1 Introduction
In this work, we study the multiplicity of solutions to the following fractional elliptic system:
(1.1)
{
(
-
Δ
)
p
s
u
=
λ
|
u
|
q
-
2
u
+
2
α
α
+
β
|
u
|
α
-
2
u
|
v
|
β
in
Ω
,
(
-
Δ
)
p
s
v
=
μ
|
v
|
q
-
2
v
+
2
β
α
+
β
|
u
|
α
|
v
|
β
-
2
v
in
Ω
,
u
=
v
=
0
in
ℝ
n
∖
Ω
,
where Ω is a smooth bounded set in ℝn, n>ps with s∈(0,1), λ,μ>0 are two parameters, 1<q<p and α>1,β>1 satisfy α+β=ps∗, where ps∗=npn-ps is the fractional critical Sobolev exponent, and (-Δ)ps is the fractional p-Laplacian operator, defined on smooth functions as
(
-
Δ
)
p
s
u
(
x
)
=
2
lim
ϵ
→
0
∫
ℝ
n
∖
B
ϵ
(
x
)
|
u
(
y
)
-
u
(
x
)
|
p
-
2
(
u
(
y
)
-
u
(
x
)
)
|
x
-
y
|
n
+
p
s
𝑑
y
,
x
∈
ℝ
n
.
This definition is consistent, up to a normalization constant depending on n and s, with the linear fractional Laplacian (-Δ)s for the case p=2. If we set α=β, α+β=r, λ=μ and u=v, then system (1.1) reduces to the following fractional equation with concave-convex nonlinearities:
(1.2)
{
(
-
Δ
)
p
s
u
=
λ
|
u
|
q
-
2
u
+
|
u
|
r
-
2
u
in
Ω
,
u
=
0
in
ℝ
n
∖
Ω
,
where 1<q<p and p<r<ps∗. In [14] Goyal and Sreenadh studied the existence and multiplicity of nonnegative solutions to the nonlocal problem (1.2) for subcritical concave-convex nonlinearities. For the fractional p-Laplacian, consider the following general problem:
{
(
-
Δ
)
p
s
u
=
f
(
x
,
u
)
in
Ω
,
u
=
0
in
ℝ
n
∖
Ω
.
So far various results have been obtained for these kind of problems. Lindgren and Lindqvist [19] considered the eigenvalue problem associated with (-Δ)ps and obtained some properties of the first and of higher (variational) eigenvalues. Some results about the existence of solutions have been considered in [13, 18, 21], see also the references therein. Let us also mention [22] where, by using variational methods and topological degree theory, Pucci, Xiang and Zhang proved multiplicity results for fractional p-Kirchhoff equations.
On the other hand, the fractional problems for p=2 have been investigated by many researchers, see, for example, [2, 23, 6] for the critical case and [11] for the fractional Kirchhoff type problem. In particular, Brändle et al. [3] studied the fractional Laplacian equation involving a concave-convex nonlinearity in the subcritical case. The existence and multiplicity of solutions for system (1.1), when s=1, were considered by many authors, see [16, 17, 24] and references therein. In particular, Hsu [16] obtained multiple solutions for the following critical elliptic system:
{
-
Δ
p
u
=
λ
|
u
|
q
-
2
u
+
2
α
α
+
β
|
u
|
α
-
2
u
|
v
|
β
in
Ω
,
-
Δ
p
v
=
μ
|
v
|
q
-
2
v
+
2
β
α
+
β
|
u
|
α
|
v
|
β
-
2
v
in
Ω
,
u
=
v
=
0
on
∂
Ω
,
where q<p and α>1,β>1 satisfy α+β=npn-p. For system (1.1) with p=2, we mention [10, 15]. Moreover, Giacomoni, Mishra and Sreenadh [12] showed the existence of multiple solutions for critical growth fractional elliptic systems with exponential nonlinearity by analyzing the fibering maps.
However, as far as we know, there are a few results on the case p≠2 with concave-convex critical nonlinearities. Recently, Chen and Deng [7] studied system (1.1) with a subcritical concave-convex type nonlinearity, i.e., when α+β<ps*. Motivated by the above results, in the present paper, we are interested in the multiplicity of solutions for the critical fractional p-Laplacian system (1.1), i.e., when
α
+
β
=
p
s
∗
.
We denote by Ws,p(Ω) the usual fractional Sobolev space endowed with the norm
∥
u
∥
W
s
,
p
(
Ω
)
:=
∥
u
∥
L
p
(
Ω
)
+
(
∫
Ω
×
Ω
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
)
1
p
.
Set Q:=ℝ2n∖(𝒞Ω×𝒞Ω) with 𝒞Ω=ℝn∖Ω. We define
X
:=
{
u
:
ℝ
n
→
ℝ
measurable
,
u
|
Ω
∈
L
p
(
Ω
)
and
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
<
∞
}
.
The space X is endowed with the norm
∥
u
∥
X
:=
∥
u
∥
L
p
(
Ω
)
+
(
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
)
1
p
.
The space X0 is defined as X0:={u∈X:u=0 on 𝒞Ω} or, equivalently, as C0∞(Ω)¯X and, for any p>1, it is a uniformly convex Banach space endowed with the norm
(1.3)
∥
u
∥
X
0
=
(
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
)
1
p
.
Since u=0 in ℝn∖Ω, the integral in (1.3) can be extended to all ℝn. The embedding X0↪Lr(Ω) is continuous for any r∈[1,ps∗] and compact for r∈[1,ps∗). We set E:=X0×X0, with the norm
∥
(
u
,
v
)
∥
=
(
∥
u
∥
X
0
p
+
∥
v
∥
X
0
p
)
1
p
=
(
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
+
∫
Q
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
)
1
p
.
For convenience, we define
(1.4)
𝒜
(
u
,
ϕ
)
:=
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
(
ϕ
(
x
)
-
ϕ
(
y
)
)
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
.
Definition 1.1
We say that (u,v)∈E is a weak solution of problem (1.1) if
𝒜
(
u
,
ϕ
)
+
𝒜
(
v
,
ψ
)
=
∫
Ω
(
λ
|
u
|
q
-
2
u
ϕ
+
μ
|
v
|
q
-
2
v
ψ
)
d
x
+
2
α
α
+
β
∫
Ω
|
u
|
α
-
2
u
|
v
|
β
ϕ
d
x
+
2
β
α
+
β
∫
Ω
|
u
|
α
|
v
|
β
-
2
v
ψ
d
x
for all (ϕ,ψ)∈E.
In the sequel we omit the term weak when referring to solutions which satisfy Definition 1.1. Let s∈(0,1), p>1 and let Ω be a bounded domain of ℝn. The next theorem is our main result.
Theorem 1.2
Assume that
(1.5)
p
2
s
<
n
<
{
∞
if
p
≥
2
,
p
s
2
-
p
if
p
<
2
,
n
(
p
-
1
)
n
-
p
s
≤
q
<
p
,
α
+
β
=
n
p
n
-
p
s
.
Then there exists a positive constant Λ∗=Λ*(p,q,s,n,|Ω|) such that for λ, μ satisfying
0
<
λ
p
p
-
q
+
μ
p
p
-
q
<
Λ
∗
,
system (1.1) admits at least two nontrivial solutions.
For the critical case, since the embedding X0↪Lps∗(ℝn) fails to be compact, the energy functional does not satisfy the Palais–Smale condition globally, but that holds true when the energy level falls inside a suitable range related to the best fractional critical Sobolev constant S, namely,
(1.6)
S
:=
inf
u
∈
X
0
∖
{
0
}
∫
ℝ
2
n
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
u
(
x
)
|
n
p
n
-
p
s
d
x
)
n
-
p
s
n
.
For the critical fractional case with p≠2, the main difficulty is the lack of an explicit formula for minimizers of S, which is very often a key tool to handle the estimates leading to the compactness range of the functional. It was conjectured that, up to a multiplicative constant, all minimizers are of the form U(x-x0ϵ), with
U
(
x
)
=
(
1
+
|
x
|
p
p
-
1
)
-
n
-
p
s
p
,
x
∈
ℝ
n
.
This conjecture was proved in [8] for p=2, but for p≠2, it is not even known if these functions are minimizers of S. On the other hand, as in [20], we can overcome this difficulty by the optimal asymptotic behavior of minimizers, which was recently obtained in [4]. This will allow us to prove Lemma 4.10, related to the Palais–Smale condition. That is the only point where the restriction (1.6) on p,q,n comes into play. On the other hand, we point out that, as detected in [20], n=p2s corresponds to the critical dimension for the nonlocal Brézis–Nirenberg problem.
This paper is organized as follows. In Section 2, we give some notations and preliminaries for the Nehari manifold and fibering maps. In Section 3, we show that the (PS)c condition holds for Jλ,μ, with c in certain interval. In Sections 4 and 5, we complete the proof of Theorem 1.2.
2 The Fibering Properties
In this section, we give some notations and preliminaries for the Nehari manifold and the analysis of the fibering maps. Being a weak solution (u,v)∈E is equivalent to being a critical point of the following C1 functional on E:
J
λ
,
μ
(
u
,
v
)
:=
1
p
∫
Q
|
u
(
x
)
-
u
(
y
)
|
p
|
x
-
y
|
n
+
p
s
d
x
d
y
+
1
p
∫
Q
|
v
(
x
)
-
v
(
y
)
|
p
|
x
-
y
|
n
+
p
s
d
x
d
y
-
1
q
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
α
+
β
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
By a direct calculation, we have that Jλ,μ∈C1(E,ℝ) and
〈
J
λ
,
μ
′
(
u
,
v
)
,
(
ϕ
,
ψ
)
〉
=
𝒜
(
u
,
ϕ
)
+
𝒜
(
v
,
ψ
)
-
∫
Ω
(
λ
|
u
|
q
-
2
u
ϕ
+
μ
|
v
|
q
-
2
v
ψ
)
𝑑
x
-
2
α
α
+
β
∫
Ω
|
u
|
α
-
2
u
|
v
|
β
ϕ
d
x
-
2
β
α
+
β
∫
Ω
|
u
|
α
|
v
|
β
-
2
v
ψ
d
x
for any (ϕ,ψ)∈E. We will study critical points of the functional Jλ,μ on E. Consider the Nehari manifold
𝒩
λ
,
μ
=
{
(
u
,
v
)
∈
E
∖
{
(
0
,
0
)
}
:
〈
J
λ
,
μ
′
(
u
,
v
)
,
(
u
,
v
)
〉
=
0
}
.
