2 Mathematical background and hypotheses
Let X be a Banach space and
X
*
its topological dual. By
〈
⋅
,
⋅
〉
we denote the duality brackets for the pair
(
X
*
,
X
)
.
Given
φ
∈
C
1
(
X
,
ℝ
)
, we say that φ satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds:
Every sequence
{
u
n
}
n
≥
1
⊆
X
such that
{
φ
(
u
n
)
}
n
≥
1
⊆
ℝ
is bounded and
(
1
+
∥
u
n
∥
X
)
φ
′
(
u
n
)
→
0
in
X
*
as
n
→
∞
admits a strongly convergent subsequence.
This is a compactness-type condition on the functional φ, more general than the usual Palais–Smale condition.
It leads to a deformation theorem from which
one can deduce the minimax theory of the critical values of φ.
One of the main results in that theory is the so-called “mountain pass theorem”, which we recall here.
Theorem 2.1.
If X is a Banach space,
φ
∈
C
1
(
X
,
R
)
satisfies the C-condition,
u
0
,
u
1
∈
X
,
∥
u
1
-
u
0
∥
X
>
ρ
>
0
,
max
{
φ
(
u
0
)
,
φ
(
u
1
)
}
<
inf
[
φ
(
u
)
:
∥
u
-
u
0
∥
X
=
ρ
]
=
η
ρ
,
and
c
=
inf
γ
∈
Γ
max
0
≤
t
≤
1
φ
(
γ
(
t
)
)
,
with
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
u
0
,
γ
(
1
)
=
u
1
}
,
then
c
≥
η
ρ
and c is a critical value of φ.
In the analysis of problem (1.1), we will use the Sobolev spaces
W
0
1
,
p
(
Ω
)
and
H
0
1
(
Ω
)
.
Since
2
<
p
, we have that
W
0
1
,
p
(
Ω
)
↪
H
0
1
(
Ω
)
densely.
By
∥
⋅
∥
we denote the norm of
W
0
1
,
p
(
Ω
)
.
By the Poincaré inequality, we have
∥
u
∥
=
∥
D
u
∥
p
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Also we will use the Banach space
C
0
1
(
Ω
¯
)
=
{
u
∈
C
1
(
Ω
¯
)
:
u
|
∂
Ω
=
0
}
.
This is an ordered Banach space with positive (order) cone
C
+
=
{
u
∈
C
0
1
(
Ω
¯
)
:
u
(
z
)
≥
0
for all
z
∈
Ω
¯
}
.
This cone has a nonempty interior given by
int
C
+
=
{
u
∈
C
+
:
u
(
z
)
>
0
for all
z
∈
Ω
,
∂
u
∂
n
|
∂
Ω
<
0
}
.
Here
∂
u
∂
n
=
(
D
u
,
n
)
ℝ
N
, with
n
(
⋅
)
being the outward unit normal on
∂
Ω
.
For every
r
∈
(
1
,
+
∞
)
, let
A
r
:
W
0
1
,
r
(
Ω
)
→
W
-
1
,
r
′
(
Ω
)
=
W
0
1
,
r
(
Ω
)
*
(
1
r
+
1
r
′
=
1
) be the map defined by
〈
A
r
(
u
)
,
h
〉
=
∫
Ω
|
D
u
|
r
-
2
(
D
u
,
D
h
)
ℝ
N
𝑑
z
for all
u
,
h
∈
W
0
1
,
r
(
Ω
)
.
For this map, we have the following result (see [18, p. 40]).
Proposition 2.2.
The map
A
r
(
⋅
)
is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone, too) and of type
(
S
)
+
, that is, if
u
n
→
𝑤
u
in
W
0
1
,
r
(
Ω
)
and
lim sup
n
→
∞
〈
A
r
(
u
n
)
,
u
n
-
u
〉
≤
0
,
then
u
n
→
u
in
W
0
1
,
r
(
Ω
)
.
If
r
=
2
, then
A
2
=
A
∈
L
(
H
0
1
(
Ω
)
,
H
-
1
(
Ω
)
)
.
Suppose that
f
0
:
Ω
×
ℝ
→
ℝ
is a Caratheodory function such that
|
f
0
(
z
,
x
)
|
≤
a
0
(
z
)
[
1
+
|
x
|
r
-
1
]
for a.a.
z
∈
Ω
and all
x
∈
ℝ
,
with
a
0
∈
L
∞
(
Ω
)
+
and
1
<
r
≤
p
*
=
{
N
p
N
-
p
if
p
<
N
,
+
∞
if
N
≤
p
(the critical Sobolev exponent for p).
We set
F
0
(
z
,
x
)
=
∫
0
x
f
0
(
z
,
s
)
𝑑
s
and we consider the
C
1
-functional
φ
0
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
0
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
0
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
The next result can be found in [18, p. 409].
Proposition 2.3.
If
u
0
∈
W
0
1
,
p
(
Ω
)
is a local
C
0
1
(
Ω
¯
)
-minimizer of
φ
0
, that is, there exists
ρ
0
>
0
such that
φ
0
(
u
0
)
≤
φ
0
(
u
0
+
h
)
for all
h
∈
C
0
1
(
Ω
¯
)
, with
∥
h
∥
C
0
1
(
Ω
¯
)
≤
ρ
0
,
then
u
0
∈
C
0
1
,
α
(
Ω
¯
)
for some
α
∈
(
0
,
1
)
and
u
0
is also a local
W
0
1
,
p
(
Ω
)
-minimizer of
φ
0
, that is, there exists
ρ
1
>
0
such that
φ
0
(
u
0
)
≤
φ
0
(
u
0
+
h
)
for all
h
∈
W
0
1
,
p
(
Ω
)
, with
∥
h
∥
≤
ρ
1
.
To make an effective use of Proposition 2.3, we need a strong comparison principle, which is provided by the next proposition and is a particular case of [9, Proposition 3].
Given
h
1
,
h
2
∈
L
∞
(
Ω
)
we write
h
1
≺
h
2
if for every
K
⊆
Ω
compact, we can find
ε
=
ε
(
K
)
>
0
such that
h
1
(
z
)
+
ε
≤
h
2
(
z
)
for a.a.
z
∈
K
.
Evidently, if
h
1
,
h
2
∈
C
(
Ω
)
, and
h
1
(
z
)
<
h
2
(
z
)
for all
z
∈
Ω
, then
h
1
≺
h
2
.
Proposition 2.4.
If
ξ
,
h
1
,
h
2
∈
L
∞
(
Ω
)
,
ξ
(
z
)
≥
0
for a.a.
z
∈
Ω
,
h
1
≺
h
2
,
u
∈
C
0
1
(
Ω
¯
)
∖
{
0
}
,
v
∈
int
C
+
and
-
Δ
p
u
(
z
)
-
Δ
u
(
z
)
+
ξ
(
z
)
|
u
(
z
)
|
p
-
2
u
(
z
)
=
h
1
(
z
)
for a.a.
z
∈
Ω
,
-
Δ
p
v
(
z
)
-
Δ
v
(
z
)
+
ξ
(
z
)
v
(
z
)
p
-
1
=
h
2
(
z
)
for a.a.
z
∈
Ω
,
then
v
-
u
∈
int
C
+
.
Next we recall some basic facts about the spectrum of the Dirichlet p-Laplacian. So, we consider the following nonlinear eigenvalue problem:
(2.1)
-
Δ
p
u
(
z
)
=
λ
^
|
u
(
z
)
|
p
-
2
u
(
z
)
in
Ω
,
u
|
∂
Ω
=
0
.
We say that
λ
^
∈
ℝ
is an eigenvalue of
(
-
Δ
p
,
W
0
1
,
p
(
Ω
)
)
if problem (2.1) admits a nontrivial solution
u
^
∈
W
0
1
,
p
(
Ω
)
, known as an eigenfunction corresponding to the eigenvalue
λ
^
.
