1 Introduction
This work is devoted to the analysis of positive solutions to the semilinear boundary value problem
(1.1)
{
-
Δ
u
=
λ
u
q
(
x
)
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
,
where Ω⊂ℝN is a bounded smooth domain and λ>0 is a bifurcation parameter. The exponent q(x) is assumed to be a positive function q∈Cα(Ω¯), 0<α<1, such that
q
(
x
)
>
1
,
x
∈
Ω
.
Thus, the reaction term in problem (1.1) exhibits a variable order and a convex profile (since q is greater than one in Ω). However, we are also assuming that q achieves the value one somewhere on the boundary. Specifically, and to simplify the exposition, we will assume that
(1.2)
q
(
x
)
=
1
,
x
∈
∂
Ω
.
As will be seen later, this behavior of q turns out to be critical with respect to several technical points (for instance, the applicability of the method of sub- and supersolutions, checking Palais–Smale-type conditions, the moving planes method or Pohozaev-type relations). More importantly, the standard rescaling technique in [14] can not be employed to obtain a priori estimates for (1.1). Indeed, a critical case arises when the points where a possible blowing-up sequence of solutions attains its maximum, accumulate at a point x0 on the boundary ∂Ω where q(x0)=1 (see [13]).
The study of problem (1.1) under the limiting case (1.2) of the exponent q remained open in [13] where q was allowed to take values both greater and smaller than one, but q was restricted to satisfy q(x)>1 in those common components (if any) of ∂Ω and ∂Ω+, where Ω+={x∈Ω:q(x)>1}. In addition, the growth of q was constrained in [13] to fulfill q(x)<NN-2. We are also relaxing this seemingly “technical” restriction to permit the more natural subcritical growth
(1.3)
q
(
x
)
<
N
+
2
N
-
2
.
In fact, we are assuming henceforth that N≥3. An analysis similar to the one developed here can be used to handle the case N=2 without further restriction on the size of q.
The subject of reaction-diffusion equations with constant-order reactions has been widely studied (see [9, 17, 26, 22, 21], to quote some few standards on the topic). However, the variable exponent case is not yet completely understood. We refer to the pioneering work [18] dealing with a sublinear population dynamics model, [7, 19] on diffusion through a porous medium, [12] extending the results in [18] and the already mentioned [13] on a variable exponent problem of concave-convex nature. In addition, [11, 10] address the existence of large solutions for two different kinds of variable exponent nonlinearities. Finally, see [25] for an updated review on problems in the spirit of [13]. This is just a minimal sample of references rather than an exhaustive account on the subject.
It should be remarked that problem (1.1) under the more restrictive assumption q(x)≥q0>1 in Ω¯ was analyzed in [19, Theorem 2.1]. The present paper generalizes in several aspects this result. In particular, it covers the case q=1 on ∂Ω with a linear decay (see further comments in Section 2).
Our main result is the following.
Theorem 1.1.
Let Ω⊂RN be a bounded smooth domain, and let q∈Cα(Ω¯) satisfy q(x)>1 in Ω, q≡1 on ∂Ω and the growth condition (1.3). Assume moreover that there exist a small η>0 and a constant C0>0 such that
(1.4)
q
(
x
)
≥
1
+
C
0
d
(
x
)
if
d
(
x
)
<
η
,
where d(x)=dist(x,∂Ω). Then the boundary value problem
(1.5)
{
-
Δ
u
=
λ
|
u
|
q
(
x
)
-
1
u
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
,
exhibits the following features:
For each
λ
>
0
, problem (
1.5
) possesses a positive solution
u
λ
∈
C
2
,
β
(
Ω
¯
)
. Moreover, this positive solution is linearly unstable.
If
Ω
=
B
is an open ball and
q
is a radially symmetric function, then for every
λ
>
0
the positive solution given by (i) can be chosen to be radially symmetric.
Any family
u
λ
of positive solutions satisfies
∥
u
λ
∥
∞
→
∞
as
λ
→
0
+
. Moreover,
(1.6)
lim
¯
λ
→
0
+
λ
1
q
+
-
1
∥
u
λ
∥
∞
>
0
,
where
q
+
=
max
Ω
q
.
Let
u
λ
be either of the families of positive solutions to (
1.5
) introduced in (i) and (ii). Then
lim
λ
→
∞
∥
u
λ
∥
C
2
,
β
(
Ω
¯
)
=
0
.
Moreover,
∥
u
λ
∥
C
2
,
β
(
Ω
¯
)
decays exponentially to zero as
λ
→
∞
. More precisely, there exist
C
1
,
C
2
,
λ
0
>
0
such that
∥
u
λ
∥
C
2
,
β
(
Ω
¯
)
≤
C
1
e
-
C
2
λ
3
for
λ
≥
λ
0
.
Remark 1.2.