Then (u,v)∈𝒩λ,μ if and only if (u,v)≠(0,0) and
∥
(
u
,
v
)
∥
p
=
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
+
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
The Nehari manifold 𝒩λ,μ is closely linked to the behavior of a function of the form φu,v:t↦Jλ,μ(tu,tv) for t>0, defined by
φ
u
,
v
(
t
)
:=
J
λ
,
μ
(
t
u
,
t
v
)
=
t
p
p
∥
(
u
,
v
)
∥
p
-
t
q
q
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
t
α
+
β
α
+
β
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
Such maps are known as fibering maps and were introduced by Drabek and Pohozaev in [9].
Lemma 2.1
Lemma 2.1 (Fibering Map)
Let (u,v)∈E∖{(0,0)}. Then (tu,tv)∈Nλ,μ if and only if φu,v′(t)=0.
Proof.
The result is a consequence of the fact that φu,v′(t)=〈Jλ,μ′(tu,tv),(u,v)〉. ∎
We note that
(2.1)
φ
u
,
v
′
(
t
)
=
t
p
-
1
∥
(
u
,
v
)
∥
p
-
t
q
-
1
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
t
α
+
β
-
1
∫
Ω
|
u
|
α
|
v
|
β
d
x
and
φ
u
,
v
′′
(
t
)
=
(
p
-
1
)
t
p
-
2
∥
(
u
,
v
)
∥
p
-
(
q
-
1
)
t
q
-
2
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
(
α
+
β
-
1
)
t
α
+
β
-
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
By Lemma 2.1, (u,v)∈𝒩λ,μ if and only if φu,v′(1)=0. Hence, for (u,v)∈𝒩λ,μ, (2.1) yields
φ
u
,
v
′′
(
1
)
=
(
p
-
1
)
∥
(
u
,
v
)
∥
p
-
(
q
-
1
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
(
α
+
β
-
1
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
=
2
(
p
-
(
α
+
β
)
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
+
(
p
-
q
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
=
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
(
α
+
β
)
-
q
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
(2.2)
=
(
p
-
(
α
+
β
)
)
∥
(
u
,
v
)
∥
p
+
(
(
α
+
β
)
-
q
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
.
Thus, it is natural to split 𝒩λ,μ into three parts corresponding to local minima, local maxima and points of inflection of φu,v, namely,
𝒩
λ
,
μ
+
:=
{
(
u
,
v
)
∈
𝒩
λ
,
μ
:
φ
u
,
v
′′
(
1
)
>
0
}
,
𝒩
λ
,
μ
-
:=
{
(
u
,
v
)
∈
𝒩
λ
,
μ
:
φ
u
,
v
′′
(
1
)
<
0
}
,
𝒩
λ
,
μ
0
:=
{
(
u
,
v
)
∈
𝒩
λ
,
μ
:
φ
u
,
v
′′
(
1
)
=
0
}
.
We will prove the existence of solutions of problem (1.1) by investigating the existence of minimizers of the functional Jλ,μ on 𝒩λ,μ. Although 𝒩λ,μ is a subset of E, we can see that the local minimizers on the Nehari manifold 𝒩λ,μ are usually critical points of Jλ,μ. We have the following lemma.
Lemma 2.2
Lemma 2.2 (Natural Constraint)
Suppose that (u0,v0) is a local minimizer of the functional Jλ,μ on Nλ,μ and that (u0,v0)∉Nλ,μ0. Then (u0,v0) is a critical point of Jλ,μ.
Proof.
The proof is a standard corollary of the lagrange multiplier rule, where the constraint is
Q
(
u
,
v
)
=
∥
(
u
,
v
)
∥
p
-
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
,
after observing that, for (u,v)∈𝒩λ,μ,
〈
Q
′
(
u
,
v
)
,
(
u
,
v
)
〉
=
p
∥
(
u
,
v
)
∥
p
-
q
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
(
α
+
β
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
=
(
p
-
1
)
∥
(
u
,
v
)
∥
p
-
(
q
-
1
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
(
α
+
β
-
1
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
=
φ
u
,
v
′′
(
1
)
≠
0
,
by the assumption that (u,v)∉𝒩λ,μ0. ∎
In order to understand the Nehari manifold and the fibering maps, we consider Ψu,v:ℝ+→ℝ defined by
Ψ
u
,
v
(
t
)
:=
t
p
-
(
α
+
β
)
∥
(
u
,
v
)
∥
p
-
t
q
-
(
α
+
β
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
.
By simple computations, we have the following results.
Lemma 2.3
Lemma 2.3 (Properties of Ψu,v)
Let (u,v)∈E∖{(0,0)}. Then Ψu,v satisfies the following properties:
Ψ
u
,
v
(
t
)
has a unique critical point at
t
max
(
u
,
v
)
:=
(
(
α
+
β
-
q
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
(
α
+
β
-
p
)
∥
(
u
,
v
)
∥
p
)
1
p
-
q
>
0
,
Ψu,v(t) is strictly increasing on (0,tmax(u,v)) and strictly decreasing on (tmax(u,v),+∞),
limt→0+Ψu,v(t)=-∞ and limt→+∞Ψu,v(t)=0.
Lemma 2.4
Lemma 2.4 (Characterization of Nλ,μ±)
We have (tu,tv)∈Nλ,μ± if and only if ±Ψu,v′(t)>0.
Proof.
It is clear that for t>0, (tu,tv)∈𝒩λ,μ if and only if
(2.3)
Ψ
u
,
v
(
t
)
=
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
Moreover,
Ψ
u
,
v
′
(
t
)
=
(
p
-
(
α
+
β
)
)
t
p
-
(
α
+
β
)
-
1
∥
(
u
,
v
)
∥
p
-
(
q
-
(
α
+
β
)
)
t
q
-
(
α
+
β
)
-
1
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
,
and if (tu,tv)∈𝒩λ,μ, then
(2.4)
t
α
+
β
-
1
Ψ
u
,
v
′
(
t
)
=
φ
u
,
v
′′
(
t
)
=
t
-
2
φ
t
u
,
t
v
′′
(
1
)
.
Hence, (tu,tv)∈𝒩λ,μ+ (resp. 𝒩λ,μ-) if and only if Ψu,v′(t)>0 (resp. <0). ∎
Lemma 2.5
Lemma 2.5 (Elements of Nλ,μ±)
Let us set
(2.5)
Λ
1
=
(
p
-
q
2
(
α
+
β
-
q
)
)
p
α
+
β
-
p
(
α
+
β
-
q
α
+
β
-
p
|
Ω
|
α
+
β
-
q
α
+
β
)
-
p
p
-
q
S
α
+
β
α
+
β
-
p
+
q
p
-
q
,
with S being the best constant for the Sobolev embedding of X0 into Lps*(Rn). If (u,v)∈E∖{(0,0)}, then for any λ, μ satisfying
0
<
λ
p
p
-
q
+
μ
p
p
-
q
<
Λ
1
,
there exist unique t1,t2>0 such that t1<tmax(u,v)<t2 and
(
t
1
u
,
t
1
v
)
∈
𝒩
λ
,
μ
+
,
(
t
2
u
,
t
2
v
)
∈
𝒩
λ
,
μ
-
.
Moreover,
J
λ
,
μ
(
t
1
u
,
t
1
v
)
=
inf
0
≤
t
≤
t
max
J
λ
,
μ
(
t
u
,
t
v
)
,
J
λ
,
μ
(
t
2
u
,
t
2
v
)
=
sup
t
≥
0
J
λ
,
μ
(
t
u
,
t
v
)
.
Proof.
As ∫Ω|u|α|v|βdx>0, we know that (2.3) has no solution if and only if λ and μ satisfy the condition
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
>
Ψ
u
,
v
(
t
max
(
u
,
v
)
)
.
By Lemma 2.3, we have
Ψ
u
,
v
(
t
max
(
u
,
v
)
)
=
[
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
-
(
α
+
β
-
q
α
+
β
-
p
)
q
-
(
α
+
β
)
p
-
q
]
(
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
)
p
-
(
α
+
β
)
p
-
q
∥
(
u
,
v
)
∥
p
(
q
-
(
α
+
β
)
)
p
-
q
=
p
-
q
α
+
β
-
q
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
(
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
)
p
-
(
α
+
β
)
p
-
q
∥
(
u
,
v
)
∥
p
(
q
-
(
α
+
β
)
)
p
-
q
.
By Hölder’s inequality and the definition of S, we find
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
≤
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
,
v
)
∥
q
.
Then, since q<p<α+β=ps∗, we have
Ψ
u
,
v
(
t
max
(
u
,
v
)
)
≥
p
-
q
α
+
β
-
q
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
[
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
,
v
)
∥
q
]
p
-
(
α
+
β
)
p
-
q
∥
(
u
,
v
)
∥
p
(
q
-
(
α
+
β
)
)
p
-
q
(2.6)
=
p
-
q
α
+
β
-
q
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
[
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
]
p
-
(
α
+
β
)
p
-
q
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
(
α
+
β
)
p
∥
(
u
,
v
)
∥
α
+
β
.
On the other hand, using Young’s inequality and the definition of S, we have
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
≤
2
(
α
α
+
β
∫
Ω
|
u
|
α
+
β
d
x
+
β
α
+
β
∫
Ω
|
v
|
α
+
β
d
x
)
≤
2
S
-
α
+
β
p
∥
(
u
,
v
)
∥
α
+
β
.
For any λ,μ satisfying 0<λpp-q+μpp-q<Λ1, with Λ1 given in (2.5), we have
(2.7)
2
S
-
α
+
β
p
≤
p
-
q
α
+
β
-
q
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
[
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
]
p
-
(
α
+
β
)
p
-
q
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
(
α
+
β
)
p
.
Thus, from (2.6) and (2.7), if λ,μ satisfy 0<λpp-q+μpp-q<Λ1, we have
0
<
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
≤
2
S
-
α
+
β
p
∥
(
u
,
v
)
∥
α
+
β
≤
p
-
q
α
+
β
-
q
(
α
+
β
-
q
α
+
β
-
p
)
p
-
(
α
+
β
)
p
-
q
[
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
]
p
-
(
α
+
β
)
p
-
q
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
(
α
+
β
)
p
∥
(
u
,
v
)
∥
α
+
β
<
Ψ
u
,
v
(
t
max
(
u
,
v
)
)
.