By
σ
^
(
p
)
⊆
ℝ
we denote the spectrum of
(
-
Δ
p
,
W
0
1
,
p
(
Ω
)
)
(that is, the set of eigenvalues). We know that there exists a smallest eigenvalue
λ
^
1
(
p
)
∈
ℝ
, which has the following properties:
λ
^
1
(
p
)
>
0
and it is isolated in the spectrum
σ
^
(
p
)
(that is, there exists
ε
>
0
such that
(
λ
^
1
(
p
)
,
λ
^
1
(
p
)
+
ε
)
∩
σ
^
(
p
)
=
∅
)
,
λ
^
1
(
p
)
is simple (that is, if
u
^
1
,
u
^
2
are eigenfunctions corresponding to
λ
^
1
(
p
)
, then
u
^
1
=
ξ
u
^
2
with
ξ
≠
0
),
we have
(2.2)
λ
^
1
(
p
)
=
inf
[
∥
D
u
∥
p
p
∥
u
∥
p
p
:
u
∈
W
0
1
,
p
(
Ω
)
,
u
≠
0
]
.
The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace.
Evidently, the elements of this eigenspace have constant sign.
By
u
^
1
(
p
)
we denote the positive
L
p
-normalized (that is,
∥
u
^
1
(
p
)
∥
p
=
1
) eigenfunction corresponding to the eigenvalue
λ
^
1
(
p
)
>
0
.
The nonlinear regularity theory and the nonlinear maximum principle (see [10, pp. 737–738]) imply that
u
^
1
∈
int
C
+
.
It is easy to see that
σ
^
(
p
)
⊆
(
0
,
+
∞
)
is closed and using the Ljusternik–Schnirelmann minimax scheme, we produce a whole sequence of distinct eigenvalues
{
λ
^
k
(
p
)
}
k
∈
ℕ
of
(
-
Δ
p
,
W
0
1
,
p
(
Ω
)
)
such that
λ
^
k
(
p
)
→
+
∞
as
k
→
+
∞
.
These eigenvalues are known as “variational eigenvalues” and we do not know if they exhaust
σ
^
(
p
)
.
This is the case if
N
=
1
(scalar eigenvalue problem) or if
p
=
2
(linear eigenvalue problem).
All eigenvalues
λ
^
≠
λ
^
1
(
p
)
have nodal eigenfunctions.
In what follows, for notational simplicity, we write
λ
^
1
=
λ
^
1
(
p
)
and
u
^
1
=
u
^
1
(
p
)
∈
int
C
+
.
We will also encounter a weighted version of the eigenvalue problem (2.1). Namely, let
η
∈
L
∞
(
Ω
)
, with
η
(
z
)
≥
0
for a.a.
z
∈
Ω
,
η
≢
0
.
We consider the following nonlinear eigenvalue problem:
-
Δ
p
u
(
z
)
=
λ
~
η
(
z
)
|
u
(
z
)
|
p
-
2
u
(
z
)
in
Ω
,
u
|
∂
Ω
=
0
.
The same results are true for this problem. So, there exists a smallest eigenvalue
λ
~
1
(
p
,
η
)
>
0
, which is isolated, simple and admits the following variational characterization:
λ
~
1
(
p
,
η
)
=
inf
[
∥
D
u
∥
p
p
∫
Ω
η
(
z
)
|
u
|
p
𝑑
z
:
u
∈
W
0
1
,
p
(
Ω
)
,
u
≠
0
]
.
These properties lead to the following monotonicity property for the map
η
↦
λ
~
1
(
p
,
η
)
(see [18, p. 250]).
Proposition 2.5.
If
η
,
η
^
∈
L
∞
(
Ω
)
,
0
≤
η
(
z
)
≤
η
^
(
z
)
for a.a.
z
∈
Ω
,
η
≢
0
,
η
^
≢
η
, then
λ
~
1
(
p
,
η
^
)
<
λ
~
1
(
p
,
η
)
.
Next let us recall some basic definitions and facts from the theory of critical groups (Morse theory).
So, let X be a Banach space,
φ
∈
C
1
(
X
,
ℝ
)
and
c
∈
ℝ
.
We introduce the following sets:
K
φ
=
{
u
∈
X
:
φ
′
(
u
)
=
0
}
(
the critical set of
φ
)
,
K
φ
c
=
{
u
∈
K
φ
:
φ
(
u
)
=
c
}
(
the critical set of
φ
at the level
c
)
,
φ
c
=
{
u
∈
X
:
φ
(
u
)
≤
c
}
(
the sublevel set of
φ
at
c
)
.
Let
(
Y
1
,
Y
2
)
be a topological pair such that
Y
2
⊆
Y
1
⊆
X
. For any
k
∈
ℕ
0
, by
H
k
(
Y
1
,
Y
2
)
we denote the kth-relative singular homology group with integer coefficients for the pair
(
Y
1
,
Y
2
)
.
Recall that
H
k
(
Y
1
,
Y
2
)
=
0
for all
k
∈
-
ℕ
.
If
u
∈
K
φ
c
is isolated, then the critical groups of φ at u are defined by
C
k
(
φ
,
u
)
=
H
k
(
φ
c
∩
U
,
φ
c
∩
U
∖
{
u
}
)
for all
k
∈
ℕ
0
,
with U being a neighborhood of u such that
K
φ
∩
φ
c
∩
U
=
{
u
}
.
The excision property of singular homology implies that this definition of critical groups above is independent of the choice of the neighborhood U.
If
u
∈
K
φ
is isolated and of mountain pass-type (see Theorem 2.1), then
C
1
(
φ
,
u
)
≠
0
. Moreover, if
φ
∈
C
2
(
X
,
ℝ
)
, then from [21] we know that
C
k
(
φ
,
u
)
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
,
with
δ
k
,
m
being the Kronecker symbol defined by
δ
k
,
m
=
{
1
if
k
=
m
,
0
if
k
≠
m
,
k
,
m
∈
ℕ
0
.
Next we fix our notation. Given
x
∈
ℝ
, we set
x
±
=
max
{
±
x
,
0
}
.
Then for
u
∈
W
0
1
,
p
(
Ω
)
, we define
u
±
(
⋅
)
=
u
(
⋅
)
±
.
We know that
u
±
∈
W
0
1
,
p
(
Ω
)
,
u
=
u
+
-
u
-
,
|
u
|
=
u
+
+
u
-
. Also, given a measurable function
g
:
Ω
×
ℝ
→
ℝ
(for example, a Caratheodory function), we define
N
g
(
u
)
(
⋅
)
=
g
(
⋅
,
u
(
⋅
)
)
for all
u
∈
W
0
1
,
p
(
Ω
)
,
which is the Nemytskii operator corresponding to g. By
|
⋅
|
N
we denote the Lebesgue measure on
ℝ
N
, and if
1
<
r
<
+
∞
, then
1
r
+
1
r
′
=
1
.
If
u
,
u
^
∈
W
1
,
p
(
Ω
)
and
u
≤
u
^
, then we define
[
u
,
u
^
]
=
{
y
∈
W
0
1
,
p
(
Ω
)
:
u
(
z
)
≤
y
(
z
)
≤
u
^
(
z
)
for a.a.
z
∈
Ω
}
.
By
int
C
0
1
(
Ω
¯
)
[
u
,
u
^
]
, we denote the interior of
[
u
,
u
^
]
∩
C
0
1
(
Ω
¯
)
in the
C
0
1
(
Ω
¯
)
-norm.
Also, if
k
:
(
0
,
+
∞
)
→
C
0
1
(
Ω
¯
)
, then we say that
k
(
⋅
)
is strictly increasing if
λ
<
λ
′
implies that
k
(
λ
′
)
-
k
(
λ
)
∈
int
C
+
.
Finally, if
u
∈
W
1
,
p
(
Ω
)
, then
[
u
)
=
{
y
∈
W
0
1
,
p
(
Ω
)
:
u
(
z
)
≤
y
(
z
)
for a.a.
z
∈
Ω
}
.