(a) When q(x)=q+>1 is a constant, a scaling argument shows that any family uλ of nontrivial solutions can be written as
(1.7)
u
λ
=
λ
-
1
q
+
-
1
u
1
,
u
1
being a solution corresponding to λ=1. Thus the limit in (1.6) achieves a finite value in this case. This may be false when dealing with variable exponents (see Remark 3.1).
(b) Relation (1.7) shows that in the case q(x)=q+ all families of positive solutions uλ to (1.5) decay to zero as a negative power of λ as λ→∞. This has to be contrasted with the variable exponent case where the convergence to zero is exponential, as shown in Theorem 1.1 (iv) (see Section 4 for details).
The rest of the paper is organized as follows: Section 2 introduces the proper variational tools required to handle our problem, i.e. in the critical framework where q=1 on ∂Ω (here the results in [16] deserve a special mention). The proof of Theorem 1.1 is contained in Section 3, while some key uniform estimates are postponed to Section 4.
2 Background Results
By a solution to problem (1.5) we understand a function u∈H01(Ω) solving (1.5) in the weak sense. Since the growth condition (1.3) leads to the estimate
(2.1)
|
u
|
q
(
x
)
≤
C
(
1
+
|
u
|
q
+
)
,
which holds for every u∈ℝ, x∈Ω¯, with q+=maxΩ¯q satisfying
q
+
<
N
+
2
N
-
2
,
every weak solution u∈H01(Ω) is indeed a classical solution (see [13, 27]). Moreover, since q∈Cα(Ω¯), this solution lies in C2,β(Ω¯) for some β∈(0,1) (see [15]).
Solutions u∈H01(Ω) to (1.5) are characterized as the critical points of the functional
(2.2)
J
λ
(
u
)
=
1
2
∫
Ω
|
∇
u
|
2
𝑑
x
-
λ
𝒫
(
u
)
=
1
2
∥
u
∥
H
0
1
(
Ω
)
2
-
λ
𝒫
(
u
)
,
with
(2.3)
𝒫
(
u
)
=
∫
Ω
|
u
|
q
(
x
)
+
1
q
(
x
)
+
1
𝑑
x
,
𝒫
referring to “potential”. Here we are following a variational approach to get the existence of nontrivial solutions to (1.5).
At this point, it is worth to mention that it is not possible to find a positive solution u to (1.5) by the method of sub- and supersolutions (as in the case where q is constant). Otherwise this approach would imply the existence of a minimal positive solution u+∈H01(Ω) satisfying
0
<
u
+
≤
u
.
However, if σ=σ1(-Δ-λqu+q-1) stands for the first eigenvalue of the linearization of problem (1.5) at u+, i.e.,
(2.4)
{
-
Δ
u
-
λ
q
u
+
q
-
1
u
=
σ
u
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
,
then it is well known that σ1(-Δ-λqu+q-1)≥0. We refer to [13, 1] for a proof of this fact, and to [13] for further properties of the eigenvalue problem (2.4).
On the other hand, that u+ solves (1.5) means that the principal eigenvalue σ1(-Δ-λu+q-1) equals zero (here and above σ1(-Δ+V(x)) stands for the first eigenvalue of the operator -Δ+V(x) in H01(Ω)). Since
(2.5)
σ
1
(
-
Δ
-
λ
q
u
+
q
-
1
)
<
σ
1
(
-
Δ
-
λ
u
+
q
-
1
)
=
0
,
we get a contradiction. So, the sub- and supersolutions method does not provide positive solutions to (1.5).
Let us check now some variational properties of Jλ. First, Jλ:H01(Ω)→ℝ is a C1 functional. Moreover, 𝒫:H01(Ω)→ℝ is C1 and its derivative
D
𝒫
:
H
0
1
(
Ω
)
→
H
-
1
(
Ω
)
(H-1(Ω) stands for the dual space of H01(Ω)) defines a completely continuous operator. In fact, for η>0 satisfying
q
+
≤
2
*
-
η
2
*
(
2
*
-
1
)
,
2
*
=
2
N
N
-
2
,
the Nemytskii operator 𝒩:H01(Ω)→L2*-ηq++1(Ω), given by
𝒩
(
u
)
=
|
u
|
q
(
x
)
+
1
q
(
x
)
+
1
,
and its derivative
D
𝒩
(
u
)
:
H
0
1
(
Ω
)
→
L
2
*
-
η
q
+
(
Ω
)
,
D
𝒩
(
u
)
v
=
|
u
|
q
(
x
)
-
1
u
v
,
are completely continuous operators. Finally,
D
𝒫
(
u
)
v
=
∫
Ω
D
𝒩
(
u
)
v
𝑑
x
.