Then, there exist unique t1>0 and t2>0, with t1<tmax(u,v)<t2, such that
Ψ
u
,
v
(
t
1
)
=
Ψ
u
,
v
(
t
2
)
=
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
,
Ψ
u
,
v
′
(
t
1
)
>
0
,
Ψ
u
,
v
′
(
t
2
)
<
0
.
In turn, (2.1) and (2.3) give that φu,v′(t1)=φu,v′(t2)=0. By (2.4), we have that φu,v′′(t1)>0 and φu,v′′(t2)<0. These facts imply that φu,v has a local minimum at t1 and a local maximum at t2 such that (t1u,t1v)∈𝒩λ,μ+ and (t2u,t2v)∈𝒩λ,μ-. Since φu,v(t)=Jλ,μ(tu,tv), we have Jλ,μ(t2u,t2v)≥Jλ,μ(tu,tv)≥Jλ,μ(t1u,t1v) for each t∈[t1,t2] and Jλ,μ(t1u,t1v)≤Jλ,μ(tu,tv) for each t∈[0,t1]. Thus,
J
λ
,
μ
(
t
1
u
,
t
1
v
)
=
inf
0
≤
t
≤
t
max
J
λ
,
μ
(
t
u
,
t
v
)
,
J
λ
,
μ
(
t
2
u
,
t
2
v
)
=
sup
t
≥
0
J
λ
,
μ
(
t
u
,
t
v
)
.
The graphs of Ψu,v and φu,v can be seen in Figure 1. ∎
3 The Palais–Smale Condition
In this section, we show that the functional Jλ,μ satisfies the (PS)c condition.
Definition 3.1
Let c∈ℝ, let E be a Banach space and let Jλ,μ∈C1(E,ℝ). We say that {(uk,vk)}k∈ℕ is a (PS)c sequence in E for Jλ,μ if Jλ,μ(uk,vk)=c+o(1) and Jλ,μ′(uk,vk)=o(1) strongly in E∗ as k→∞. We say that Jλ,μ satisfies the (PS)c condition if any (PS)c sequence {(uk,vk)}k∈ℕ for Jλ,μ in E admits a convergent subsequence.
Lemma 3.2
Lemma 3.2 (Boundedness of (PS)c Sequences)
If {(uk,vk)}k∈N⊂E is a (PS)c sequence for Jλ,μ, then it follows that {(uk,vk)}k∈N is bounded in E.
Proof.
If {(uk,vk)}⊂E is a (PS)c sequence for Jλ,μ, then we have
J
λ
,
μ
(
u
k
,
v
k
)
→
c
,
J
λ
,
μ
′
(
u
k
,
v
k
)
→
0
in
E
∗
as
k
→
∞
.
That is,
(3.1)
1
p
∥
(
u
k
,
v
k
)
∥
p
-
1
q
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
d
x
-
2
α
+
β
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
c
+
o
k
(
1
)
,
(3.2)
∥
(
u
k
,
v
k
)
∥
p
-
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
d
x
-
2
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
o
k
(
∥
(
u
k
,
v
k
)
∥
)
as
k
→
∞
.
We shall show that (uk,vk) is bounded in E by contradiction. Assume that ∥(uk,vk)∥→∞, and set
u
~
k
:=
u
k
∥
(
u
k
,
v
k
)
∥
,
v
~
k
:=
v
k
∥
(
u
k
,
v
k
)
∥
.
Then ∥(u~k,v~k)∥=1. There is a subsequence, still denoted by (u~k,v~k), with (u~k,v~k)⇀(u~,v~)∈E and
u
~
k
→
u
~
,
v
~
k
→
v
~
in
L
r
(
ℝ
n
)
,
u
~
k
→
u
~
,
v
~
k
→
v
~
a.e. in
ℝ
n
,
for any 1≤r<ps∗=npn-ps. Then, the Dominated Convergence Theorem yields
(3.3)
∫
Ω
(
λ
|
u
~
k
|
q
+
μ
|
v
~
k
|
q
)
𝑑
x
→
∫
Ω
(
λ
|
u
~
|
q
+
μ
|
v
~
|
q
)
𝑑
x
as
k
→
∞
.
Moreover, from (3.1) and (3.2), we find that (u~k,v~k) satisfy
1
p
∥
(
u
~
k
,
v
~
k
)
∥
p
-
∥
(
u
k
,
v
k
)
∥
q
-
p
q
∫
Ω
(
λ
|
u
~
k
|
q
+
μ
|
v
~
k
|
q
)
d
x
-
2
∥
(
u
k
,
v
k
)
∥
α
+
β
-
p
α
+
β
∫
Ω
|
u
~
k
|
α
|
v
~
k
|
β
d
x
=
o
k
(
1
)
,
∥
(
u
~
k
,
v
~
k
)
∥
p
-
∥
(
u
k
,
v
k
)
∥
q
-
p
∫
Ω
(
λ
|
u
~
k
|
q
+
μ
|
v
~
k
|
q
)
d
x
-
2
∥
(
u
k
,
v
k
)
∥
α
+
β
-
p
∫
Ω
|
u
~
k
|
α
|
v
~
k
|
β
d
x
=
o
k
(
1
)
.
From the above two equalities and (3.3), we obtain
∥
(
u
~
k
,
v
~
k
)
∥
p
=
p
(
α
+
β
-
q
)
q
(
α
+
β
-
p
)
∥
(
u
k
,
v
k
)
∥
q
-
p
∫
Ω
(
λ
|
u
~
k
|
q
+
μ
|
v
~
k
|
q
)
𝑑
x
+
o
k
(
1
)
=
p
(
α
+
β
-
q
)
q
(
α
+
β
-
p
)
∥
(
u
k
,
v
k
)
∥
q
-
p
∫
Ω
(
λ
|
u
~
|
q
+
μ
|
v
~
|
q
)
𝑑
x
+
o
k
(
1
)
.
Since 1<q<p and ∥(uk,vk)∥→∞, we get ∥(u~k,v~k)∥p→0, which contradicts ∥(u~k,v~k)∥=1. ∎
Lemma 3.3
Lemma 3.3 (Uniform Lower Bound)
If {(uk,vk)}k∈N is a (PS)c sequence for Jλ,μ with (uk,vk)⇀(u,v) in E, then Jλ,μ′(u,v)=0, and there exists a positive constant C0 depending on p,q,s,n,S and |Ω| such that
(3.4)
J
λ
,
μ
(
u
,
v
)
≥
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
,
where we have set
(3.5)
C
0
:=
p
-
q
p
q
p
s
*
(
p
s
∗
-
q
)
p
p
-
q
(
p
s
∗
-
p
)
q
p
-
q
|
Ω
|
p
(
p
s
∗
-
q
)
p
s
∗
(
p
-
q
)
S
-
q
p
-
q
,
with S being the best constant for the Sobolev embedding of X0 into Lps*(Rn).
Proof.
Assume that {(uk,vk)}⊂E is a (PS)c sequence for Jλ,μ with (uk,vk)⇀(u,v) in E. That is,
J
λ
,
μ
′
(
u
k
,
v
k
)
=
o
(
1
)
strongly in
E
∗
as
k
→
∞
.
Let (ϕ,ψ)∈E. Then we have
〈
J
λ
,
μ
′
(
u
k
,
v
k
)
-
J
λ
,
μ
′
(
u
,
v
)
,
(
ϕ
,
ψ
)
〉
=
𝒜
(
u
k
,
ϕ
)
-
𝒜
(
u
,
ϕ
)
+
𝒜
(
v
k
,
ψ
)
-
𝒜
(
v
,
ψ
)
-
λ
∫
Ω
(
|
u
k
|
q
-
2
u
k
-
|
u
|
q
-
2
u
)
ϕ
𝑑
x
-
μ
∫
Ω
(
|
v
k
|
q
-
2
v
k
-
|
v
|
q
-
2
v
)
ψ
𝑑
x
-
2
α
α
+
β
∫
Ω
(
|
u
k
|
α
-
2
u
k
|
v
k
|
β
-
|
u
|
α
-
2
u
|
v
|
β
)
ϕ
𝑑
x
-
2
β
α
+
β
∫
Ω
(
|
u
k
|
α
|
v
k
|
β
-
2
v
k
-
|
u
|
α
|
v
|
β
-
2
v
)
ψ
𝑑
x
,
where 𝒜 is defined in (1.4). We claim that, from (uk,vk)⇀(u,v) in E, we have
lim
k
𝒜
(
u
k
,
ϕ
)
=
𝒜
(
u
,
ϕ
)
,
lim
k
𝒜
(
v
k
,
ψ
)
=
𝒜
(
v
,
ψ
)
for any
ϕ
,
ψ
∈
X
0
as
k
→
∞
.
In fact, the sequences
{
|
u
k
(
x
)
-
u
k
(
y
)
|
p
-
2
(
u
k
(
x
)
-
u
k
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
}
k
∈
ℕ
,
{
|
v
k
(
x
)
-
v
k
(
y
)
|
p
-
2
(
v
k
(
x
)
-
v
k
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
}
k
∈
ℕ
are bounded in Lp′(ℝn) and by the pointwise converge uk→u and vk→v, we have
|
u
k
(
x
)
-
u
k
(
y
)
|
p
-
2
(
u
k
(
x
)
-
u
k
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
⇀
L
p
′
(
ℝ
n
)
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
,
and
|
v
k
(
x
)
-
v
k
(
y
)
|
p
-
2
(
v
k
(
x
)
-
v
k
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
⇀
L
p
′
(
ℝ
n
)
|
v
(
x
)
-
v
(
y
)
|
p
-
2
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
n
+
p
s
p
′
.
Since
ϕ
(
x
)
-
ϕ
(
y
)
|
x
-
y
|
n
+
p
s
p
∈
L
p
(
ℝ
n
)
,
ψ
(
x
)
-
ψ
(
y
)
|
x
-
y
|
n
+
p
s
p
∈
L
p
(
ℝ
n
)
,
the claim follows. The sequences uk and vk are bounded in X0, and then in Lps∗(Ω). Then uk→u and vk→v weakly in Lps∗(ℝn). Furthermore, we obtain
|
u
k
|
q
-
2
u
k
⇀
L
q
′
(
Ω
)
|
u
|
q
-
2
u
,
|
v
k
|
q
-
2
v
k
⇀
L
q
′
(
Ω
)
|
v
|
q
-
2
v
,
|
u
k
|
α
-
2
u
k
|
v
k
|
β
⇀
L
α
+
β
α
+
β
-
1
(
Ω
)
|
u
|
α
-
2
u
|
v
|
β
,
|
u
k
|
α
|
v
k
|
β
-
2
v
k
⇀
L
α
+
β
α
+
β
-
1
(
Ω
)
|
u
|
α
|
v
|
β
-
2
v
.
