r
|
∇
w
|
p
)
𝑑
x
(
∫
Ω
|
u
|
q
|
w
|
r
𝑑
x
)
p
q
+
r
≥
t
r
q
+
r
∫
Ω
|
∇
u
|
p
𝑑
x
(
∫
Ω
|
u
|
q
+
r
𝑑
x
)
p
q
+
r
+
t
-
q
q
+
r
∫
Ω
|
∇
w
|
p
𝑑
x
(
∫
Ω
|
w
|
q
+
r
𝑑
x
)
p
q
+
r
≥
[
t
r
q
+
r
+
t
-
q
q
+
r
]
S
q
+
r
(
Ω
)
.
The last expression is greater than or equal to the right-hand side of (5.1), so minimizing over (u,v) gives the reverse inequality. ∎
By Proposition 5.1,
(5.2)
S
a
,
b
(
Ω
)
=
p
(
a
a
b
b
)
1
p
λ
1
(
Ω
)
,
S
α
,
β
=
p
∗
(
α
α
β
β
)
1
p
∗
S
,
where λ1(Ω)>0 is the first Dirichlet eigenvalue of -Δp in Ω. When (H2) is satisfied, we will obtain a nontrivial nonnegative solution of system (1.1) for λ<Sa,b(Ω). Consider the C1-functional
Φ
(
w
)
=
1
p
∫
Ω
[
|
∇
u
|
p
+
|
∇
v
|
p
-
λ
(
u
+
)
a
(
v
+
)
b
]
𝑑
x
-
1
p
∗
∫
Ω
(
u
+
)
α
(
v
+
)
β
𝑑
x
,
w
∈
W
,
where W=D01,p(Ω)×D01,p(Ω) with the norm given by ∥w∥p=|∇u|pp+|∇v|pp for w=(u,v), |⋅|p denotes the norm in Lp(Ω) and u±(x)=max{±u(x),0} are the positive and negative parts of u, respectively. If w is a critical point of Φ,
0
=
Φ
′
(
w
)
(
u
-
,
v
-
)
=
∫
Ω
(
|
∇
u
-
|
p
+
|
∇
v
-
|
p
)
𝑑
x
,
and hence (u-,v-)=0, so w=(u+,v+) is a nonnegative weak solution of (1.1) with (H2) holding.
Proposition 5.2.
If 0≠c<Sα,βN/p/N and λ<Sa,b(Ω), then every (PS)c sequence of Φ has a subsequence that converges weakly to a nontrivial critical point of Φ.
Proof.
Let {wj} be a (PS)c sequence. Then
Φ
(
w
j
)
=
1
p
∫
Ω
[
|
∇
u
j
|
p
+
|
∇
v
j
|
p
-
λ
(
u
j
+
)
a
(
v
j
+
)
b
]
𝑑
x
-
1
p
∗
∫
Ω
(
u
j
+
)
α
(
v
j
+
)
β
𝑑
x
=
c
+
o
(
1
)
and
Φ
′
(
w
j
)
w
j
=
∫
Ω
[
|
∇
u
j
|
p
+
|
∇
v
j
|
p
-
λ
(
u
j
+
)
a
(
v
j
+
)
b
]
𝑑
x
-
∫
Ω
(
u
j
+
)
α
(
v
j
+
)
β
𝑑
x
(5.3)
=
o
(
∥
w
j
∥
)
,
so
(5.4)
1
N
∫
Ω
[
|
∇
u
j
|
p
+
|
∇
v
j
|
p
-
λ
(
u
j
+
)
a
(
v
j
+
)
b
]
𝑑
x
=
c
+
o
(
∥
w
j
∥
+
1
)
.
Since the integral on the left-hand side is greater than or equal to (1-λSa,b(Ω))∥wj∥p, λ<Sa,b(Ω) and p>1, it follows that {wj} is bounded in W. So a renamed subsequence converges to some w weakly in W, strongly in Ls(Ω)×Lt(Ω) for all 1≤s, t<p∗ and a.e. in Ω. Then wj→w strongly in W01,q(Ω)×W01,r(Ω) for all 1≤q, r<p by Boccardo and Murat [6, Theorem 2.1], and hence ∇wj→∇w a.e. in Ω for a further subsequence. It then follows that w is a critical point of Φ.
Suppose w=0. Since {wj} is bounded in W and converges to zero in Lp(Ω)×Lp(Ω), equation (5.3) and the Hölder inequality give
o
(
1
)
=
∫
Ω
(
|
∇
u
j
|
p
+
|
∇
v
j
|
p
)
𝑑
x
-
∫
Ω
(
u
j
+
)
α
(
v
j
+
)
β
𝑑
x
≥
∥
w
j
∥
p
(
1
-
∥
w
j
∥
p
∗
-
p
S
α
,
β
p
∗
p
)
.
