1 Introduction, Previous Results and Main Theorems
In this paper, we will consider the boundary value problem with nonlinear mixed boundary conditions and spatial heterogeneities given by
(1.1)
{
-
Δ
u
=
λ
u
-
a
(
x
)
u
p
in
Ω
,
p
>
1
,
u
=
0
on
Γ
0
,
∂
u
+
V
(
x
)
u
=
γ
b
(
x
)
u
q
on
Γ
1
,
q
>
1
,
where
Ω is a bounded domain of ℝN, N≥1, of class 𝒞2, with boundary ∂Ω=Γ0∪Γ1, where Γ0 and Γ1 are disjoint open and closed subsets of ∂Ω,
-
Δ
stands for the minus Laplacian operator in ℝN and λ∈ℝ,
a
∈
𝒞
(
Ω
¯
)
with a≥0 in Ω measures the spatial heterogeneities in Ω,
the potentials V,b∈𝒞(Γ1), where b⪈0 on Γ1 and V possesses arbitrary sign in each point x∈Γ1, represent the spatial heterogeneities on the boundary,
γ
∈
ℝ
will be the bifurcation-continuation parameter, and ∂u(x) stands for the outer normal derivative of u at x∈Γ1.
This work is devoted to ascertain the global structure of the set of positive solutions of (1.1) considering γ as the bifurcation-continuation parameter, for some fixed λ in a suitable interval. The previous works [5, 6, 7, 9, 11] play a crucial role in obtaining the results of this paper. We will distinguish the following cases:
Case (1): a=0 in Ω and
(1.2)
b
(
x
)
≥
b
¯
>
0
for all
x
∈
Γ
1
.
In this case, we set Ω0:=Ω.
Case (2): a⪈0 in Ω, b⪈0 on Γ1,
(1.3)
Ω
0
:
=
int
{
x
∈
Ω
:
a
(
x
)
=
0
}
≠
∅
,
Ω
0
∈
𝒞
2
,
Ω
¯
0
⊂
Ω
and
a
is bounded away from zero in any compact subset of
(
Ω
∖
Ω
¯
0
)
∪
Γ
1
.
By a⪈0 in Ω, it is meant that a(x)≥0 for all x∈Ω and a≠0. Similarly, b⪈0 on Γ1 means that b(x)≥0 for all x∈Γ1 and b≠0.
The case when b(x) changes sign on Γ1 and/or a(x) changes sign in Ω remains outside the general scope of this work. It will be analyzed elsewhere.
The global structure of the diagram of positive solutions of (1.1) considering λ as bifurcation parameter, for some fixed γ, was already analyzed in [11, 5] (for γ=0) and in [6] (for γ≠0), in the particular case when a⪈0 in Ω vanishes in Ω0 with Ω¯0⊂Ω, and in [7], in the special case when a=0 in Ω, that is, when Ω0=Ω. In this work, this perspective will be completed by analyzing the structure of the diagram of positive solutions of (1.1) regarding to γ as the main bifurcation-continuation parameter for a fixed value of λ in a suitable interval.
In both cases, (1) and (2), we analyze the existence, uniqueness, asymptotic behavior and stability of the positive solutions of (1.1) and ascertain the exact decay rate to zero as γ↓-∞ of the positive solutions of (1.1), in terms of the parameter γ in the nonlinear mixed boundary conditions. In both cases, it will be shown that the decay rate in L∞(Ω) as γ↓-∞ of the positive solutions of (1.1) is, up multiplicative constants, of order (-γ)-1q-1. Also, in Case (1), we will ascertain the exact blow-up rate to infinity, as γ↑0, of the continuum of positive solutions of (1.1), in terms of the parameter γ. Up to a multiplicative constant, it will be established that such blow-up rate also is of order (-γ)-1q-1 in L∞(Ω).
By a positive solution of (1.1), it is meant any couple (λ,u)∈ℝ×Wr2(Ω) for some r>N, with u>0 in Ω satisfying (1.1). It should be noted that Wr2(Ω)⊂𝒞2-Nr(Ω¯) and that any function u∈Wr2(Ω), r>N, is a.e. twice differentiable (cf. [18, Theorem VIII.1]). We will say that a positive solution (λ,u) of (1.1) is strongly positive in Ω, and we will denote it by u≫0, if u(x)>0 for all x∈Ω∪Γ1 and ∂u(x)<0 for all x∈Γ0.
Hereafter, for each γ∈ℝ, we will denote
Λ
(
γ
)
:
=
{
λ
∈
ℝ
:
(1.1) possesses a positive solution
}
,
by 𝒞λ+(λ=λ~,γ∈I) the continuum of positive solutions of (1.1) which, considering λ as bifurcation parameter, emanates from the trivial branch (λ,u)=(λ,0) at λ=λ~ for some fixed γ in the interval I⊂ℝ, and by 𝒞γ+(γ=γ~,λ∈J) the continuum of positive solutions of (1.1) which, considering γ as bifurcation parameter, emanates from the trivial branch (γ,u)=(γ,0) at γ=γ~ for some fixed λ in the interval J⊂ℝ.
Also, for each V∈𝒞(Γ1), 𝔅(V(x)) stands for the boundary operator
𝔅
(
V
(
x
)
)
u
:
=
{
u
on
Γ
0
,
∂
u
+
V
(
x
)
u
on
Γ
1
,
and 𝔇 stands for the Dirichlet operator on ∂Ω. It is known (cf. [3]) that
𝔅
(
V
(
x
)
)
∈
ℒ
(
W
r
2
(
Ω
)
,
W
r
2
-
1
r
(
Γ
0
)
×
W
r
1
-
1
r
(
Γ
1
)
)
.
Owing to the results in [3] and [16, Sections 7.2, 7.4], we have that, for any K∈𝒞(Ω¯) and V∈𝒞(Γ1), the boundary eigenvalue problem
(1.4)
{
(
-
Δ
+
K
(
x
)
)
φ
=
σ
φ
in
Ω
,
𝔅
(
V
(
x
)
)
φ
=
0
¯
on
∂
Ω
possesses a unique eigenvalue that possesses a positive eigenfunction, unique up to a multiplicative constant, named principal eigenvalue of (1.4). In the sequel, we will denote it by σ1Ω[-Δ+K(x),𝔅(V(x))]. Also, the principal eigenvalue of (1.4) is simple and dominant in the sense that any other eigenvalue λ of (1.4) satisfies
Re
(
λ
)
>
σ
1
Ω
[
-
Δ
+
K
(
x
)
,
𝔅
(
V
(
x
)
)
]
,
where Re(λ) stands for the real part of λ. In [16, Theorem 7.8], the dominance of the principal eigenvalue σ1Ω[-Δ+K(x),𝔅(V(x))] is proved, and using the generalized Krein–Rutman theorem given in [16, Theorem 6.3], the dominance of it among the real numbers is definitely proved.
In addition, if φ* stands for the positive eigenfunction of (1.4) associated to σ1Ω[-Δ+K(x),𝔅(V(x))], unique up to a multiplicative constant, then
(1.5)
φ
*
≫
0
in
Ω
,
and
(1.6)
φ
*
∈
W
2
(
Ω
)
:
=
⋂
r
>
N
W
r
2
(
Ω
)
⊂
𝒞
1
+
α
(
Ω
¯
)
for all
α
∈
(
0
,
1
)
.
Hereafter, we will denote σ1:=σ1Ω[-Δ,𝔅(V(x))], and by φ1>0 the principal eigenfunction associated to the principal eigenvalue σ1, normalized so that ∥φ1∥L∞(Ω)=1. By (1.5) and (1.6), we get
φ
1
≫
0
in
Ω
and
φ
1
∈
𝒞
1
+
α
(
Ω
¯
)
for all
α
∈
(
0
,
1
)
.
Also, we will denote
(1.7)
σ
0
*
:
=
σ
1
Ω
[
-
Δ
,
𝔇
]
,
σ
0
:
=
σ
1
Ω
0
[
-
Δ
,
𝔇
]
,
that is, the principal eigenvalue of the -Δ operator in the domains Ω0 and Ω, respectively, under Dirichlet boundary conditions. Owing to [8, Propositions 3.1, 3.2], we know that, in the case when Ω¯0⊂Ω, then
(1.8)
σ
1
<
σ
0
*
<
σ
0
,
and in the case when Ω0=Ω, then σ1<σ0*=σ0. Therefore, in any of the previous cases, σ1<σ0.
Hereafter, we will denote
𝒞
Γ
0
1
(
Ω
¯
)
:
=
{
u
∈
𝒞
1
(
Ω
¯
)
:
u
|
Γ
0
=
0
}
,
W
r
,
Γ
0
2
(
Ω
)
:
=
{
u
∈
W
r
2
(
Ω
)
:
u
|
Γ
0
=
0
}
.