Since ϕ,ψ∈X0⊂Lq(Ω)∩Lα+β(Ω), it follows that, as k→∞,
∫
Ω
(
|
u
k
|
q
-
2
u
k
-
|
u
|
q
-
2
u
)
ϕ
𝑑
x
→
0
,
∫
Ω
(
|
v
k
|
q
-
2
v
k
-
|
v
|
q
-
2
v
)
ψ
𝑑
x
→
0
,
and
∫
Ω
(
|
u
k
|
α
-
2
u
k
|
v
k
|
β
-
|
u
|
α
-
2
u
|
v
|
β
)
ϕ
𝑑
x
→
0
,
∫
Ω
(
|
u
k
|
α
|
v
k
|
β
-
2
v
k
-
|
u
|
α
|
v
|
β
-
2
v
)
ψ
𝑑
x
→
0
.
Hence,
〈
J
λ
,
μ
′
(
u
k
,
v
k
)
-
J
λ
,
μ
′
(
u
,
v
)
,
(
ϕ
,
ψ
)
〉
→
0
for all
(
ϕ
,
ψ
)
∈
E
,
which yields Jλ,μ′(u,v)=0. In particular, we get
〈
J
λ
,
μ
′
(
u
,
v
)
,
(
u
,
v
)
〉
=
0
,
i.e.,
2
∫
Ω
|
u
|
α
|
v
|
β
d
x
=
∥
(
u
,
v
)
∥
p
-
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
.
Then
J
λ
,
μ
(
u
,
v
)
=
(
1
p
-
1
p
s
∗
)
∥
(
u
,
v
)
∥
p
-
(
1
q
-
1
p
s
∗
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
(3.6)
=
s
n
∥
(
u
,
v
)
∥
p
-
(
1
q
-
1
p
s
∗
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
.
By Hölder’s inequality, the Sobolev embedding, (1.6) and Young’s inequality, we have
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
≤
|
Ω
|
p
s
∗
-
q
p
s
∗
S
-
q
p
(
λ
∥
u
∥
X
0
q
+
μ
∥
v
∥
X
0
q
)
=
(
[
p
q
s
n
(
1
q
-
1
p
s
∗
)
-
1
]
q
p
∥
u
∥
X
0
q
)
(
[
p
q
s
n
(
1
q
-
1
p
s
∗
)
-
1
]
-
q
p
|
Ω
|
p
s
∗
-
q
p
s
∗
S
-
q
p
λ
)
+
(
[
p
q
s
n
(
1
q
-
1
p
s
∗
)
-
1
]
q
p
∥
v
∥
X
0
q
)
(
[
p
q
s
n
(
1
q
-
1
p
s
∗
)
-
1
]
-
q
p
|
Ω
|
p
s
∗
-
q
p
s
∗
S
-
q
p
μ
)
≤
s
n
(
1
q
-
1
p
s
∗
)
-
1
(
∥
u
∥
X
0
p
+
∥
v
∥
X
0
p
)
+
C
^
(
λ
p
p
-
q
+
μ
p
p
-
q
)
(3.7)
=
s
n
(
1
q
-
1
p
s
∗
)
-
1
∥
(
u
,
v
)
∥
p
+
C
^
(
λ
p
p
-
q
+
μ
p
p
-
q
)
,
with
C
^
=
p
-
q
p
(
[
p
q
s
n
(
1
q
-
1
p
s
∗
)
-
1
]
-
q
p
|
Ω
|
p
s
∗
-
q
p
s
∗
S
-
q
p
)
p
p
-
q
=
p
-
q
p
(
p
s
∗
-
q
p
s
∗
-
p
)
q
p
-
q
|
Ω
|
p
(
p
s
∗
-
q
)
p
s
∗
(
p
-
q
)
S
-
q
p
-
q
.
Then (3.4) follows from (3.6) and (3.7) with C0=(1q-1ps∗)C^. ∎
Let us set
(3.8)
S
α
,
β
:=
inf
(
u
,
v
)
∈
E
∖
{
0
}
∥
(
u
,
v
)
∥
p
(
∫
Ω
|
u
|
α
|
v
|
β
d
x
)
p
α
+
β
.
We have the following result which provides a connection between Sα,β and S. The proof essentially follows by the line of arguments used in [1] but, for the sake of self-containedness, we include it.
Lemma 3.4
Lemma 3.4 (Sα,β versus S)
We have
(3.9)
S
α
,
β
=
[
(
α
β
)
β
α
+
β
+
(
β
α
)
α
α
+
β
]
S
.
Proof.
Let {ωn}n∈ℕ⊂X0 be a minimization sequence for S. Let s,t>0 be chosen later and consider the sequences un:=sωn and vn:=tωn in X0. By the definition of Sα,β, we have
(3.10)
s
p
+
t
p
(
s
α
t
β
)
p
p
s
*
∫
ℝ
2
n
|
ω
n
(
x
)
-
ω
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
ω
n
|
p
s
*
d
x
)
p
p
s
*
≥
S
α
,
β
.
Observe that
s
p
+
t
p
(
s
α
t
β
)
p
p
s
*
=
(
s
t
)
p
β
p
s
∗
+
(
s
t
)
-
p
α
p
s
∗
.
Let us consider the function g:ℝ+→ℝ+ defined by
g
(
x
)
:=
x
p
β
p
s
*
+
x
-
p
α
p
s
*
.
Then we have
s
p
+
t
p
(
s
α
t
β
)
p
p
s
*
=
g
(
s
t
)
,
and the function g achieves its minimum at point x0=(αβ)1p with minimum value
min
x
∈
ℝ
+
g
(
x
)
=
(
α
β
)
β
p
s
*
+
(
β
α
)
α
p
s
*
.
Choosing s,t in (3.10) such that st=(αβ)1p and letting n→∞ yields
(3.11)
[
(
α
β
)
β
p
s
*
+
(
β
α
)
α
p
s
*
]
S
≥
S
α
,
β
.
On the other hand, let {(un,vn)}n∈ℕ⊂E∖{(0,0)} be a minimizing sequence for Sα,β. Set zn:=snvn for sn>0 with ∫Ω|un|ps*dx=∫Ω|zn|ps*dx. Then Young’s inequality implies
∫
Ω
|
u
n
|
α
|
z
n
|
β
d
x
≤
α
α
+
β
∫
Ω
|
u
n
|
α
+
β
d
x
+
β
α
+
β
∫
Ω
|
z
n
|
α
+
β
d
x
=
∫
Ω
|
z
n
|
α
+
β
d
x
=
∫
Ω
|
u
n
|
α
+
β
d
x
.
Then we have
∫
ℝ
2
n
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
+
∫
ℝ
2
n
|
v
n
(
x
)
-
v
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
u
n
|
α
|
v
n
|
β
d
x
)
p
α
+
β
=
s
n
p
β
α
+
β
(
∫
ℝ
2
n
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
+
∫
ℝ
2
n
|
v
n
(
x
)
-
v
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
)
(
∫
Ω
|
u
n
|
α
|
z
n
|
β
d
x
)
p
α
+
β
≥
s
n
p
β
α
+
β
∫
ℝ
2
n
|
u
n
(
x
)
-
u
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
u
n
|
α
+
β
d
x
)
p
α
+
β
+
s
n
p
β
α
+
β
s
n
-
p
∫
ℝ
2
n
|
z
n
(
x
)
-
z
n
(
y
)
|
p
|
x
-
y
|
n
+
p
s
𝑑
x
𝑑
y
(
∫
Ω
|
z
n
|
α
+
β
d
x
)
p
α
+
β
≥
g
(
s
n
)
S
≥
[
(
α
β
)
β
p
s
*
+
(
β
α
)
α
p
s
*
]
S
.
In the last inequality, passing to the limit as n→∞, we obtain
(3.12)
[
(
α
β
)
β
p
s
*
+
(
β
α
)
α
p
s
*
]
S
≤
S
α
,
β
.
Thus, (3.9) follows from (3.11) and (3.12). ∎
Lemma 3.5
Lemma 3.5 (Palais–Smale Range)
The functional Jλ,μ satisfies the (PS)c condition with c satisfying
(3.13)
-
∞
<
c
<
c
∞
=
2
s
n
(
S
α
,
β
2
)
n
p
s
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
,
where C0 is the positive constant defined in (3.5).
Proof.
Let {(uk,vk)}k∈ℕ be a (PS)c sequence of Jλ,μ in E. Then
(3.14)
1
p
∥
(
u
k
,
v
k
)
∥
p
-
1
q
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
d
x
-
2
p
s
∗
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
c
+
o
k
(
1
)
,
(3.15)
∥
(
u
k
,
v
k
)
∥
p
-
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
d
x
-
2
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
o
k
(
1
)
.
We know, by Lemma 3.2, that {(uk,vk)}k∈ℕ is bounded in E. Then, up to a subsequence, (uk,vk)⇀(u,v) in E and, by Lemma 3.3, we have that (u,v) is a critical point of Jλ,μ.
Next we show that (uk,vk) converges strongly to (u,v) as k→∞ in E. Since uk→u and vk→v in Lr(ℝn), we obtain
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
→
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
as
k
→
∞
.
Moreover, by variants of the Brezis–Lieb Lemma, we can easily get (cf. [5, Lemma 2.2])
(3.16)
∥
(
u
k
,
v
k
)
∥
p
=
∥
(
u
k
-
u
,
v
k
-
v
)
∥
p
+
∥
(
u
,
v
)
∥
p
+
o
k
(
1
)
and
(3.17)
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
∫
Ω
|
u
k
-
u
|
α
|
v
k
-
v
|
β
d
x
+
∫
Ω
|
u
|
α
|
v
|
β
d
x
+
o
k
(
1
)
.
Taking (3.16) and (3.17) into (3.14) and (3.15), we find that
(3.18)
1
p
∥
(
u
k
-
u
,
v
k
-
v
)
∥
p
-
2
p
s
∗
∫
Ω
|
u
k
-
u
|
α
|
v
k
-
v
|
β
d
x
=
c
-
J
λ
,
μ
(
u
,
v
)
+
o
k
(
1
)
and
∥
(
u
k
-
u
,
v
k
-
v
)
∥
p
=
2
∫
Ω
|
u
k
-
u
|
α
|
v
k
-
v
|
β
d
x
+
o
k
(
1
)
.