Now we can introduce the hypotheses on the perturbation term
f
(
z
,
x
)
.
Hypothesis 2.6.
f
:
Ω
×
ℝ
→
ℝ
is a Caratheodory function such that
f
(
z
,
0
)
=
0
for a.a.
z
∈
Ω
, which satisfies the following properties:
For every
ρ
>
0
, there exists
a
ρ
∈
L
∞
(
Ω
)
such that
|
f
(
z
,
x
)
|
≤
a
ρ
(
z
)
for a.a.
z
∈
Ω
and all
|
x
|
≤
ρ
.
We have
λ
^
1
≤
lim inf
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
-
2
x
≤
lim sup
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
-
2
x
≤
η
^
uniformly for a.a.
z
∈
Ω
.
If
F
(
z
,
x
)
=
∫
0
x
f
(
z
,
s
)
𝑑
s
, then
lim
x
→
±
∞
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
x
2
=
-
∞
uniformly for a.a.
z
∈
Ω
.
There exist functions
w
±
∈
W
1
,
p
(
Ω
)
∩
L
∞
(
Ω
)
, with
Δ
p
w
±
,
Δ
w
±
∈
L
p
′
(
Ω
)
, and
θ
±
>
0
such that, for a.a.
z
∈
Ω
,
w
-
(
z
)
≤
-
c
-
<
0
<
c
+
≤
w
+
(
z
)
,
-
Δ
p
w
-
(
z
)
-
Δ
w
-
(
z
)
≤
0
≤
-
Δ
p
w
+
(
z
)
-
Δ
w
+
(
z
)
,
θ
+
w
+
(
z
)
q
-
1
+
f
(
z
,
w
+
(
z
)
)
≤
-
c
^
+
<
0
≤
c
^
-
≤
θ
-
|
w
-
(
z
)
|
q
-
2
w
-
(
z
)
+
f
(
z
,
w
-
(
z
)
)
.
If
c
0
=
min
{
c
+
,
c
-
}
>
0
, then we can find
δ
0
∈
(
0
,
c
0
)
such that for all
K
⊆
Ω
compact and all
0
<
m
<
δ
0
, we have
0
<
c
K
m
≤
f
(
z
,
x
)
x
≤
q
F
(
z
,
x
)
for a.a.
z
∈
K
and all
m
≤
|
x
|
≤
δ
0
.
For every
ρ
>
0
, there exists
ξ
^
ρ
>
0
such that for a.a.
z
∈
Ω
, the function
x
↦
f
(
z
,
x
)
+
ξ
^
ρ
|
x
|
p
-
2
x
is nondecreasing on
[
-
ρ
,
ρ
]
.
Remark 2.7.
Hypotheses 2.6 (i), (ii) imply that
|
f
(
z
,
x
)
|
≤
c
1
[
1
+
|
x
|
p
-
1
]
for a.a.
z
∈
Ω
and all
x
∈
ℝ
,
with
c
1
>
0
.
Hypothesis 2.6 (ii) says that at
±
∞
we can have resonance with respect to the principal eigenvalue
λ
^
1
>
0
. In the process of the proof we shall see that Hypothesis 2.6 (iii) implies that the resonance occurs from the right of
λ
^
1
>
0
in the sense that
λ
^
1
|
x
|
p
-
p
F
(
z
,
x
)
→
-
∞
uniformly for a.a.
z
∈
Ω
as
x
→
±
∞
.
This makes the problem noncoercive and so the direct method of the calculus of variations is not directly applicable on (1.1).
Hypothesis 2.6 (iv) is satisfied if we can find
t
-
<
0
<
t
+
such that
f
(
z
,
t
+
)
≤
-
c
^
+
<
0
<
c
^
-
≤
f
(
z
,
t
-
)
for a.a.
z
∈
Ω
.
Therefore, Hypotheses 2.6 (iv), (v) dictate an oscillatory behavior for
f
(
z
,
⋅
)
near zero.
Hypothesis 2.6 (vi) is satisfied if, for example, for a.a.
z
∈
Ω
,
f
(
z
,
⋅
)
is differentiable, and for every
ρ
>
0
, we can find
ξ
^
ρ
>
0
such that
f
x
′
(
z
,
x
)
x
2
≥
-
ξ
^
ρ
|
x
|
p
for a.a.
z
∈
Ω
,
|
x
|
≤
ρ
.
Example 2.8.
The following function satisfies Hypothesis 2.6 (for the sake of simplicity, we drop the z-dependence):
f
(
x
)
=
{
|
x
|
τ
-
2
x
-
2
|
x
|
η
-
2
x
if
|
x
|
≤
1
,
β
|
x
|
p
-
2
x
+
|
x
|
μ
-
2
x
+
(
β
+
2
)
|
x
|
γ
-
2
x
if
|
x
|
>
1
,
with
1
<
τ
<
η
<
∞
,
β
≥
λ
^
1
and
1
<
γ
≤
2
<
μ
<
p
.
3 Solutions of constant sign
In this section we produce solutions of constant sign (positive and negative solutions) of problem (1.1).
We start by considering the following auxiliary nonlinear parametric Dirichlet
(
p
,
2
)
-equation:
(3.1)
-
Δ
p
u
(
z
)
-
Δ
u
(
z
)
=
λ
|
u
(
z
)
|
q
-
2
u
(
z
)
in
Ω
,
λ
>
0
,
u
|
∂
Ω
=
0
.
Proposition 3.1.
For every
λ
>
0
, problem (3.1) has a unique positive solution
u
~
λ
∈
int
C
+
, and since (3.1) is odd,
v
~
λ
=
-
u
~
λ
∈
-
int
C
+
is the unique negative solution of (3.1); moreover,
λ
↦
u
~
λ
is strictly increasing and
∥
u
~
λ
∥
C
0
1
(
Ω
¯
)
→
0
+
as
λ
→
0
+
.
Proof.
The existence of a positive solution for problem (3.1) can be established using the direct method of the calculus of variations.
More precisely, let
ψ
λ
:
W
0
1
,
p
(
Ω
)
→
ℝ
be the
C
1
-functional defined by
ψ
λ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
λ
q
∥
u
+
∥
q
q
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Since
1
<
q
<
2
<
p
, we see that
ψ
λ
(
⋅
)
is coercive.
Also, using the Sobolev embedding theorem, we show that
ψ
λ
(
⋅
)
is sequentially weakly lower semicontinuous.
So, using the Weierstrass–Tonelli theorem, we can find
u
~
λ
∈
W
0
1
,
p
(
Ω
)
such that
(3.2)
ψ
λ
(
u
~
λ
)
=
inf
[
ψ
λ
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
.
Since
q
<
2
<
p
, for
t
∈
(
0
,
1
)
small, we have
ψ
λ
(
t
u
^
1
)
<
0
, hence
ψ
λ
(
u
λ
)
<
0
=
ψ
λ
(
0
)
(see (3.2)), and thus
u
λ
≠
0
.
From (3.2) we have
ψ
λ
′
(
u
~
λ
)
=
0
,
that is,
(3.3)
〈
A
p
(
u
~
λ
)
,
h
〉
+
〈
A
(
u
~
λ
)
,
h
〉
=
λ
∫
Ω
(
u
~
λ
+
)
q
-
1
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.3) we choose
h
=
-
u
~
λ
-
∈
W
0
1
,
p
(
Ω
)
.
Then
∥
D
u
~
λ
-
∥
p
p
+
∥
D
u
~
λ
-
∥
2
2
=
0
, and thus
u
~
λ
≥
0
and
u
~
λ
≠
0
.
From (3.3) it follows that
-
Δ
p
u
~
λ
(
z
)
-
Δ
u
~
λ
(
z
)
=
λ
u
~
λ
(
z
)
q
-
1
for a.a.
z
∈
Ω
,
u
~
λ
|
∂
Ω
=
0
,
therefore
u
~
λ
is a positive solution of (3.1).
From [18, Corollary 8.6, p. 208], we have
u
~
λ
∈
L
∞
(
Ω
)
.
