1 Introduction
Let Ω⊂ℝN be a bounded smooth domain, s∈(0,1) and N>sp. In this paper, we investigate the existence of multiple solutions for the nonlocal problem
(1.1)
{
(
-
Δ
)
p
s
u
=
-
λ
|
u
|
q
-
2
u
+
a
|
u
|
p
-
2
u
+
b
(
u
+
)
p
s
*
-
1
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
where (-Δ)ps is the fractional p-Laplacian operator defined by
(
-
Δ
)
p
s
u
(
x
)
=
2
lim
ε
→
0
∫
ℝ
N
∖
B
(
x
,
ε
)
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
|
x
-
y
|
N
+
s
p
d
y
,
x
∈
ℝ
N
.
In (1.1), λ>0 is a parameter, a,b>0 are real constants, 1<q<p, ps*=pNN-sp is the critical Sobolev exponent for (-Δ)ps and u+:=max{0,u} denotes the positive part of u, while the negative part of u will be denoted u-=min{0,u}. Consequently, u=u++u-.
The term (u+) appears for the first time in the classical paper by Ruf and Srikanth [34]. But our problem is also related to two classical local problems, namely, the Ambrosetti–Prodi and the Brezis–Nirenberg problems. On these subjects, see the excellent books [22] and [39], respectively.
In 1972, Ambrosetti and Prodi [4] considered the Dirichlet boundary value problem
(AP)
-
Δ
u
=
g
(
u
)
+
f
(
x
)
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω⊂ℝN is a bounded smooth domain, -Δ denotes the Laplacian operator, f:ℝN→ℝ is a C1 function, g:ℝ→ℝ is C2, convex and satisfies
0
<
g
-
=
lim
t
→
-
∞
g
′
(
t
)
<
λ
1
<
g
+
=
lim
t
→
+
∞
g
′
(
t
)
<
λ
2
,
with λ1 and λ2 denoting the first and second eigenvalues of (-Δ,H01(Ω)). They proved the existence of a C1 manifold M in C0,α(Ω¯), which splits the space into two open sets O0 and O2 with the following properties:
if f∈O0, problem (AP) has no solution;
if f∈M, problem (AP) has exactly one solution;
if f∈O2, problem (AP) has exactly two solutions.
Many authors have extended this result in different ways, and we would like to apologize if we omit some important contributions, but we cite, e.g., the papers [2, 6, 8, 9, 14, 18, 19, 27, 34, 37] and references therein. All these results show the role of the interaction between g± and the eigenvalues of (-Δ,H01(Ω)).
On the other hand, in 1983, Brezis and Nirenberg [11] studied the Dirichlet boundary problem
(BN)
-
Δ
u
=
a
u
+
|
u
|
2
*
-
2
u
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω⊂ℝN is a bounded smooth domain, a>0 and 2*=2NN-2 (N≥3) is the Sobolev critical exponent. As before, denoting by λj (j=1,2,…) the eigenvalues of -Δ, the authors proved that there exists a0>0 such that
if a0<a<λ1, N=3 and Ω=B1(0), problem (BN) has at least one positive solution;
if a<λ1 and N≥4, problem (BN) has at least one positive solution.
Among many works extending or complementing the above result for both local and nonlocal operators, we mention, e.g., [3, 4, 7, 10, 14, 16, 28, 30, 32, 36]. But we would like to highlight Capozzi, Fortunato and Palmieri [15], where the authors proved that problem (BN) has at least one nontrivial solution for all a>0 if N≥5 and for all a≠λj if N=4.
In the interesting work [21], de Paiva and Presoto established a multiplicity result for the Dirichlet boundary problem
-
Δ
u
=
-
λ
|
u
|
q
-
2
u
+
a
u
+
(
u
+
)
p
-
1
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω⊂ℝN is a bounded smooth domain, a,λ>0 and 1<q<2<p≤2*. (See also [20] for the subcritical case and [24, 31] for related problems.) The main goal of the present paper is to prove the result obtained in [21] for the fractional p-Laplacian operator extending the results obtained in [29, 13].
3
C
δ
0
versus Ws,p Minimization for Polynomial Growth
The main result of this section is a local minimization equivalence for functionals defined in the fractional Sobolev space Xps with polynomial growth nonlinearity, following ideas developed by Barrios, Colorado, de Pablo and Sanchéz [7], Giacomoni, Prashanth and Sreenadh [23] and Iannizzotto, Mosconi and Squassina [25]. The result we prove is more general than those found in [7] and [25] since we allow p>1.
We start showing a regularization result that will be useful in the proof of Theorem 2. Its proof is similar to that of [13, Lemma 3.1].
Proposition 3.1.
Suppose |g(t)|≤C(1+|t|q-1) for some 1≤q≤ps* and C>0. Let (vε)ε∈(0,1)⊆Xps be a bounded family of solutions in Xps to the problem
(3.1)
{
(
-
Δ
)
p
s
u
=
(
1
1
-
ξ
ε
)
g
(
u
)
𝑖𝑛
Ω
,
u
=
0
𝑖𝑛
ℝ
N
∖
Ω
,
with ξε≤0. Then supε∈(0,1)∥vε∥L∞(Ω)<∞.
Proof.
For 0<k∈ℕ, we define
T
k
(
s
)
=
{
s
+
k
if
s
≤
-
k
,
0
if
-
k
<
s
<
k
,
s
-
k
if
s
≥
k
,
Ω
k
=
{
x
∈
Ω
:
|
v
ε
(
x
)
|
≥
k
}
.
Observe that Tk(vε)∈Xps and ∥Tk(vε)∥Xpsp≤Cp∥vε∥Xpsp<∞ for a constant C>0.
Taking Tk(vε) as a test-function, we obtain
A
(
v
ε
)
⋅
T
k
(
v
ε
)
=
∫
Ω
(
1
1
-
ξ
ε
)
g
(
v
ε
)
T
k
(
v
ε
)
d
x
≤
∫
Ω
k
C
|
T
k
(
v
ε
)
|
d
x
+
C
∫
Ω
k
|
v
ε
|
p
s
*
-
1
|
T
k
(
v
ε
)
|
d
x
.
Now consider 1<θ3<θ2<p, p<θ2θ3+1 and (ps*-1)θ1<ps* such that θ1-1+θ2-1+θ3-1=1. Therefore, by applying Hölder’s inequality, we obtain
(3.2)
A
(
v
ε
)
⋅
T
k
(
v
ε
)
≤
C
(
∫
Ω
|
T
k
(
v
ε
)
|
θ
2
d
x
)
1
θ
2
|
Ω
k
|
1
θ
3
.
Denote
T
(
x
,
y
)
=
|
v
ε
(
x
)
-
v
ε
(
y
)
|
p
-
2
(
v
ε
(
x
)
-
v
ε
(
y
)
)
(
T
k
(
v
ε
)
(
x
)
-
T
k
(
v
ε
)
(
y
)
)
|
x
-
y
|
N
+
s
p
.
Noting that the inequality
|
s
-
t
|
p
-
2
(
s
-
t
)
(
T
k
(
s
)
-
T
k
(
t
)
)
≥
|
T
k
(
s
)
-
T
k
(
t
)
|
p
for all
s
,
t
∈
ℝ
,
holds since both Tk(s) and s-Tk(s) are non-decreasing functions, we obtain
T
(
x
,
y
)
≥
|
T
k
(
v
ε
)
(
x
)
-
T
k
(
v
ε
)
(
y
)
|
p
|
x
-
y
|
N
+
s
p
.
Therefore, we have the estimate
A
(
v
ε
)
⋅
T
k
(
v
ε
)
≥
∫
ℝ
2
N
|
T
k
(
v
ε
)
(
x
)
-
T
k
(
v
ε
)
(
y
)
|
p
|
x
-
y
|
N
+
s
p
d
x
d
y
=
∥
T
k
(
v
ε
)
∥
X
p
s
p
.