For each fixed λ in a suitable interval, the solutions of (1.1) are the roots of the operator 𝔉λ:ℝ×𝒰↦𝒰 defined by
(1.9)
𝔉
λ
(
γ
,
u
)
:
=
u
-
𝔎
~
1
(
(
λ
+
ω
)
u
-
a
(
x
)
u
p
)
-
𝔎
~
2
(
γ
b
u
q
)
,
for 𝒰:=𝒞Γ01(Ω¯), where 𝔎~1,𝔎~2:𝒰↦𝒰 are the compact operators defined by
𝔎
~
1
:
=
i
∘
K
1
∘
j
:
𝒰
↦
𝒰
,
𝔎
~
2
:
=
i
∘
K
2
∘
i
*
∘
t
1
:
𝒰
↦
𝒰
,
where, for each r>N, f∈Lr(Ω) and g∈Wr1-1r(Γ1), u1:=K1(f)∈Wr,Γ02(Ω) and u2:=K2(g)∈Wr,Γ02(Ω) stand for the unique solutions of the problems
{
(
-
Δ
+
ω
)
u
1
=
f
in
Ω
,
𝔅
(
V
(
x
)
)
u
1
=
0
on
∂
Ω
,
{
(
-
Δ
+
ω
)
u
2
=
0
in
Ω
,
u
2
=
0
on
Γ
0
,
∂
u
2
+
V
(
x
)
u
2
=
g
on
Γ
1
,
respectively, for some fixed
(1.10)
ω
>
-
σ
1
,
i
,
j
and i* being the inclusion operators
i
:
W
r
,
Γ
0
2
(
Ω
)
↦
𝒰
,
i
*
:
𝒞
1
(
Γ
1
)
↦
W
r
1
-
1
r
(
Γ
1
)
,
j
:
𝒰
↦
L
r
(
Ω
)
and t1:𝒰↦𝒞1(Γ1) the trace operator on Γ1.
Hereafter, given a solution u0 of (1.1) for the values (λ,γ)=(λ0,γ0) of the parameters, we will denote by (ℒ(λ0,u0),ℬ(γ0,u0)) the linearization of (1.1) around u0, where ℒ(λ0,u0) and ℬ(γ0,u0) stand for the linear operators
(1.11)
ℒ
(
λ
0
,
u
0
)
:
=
-
Δ
-
λ
0
+
p
a
(
x
)
u
0
p
-
1
,
ℬ
(
γ
0
,
u
0
)
:
=
𝔅
(
V
(
x
)
-
q
γ
0
b
(
x
)
u
0
q
-
1
)
.
Owing to (1.10), we get
(1.12)
Ker
(
D
u
𝔉
λ
0
(
γ
0
,
u
0
)
)
=
Ker
(
ℒ
(
λ
0
,
u
0
)
,
ℬ
(
γ
0
,
u
0
)
)
.
Also, we will say that a non-negative solution u0 of (1.1) for the values (λ,γ)=(λ0,γ0) is
linearly asymptotically stable if σ1Ω[ℒ(λ0,u0),ℬ(γ0,u0)]>0,
linearly unstable if σ1Ω[ℒ(λ0,u0),ℬ(γ0,u0)]<0,
neutrally stable if σ1Ω[ℒ(λ0,u0),ℬ(γ0,u0)]=0,
linearly stable if σ1Ω[ℒ(λ0,u0),ℬ(γ0,u0)]≥0.
Owing to [6, Proposition 2.1], it is known that if uλ,γ is a positive solution of (1.1) for the values (λ,γ) of the parameters, then
(1.13)
λ
=
σ
1
Ω
[
-
Δ
+
a
(
x
)
u
λ
,
γ
p
-
1
,
𝔅
(
V
(
x
)
-
γ
b
(
x
)
u
λ
,
γ
q
-
1
)
]
,
u
λ
,
γ
∈
𝒞
1
+
α
(
Ω
¯
)
for all α∈(0,1) and uλ,γ is strongly positive in Ω.
1.1 Case (1): Previous Results and Main Theorem
In Case (1), (1.1) becomes
(1.14)
{
-
Δ
u
=
λ
u
in
Ω
,
u
=
0
on
Γ
0
,
∂
u
+
V
(
x
)
u
=
γ
b
(
x
)
u
q
on
Γ
1
,
q
>
1
,
Ω
0
:
=
Ω
, σ0=σ0*=σ1Ω[-Δ,𝔇], and owing to the results in [3, 7], the following is known:
If γ=0, then (1.14) becomes
(1.15)
{
-
Δ
u
=
λ
u
in
Ω
,
𝔅
(
V
(
x
)
)
u
=
0
on
∂
Ω
,
which possesses a positive solution if and only if λ=σ1, and in this case, all the positive solutions of (1.15) are a positive multiple of φ1, where φ1 is the principal eigenfunction associated to the principal eigenvalue σ1 of (1.15), normalized so that ∥φ1∥L∞(Ω)=1. That is, Λ(0)={σ1}. In this case, for γ=0, considering λ as bifurcation parameter, we get a genuine vertical bifurcation to positive solutions from the trivial branch (λ,u)=(λ,0) at λ=σ1, and the global bifurcation diagram of positive solutions of (1.14) is like shown in Figure 1, where ℭ+:=𝒞λ+(λ=σ1,γ=0).
For any γ<0, independently of its size, Λ(γ)=(σ1,σ0), and hence (cf. [7, Theorem 3.1])
(1.16)
(
σ
1
,
σ
0
)
=
⋂
γ
<
0
Λ
(
γ
)
.
In this case, owing to the previous results in [7], we know that the global bifurcation diagram of positive solutions is like shown in Figure 2, where ℭ+:=𝒞λ+(λ=σ1,γ<0).
If γ>0, then Λ(γ)⊂(-∞,σ1) and Λ(γ) might depend on γ in the sense that Λ(γ1)≠Λ(γ2) if γ1>0, γ2>0 with γ1≠γ2, and it might occur that ⋂γ>0Λ(γ)=∅ (cf. [7, Theorem 4.5]).
Then, in order to fix λ in a suitable interval for considering γ as the main bifurcation-continuation parameter, owing to the previous results, we get in Case (1):
If λ≥σ0, then λ∉Λ(γ) for any γ∈ℝ,
If λ=σ1, then (1.14) possesses a positive solution if and only if γ=0, and in this case, all the positive solutions are positive multiples of φ1. Then, considering γ as bifurcation parameter at λ=σ1, we get a vertical bifurcation to positive solutions from the trivial branch (γ,u)=(γ,0) at γ=0, and the global bifurcation diagram of positive solutions in the γ parameter of (1.14) looks like shown in Figure 3, where ℭ+:=𝒞γ+(γ=0,λ=σ1)
If λ∈(σ1,σ0), then the structure of the set of positive solutions of (1.14), as well as their point-wise behavior regarding to γ as the main bifurcation parameter, are given by Theorem 1.1, which is the main result of this paper for Case (1).
Hereafter, we will set
γ
~
:
=
-
γ
and
b
¯
:
=
∥
b
∥
L
∞
(
Γ
1
)
>
0
.
Theorem 1.1.
Assume that
(1.17)
λ
∈
(
σ
1
,
σ
0
)
and (1.2) holds.
Problem (
1.14
) possesses a positive solution if and only if
γ
<
0
, and it is unique if it exists. In the sequel, we will denote it by
u
γ
.
Any positive solution of (
1.14
) is linearly asymptotically stable.
The map
(
-
∞
,
0
)
→
𝒞
Γ
0
1
(
Ω
¯
)
, γ→uγ is differentiable, and
(1.18)
u
˙
γ
:
=
d
u
γ
d
γ
≫
0
𝑖𝑛
Ω
.
In particular, we get
u
˙
γ
(
x
)
>
0
for all
x
∈
Ω
∪
Γ
1
, and therefore the map
(
-
∞
,
0
)
→
𝒞
Γ
0
(
Ω
¯
)
, γ→uγ is strictly increasing.
All the positive solutions of (
1.14
) belong to the branch
(1.19)
Γ
+
(
λ
)
:
=
{
(
γ
,
u
γ
)
:
γ
<
0
}
,
which is increasing with respect to γ in L∞(Ω). Moreover, there exist C(λ)>0 and D(λ)>0 such that, for all γ<0, the following holds:
(1.20)
C
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
≤
∥
u
γ
∥
L
∞
(
Ω
)
≤
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
.
In particular,
(1.21)
lim
γ
↓
-
∞
∥
u
γ
∥
L
∞
(
Ω
)
=
0
𝑎𝑛𝑑
lim
γ
↑
0
∥
u
γ
∥
L
∞
(
Ω
)
=
∞
,
that is, the branch
Γ
+
(
λ
)
bifurcates to positive solutions from the trivial solution as
γ
↓
-
∞
, while it bifurcates to positive solutions from infinity as
γ
↑
0
.