Then, invoking [15, Theorem 1], we infer that
u
~
λ
∈
C
+
∖
{
0
}
.
The nonlinear strong maximum principle of Pucci and Serrin [26, pp. 111, 120] implies that
u
~
λ
∈
int
C
+
.
Next we show that this positive solution is unique. To this end, we introduce the integral functional
j
:
L
1
(
Ω
)
→
ℝ
¯
=
ℝ
∪
{
+
∞
}
defined by
j
(
u
)
=
{
1
p
∥
D
u
1
/
2
∥
p
p
+
1
2
∥
D
u
1
/
2
∥
2
2
if
u
≥
0
,
u
1
/
2
∈
W
0
1
,
p
(
Ω
)
,
+
∞
otherwise
.
From [7, Lemma 1], we know that
j
(
⋅
)
is convex.
Suppose that
y
~
λ
∈
W
0
1
,
p
(
Ω
)
is another positive solution of (3.1). Again we have
y
~
λ
∈
int
C
+
.
Then for any
h
∈
C
0
1
(
Ω
¯
)
and for
|
t
|
<
1
small, we have
u
~
λ
+
t
h
∈
dom
j
and
y
~
λ
+
t
h
∈
dom
j
,
where
dom
j
=
{
u
∈
L
1
(
Ω
)
:
j
(
u
)
<
+
∞
}
(the effective domain of
j
(
⋅
)
). We can easily see that
j
(
⋅
)
is Gateaux differentiable at
u
~
λ
2
and at
y
~
λ
2
in the direction of h. Moreover, the chain rule and Green’s identity (see [10, p. 211]) imply that
j
λ
′
(
u
~
λ
2
)
(
h
)
=
1
2
∫
Ω
-
Δ
p
u
~
λ
-
Δ
u
~
λ
u
~
λ
h
𝑑
z
,
j
λ
′
(
y
~
λ
2
)
(
h
)
=
1
2
∫
Ω
-
Δ
p
y
~
λ
-
Δ
y
~
λ
y
~
λ
h
𝑑
z
for all
h
∈
C
0
1
(
Ω
¯
)
.
The convexity of
j
(
⋅
)
implies the monotonicity of
j
′
(
⋅
)
.
Therefore,
0
≤
∫
Ω
(
-
Δ
p
u
~
λ
-
Δ
u
~
λ
u
~
λ
-
Δ
p
y
~
λ
-
Δ
y
~
λ
y
~
λ
)
(
u
~
λ
-
y
~
λ
)
𝑑
z
=
∫
Ω
λ
[
1
u
~
λ
2
-
q
-
1
y
~
λ
2
-
q
]
(
u
~
λ
-
y
~
λ
)
𝑑
z
,
and hence
u
~
λ
=
y
~
λ
(since
q
<
2
).
This proves the uniqueness of the positive solution of problem (3.1).
Let
0
<
η
<
λ
and consider the Caratheodory function
(3.4)
k
η
(
z
,
x
)
=
{
η
(
x
+
)
q
-
1
if
x
≤
u
~
λ
(
z
)
,
η
u
~
λ
(
z
)
q
-
1
if
u
~
λ
(
z
)
<
x
.
We set
K
η
(
z
,
x
)
=
∫
0
x
k
η
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
ψ
^
η
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
ψ
^
η
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
K
η
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Evidently
ψ
^
η
(
⋅
)
is coercive (see (3.4)) and sequentially weakly lower semicontinuous.
So, we can find
u
¯
η
∈
W
0
1
,
p
(
Ω
)
such that
(3.5)
ψ
^
η
(
u
¯
η
)
=
inf
[
ψ
^
η
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
.
Since
1
<
q
<
2
<
p
, we see that
ψ
^
η
(
u
¯
η
)
<
0
=
ψ
^
η
(
0
)
, therefore
u
¯
η
≠
0
.
From (3.5) we have
ψ
^
η
′
(
u
¯
η
)
=
0
,
that is,
(3.6)
〈
A
p
(
u
¯
λ
)
,
h
〉
+
〈
A
(
u
¯
λ
)
,
h
〉
=
∫
Ω
k
λ
(
z
,
u
¯
η
)
,
d
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.6), first we choose
h
=
-
u
¯
η
-
∈
W
0
1
,
p
(
Ω
)
.
Then
∥
D
u
¯
η
-
∥
p
p
+
∥
D
u
¯
η
-
∥
2
2
=
0
(see (3.4)),
hence
u
¯
η
≥
0
,
u
¯
η
≠
0
.
Also, in (3.6), we choose
h
=
(
u
¯
η
-
u
~
λ
)
+
∈
W
0
1
,
p
(
Ω
)
.
Then (from (3.4) and since
η
<
λ
)
〈
A
p
(
u
¯
η
)
,
(
u
¯
η
-
u
~
λ
)
+
〉
+
〈
A
(
u
¯
η
)
,
(
u
¯
η
-
u
~
λ
)
+
〉
=
∫
Ω
η
u
~
λ
q
-
1
(
u
¯
η
-
u
~
λ
)
+
𝑑
z
≤
∫
Ω
λ
u
~
λ
q
-
1
(
u
¯
η
-
u
~
λ
)
+
𝑑
z
=
〈
A
p
(
u
~
λ
)
,
(
u
¯
η
-
u
~
λ
)
+
〉
+
〈
A
(
u
~
λ
)
,
(
u
¯
η
-
u
~
λ
)
+
〉
,
hence
u
¯
η
≤
u
~
λ
(see Proposition 2.2).
So, we have proved that
(3.7)
u
¯
η
∈
[
0
,
u
~
λ
]
,
u
¯
η
≠
0
.
From (3.4), (3.6) and (3.7), we infer that
u
¯
η
=
u
~
η
∈
int
C
+
, thus
(3.8)
u
~
η
≤
u
~
λ
.
We have (see (3.8) and recall that
η
<
λ
,
u
~
η
∈
int
C
+
)
(3.9)
-
Δ
p
u
~
η
-
Δ
u
~
η
=
η
u
~
η
q
-
1
=
λ
u
~
η
q
-
1
-
(
λ
-
η
)
u
~
η
q
-
1
<
λ
u
~
λ
q
-
1
=
-
Δ
p
u
~
λ
-
Δ
u
~
λ
for a.a.
z
∈
Ω
.
Let
h
1
=
η
u
~
η
q
-
1
and
h
2
=
λ
u
~
λ
q
-
1
. Evidently
h
1
,
h
2
∈
C
0
1
(
Ω
¯
)
.
For all
K
⊆
Ω
compact, let
c
K
=
min
K
u
~
η
>
0
(recall
u
~
η
∈
int
C
+
).
Then (see (3.8))
h
2
(
z
)
-
h
1
(
z
)
=
λ
u
~
λ
(
z
)
q
-
1
-
η
u
~
η
(
z
)
q
-
1
≥
(
λ
-
η
)
u
~
η
(
z
)
q
-
1
≥
(
λ
-
η
)
c
K
>
0
for a.a.
z
∈
K
.
From (3.9) and Proposition 2.4 it follows that
u
~
λ
-
u
~
η
∈
int
C
+
, therefore
λ
↦
u
~
λ
is strictly increasing from
(
0
,
+
∞
)
into
C
0
1
(
Ω
¯
)
.
Finally, let
λ
>
0
and let
u
~
λ
∈
int
C
+
be the unique solution of (3.1).
Then we have
〈
A
p
(
u
~
λ
)
,
h
〉
+
〈
A
(
u
~
λ
)
,
h
〉
=
λ
∫
Ω
u
~
λ
q
-
1
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
Choosing
h
=
u
~
λ
∈
W
0
1
,
p
(
Ω
)
, we obtain
∥
D
u
~
λ
∥
p
p
≤
λ
∥
u
~
λ
∥
q
q
, hence
∥
u
~
λ
∥
p
-
q
≤
λ
c
2
for some
c
2
>
0
(recall that
q
<
p
).