It follows from the continuous immersion Xps↪Lθ2(Ω) that (for a constant C1>0)
(3.3)
C
1
(
∫
Ω
|
T
k
(
v
ε
)
|
θ
2
d
x
)
p
θ
2
≤
A
(
v
ε
)
⋅
T
k
(
v
ε
)
.
So (3.2) and (3.3) guarantee the existence of C>0 such that
∫
Ω
|
T
k
(
v
ε
)
|
θ
2
d
x
≤
C
|
Ω
k
|
θ
2
θ
3
(
p
-
1
)
=
C
|
Ω
k
|
β
β
-
1
,
where β=θ2θ2-θ3(p-1)>1, the last inequality being a consequence of p<θ2θ3+1.
Since, for all s∈ℝ, we have |Tk(s)|=(|s|-k)(1-χ[-k,k](s)), we conclude that, if 0<k<h∈ℕ, then Ωh⊂Ωk. Therefore,
∫
Ω
|
T
k
(
v
ε
)
|
θ
2
d
x
=
∫
Ω
k
(
|
v
ε
|
-
k
)
θ
2
d
x
≥
∫
Ω
h
(
|
v
ε
|
-
k
)
θ
2
d
x
≥
(
h
-
k
)
θ
2
|
Ω
h
|
.
Defining, for 0<k∈ℕ, ϕ(k)=|Ωk|, it follows
ϕ
(
h
)
≤
C
(
h
-
k
)
-
θ
2
ϕ
(
k
)
β
β
-
1
,
0
<
k
<
h
∈
ℕ
.
Considering the sequence (kn) defined by k0=0 and kn=kn-1+d2n, where d=2βC1θ2|Ω|1(p-1)θ2, we have 0≤ϕ(kn)≤ϕ(0)2nr(β-1) for all n∈ℕ. Thus, limn→∞ϕ(kn)=0. Since ϕ(kn)≥ϕ(d) implies ϕ(d)=0, we have |vε(x)|≤d a.e. in Ω for all ε∈(0,1). We are done. ∎
We recall the definitions of the spaces Cδ0(Ω¯) and Cδ0,α(Ω¯). Let δ:Ω¯→ℝ+ be given by δ(x)=dist(x,ℝN∖Ω). Then, for 0<α<1, we have
C
δ
0
(
Ω
¯
)
=
{
u
∈
C
0
(
Ω
¯
)
:
u
δ
s
has a continuous extension to
Ω
¯
}
,
C
δ
0
,
α
(
Ω
¯
)
=
{
u
∈
C
0
(
Ω
¯
)
:
u
δ
s
has a
α
-Hölder extension to
Ω
¯
}
with the respective norms
∥
u
∥
0
,
δ
=
∥
u
δ
s
∥
L
∞
(
Ω
)
and
∥
u
∥
α
,
δ
=
∥
u
∥
0
,
δ
+
sup
x
,
y
∈
Ω
¯
,
x
≠
y
.
|
u
(
x
)
δ
(
x
)
s
-
u
(
y
)
δ
(
y
)
s
|
|
x
-
y
|
α
.
Let us consider the Dirichlet problem
(3.4)
{
(
-
Δ
)
p
s
u
=
f
(
u
)
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
where Ω⊂ℝN (N>1) is a bounded, smooth domain, s∈(0,1), p>1 and f∈L∞(Ω).
The next two results can be found in Iannizzotto, Mosconi and Squassina [26, Theorems 1.1 and 4.4], respectively. They will play a major role in the proof of Theorem 2.
Proposition 3.2.
There exist α∈(0,s] and CΩ>0 depending only on N, p, s, with CΩ also depending on Ω, such that, for all weak solution u∈Xps of (3.4), u∈Cα(Ω¯) and
∥
u
∥
C
α
(
Ω
¯
)
≤
C
Ω
∥
f
∥
L
∞
(
Ω
)
1
p
-
1
.
Proposition 3.3.
Let u∈Xps satisfy |(-Δ)psu|≤K weakly in Ω for some K>0. Then |u|≤(CΩK)1p-1δs a.e. in Ω for some CΩ=C(N,p,s,Ω).
The proof of the next result is similar to that of [13, Theorem 1]. We emphasize that, only in this section, δ represents the function defined by δ(x)=dist(x,∂Ω).
Theorem 2.
Suppose that g∈C(Ω) satisfies
(3.5)
|
g
(
t
)
|
≤
C
(
1
+
|
t
|
q
-
1
)
for some
1
≤
q
≤
p
s
*
𝑎𝑛𝑑
C
>
0
,
and consider the functional Φ:Xps→R defined by
Φ
(
u
)
=
1
p
∥
u
∥
X
p
s
p
-
∫
Ω
G
(
u
)
d
x
,
where G(t)=∫0tg(s)ds.
If 0 is a local minimum of Φ in Cδ0(Ω¯), that is, there exists r1>0 such that
Φ
(
0
)
≤
Φ
(
z
)
for all
z
∈
X
p
s
∩
C
δ
0
(
Ω
¯
)
,
∥
z
∥
0
,
δ
≤
r
1
,
then 0 is a local minimum of Φ in Xps, that is, there exists r2>0 such that
Φ
(
0
)
≤
Φ
(
z
)
for all
z
∈
X
p
s
,
∥
z
∥
X
p
s
≤
r
2
.
Proof.
Let us consider initially the subcritical caseq<ps*. By contradiction, denoting
B
¯
ε
=
{
z
∈
X
p
s
:
∥
z
∥
X
p
s
≤
ε
}
,
let us suppose that, for any ε>0, there exists uε∈B¯ε such that
(3.6)
Φ
(
u
ε
)
<
Φ
(
0
)
.
Since Φ:B¯ε→ℝ is weakly lower semicontinuous, there exists vε∈B¯ε such that infu∈B¯εΦ(u)=Φ(vε). It follows from (3.6) that
Φ
(
v
ε
)
=
inf
u
∈
B
¯
ε
Φ
(
u
)
≤
Φ
(
u
ε
)
<
Φ
(
0
)
.
We will show that vε→0 in Cδ0(Ω¯) as ε→0 since this implies, for r1>0, the existence of z∈Cδ0(Ω¯) such that ∥z∥0,δ<r1 and Φ(z)<Φ(0), contradicting our hypothesis. Since vε is a critical point of Φ in Xps, by Lagrange multipliers, it is not difficult to verify that
(3.7)
Φ
′
(
v
ε
)
=
ξ
ε
A
(
v
ε
)
implies ξε≤0. Thus, it follows from (3.7) that vε satisfies
(Pv)
{
(
-
Δ
)
p
s
v
ε
=
(
1
1
-
ξ
ε
)
g
(
v
ε
)
=
:
g
ε
(
v
ε
)
in
Ω
,
v
ε
=
0
in
ℝ
N
∖
Ω
,
If ∥vε∥Xps≤ε<1, Proposition 3.1 shows the existence of a constant C1>0, not depending on ε, such that
(3.8)
∥
v
ε
∥
L
∞
(
Ω
)
≤
C
1
.
Since ξε≤0, (3.5) and (3.8) imply that ∥gε(vε)∥L∞(0,1)≤C2 for some constant C2>0. Proposition 3.2 yields ∥vε∥C0,β(Ω¯)≤C3 for 0<β≤s and a constant C3 not depending on ε. Now, it follows from the Arzelà–Ascoli theorem the existence of a sequence(vε) such that vε→0 uniformly as ε→0. Passing to a subsequence, we can suppose that vε→0 a.e. in Ω and, therefore, vε→0 uniformly in Ω¯. But now it follows from Proposition 3.3 that
∥
v
ε
∥
0
,
δ
=
∥
v
ε
δ
s
∥
L
∞
(
Ω
)
≤
C
sup
x
∈
(
0
,
1
)
|
g
ε
(
v
ε
(
x
)
)
|
for a constant C>0. The proof of the subcritical case is complete.
