1 Introduction
The results we present in this paper are concerned with existence or nonexistence of nontrivial solutions for Dirichlet problems of the form
(1.1)
{
Δ
u
+
f
(
u
)
=
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a bounded domain of ℝn, n≥3 and f has supercritical growth from the viewpoint of the Sobolev embedding.
Let us consider, for example, the case where f(t)=|t|p-2t for all t∈ℝ (this function obviously satisfies condition (2.2) we use in this paper). In this case, a well-known nonexistence result of Pohozaev (see [28]) says that the Dirichlet problem
(1.2)
{
Δ
u
+
|
u
|
p
-
2
u
=
0
in
Ω
,
u
=
0
on
∂
Ω
,
has only the trivial solution u≡0 when Ω is star-shaped and p≥2nn-2 (the critical Sobolev exponent).
On the other hand, if Ω is an annulus, it is easy to find infinitely many radial solutions for all p>2 (as pointed out by Kazdan and Werner in [9]). Thus, it is natural to ask whether or not the nonexistence result of Pohozaev can be extended to non-star-shaped domains and the existence result in the annulus can be extended, for example, to all noncontractible domains of ℝn.
Answering some stimulating questions pointed out by Brezis, Nirenberg, Rabinowitz, etc. (see [2, 3]), many results have been obtained, relating nonexistence, existence and multiplicity of nontrivial solutions to the shape of Ω (see [1, 4, 5, 7, 8, 6, 17, 18, 21, 23, 20, 19, 25, 26, 27, 14, 13, 11, 10, 29, 30], etc.).
In the present paper our aim is to show that, even if the Pohozaev nonexistence result cannot be extended to all the contractible domains of ℝn, one can prove that there exist contractible non-star-shaped domains Ω, which may be very different from the star-shaped ones and even arbitrarily close to noncontractible domains, such that the Dirichlet problem (1.2) has only the trivial solution u≡0 for all p>2nn-2.
In order to construct such domains, we use suitable Pohozaev-type integral identities in tubular domains Ω=Tε(Γk) with thickness ε>0 and center Γk, where Γk is a k-dimensional, compact, smooth submanifold of ℝn.
If k=1, Γk is contractible in itself and p>2nn-2, we prove that there exists ε¯>0 such that, for all ε∈(0,ε¯), the Dirichlet problem (1.2) with Ω=Tε(Γk) does not have any nontrivial solution (this nonexistence result follows, as a particular case, from Theorem 2.2).
Let us point out that, if k=1 but Γk is noncontractible in itself or if k>1, a nonexistence result analogous to Theorem 2.2 cannot hold under the assumption p>2nn-2. In fact, the method we use in Theorem 2.2 fails when k=1 and Γk is noncontractible because the multipliers to be used in the Pohozaev-type integral identity are not well defined. Using other multipliers, we obtain a weaker nonexistence result which holds only when n≥4 and p>2(n-1)n-3 (it follows from Theorem 2.5). On the other hand, this weaker result is sharp because, if Γk is for example a circle of radius R (that is, Tε(Γk) is a solid torus), one can easily obtain infinitely many solutions for all ε∈(0,R) when n=3 and p>2 or n≥4 and p∈(2,2(n-1)n-3).
Propositions 3.2, 3.3 and 3.4 give examples of existence and multiplicity results of positive and sign changing solutions for some p>2nn-2 in tubular domains Tε(Γk) with k≥2 and Γk contractible in itself. These examples explain why Theorem 2.2 cannot be extended to the case k>1 under the assumption p>2nn-2.
However, in the case k>1, with Γk contractible or not, we prove a weaker nonexistence result (given by Theorem 3.5) which holds only when n>k+2 and p>2(n-k)n-k-2.
Some existence and multiplicity results, when n≤k+2 and p>2 or n>k+2 and 2<p<2(n-k)n-k-2, in tubular domains Tε(Γk) with k≥2 and ε non-necessarily small, show that also the nonexistence result given by Theorem 3.5 is sharp.
Finally, let us point out that if in the equation Δu+f(u)=0 we replace the Laplace operator Δu by the operator div(|Du|q-2Du) with 1<q<2, then critical and supercritical nonlinearities arise also for n=2 and produce analogous nonexistence results (see [16, 15]). These results suggest that if n=2, 1<q<2 and p>2q2-q, the Pohozaev nonexistence result for star-shaped domains can be extended to all the contractible domains of ℝ2. On the contrary, this extension is not possible, for example, if n≥3, q=2 and p≥2nn-2 because of Propositions 3.2, 3.3 and 3.4 (see Remark 3.7).
2 Integral Identities and Nonexistence Results
In order to obtain nonexistence results for nontrivial solutions of problem (1.1), we use the Pohozaev-type integral identity given in the following lemma.