If ∥wj∥→0, then Φ(wj)→0, contradicting c≠0, so this implies
∥
w
j
∥
p
≥
S
α
,
β
N
p
+
o
(
1
)
for a renamed subsequence. Then (5.4) gives
c
=
∥
w
j
∥
p
N
+
o
(
1
)
≥
S
α
,
β
N
p
N
+
o
(
1
)
,
contradicting c<Sα,βN/p/N. ∎
Recall (1.4) and (1.5) and let η:[0,∞)→[0,1] be a smooth cut-off function such that η(s)=1 for s≤14 and η(s)=0 for s≥12; set
u
ε
,
ρ
(
x
)
=
η
(
|
x
|
ρ
)
U
ε
,
0
(
x
)
for ρ>0. We have the following estimates for uε,ρ (see [15, Lemma 3.1]):
(5.5)
∫
ℝ
N
|
∇
u
ε
,
ρ
|
p
𝑑
x
≤
S
N
p
+
C
(
ε
ρ
)
N
-
p
p
-
1
,
(5.6)
∫
ℝ
N
u
ε
,
ρ
p
𝑑
x
≥
{
1
C
ε
p
log
(
ρ
ε
)
-
C
ε
p
if
N
=
p
2
,
1
C
ε
p
-
C
ρ
p
(
ε
ρ
)
N
-
p
p
-
1
if
N
>
p
2
,
(5.7)
∫
ℝ
N
u
ε
,
ρ
p
∗
𝑑
x
≥
S
N
p
-
C
(
ε
ρ
)
N
p
-
1
,
where C=C(N,p). We will make use of these estimates in the proof of our last theorem.
Proof of Theorem 1.5.
In view of (5.2),
Φ
(
w
)
≥
1
p
(
1
-
λ
S
a
,
b
(
Ω
)
)
∥
w
∥
p
-
1
p
∗
S
α
,
β
p
∗
p
∥
w
∥
p
∗
,
so the origin is a strict local minimizer of Φ. We may assume without loss of generality that 0∈Ω. Fix ρ>0 so small that Ω⊃Bρ(0)⊃suppuε,ρ, and let wε=(α1/puε,ρ,β1/puε,ρ)∈W. Note that
Φ
(
R
w
ε
)
=
R
p
p
(
p
∗
|
∇
u
ε
,
ρ
|
p
p
-
λ
α
a
p
β
b
p
|
u
ε
,
ρ
|
p
p
)
-
R
p
∗
p
∗
α
α
p
β
β
p
|
u
ε
,
ρ
|
p
∗
p
∗
→
-
∞
as R→+∞ and fix R0>0 so large that Φ(R0wε)<0. Then let
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
W
)
:
γ
(
0
)
=
0
,
γ
(
1
)
=
R
0
w
ε
}
and set
c
:=
inf
γ
∈
Γ
max
t
∈
[
0
,
1
]
Φ
(
γ
(
t
)
)
>
0
.
By the mountain pass theorem, Φ has a (PS)c sequence {wj}.
Since t↦tR0wε is a path in Γ,
(5.8)
c
≤
max
t
∈
[
0
,
1
]
Φ
(
t
R
0
w
ε
)
=
1
N
(
p
∗
|
∇
u
ε
,
ρ
|
p
p
-
λ
(
α
a
β
b
)
1
p
|
u
ε
,
ρ
|
p
p
(
α
α
β
β
)
1
p
∗
|
u
ε
,
ρ
|
p
∗
p
)
N
p
=
:
1
N
S
ε
N
p
.
By (5.5)–(5.7),
S
ε
≤
p
∗
S
p
+
λ
(
α
a
β
b
)
1
p
C
ε
p
log
ε
+
O
(
ε
p
)
(
α
α
β
β
)
1
p
∗
(
S
p
+
O
(
ε
p
2
p
-
1
)
)
p
-
1
p
=
S
α
,
β
-
(
λ
α
a
p
-
α
p
∗
β
b
p
-
β
p
∗
C
S
p
-
1
|
log
ε
|
+
O
(
1
)
)
ε
p
if N=p2, and
S
ε
≤
p
∗
S
N
p
-
λ
(
α
a
β
b
)
1
p
C
ε
p
+
O
(
ε
N
-
p
p
-
1
)
(
α
α
β
β
)
1
p
∗
(
S
N
p
+
O
(
ε
N
p
-
1
)
)
N
-
p
N
=
S
α
,
β
-
(
λ
α
a
p
-
α
p
∗
β
b
p
-
β
p
∗
C
S
N
-
p
p
+
O
(
ε
N
-
p
2
p
-
1
)
)
ε
p
if N>p2, so Sε<Sα,β if ε>0 is sufficiently small. So c<Sα,βN/p/N by (5.8), and hence a subsequence of {wj} converges weakly to a nontrivial critical point of Φ by Proposition 5.2, which then is a nontrivial nonnegative solution of (1.1) with (H2) holding. ∎
und
Wenming Zou