We now consider the critical case q=ps*. As before, we argument by contradiction. For this, we define gk,Gk:ℝ→ℝ by
g
k
(
s
)
=
g
(
t
k
(
s
)
)
and
G
k
(
t
)
=
∫
0
t
g
k
(
s
)
d
s
,
with tk given by
t
k
(
s
)
=
{
-
k
,
if
s
≤
-
k
,
s
if
-
k
<
s
<
k
,
k
if
s
≥
k
.
Considering Φk∈C1(Xps,ℝ) given by
Φ
k
(
u
)
=
∥
u
∥
X
p
s
p
p
-
∫
Ω
G
k
(
t
)
d
t
,
it follows Φk(u)→Φ(u) as k→∞. Thus, for any ε∈(0,1), there exists kε≥1 such that Φkε(wε)<Φ(0), and the subcritical growth of gk guarantees the existence of uε∈B¯ε such that
Φ
k
ε
(
u
ε
)
=
inf
u
∈
B
¯
ε
Φ
k
ε
(
u
)
≤
Φ
k
ε
(
w
ε
)
<
Φ
(
0
)
.
As in the subcritical case, we find ξε≤0 such that uε is a weak solution to problem (Pv) with uε instead of vε. From the definition of gk and since ∥uε∥Xps≤ε<1, by applying Proposition 3.1, we obtain ∥gkεε(uε)∥L∞(Ω)≤C2 for a constant C2>0. It follows from Proposition 3.2 that ∥uε∥C0,β(Ω¯)≤C3 for 0<β≤s, the constant C3 not depending on ε. By applying the Arzelà–Ascoli theorem, the conclusion is now obtained as in the subcritical case. ∎
Remark 3.4.
If 0 is a strict local minimum in Cδ0(Ω¯), it follows that 0 is also a strict local minimum in Xps.
4 Positive and Negative Solutions
Most of the results in this section are standard. Therefore, our presentation will be only schematic.
We denote
(4.1)
S
p
,
s
=
inf
{
∥
u
∥
X
p
s
p
(
∫
Ω
|
u
|
p
s
*
d
x
)
p
p
s
*
;
u
∈
X
p
s
,
u
≠
0
}
the best constant of the immersion Xps↪Lps*(Ω); see [30].
If we take care of the operator A, the proof of the next two results is similar to that exposed in [21].
Lemma 4.1.
If a>λ1, b>0, 1<q<p and λ>0, then any (PS)-sequence of Iλ,s is bounded in Xps.
Lemma 4.2.
If a>λ1, b>0, 1<q<p and λ>0, then Iλ,s satisfies the (PS)-condition at any level C such that
C
<
s
N
b
s
p
-
N
s
p
S
p
,
s
N
s
p
.
We now consider the positive part of the functional Iλ,s. That is, Iλ,s+:Xps→ℝ given by
I
λ
,
s
+
(
u
)
=
1
p
∥
u
∥
X
p
s
p
+
λ
q
∫
Ω
|
u
+
|
q
d
x
-
a
p
∫
Ω
|
u
+
|
p
d
x
-
b
p
s
*
∫
Ω
(
u
+
)
p
s
*
d
x
.
Of course, Iλ,s+∈C1(Xps,ℝ), and it holds, for all u,h∈Xps,
(
I
λ
,
s
+
)
′
(
u
)
⋅
h
=
A
(
u
)
⋅
h
+
λ
∫
Ω
|
u
+
|
q
-
1
h
d
x
-
a
∫
Ω
|
u
+
|
p
-
1
h
d
x
-
b
∫
Ω
(
u
+
)
p
s
*
-
1
h
d
x
.
Furthermore, critical points of Iλ,s+ are weak solutions to the problem
{
(
-
Δ
)
p
s
u
=
-
λ
|
u
+
|
q
-
1
+
a
|
u
+
|
p
-
1
+
b
(
u
+
)
p
s
*
-
1
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
,
where a,b,λ>0, 1<q<p and u+=max{u,0}.
We now recall the following elementary inequality, which has a straightforward proof.
Lemma 4.3.
For all u:RN→R, p>1 and x,y∈RN, it holds
|
u
+
(
x
)
-
u
+
(
y
)
|
p
≤
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
(
u
+
(
x
)
-
u
+
(
y
)
)
,
|
u
-
(
x
)
-
u
-
(
y
)
|
p
≤
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
(
u
-
(
x
)
-
u
-
(
y
)
)
.
If u is a critical point of Iλ,s+, then (Iλ,s+)′(u)⋅h=0 for all h∈Xps. Taking h=u-, it follows from Lemma 4.3 that
0
=
(
I
λ
,
s
+
)
′
(
u
)
⋅
u
-
=
A
(
u
)
⋅
u
-
≥
∫
ℝ
2
N
|
u
-
(
x
)
-
u
-
(
y
)
|
p
|
x
-
y
|
N
+
s
p
d
x
d
y
=
∥
u
-
∥
X
p
s
p
and therefore u-=0. Thus, a critical point u of Iλ,s+ satisfies u=u+≥0.
As in the proof of Lemma 4.2, we have that Iλ,s+ satisfy the (PS)-condition at any level
(4.2)
C
<
s
N
b
s
p
-
N
s
p
S
p
,
s
N
s
p
for any λ>0. The definition of the functional J was removed.
Lemma 4.4.
If a,b>0 and 1<q<p, then the trivial solution u=0 is a strict local minimizer of Iλ,s for any λ>0.
Proof.
According to Remark 3.4, it is enough to show that u=0 is a strict local minimum of Iλ,s+ in Cδ0(Ω¯). Take u∈Cδ0(Ω¯)∖{0}, and consider the functional
I
λ
,
s
+
(
u
)
=
1
p
∥
u
∥
X
p
s
p
+
λ
q
∫
Ω
|
u
+
|
q
d
x
-
a
p
∫
Ω
|
u
+
|
p
d
x
-
b
p
s
*
∫
Ω
|
u
+
|
p
s
*
d
x
.
So, for positive constants C1 and C2, we have
I
λ
,
s
+
(
u
)
≥
1
p
∥
u
∥
X
p
s
p
+
(
λ
q
-
a
C
1
p
∥
u
∥
0
,
δ
p
-
q
-
b
C
2
p
s
*
∥
u
∥
0
,
δ
p
s
*
-
q
)
∫
Ω
|
u
+
|
q
d
x
This implies for all u≠0 with ∥u∥0,δ sufficiently small,
I
λ
,
s
+
(
u
)
>
0
=
I
λ
,
s
+
(
0
)
.
∎
Lemma 4.5.
If λ1<a, 1<q<p and b>0, then, for any fixed Λ>0, there exists t0=t0(Λ)>0 such that Iλ,s+(tφ1)<0 for all t≥t0 and λ<Λ.
Proof.
For a fixed Λ>0, our hypotheses guarantee that we can choose t0=t0(Λ)>0 such that, if t≥t0 and λ<Λ, it follows Iλ,s+(tφ1)<0. ∎
Now we prove that (1.1) has at least one positive solution.
Proposition 4.6.
Suppose that λ>0, 1<q<p, λ1<a and b>0. There exists λ0>0 such that, if 0<λ<λ0, then problem (1.1) has at least one positive solution.
Proof.
We observe that a non-negative weak solution of (1.1) is a critical point of the functional Iλ,s+. We now apply the mountain pass theorem. The geometric conditions of this theorem are consequences of Lemmas 4.4 and 4.5. We now prove the existence of λ0>0 such that, if 0<λ<λ0, then Iλ,s+ satisfies the (PS)-condition at level
C
λ
+
=
inf
g
∈
Γ
+
max
u
∈
g
(
[
0
,
1
]
)
I
λ
,
s
+
(
u
)
,
where Γ+={g∈C([0,1],Xps):g(0)=0,g(1)=t0φ1}, with t0 obtained in Lemma 4.5.