Hence, we may assume that
(3.19)
∥
(
u
k
-
u
,
v
k
-
v
)
∥
p
→
m
,
2
∫
Ω
|
u
k
-
u
|
α
|
v
k
-
v
|
β
d
x
→
m
as
k
→
∞
.
If m=0, we are done. Suppose m>0. Then, from (3.19) and the definition of Sα,β in (3.8), we have
S
α
,
β
(
m
2
)
p
p
s
∗
=
S
α
,
β
lim
k
→
∞
(
∫
Ω
|
u
k
-
u
|
α
|
v
k
-
v
|
β
d
x
)
p
p
s
∗
≤
lim
k
→
∞
∥
(
u
k
-
u
,
v
k
-
v
)
∥
p
=
m
,
which yields m≥2(Sα,β2)nps. From (3.18), we obtain
c
=
s
n
m
+
J
λ
,
μ
(
u
,
v
)
.
By Lemma 3.3 and for m≥2(Sα,β2)nps, we find
c
≥
2
s
n
(
S
α
,
β
2
)
n
p
s
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
,
which is impossible for
-
∞
<
c
<
2
s
n
(
S
α
,
β
2
)
n
p
s
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
.
∎
4 Existence of Solutions
We start with some lemmas.
Lemma 4.1
Lemma 4.1 (Nλ,μ0 is Empty)
Let λ,μ be such that 0<λpp-q+μpp-q<Λ1, where Λ1 is as in (2.5). Then Nλ,μ0=∅.
Proof.
From the proof of Lemma 2.5, we have that there exist exactly two numbers t2>t1>0 such that φu,v′(t1)=φu,v′(t2)=0. Furthermore, φu,v′′(t1)>0>φu,v′′(t2). If, by contradiction, (u,v)∈𝒩λ,μ0, then we have that φu,v′(1)=0 with φu,v′′(1)=0. Then, either t1=1 or t2=1. In turn, either φu,v′′(1)>0 or φu,v′′(1)<0, a contradiction. ∎
Lemma 4.2
Lemma 4.2 (Coercivity)
The functional Jλ,μ is coercive and bounded from below on Nλ,μ for all λ>0 and μ>0.
Proof.
Let λ>0 and μ>0 and pick (u,v)∈𝒩λ,μ. Then, we have
J
λ
,
μ
(
u
,
v
)
=
(
1
p
-
1
α
+
β
)
∥
(
u
,
v
)
∥
p
-
(
1
q
-
1
α
+
β
)
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
≥
(
1
p
-
1
α
+
β
)
∥
(
u
,
v
)
∥
p
-
(
1
q
-
1
α
+
β
)
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
,
v
)
∥
q
,
which yields the assertion. ∎
By Lemmas 4.1 and 4.2, for any λ,μ satisfying 0<λpp-q+μpp-q<Λ1, we have
𝒩
λ
,
μ
=
𝒩
λ
,
μ
+
∪
𝒩
λ
,
μ
-
,
and Jλ,μ is coercive and bounded from below on 𝒩λ,μ+ and 𝒩λ,μ-. Therefore, we may define
c
λ
,
μ
:=
inf
𝒩
λ
,
μ
J
λ
,
μ
,
c
λ
,
μ
±
:=
inf
𝒩
λ
,
μ
±
J
λ
,
μ
.
Of course, by Lemma 4.2, we have cλ,μ,cλ,μ±>-∞. The following result is valid.
Lemma 4.3
Lemma 4.3 (cλ,μ+<0 and cλ,μ->0)
Let Λ1 be as in (2.5). Then the following facts hold:
if 0<λpp-q+μpp-q<Λ1, then cλ,μ≤cλ,μ+<0,
if 0<λpp-q+μpp-q<(qp)pp-qΛ1, then cλ,μ->d0 for some d0=d0(λ,μ,p,q,n,s,|Ω|)>0.
Proof.
Let us prove (i). Let (u,v)∈𝒩λ,μ+. Then we have φu,v′′(1)>0, which combined with (2.2) yields
p
-
q
2
(
α
+
β
-
q
)
∥
(
u
,
v
)
∥
p
>
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
Therefore,
J
λ
,
μ
(
u
,
v
)
=
(
1
p
-
1
q
)
∥
(
u
,
v
)
∥
p
+
2
(
1
q
-
1
α
+
β
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
<
[
(
1
p
-
1
q
)
+
(
1
q
-
1
α
+
β
)
p
-
q
α
+
β
-
q
]
∥
(
u
,
v
)
∥
p
=
-
(
p
-
q
)
(
α
+
β
-
p
)
p
q
(
α
+
β
)
∥
(
u
,
v
)
∥
p
<
0
.
Therefore, cλ,μ≤cλ,μ+<0 follows from the definitions of cλ,μ and cλ,μ+.
Let us now come to (ii). Let (u,v)∈𝒩λ,μ-. Then, we have φu,v′′(1)<0, which combined with (2.2) yields
p
-
q
2
(
α
+
β
-
q
)
∥
(
u
,
v
)
∥
p
<
∫
Ω
|
u
|
α
|
v
|
β
d
x
.
By Young’s inequality and the definition of S, we obtain
∫
Ω
|
u
|
α
|
v
|
β
d
x
≤
α
α
+
β
∫
Ω
|
u
|
α
+
β
d
x
+
β
α
+
β
∫
Ω
|
v
|
α
+
β
d
x
≤
S
-
α
+
β
p
∥
(
u
,
v
)
∥
α
+
β
.
Thus,
∥
(
u
,
v
)
∥
>
(
p
-
q
2
(
α
+
β
-
q
)
)
1
α
+
β
-
p
S
α
+
β
p
(
α
+
β
-
p
)
.
Moreover, by Hölder’s inequality and the definition of S, we find
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
𝑑
x
≤
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
,
v
)
∥
q
.
Therefore, if 0<λpp-q+μpp-q<(qp)pp-qΛ1, then we have
J
λ
,
μ
(
u
,
v
)
≥
∥
(
u
,
v
)
∥
q
[
(
1
p
-
1
α
+
β
)
∥
(
u
,
v
)
∥
p
-
q
-
(
1
q
-
1
α
+
β
)
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
]
>
∥
(
u
,
v
)
∥
q
[
(
1
p
-
1
α
+
β
)
(
p
-
q
2
(
α
+
β
-
q
)
)
p
-
q
α
+
β
-
p
S
(
α
+
β
)
(
p
-
q
)
p
(
α
+
β
-
p
)
-
(
1
q
-
1
α
+
β
)
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
]
≥
d
0
>
0
.
∎
4.1 The First Solution
We now prove the existence of a first solution (u1,v1) to (1.1). First, we need some preliminary results.
Lemma 4.4
Lemma 4.4 (Curves into Nλ,μ)
Let Λ1 be as in (2.5) and assume that 0<λpp-q+μpp-q<Λ1. Then, for any z=(u,v)∈Nλ,μ, there exists ϵ>0 and a differentiable map ξ:B(0,ϵ)⊂E→R+, with ξ(0)=1, such that ξ(ω)(z-ω)∈Nλ,μ and
(4.1)
〈
ξ
′
(
0
)
,
ω
〉
=
-
p
𝒜
(
u
,
ω
1
)
+
p
𝒜
(
v
,
ω
2
)
-
K
λ
,
μ
(
z
,
ω
)
-
2
∫
Ω
(
α
|
u
|
α
-
2
u
ω
1
|
v
|
β
+
β
|
u
|
α
|
v
|
β
-
2
v
ω
2
)
𝑑
x
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
|
α
|
u
|
β
d
x
for all ω=(ω1,ω2)∈E, where
K
λ
,
μ
(
z
,
ω
)
=
q
∫
Ω
(
λ
|
u
|
q
-
2
u
ω
1
+
μ
|
v
|
q
-
2
v
ω
2
)
𝑑
x
.
Proof.
For z=(u,v)∈𝒩λ,μ, define a function Fz:ℝ+×E→ℝ by
F
z
(
ξ
,
ω
)
:=
〈
J
λ
,
μ
′
(
ξ
(
z
-
ω
)
)
,
ξ
(
z
-
ω
)
〉
=
ξ
p
(
𝒜
(
u
-
ω
1
,
u
-
ω
1
)
+
𝒜
(
v
-
ω
2
,
v
-
ω
2
)
)
-
ξ
q
∫
Ω
(
λ
|
u
-
ω
1
|
q
+
μ
|
v
-
ω
2
|
q
)
𝑑
x
-
2
ξ
α
+
β
∫
Ω
|
u
-
ω
1
|
α
|
v
-
ω
2
|
β
d
x
,
ξ
∈
ℝ
+
,
ω
∈
E
.
Then Fz(1,0)=〈Jλ,μ′(z),z〉=0 and, by Lemma 4.1, we have
d
d
ξ
F
z
(
1
,
(
0
,
0
)
)
=
p
∥
(
u
,
v
)
∥
p
-
q
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
-
2
(
α
+
β
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
=
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
|
α
|
u
|
β
d
x
≠
0
.
By the Implicit Function Theorem, there exist ϵ>0 and a C1 map ξ:B(0,ϵ)⊂E→ℝ+, with ξ(0)=1, such that
〈
ξ
′
(
0
)
,
ω
〉
=
-
p
𝒜
(
u
,
ω
1
)
+
p
𝒜
(
v
,
ω
2
)
-
K
λ
,
μ
(
z
,
ω
)
-
2
∫
Ω
(
α
|
u
|
α
-
2
u
ω
1
|
v
|
β
+
β
|
u
|
α
|
v
|
β
-
2
v
ω
2
)
𝑑
x
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
|
α
|
u
|
β
d
x
and Fz(ξ(ω),ω)=0 for all ω∈B(0,ϵ), which is equivalent to
〈
J
λ
,
μ
′
(
ξ
(
ω
)
(
z
-
ω
)
)
,
ξ
(
ω
)
(
z
-
ω
)
〉
=
0
for all
ω
∈
B
(
0
,
ϵ
)
,
i.e., ξ(ω)(z-ω)∈𝒩λ,μ. ∎
Lemma 4.5
Lemma 4.5 (Curves into Nλ,μ-)
Let Λ1 be as in (2.5) and assume 0<λpp-q+μpp-q<Λ1. Then, for each z∈Nλ,μ-, there exist ϵ>0 and a differentiable map ξ-:B(0,ϵ)⊂E→R+, with ξ-(0)=1, such that ξ-(ω)(z-ω)∈Nλ,μ- and
〈
(
ξ
-
)
′
(
0
)
,
ω
〉
=
-
p
𝒜
(
u
,
ω
1
)
+
p
𝒜
(
v
,
ω
2
)
-
K
λ
,
μ
(
z
,
ω
)
-
2
∫
Ω
(
α
|
u
|
α
-
2
u
ω
1
|
v
|
β
+
β
|
u
|
α
|
v
|
β
-
2
v
ω
2
)
𝑑
x
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
|
α
|
u
|
β
d
x
for every ω∈B(0,ϵ).