Lemma 2.1.
Let Ω be a piecewise smooth bounded domain of Rn, n≥3, let v=(v1,…,vn)∈C1(Ω¯,Rn) be a vector field in Ω¯ and let f be a continuous function in R. Then every solution of problem (1.1) satisfies the following integral identity:
(2.1)
1
2
∫
∂
Ω
|
D
u
|
2
v
⋅
𝐧
𝑑
σ
=
∫
Ω
𝑑
v
[
D
u
]
⋅
D
u
𝑑
x
+
∫
Ω
div
v
(
F
(
u
)
-
1
2
|
D
u
|
2
)
d
x
,
where n denotes the outward normal to ∂Ω,
d
v
[
ξ
]
=
∑
i
=
1
n
D
i
v
ξ
i
for all
ξ
=
(
ξ
1
,
…
,
ξ
n
)
∈
ℝ
n
and
F
(
t
)
=
∫
0
t
f
(
τ
)
𝑑
τ
for all
t
∈
ℝ
.
For the proof it is sufficient to apply the Gauss–Green formula to the function v⋅DuDu and argue as in [28]. Notice that the Pohozaev identity is obtained for v(x)=x.
Now, our aim is to find suitable domains Ω and vector fields v∈𝒞1(Ω¯,ℝn) such that identity (2.1) can be satisfied only by the trivial solution of problem (1.1). In order to construct Ω and v with this property, let us consider a curve γ∈𝒞3([a,b],ℝn) such that γ′(t)≠0 for all t∈[a,b] and γ(t1)≠γ(t2) if t1≠t2, t1,t2∈[a,b]. For all t∈[a,b] and r>0, let us set N(t)={ξ∈ℝn:ξ⋅γ′(t)=0} and Nr(t)={ξ∈N(t):|ξ|<r}. Notice that there exists ε¯1>0 such that, for all ε∈(0,ε¯1],
[
γ
(
t
1
)
+
N
¯
ε
(
t
1
)
]
∩
[
γ
(
t
2
)
+
N
¯
ε
(
t
2
)
]
=
∅
if
t
1
≠
t
2
,
t
1
,
t
2
∈
[
a
,
b
]
.
For all ε∈(0,ε¯1) let us consider the open, piecewise smooth, bounded domain Tεγ defined by
T
ε
γ
=
⋃
t
∈
(
a
,
b
)
[
γ
(
t
)
+
N
ε
(
t
)
]
.
Then the following nonexistence result holds for the nontrivial solutions in the domain Ω=Tεγ.
Theorem 2.2.
Assume that the continuous function f satisfies the condition
(2.2)
t
f
(
t
)
≥
p
∫
0
t
f
(
τ
)
𝑑
τ
≥
0
for all
t
∈
ℝ
for a suitable p>2nn-2. Then there exists ε¯>0 such that for all ε∈(0,ε¯) the Dirichlet problem (1.1) has only the trivial solution u≡0 in the domain Ω=Tεγ.
It is clear that condition (2.2) implies f(0)=0, so the function u≡0 in Tεγ is a trivial solution for all ε∈(0,ε¯1). In order to prove that it is the unique solution for ε small enough, we need some preliminary results.
Notice that if ε∈(0,ε¯1), the following property holds: for all x∈Tεγ there exists a unique t(x)∈(a,b) such that dist(x,Γ)=|x-γ(t(x))|, where
Γ
=
{
γ
(
t
)
:
t
∈
[
a
,
b
]
}
.
If we set ξ(x)=x-γ(t(x)), we have ξ(x)⋅γ′(t(x))=0 for all x∈Tεγ. Therefore, for all x∈Tεγ there exists a unique pair (t(x),ξ(x)) such that t(x)∈[a,b], ξ(x)∈Nε(t(x)) and x=γ(t(x))+ξ(x).
Without any loss of generality, we can assume in addition that a≤0≤b and |γ′(t)|=1 for all t∈[a,b].
For all ξ∈Nε(0) let us consider the function τ↦x(ξ,τ) which solves the Cauchy problem
{
∂
x
∂
τ
=
γ
′
(
t
(
x
)
)
,
x
(
ξ
,
0
)
=
γ
(
0
)
+
ξ
.
Note that dist(x(ξ,τ),Γ)=|ξ| for all τ∈[a,b]. Moreover, for all ξ∈Nε(0), the function given by τ↦t(x(ξ,τ)) is increasing. As a consequence, we can consider the inverse function given by t↦τ(ξ,t) which satisfies t(x(ξ,τ(ξ,t)))=t for all t∈[a,b].