The following holds:
(1.22)
lim
γ
↑
0
u
γ
(
x
)
=
∞
,
uniformly in compact subsets
K
⊂
Ω
∪
Γ
1
.
For each
γ
<
0
, the unique positive solution
u
γ
of (
1.14
) is globally asymptotically stable as steady-state solution of the parabolic problem associated to (
1.14
).
Owing to Theorem 1.1, we get that, for each fixed λ∈(σ1,σ0), the global bifurcation diagram of positive solutions of (1.14) in the γ-parameter is like shown in Figure 4, where Γ+(λ) stands for the branch of positive solutions defined by (1.19). As illustrated by Figure 4, Γ+(λ) bifurcates to positive solutions from the trivial branch (γ,u)=(γ,0) as γ↓-∞, and from infinity as γ↑0, with exactly the same rate γ~-1q-1, up to some multiplicative constant.
1.2 Case (2): Previous Results and Main Theorem
In Case (2), owing to the previous results of [11, 5, 9, 6], the following is known:
If γ=0, then (1.1) becomes the boundary value problem with linear mixed boundary conditions
{
-
Δ
u
=
λ
u
-
a
(
x
)
u
p
in
Ω
,
𝔅
(
V
(
x
)
)
u
=
0
on
∂
Ω
,
which has been widely analyzed in [11, 5, 9], and recently in [10] in a much more general framework with no special restriction on the structure of a-1(0)=Ω0. In this case, it is well known that the problem possesses a positive solution if and only if λ∈(σ1,σ0), that is, Λ(0)=(σ1,σ0) (cf. [11, Theorem 3.5], [5, Theorem 4.2]). The global bifurcation diagram of positive solutions in this case, considering λ as bifurcation parameter, is like shown in Figure 5, where ℭγ=0+:=𝒞λ+(λ=σ1,γ=0).
For any γ<0, independently of its size, Λ(γ)=(σ1,σ0), that is, ⋂γ<0Λ(γ)=(σ1,σ0), (cf. [6, Theorem 3.1]), and its global bifurcation diagram of positive solutions, considering λ as bifurcation parameter, is like shown in Figure 5, where ℭγ<0+:=𝒞λ+(λ=σ1,γ<0).
If γ>0 and p>2q-1, then (cf. [6, Theorem 4.7])
(1.23)
(
σ
1
,
σ
0
)
⊂
Λ
(
γ
)
,
and its global bifurcation diagram of positive solutions, considering λ as bifurcation parameter, should be like shown in Figure 6, where ℭ+:=𝒞λ+(λ=σ1,γ>0).
Therefore, owing to the previous results, in Case (2), we find that ⋂γ≤0Λ(γ)=(σ1,σ0), and if p>2q-1, then (1.23) holds, and hence ⋂γ∈ℝΛ(γ)=(σ1,σ0). Then, to get the structure of the set of positive solutions of (1.1), considering γ as bifurcation-continuation parameter, we will fix λ∈(σ1,σ0). Next follows the main theorem in Case (2).
Theorem 1.2.
Assume that (1.17) holds, and let us denote by u0 the unique positive solution of (1.1) for γ=0.
For each
γ
≤
0
, (
1.1
) possesses a unique positive solution
u
γ
, which is linearly asymptotically stable. Moreover, the map
(
-
∞
,
0
]
→
𝒞
Γ
0
1
(
Ω
¯
)
, γ→uγ is differentiable and
u
˙
γ
:
=
d
u
γ
d
γ
≫
0
𝑖𝑛
Ω
,
and in particular, the map
(1.24)
(
-
∞
,
0
]
↦
𝒞
Γ
0
(
Ω
¯
)
,
γ
↦
u
γ
is strictly increasing.
If (
1.2
) holds and
λ
∈
(
σ
1
,
σ
0
*
)
(cf. (
1.7
)), then there exists
D
(
λ
)
>
0
such that
(1.25)
∥
u
γ
∥
L
∞
(
Ω
)
≤
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
for all
γ
<
0
.
In particular,
(1.26)
lim
γ
↓
-
∞
∥
u
γ
∥
L
∞
(
Ω
)
=
0
,
that is, the problem exhibits bifurcation to positive solutions from the trivial branch
(
γ
,
u
)
=
(
γ
,
0
)
when
γ
↓
-
∞
.
There exists
ε
0
>
0
and a differentiable map
u
:
(
-
ε
0
,
ε
0
)
→
𝒞
Γ
0
1
(
Ω
¯
)
,
γ
↦
u
γ
*
such that
u
γ
*
=
u
γ
for all
γ
∈
(
-
ε
0
,
0
]
, there exists a neighborhood
𝔘
of
(
γ
,
u
)
=
(
0
,
u
0
)
in
(
-
ε
0
,
ε
0
)
×
𝒞
Γ
0
1
(
Ω
¯
)
such that if
(
γ
,
u
~
)
∈
𝔘
is a positive solution of (
1.1
), then
u
~
=
u
γ
*
, and in addition,
(
γ
,
u
γ
*
)
is a positive linearly asymptotically stable solution of (
1.1
) for all
γ
∈
(
-
ε
0
,
ε
0
)
. Moreover,
u
˙
γ
*
:
=
d
u
γ
*
d
γ
≫
0
𝑖𝑛
Ω
,
and in particular, the map
(
-
ε
0
,
ε
0
)
→
𝒞
Γ
0
(
Ω
¯
)
, γ↦uγ* is strictly increasing.
Any positive solution
u
^
γ
of (
1.1
) for
γ
>
0
satisfies
(1.27)
u
^
γ
≫
u
0
If
p
>
2
q
-
1
, then the following hold:
For each
γ
>
0
, (
1.1
) possesses at least a positive solution.
For each
γ
>
0
, (
1.1
) possesses a minimal positive solution
u
γ
min
satisfying
(1.28)
u
γ
min
≫
u
0
and
(1.29)
u
γ
min
=
u
γ
*
𝑓𝑜𝑟
γ
∈
(
0
,
ε
0
)
,
where
ε
0
and
u
γ
*
are defined by (iii).
There exist uniform
L
∞
(
Ω
)
bounds for the positive solutions of (
1.1
) in compact intervals of values of γ.
Theorem 1.2 establishes in Case (2) that, for each fixed λ∈(σ1,σ0), the global bifurcation diagram in the γ-parameter of the positive solutions of (1.1) should be like shown by Figure 7, where the continuous line stands for the exact structure of the set of positive solutions for γ<ε0 (cf. Theorem 1.2 (iii)) and the dashed line stands for a possible configuration of the set of positive solutions of (1.1) for γ>ε0.
The distribution of the rest of this paper is the following. Section 2 contains some results about the local structure of the bifurcation diagram of positive solutions of (1.1) in a neighborhood of the positive linearly asymptotically stable or neutrally stable solutions of (1.1). Section 3 contains the proof of Theorem 1.1. To discuss in a particular example the results established by Theorem 1.1, Section 4 analyzes a one-dimensional example of boundary value problem of Case (1), which is explicitly solved for every γ<0 obtaining in addition its branch of positive solutions. The explicit branch of positive solutions obtained, blows up to infinity in L∞(Ω) as γ↑0, and decays to zero in L∞(Ω) as γ↓-∞, with the rate provided by Theorem 1.1, that is, with rate γ~-1q-1 in both cases. Finally, Section 5 contains the proof of Theorem 1.2.
2 Local Structure of the Bifurcation Diagram of Positive Solutions of (1.1)
This section analyzes the local structure of the bifurcation diagram of positive solutions of (1.1) in a neighborhood of the positive linearly asymptotically stable and positive neutrally stable solutions of (1.1), assuming that λ∈(σ1,σ0) is fixed. The results of this section are based on [14, Sections 2, 3], [15, Section 2] and [17, Chapter 2], though here the underlying technicalities are substantially more sophisticated by the intrinsic nature of our problem.
As it will be shown in the proofs of Theorems 1.1 and 1.2, when λ∈(σ1,σ0) and a=0 in Ω (Case (1)), then the positive solutions of (1.14) exist only for γ<0 and all of them are linearly asymptotically stable, while if a⪈0 in Ω (Case (2)), all the positive solutions for γ≤0 are linearly asymptotically stable and only may exist neutrally stable solutions if γ>0.
The next result provides us the local structure of the bifurcation diagram of positive solutions of (1.1) around a positive linearly asymptotically stable solution.
Proposition 2.1.
Let (γ0,u0) be a positive linearly asymptotically stable solution of (1.1). Then there exists ε>0 and a differentiable function
u
:
(
γ
0
-
ε
,
γ
0
+
ε
)
→
𝒞
Γ
0
1
(
Ω
¯
)
,
γ
↦
u
γ
such that uγ0=u0, there exists a neighborhood U of (γ0,u0) in (γ0-ε,γ0+ε)×CΓ01(Ω¯) such that if (γ,u~)∈U is a positive solution of (1.1), then u~=uγ, and in addition, (γ,uγ) is a positive linearly asymptotically stable solution of (1.1) for all γ∈(γ0-ε,γ0+ε). Moreover,
(2.1)
u
˙
γ
:
=
d
u
γ
d
γ
≫
0
𝑖𝑛
Ω
,
and in particular, the map
(2.2)
(
γ
0
-
ε
,
γ
0
+
ε
)
→
𝒞
Γ
0
(
Ω
¯
)
,
γ
↦
u
γ
is strictly increasing.