To deal with the existence of nontrivial solutions of (1.5) some further ingredients of critical point theory are necessary. Let X be a Banach space and J:X→ℝ a C1 functional. It is said that J exhibits a Mountain Pass (MP for short) geometry near u=0 if there exist R,η>0 and ψ∈X∖{0} with ∥ψ∥X>R satisfying
(2.6)
J
(
u
)
≥
η
>
max
{
J
(
0
)
,
J
(
ψ
)
}
for all u∈X, ∥u∥X=R. In this case, the number
c
=
inf
Γ
max
t
∈
[
0
,
1
]
J
∘
γ
(
t
)
,
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
0
,
γ
(
1
)
=
ψ
}
,
is called an MP level. On the other hand, a sequence {un}⊂X is a Palais–Smale (PS) sequence at level c0 if J(un)→c0 and for the sequence of derivatives DJ(un)→0 holds in X* (the dual space of X). Finally, J is said to verify the PS condition at level c0 if it is possible to extract a convergent subsequence from every PS sequence {un} at level c0. If this condition is satisfied regardless of the value of c0, we say that J satisfies the PS condition.
The MP theorem (see [2] for the original source and also [24, 27] for a general survey on the subject) asserts that every C1 functional J having an MP geometry near u=0 and satisfying the PS condition possesses a critical point u at the MP level c given by (2.6).
However, the MP theorem can not be directly applied to problem (1.5) due to the behavior (1.2) of q on the boundary. While it will be shown in Section 3 that the functional Jλ defined in (2.2) has an MP geometry near u=0, to check the PS condition is not an easy task. A weaker statement, whose proof is standard, and therefore omitted (we refer to [24, 27]), is the following.
Lemma 2.1.
Every bounded PS sequence {un}⊂H01(Ω) of Jλ admits a convergent subsequence in H01(Ω).
In the case where
(2.7)
q
(
x
)
≥
q
0
>
1
,
x
∈
Ω
¯
,
the so-called Ambrosetti–Rabinowitz relation (see [2])
(2.8)
1
q
(
x
)
+
1
|
u
|
q
(
x
)
+
1
≤
θ
|
u
|
q
(
x
)
+
1
,
u
∈
ℝ
,
holds for a certain θ∈[0,12). Based upon (2.8), it can be shown that every PS sequence is indeed a bounded PS sequence (BPS in the sequel); see [2, Lemma 3.6] and [24, 27]. Therefore, Jλ verifies the PS condition provided (2.7) is satisfied and problem (1.1) admits, for each λ>0, a positive solution under this restrictive condition on q. This argument provides an alternative proof of [19, Theorem 2.1].
However, (2.8) fails near ∂Ω in our case. To circumvent the problem we are employing an alternative approach using ideas from [16]. To this purpose consider a family Jλ:X→ℝ, λ∈I (where I is a real interval) of C1 functionals. We say that Jλ has an MP geometry at u=0 which is uniform with respect to λ∈I if R,η>0 and ψ∈X∖{0} can be chosen independent of λ∈I in the previous definition. In this case, the reference MP level will be designated as
(2.9)
c
λ
=
inf
Γ
max
J
λ
∘
γ
,
with
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
0
,
γ
(
1
)
=
ψ
}
.
The next result is a shortened version of [16, Theorem 1.1 and Corollary 1.2]. They are stated there under less restrictive hypotheses. However, we have narrowed the scope of the assertions to confine ourselves to the setting of problem (1.5).
Theorem 2.2 ([16]).
Let I be a real interval, and let Jλ:X→R, λ∈I, be a family of C1 functionals of the form
J
λ
(
u
)
=
A
(
u
)
-
λ
𝒫
(
u
)
,
where P(u)≥0 for all u∈X and A(u)→∞ as ∥u∥X→∞. Assume that Jλ has a uniform MP geometry at u=0 when λ∈I. Then the following features hold:
For almost all
λ
∈
I
there exists a BPS at level
c
λ
, with
c
λ
defined by (
2.9
).
Assume that both
𝒫
and
D
𝒫
are bounded on bounded sets of
X
, that any BPS sequence at the level
c
λ
admits a convergent subsequence in
X
, and let
λ
0
belong to the interior of
I
. Then there exist an increasing sequence
λ
n
with
λ
n
→
λ
0
, and
{
u
n
}
⊂
X
such that
J
λ
n
(
u
n
)
=
c
λ
n
,
c
λ
n
→
c
λ
0
,
D
J
λ
n
(
u
n
)
=
0
for all
n
. Moreover, provided that
u
n
is bounded, it becomes a PS sequence for
J
λ
0
at the level
c
λ
0
.
Theorem 2.2 is our main tool to establish the existence of nontrivial solutions to problem (1.5).
3 Proof of Theorem 1.1
We are first discussing the geometry of Jλ near u=0. A preliminary fact is
∫
Ω
1
q
(
x
)
+
1
|
u
|
q
(
x
)
+
1
𝑑
x
=
o
(
∥
u
∥
H
0
1
(
Ω
)
2
)
as
∥
u
∥
H
0
1
(
Ω
)
→
0
.