Notice that τ(ξ,0)=0 for all ξ∈Nε(0) because t(x(ξ,0))=0. For all ξ∈Nε(0), let us set
ψ
(
ξ
,
t
)
=
x
(
ξ
,
τ
(
ξ
,
t
)
)
-
γ
(
t
)
.
Then ψ(ξ,t)∈Nε(t) and |ψ(ξ,t)|=|ξ| for all ξ∈Nε(0) and for all t∈[a,b]. Moreover, for all x∈Tεγ there exists a unique ξ∈Nε(0) such that ξ(x)=ψ(ξ,t(x)) and the function ξ↦ψ(ξ,t) is a one-to-one function between Nε(0) and Nε(t), satisfying |ψ(ξ1,t)-ψ(ξ2,t)|=|ξ1-ξ2| for all ξ1,ξ2∈Nε(0), for all t∈[a,b].
Now, let us consider the vector field v defined by
(2.3)
v
(
γ
(
t
)
+
ψ
(
ξ
,
t
)
)
=
t
γ
′
(
t
)
[
1
-
ψ
(
ξ
,
t
)
⋅
γ
′′
(
t
)
]
+
ψ
(
ξ
,
t
)
for all
t
∈
(
a
,
b
)
and all
ξ
∈
N
ε
(
0
)
.
Since γ∈𝒞3([a,b],ℝn), we have v∈𝒞1(T¯εγ,ℝn), so the integral identity (2.1) holds.
In the following lemma we establish some properties of the vector field v.
Lemma 2.3.
In the domain T¯ε¯1γ, let us consider the vector field v∈C1(T¯ε¯1γ,Rn) defined in (2.3). Then we have:
v
⋅
𝐧
>
0
on
∂
T
¯
ε
γ
for all
ε
∈
(
0
,
ε
¯
1
)
,
lim
ε
→
0
sup
{
|
n
-
div
v
(
x
)
|
:
x
∈
T
ε
γ
}
=
0
,
lim
ε
→
0
sup
{
|
1
-
d
v
(
x
)
[
η
]
⋅
η
|
:
x
∈
T
ε
γ
,
η
∈
ℝ
n
,
|
η
|
=
1
}
=
0
.
Proof.
Taking into account the choice of ε¯1, since we are assuming |γ′(t)|=1 for all t∈[a,b], we have [1-ψ(ξ,t)⋅γ′′(t)]≥0 for all t∈[a,b]. Therefore, since we are also assuming a≤0≤b, property (a) is a direct consequence of the definition of Tεγ and v.
In order to prove (b), notice that, since v∈𝒞1(T¯εγ,ℝn) for all ε∈(0,ε¯1), there exist tε∈[a,b] and ξε∈Nε(0)¯ such that
|
n
-
div
v
(
γ
(
t
ε
)
+
ψ
(
ξ
ε
,
t
ε
)
)
|
=
max
{
|
n
-
div
v
(
x
)
|
:
x
∈
T
¯
ε
γ
}
for all
ε
∈
(
0
,
ε
¯
1
)
.
When ε→0, we obtain (up to a subsequence) tε→t0 for a suitable t0∈[a,b] while ξε→0 (because |ξε|≤ε) and, as a consequence, also ψ(ξε,tε)→0 (because |ψ(ξε,tε)|=|ξε|). Therefore, we get
lim
ε
→
0
max
{
|
n
-
div
v
(
x
)
|
:
x
∈
T
¯
ε
γ
}
=
|
n
-
div
v
(
γ
(
t
0
)
)
|
.
Now, notice that
d
v
(
γ
(
t
0
)
)
[
γ
′
(
t
0
)
]
=
γ
′
(
t
0
)
+
t
0
γ
′′
(
t
0
)
and
d
v
(
γ
(
t
0
)
)
[
ψ
]
=
-
t
0
[
ψ
⋅
γ
′′
(
t
0
)
]
γ
′
(
t
0
)
+
ψ
for all
ψ
∈
N
(
t
0
)
.
It follows that divv(γ(t0))=n, so property (b) holds.
In a similar way we can prove property (c). In fact, since v∈𝒞1(T¯εγ,ℝn) for all ε∈(0,ε¯1), there exist t¯ε∈[a,b], ξ¯ε∈Nε(0)¯ and η¯ε∈ℝn such that |η¯ε|=1 and
|
1
-
d
v
(
γ
(
t
¯
ε
)
+
ψ
(
ξ
¯
ε
,
t
¯
ε
)
)
[
η
¯
ε
]
⋅
η
¯
ε
|
=
max
{
|
1
-
d
v
(
x
)
[
η
]
⋅
η
|
:
x
∈
T
¯
ε
γ
,
η
∈
ℝ
n
,
|
η
|
=
1
}
.