Proof.
Since the solutions of (1.1) are the roots of operator (1.9), we get 𝔉λ(γ0,u0)=0. Also, since (γ0,u0) is a positive linearly asymptotically stable solution of (1.1), this yields
(2.3)
σ
1
Ω
[
ℒ
(
λ
,
u
0
)
,
ℬ
(
γ
0
,
u
0
)
]
>
0
,
which, owing to (1.12), implies that Du𝔉λ(γ0,u0) is a topological isomorphism. Then the existence, local uniqueness and regularity of the branch of positive solutions
(2.4)
{
(
γ
,
u
γ
)
:
γ
∈
(
γ
0
-
ε
,
γ
0
+
ε
)
}
follow from the implicit function theorem applied to the operator 𝔉λ(γ,u) at (γ0,u0). On the other hand, owing to (1.11), to the continuous dependence of the principal eigenvalue with respect to the potential in the differential operator and on the boundary conditions (cf.[8, Corollary 3.4, Theorem 8.2]), to the regularity of the branch (2.4) given by the implicit function theorem and to (2.3), we get
lim
γ
→
γ
0
σ
1
Ω
[
ℒ
(
λ
,
u
γ
)
,
ℬ
(
γ
,
u
γ
)
]
=
σ
1
Ω
[
ℒ
(
λ
,
u
0
)
,
ℬ
(
γ
0
,
u
0
)
]
>
0
,
and hence, shortening ε>0 if it is necessary, we have
(2.5)
σ
1
Ω
[
ℒ
(
λ
,
u
γ
)
,
ℬ
(
γ
,
u
γ
)
]
>
0
for all
γ
∈
(
γ
0
-
ε
,
γ
0
+
ε
)
,
which proves that (γ,uγ) is linearly asymptotically stable for all γ∈(γ0-ε,γ0+ε).
We now prove (2.1). Indeed, differentiating (1.1) with respect to the γ-parameter along the branch (2.4) yields
(2.6)
{
ℒ
(
λ
,
u
γ
)
u
˙
γ
=
0
in
Ω
,
ℬ
(
γ
,
u
γ
)
u
˙
γ
=
(
0
,
b
(
x
)
u
γ
q
)
>
0
on
∂
Ω
.
Thus, by (2.5), it follows from the characterization of the strong maximum principle that (2.6) satisfies the strong maximum principle, and therefore (2.1) holds. Now (2.2) is straightforward from (2.1). ∎
The following result will be used to prove Proposition 2.3.
Lemma 2.2.
Assume a⪈0 in Ω; let (γ0,u0) be a positive solution of (1.1) with γ0>0 such that
σ
1
Ω
[
ℒ
(
λ
,
u
0
)
,
ℬ
(
γ
0
,
u
0
)
]
=
0
,
and let ϕ denote a principal eigenfunction associated to σ1Ω[L(λ,u0),B(γ0,u0)]. Then the following holds:
(2.7)
(
p
-
1
)
∫
Ω
a
(
x
)
u
0
p
-
2
ϕ
3
=
γ
0
(
q
-
1
)
∫
Γ
1
b
(
x
)
ϕ
3
u
0
q
-
2
-
2
∫
Ω
ϕ
u
0
|
∇
ϕ
-
ϕ
u
0
∇
u
0
|
2
.
Proof.
By construction, we have
(2.8)
∫
Ω
(
ϕ
u
0
)
2
(
-
ϕ
Δ
u
0
+
u
0
Δ
ϕ
)
=
(
p
-
1
)
∫
Ω
a
(
x
)
u
0
p
-
2
ϕ
3
.
Also, since ϕ≫0, u0≫0 in Ω, we get ∂ϕ(x)<0, ∂u0(x)<0 for all x∈Γ0 and ϕu0∈𝒞1(Ω¯). Then, since
ϕ
3
u
0
2
∂
u
0
|
Γ
0
=
(
ϕ
u
0
)
2
ϕ
∂
u
0
|
Γ
0
=
0
and
ϕ
2
u
0
∂
ϕ
|
Γ
0
=
(
ϕ
u
0
)
ϕ
∂
ϕ
|
Γ
0
=
0
,
integrating by parts on the left-hand side of (2.8) and taking into account
〈
∇
(
ϕ
3
u
0
2
)
,
∇
u
0
〉
=
3
ϕ
2
u
0
2
〈
∇
ϕ
,
∇
u
0
〉
-
2
ϕ
3
u
0
3
|
∇
u
0
|
2
,
〈
∇
(
ϕ
2
u
0
)
,
∇
ϕ
〉
=
2
ϕ
u
0
|
∇
ϕ
|
2
-
(
ϕ
u
0
)
2
〈
∇
ϕ
,
∇
u
0
〉
,
we get
(2.9)
∫
Ω
(
ϕ
u
0
)
2
(
-
ϕ
Δ
u
0
+
u
0
Δ
ϕ
)
=
γ
0
(
q
-
1
)
∫
Γ
1
b
(
x
)
ϕ
3
u
0
q
-
2
-
2
∫
Ω
ϕ
u
0
|
∇
ϕ
-
ϕ
u
0
∇
u
0
|
2
.
Now (2.8) and (2.9) imply (2.7). ∎
The next result provides us the local structure of the bifurcation diagram of positive solutions of (1.1) around any positive neutrally stable solution.
Proposition 2.3.
Assume a⪈0 in Ω; let (γ0,u0) be a positive neutrally solution of (1.1) for γ0>0, and let ϕ denote a principal eigenfunction associated to σ1Ω[L(λ,u0),B(γ0,u0)]. Then there exist ε>0 and a differentiable function
(
γ
,
u
)
:
(
-
ε
,
ε
)
→
ℝ
×
𝒞
Γ
0
1
(
Ω
¯
)
,
s
↦
(
γ
(
s
)
,
u
(
s
)
)
such that (γ(0),u(0))=(γ0,u0), (γ(s),u(s)) is a positive solution of (1.1) for each s∈(-ε,ε), there exists a neighborhood U of (γ0,u0) in R×CΓ01(Ω¯) such that if (γ,u)∈U is a positive solution of (1.1), then
(
γ
,
u
)
=
(
γ
(
s
)
,
u
(
s
)
)
for some
s
∈
(
-
ε
,
ε
)
,
and in addition,
(2.10)
γ
(
s
)
=
γ
0
+
β
s
2
+
O
(
s
3
)
,
u
(
s
)
=
u
0
+
s
ϕ
+
v
(
s
)
,
where v(s)=O(s2) as s→0, ∫Ωv(s)ϕ=0 for each s∈(-ε,ε) and
(2.11)
β
=
p
(
p
-
1
)
∫
Ω
a
(
x
)
u
0
p
-
2
ϕ
3
-
γ
0
q
(
q
-
1
)
∫
Γ
1
b
(
x
)
u
0
q
-
2
ϕ
3
2
∫
Γ
1
b
(
x
)
u
0
q
ϕ
and also,
(2.12)
β
=
γ
0
(
q
-
1
)
(
p
-
q
)
∫
Γ
1
b
(
x
)
u
0
q
-
2
ϕ
3
-
2
p
∫
Ω
ϕ
u
0
|
∇
ϕ
-
ϕ
u
0
∇
u
0
|
2
2
∫
Γ
1
b
(
x
)
u
0
q
ϕ
.
Moreover,
(2.13)
sign
(
γ
′
(
s
)
)
=
sign
(
σ
1
Ω
[
ℒ
(
λ
,
u
(
s
)
)
,
ℬ
(
γ
(
s
)
,
u
(
s
)
)
]
)
.
In particular, if p≤q, then β<0,
(2.14)
∫
Ω
a
(
x
)
u
0
p
-
2
ϕ
3
∫
Γ
1
b
(
x
)
u
0
q
-
2
ϕ
3
<
γ
0
q
(
q
-
1
)
p
(
p
-
1
)
,
and the set of positive solutions of (1.1) around (γ0,u0) possesses the structure of a quadratic subcritical turning point, and there exists δ>0 small enough such that (1.1) possesses at least two positive solutions for each γ∈(γ0-δ,γ0), one of them linearly asymptotically stable in the lower branch, and the other linearly unstable in the upper branch.
Proof.
The existence of the branch of positive solutions of (1.1)
{
(
γ
(
s
)
,
u
(
s
)
)
:
s
∈
(
-
ε
,
ε
)
}
,
its local uniqueness in a neighborhood 𝔘 of (γ0,u0) and the structure of the expansions (2.10) and (2.11) follow from [1, Theorem 2.1], applied to the operator 𝔉λ(γ,u) at (γ,u)=(γ0,u0) (cf. (1.9)).