In fact, let un∈H01(Ω) be any sequence such that tn=∥un∥H01(Ω)→0. By setting un=tnvn, a subsequence vn′ of vn can be extracted such that (we write vn instead of vn′ in what follows) vn→v weakly in H01(Ω) and strongly in Lq++1(Ω). In addition, vn→v a.e. in Ω, while, by the results in [3], there exists h∈Lq++1(Ω) such that |vn(x)|≤h(x) a.e. in Ω. By using (2.1), we obtain
lim
n
→
∞
1
∥
u
n
∥
H
0
1
(
Ω
)
2
∫
Ω
1
q
(
x
)
+
1
|
u
n
|
q
(
x
)
+
1
𝑑
x
=
lim
n
→
∞
∫
Ω
t
n
q
(
x
)
-
1
q
(
x
)
+
1
|
v
n
|
q
(
x
)
+
1
𝑑
x
=
0
.
This proves the assertion. As a consequence, given any bounded interval I⊂ℝ+, there exist ε>0 and C>0, both independent of λ∈I, such that
(3.1)
J
λ
(
u
)
≥
C
∥
u
∥
H
0
1
(
Ω
)
2
for all u∈H01(Ω) satisfying ∥u∥H01(Ω)≤ε.
To check that the functional Jλ exhibits an MP geometry at u=0 which is “uniform” when λ varies in bounded intervals I⊂ℝ+, we look for a function ψ∈H01(Ω) so that
(3.2)
J
λ
(
ψ
)
<
0
for all λ∈I, where ψ does not depend on λ∈I. Set ψ=tϕ1, where ϕ1>0 is an eigenfunction associated to the first Dirichlet eigenvalue λ1 and t>0 is to be determined. Then
J
λ
(
t
ϕ
1
)
≤
∫
d
(
x
)
≤
d
0
(
B
(
t
ϕ
1
)
2
-
E
(
t
ϕ
1
)
q
(
x
)
+
1
)
𝑑
x
+
∫
d
(
x
)
>
d
0
(
B
(
t
ϕ
1
)
2
-
E
(
t
ϕ
1
)
q
(
x
)
+
1
)
𝑑
x
,
where d(x)=dist(x,∂Ω), d0>0 is a suitably small constant,
E
=
λ
q
+
+
1
and
B
=
λ
1
2
.
Now choose x¯∈Ω so that q(x¯)=q+=maxΩ¯q, and take a small δ with the property that
B
(
x
¯
,
δ
)
⊂
{
d
(
x
)
>
d
0
}
and
q
(
x
)
>
q
+
-
η
>
1
in B(x¯,δ). We have
(3.3)
J
λ
(
t
ϕ
1
)
≤
B
t
2
∫
d
(
x
)
≤
d
0
ϕ
1
2
𝑑
x
-
∫
B
(
x
¯
,
δ
)
(
E
(
t
ϕ
1
)
q
+
-
η
+
1
-
B
(
t
ϕ
1
)
2
)
𝑑
x
if t>0 is taken so large such that (tϕ1)q(x)-1≥BE in d(x)>d0. Since q+-η+1>2, it follows from (3.3) that a value of t0>0 can be found (independent of λ∈I) so that (3.2) holds for ψ=t0ϕ1.
Set now
A
(
u
)
=
1
2
∥
u
∥
H
0
1
(
Ω
)
2
and 𝒫(u) as defined in (2.3) and fix a bounded interval I⊂ℝ+. Then Theorem 2.2 (i) can be applied to show the existence, for almost all λ∈I, of a BPS sequence vn∈H01(Ω) of Jλ at the MP level cλ. Since Lemma 2.1 permits to extract a convergent subsequence vn′→v in H01(Ω) and cλ>0 (after a proper choice of 0<R≤ε in (3.1)), we get a nontrivial solution v to (1.5). Moreover, this argument proves the existence of a nontrivial solution to (1.5) for almost all λ>0.
Let us show next that existence actually holds for allλ>0. First, notice that (2.1) implies that both
N
(
u
)
=
|
u
|
q
(
x
)
q
(
x
)
+
1
and
D
N
(
u
)
=
|
u
|
q
(
x
)
-
1
u
remain bounded on bounded sets of H01(Ω), and so do both 𝒫 and D𝒫. On the other hand, Lemma 2.1 says that every BPS sequence of Jλ admits a convergent subsequence in H01(Ω). Thus, by applying Theorem 2.2 (ii) at a fixed λ0>0, we obtain sequences λn→λ0, λn increasing, and un∈H01(Ω) such that
J
λ
n
(
u
n
)
=
c
λ
n
,
D
J
λ
n
(
u
n
)
=
0
,
c
λ
n
→
c
λ
0
.