Proof.
Arguing as in the proof of Lemma 4.4, there exist ϵ>0 and a differentiable map ξ-:B(0,ϵ)⊂E→ℝ+ such that ξ-(0)=1, ξ-(ω)(z-ω)∈𝒩λ,μ for all ω∈B(0,ϵ) and satisfying (4.1). Since
φ
u
,
v
′′
(
1
)
=
(
p
-
q
)
∥
(
u
,
v
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
|
α
|
v
|
β
d
x
<
0
,
by continuity, we have
φ
ξ
-
(
ω
)
(
u
-
ω
1
)
,
ξ
-
(
ω
)
(
v
-
ω
2
)
′′
(
1
)
=
(
p
-
q
)
∥
(
ξ
-
(
ω
)
(
u
-
ω
1
)
,
ξ
-
(
ω
)
(
v
-
ω
2
)
)
∥
p
-
2
(
(
α
+
β
)
-
q
)
∫
Ω
|
ξ
-
(
ω
)
(
u
-
ω
1
)
|
α
|
ξ
-
(
ω
)
(
v
-
ω
2
)
|
β
d
x
<
0
for ϵ sufficiently small, which implies ξ-(ω)(z-ω)∈𝒩λ,μ-. ∎
Proposition 4.6
Proposition 4.6 ((PS)cλ,μ-Sequences)
The following facts hold:
If 0<λpp-q+μpp-q<Λ1, then there exists a (PS)cλ,μ-sequence {(uk,vk)}⊂𝒩λ,μ for Jλ,μ.
If 0<λpp-q+μpp-q<(qp)pp-qΛ1, then there exists a (PS)cλ,μ--sequence {(uk,vk)}⊂𝒩λ,μ- for Jλ,μ.
Proof.
(i) By Ekeland’s Variational Principle, there exists a minimizing sequence {(uk,vk)}⊂𝒩λ,μ such that
(4.2)
J
λ
,
μ
(
u
k
,
v
k
)
<
c
λ
,
μ
+
1
k
,
J
λ
,
μ
(
u
k
,
v
k
)
<
J
λ
,
μ
(
w
1
,
w
2
)
+
1
k
∥
(
w
1
,
w
2
)
-
(
u
k
,
v
k
)
∥
for each (w1,w2)∈𝒩λ,μ. Taking k large and using cλ,μ<0, we have
(4.3)
J
λ
,
μ
(
u
k
,
v
k
)
=
(
1
p
-
1
α
+
β
)
∥
(
u
k
,
v
k
)
∥
p
-
(
1
q
-
1
α
+
β
)
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
<
c
λ
,
μ
2
.
This yields
(4.4)
-
q
(
α
+
β
)
2
(
α
+
β
-
q
)
c
λ
,
μ
<
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
≤
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
k
,
v
k
)
∥
q
.
Consequently, (uk,vk)≠0 and, by combining it with (4.3) and (4.4), and using Hölder’s inequality, we have
∥
(
u
k
,
v
k
)
∥
>
[
-
q
(
α
+
β
)
2
(
α
+
β
-
q
)
c
λ
,
μ
S
q
p
|
Ω
|
-
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
q
-
p
p
]
1
q
,
(4.5)
∥
(
u
k
,
v
k
)
∥
<
[
p
(
α
+
β
-
q
)
q
(
α
+
β
-
p
)
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
]
1
p
-
q
.
Now we prove that ∥Jλ,μ′(uk,vk)∥E-1→0 as k→∞. Fix k∈ℕ. By applying Lemma 4.4 to zk=(uk,vk), we obtain a function ξk:B(0,ϵk)→ℝ+, for some ϵk>0, such that ξk(h)(zk-h)∈𝒩λ,μ. Take 0<ρ<ϵk. Let w∈E with w≢0 and put h*=ρw∥w∥. We set hρ=ξk(h*)(zk-h*). Then hρ∈𝒩λ,μ and from (4.2) we have
J
λ
,
μ
(
h
ρ
)
-
J
λ
,
μ
(
z
k
)
≥
-
1
k
∥
h
ρ
-
z
k
∥
.
By the Mean Value Theorem, we get
〈
J
λ
,
μ
′
(
z
k
)
,
h
ρ
-
z
k
〉
+
o
(
∥
h
ρ
-
z
k
∥
)
≥
-
1
k
∥
h
ρ
-
z
k
∥
.
Thus, we have
〈
J
λ
,
μ
′
(
z
k
)
,
-
h
*
〉
+
(
ξ
k
(
h
*
)
-
1
)
〈
J
λ
,
μ
′
(
z
k
)
,
z
k
-
h
*
〉
≥
-
1
k
∥
h
ρ
-
z
k
∥
+
o
(
∥
h
ρ
-
z
k
∥
)
.
Whence, from the fact that ξk(h*)(zk-h*)∈𝒩λ,μ, it follows that
-
ρ
〈
J
λ
,
μ
′
(
z
k
)
,
w
∥
w
∥
〉
+
(
ξ
k
(
h
*
)
-
1
)
〈
J
λ
,
μ
′
(
z
k
)
-
J
λ
,
μ
′
(
h
ρ
)
,
z
k
-
h
*
〉
≥
-
1
k
∥
h
ρ
-
z
k
∥
+
o
(
∥
h
ρ
-
z
k
∥
)
.
Hence, we get
(4.6)
〈
J
λ
,
μ
′
(
z
k
)
,
w
∥
w
∥
〉
≤
1
k
ρ
∥
h
ρ
-
z
k
∥
+
o
(
∥
h
ρ
-
z
k
∥
)
ρ
+
(
ξ
k
(
h
*
)
-
1
)
ρ
〈
J
λ
,
μ
′
(
z
k
)
-
J
λ
,
μ
′
(
h
ρ
)
,
z
k
-
h
*
〉
.
Since
∥
h
ρ
-
z
k
∥
≤
ρ
|
ξ
k
(
h
*
)
|
+
|
ξ
k
(
h
*
)
-
1
|
∥
z
k
∥
and
lim
ρ
→
0
|
ξ
k
(
h
*
)
-
1
|
ρ
≤
∥
ξ
k
′
(
0
)
∥
,
for k∈ℕ fixed, if ρ→0 in (4.6), then, by virtue of (4.5), we can choose C>0 independent of ρ such that
〈
J
λ
,
μ
′
(
z
k
)
,
w
∥
w
∥
〉
≤
C
k
(
1
+
∥
ξ
k
′
(
0
)
∥
)
.
Thus, we are done if supk∈ℕ∥ξk′(0)∥E*<∞. By (4.1), (4.5) and Hölder’s inequality, we have
|
〈
ξ
k
′
(
0
)
,
h
〉
|
≤
C
1
∥
h
∥
|
(
p
-
q
)
∥
(
u
k
,
v
k
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
k
|
α
v
k
|
β
d
x
|
for some C1>0. We only need to prove that
|
(
p
-
q
)
∥
(
u
k
,
v
k
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
|
≥
C
2
for some C2>0 and k large. By contradiction, suppose that there exists a subsequence {(uk,vk)}k∈ℕ with
(4.7)
(
p
-
q
)
∥
(
u
k
,
v
k
)
∥
p
-
2
(
α
+
β
-
q
)
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
=
o
k
(
1
)
.
By (4.7) and the fact that (uk,vk)∈𝒩λ,μ, we have
(4.8)
∥
(
u
k
,
v
k
)
∥
p
=
2
(
α
+
β
-
q
)
p
-
q
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
+
o
k
(
1
)
,
(4.9)
∥
(
u
k
,
v
k
)
∥
p
=
α
+
β
-
q
α
+
β
-
p
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
+
o
k
(
1
)
.
By Young’s inequality, it follows that
∫
Ω
|
u
k
|
α
|
v
k
|
β
d
x
≤
S
-
α
+
β
p
∥
(
u
k
,
v
k
)
∥
α
+
β
.
By this and (4.8), we get
(4.10)
∥
(
u
k
,
v
k
)
∥
≥
(
p
-
q
2
(
α
+
β
-
q
)
S
α
+
β
p
)
1
α
+
β
-
p
+
o
k
(
1
)
.
Moreover, from (4.9) and by Hölder’s inequality, we obtain
∥
(
u
k
,
v
k
)
∥
p
≤
α
+
β
-
q
α
+
β
-
p
|
Ω
|
α
+
β
-
q
α
+
β
S
-
q
p
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
q
p
∥
(
u
k
,
v
k
)
∥
q
+
o
k
(
1
)
.
Thus,
(4.11)
∥
(
u
k
,
v
k
)
∥
≤
(
α
+
β
-
q
α
+
β
-
p
S
-
q
p
|
Ω
|
α
+
β
-
q
α
+
β
)
1
p
-
q
(
λ
p
p
-
q
+
μ
p
p
-
q
)
1
p
+
o
k
(
1
)
.
From (4.10) and (4.11), and for k large enough, we get
λ
p
p
-
q
+
μ
p
p
-
q
≥
(
p
-
q
2
(
α
+
β
-
q
)
)
p
α
+
β
-
p
(
α
+
β
-
q
α
+
β
-
p
|
Ω
|
α
+
β
-
q
α
+
β
)
-
p
p
-
q
S
α
+
β
α
+
β
-
p
+
q
p
-
q
=
Λ
1
,
which contradicts 0<λpp-q+μpp-q<Λ1. Therefore,
〈
J
λ
,
μ
′
(
u
k
,
v
k
)
,
∥
w
∥
-
1
w
〉
≤
C
k
.
This proves (i). By Lemma 4.5, using the same arguments, we can get (ii). ∎
Here is the main result of the section.
Proposition 4.7
Proposition 4.7 (Existence of the First Solution)
Let Λ1 be as in (2.5). Assume that 0<λpp-q+μpp-q<Λ1. Then there exists (u1,v1)∈Nλ,μ+ with the following properties:
Jλ,μ(u1,v1)=cλ,μ=cλ,μ+<0,
(
u
1
,
v
1
)
is a solution of problem (
1.1
).
Proof.