Therefore, given
μ
>
0
, we see that
(3.10)
{
u
~
λ
}
λ
>
0
⊆
W
0
1
,
p
(
Ω
)
is bounded
,
∥
u
~
λ
∥
→
0
as
λ
→
0
+
.
Invoking [18, Corollary 8.6, p. 208], we can find
c
3
>
0
such that
∥
u
~
λ
∥
∞
≤
c
3
for all
0
<
λ
≤
μ
.
Then [15, Theorem 1] implies that there exist
α
∈
(
0
,
1
)
and
c
4
>
0
such that
(3.11)
u
~
λ
∈
C
0
1
,
α
(
Ω
¯
)
,
∥
u
~
λ
∥
C
0
1
,
α
(
Ω
¯
)
≤
c
4
for all
0
<
λ
≤
μ
.
From (3.10), (3.11) and the compact embedding of
C
0
1
,
α
(
Ω
¯
)
into
C
0
1
(
Ω
¯
)
, we conclude that
u
~
λ
→
0
in
C
0
1
(
Ω
¯
)
as
λ
→
0
+
.
Since problem (3.1) is odd,
v
~
λ
=
-
u
~
λ
∈
-
int
C
+
is the unique negative solution of (3.1) for all
λ
>
0
.
Also,
λ
↦
v
~
λ
is strictly decreasing from
(
0
,
+
∞
)
into
C
0
1
(
Ω
¯
)
and
∥
v
~
λ
∥
C
0
1
(
Ω
¯
)
→
0
as
λ
→
0
+
.
∎
Let
δ
0
>
0
be as postulated by Hypothesis 2.6 (v).
On account of Proposition 2.5, we can find
λ
±
>
0
such that
u
~
λ
(
z
)
∈
[
0
,
δ
0
]
for all
z
∈
Ω
¯
,
0
<
λ
≤
λ
+
,
(3.12)
v
~
λ
(
z
)
∈
[
-
δ
0
,
0
]
for all
z
∈
Ω
¯
,
0
<
λ
≤
λ
-
.
With
θ
±
>
0
as in Hypothesis 2.6 (iv), we set
λ
+
*
=
min
{
λ
+
,
θ
+
}
and
λ
-
*
=
min
{
λ
-
,
θ
-
}
.
Proposition 3.2.
If Hypothesis 2.6 holds, then
for all
0
<
λ
≤
λ
+
*
, problem (
1.1
) admits a positive solution
u
0
∈
int
C
0
1
(
Ω
¯
)
[
u
~
λ
,
w
+
]
,
for all
0
<
λ
≤
λ
-
*
, problem (
1.1
) admits a negative solution
v
0
∈
int
C
0
1
(
Ω
¯
)
[
w
-
,
v
~
λ
]
.
Proof.
(a) Recall that
δ
0
<
c
0
(see Hypothesis 2.6 (v)).
This fact and (3.12) permit the definition of the Caratheodory function
(3.13)
e
λ
+
(
z
,
x
)
=
{
λ
u
~
λ
(
z
)
q
-
1
+
f
(
z
,
u
~
λ
(
z
)
)
if
x
<
u
~
λ
(
z
)
,
λ
x
q
-
1
+
f
(
z
,
x
)
if
u
~
λ
(
z
)
≤
x
≤
w
+
(
z
)
,
λ
w
+
(
z
)
q
-
1
+
f
(
z
,
w
+
(
z
)
)
if
w
+
(
z
)
<
x
.
We set
E
λ
+
(
z
,
x
)
=
∫
0
x
e
λ
+
(
z
,
x
)
𝑑
s
and consider the
C
1
-functional
φ
^
λ
+
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
^
λ
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
E
λ
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From (3.13) it is clear that
φ
^
λ
+
(
⋅
)
is coercive. Also, it is sequentially weakly lower semicontinuous.
So, we can find
u
0
∈
W
0
1
,
p
(
Ω
)
such that
φ
^
λ
+
(
u
0
)
=
inf
[
φ
^
λ
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
,
hence
(
φ
^
λ
+
)
′
(
u
0
)
=
0
, and therefore
(3.14)
〈
A
p
(
u
0
)
,
h
〉
+
〈
A
(
u
0
)
,
h
〉
=
∫
Ω
e
λ
+
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.14), first we choose
h
=
(
u
~
λ
-
u
0
)
+
∈
W
0
1
,
p
(
Ω
)
. Then (see (3.12), (3.13), Hypothesis 2.6 (v) and Proposition 3.1)
〈
A
p
(
u
0
)
,
(
u
~
λ
-
u
0
)
+
〉
+
〈
A
(
u
0
)
,
(
u
~
λ
-
u
0
)
+
〉
=
∫
Ω
[
λ
u
~
λ
q
-
1
+
f
(
z
,
u
~
λ
)
]
(
u
~
λ
-
u
0
)
+
𝑑
z
≥
∫
Ω
λ
u
~
λ
q
-
1
(
u
~
λ
-
u
0
)
+
𝑑
z
=
〈
A
p
(
u
~
λ
)
,
(
u
~
λ
-
u
0
)
+
〉
+
〈
A
(
u
~
λ
)
,
(
u
~
λ
-
u
0
)
+
〉
,
hence
u
~
λ
≤
u
0
(see Proposition 2.2).
Next in (3.14) we choose
h
=
(
u
0
-
w
+
)
+
∈
W
0
1
,
p
(
Ω
)
.
Then (see (3.13), Hypothesis 2.6 (iv) and recall that
0
<
λ
≤
λ
+
*
≤
θ
+
)
〈
A
p
(
u
0
)
,
(
u
0
-
w
+
)
+
〉
+
〈
A
(
u
0
)
,
(
u
0
-
w
+
)
+
〉
=
∫
Ω
[
λ
w
+
q
-
1
+
f
(
z
,
w
+
)
]
(
u
0
-
w
+
)
+
𝑑
z
≤
∫
Ω
[
θ
+
w
+
q
-
1
+
f
(
z
,
w
+
)
]
(
u
0
-
w
+
)
+
𝑑
x
≤
〈
A
p
(
w
+
)
,
(
u
0
-
w
+
)
+
〉
+
〈
A
(
w
+
)
,
(
u
0
-
w
+
)
+
〉
,
thus
u
0
≤
w
+
(see Proposition 2.2).
We have proved that
(3.15)
u
0
∈
[
u
~
λ
,
w
+
]
.
From (3.13), (3.14), (3.15), it follows that
(3.16)
-
Δ
p
u
0
(
z
)
-
Δ
u
0
(
z
)
=
λ
u
0
(
z
)
q
-
1
+
f
(
z
,
u
0
(
z
)
)
for a.a.
z
∈
Ω
,
u
0
|
∂
Ω
=
0
.
From (3.15), (3.16) and [15, Theorem 1], we infer that
(3.17)
u
0
∈
[
u
~
λ
,
w
+
]
∩
int
C
+
.
Now let
ρ
=
∥
u
0
∥
∞
and let
ξ
^
ρ
>
0
be as postulated by Hypothesis 2.6 (vi).
Then we have (see (3.12), (3.16), (3.17), Hypotheses 2.6 (v)–(vi) and Proposition 3.1)
-
Δ
p
u
0
(
z
)
-
Δ
u
0
(
z
)
+
ξ
^
ρ
u
0
(
z
)
p
-
1
=
λ
u
0
(
z
)
q
-
1
+
f
(
z
,
u
0
(
z
)
)
+
ξ
^
ρ
u
0
(
z
)
p
-
1
≥
λ
u
~
λ
(
z
)
q
-
1
+
f
(
z
,
u
~
λ
(
z
)
)
+
ξ
^
ρ
u
~
λ
(
z
)
p
-
1
≥
λ
u
~
λ
(
z
)
q
-
1
+
ξ
^
ρ
u
~
λ
(
z
)
p
-
1
(3.18)
=
-
Δ
p
u
~
λ
(
z
)
-
Δ
u
~
λ
(
z
)
+
ξ
^
ρ
u
~
λ
(
z
)
p
-
1
for a.a.
z
∈
Ω
.