Since |ψ(ξ¯ε,t¯ε)|=|ξ¯ε|≤ε for all ε∈(0,ε¯1), we have limε→0ψ(ξ¯ε,t¯ε)=0. Moreover, there exist t¯0∈[a,b] and η¯0∈ℝn such that (up to a subsequence) t¯ε→t¯0 and η¯ε→η¯0 as ε→0. It follows that
lim
ε
→
0
max
{
|
1
-
d
v
(
x
)
[
η
]
⋅
η
|
:
x
∈
T
¯
ε
γ
,
η
∈
ℝ
n
,
|
η
|
=
1
}
=
|
1
-
d
v
(
γ
(
t
¯
0
)
)
[
η
¯
0
]
⋅
η
¯
0
|
.
Now, let us set ψ¯0=η¯0-η¯0⋅γ′(t¯0)γ′(t¯0) and notice that ψ¯0∈N(t¯0). Therefore we have
d
v
(
γ
(
t
¯
0
)
)
[
ψ
¯
0
]
=
ψ
¯
0
-
t
¯
0
ψ
¯
0
⋅
γ
′′
(
t
¯
0
)
γ
′
(
t
¯
0
)
.
Thus, since
d
v
(
γ
(
t
¯
0
)
)
[
γ
′
(
t
¯
0
)
]
=
γ
′
(
t
¯
0
)
+
t
¯
0
γ
′′
(
t
¯
0
)
and γ′(t¯0)⋅γ′′(t¯0)=0, we obtain
d
v
(
γ
(
t
¯
0
)
)
[
η
¯
0
]
⋅
η
¯
0
=
d
v
(
γ
(
t
¯
0
)
)
[
η
¯
0
⋅
γ
′
(
t
¯
0
)
γ
′
(
t
¯
0
)
+
ψ
¯
0
]
⋅
(
η
¯
0
⋅
γ
′
(
t
¯
0
)
γ
′
(
t
¯
0
)
+
ψ
¯
0
)
=
{
η
¯
0
⋅
γ
′
(
t
¯
0
)
[
γ
′
(
t
¯
0
)
+
t
¯
0
γ
′′
(
t
¯
0
)
]
+
ψ
¯
0
-
t
¯
0
ψ
¯
0
⋅
γ
′′
(
t
¯
0
)
γ
′
(
t
¯
0
)
}
⋅
(
η
¯
0
⋅
γ
′
(
t
¯
0
)
γ
′
(
t
¯
0
)
+
ψ
¯
0
)
=
[
η
¯
0
⋅
γ
′
(
t
¯
0
)
]
2
+
|
ψ
¯
0
|
2
=
|
η
¯
0
|
2
=
1
,
which implies property (c). ∎
Corollary 2.4.
Let f and F be as in Lemma 2.1. Let Tεγ and v∈C1(T¯εγ,Rn) be as in Lemma 2.3. Then every solution uε of the Dirichlet problem (1.1) in Ω=Tεγ satisfies the inequality
0
≤
[
1
-
n
2
+
μ
(
ε
)
]
∫
T
ε
γ
|
D
u
ε
|
2
𝑑
x
+
∫
T
ε
γ
(
div
v
)
F
(
u
ε
)
𝑑
x
,
where μ(ε)→0 as ε→0.
The proof follows directly from Lemmas 2.1 and 2.3.
Proof of Theorem 2.2.
In order to prove that the trivial solution u≡0 in Tεγ is the unique solution for ε small enough, for every ε∈(0,ε¯1], let us consider a solution uε of problem (1.1) in Ω=Tεγ. Thus, taking into account Lemma 2.1 and condition (2.2), from Lemma 2.3 and Corollary 2.4 we obtain
0
≤
[
1
-
n
2
+
μ
(
ε
)
]
∫
T
ε
γ
|
D
u
ε
|
2
𝑑
x
+
[
n
+
μ
¯
(
ε
)
]
1
p
∫
T
ε
γ
u
ε
f
(
u
ε
)
𝑑
x
,
where μ¯(ε)→0 as ε→0. On the other hand, since uε is a solution of problem (1.1) in Ω=Tεγ, we have
∫
T
ε
γ
u
ε
f
(
u
ε
)
𝑑
x
=
∫
T
ε
γ
|
D
u
ε
|
2
𝑑
x
.
Therefore, we obtain
0
≤
[
1
-
n
2
+
n
p
+
μ
(
ε
)
+
μ
¯
(
ε
)
]
∫
T
ε
γ
|
D
u
ε
|
2
𝑑
x
.