In order to do that, we observe that, for all 0≤t≤1, our hypotheses imply that
inf
g
∈
Γ
+
max
u
∈
g
(
[
0
,
1
]
)
I
λ
,
s
+
(
u
)
≤
max
u
∈
g
0
(
[
0
,
1
]
)
I
λ
,
s
+
(
u
)
=
max
t
∈
[
0
,
1
]
I
λ
,
s
+
(
g
0
(
t
)
)
≤
λ
t
0
q
q
∥
φ
1
∥
L
q
(
Ω
)
q
,
from which the existence of λ0>0 follows such that
0
≤
C
λ
+
<
s
N
b
s
p
-
N
s
p
S
p
,
s
N
s
p
for all
0
<
λ
<
λ
0
<
Λ
,
with Λ as in Lemma 4.5. The proof is complete as a consequence of (4.2). ∎
In order to show the existence of a negative solution for Iλ,s, we consider Iλ,s-:Xps→ℝ given by
I
λ
,
s
-
(
u
)
=
1
p
∥
u
∥
X
p
s
p
+
λ
q
∫
Ω
|
u
-
|
q
d
x
-
a
p
∫
Ω
|
u
-
|
p
d
x
,
where u-=min{u,0}.
Of course, Iλ,s-∈C1(Xps,ℝ) and critical points of Iλ,s- are weak solutions to the problem
(4.3)
{
(
-
Δ
)
p
s
u
=
-
λ
|
u
-
|
q
-
1
+
a
|
u
-
|
p
-
1
in
Ω
,
u
=
0
in
ℝ
N
∖
Ω
.
As before, a critical point u of Iλ,s- satisfies u=u-≤0.
Observe that Iλ,s- satisfies the (PS)-condition at all levels for any λ>0 since the nonlinearity in (4.3) does not have fractional critical power.
Lemma 4.7.
If 0<a and 1<q<p, then u=0 is a strict local minimizer of Iλ,s- for all λ>0.
Proof.
According to Remark 3.4, it is enough to show that u=0 is a strict local minimum of Iλ,s- in Cδ0(Ω¯). It is not difficult to verify that, for all u∈Cδ0(Ω¯)∖{0} and a fixed λ>0, we have
I
λ
,
s
-
(
u
)
≥
1
p
∥
u
∥
X
p
s
p
+
(
λ
q
-
a
p
∥
u
∥
0
,
δ
p
-
q
)
∫
Ω
|
u
-
|
q
d
x
.
Taking R=(λpqa)1p-q, it follows a∥u∥0,δp-qp<λq and for all u≠0, Iλ,s-(u)>0=Iλ,s-(0) if ∥u∥0,δ<R. ∎
The proof of the next result is analogous to that of Lemma 4.5.
Lemma 4.8.
If λ1<a and 1<q<p, then, for any fixed Λ>0, there exists t0′=t0′(Λ)>0 such that
I
λ
,
s
-
(
-
t
φ
1
)
<
0
for all
t
≥
t
0
′
𝑎𝑛𝑑
λ
<
Λ
.
The proof of the next result is similar to that of Proposition 4.6. In the proof, the inequality
0
≤
C
λ
-
≤
λ
(
t
0
′
)
q
q
∥
φ
1
∥
L
q
(
Ω
)
q
for all
λ
>
0
plays an essential role, with Cλ- defined analogously to Cλ+.
Proposition 4.9.
Suppose that λ>0, 1<q<p, λ1<a and b>0. There exists λ0>0 such that, if 0<λ<λ0, then problem (1.1) has at least one negative solution.
5 A Third Solution via Linking Theorem
In contrast to the previous section, the proof of the existence of a third solution to (1.1) is much more intricate and also technical. We obtain a third solution by applying the linking theorem and a series of previously obtained results that will be useful in our proof.
We suppose 0<s<1, N>sp, λ>0, λ1<a<λ* and b>0. The proof of our first result is simple.
Proposition 5.1.
If span{φ1} denotes the space generated by the first (positive, Lp-normalized) eigenfunction of (-Δ)ps, then Xps=W⊕span{φ1}.
We recall that W={u∈Xps:A(φ1)⋅u=0} and λ*=inf{∥u∥Xpsp:u∈W,∥u∥Lp(Ω)p=1}.
The next result will be used to prove that the geometric conditions of the linking theorem are satisfied.
Proposition 5.2.
Suppose that a<λ*. Then there exist α>0 and ρ>0 such that Iλ,s(u)≥α for any u∈W with ∥u∥Xps=ρ.
Proof.
Since a>0, the immersion Xps↪Lr(Ω) for r∈[p,ps*] combined with the definition of λ* implies (after some calculations) that
I
λ
,
s
(
u
)
≥
1
p
(
1
-
a
λ
*
)
∥
u
∥
X
p
s
p
-
b
C
p
s
*
p
s
*
∥
u
∥
X
p
s
p
s
*
≥
∥
u
∥
X
p
s
p
(
A
-
B
∥
u
∥
X
p
s
p
s
*
-
p
)
,
where A=1p(1-aλ*)>0 and B=bCps*ps*>0. If 0<ρ<(AB)1ps*-p, then Iλ,s(u)≥ρp(A-Bρps*-p)=α>0, and we conclude that Iλ,s(u)≥α for all u∈W satisfying ∥u∥Xps=ρ. ∎
In order to apply the linking theorem with respect to the decomposition given in Proposition 5.1, we need to prove the existence of a vector e∈W satisfying the hypotheses of that result. We recall that Sp,s was defined in (4.1).
We now state the following result, which can be found in [30, Proposition 2.1]. See also [10].
Proposition 5.3.
Let 1<p<∞, s∈(0,1), N>sp.
There exists a minimizer for
S
p
,
s
.
For every minimizer
U
, there exist
x
0
∈
ℝ
N
and a constant sign monotone function
u
:
[
0
,
∞
)
→
ℝ
such that
U
(
x
)
=
u
(
|
x
-
x
0
|
)
.
For every non-negative minimizer
U
∈
X
p
s
and
v
∈
X
p
s
, we have
∫
ℝ
2
N
|
U
(
x
)
-
U
(
y
)
|
p
-
2
(
U
(
x
)
-
U
(
y
)
)
(
v
(
x
)
-
v
(
y
)
)
|
x
-
y
|
N
+
s
p
d
x
d
y
=
S
p
,
s
∫
ℝ
2
N
U
p
s
*
-
1
v
d
x
.
Applying Proposition 5.3, we fix a radially symmetric non-negative decreasing minimizer U=U(r) of Sp,s. Multiplying U by a positive constant, we may assume that
(5.1)
(
-
Δ
)
p
s
U
=
U
p
s
*
-
1
.
It follows from (4.1) that
(5.2)
∥
U
∥
X
p
s
p
=
∥
U
∥
L
p
s
*
(
ℝ
N
)
p
s
*
=
(
S
p
,
s
)
N
s
p
.
For any ε>0, the function
(5.3)
U
ε
(
x
)
=
1
ε
N
-
s
p
p
U
(
|
x
|
ε
)
is also a minimizer of Sp,s satisfying (5.1) and (5.2), so after a rescaling, we may assume that U(0)=1. Henceforth, U will denote such a function and {Uε}ε>0 the associated family of minimizers given by (5.3).
Since an explicit formula for a minimizer of Sp,s is unknown, we make use of some asymptotic estimates obtained by Brasco, Mosconi and Squassina [10]; see also [30, Lemma 2.2].
Lemma 5.4.
There exist constants c1,c2>0 and θ>1 such that, for all r≥1,
c
1
r
N
-
s
p
p
-
1
≤
U
(
r
)
≤
c
2
r
N
-
s
p
p
-
1
𝑎𝑛𝑑
U
(
θ
r
)
U
(
r
)
≤
1
2
.
In order to apply the linking theorem with respect to the decomposition given by Proposition 5.1, we consider the family of functions {Uε}ε>0 as defined in (5.3). Without loss of generality, we suppose that 0∈Ω (boldface removed). From now on, let us consider θ as given in Lemma 5.4. For ε,δ>0, we define, as in Chen, Mosconi and Squassina [17],
m
ε
,
δ
=
U
ε
(
δ
)
U
ε
(
δ
)
-
U
ε
(
θ
δ
)
and also
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
)
,
Since
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
)
,
it is not difficult to verify that the functions gε,δ and Gε,δ are non-decreasing and absolutely continuous. We now define the non-increasing, absolutely continuous and radially symmetric function uε,δ(r)=Gε,δ(Uε(r)) which satisfies
(5.4)
u
ε
,
δ
(
r
)
=
{
U
ε
(
r
)
if
r
≤
δ
,
0
if
r
≥
θ
δ
.