We will prove in Section 4 that such a sequence is bounded in H01(Ω) (see Lemma 4.2). Therefore, Theorem 2.2 (ii) allows us to conclude that un is a BPS sequence of Jλ0 at the level cλ0 and, as above, it furnishes a nontrivial solution u∈H01(Ω) to (1.5) satisfying Jλ0(u)=cλ0.
Observe also that the entire analysis of this section and the corresponding one in Section 4 can still be carried out if Jλ is replaced by the functional
J
λ
,
+
(
u
)
=
1
2
∥
u
∥
H
0
1
(
Ω
)
2
-
λ
∫
Ω
u
+
q
(
x
)
+
1
q
(
x
)
+
1
𝑑
x
,
where u+=max{u,0}. In fact, it can be checked that a nontrivial critical point u∈H01(Ω) of Jλ+ defines a positive solution to (1.5). On the other hand, relation (2.5) holds for u+ replaced by a positive solution u to (1.1). This permits us to conclude that such a solution u is linearly unstable. Thus the proof of (i) is completed.
Assertion (ii) is readily attained if the full analysis is performed in the subspace of H01(Ω) which consists of radially symmetric functions.
The statement in (iii) is a consequence of standard Lp estimates [15]. Indeed, if un:=uλn is a sequence of positive solutions with λn→0 and λntnq+-1→0 (here tn=∥un∥∞), then vn=un/tn solves -Δvn=λntnq(x)-1vn. Since the right-hand side converges to zero in L∞(Ω), a subsequence vn′ can be found so that vn′→0 in C1(Ω¯), which is not possible.
A proof of (iv) is postponed until the end of Section 4.
Remark 3.1.
The limit in (1.6) could be infinite. Indeed, let Ω=B be a ball centered at x=0 with radius R>0 and assume that q>1 is radially symmetric and such that {r:q(r)=q+} is a set of measure zero. Suppose that un:=uλn is a family of positive radial solutions with λn→0. As already shown, tn:=∥un∥∞=un(0)→∞. Setting un=tnvn, a straightforward computation yields the relation
t
n
=
λ
n
N
-
2
∫
0
R
s
(
1
-
(
s
R
)
N
-
2
)
v
n
(
s
)
q
(
s
)
t
n
q
(
s
)
𝑑
s
.
Thus,
1
=
λ
n
t
n
q
+
-
1
N
-
2
∫
0
R
s
(
1
-
(
s
R
)
N
-
2
)
v
n
(
s
)
q
(
s
)
t
n
-
(
q
+
-
q
(
s
)
)
𝑑
s
,
whence λntnq+-1→∞ since the integral goes to zero.
4 Uniform Bounds
This section is dedicated to show the a priori bounds required in the proof of Theorem 1.1 (i) and also the statement in Theorem 1.1 (iv). We will assume throughout that a “reference” sequence (λn,un)∈(0,+∞)×H01(Ω) satisfies the following hypothesis:
The sequence λn is increasing and λn→λ0 for some positive λ0, while Jλn(un)=cλn with
c
λ
n
=
inf
γ
∈
Γ
max
J
λ
n
∘
γ
,
where
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
H
0
1
(
Ω
)
)
:
γ
(
0
)
=
0
,
γ
(
1
)
=
ψ
}
,
ψ
∈
H
0
1
(
Ω
)
does not depend on n and cλn>max{0,Jλn(ψ)}. In addition,
D
J
λ
n
(
u
n
)
=
0
for all n∈ℕ.
Through a series of steps we will show that under condition (HJ) the sequence un is bounded in H01(Ω). Notice that cλn is bounded since Jλ is non-increasing in λ.
In what follows, C will designate different constants whose explicit values are irrelevant for the purposes of the proofs.
Lemma 4.1.
Let {(λn,un)}⊂(0,+∞)×H01(Ω) be a sequence satisfying (HJ). Then there exists C>0 such that
(4.1)
∥
|
u
n
|
q
(
x
)
+
1
∥
L
1
(
Ω
,
d
μ
)
≤
C
,
where dμ=dist(x,∂Ω)dx. Furthermore, for every 1≤p<NN-1 there exists another constant C such that
(4.2)
∥
u
n
∥
L
p
(
Ω
)
≤
C
.
Proof.
First observe that Jλn(un) is bounded from above since cλn is non-increasing. On the other hand,
∥
u
n
∥
H
0
1
(
Ω
)
2
-
λ
n
∫
Ω
1
q
(
x
)
+
1
|
u
n
|
q
(
x
)
+
1
𝑑
x
=
0
for all n. This implies that
λ
n
∫
Ω
q
(
x
)
-
1
q
(
x
)
+
1
|
u
n
|
q
(
x
)
+
1
𝑑
x
=
O
(
1
)
as n→∞. Using hypothesis (1.4), we conclude that
∫
Ω
d
(
x
)
|
u
n
|
q
(
x
)
+
1
𝑑
x
≤
C
,
which proves (4.1). In particular, λn|un|q(x) is bounded in L1(Ω,dμ) since λn remains bounded. We use now the crucial fact that (λn,un) solves (1.5) together with the estimates in [5, Lemma 2.1] (see also [23]) to achieve (4.2). ∎
Lemma 4.2.