Since 1-n2+np<0 for p>2nn-2, there exists ε¯∈(0,ε¯1) such that 1-n2+np+μ(ε)+μ¯(ε)<0 for all ε∈(0,ε¯). Therefore, for all ε∈(0,ε¯), we must have
∫
T
ε
γ
|
D
u
ε
|
2
𝑑
x
=
0
which implies uε≡0 in Tεγ and completes the proof. ∎
Taking into account the definition of the vector field v used in the proof of Theorem 2.2, one can verify by direct computations that this theorem still holds if the domain Ω=Tεγ is replaced by a more general domain Ω=Σεγ defined as follows:
Σ
ε
γ
=
{
γ
(
t
)
+
ψ
(
ξ
,
t
)
:
t
∈
[
a
,
b
]
,
ξ
∈
ε
Σ
}
,
where Σ is a domain of N(0), star-shaped with respect to the origin.
Notice that if in Theorem 2.2 we replace the vector field v defined in (2.3) by the vector field v~ defined as
v
~
(
γ
(
t
)
+
ψ
(
ξ
,
t
)
)
=
ψ
(
ξ
,
t
)
for all
t
∈
(
a
,
b
)
and all
ξ
∈
N
ε
(
0
)
¯
,
we obtain a nonexistence result for n≥4 and p>2(n-1)n-3 (the critical Sobolev exponent in dimension n-1, which is greater than 2nn-2).
Let us point out that the vector field v~ is well defined also when γ is a smooth circuit, that is, γ(a)=γ(b) and Ω is the interior of T¯εγ. Therefore, also in these domains we can prove nonexistence results for n≥4 and p>2(n-1)n-3 (see Theorem 2.5). On the contrary, in these domains the vector field v could not be well defined because
v
(
γ
(
a
)
+
ψ
(
ξ
,
a
)
)
≠
v
(
γ
(
b
)
+
ψ
(
ξ
,
b
)
)
for all
ξ
∈
N
ε
(
0
)
¯
,
while γ(a)+ψ(ξ,a)=γ(b)+ψ(ξ,b) when γ(a)=γ(b) and γ′(a)=γ′(b).
On the other hand, in these domains one cannot expect to obtain nonexistence results for p>2nn-2 since it is possible that there exist nontrivial solutions when n≥4 and 2nn-2<p<2(n-1)n-3 while they do not exist for p≥2(n-1)n-3, which happens for example in the case of a solid torus (see [22, 24, 11]).
In the next theorem we consider the case where Ω is a tubular domain near a circuit, n≥4 and condition (2.2) holds with p>2(n-1)n-3 (see Theorem 3.5 for an extension to more general tubular domains).
Theorem 2.5.
Assume that γ~:[a,b]→Rn is a smooth curve which satisfies γ~′(t)≠0, for all t∈[a,b], and γ~(a)=γ~(b), γ~′(a)=γ~′(b), γ~(t1)≠γ~(t2) if t1,t2∈(a,b) and t1≠t2. Let us set
Γ
~
=
{
γ
~
(
t
)
:
t
∈
[
a
,
b
]
}
𝑎𝑛𝑑
T
~
ε
(
Γ
~
)
=
{
x
∈
ℝ
n
:
dist
(
x
,
Γ
~
)
<
ε
}
for all
ε
>
0
.
Moreover, assume that n≥4 and condition (2.2) holds with p>2(n-1)n-3. Then there exists ε~>0 such that, for all ε∈(0,ε~), the Dirichlet problem (1.1) has only the trivial solution u≡0 in the smooth bounded domain Ω=T~ε(Γ~).
Proof.
First notice that there exists ε¯1>0 such that for all ε∈(0,ε¯1) and x∈T~ε(Γ~) there exists a unique y∈γ~ such that dist(x,Γ~)=|x-y|. Let us denote this y by p(x) and consider in T~ε(Γ~) the vector field v~ defined by v~(x)=x-p(x).
One can verify by direct computation that
d
v
~
(
γ
~
(
t
)
)
[
γ
~
′
(
t
)
]
=
0
,
d
v
~
(
γ
~
(
t
)
)
[
ψ
]
=
ψ
for all
t
∈
[
a
,
b
]
and all
ψ
∈
ℝ
n
such that
ψ
⋅
γ
~
′
(
t
)
=
0
and, as a consequence,
(2.4)
div
v
~
(
γ
~
(
t
)
)
=
n
-
1
for all
t
∈
[
a
,
b
]
,
(2.5)
d
v
~
(
γ
~
(
t
)
)
[
η
]
⋅
η
=
|
η
|
2
-
[
η
⋅
γ
~
′
(
t
)
]
2
|
γ
~
′
(
t
)
|
2
for all
t
∈
[
a
,
b
]
and all
η
∈
ℝ
n
.