According to Ambrosio and Isernia [4, p. 17], for any δ≤r≤θδ, it holds
0
≤
m
ε
,
δ
(
U
ε
(
r
)
-
U
ε
(
θ
δ
)
)
=
U
ε
(
δ
)
[
U
ε
(
r
)
-
U
ε
(
θ
δ
)
U
ε
(
δ
)
-
U
ε
(
θ
δ
)
]
≤
U
ε
(
δ
)
.
Therefore, from the definition of Gε,δ and (5.4), it follows that
u
ε
,
δ
(
r
)
≤
{
U
ε
(
r
)
if
r
<
θ
δ
,
0
if
r
≥
θ
δ
.
We denote by P1s,P2s the projections of Xps in span{φ1} and W, respectively, and define eε,δ=P2suε,δ∈W and claim that eε,δ is a continuous function. As shown in [10], we know that U∈L∞(ℝN)∩C0(ℝN). Since eε,δ=uε,δ-P1suε,δ, our claim is proved.
We now want to show that we can take e=eε,δ in the linking theorem. So we need to show that eε,δ(0)>0 for ε>0 sufficiently small. In order to do that, we obtain the inequalities of the next result using arguments similar to those used in [16], with the exception of (5.7). Estimate (5.7) follows from [4, Lemma 2.4]. Therefore, we state the following result.
Lemma 5.5.
(5.5)
∥
P
1
s
u
ε
,
δ
∥
L
∞
(
Ω
)
≤
{
∥
φ
1
∥
L
∞
(
Ω
)
p
C
1
ε
N
p
|
log
(
ε
δ
)
|
𝑖𝑓
p
=
2
N
N
+
s
,
∥
φ
1
∥
L
∞
(
Ω
)
p
C
1
ε
N
-
N
-
s
p
p
𝑖𝑓
1
<
p
<
2
N
N
+
s
,
∥
φ
1
∥
L
∞
(
Ω
)
p
C
1
δ
N
-
N
-
s
p
p
-
1
ε
N
-
s
p
p
(
p
-
1
)
𝑖𝑓
p
>
2
N
N
+
s
,
(5.6)
∥
e
ε
,
δ
∥
L
1
(
Ω
)
≤
{
C
0
ε
N
p
|
log
(
ε
δ
)
|
𝑖𝑓
p
=
2
N
N
+
s
,
C
0
ε
N
-
N
-
s
p
p
𝑖𝑓
1
<
p
<
2
N
N
+
s
,
C
0
δ
N
-
N
-
s
p
p
-
1
ε
N
-
s
p
p
(
p
-
1
)
𝑖𝑓
p
>
2
N
N
+
s
,
(5.7)
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
≤
{
K
1
ε
s
p
|
log
(
ε
δ
)
|
p
𝑖𝑓
p
=
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
K
1
ε
s
p
𝑖𝑓
1
<
p
<
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
K
1
ε
s
p
𝑖𝑓
p
>
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
(5.8)
∥
e
ε
,
δ
∥
L
p
s
*
-
1
(
Ω
)
p
s
*
-
1
≤
{
K
2
ε
N
-
s
p
p
|
log
(
ε
δ
)
|
p
s
*
-
1
𝑖𝑓
p
=
2
N
N
+
s
,
K
2
ε
N
-
s
p
p
𝑖𝑓
1
<
p
<
2
N
N
+
s
,
K
2
ε
N
-
s
p
p
𝑖𝑓
p
>
2
N
N
+
s
,
|
∫
Ω
(
|
e
ε
,
δ
|
p
s
*
-
|
u
ε
,
δ
|
p
s
*
)
d
x
|
≤
{
K
3
ε
N
|
log
(
ε
δ
)
|
p
s
*
𝑖𝑓
p
=
2
N
N
+
s
,
K
3
ε
N
+
K
3
ε
N
(
p
s
*
-
1
)
𝑖𝑓
1
<
p
<
2
N
N
+
s
,
K
3
ε
N
-
s
p
p
-
1
𝑖𝑓
p
>
2
N
N
+
s
.
We now fix K>0.
Lemma 5.6.
There exist ε(K)>0 and σ>0 such that Bσ(0)⊂{x∈Ω:eε,δ(x)>K}:=Ωε,K for all 0<ε≤ε(K). As a consequence,
|
∫
Ω
ε
,
K
|
e
ε
,
δ
|
p
s
*
d
x
-
∫
Ω
|
u
ε
,
δ
|
p
s
*
d
x
|
≤
{
K
4
ε
N
|
log
(
ε
δ
)
|
p
s
*
𝑖𝑓
p
=
2
N
N
+
s
,
K
4
ε
N
+
K
4
ε
N
(
p
s
*
-
1
)
𝑖𝑓
1
<
p
<
2
N
N
+
s
,
K
4
ε
N
-
s
p
p
-
1
𝑖𝑓
p
>
2
N
N
+
s
.
Proof.
It follows from the definition of uε,δ , Lemma 5.4 and (5.5) that
(5.9)
e
ε
,
δ
(
0
)
≥
1
ε
N
-
s
p
p
-
∥
P
1
s
u
ε
,
δ
∥
L
∞
(
Ω
)
→
+
∞
as ε→0; the proof of the claim is complete because eε,δ is continuous. The proof of the estimates is obtained by applying [19, Lemma 2.4]. ∎
It follows from Lemma 5.6 that there exists ε0>0 such that eε,δ≠0, 0<ε≤ε0. Thus, in the linking theorem, we can take e=eε,δ. If ε∈(0,ε0], take R1,R2>0, and define
(5.10)
Q
ε
,
R
1
,
R
2
=
{
u
∈
X
p
s
:
u
=
u
1
+
r
e
ε
,
δ
,
u
1
∈
span
{
φ
1
}
∩
B
¯
R
1
(
0
)
,
0
≤
r
≤
R
2
}
.
Let ∂Qε,R1,R2 be the boundary of Qε,R1,R2 in the finite-dimensional space span{φ1}⊕span{eε,δ}. We denote O(εω) for ω≥0 if |O(εω)|≤Cεω for some C>0 not depending on ε>0. We remark that O(εω) is not always positive.
The next elementary result can be found in Mosconi, Perera, Squassina and Yang [30, p. 17].
Lemma 5.7.
Given k>1 and p-1<τ<p, there exists a constant C=C(k,q)>0 such that
|
a
+
b
|
p
≤
k
|
a
|
p
+
|
b
|
p
+
C
|
a
|
p
-
τ
|
b
|
τ
for all
a
,
b
∈
ℝ
.
As an immediate consequence of Lemma 5.7, we have the following result.
Lemma 5.8.
There exist constants C1,C2>0 such that
|
a
+
b
|
p
≤
C
1
|
a
|
p
+
C
2
|
b
|
p
for all
a
,
b
∈
ℝ
.
The proof of the next result is obtained by applying Lemma 5.7 with a=∥reε,δ∥Lp(Ω) and b=∥u1+reε,δ∥Lp(Ω) and considering the cases 0<τ<1 and τ>1. In the latter case, we then apply Lemma 5.8 and once again Lemma 5.7 with a=∥reε,δ∥ and b=∥u1∥Xps.
Lemma 5.9.
Suppose that u1∈span{φ1} and τ∈(p-1,p). Then there exists a constant C*>0 such that
1
p
∥
u
1
+
r
e
ε
,
δ
∥
X
p
s
p
-
a
p
∥
u
1
+
r
e
ε
,
δ
∥
L
p
(
Ω
)
p
≤
1
p
∥
u
1
∥
X
p
s
p
-
a
p
∥
u
1
∥
L
p
(
Ω
)
p
+
C
*
r
p
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
+
C
*
r
p
-
τ
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
-
τ
∥
u
1
∥
X
p
s
τ
.