Assume that {(λn,un)}⊂(0,+∞)×H01(Ω) satisfies hypothesis (HJ). Then the sequence un is bounded in H01(Ω).
Proof.
Using once again that (λn,un) solves (1.5), we get
∫
Ω
|
∇
u
n
|
2
𝑑
x
=
λ
n
∫
Ω
|
u
n
|
q
(
x
)
+
1
𝑑
x
,
that is, to get a bound for ∥un∥H01(Ω) we need to obtain a uniform bound for the integral in the right-hand side of the above equality. So set Ωδ={x∈Ω:d(x)<δ}, with δ>0 and suitably small, and Qδ=Ω∖Ωδ. Writing
∫
Ω
|
u
n
|
q
(
x
)
+
1
𝑑
x
=
∫
Ω
δ
|
u
n
|
q
(
x
)
+
1
𝑑
x
+
∫
Q
δ
|
u
n
|
q
(
x
)
+
1
𝑑
x
,
we observe that (4.1) entails that the third integral in the equality is uniformly bounded. To estimate the second integral we select a value q1 verifying
1
<
q
1
<
N
N
-
1
.
Then we obtain
|
u
|
q
(
x
)
≤
1
+
|
u
|
q
1
for all x∈Ω¯δ if δ is chosen small enough.
On the other hand, using Lemma 4.1, we obtain
∫
Ω
δ
|
u
n
|
q
(
x
)
+
1
𝑑
x
≤
∫
Ω
δ
|
u
n
|
𝑑
x
+
∫
Ω
δ
|
u
n
|
q
1
+
1
𝑑
x
≤
C
+
∫
Ω
δ
|
u
n
|
q
1
+
1
𝑑
x
.
We now borrow ideas from [4] to estimate the last integral. First, observe that
∫
Ω
δ
|
u
n
|
q
1
+
1
𝑑
x
=
∫
Ω
δ
(
d
|
u
n
|
q
1
)
θ
(
|
u
n
|
q
1
)
1
-
θ
|
u
n
|
d
θ
𝑑
x
,
where
θ
=
2
N
+
1
<
1
.
By using Hölder’s inequality, we get
(4.3)
∫
Ω
δ
|
u
n
|
q
1
+
1
𝑑
x
≤
(
∫
Ω
δ
d
|
u
n
|
q
1
𝑑
x
)
θ
(
∫
Ω
δ
|
u
n
|
q
1
|
u
n
|
1
1
-
θ
d
θ
1
-
θ
𝑑
x
)
1
-
θ
≤
C
(
∫
Ω
δ
|
u
n
|
q
1
|
u
n
|
1
1
-
θ
d
θ
1
-
θ
𝑑
x
)
1
-
θ
,
where Lemma 4.1 has been used to estimate the second integral in the first line since q1<NN-1. We next observe that for arbitrarily small ε>0 a positive constant Cε can be chosen so that
|
u
|
q
1
≤
ε
|
u
|
q
B
T
+
C
ε
for all u≥0, where qBT=N+1N-1 is the well-known Brezis–Turner exponent (cf. [4]). Thus, the last integral in (4.3) can be estimated as
(
∫
Ω
δ
|
u
n
|
q
1
|
u
n
|
1
1
-
θ
d
θ
1
-
θ
𝑑
x
)
1
-
θ
≤
ε
1
-
θ
(
∫
Ω
δ
|
u
n
|
2
1
-
θ
d
θ
1
-
θ
𝑑
x
)
1
-
θ
+
C
ε
(
∫
Ω
δ
|
u
n
|
1
1
-
θ
d
θ
1
-
θ
𝑑
x
)
1
-
θ
(4.4)
≤
ε
1
-
θ
∥
u
n
d
θ
2
∥
L
2
1
-
θ
(
Ω
)
2
+
C
ε
∥
u
n
d
θ
∥
L
1
1
-
θ
(
Ω
)
,
where it has been used that
q
B
T
=
1
1
-
θ
.
We now recall the next variant of the Hardy inequality [4, Lemma 2.2]. It states that for every u∈H01(Ω) and 0≤s≤1 a positive constant C exists so that the inequality
∥
v
d
s
∥
L
p
(
Ω
)
≤
C
∥
v
∥
H
0
1
(
Ω
)
holds true for every v∈H01(Ω), provided that
1
p
=
1
2
-
1
-
s
N
.