By Proposition 4.6 (i), there exists a bounded minimizing sequence {(uk,vk)}⊂𝒩λ,μ such that
lim
k
→
∞
J
λ
,
μ
(
u
k
,
v
k
)
=
c
λ
,
μ
≤
c
λ
,
μ
+
<
0
,
J
λ
,
μ
′
(
u
k
,
v
k
)
=
o
k
(
1
)
in
E
∗
.
Then there exists (u1,v1)∈E such that, up to a subsequence, uk⇀u1, vk⇀v1 in X0 as well as uk→u1 and vk→v1 strongly in Lr(Ω) for any 1≤r<p∗. Then, the Dominated Convergence Theorem yields
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
→
∫
Ω
(
λ
|
u
1
|
q
+
μ
|
v
1
|
q
)
𝑑
x
as
k
→
∞
.
It is easy to get that (u1,v1) is a weak solution of (1.1), cf. Lemma 3.3. Now, since (uk,vk)∈𝒩λ,μ, we have
J
λ
,
μ
(
u
k
,
v
k
)
=
α
+
β
-
p
p
(
α
+
β
)
∥
(
u
k
,
v
k
)
∥
p
-
α
+
β
-
q
q
(
α
+
β
)
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
≥
-
α
+
β
-
q
q
(
α
+
β
)
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
.
Then, from cλ,μ<0, we get
∫
Ω
(
λ
|
u
1
|
q
+
μ
|
v
1
|
q
)
𝑑
x
≥
-
q
(
α
+
β
)
α
+
β
-
q
c
λ
,
μ
>
0
.
Therefore, (u1,v1)∈𝒩λ,μ is a nontrivial solution of (1.1).
Next, we show that (uk,vk)→(u1,v1) strongly in E and Jλ,μ(u1,v1)=cλ,μ+. In fact, since (u1,v1)∈𝒩λ,μ, in light of Fatou’s lemma, we get
c
λ
,
μ
≤
J
λ
,
μ
(
u
1
,
v
1
)
=
α
+
β
-
p
p
(
α
+
β
)
∥
(
u
1
,
v
1
)
∥
p
-
α
+
β
-
q
q
(
α
+
β
)
∫
Ω
(
λ
|
u
1
|
q
+
μ
|
v
1
|
q
)
𝑑
x
≤
lim inf
k
→
∞
(
α
+
β
-
p
p
(
α
+
β
)
∥
(
u
k
,
v
k
)
∥
p
-
α
+
β
-
q
q
(
α
+
β
)
∫
Ω
(
λ
|
u
k
|
q
+
μ
|
v
k
|
q
)
𝑑
x
)
=
lim inf
k
→
∞
J
λ
,
μ
(
u
k
,
v
k
)
=
c
λ
,
μ
.
This implies that Jλ,μ(u1,v1)=cλ,μ and ∥(uk,vk)∥p→∥(u1,v1)∥p. We also have
∥
(
u
k
-
u
1
,
v
k
-
v
1
)
∥
p
=
∥
(
u
k
,
v
k
)
∥
p
-
∥
(
u
1
,
v
1
)
∥
p
+
o
k
(
1
)
.
Therefore (uk,vk)→(u1,v1) strongly in E. We claim that (u1,v1)∈𝒩λ,μ+, which yields cλ,μ=cλ,μ+. Assume, by contradiction, that (u1,v1)∈𝒩λ,μ-. By Lemma 2.5, there exist unique t2>t1>0 such that
(
t
1
u
1
,
t
1
v
1
)
∈
𝒩
λ
,
μ
+
,
(
t
2
u
1
,
t
2
v
1
)
∈
𝒩
λ
,
μ
-
.
In particular, we have t1<t2=1. Since
d
d
t
J
λ
,
μ
(
t
1
u
1
,
t
1
v
1
)
=
0
,
d
2
d
t
2
J
λ
,
μ
(
t
1
u
1
,
t
1
v
1
)
>
0
,
there exists t∗∈(t1,1] such that Jλ,μ(t1u1,t1v1)<Jλ,μ(t∗u1,t∗v1). Then
c
λ
,
μ
≤
J
λ
,
μ
(
t
1
u
1
,
t
1
v
1
)
<
J
λ
,
μ
(
t
∗
u
1
,
t
∗
v
1
)
≤
J
λ
,
μ
(
u
1
,
v
1
)
=
c
λ
,
μ
,
which is a contradiction. Hence, (u1,v1)∈𝒩λ,μ+. ∎
4.2 The Second Solution
We next establish the existence of a minimum for Jλ,μ|𝒩λ,μ-. Let S be as in (1.6). From [4], we know that for 1<p<∞, s∈(0,1), n>ps, there exists a minimizer for S, and for every minimizer U, there exist x0∈ℝn and a constant sign monotone function u:ℝ→ℝ such that U(x)=u(|x-x0|). In the following, we shall fix a radially symmetric nonnegative decreasing minimizer U=U(r) for S. Multiplying U by a positive constant if necessary, we may assume that
(4.12)
(
-
Δ
)
p
s
U
=
U
p
s
∗
-
1
in
ℝ
n
.
For any ϵ>0, we note that the function
U
ϵ
(
x
)
=
1
ϵ
n
-
p
s
p
U
(
|
x
|
ϵ
)
is also a minimizer for S satisfying (4.12). In [4], the following asymptotic estimates for U were provided.
Lemma 4.8
Lemma 4.8 (Optimal Decay)
There exist c1,c2>0 and θ>1 such that for all r>1,
c
1
r
n
-
p
s
p
-
1
≤
U
(
r
)
≤
c
2
r
n
-
p
s
p
-
1
,
U
(
θ
r
)
U
(
r
)
≤
1
2
.
Assume, without loss of generality, that 0∈Ω. For ϵ,δ>0, let
m
ϵ
,
δ
=
U
ϵ
(
δ
)
U
ϵ
(
δ
)
-
U
ϵ
(
θ
δ
)
,
g
ϵ
,
δ
(
t
)
=
{
0
if
0
≤
t
≤
U
ϵ
(
θ
δ
)
,
m
ϵ
,
δ
p
(
t
-
U
ϵ
(
θ
δ
)
)
if
U
ϵ
(
θ
δ
)
≤
t
≤
U
ϵ
(
δ
)
,
t
+
U
ϵ
(
δ
)
(
m
ϵ
,
δ
p
-
1
-
1
)
if
t
≥
U
ϵ
(
δ
)
,
and
G
ϵ
,
δ
(
t
)
=
∫
0
t
g
ϵ
,
δ
′
(
τ
)
1
p
𝑑
τ
=
{
0
if
0
≤
t
≤
U
ϵ
(
θ
δ
)
,
m
ϵ
,
δ
(
t
-
U
ϵ
(
θ
δ
)
)
if
U
ϵ
(
θ
δ
)
≤
t
≤
U
ϵ
(
δ
)
,
t
if
t
≥
U
ϵ
(
δ
)
.
The functions gϵ,δ and Gϵ,δ are nondecreasing and absolutely continuous. Consider the radially symmetric nonincreasing function
(4.13)
u
ϵ
,
δ
(
r
)
=
G
ϵ
,
δ
(
U
ϵ
(
r
)
)
,
which satisfies
u
ϵ
,
δ
(
r
)
=
{
U
ϵ
(
r
)
if
r
≤
δ
,
0
if
r
≥
θ
δ
.
We have the following estimates for uϵ,δ, which were proved in [20, Lemma 2.7].
Lemma 4.9
Lemma 4.9 (Norm Estimates)
There exists a constant C=C(n,p,s)>0 such that for any 0<ϵ≤δ2, the following estimates hold:
∫
ℝ
2
n
|
u
ϵ
,
δ
(
x
)
-
u
ϵ
,
δ
(
y
)
|
p
|
x
-
y
|
n
+
p
s
d
x
d
y
≤
S
n
p
s
+
𝒪
(
(
ϵ
δ
)
n
-
p
s
p
-
1
)
,
∫
ℝ
n
|
u
ϵ
,
δ
(
x
)
|
p
s
∗
d
x
≥
S
n
p
s
-
C
(
(
ϵ
δ
)
n
p
-
1
)
.
Next, we prove an important technical lemma. This is the only point where we use conditions (1.5) on p,s,q,n.
Lemma 4.10
Lemma 4.10 (cλ,μ-<c∞)
Assume that conditions (1.5) hold. Then there exists Λ2>0 such that, for λ,μ satisfying 0<λpp-q+μpp-q<Λ2, there exists (u,v)∈E∖{(0,0)}, with u≥0,v≥0, such that
sup
t
≥
0
J
λ
,
μ
(
t
u
,
t
v
)
<
c
∞
,
where c∞ is the constant given in (3.13). In particular, cλ,μ-<c∞ for all λ,μ satisfying 0<λpp-q+μpp-q<Λ2.
Proof.
Write Jλ,μ(u,v)=J(u,v)-K(u,v) where the functions J:E→ℝ and K:E→ℝ are defined by
J
(
u
,
v
)
=
1
p
∥
(
u
,
v
)
∥
p
-
2
α
+
β
∫
Ω
|
u
|
α
|
v
|
β
d
x
,
K
(
u
,
v
)
=
1
q
∫
Ω
(
λ
|
u
|
q
+
μ
|
v
|
q
)
d
x
.
Set u0:=α1puϵ,δ, v0:=β1puϵ,δ, where uϵ,δ is defined by (4.13). The map h(t):=J(tu0,tv0) satisfies h(0)=0, h(t)>0 for t>0 small, and h(t)<0 for t>0 large. Moreover, h maximizes at the point
t
∗
:=
(
∥
(
u
0
,
v
0
)
∥
p
2
∫
Ω
|
u
0
|
α
|
v
0
|
β
d
x
)
1
α
+
β
-
p
.
Thus, we have
sup
t
≥
0
J
(
t
u
0
,
t
v
0
)
=
h
(
t
∗
)
=
t
∗
p
p
∥
(
u
0
,
v
0
)
∥
p
-
2
t
∗
α
+
β
α
+
β
∫
Ω
|
u
0
|
α
|
v
0
|
β
d
x
=
(
1
p
-
1
α
+
β
)
∥
(
u
0
,
v
0
)
∥
p
(
α
+
β
)
α
+
β
-
p
(
2
∫
Ω
|
u
0
|
α
|
v
0
|
β
d
x
)
p
α
+
β
-
p
=
(
1
p
-
1
α
+
β
)
(
α
+
β
)
α
+
β
α
+
β
-
p
2
p
α
+
β
-
p
α
α
α
+
β
-
p
β
β
α
+
β
-
p
∥
u
ϵ
,
δ
∥
X
0
p
(
α
+
β
)
α
+
β
-
p
(
∫
Ω
|
u
ϵ
,
δ
|
α
+
β
d
x
)
p
α
+
β
-
p
=
s
n
1
2
n
-
p
s
p
s
[
(
α
β
)
β
α
+
β
+
(
β
α
)
α
α
+
β
]
n
p
s
[
∥
u
ϵ
,
δ
∥
X
0
p
(
∫
Ω
|
u
ϵ
,
δ
|
p
s
∗
d
x
)
p
p
s
∗
]
n
p
s
.