The next result follows immediately by considering f:[0,∞)→ℝ given by f(t)=Btpp-Ctps*ps*.
Lemma 5.10.
For constants B>0 and C>0, we have
max
t
≥
0
(
B
t
p
p
-
C
t
p
s
*
p
s
*
)
=
s
N
(
B
C
p
p
s
*
)
N
s
p
.
To estimate Iλ,s on ∂Qε,R1,R2, we apply Lemmas 5.7 to 5.10 and the two next results. The first is a special case of the one proved by Chen, Mosconi and Squassina [17, Lemma 2.11], while the proof of the second can be found in [30, Lemma 2.7].
Lemma 5.11.
For any β>0 and 0<2ε≤δ<θ-1dist(0,∂Ω), we have
∥
u
ε
,
δ
∥
L
β
(
Ω
)
≤
{
C
β
ε
N
-
N
-
p
s
p
β
|
log
(
ε
δ
)
|
,
𝑖𝑓
β
=
p
s
*
p
′
,
C
β
ε
N
-
s
p
p
(
p
-
1
)
β
δ
N
-
N
-
s
p
p
-
1
β
𝑖𝑓
β
<
p
s
*
p
′
,
C
β
δ
N
-
N
-
s
p
p
β
𝑖𝑓
β
>
p
s
*
p
′
,
where p′=pp-1.
Observe that, taking β=1, we obtain
(5.11)
∥
u
ε
,
δ
∥
L
1
(
Ω
)
≤
{
C
1
ε
N
p
|
log
(
ε
δ
)
|
if
p
=
2
N
N
+
s
,
C
1
ε
N
-
N
-
s
p
p
if
1
<
p
<
2
N
N
+
s
,
C
1
δ
N
-
N
-
s
p
p
-
1
ε
N
-
s
p
p
(
p
-
1
)
if
p
>
2
N
N
+
s
.
Since (ps*-1)pp-1>ps*, it also follows from Lemma 5.11 by taking β=ps*-1 that
(5.12)
∥
u
ε
,
δ
∥
L
p
s
*
-
1
(
Ω
)
p
s
*
-
1
≤
C
p
s
*
-
1
ε
N
-
s
p
p
.
Lemma 5.12.
There exists a constant C=C(N,p,s)>0 such that, for any 0<ε≤δ2,
(5.13)
∥
u
ε
,
δ
∥
W
0
s
,
p
p
≤
S
p
,
s
N
s
p
+
C
(
ε
δ
)
N
-
s
p
p
-
1
,
∥
u
ε
,
δ
∥
L
p
(
Ω
)
p
≥
{
1
C
ε
s
p
log
(
δ
ε
)
,
N
=
s
p
2
,
1
C
ε
s
p
,
N
>
s
p
2
,
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
s
*
≥
S
p
,
s
N
s
p
-
C
(
ε
δ
)
N
p
-
1
.
In our development, we need an auxiliary result that was obtained in [19, Lemma 2.5] when s=1 and p=2. It is easily adapted for s∈(0,1) and p>1.
Lemma 5.13.
Let u,v∈Lr(Ω) with p≤r≤ps*. If ω⊂Ω and u+v>0 on ω, then
|
∫
ω
(
u
+
v
)
r
d
x
-
∫
ω
|
u
|
r
d
x
-
∫
ω
|
v
|
r
d
x
|
≤
C
∫
ω
(
|
u
|
r
-
1
|
v
|
+
|
u
|
|
v
|
r
-
1
)
d
x
,
where C depends only on r.
Adapting ideas from Miyagaki, Motreanu and Pereira [29] and de Figueiredo and Yang [19], we obtain the desired estimate Iλ,s on ∂Qε,R1,R2.
Proposition 5.14.
Consider Qε,R1,R2 defined in (5.10). There exist R1>0 and R2>0 large enough such that
I
λ
,
s
(
u
)
≤
λ
q
∫
Ω
|
u
|
q
d
x
for all
u
∈
∂
Q
ε
,
R
1
,
R
2
,
for all ε>0 small enough and all λ>0.
Proof.
We write ∂Qε,R1,R2=Γ1∪Γ2∪Γ3, with
Γ
1
=
B
R
1
∩
span
{
φ
1
}
,
Γ
2
=
{
u
∈
X
p
s
:
u
=
u
1
+
r
e
ε
,
δ
,
u
1
∈
span
{
φ
1
}
,
∥
u
1
∥
X
p
s
=
R
1
,
0
≤
r
≤
R
2
}
,
Γ
3
=
{
u
∈
X
p
s
:
u
=
u
1
+
R
2
e
ε
,
δ
,
u
1
∈
span
{
φ
1
}
,
∥
u
1
∥
X
p
s
≤
R
1
}
.
We consider Iλ,s in the three parts of the boundary Qδ,R1,R2. If u∈Γ1, then u=tφ1, and we obtain
I
λ
,
s
(
u
)
≤
|
t
|
p
p
(
λ
1
-
a
)
∫
Ω
|
φ
1
|
p
d
x
+
λ
q
∫
Ω
|
t
φ
1
|
q
d
x
≤
λ
q
∫
Ω
|
u
|
q
d
x
for all
u
∈
Γ
1
,
as desired. If u∈Γ2, then u=u1+reε,δ∈span{φ1}⊕span{eε,δ}, with ∥u1∥Xps=R1. Since P2s is bounded in Xps, there exist C1>0 such that ∥eε,δ∥Xps=∥P2suε,δ∥Xps≤C1∥uε,δ∥Xps. It follows then from (5.13) the existence of C>0 such that, for all 0<ε≤ε0^=min{δ2,ε0}, we have
(5.14)
∥
e
ε
,
δ
∥
X
p
s
p
≤
C
1
p
∥
u
ε
,
δ
∥
X
p
s
p
≤
C
1
p
S
p
,
s
N
s
p
+
C
1
p
C
(
ε
δ
)
N
-
s
p
p
-
1
.
We conclude that η:=sup0<ε≤ε0^∥eε,δ∥Xps is finite. In order to satisfy the condition R2∥eε,δ∥>ρ in the linking theorem for 0<ε≤ε0^ small enough, with δ<θ-1dist(0,∂Ω)2 as in Lemma 5.11 and ρ>0 as in Proposition 5.2, we must have R2η≥R2∥eε,δ∥Xps>ρ, showing that ρη is a lower bound for R2. Thus, we define r0=max{ρη,1} and consider two cases.
(a) 0≤r≤r0. Since all norms are equivalent on finite-dimensional spaces,
I
λ
,
s
(
u
)
≤
1
p
(
1
-
a
λ
1
)
R
1
p
+
C
*
r
0
p
η
p
+
C
*
r
0
p
-
τ
η
p
-
τ
R
1
τ
+
λ
q
∫
Ω
|
u
|
q
d
x
.
Since a>λ1, the result is obtained in this case for all R1>0 large enough to be fixed later.
(b) r>r0. We suppose R1≥1 and denote
K
(
R
1
)
:
=
1
r
0
sup
{
∥
u
1
∥
L
∞
(
Ω
)
:
u
1
∈
span
{
φ
1
}
,
∥
u
1
∥
X
p
s
=
R
1
}
∈
[
c
0
R
1
,
c
1
R
1
]
,
with positive constants c0,c1. We introduce the open set Ωε,δ={x∈Ω:eε,δ(x)>K(R1)}. It follows from (5.9) that 0∈Ωε,δ if ε∈(0,ε0^]. If x∈Ωε,δ and r>r0, we have eε,δ(x)>|u1(x)r|≥-u1(x)r a.e. in Ω. That is,
(5.15)
e
ε
,
δ
(
x
)
+
u
1
(
x
)
r
>
0
for all x∈Ωε,δ, r>r0 and u1∈span{φ1} with ∥u1∥Xps=R1. We now take R2=2R1. By Lemma 5.9 and using estimate (5.14), we can write
I
λ
,
s
(
u
)
≤
1
p
(
1
-
a
λ
1
)
R
1
p
+
C
*
r
p
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
+
2
p
-
τ
C
*
R
1
p
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
-
τ
-
b
p
s
*
r
p
s
*
∫
Ω
ε
,
δ
(
u
1
r
+
e
ε
,
δ
)
p
s
*
+
λ
q
∫
Ω
|
u
|
q
d
x
.