It follows that
lim
ε
→
0
sup
{
|
n
-
1
-
div
v
~
(
x
)
|
:
x
∈
T
~
ε
(
Γ
~
)
}
=
0
as one can easily obtain from (2.4) arguing as in the proof of assertion (b) of Lemma 2.3. Moreover, from (2.5) we obtain
(2.6)
lim
ε
→
0
sup
{
d
v
~
(
x
)
[
η
]
⋅
η
:
x
∈
T
~
ε
(
Γ
~
)
,
η
∈
ℝ
n
,
|
η
|
=
1
}
=
1
.
In fact, for all ε∈(0,ε¯1), choose xε∈T~ε(Γ~) and ηε∈ℝn such that |ηε|=1 and sε-ε≤dv~(xε)[ηε]⋅ηε, where
s
ε
=
sup
{
d
v
~
(
x
)
[
η
]
⋅
η
:
x
∈
T
~
ε
(
Γ
~
)
,
η
∈
ℝ
n
,
|
η
|
=
1
}
.
Since dist(xε,Γ~)→0 as ε→0, and Γ~ is a compact manifold, from (2.5) we infer that lim supε→0sε≤1. On the other hand, (2.5) implies sε≥1 for all ε∈(0,ε¯1), so (2.6) is proved.
Furthermore, one can easily verify that v~⋅𝐧>0 on ∂T~ε(Γ~) for all ε∈(0,ε¯1). Thus, taking also into account condition (2.2), from Lemma 2.1 we infer that every solution u~ε of problem (1.1) in the domain T~ε(Γ~) satisfies
0
≤
[
1
-
n
-
1
2
+
μ
~
(
ε
)
]
∫
T
~
ε
(
Γ
~
)
|
D
u
~
ε
|
2
𝑑
x
+
[
n
-
1
p
+
μ
~
(
ε
)
]
∫
T
~
ε
(
Γ
~
)
u
~
ε
f
(
u
~
ε
)
𝑑
x
,
where μ~(ε)→0 as ε→0. Since
∫
T
~
ε
(
Γ
~
)
u
~
ε
f
(
u
~
ε
)
𝑑
x
=
∫
T
~
ε
(
Γ
~
)
|
D
u
~
ε
|
2
𝑑
x
(because u~ε solves problem (1.1) in T~ε(Γ~)), we obtain
(2.7)
0
≤
[
1
-
n
-
1
2
+
n
-
1
p
+
2
μ
~
(
ε
)
]
∫
T
~
ε
(
Γ
~
)
|
D
u
~
ε
|
2
𝑑
x
,
where 1-n-12+n-1p<0 because n≥4 and p>2(n-1)n-3. Therefore, there exists ε~∈(0,ε¯1) such that, for all ε∈(0,ε~), (2.7) implies u~ε≡0 in T~ε(Γ~). So the proof is complete. ∎
3 Tubular Domains of Higher Dimension and Final Remarks
The nonexistence results presented in Section 2 are concerned with domains Ω which are thin neighborhoods of a 1-dimensional manifold (with boundary and contractible in Theorem 2.2, without boundary and noncontractible in Theorem 2.5). In this section we consider the case where Ω is a thin neighborhood of a k-dimensional, smooth, compact manifold Γk with k>1.
If Γk is a submanifold of ℝn with n>k, for all x∈Γk we set N(x)=T⊥(x) and Nε(x)={x∈N(x):|x|<ε}, where T(x) is the tangent space to Γk in x and N(x) is the normal space. Since Γk is a compact smooth submanifold, there exists ε¯1>0 such that, for all ε∈(0,ε¯1], we have [x1+N¯ε(x1)]∩[x2+N¯ε(x2)]=∅ for all x1 and x2 in Γk such that x1≠x2. Then, for all ε∈(0,ε¯1), we consider the piecewise smooth, bounded domain Tε(Γk) defined as the interior of the set ⋃x∈Γk[x+Nε(x)] (we say that Tε(Γk) is the tubular domain with thickness ε and center Γk). Our aim is to study existence and nonexistence of nontrivial solutions of problem (1.1) in the domain Ω=Tε(Γk).
Let us point out that when k>1 the method used to prove Theorem 2.2 does not work. The reason is explained by the existence result holding in the contractible domains introduced in the following example.
Example 3.1.
For all n≥k+1, let us consider the function γk:ℝk→ℝn defined as follows:
γ
k
,
i
(
x
1
,
…
,
x
k
)
=
2
x
i
|
x
|
2
+
1
for
i
=
1
,
…
,
k
,
γ
k
,
k
+
1
(
x
1
,
…
,
x
k
)
=
|
x
|
2
-
1
|
x
|
2
+
1
γ
k
,
i
(
x
1
,
…
,
x
k
)
=
0
for
i
=
k
+
2
,
…
,
n
(γk is the stereographic projection of ℝk on a k-dimensional sphere of ℝn). Moreover, for all r>0, let us set Γkr={γk(x):x∈ℝk,|x|<r}.