Set
h
1
(
z
)
=
λ
u
~
λ
(
z
)
q
-
1
+
ξ
^
ρ
u
~
λ
(
z
)
p
-
1
,
h
2
(
z
)
=
λ
u
0
(
z
)
q
-
1
+
f
(
z
,
u
0
(
z
)
)
+
ξ
^
ρ
u
0
(
z
)
p
-
1
.
Evidently
h
1
,
h
2
∈
L
∞
(
Ω
)
. Also, for
K
⊆
Ω
compact, we have (recall that
u
0
∈
int
C
+
)
0
<
m
K
≤
u
0
(
z
)
for all
z
∈
K
.
Therefore (see (3.17) and Hypothesis 2.6 (v)),
h
2
(
z
)
-
h
1
(
z
)
≥
f
(
z
,
u
0
(
z
)
)
≥
c
K
>
0
for a.a.
z
∈
Ω
.
Then (3.18) and Proposition 2.4 imply that
u
0
-
u
~
λ
∈
int
C
+
.
Also, we have (see (3.16), (3.17), Hypotheses 2.6 (iv), (vi), and recall that
0
<
λ
≤
θ
+
)
-
Δ
p
u
0
(
z
)
-
Δ
u
0
(
z
)
+
ξ
^
ρ
u
0
(
z
)
p
-
1
=
λ
u
0
(
z
)
q
-
1
+
f
(
z
,
u
0
(
z
)
)
+
ξ
^
ρ
u
0
(
z
)
p
-
1
≤
θ
+
w
+
(
z
)
q
-
1
+
f
(
z
,
w
+
(
z
)
)
+
ξ
^
ρ
w
+
(
z
)
p
-
1
≤
-
c
^
1
+
ξ
^
ρ
w
+
(
z
)
p
-
1
(3.19)
≤
-
Δ
p
w
+
(
z
)
-
Δ
w
+
(
z
)
+
ξ
^
ρ
w
+
(
z
)
p
-
1
for a.a.
z
∈
Ω
.
From (3.19) and Proposition 2.4, we have
w
+
-
u
0
∈
int
C
+
.
We conclude that
u
0
∈
int
C
0
1
(
Ω
¯
)
[
u
~
λ
,
w
+
]
.
(b) In the negative semiaxis, we consider the Caratheodory function
(3.20)
e
λ
-
(
z
,
x
)
=
{
λ
|
w
-
(
z
)
|
q
-
2
w
-
(
z
)
+
f
(
z
,
w
-
(
z
)
)
if
x
<
w
-
(
z
)
,
λ
|
x
|
q
-
2
x
+
f
(
z
,
x
)
if
w
-
(
z
)
≤
x
≤
v
~
λ
(
z
)
,
λ
|
v
~
λ
|
q
-
2
v
~
λ
(
z
)
+
f
(
z
,
v
~
λ
(
z
)
)
if
v
~
λ
(
z
)
<
x
.
We set
E
λ
-
(
z
,
x
)
=
∫
0
x
e
λ
-
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
φ
^
λ
-
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
^
λ
-
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
E
λ
-
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Arguing as in part (a), using this time the functional
φ
^
λ
-
and (3.20), we produce a negative solution
v
0
for problem (1.1), with
λ
∈
(
0
,
λ
-
*
]
, such that
v
0
∈
int
C
0
1
(
Ω
¯
)
[
w
-
,
v
~
λ
]
.
∎
Next, using
u
0
∈
int
C
+
,
v
0
∈
-
int
C
+
together with suitable variational, truncation and comparison arguments, we will generate a second pair of constant sign smooth solutions.
Proposition 3.3.
If Hypothesis 2.6 holds, then
for all
0
<
λ
≤
λ
+
*
, problem (
1.1
) has a second positive solution
u
^
∈
int
C
+
, with
u
^
≠
u
0
,
u
^
-
u
~
λ
∈
int
C
+
;
for all
0
<
λ
≤
λ
-
*
, problem (
1.1
) has a second negative solution
v
^
∈
-
int
C
+
, with
v
^
≠
v
0
,
v
~
λ
-
v
^
∈
int
C
+
.
Proof.
(a) Let
u
~
λ
∈
int
C
+
be the unique positive solution of (3.1) (see Proposition 3.1). We introduce the following truncation of the reaction in problem (1.1):
(3.21)
r
λ
+
(
z
,
x
)
=
{
λ
u
~
λ
(
z
)
q
-
1
+
f
(
z
,
u
~
λ
(
z
)
)
if
x
≤
u
~
λ
(
z
)
,
λ
x
q
-
1
+
f
(
z
,
x
)
if
u
~
λ
(
z
)
<
x
.
This is a Caratheodory function.
Set
R
λ
+
(
z
,
x
)
=
∫
0
x
r
λ
+
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
φ
λ
+
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
λ
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
R
λ
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From (3.13) and (3.21), it is clear that
(3.22)
φ
λ
+
|
[
u
~
λ
,
w
+
]
=
φ
^
λ
+
|
[
u
~
λ
,
w
+
]
.
Let
u
0
∈
int
C
+
be the positive solution of problem (1.1) produced in Proposition 3.2.
From the proof of that proposition, we know that
u
0
∈
int
C
+
is a minimizer of
φ
^
λ
+
, and
u
0
∈
int
C
0
1
(
Ω
¯
)
[
u
~
λ
,
w
+
]
.
Hence, from (3.22), it follows that
u
0
∈
int
C
+
is a local
C
0
1
(
Ω
¯
)
-minimizer of
φ
λ
+
, and thus (see Proposition 2.3)
(3.23)
u
0
∈
int
C
+
is a local
W
0
1
,
p
(
Ω
)
-minimizer of
φ
λ
+
.
Claim 1:
K
φ
λ
+
⊆
[
u
~
λ
)
∩
int
C
+
and
u
-
u
~
λ
∈
int
C
+
for all
u
∈
K
φ
λ
+
.
Let
u
∈
K
φ
λ
+
. Then
(3.24)
〈
A
p
(
u
)
,
h
〉
+
〈
A
(
u
)
,
h
〉
=
∫
Ω
r
λ
+
(
z
,
u
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.24), we choose
h
=
(
u
~
λ
-
u
)
+
∈
W
0
1
,
p
(
Ω
)
.
Then (see (3.21), (3.12), Hypothesis 2.6 (v) and Proposition 3.1)
〈
A
p
(
u
)
,
(
u
~
λ
-
u
)
+
〉
+
〈
A
(
u
)
,
(
u
~
λ
-
u
)
+
〉
=
∫
Ω
[
λ
u
~
λ
q
-
1
+
f
(
z
,
u
~
λ
)
]
(
u
~
λ
-
u
)
+
𝑑
z
≥
∫
Ω
λ
u
~
λ
q
-
1
(
u
~
λ
-
u
)
+
𝑑
z
=
〈
A
p
(
u
~
λ
)
,
(
u
~
λ
-
u
)
+
〉
+
〈
A
(
u
~
λ
)
,
(
u
~
λ
-
u
)
+
〉
,
thus
u
~
λ
≤
u
(see Proposition 2.2).
The nonlinear regularity theory of Lieberman [15] implies
u
∈
int
C
+
. In addition, as in the proof of Proposition 3.2, using Proposition 2.4, we show that
u
-
u
~
λ
∈
int
C
+
.
This proves claim 1.
On account of claim 1, we may assume that
(3.25)
K
φ
λ
+
is finite
.
Otherwise, we already have an infinity of smooth solutions of problem (1.1) satisfying
u
-
u
~
λ
∈
int
C
+
for all the solutions u (see (3.21) and claim 1). Hence, we are done.
From (3.23) and (3.25) it follows that we can find
ρ
∈
(
0
,
1
)
small such that
(3.26)
φ
λ
+
(
u
0
)
<
inf
[
φ
λ
+
(
u
)
:
∥
u
-
u
0
∥
=
ρ
]
=
m
λ
+
(see the proof of [1, Proposition 29]).