From Lemma 4.9 and (3.9), we have
sup
t
≥
0
J
(
t
u
0
,
t
v
0
)
≤
s
n
1
2
n
-
p
s
p
s
[
(
α
β
)
β
α
+
β
+
(
β
α
)
α
α
+
β
]
n
p
s
[
S
n
p
s
+
𝒪
(
(
ϵ
δ
)
n
-
p
s
p
-
1
)
(
S
n
p
s
-
C
(
(
ϵ
δ
)
n
p
-
1
)
)
p
p
s
∗
]
n
p
s
(4.14)
≤
2
s
n
(
S
α
,
β
2
)
n
p
s
+
𝒪
(
(
ϵ
δ
)
n
-
p
s
p
-
1
)
.
Let δ1>0 be such that for all λ,μ satisfying 0<λpp-q+μpp-q<δ1, the following holds:
c
∞
=
2
s
n
(
S
α
,
β
2
)
n
p
s
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
>
0
.
We have
J
λ
,
μ
(
t
u
0
,
t
v
0
)
≤
t
p
p
∥
(
u
0
,
v
0
)
∥
p
≤
C
t
p
for
t
≥
0
and
λ
,
μ
>
0
.
Thus, there exists t0∈(0,1) such that
sup
0
≤
t
≤
t
0
J
λ
,
μ
(
t
u
0
,
t
v
0
)
<
c
∞
for all λ,μ satisfying 0<λpp-q+μpp-q<δ1. Since α,β>1, from (4.13) and (4.14), it follows that
sup
t
≥
t
0
J
λ
,
μ
(
t
u
0
,
t
v
0
)
=
sup
t
≥
t
0
[
J
(
t
u
0
,
t
v
0
)
-
K
(
t
u
0
,
t
v
0
)
]
≤
2
s
n
(
S
α
,
β
2
)
n
p
s
+
𝒪
(
(
ϵ
δ
)
n
-
p
s
p
-
1
)
-
t
0
q
q
(
λ
α
q
p
+
μ
β
q
p
)
∫
B
(
0
,
δ
)
|
u
ϵ
,
δ
|
q
d
x
≤
2
s
n
(
S
α
,
β
2
)
n
p
s
+
𝒪
(
(
ϵ
δ
)
n
-
p
s
p
-
1
)
-
t
0
q
q
(
λ
+
μ
)
∫
B
(
0
,
δ
)
|
u
ϵ
,
δ
|
q
d
x
.
Fix now δ>0 sufficiently small such that Bθδ(0)⋐Ω (we assume without loss of generality that 0∈Ω), so that supp(uϵ,δ)⊂Ω, according to formula (4.13). By means of Lemma 4.8, for any 0<ϵ≤δ2, we have
∫
B
(
0
,
δ
)
|
u
ϵ
,
δ
(
x
)
|
q
d
x
=
∫
B
(
0
,
δ
)
|
U
ϵ
(
x
)
|
q
d
x
=
ϵ
n
-
n
-
p
s
p
q
∫
B
(
0
,
δ
ϵ
)
|
U
(
x
)
|
q
d
x
≥
ϵ
n
-
n
-
p
s
p
q
ω
n
-
1
∫
1
δ
/
ϵ
U
(
r
)
q
r
n
-
1
𝑑
r
≥
ϵ
n
-
n
-
p
s
p
q
ω
n
-
1
c
1
q
∫
1
δ
/
ϵ
r
n
-
n
-
p
s
p
-
1
q
-
1
𝑑
r
≃
C
{
ϵ
n
-
n
-
p
s
p
q
if
q
>
n
(
p
-
1
)
n
-
p
s
,
ϵ
n
-
n
-
p
s
p
q
|
log
ϵ
|
if
q
=
n
(
p
-
1
)
n
-
p
s
,
ϵ
(
n
-
p
s
)
q
p
(
p
-
1
)
if
q
<
n
(
p
-
1
)
n
-
p
s
.
Therefore, taking into account conditions (1.5), we have
sup
t
≥
t
0
J
λ
,
μ
(
t
u
0
,
t
v
0
)
≤
2
s
n
(
S
α
,
β
2
)
n
p
s
+
C
(
ϵ
n
-
p
s
p
-
1
)
-
C
(
λ
+
μ
)
{
ϵ
n
-
n
-
p
s
p
q
if
q
>
n
(
p
-
1
)
n
-
p
s
,
ϵ
n
-
n
-
p
s
p
q
|
log
ϵ
|
if
q
=
n
(
p
-
1
)
n
-
p
s
.
For ϵ=(λpp-q+μpp-q)p-1n-ps∈(0,δ2), we get
sup
t
≥
t
0
J
λ
,
μ
(
t
u
0
,
t
v
0
)
≤
2
s
n
(
S
α
,
β
2
)
n
p
s
+
C
(
λ
p
p
-
q
+
μ
p
p
-
q
)
-
C
(
λ
+
μ
)
{
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
1
n
-
p
s
(
n
-
n
-
p
s
p
q
)
if
q
>
n
(
p
-
1
)
n
-
p
s
,
(
λ
p
p
-
q
+
μ
p
p
-
q
)
n
(
p
-
1
)
p
(
n
-
p
s
)
|
log
(
λ
p
p
-
q
+
μ
p
p
-
q
)
|
if
q
=
n
(
p
-
1
)
n
-
p
s
.
If q>n(p-1)n-ps, we can choose δ2>0 such that for λ,μ satisfying 0<λpp-q+μpp-q<δ2,
(4.15)
C
(
λ
p
p
-
q
+
μ
p
p
-
q
)
-
C
(
λ
+
μ
)
(
λ
p
p
-
q
+
μ
p
p
-
q
)
p
-
1
n
-
p
s
(
n
-
n
-
p
s
p
q
)
<
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
,
where C0 is the positive constant defined in (3.5). In fact, (4.15) holds if
1
+
p
p
-
q
p
-
1
n
-
p
s
(
n
-
n
-
p
s
p
q
)
<
p
p
-
q
⇔
q
>
n
(
p
-
1
)
n
-
p
s
.
If instead q=n(p-1)n-ps, we can choose δ3>0 such that for λ,μ satisfying 0<λpp-q+μpp-q<δ3,
C
(
λ
p
p
-
q
+
μ
p
p
-
q
)
-
C
(
λ
+
μ
)
(
λ
p
p
-
q
+
μ
p
p
-
q
)
n
(
p
-
1
)
p
(
n
-
p
s
)
|
log
(
λ
p
p
-
q
+
μ
p
p
-
q
)
|
<
-
C
0
(
λ
p
p
-
q
+
μ
p
p
-
q
)
as |log(λpp-q+μpp-q)|→+∞ for λ,μ→0, and
(
λ
+
μ
)
(
λ
p
p
-
q
+
μ
p
p
-
q
)
n
(
p
-
1
)
p
(
n
-
p
s
)
≃
(
λ
p
p
-
q
+
μ
p
p
-
q
)
.
Then, taking
Λ
2
=
min
{
δ
1
,
δ
2
,
δ
3
,
(
δ
2
)
n
-
p
s
p
-
1
}
>
0
,
for all λ,μ satisfying 0<λpp-q+μpp-q<Λ2, we have
(4.16)
sup
t
≥
0
J
λ
,
μ
(
t
u
,
t
v
)
<
c
∞
.
Since (u0,v0)≠(0,0), from Lemma 2.5 and (4.16), there exists t2>0 such that (t2u0,t2v0)∈𝒩λ,μ- and
c
λ
,
μ
-
≤
J
λ
,
μ
(
t
2
u
0
,
t
2
v
0
)
≤
sup
t
≥
0
J
λ
,
μ
(
t
u
0
,
t
v
0
)
<
c
∞
for all λ,μ satisfying 0<λpp-q+μpp-q<Λ2. This concludes the proof. ∎
Proposition 4.11
Proposition 4.11 (Existence of the Second Solution)
There exists a positive constant Λ3>0, such that for λ,μ satisfying 0<λpp-q+μpp-q<Λ3, the functional Jλ,μ has a minimizer (u2,v2) in Nλ,μ- with the following properties:
(
u
2
,
v
2
)
is a solution of problem (
1.1
).
Proof.
Let Λ2 be as in Lemma 4.10, and set
Λ
3
:=
{
Λ
2
,
(
q
p
)
p
p
-
q
Λ
1
}
.
By means of Proposition 4.6 (ii), for all λ,μ satisfying 0<λpp-q+μpp-q<Λ3, there exists a bounded (PS)cλ,μ- sequence (u~k,v~k)}⊂𝒩λ,μ- for Jλ,μ. By the same argument used in the proof of Proposition 4.7, there exists (u2,v2)∈E such that, up to a subsequence, u~k→u2, v~k→v2 strongly in E and Jλ,μ(u2,v2)=cλ,μ-. Moreover, (u2,v2) is a solution of problem (1.1).
Next we show that (u2,v2)∈𝒩λ,μ-. In fact, since (u~k,v~k)∈𝒩λ,μ-, we have
φ
u
~
k
,
v
~
k
′′
(
1
)
=
(
p
-
q
)
∥
(
u
~
k
,
v
~
k
)
∥
p
-
2
(
(
α
+
β
)
-
q
)
∫
Ω
|
u
~
k
|
α
|
v
~
k
|
β
d
x
<
0
.
Since u~k→u2, v~k→v2 strongly in E, passing to the limit, we obtain
φ
u
2
,
v
2
′′
(
1
)
=
(
p
-
q
)
∥
(
u
2
,
v
2
)
∥
p
-
2
(
(
α
+
β
)
-
q
)
∫
Ω
|
u
2
|
α
|
v
2
|
β
d
x
≤
0
.
Since 𝒩λ,μ0=∅, we conclude that φu2,v2′′(1)<0, i.e., (u2,v2)∈𝒩λ,μ-. ∎