Taking into account that p=21-θ for s=θ2, while the corresponding p associated to s=θ satisfies p>11-θ, we conclude from (4.3) and (4.4) (after possibly diminishing ε) that
∫
Ω
δ
|
u
n
|
q
(
x
)
+
1
𝑑
x
≤
ε
1
-
θ
∥
u
∥
H
0
1
(
Ω
)
2
+
C
ε
∥
u
∥
H
0
1
(
Ω
)
+
C
.
Thus,
∥
u
n
∥
H
0
1
(
Ω
)
2
≤
ε
1
-
θ
∥
u
n
∥
H
0
1
(
Ω
)
2
+
C
ε
∥
u
n
∥
H
0
1
(
Ω
)
+
C
,
which certainly implies that ∥un∥H01(Ω) is bounded if ε is properly chosen. ∎
Proof of Theorem 1.1 (iv).
We first show that the MP level cλ involved in Section 3 and associated to the solution uλ satisfies
lim
λ
→
∞
c
λ
=
0
.
Let γλ be the path γλ(τ)=τψ, 0≤τ≤1, which joins u=0 with the function ψ=t0ϕ1 computed in Section 3 (notice that now t0 depends on λ since λ→∞). Set
c
~
λ
=
max
J
λ
∘
γ
λ
=
max
0
≤
t
≤
t
0
J
λ
(
t
ϕ
1
)
.
Since 0<cλ≤c~λ, it suffices to show that
(4.5)
lim
λ
→
∞
c
~
λ
=
0
.
Take h(t)=Jλ(tϕ1). It is clear that h achieves its maximum at a value t=tλ satisfying
(4.6)
∫
Ω
λ
1
ϕ
1
2
𝑑
x
=
λ
∫
Ω
t
λ
q
(
x
)
-
1
ϕ
1
q
(
x
)
+
1
𝑑
x
.
Hence, tλ→0 as λ→∞. From
(4.7)
c
~
λ
=
J
λ
(
t
λ
ϕ
1
)
≤
λ
1
t
λ
2
2
∫
Ω
ϕ
1
2
𝑑
x
assertion (4.5) follows.
Next, we use both (4.6) and (4.7) to provide a better estimate of c~λ. For a small δ>0 there exists k0>0 such that
ϕ
1
(
x
)
≥
k
0
d
(
x
)
for all x∈Ωδ. For the sake of simplicity we assume that q is Lipschitz in Ω¯δ and so a constant k1 exists so that q satisfies
1
<
q
(
x
)
<
1
+
k
1
d
(
x
)
for all x∈Ω¯δ (the case q∈Cα can be handled analogously). Denoting temporarily ε=tλ, we obtain
∫
Ω
ε
q
(
x
)
-
1
ϕ
1
q
(
x
)
+
1
𝑑
x
≥
∫
Ω
δ
ε
q
(
x
)
-
1
ϕ
1
q
(
x
)
+
1
𝑑
x
≥
∫
Ω
δ
ε
k
1
d
(
k
0
d
)
2
+
k
1
d
𝑑
x
,
while
∫
Ω
δ
ε
k
1
d
(
k
0
d
)
2
+
k
1
d
𝑑
x
≥
|
∂
Ω
|
∫
0
δ
ε
k
1
s
(
k
0
s
)
2
+
k
1
s
𝑑
s
.
On the other hand,
∫
0
δ
ε
k
1
s
(
k
0
s
)
2
+
k
1
s
𝑑
s
≥
(
1
-
η
)
∫
0
δ
ε
k
1
s
(
k
0
s
)
2
𝑑
s
,
where 0<η<1 provided that δ is chosen suitably small. A direct computation reveals that
(4.8)
∫
0
δ
ε
k
1
s
(
k
0
s
)
2
𝑑
s
=
-
C
ln
3
ε
(
1
+
o
(
1
)
)
,
as ε→0+, for a certain positive constant C. Going back to (4.6) and using (4.8), we arrive at
∫
Ω
λ
1
ϕ
2
𝑑
x
≥
-
C
λ
ln
3
ε
,
which yields, after restoring tλ instead of ε,
t
λ
=
O
(
e
-
C
λ
3
)
as λ→∞.
Using (4.7) and taking into account that cλ≤c~λ, we finally get that
c
λ
=
O
(
e
-
C
λ
3
)
as λ→∞.
Now we proceed to obtain exponentially small uniform bounds of uλ as λ→∞. Just as before the relation
λ
∫
Ω
q
(
x
)
-
1
q
(
x
)
+
1
|
u
λ
|
q
(
x
)
+
1
𝑑
x
=
O
(
e
-
C
λ
3
)
holds as λ→∞, and so
(4.9)
∫
Ω
|
u
λ
|
q
(
x
)
+
1
𝑑
μ
=
o
(
e
-
C
λ
3
)
as
λ
→
∞
,
where dμ=d(x)dx.