Now (2.12) follows from (2.11) and Lemma 2.2.
To prove (2.13), we will adapt the arguments given in [2, Proposition 20.8] and [17, Proposition 2.8] to the current framework. Indeed, taking into account that (γ(s),u(s)) is a positive solution of (1.1) for s∈(-ε,ε), differentiating (1.1) with respect to s yields
(2.15)
{
ℒ
(
λ
,
u
(
s
)
)
u
′
(
s
)
=
0
in
Ω
,
ℬ
(
γ
(
s
)
,
u
(
s
)
)
u
′
(
s
)
=
(
0
,
γ
′
(
s
)
b
(
x
)
u
q
(
s
)
)
on
∂
Ω
.
Now consider ϕ(s) a principal eigenfunction associated to the principal eigenvalue σ1Ω[ℒ(λ,u(s)),ℬ(γ(s),u(s))]. By definition, the following holds:
(2.16)
{
ℒ
(
λ
,
u
(
s
)
)
ϕ
(
s
)
=
σ
1
Ω
[
ℒ
(
λ
,
u
(
s
)
)
,
ℬ
(
γ
(
s
)
,
u
(
s
)
)
]
ϕ
(
s
)
in
Ω
,
ℬ
(
γ
(
s
)
,
u
(
s
)
)
ϕ
(
s
)
=
0
on
∂
Ω
.
Multiplying the differential equation of (2.15) by ϕ(s) and the differential equation of (2.16) by u′(s), and integrating by parts in Ω, the following holds:
σ
1
Ω
[
ℒ
(
λ
,
u
(
s
)
)
,
ℬ
(
γ
(
s
)
,
u
(
s
)
)
]
∫
Ω
ϕ
(
s
)
u
′
(
s
)
=
γ
′
(
s
)
∫
Γ
1
b
(
x
)
u
q
(
s
)
ϕ
(
s
)
.
Now, since u′(0)=ϕ≫0 in Ω, by continuity and owing to (2.10), we get u′(s)≫0 in Ω for s≈0, and hence ∫Ωϕ(s)u′(s)>0 for s≈0. Then, since ∫Γ1b(x)uq(s)ϕ(s)>0, we obtain that (2.13) holds.
Finally, the fact that β<0 if p≤q follows from (2.12), taking into account that γ0>0, and (2.14) follows from (2.11), taking into account that β<0. The remaining assertions follow from (2.10) and (2.13). ∎
3 Proof of Theorem 1.1
Along this section, it will be assumed that a=0 in Ω. This section delivers the proof of Theorem 1.1. Some preliminaries are imperative for it.
Lemma 3.1.
Assume (1.17). Then there exist C(λ)>0 and, for each γ<0, a positive strict subsolution φ~λ,γ of (1.14) such that
(3.1)
∥
φ
~
λ
,
γ
∥
L
∞
(
Ω
)
=
C
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
and
(3.2)
lim
γ
↑
0
φ
~
λ
,
γ
(
x
)
=
∞
for all
x
∈
Ω
∪
Γ
1
.
Moreover, (3.2) holds uniformly in compact subsets contained in Ω∪Γ1.
Proof.
For each ε>0, let us denote σ1ε:=σ1Ω[-Δ,𝔅(V(x)+ε)]. Owing to [8, Theorem 8.2, Remark 8.3] and [8, Proposition 3.5], we have
(3.3)
σ
1
ε
>
σ
1
for all
ε
>
0
and
lim
ε
↓
0
σ
1
ε
=
σ
1
.
Since λ∈(σ1,σ0) and (3.3) holds, there exists a sufficiently small ε1=ε1(λ)>0 such that
(3.4)
σ
1
<
σ
1
ε
<
λ
for all
ε
∈
(
0
,
ε
1
)
.
Fix ε:=ε(λ)>0 satisfying (3.4), and let us denote by φε the principal eigenfunction associated to σ1ε normalized so that
(3.5)
∥
φ
ε
∥
L
∞
(
Ω
)
=
1
.
Now let us consider the function φ~λ,γ:=μ(λ,γ)φε, where
(3.6)
μ
(
λ
,
γ
)
:
=
C
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
>
0
,
C
(
λ
)
=
C
(
ε
(
λ
)
)
:
=
(
ε
2
)
1
q
-
1
.
We are going to prove that, for each γ<0, φ~λ,γ is a positive strict subsolution of (1.14). Indeed, by construction and owing to (3.4), the following holds in Ω:
(3.7)
(
-
Δ
-
λ
)
φ
~
λ
,
γ
=
μ
(
λ
,
γ
)
(
σ
1
ε
-
λ
)
φ
ε
<
0
.
As for the boundary conditions,
(3.8)
φ
~
λ
,
γ
|
Γ
0
=
μ
(
λ
,
γ
)
φ
ε
|
Γ
0
=
0
,
and since γ~>0, b⪈0, φε>0 and (3.5) holds, by construction and taking into account (3.6), the following holds on Γ1:
(3.9)
∂
φ
~
λ
,
γ
+
V
(
x
)
φ
~
λ
,
γ
+
γ
~
b
(
x
)
φ
~
λ
,
γ
q
=
μ
(
λ
,
γ
)
φ
ε
(
-
ε
+
γ
~
(
μ
(
λ
,
γ
)
)
q
-
1
b
(
x
)
φ
ε
q
-
1
)
≤
μ
(
λ
,
γ
)
φ
ε
(
-
ε
+
γ
~
μ
(
λ
,
γ
)
q
-
1
b
¯
)
=
-
ε
2
μ
(
λ
,
γ
)
φ
ε
<
0
.
Moreover, owing to (3.5), we find that
(3.10)
∥
φ
~
λ
,
γ
∥
L
∞
(
Ω
)
=
μ
(
λ
,
γ
)
=
C
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
.
Then (3.7), (3.8), (3.9) and (3.10) imply that φ~λ,γ is a positive strict subsolution of (1.14) satisfying (3.1).
On the other hand, since φε is strongly positive in Ω, we get φε(x)>0 for all x∈Ω∪Γ1, and hence
lim
γ
↑
0
φ
~
λ
,
γ
(
x
)
=
lim
γ
~
↓
0
φ
~
λ
,
γ
(
x
)
=
lim
γ
~
↓
0
(
ε
2
)
1
q
-
1
(
1
b
¯
γ
~
)
1
q
-
1
φ
ε
(
x
)
=
∞
,
which proves (3.2). To prove that (3.2) holds uniformly in compact subsets contained in Ω∪Γ1, let K be a compact subset contained in Ω∪Γ1. Since φε is bounded away from zero in any compact subset K⊂Ω∪Γ1, we get mK:=minx∈Kφε(x)>0. Then, for each x∈K,
(3.11)
φ
~
λ
,
γ
(
x
)
=
μ
(
λ
,
γ
)
φ
ε
(
x
)
≥
μ
(
λ
,
γ
)
m
K
=
(
ε
2
)
1
q
-
1
(
1
b
¯
γ
~
)
1
q
-
1
m
K
>
0
,
and since q>1, taking limits in (3.11) when γ↑0, we get that (3.2) holds uniformly in K. ∎
Lemma 3.2.
Assume (1.17) and (1.2). Then there exist D(λ)>0 and, for each γ<0, a positive strict supersolution φ¯λ,γ of (1.14) satisfying
(3.12)
∥
φ
¯
λ
,
γ
∥
L
∞
(
Ω
)
=
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
and
(3.13)
φ
¯
λ
,
γ
≥
φ
~
λ
,
γ
,
where φ~λ,γ is the positive strict subsolution of (1.14) built in Lemma 3.1.
Proof.
For each n∈𝐍, let us denote σ1n:=σ1Ω[-Δ,𝔅(V(x)+n)], and by φn the principal eigenfunction associated to σ1n, normalized so that
(3.14)
∥
φ
n
∥
L
∞
(
Ω
)
=
1
.
Since φn is strongly positive in Ω, φn(x)>0 for all x∈Γ1. Let us denote
(3.15)
m
n
:
=
min
x
∈
Γ
1
φ
n
(
x
)
>
0
,
g
(
n
)
:
=
(
n
+
1
)
1
q
-
1
m
n
>
0
Owing to [8, Proposition 3.1] and [8, Theorem 9.1], we have
(3.16)
σ
1
n
<
σ
0
for all
n
∈
ℕ
and
lim
n
↑
∞
σ
1
n
=
σ
0
.
Since λ∈(σ1,σ0) and (3.16) holds, there exists n=n(λ)∈𝐍 large enough such that
(3.17)
σ
1
<
λ
<
σ
1
n
<
σ
0
.