Now, applying Lemma 5.13 for u=u1r, v=eε,δ and ω=Ωε,δ, we obtain
I
λ
,
s
(
u
)
≤
1
p
(
1
-
a
λ
1
)
R
1
p
+
C
*
r
p
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
+
2
p
-
τ
C
*
R
1
p
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
-
τ
-
b
p
s
*
r
p
s
*
[
∥
e
ε
,
δ
∥
L
p
s
*
(
Ω
ε
,
δ
)
p
s
*
-
C
2
(
∥
e
ε
,
δ
∥
L
1
(
Ω
)
R
1
p
s
*
-
1
+
∥
e
ε
,
δ
∥
L
p
s
*
-
1
(
Ω
)
p
s
*
-
1
R
1
)
]
+
λ
q
∫
Ω
|
u
|
q
d
x
for a positive constant C2>0.
We consider the case where p>2NN+s and N>sp((p-1)2+p)>sp2 since the case 1<p≤2NN+s and N>sp2 is analogous. Estimates (5.6), (5.7), (5.8) and Lemma 5.6 combined with Lemma 5.12 imply that
I
λ
,
s
(
u
)
≤
1
p
[
(
1
-
a
λ
1
)
+
2
p
-
τ
C
*
ε
s
(
p
-
τ
)
]
R
1
p
+
C
*
r
p
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
-
b
p
s
*
r
p
s
*
[
S
p
,
s
N
s
p
+
O
(
ε
N
-
s
p
p
-
1
)
-
C
2
ε
N
-
s
p
p
(
p
-
1
)
-
γ
(
p
s
*
-
1
)
-
C
2
ε
N
-
s
p
p
-
γ
]
+
λ
q
∫
Ω
|
u
|
q
d
x
,
where R1=ε-γ with γ chosen so that 0<γ<min{N-spp(p-1)(ps*-1),N-spp}. Taking
A
=
(
1
-
a
λ
1
)
+
2
p
-
τ
C
*
ε
s
(
p
-
τ
)
,
B
=
C
*
p
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
,
C
=
b
S
p
,
s
N
s
p
+
O
(
ε
N
-
s
p
p
-
1
)
-
b
C
2
ε
N
-
s
p
p
(
p
-
1
)
-
γ
(
p
s
*
-
1
)
-
b
C
2
ε
N
-
s
p
p
-
γ
,
it is easy to see that there exists ε1>0 small enough such that, for all 0<ε<ε1<ε0^, we have A<0 and C>0, so
I
λ
,
s
(
u
)
≤
A
p
R
1
p
+
B
p
r
p
-
C
p
s
*
r
p
s
*
+
λ
q
∫
Ω
|
u
|
q
d
x
.
By applying Lemma 5.10 to the function h(r)=Brpp-Crps*ps*, we obtain
I
λ
,
s
(
u
)
≤
A
p
R
1
p
+
λ
q
∫
Ω
|
u
|
q
d
x
+
1
N
(
C
*
p
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
[
b
S
p
,
s
N
s
p
+
O
(
ε
N
-
s
p
p
-
1
)
-
b
C
2
ε
N
-
s
p
p
(
p
-
1
)
-
γ
(
p
s
*
-
1
)
-
b
C
2
ε
N
-
s
p
p
-
γ
]
p
p
s
*
)
N
s
p
.
Therefore, since A<0 and there exists ε1^>0 small enough such that, if 0<ε<ε1^<ε1, then R1>0 is large enough, we have the result for u∈Γ2. Finally, if u∈Γ3, then u=u1+R2eε,δ∈span{φ1}⊕span{eε,δ}, with ∥u1∥Xps≤R1. We recall that R2=2R1. By (5.14), since ∥eε,δ∥Xps is bounded, we have
(5.16)
I
λ
,
s
(
u
)
≤
[
C
*
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
+
C
*
C
2
2
τ
]
R
2
p
-
b
p
s
*
R
2
p
s
*
∫
Ω
(
(
u
1
R
2
+
e
ε
,
δ
)
+
)
p
s
*
+
λ
q
∫
Ω
|
u
|
q
d
x
.
Since the norms are equivalent on finite-dimensional spaces, there exists a constant C3>0 such that, if ∥u1∥Xps≤R1, then ∥u1∥L∞(Ω)≤C3∥u1∥Xps≤C3R1. Furthermore,
(5.17)
Ω
ε
′
:
=
{
x
∈
Ω
:
e
ε
,
δ
(
x
)
>
C
3
2
+
1
}
⊃
{
x
∈
Ω
:
e
ε
,
δ
(
x
)
>
C
3
+
1
}
:
=
D
ε
,
with |Dε|>0. So, for all x∈Ωε′, it follows from (5.15), (5.17) and R2=2R1 that
(5.18)
u
1
(
x
)
R
2
+
e
ε
,
δ
(
x
)
>
u
1
(
x
)
R
2
+
C
3
R
1
R
2
+
1
≥
u
1
(
x
)
R
2
+
∥
u
1
∥
L
∞
(
Ω
)
R
2
+
1
≥
1
.
Substituting (5.17) and (5.18) into (5.16), we obtain
I
λ
,
s
(
u
)
≤
[
C
*
(
C
1
p
S
p
,
s
N
s
p
+
C
0
)
+
C
*
C
2
2
τ
]
R
2
p
-
b
p
s
*
R
2
p
s
*
|
D
ε
|
+
λ
q
∫
Ω
|
u
|
q
d
x
for all
u
∈
Γ
3
.
So, by choosing ε>0 small enough, we obtain R2 (and also R1) large enough, and our proof is complete. ∎
Remark 5.15.
It is noteworthy to stress that R1, R2 and ε in Proposition 5.14 do not depend on λ.
We also recall the following elementary inequality (see [38, p. 122]).
Lemma 5.16.
For p>1, there exists a positive constant Kp, depending on p, such that, for all a,b∈R,
|
|
b
|
p
-
|
a
|
p
-
|
b
-
a
|
p
|
≤
K
p
(
|
b
-
a
|
p
-
1
|
a
|
+
|
a
|
p
-
1
|
b
-
a
|
)
.
By adapting the proof of [29, Lemma 6.1], we obtain the following result. Some details are to be found there.
Lemma 5.17.
Suppose that N>sp. For ε>0 small enough, the following estimate is true.
I
λ
,
s
(
u
)
≤
s
N
(
1
b
)
N
-
s
p
s
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
)
N
s
p
+
λ
q
∫
Ω
|
u
|
q
d
x
+
{
K
0
ε
s
(
p
-
1
)
|
log
(
ε
δ
)
|
p
s
*
𝑖𝑓
p
=
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
K
0
ε
s
(
p
-
1
)
𝑖𝑓
1
<
p
<
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
K
0
ε
s
(
p
-
1
)
+
K
0
ε
s
𝑖𝑓
p
>
2
N
N
+
s
𝑎𝑛𝑑
N
>
s
p
2
,
for all u∈Qε,R1,R2, where the constant K0>0 does not depend on ε.
Proof.
By applying Lemma 5.16 for a=u1(x) and b=u1(x)+reε,δ(x), the equivalences of norms in finite-dimensional spaces guarantees that
1
p
∥
u
1
+
r
e
ε
,
δ
∥
X
p
s
p
-
a
p
∥
u
1
+
r
e
ε
,
δ
∥
L
p
(
Ω
)
p
≤
1
p
(
∥
u
1
∥
X
p
s
p
-
a
∥
u
1
∥
L
p
(
Ω
)
p
)
+
r
p
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
)
+
C
R
2
p
-
1
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
-
1
∥
u
1
∥
X
p
s
+
C
R
2
∥
u
1
∥
X
p
s
p
-
1
∥
e
ε
,
δ
∥
L
p
(
Ω
)
.