From Hypothesis 2.6 (iii), we see that given
β
>
0
, we can find
M
1
=
M
1
(
β
)
>
0
such that
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
≤
-
β
x
2
for a.a.
z
∈
Ω
,
|
x
|
≥
M
1
.
Then we have
d
d
x
[
F
(
z
,
x
)
|
x
|
p
]
=
f
(
z
,
x
)
|
x
|
p
-
p
|
x
|
p
-
2
x
F
(
z
,
x
)
|
x
|
2
p
=
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
|
x
|
p
x
{
≤
-
β
x
p
-
1
if
x
≥
M
1
,
≥
β
|
x
|
p
-
1
if
x
≤
-
M
1
,
hence
(3.27)
F
(
z
,
x
)
|
x
|
p
-
F
(
z
,
y
)
|
y
|
p
≤
β
p
-
2
[
1
|
x
|
p
-
2
-
1
|
y
|
p
-
2
]
for a.a.
z
∈
Ω
,
|
x
|
≥
|
y
|
≥
M
1
.
From Hypothesis 2.6 (ii), we have
(3.28)
λ
^
1
p
≤
lim inf
x
→
±
∞
F
(
z
,
x
)
|
x
|
p
≤
lim sup
x
→
±
∞
F
(
z
,
x
)
|
x
|
p
≤
η
^
p
uniformly for a.a.
z
∈
Ω
.
So, if in (3.27) we let
x
→
±
∞
and use (3.28), then
λ
^
1
p
-
F
(
z
,
y
)
|
y
|
p
≤
-
β
p
-
2
1
|
y
|
p
-
2
,
and thus
λ
^
1
|
y
|
p
-
p
F
(
z
,
y
)
y
2
≤
-
β
p
-
2
≤
-
β
for a.a.
z
∈
Ω
,
|
y
|
≥
M
1
.
Since
β
>
0
is arbitrary, we conclude that
(3.29)
λ
^
1
|
x
|
p
-
p
F
(
z
,
x
)
x
2
→
-
∞
uniformly for a.a.
z
∈
Ω
as
x
→
±
∞
.
Claim 2:
φ
λ
+
(
t
u
^
1
)
→
-
∞
as
t
→
+
∞
.
For
t
>
0
, we have
p
φ
λ
+
(
t
u
^
1
)
=
t
p
λ
^
1
+
p
t
2
2
∥
D
u
^
1
∥
2
2
-
∫
Ω
R
λ
+
(
z
,
t
u
^
1
)
𝑑
z
≤
t
p
λ
^
1
+
p
t
2
2
∥
D
u
^
1
∥
2
2
-
p
λ
t
q
q
∥
D
u
^
1
∥
q
q
-
∫
Ω
p
F
(
z
,
t
u
^
1
)
𝑑
z
+
c
4
=
∫
Ω
[
λ
^
1
(
t
u
^
1
)
p
-
p
F
(
z
,
t
u
^
1
)
]
𝑑
z
+
p
t
2
2
∥
D
u
^
1
∥
2
2
-
p
q
∥
D
u
^
1
∥
q
q
+
c
4
for some
c
4
>
0
(see (3.21)), therefore
p
φ
λ
+
(
t
u
^
1
)
t
2
=
∫
Ω
[
λ
^
1
(
t
u
^
1
)
p
-
p
F
(
z
,
t
u
^
1
)
]
(
t
u
^
1
)
2
u
^
1
2
𝑑
z
+
p
2
∥
D
(
t
u
^
1
)
∥
2
2
-
p
q
1
λ
2
-
q
∥
D
u
^
1
∥
q
q
+
c
4
t
2
.
Passing to the limit as
t
→
+
∞
and using (3.29), Fatou’s lemma and the fact that
q
<
2
, we obtain
p
φ
λ
+
(
t
u
^
1
)
t
2
→
-
∞
as
t
→
+
∞
,
hence
φ
λ
+
(
t
u
^
1
)
→
-
∞
as
t
→
+
∞
.
This proves claim 2.
Claim 3:
φ
λ
+
satisfies the C-condition.
We consider a sequence
{
u
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
such that
(3.30)
|
φ
λ
+
(
u
n
)
|
≤
M
2
for some
M
2
>
0
and all
n
∈
ℕ
and
(3.31)
(
1
+
∥
u
n
∥
)
(
φ
λ
+
)
′
(
u
n
)
→
0
in
W
-
1
,
p
′
(
Ω
)
as
n
→
∞
.
From (3.31) we have
(3.32)
|
〈
A
p
(
u
n
)
,
h
〉
+
〈
A
(
u
n
)
,
h
〉
-
∫
Ω
r
λ
+
(
z
,
u
n
)
h
𝑑
z
|
≤
ε
n
∥
h
∥
1
+
∥
u
n
∥
for all
h
∈
W
0
1
,
p
(
Ω
)
, with
ε
n
→
0
+
.
In (3.32), we choose
h
=
-
v
n
-
∈
W
0
1
,
p
(
Ω
)
. Using (3.21), we have
∥
D
u
n
-
∥
p
p
+
∥
D
u
n
-
∥
2
2
-
∫
Ω
[
λ
u
~
λ
q
-
1
+
f
(
z
,
u
~
λ
)
]
(
-
u
n
-
)
𝑑
z
≤
ε
n
for all
n
∈
ℕ
,
which implies
∥
u
n
-
∥
p
≤
c
5
∥
u
n
-
∥
for some
c
5
>
0
and all
n
∈
ℕ
(see Hypothesis 2.6 (i)), hence
(3.33)
{
u
n
-
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded.
Next we show that
{
u
n
+
}
⊆
W
0
1
,
p
(
Ω
)
is bounded.
Arguing by contradiction, we assume that at least for a subsequence, we have
∥
u
n
+
∥
→
+
∞
as
n
→
∞
.
We let
y
n
=
u
n
+
∥
u
n
+
∥
,
n
∈
ℕ
.
Then
∥
y
n
∥
=
1
,
y
n
≥
0
for all
n
∈
ℕ
.
We may assume that
y
n
→
𝑤
y
in
W
0
1
,
p
(
Ω
)
,
y
n
→
y
in
L
p
(
Ω
)
,
y
≥
0
.
Multiplying (3.32) with
1
∥
u
n
+
∥
p
-
1
and using (3.33), we obtain
|
〈
A
p
(
y
n
)
,
h
〉
+
〈
A
(
y
n
)
,
h
〉
∥
u
n
+
∥
p
-
2
-
∫
Ω
r
λ
+
(
z
,
u
n
+
)
∥
u
n
+
∥
p
-
1
h
𝑑
z
|
≤
ε
n
′
∥
h
∥
for all
h
∈
W
0
1
,
p
(
Ω
)
, with
ε
n
′
→
0
+
.
Using (3.21), we have
(3.34)
|
〈
A
p
(
y
n
)
,
h
〉
+
〈
A
(
y
n
)
,
h
〉
∥
u
n
+
∥
p
-
2
-
∫
Ω
[
λ
(
u
n
+
)
q
-
1
+
f
(
z
,
u
n
+
)
]
∥
u
n
+
∥
p
-
1
h
𝑑
z
|
≤
ε
n
′′
∥
h
∥
for all
h
∈
W
0
1
,
p
(
Ω
)
, with
ε
n
′′
→
0
+
.
On account of (3.2), we see that
{
λ
(
u
n
+
)
q
-
1
+
f
(
z
,
u
n
+
)
∥
u
n
+
∥
p
-
1
}
n
≥
1
⊆
L
p
′
(
Ω
)
is bounded.