Fix n:=n(λ)∈𝐍 satisfying (3.17), and let us denote D~(λ):=g(n(λ))>0. Let φε and C(λ) be the principal eigenfunction associated to σ1ε and the expression C(λ) considered in Lemma 3.1, respectively. Since φε and φn are strongly positive in Ω, there exists K(λ)≥1 such that
(3.18)
C
(
λ
)
φ
ε
≤
K
(
λ
)
D
~
(
λ
)
φ
n
.
Now let us denote
(3.19)
D
(
λ
)
:
=
K
(
λ
)
D
~
(
λ
)
,
α
(
λ
,
γ
)
:
=
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
>
0
,
and let us consider the function φ¯λ,γ=α(λ,γ)φn. We are going to prove that, for each γ<0, φ¯λ,γ is a positive strict supersolution of (1.14) satisfying (3.12) and (3.13). Indeed, by construction, owing to (3.17) and (3.19), the following holds in Ω:
(3.20)
(
-
Δ
-
λ
)
φ
¯
λ
,
γ
=
α
(
λ
,
γ
)
(
σ
1
n
-
λ
)
φ
n
>
0
.
As for the boundary conditions,
(3.21)
φ
¯
λ
,
γ
|
Γ
0
=
α
(
λ
,
γ
)
φ
n
|
Γ
0
=
0
,
and since γ~>0, b(x)≥b¯>0, φn>0, k(λ)≥1, q>1 and taking into account the definition of α(λ,γ) (cf. (3.15), (3.19)), by construction, the following holds on Γ1:
(3.22)
∂
φ
¯
λ
,
γ
+
V
(
x
)
φ
¯
λ
,
γ
+
γ
~
b
(
x
)
φ
¯
λ
,
γ
q
=
α
(
λ
,
γ
)
φ
n
(
γ
~
b
(
x
)
(
α
(
λ
,
γ
)
)
q
-
1
φ
n
q
-
1
-
n
)
≥
α
(
λ
,
γ
)
φ
n
(
γ
~
b
¯
(
α
(
λ
,
γ
)
)
q
-
1
m
n
q
-
1
-
n
)
=
α
(
λ
,
γ
)
φ
n
(
(
n
+
1
)
(
k
(
λ
)
)
q
-
1
-
n
)
≥
α
(
λ
,
γ
)
m
n
>
0
.
On the other hand, by construction and owing to (3.14), we obtain
(3.23)
∥
φ
¯
λ
,
γ
∥
L
∞
(
Ω
)
=
α
(
λ
,
γ
)
=
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
.
Then (3.20)–(3.23) prove that φ¯λ,γ is a positive strict supersolution of (1.14) satisfying (3.12).
Finally, (3.13) follows by construction, taking into account (3.18). ∎
Now we are going to prove Theorem 1.1.
Proof of Theorem 1.1.
(i) To prove the necessary condition, we will argue by contradiction. Assume there exists uγ, a positive solution of (1.14) for some γ≥0. Then (1.13) holds (for a=0), and owing to the monotonicity of the principal eigenvalue with respect to the potential on the boundary (cf. [8, Proposition 3.5]) and since b⪈0, we find that
λ
=
σ
1
Ω
[
-
Δ
,
𝔅
(
V
(
x
)
-
γ
b
(
x
)
u
γ
q
-
1
)
]
≤
σ
1
Ω
[
-
Δ
,
𝔅
(
V
(
x
)
)
]
=
σ
1
,
which contradicts (1.17). The sufficient condition follows from the fact that Λ(γ)=(σ1,σ0) for each γ<0 (cf. (1.16), [7, Theorem 3.1]). The proof of the uniqueness of the positive solution of (1.14) for each γ<0 follows arguing in the same way as in the proof of [7, Theorem 3.1 (ii)].
(ii) Let γ<0, and let uγ be the unique positive solution of (1.14) for such a value γ<0 (guaranteed by (i)). The linearization of (1.14) in uγ is given by
(
ℒ
(
λ
,
u
γ
)
,
𝔅
(
γ
,
u
γ
)
)
:
=
(
-
Δ
-
λ
,
𝔅
(
V
(
x
)
+
q
γ
~
b
(
x
)
u
γ
q
-
1
)
)
.
Now, since, by (1.13),
(3.24)
λ
=
σ
1
Ω
[
-
Δ
,
𝔅
(
V
(
x
)
+
γ
~
b
(
x
)
u
γ
q
-
1
)
]
,
owing to the facts that b⪈0, q>1 and γ~>0, it follows from the monotonicity of the principal eigenvalue with respect to the potential on the boundary ([8, Proposition 3.5]) and (3.24) that
σ
1
Ω
[
ℒ
(
λ
,
u
γ
)
,
𝔅
(
γ
,
u
γ
)
]
>
σ
1
Ω
[
-
Δ
-
λ
,
𝔅
(
V
(
x
)
+
γ
~
b
(
x
)
u
γ
q
-
1
)
]
=
0
,
which proves that uγ is linearly asymptotically stable.
(iii) It follows from the existence and uniqueness of positive solution uγ of (1.14) for any γ<0, and the fact that it is linearly asymptotically stable for any γ<0 (guaranteed by (i), (ii)) and Proposition 2.1.
(iv) The existence of the branch of positive solutions Γ+(λ) and the fact that all the positive solutions of (1.14) for λ∈(σ1,σ0) belong to Γ+(λ) follow from (i). The increasing character of the branch Γ+(λ) in L∞(Ω)-norm with respect to the parameter γ<0 follows from (1.18). On the other hand, to prove (1.20) and (1.21), let us consider, for each γ<0, the unique positive solution uγ of (1.14), the positive strict supersolution φ¯λ,γ of (1.14) given by Lemma 3.2 and the positive strict subsolution φ~λ,γ of (1.14) given by Lemma 3.1 satisfying (3.13). Owing to the sub-supersolution method (cf. [2]) and the uniqueness of positive solution of (1.14), for each γ<0, we get
(3.25)
φ
~
λ
,
γ
<
u
γ
<
φ
¯
λ
,
γ
,
and hence
(3.26)
∥
φ
~
λ
,
γ
∥
L
∞
(
Ω
)
≤
∥
u
γ
∥
L
∞
(
Ω
)
≤
∥
φ
¯
λ
,
γ
∥
L
∞
(
Ω
)
.
Now (3.1), (3.12) and (3.26) imply (1.20). Finally, taking limits in (1.20) as γ↑0 and as γ↓-∞, we obtain, respectively,
lim
γ
↑
0
∥
u
γ
∥
L
∞
(
Ω
)
=
∞
,
lim
γ
↓
-
∞
∥
u
γ
∥
L
∞
(
Ω
)
=
0
,
which prove (1.21). This completes the proof of (iv).
(v) It follows from (3.25), taking into account that, owing to Lemma 3.1, (3.2) holds uniformly in compact subsets contained in Ω∪Γ1.
Finally, (vi) is [7, Theorem 3.1 (ii), Theorem 3.8 (ii)]. ∎
4 A One-Dimensional Example of Case (1)
In this section, we will consider the one-dimensional mixed boundary value problem
(4.1)
{
-
u
′′
(
x
)
=
λ
u
(
x
)
,
0
<
x
<
1
,
u
(
0
)
=
0
,
u
′
(
1
)
-
u
(
1
)
=
γ
b
(
u
(
1
)
)
q
,
q
>
1
,
where γ<0 and b>0. This problem fits into the framework of (1.14) with
N
=
1
,
Ω
=
(
0
,
1
)
,
Γ
0
=
{
x
0
=
0
}
,
Γ
1
=
{
x
1
=
1
}
,
b
¯
=
b
¯
=
b
,
V
=
-
1
It can be explicitly solved, and the main goal of this section is to show, through the explicit solution of (4.1), the consequences of Theorem 1.1 in the general case. In this framework, σ1=0 is the principal eigenvalue associated to the eigenvalue problem
{
-
u
(
x
)
′′
=
λ
u
(
x
)
,
0
<
x
<
1
,
u
(
0
)
=
0
,
u
′
(
1
)
-
u
(
1
)
=
0
,
and σ0=π2 is the principal eigenvalue associated to the eigenvalue problem
{
-
u
(
x
)
′′
=
λ
u
(
x
)
,
0
<
x
<
1
,
u
(
0
)
=
0
,
u
(
1
)
=
0
.