Thus, by applying Lemma 5.9 and estimate (5.7), since a>λ1, we obtain
I
λ
,
s
(
u
)
≤
r
p
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
)
-
b
p
s
*
∫
Ω
(
u
+
)
p
s
*
d
x
+
λ
q
∫
Ω
|
u
|
q
d
x
+
{
2
C
′
ε
s
(
p
-
1
)
|
log
(
ε
δ
)
|
if
p
=
2
N
N
+
s
and
N
>
s
p
2
,
2
C
′
ε
s
(
p
-
1
)
if
1
<
p
<
2
N
N
+
s
and
N
>
s
p
2
,
C
′
ε
s
(
p
-
1
)
+
C
′
ε
s
if
p
>
2
N
N
+
s
and
N
>
s
p
2
.
We control the term -bps*∥u+∥Lps*(Ω)ps* by applying estimates (5.11) and (5.12) and conclude the existence of a constant C6>0 such that
-
b
p
s
*
∫
Ω
(
u
+
)
p
s
*
d
x
≤
-
b
r
p
s
*
p
s
*
∫
Ω
(
u
ε
,
δ
)
p
s
*
d
x
+
{
C
6
ε
N
-
s
p
p
|
log
(
ε
δ
)
|
p
s
*
if
p
=
2
N
N
+
s
,
C
6
ε
N
-
s
p
p
+
C
6
ε
N
(
p
s
*
-
1
)
if
1
<
p
<
2
N
N
+
s
,
C
6
ε
N
-
s
p
p
if
p
>
2
N
N
+
s
.
Thus, there exists a constant K0>0 such that
(5.19)
I
λ
,
s
(
u
)
≤
r
p
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
)
-
b
r
p
s
*
p
s
*
∫
Ω
|
u
ε
,
δ
|
p
s
*
d
x
+
λ
q
∫
Ω
|
u
|
q
d
x
+
{
K
0
ε
s
(
p
-
1
)
|
log
(
ε
δ
)
|
p
s
*
if
p
=
2
N
N
+
s
and
N
>
s
p
2
,
K
0
ε
s
(
p
-
1
)
if
1
<
p
<
2
N
N
+
s
and
N
>
s
p
2
,
K
0
ε
s
(
p
-
1
)
+
K
0
ε
s
if
p
>
2
N
N
+
s
and
N
>
s
p
2
.
By applying Lemma 5.10, the function f:[0,+∞)→ℝ given by
f
(
t
)
=
t
p
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
)
-
b
t
p
s
*
p
s
*
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
s
*
admits its maximum at a point tM so that
f
(
t
M
)
=
s
N
(
1
b
)
N
-
s
p
s
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
)
N
s
p
.
Our result now follows from (5.19). ∎
Also, the next result adapts a similar result obtained in [29, Lemma 6.2].
Lemma 5.18.
Suppose that N>sp2 and 1<p≤2NN+s. Then we have
c
s
:
=
inf
h
∈
Γ
sup
u
∈
Q
ε
,
R
1
,
R
2
I
λ
,
s
(
h
(
u
)
)
<
s
N
b
s
p
-
N
s
p
S
p
,
s
N
s
p
for all ε>0 and λ>0 small enough, where Γ={h∈C(Q¯ε,R1,R2,Xps):h=id𝑖𝑛∂Qε,R1,R2}. The same result is also valid if N>sp((p-1)2+p) and p>2NN+s.
Proof.
Since h=idQε,R1,R2∈Γ, the compactness of Qε,R1,R2 assures that is enough to prove
(5.20)
I
λ
,
s
(
u
)
<
s
N
b
s
p
-
N
s
p
S
p
,
s
N
s
p
for all
u
∈
Q
ε
,
R
1
,
R
2
.
By applying Lemma 5.8 for a=uε,δ-P1suε,δ=eε,δ and b=P2suε,δ, for some constant c0>0, we have
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
≥
1
c
0
∥
u
ε
,
δ
∥
L
p
(
Ω
)
p
-
∥
P
1
s
u
ε
,
δ
∥
L
p
(
Ω
)
p
.
Since N>sp2, it follows from Lemma 5.12 that ∥uε,δ∥Lp(Ω)p≥c2εsp for a constant c2>0. Taking into account (5.5), we obtain
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
≥
c
2
c
0
ε
s
p
-
{
c
1
ε
N
|
log
(
ε
δ
)
|
p
if
p
=
2
N
N
+
s
and
N
>
s
p
2
,
c
1
ε
N
(
p
-
1
)
+
s
p
if
1
<
p
<
2
N
N
+
s
and
N
>
s
p
2
,
c
1
ε
N
-
s
p
p
-
1
if
p
>
2
N
N
+
s
and
N
>
s
p
2
.
But Lemma 5.12 yields
∥
u
ε
,
δ
∥
L
p
s
*
p
s
*
≥
S
p
,
s
N
s
p
+
O
(
ε
N
p
-
1
)
.
Now, mimicking [16, p. 286], we obtain
|
∥
e
ε
,
δ
∥
X
p
s
p
-
∥
u
ε
,
δ
∥
X
p
s
p
|
≤
c
3
∥
u
ε
,
δ
∥
X
p
s
p
-
1
∥
P
1
s
u
ε
,
δ
∥
L
∞
(
Ω
)
+
c
3
∥
P
1
s
u
ε
,
δ
∥
L
∞
(
Ω
)
p
.
A new application of Lemma 5.12 and estimate (5.5) give
∥
e
ε
,
δ
∥
X
p
s
p
≤
∥
u
ε
,
δ
∥
X
p
s
p
+
{
c
4
ε
N
p
|
log
(
ε
δ
)
|
if
p
=
2
N
N
+
s
c
4
ε
N
-
N
-
s
p
p
if
1
<
p
<
2
N
N
+
s
,
c
4
ε
N
-
s
p
p
(
p
-
1
)
if
p
>
2
N
N
+
s
.
Case 1: Suppose N>sp2 and p=2NN+s. By applying the mean value theorem to f(t)=(1+t)N-spN, we conclude that
|
1
-
(
1
+
S
p
,
s
-
N
s
p
O
(
ε
N
p
-
1
)
)
N
-
s
p
N
|
=
O
(
ε
N
p
-
1
)
.
Then adding and subtracting (1+Sp,s-NspO(εNp-1))N-spNSp,sNsp, we have
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
≤
S
p
,
s
+
ε
s
p
[
O
(
ε
N
p
-
1
-
s
p
)
+
O
(
ε
N
-
s
p
p
-
1
-
s
p
)
+
(
c
4
+
a
c
2
)
ε
(
N
p
-
s
p
)
|
log
(
ε
δ
)
|
-
a
c
2
c
0
]
(
S
p
,
s
N
s
p
+
O
(
ε
N
p
-
1
)
)
N
-
s
p
N
,
from which
(5.21)
∥
e
ε
,
δ
∥
X
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
<
S
p
,
s
follows if ε>0 is small enough. Therefore,
s
N
(
1
b
)
N
-
s
p
s
p
(
∥
e
ε
,
δ
∥
X
p
s
p
-
a
∥
e
ε
,
δ
∥
L
p
(
Ω
)
p
∥
u
ε
,
δ
∥
L
p
s
*
(
Ω
)
p
)
N
s
p
<
s
N
(
1
b
)
N
-
s
p
s
p
S
p
,
s
N
s
p
.
The boundedness of Qε,R1,R2 and Lemma 5.17 guarantee that (5.20) is true for ε>0 and λ>0 small enough.
Case 2:
N
>
s
p
2
and 1<p<2NN+s or N>sp((p-1)2+p) and p>2NN+s. The proofs are analogous to that of case 1. The conclusion follows once inequality (5.21) is obtained. ∎