Then one can easily verify that the domain Tε(Γkr) is contractible in itself for all r>0 and ε∈(0,1). Moreover, the following propositions hold (the detailed proofs are reported in a paper in preparation).
Proposition 3.2.
Let k≥2, n≥k+1 and ε∈(0,1). Assume that f(t)=|t|p-2t with p>2nn-2 and that p<2(n-k+1)n-k-1 if n>k+1. Then there exists r¯>0 such that, if r>r¯, problem (1.1) in the domain Ω=Tε(Γkr) has positive and sign changing solutions; moreover, for all ε∈(0,1) the number of solutions tend to infinity as r→∞.
For the proof it suffices to look for solutions having radial symmetry with respect to the first k variables and argue as in [18, 27, 19, 26, 23, 13, 14, 12].
Proposition 3.3.
Let k≥2, n≥k+1, r>1, ε∈(0,1). Moreover, assume that f(t)=|t|p-2t for all t∈R. Then there exists p¯>2nn-2 such that, if n=k+1 and p≥p¯ or if n>k+1 and p∈[p¯,2(n-k+1)n-k-1), problem (1.1) with Ω=Tε(Γkr) has nontrivial solutions.
The proof can be carried out arguing for example as in [14] in order to obtain solutions having radial symmetry with respect to the first k variables.
Proposition 3.4.
Let k≥2, n≥k+1, r>1, ε∈(0,1) and assume that f(t)=|t|p-2t for all t∈R. Then there exists p~>2nn-2 such that problem (1.1) with Ω=Tε(Γkr) has positive solutions for all p∈(2nn-2,p~). Moreover, the number of solutions tends to infinity as p→2nn-2.
The proof of the proposition is based on a Lyapunov–Schmidt-type finite-dimensional reduction method as in [11, 12], etc.
Notice that, if r<1, the domain Tε(Γkr) is star-shaped for ε close to 1, so problem (1.1) has only the trivial solution u≡0 for all p≥2nn-2.
Thus, while Theorem 2.2 gives a nonexistence result for all p>2nn-2 when k=1, Γk is contractible in itself and Ω is a thin tubular domain centered in Γk, Propositions 3.2, 3.3 and 3.4 give examples of existence results for p>2nn-2 when Ω is a tubular domain centered in a suitable k-dimensional manifold Γkr, contractible in itself but with k≥2. Let us point out that, if p>2nn-2 and Ω=Tε(Γkr), arguing as in the proof of Theorem 2.2 one can prove that for k=1 there exists ε¯∈(0,1) such that the Dirichlet problem has only the trivial solution u≡0 for all ε∈(0,ε¯) and r>0. On the contrary, Proposition 3.2 guarantees that for all k≥2, p>2nn-2 and ε∈(0,1), there exist many nontrivial solutions when r is large enough. So, if p>2nn-2, for ε∈(0,ε¯) and r>0 large enough, in the domain Ω=Tε(Γkr) we have existence of nontrivial solutions for k≥2 and nonexistence for k=1. In this sense we mean that the method used to prove Theorem 2.2 cannot work in the case k≥2 (see also Remark 3.6 for more details about the differences between the cases k=1 and k>1).
However, notice that a weaker nonexistence result holds for all k≥1 (even if Γk is noncontractible in itself) when n>k+2 and p>2(n-k)n-k-2, as we prove in the following Theorem 3.5.
If n≤k+2 and p>2 or n>k+2 and 2<p<2(n-k)n-k-2, the existence of nontrivial solutions can be proved even if Ω is a tubular domain Tε(Γk) with ε not necessarily small: for example, if Γk is a k-dimensional sphere, we can look for solutions with radial symmetry with respect to k+1 variables, so we obtain infinitely many solutions for all ε∈(0,R), where R is the radius of the sphere.
Theorem 3.5.
Let k≥1, n>k+2 and assume that Γk is a k-dimensional, compact, smooth submanifold of Rn. Moreover, assume that condition (2.2) holds with p>2(n-k)n-k-2. Then there exists ε¯>0 such that, for all ε∈(0,ε¯), the Dirichlet problem (1.1) has only the trivial solution u≡0 in the tubular domain Ω=Tε(Γk).
Proof.
Taking into account the definition of the tubular domain Tε(Γk), for all ε∈(0,ε¯1) and x∈Tε(Γk) there exists a unique y∈Γk such that x∈y+Nε(y). Then denote this y by pk(x) and set vk(x)=x-pk(x) for all x∈Tε(Γk). One can easily verify that the vector field vk satisfies vk⋅𝐧≥0 on ∂Tε(Γk) for all ε∈(0,ε¯1).