Then, taking any fixed λ∈(σ1,σ0)=(0,π2), it is straightforward to see that, for each γ<0, (4.1) possesses a unique positive solution (as Theorem 1.1 (i) establishes). Namely,
(4.2)
u
γ
(
x
)
=
H
(
λ
)
(
1
b
γ
~
)
1
q
-
1
sin
(
λ
x
)
,
where
H
(
λ
)
:
=
(
sin
(
λ
)
-
λ
cos
(
λ
)
(
sin
(
λ
)
)
q
)
1
q
-
1
>
0
It should be noted that H(λ)>0 for each λ∈(0,π2). Indeed, if λ∈(0,π24), then
sin
(
λ
)
>
0
,
cos
(
λ
)
>
0
,
tan
(
λ
)
>
λ
,
which imply H(λ)>0, and if λ∈[π24,π2), then H(λ)>0 because, in this case, sin(λ)>0 and cos(λ)≤0. It is easily seen that, for any fixed λ∈(0,π2) and any γ<0,
(4.3)
∥
u
γ
∥
L
∞
(
0
,
1
)
=
u
γ
(
x
λ
)
=
H
~
(
λ
)
(
1
b
γ
~
)
1
q
-
1
,
where
x
λ
=
{
1
if
λ
∈
(
0
,
π
2
4
]
,
π
2
λ
if
λ
∈
(
π
2
4
,
π
2
)
,
H
~
(
λ
)
=
{
H
(
λ
)
sin
(
λ
)
if
λ
∈
(
0
,
π
2
4
]
,
H
(
λ
)
if
λ
∈
(
π
2
4
,
π
2
)
.
Also, since λ∈(0,π), for any x∈(0,1], we get λx∈(0,π); hence sin(λx)>0, and therefore it follows from (4.2) that
(4.4)
lim
γ
↑
0
u
γ
(
x
)
=
lim
γ
~
↓
0
u
γ
(
x
)
=
∞
,
0
<
x
≤
1
.
According to Theorem 1.1, in this particular example, (4.3) shows that the branch Γ+(λ):={(γ,uγ):γ<0} is increasing with γ in ∥⋅∥L∞(Ω), that (1.20) holds for C(λ)=D(λ)=H~(λ), and that Γ+(λ) bifurcates to positive solutions from the trivial branch (γ,u)=(γ,0) as γ↓-∞ and from infinity as γ↑0, with rate γ~-1q-1 in both cases, up to multiplicative constants. Similarly, (4.4) and (4.2) show that (1.22) holds uniformly in compact subsets of Ω∪Γ1=(0,1].
5 Proof of Theorem 1.2
In the sequel, for each ρ>0, we set
𝒜
ρ
:
=
(
Γ
1
+
B
ρ
(
0
)
)
∩
Ω
,
Γ
0
ρ
:
=
∂
𝒜
ρ
∖
Γ
1
⊂
Ω
,
Ω
ρ
:
=
Ω
∖
𝒜
¯
ρ
,
where Bρ(0) denotes the ball of ℝN centered at the origin with radius ρ>0. The following result will be used in the proof of Theorem 1.2.
Lemma 5.1.
Let I=[α,β]⊂R with 0≤α<β, and let us assume that (1.17) holds and there exist constants ρ~>0 and C:=C(ρ~)>0 such that
(5.1)
∥
u
γ
∥
L
∞
(
𝒜
ρ
~
)
≤
C
for all
γ
∈
[
α
,
β
]
,
where uγ denotes any positive solution of (1.1) for the value γ>0 of the parameter. Then there exists C1>0 such that
(5.2)
∥
u
γ
∥
L
∞
(
Ω
)
≤
C
1
for all
γ
∈
[
α
,
β
]
.
Proof.
Let ρ~>0, and let C:=C(ρ~)>0 satisfy (5.1). Let ρ1∈(0,ρ~) be sufficiently small so that
𝒜
¯
ρ
1
∩
Ω
¯
0
=
∅
,
𝒜
¯
ρ
1
∩
Γ
0
=
∅
,
whose existence is guaranteed by the facts that Γ1∩Γ0=∅ and Ω¯0⊂Ω. Owing to (5.1), any positive solution uγ of (1.1) for γ∈I satisfies
(5.3)
{
-
Δ
u
γ
=
λ
u
γ
-
a
(
x
)
u
γ
p
in
Ω
ρ
1
,
u
γ
=
0
on
Γ
0
,
u
γ
≤
C
on
Γ
0
ρ
1
.
Also, let u0 be the unique positive solution of (1.1) for γ=0, whose existence and uniqueness are guaranteed by [11, Theorem 3.5], [5, Theorem 4.2]. Since u0 is strongly positive in Ω, we get
u
¯
0
:
=
min
x
∈
Γ
0
ρ
1
u
0
(
x
)
>
0
,
and taking
k
>
max
{
1
,
C
u
¯
0
}
,
it is easy to see that the function u¯:=ku0 satisfies
(5.4)
{
(
-
Δ
-
λ
+
a
(
x
)
u
¯
p
-
1
)
u
¯
>
0
in
Ω
ρ
1
,
u
¯
=
0
on
Γ
0
,
u
¯
>
C
>
0
on
Γ
0
ρ
1
.
Then, owing to (5.3) and (5.4), the function ψγ:=u¯-uγ satisfies
(5.5)
{
(
-
Δ
-
λ
+
a
(
x
)
u
¯
p
-
u
γ
p
u
¯
-
u
γ
)
ψ
γ
>
0
in
Ω
ρ
1
,
ψ
γ
=
0
on
Γ
0
,
ψ
γ
>
0
on
Γ
0
ρ
1
.
On the other hand, since u¯>0 fits (5.4), it follows from the characterization of the strong maximum principle (cf. [4, Theorem 2.4]) that
(5.6)
σ
1
Ω
ρ
1
[
-
Δ
-
λ
+
a
(
x
)
u
¯
p
-
1
,
𝒟
]
>
0
.
Also, since a⪈0 and
u
¯
p
-
u
γ
p
u
¯
-
u
γ
>
u
¯
p
-
1
,
it follows from the monotonicity of the principal eigenvalue with respect to the potential and (5.6) that
(5.7)
σ
1
Ω
ρ
1
[
-
Δ
-
λ
+
a
(
x
)
u
¯
p
-
u
γ
p
u
¯
-
u
γ
,
𝒟
]
>
σ
1
Ω
ρ
1
[
-
Δ
-
λ
+
a
(
x
)
u
¯
p
-
1
,
𝒟
]
>
0
.
Now, owing again to the characterization of the strong maximum principle, it follows from (5.7) that (5.5) satisfies the strong maximum principle, and hence ψγ≫0 in Ω¯ρ1, that is,
(5.8)
u
γ
≪
u
¯
in
Ω
¯
ρ
1
for all
γ
∈
[
α
,
β
]
.
Now (5.1) and (5.8) imply
∥
u
γ
∥
L
∞
(
Ω
)
≤
C
1
:
=
max
{
C
,
∥
u
¯
∥
L
∞
(
Ω
ρ
1
)
}
for all
γ
∈
[
α
,
β
]
,
which proves (5.2). ∎
Proof of Theorem 1.2.
(i) The proofs of the existence and uniqueness of positive solution for each γ≤0 follow from [11, Theorem 3.5], [5, Theorem 4.2] in the particular case when γ=0, and from [6, Theorem 3.1] when γ<0. The fact that uγ is linearly asymptotically stable for each γ≤0 follows arguing as in the proof of Theorem 1.1 (ii), taking into account that a⪈0 in Ω and using the monotonicity of the principal eigenvalue with respect to the potential in the differential operator (cf. [8, Proposition 3.3]). The remaining assertions of (i) follow from the fact that uγ is linearly asymptotically stable for each γ≤0, by Proposition 2.1. This completes the proof of (i).
(ii) Since λ∈(σ1,σ0*)⊂(σ1,σ0) (cf. (1.7), (1.8)), for each γ<0, let vγ be the unique positive solution of (1.14), whose existence and uniqueness are guaranteed in Theorem 1.1, and let uγ be the unique positive solution of (1.1), whose existence and uniqueness have been proved in (i). Owing to the fact that (1.2) holds, it follows from (1.20) the existence of D(λ)>0 such that
(5.9)
∥
v
γ
∥
L
∞
(
Ω
)
≤
D
(
λ
)
(
1
b
¯
γ
~
)
1
q
-
1
,
γ
<
0
.
On the other hand, it is easy to prove that uγ is a positive strict subsolution of (1.14), and hence, owing to the characterization of the strong maximum principle (cf. [4, Theorem 2.4]), we obtain (cf.[7, Lemma 3.5])
(5.10)
u
γ
<
v
γ
.
Now (1.25) follows from (5.9) and (5.10), and (1.26) follows letting γ↓-∞ in (1.25). This completes the proof of (ii).
(iii) It follows from the fact that u0 is linearly asymptotically stable, as it has been proved in (i), and Proposition 2.1.
(iv) Let u^γ be any positive solution of (1.1) for some γ>0. Then, by construction, the function φγ:=u^γ-u0 satisfies
(5.11)
{
(
-
Δ
-
λ
+
a
(
x
)
u
^
γ
p
-
u
0
p
u
^
γ
-
u
0
)
φ
γ
=
0
in
Ω
,
φ
γ
=
0
on
Γ
0
,
∂
φ
γ
+
V
(
x
)
φ
γ
=
γ
b
(
x
)
u
^
γ
q
>
0
on
Γ
1
.