Some few facts from the theory of Lp(x) spaces are now required. For p∈L∞(Ω), p≥1 a.e. in Ω, the modular functional ρp is defined on the set M(Ω) of measurable functions u in Ω as
ρ
p
(
u
)
=
∫
Ω
|
u
|
p
(
x
)
𝑑
μ
,
d
μ
=
d
(
x
)
d
x
.
Then the class Lp(x)(Ω):={u∈M(Ω):ρp(u)<∞} becomes a Banach space under the norm
∥
u
∥
p
(
x
)
=
inf
{
λ
>
0
:
ρ
p
(
u
λ
)
≤
1
}
.
This is, of course, the so-called Luxemburg norm. We refer to [20] for a comprehensive account on this and further abstract classes of generalized Orlicz-type spaces. Some well-known features are the following (see for instance [8] for an expeditious overview):
∥
u
∥
p
(
x
)
<
1
if and only if ρp(u)<1.
∥
u
∥
p
(
x
)
<
1
implies ∥u∥p(x)p+≤ρp(u)≤∥u∥p(x)p-, where p-=essinfu, p+=esssupu.
The embedding Lr(x)⊂Lp(x) is continuous provided that r∈L∞(Ω) fulfills r≥p a.e. in Ω.
Hence, according to (i) and (4.9),
∥
u
∥
q
(
x
)
+
1
q
+
+
1
≤
ρ
q
+
1
(
u
)
=
o
(
e
-
C
λ
3
)
as
λ
→
∞
,
while (iii) implies that
∥
u
∥
q
(
x
)
=
o
(
e
-
C
λ
3
)
as
λ
→
∞
since q-=1. Thus, it follows from (ii) that
(4.10)
∫
Ω
λ
|
u
λ
|
q
(
x
)
𝑑
μ
=
o
(
e
-
C
λ
3
)
as
λ
→
∞
.
Recall that the constant C is not the same in each particular appearance.
Using the same argument as in Lemma 4.1, we conclude that
(4.11)
∥
u
λ
∥
p
=
o
(
e
-
C
λ
3
)
as
λ
→
∞
for every 1≤p<NN-1.
To estimate ∥uλ∥H01(Ω) we follow the approach in Lemma 4.2 and arrive at the estimate
(4.12)
∫
Ω
∥
∇
u
λ
∥
2
𝑑
x
≤
λ
∫
Ω
δ
|
u
λ
|
𝑑
x
+
λ
∫
Ω
δ
|
u
λ
|
q
1
+
1
𝑑
x
+
O
(
e
-
C
λ
3
)
.
Here, δ>0 and q1 are the numbers introduced in the proof of Lemma 4.2, and (4.10) has been employed to estimate the integral
λ
∫
Q
δ
|
u
λ
|
q
(
x
)
+
1
𝑑
x
,
where
Q
δ
=
Ω
∖
Ω
δ
.
The second integral in (4.12) is O(e-Cλ3) thanks to (4.11). By using the computation in (4.4), we obtain
λ
∫
Ω
δ
|
u
λ
|
q
1
+
1
𝑑
x
≤
(
∫
Ω
δ
λ
1
θ
|
u
λ
|
q
1
𝑑
x
)
θ
{
ε
1
-
θ
∥
u
λ
∥
H
0
1
(
Ω
)
2
+
C
ε
∥
u
λ
∥
H
0
1
(
Ω
)
}
.
Thus,
(4.13)
∥
u
λ
∥
H
0
1
(
Ω
)
2
≤
O
(
e
-
C
λ
3
)
{
ε
1
-
θ
∥
u
λ
∥
H
0
1
(
Ω
)
2
+
C
ε
∥
u
λ
∥
H
0
1
(
Ω
)
}
+
O
(
e
-
C
λ
3
)
as λ→∞. This allows us to conclude first that ∥uλ∥H01(Ω)=O(1) as λ→∞. In a second instance it implies that
∥
u
λ
∥
H
0
1
(
Ω
)
=
O
(
e
-
C
λ
3
)
as λ→∞ (the interchange between “O” and “o” is obtained by modifying C). Finally, estimate (4.13) leads in a standard way to a corresponding one in C2,β(Ω¯) for a certain 0<β<1.
This completes the proof of Theorem 1.1. ∎
Remark 4.3.
When studying the asymptotic behavior as λ→∞ of a family uλ of solutions through a scaling method (cf. [14]), a critical issue is the concentration of maxima of solutions on ∂Ω. In the case where Ω is a convex domain, this possibility can be ruled out in some problems by means of the moving planes technique as in [6]. However, this approach cannot be used here since our nonlinearity uq(x) lacks the right monotonicity properties near the boundary. This is due, of course, to the fact that q=1 on ∂Ω.
Acknowledgements
The authors would like thank the referee of this paper for his/her opportune comments.
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