Therefore, from Lemma 2.1 we infer that every solution uε of problem (1.1) in Tε(Γk) satisfies
0
≤
∫
T
ε
(
Γ
k
)
𝑑
v
k
[
D
u
ε
]
⋅
D
u
ε
𝑑
x
+
∫
T
ε
(
Γ
k
)
div
v
k
(
F
(
u
ε
)
-
1
2
|
D
u
ε
|
2
)
d
x
.
Notice that
d
v
k
(
x
)
[
ϕ
]
=
0
,
d
v
k
(
x
)
[
ψ
]
=
ψ
for all
x
∈
Γ
k
, all
ϕ
∈
T
(
x
)
and all
ψ
∈
N
(
x
)
as one can verify by direct computation.
As a consequence, we obtain
div
v
k
(
x
)
=
n
-
k
,
d
v
k
(
x
)
[
ϕ
+
ψ
]
⋅
(
ϕ
+
ψ
)
=
|
ψ
|
2
for all
x
∈
Γ
k
, all
ϕ
∈
T
(
x
)
and all
ψ
∈
N
(
x
)
.
Since Γk is a compact manifold, it follows that
lim
ε
→
0
sup
{
|
n
-
k
-
div
v
k
(
x
)
|
:
x
∈
T
ε
(
Γ
k
)
}
=
0
and
lim
ε
→
0
sup
{
d
v
k
(
x
)
[
η
]
⋅
η
:
x
∈
T
ε
(
Γ
k
)
,
η
∈
ℝ
n
,
|
η
|
=
1
}
=
1
as one can infer arguing as in the proof of Theorem 2.5.
Thus, taking also into account that
∫
T
ε
(
Γ
k
)
u
ε
f
(
u
ε
)
𝑑
x
=
∫
T
ε
(
Γ
k
)
|
D
u
ε
|
2
𝑑
x
,
from condition (2.2) we infer that
0
≤
[
1
-
n
-
k
2
+
n
-
k
p
+
μ
k
(
ε
)
]
∫
T
ε
(
Γ
k
)
|
D
u
ε
|
2
𝑑
x
,
where μk(ε)→0 as ε→0. Since 1-n-k2+n-kp<0 (because n>k+2 and p>2(n-k)n-k-2), it follows that there exists ε¯∈(0,ε¯1) such that for all ε∈(0,ε¯) we have uε≡0 in Tε(Γk), so the problem has only the trivial solution u≡0. ∎
Remark 3.6.
Proposition 3.2 as well as the results reported in [18, 27, 19, 26, 23, 13, 14, 12] suggest that the existence of nontrivial solutions is related to the property that the domain Ω is obtained by removing a subset of small capacity from a domain having a different k-dimensional homology group with k≥2.
For example, in the case of domains with small holes, every hole has small capacity and changes the (n-1)-dimensional homology group.
In the case of tubular domains Tε(Γkr), the existence results for k≥2 and r large enough given by Proposition 3.2 are related to the fact that Γkr tends to a k-dimensional sphere Sk as r→∞, the capacity of Tε(Sk)∖Tε(Γkr) tends to 0 as r→∞ and the domains Tε(Sk) and Tε(Γkr) have different k-dimensional homology group.
On the contrary, when k=1, the capacity of Tε(S1)∖Tε(Γ1r) does not tend to 0 as r→∞. This fact explains the nonexistence result in the case of the domains Tε(Γ1r), when ε is small enough, for all r>0.
Remark 3.7.
If n=2, we do not have critical or supercritical phenomena for the Laplace operator. But, if we replace it by the q-Laplace operator, these phenomena arise and may produce nonexistence results for nontrivial solutions. For example, if we consider the Dirichlet problem
(3.1)
{
div
(
|
D
u
|
q
-
2
D
u
)
+
|
u
|
p
-
2
u
=
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a bounded domain of ℝ2, 1<q<2, p≥2q2-q, then one can prove nonexistence results in some bounded contractible domains which can be non-star-shaped and even arbitrarily close to noncontractible domains (see [16, 15]). For example, if Ω=Tε(Γ1r), there exists ε¯>0 such that problem (3.1) has only the trivial solution u≡0 for all r>0 and ε∈(0,ε¯).
The results obtained in [16, 15] suggest that the nonexistence of nontrivial solutions for Dirichlet problem (3.1) might be proved in all the contractible domains of ℝ2 (while it is not possible for problem (1.2) when n≥3 and p≥2nn-2 because of Proposition 3.2).