Also, due to (1.13), we obtain
(5.12)
σ
1
Ω
[
-
Δ
-
λ
+
a
(
x
)
u
0
p
-
1
,
𝔅
(
V
(
x
)
)
]
=
0
,
and since a⪈0 in Ω and
u
^
γ
p
-
u
0
p
u
^
γ
-
u
0
>
u
0
p
-
1
≫
0
,
it follows from the monotonicity of the principal eigenvalue with respect to the potential (cf. [8, Proposition 3.3]) and (5.12) that
(5.13)
σ
1
Ω
[
-
Δ
-
λ
+
a
(
x
)
u
^
γ
p
-
u
0
p
u
^
γ
-
u
0
,
𝔅
(
V
)
]
>
σ
1
Ω
[
-
Δ
-
λ
+
a
(
x
)
u
0
p
-
1
,
𝔅
(
V
)
]
=
0
.
Now, owing to the characterization of the strong maximum principle, it follows from (5.13) that
(
-
Δ
-
λ
+
a
(
x
)
u
^
γ
p
-
u
0
p
u
^
γ
-
u
0
,
𝔅
(
V
)
,
Ω
)
satisfies the strong maximum principle, and therefore (5.11) implies φγ≫0 in Ω, that is, (1.27) holds. This completes the proof of (iv).
(v) The proof of (v) (a) follows from [6, Theorem 4.7 (i)] since, in particular, it establishes that
(
σ
1
,
σ
0
)
⊂
⋂
γ
>
0
Λ
(
γ
)
.
To prove (v) (b), let u^γ be any positive solution of (1.1) for some γ>0, whose existence is guaranteed by (v) (a). Owing to (iv), (1.27) holds. On the other hand, it is easy to see that u0 is a positive strict subsolution of (1.1) for each γ>0. Then we get that [u0,u^γ] is an ordered sub-supersolution pair of (1.1) for γ>0, and hence [2, Theorem 6.1] implies the existence of a minimal solution uγmin of (1.1) in the interval (u0,u^γ] for each γ>0, which, together with (iv), proves (1.28).
We now prove (1.29). Indeed, by the definition of uγmin and owing to (1.28), we get, for each γ∈(0,ε0),
(5.14)
u
0
≪
u
γ
min
≤
u
γ
*
.
On the other hand, by the continuity of the branch Γ+(λ,ε0)={(γ,uγ*):γ∈[0,ε0)}, we find that
(5.15)
lim
γ
↓
0
∥
u
γ
*
-
u
0
∥
L
∞
(
Ω
)
=
0
,
and hence (5.14) and (5.15) imply that
(5.16)
lim
γ
↓
0
∥
u
γ
min
-
u
0
∥
L
∞
(
Ω
)
=
0
.
Now (1.29) follows from (5.16) and the fact that the unique positive solution of (1.1) for each γ∈(-ε0,ε0) in a small neighborhood 𝔘 of (γ,u)=(0,u0) in (-ε0,ε0)×𝒞Γ01(Ω¯) is given by uγ* (cf. (iii)). This ends the proof of (v) (b).
We now prove (v) (c). The existence of uniform L∞(Ω)-bounds when γ varies in compact intervals I=[α,β]⊂(-∞,0] follows from the existence and uniqueness of positive solutions of (1.1) guaranteed by (i) and the fact that the map (1.24) is increasing in L∞(Ω). In particular, ∥uγ∥L∞(Ω)≤∥u0∥L∞(Ω) for all γ≤0. To prove the existence of uniform L∞(Ω)-bounds when γ varies in compact intervals I=[α,β]⊂[0,∞), we will argue by contradiction, using rescaling and blow-up arguments, much like in [6, Theorem 4.4], but now, varying γ instead of λ. Suppose there exists a compact interval I=[α,β] with α≥0 of values of γ such that the positive solutions of (1.1) are not uniformly bounded in L∞(Ω). Then Lemma 5.1 implies the existence of a sequence (γk,uk), k≥1, of positive solutions of (1.1) with γk∈I, k≥1, a sequence of points {xk}k=1∞⊂Ω¯ and a point x∞∈Γ1 such that
(5.17)
M
k
:
=
u
k
(
x
k
)
=
sup
x
∈
Ω
u
k
(
x
)
→
∞
,
x
k
→
x
∞
∈
Γ
1
.
Since γk∈I, k≥1, taking a subsequence if it is necessary, we can assume without loss of generality that
(5.18)
lim
k
→
∞
γ
k
=
γ
∞
∈
[
α
,
β
]
.
Also, since x∞∈Γ1 and the potential a is bounded away from zero in any compact subset of (Ω∖Ω¯0)∪Γ1 (cf. (1.3)), we obtain
(5.19)
a
(
x
∞
)
>
0
Set
(5.20)
ρ
k
:
=
M
k
-
p
-
1
2
.
Since p>1, (5.17) implies
(5.21)
lim
k
→
∞
ρ
k
=
0
.
We will denote by xj the j-th coordinate of x. Since Γ1 is smooth enough, by straightening Γ1 in a neighborhood of x∞ by a non-singular change of coordinates, we can assume that
B
δ
(
x
∞
)
∩
Ω
=
{
x
∈
ℝ
N
:
x
N
>
0
}
∩
B
δ
(
x
∞
)
,
B
δ
(
x
∞
)
∩
Γ
1
=
{
x
∈
ℝ
N
:
x
N
=
0
}
∩
B
δ
(
x
∞
)
,
for some fixed δ small enough, where Bδ(x∞) stands for the ball of radius δ>0 centered at the point x∞. Since xk→x∞ as k→∞, |xk-x∞|<δ2, for k large enough, and hence |x-xk|<δ2 implies |x-x∞|<δ. Thus, for k large enough, Bδ2(xk)⊂Bδ(x∞), and hence
Ω
k
:
=
B
δ
2
(
x
k
)
∩
{
x
∈
ℝ
N
:
x
N
>
0
}
⊂
B
δ
(
x
∞
)
∩
Ω
.
Now, taking into account (5.20) and making the change of variable
(5.22)
y
=
x
-
x
k
ρ
k
,
v
k
(
y
)
:
=
u
k
(
x
)
M
k
=
u
k
(
x
k
+
ρ
k
y
)
M
k
,
Ω
k
transforms into
Ω
~
k
:
=
B
δ
2
ρ
k
(
0
)
∩
{
y
∈
ℝ
N
:
y
N
>
-
x
k
N
ρ
k
}
,
the differential equation
-
Δ
u
k
=
λ
u
k
(
x
)
-
a
(
x
)
u
k
p
(
x
)
,
x
∈
Ω
k
,
into
-
Δ
v
k
(
y
)
=
λ
ρ
k
2
v
k
(
y
)
-
a
(
x
k
+
ρ
k
y
)
v
k
p
(
y
)
,
y
∈
Ω
~
k
and the boundary condition
∂
u
k
+
V
(
x
)
u
k
(
x
)
=
γ
k
b
(
x
)
u
k
q
(
x
)
,
x
∈
Ω
¯
k
∩
Γ
1
,
into
-
∂
v
k
(
y
)
∂
y
N
+
V
(
x
k
+
ρ
k
y
)
ρ
k
v
k
(
y
)
=
γ
k
b
(
x
k
+
ρ
k
y
)
M
k
2
q
-
1
-
p
2
v
k
q
(
y
)
,
with
y
∈
{
y
∈
ℝ
N
:
y
N
=
-
x
k
N
ρ
k
}
∩
B
¯
δ
2
ρ
k
(
0
)
By construction, we get that, for k≥1 large enough, vk(y) is well defined in Ω~k, and owing to (5.22), the following holds:
(5.23)
0
<
v
k
(
y
)
≤
v
k
(
0
)
=
u
k
(
x
k
)
M
k
-
1
=
1
,
y
∈
Ω
~
k
.
Now, arguing as in [6, Theorem 4.4], distinguishing the different behaviors of the sequence xkNρk as k→∞, and taking into account that p>2q-1, (5.17)–(5.19), (5.21) and (5.23), we get that if the sequence xkNρk is not bounded away from zero, then there exists a non-negative function w∞∈Wr2(BR+(0)) for some R>0, attaining its maximum at y=0 and satisfying
-
Δ
w
∞
(
0
)
=
-
a
(
x
∞
)
w
∞
p
(
0
)
=
-
a
∞
<
0
,
which contradicts the fact that w∞(y) attains its maximum at y=0, and if the sequence xkNρk is either not bounded above, or it is bounded above and bounded away from zero, then there exists of a non-negative function v∞(y)∈Wr2(BR(0)) for some R>0, satisfying
-
Δ
v
∞
(
y
)
=
-
a
(
x
∞
)
v
∞
p
(
y
)
<
0
in
B
R
(
0
)
and attaining its maximum at y=0, which gives again a contradiction. This completes the proof of (v) (c). ∎