Therefore, we may assume that
(3.35)
λ
(
u
n
+
)
q
-
1
+
f
(
z
,
u
n
+
)
∥
u
n
+
∥
p
-
1
→
𝑤
η
0
(
z
)
y
p
-
1
in
L
p
′
(
Ω
)
,
with
λ
^
1
≤
η
(
z
)
≤
η
^
for a.a.
z
∈
Ω
(see Hypothesis 2.6 (ii) and recall that
q
<
2
<
p
).
In (3.34), we choose
h
=
y
n
-
y
∈
W
0
1
,
p
(
Ω
)
.
Passing to the limit as
n
→
∞
, we obtain (recall that
2
<
p
)
lim
n
→
∞
〈
A
p
(
y
n
)
,
y
n
-
y
〉
=
0
,
therefore (see Proposition 2.2)
(3.36)
y
n
→
y
in
W
0
1
,
p
(
Ω
)
⟹
∥
y
∥
=
1
,
y
≥
0
.
So, if in (3.34) we pass to the limit as
n
→
∞
and use (3.35), (3.36), then
〈
A
p
(
y
)
,
h
〉
=
∫
Ω
η
0
(
z
)
y
p
-
1
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
,
and hence
(3.37)
-
Δ
p
y
(
z
)
=
η
0
(
z
)
y
(
z
)
p
-
1
for a.a.
z
∈
Ω
,
y
|
∂
Ω
=
0
.
Suppose that
η
0
≢
λ
^
1
(see (3.35)). We have
λ
~
1
(
p
,
η
0
)
<
λ
~
1
(
p
,
λ
^
1
)
=
1
(see Proposition 2.5), and thus
y must be nodal (see (3.37)), which contradicts (3.36).
Next assume that
η
0
(
z
)
=
λ
^
1
for a.a.
z
∈
Ω
.
From (3.37) it follows that
y
=
ξ
u
^
1
∈
int
C
+
(
ξ
>
0
), hence
y
(
z
)
>
0
for all
z
∈
Ω
, and thus
u
n
+
(
z
)
→
+
∞
for all
z
∈
Ω
.
Therefore (see Hypothesis 2.6 (iii)),
f
(
z
,
u
n
+
(
z
)
)
u
n
+
(
z
)
-
p
F
(
z
,
u
n
+
(
z
)
)
u
n
+
(
z
)
2
→
-
∞
for a.a.
z
∈
Ω
,
and thus (by Fatou’s lemma)
(3.38)
∫
Ω
f
(
z
,
u
n
+
)
u
n
+
-
p
F
(
z
,
u
n
+
)
∥
u
n
+
∥
2
𝑑
z
→
-
∞
.
From (3.30) and (3.33), we have
(3.39)
∥
D
u
n
+
∥
p
p
+
p
2
∥
D
u
n
+
∥
2
2
-
∫
Ω
[
λ
p
q
|
u
n
+
|
q
+
p
F
(
z
,
u
n
+
)
]
𝑑
z
≥
-
M
3
for some
M
3
>
0
and all
n
∈
ℕ
(see (3.21)).
Similarly, if we use (3.33) in (3.32) and also choose
h
=
u
n
+
∈
W
0
1
,
p
(
Ω
)
, then
(3.40)
-
∥
D
u
n
+
∥
p
p
-
∥
D
u
n
+
∥
2
2
+
∫
Ω
[
λ
(
u
n
+
)
q
+
f
(
z
,
u
n
+
)
u
n
+
]
𝑑
z
≥
-
M
4
for some
M
4
>
0
and all
n
∈
ℕ
(see (3.21)).
We add (3.39) and (3.40) and obtain
(
p
2
-
1
)
∥
D
u
n
+
∥
2
2
+
∫
Ω
[
f
(
z
,
u
n
+
)
u
n
+
-
p
F
(
z
,
u
n
+
)
]
𝑑
z
≥
-
M
5
+
λ
(
p
q
-
1
)
∥
u
n
+
∥
q
q
,
with
M
5
=
M
3
+
M
4
>
0
, for all
n
∈
ℕ
, hence
(
p
2
-
1
)
∥
D
y
n
∥
2
2
+
∫
Ω
f
(
z
,
u
n
+
)
u
n
+
-
p
F
(
z
,
u
n
+
)
∥
u
n
+
∥
2
𝑑
z
≥
-
M
5
∥
u
n
+
∥
2
+
λ
(
p
q
-
1
)
∥
y
n
∥
q
q
∥
u
n
+
∥
2
-
q
.
Since
q
<
2
<
p
, passing to the limit as
n
→
∞
, we have a contradiction (see (3.38)).
Therefore,
{
u
n
+
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded, and hence
{
u
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded (see (3.33)).
We may assume that
(3.41)
u
n
→
𝑤
u
in
W
0
1
,
p
(
Ω
)
,
u
n
→
u
in
L
p
(
Ω
)
.
In (3.32) we choose
h
=
u
n
-
u
∈
W
0
1
,
p
(
Ω
)
, pass to the limit as
n
→
∞
and use (3.41).
Then
lim
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
〉
+
〈
A
(
u
n
)
,
u
n
-
u
〉
]
=
0
,
which implies (recall that
A
(
⋅
)
is monotone)
lim sup
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
〉
+
〈
A
(
u
)
,
u
n
-
u
〉
]
≤
0
.
Hence (see (3.41)),
lim sup
n
→
∞
〈
A
p
(
u
n
)
,
u
n
-
u
〉
≤
0
,
and thus (see Proposition 2.2)
u
n
→
u
in
W
0
1
,
p
(
Ω
)
.
Therefore,
φ
λ
+
satisfies the C-conditions and this proves claim 3.
Then (3.26) and claims 2 and 3, permit the use of Theorem 2.1 (the mountain pass theorem).
So, we can find
u
^
∈
W
0
1
,
p
(
Ω
)
such that
u
^
∈
K
φ
λ
+
⊆
[
u
~
λ
)
∩
int
C
+
(see claim 1) and
m
λ
+
≤
φ
λ
+
(
u
^
)
(see (3.26)).
From (3.21) and (3.26) we infer that
u
^
is a positive solution of (1.1) and
u
^
≠
u
0
.
As in the proof of Proposition 3.2, using Proposition 2.4, we can show that
u
^
-
u
~
λ
∈
int
C
+
.
(b) Let
v
~
λ
=
-
u
~
λ
∈
-
int
C
+
be the unique negative solution of (3.1) (see Proposition 3.1). We consider the following truncation of the reaction in problem (3.1):
r
λ
-
(
z
,
x
)
=
{
λ
|
x
|
q
-
2
x
+
f
(
z
,
x
)
if
x
≤
v
~
λ
(
z
)
,
λ
|
v
~
λ
(
z
)
|
q
-
2
v
~
λ
(
z
)
+
f
(
z
,
v
~
λ
(
z
)
)
if
v
~
λ
(
z
)
<
x
.
This is a Caratheodory function.
Set
R
λ
-
(
z
,
x
)
=
∫
0
x
r
λ
-
(
z
,
s
)
𝑑
s
and consider the
C
1
-function
φ
λ
-
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
λ
-
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
R
λ
-
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Working with
φ
λ
-
as in part (a), we see that for
λ
∈
(
0
,
λ
-
*
]
, we can find
v
^
∈
-
int
C
+
, a second negative solution of (1.1), such that
v
^
≠
v
0
and
v
~
λ
-
v
∈
int
C
+
.
∎
We will show that we have extremal constant sign solutions, that is, there is a smallest positive solution for problem (1.1), with
λ
∈
(
0
,
λ
+
*
]
, and a biggest negative solution for (1.1), with
λ
∈
(
0
,
λ
-
*
]
.
These extremal constant sign solutions will be used in the next section to generate nodal solutions.
We introduce the following solution sets:
S
λ
+
is the set of positive solutions for problem (1.1),
S
λ
-
is the set of negative solutions for problem (1.1).
From Proposition 3.2 we know that
for all
0
<
λ
≤
λ
+
*
,
∅
≠
S
λ
+
⊆
int
C
+
,
for all
0
<
λ
≤
λ
-
*
,
∅
≠
S
λ
-
⊆
-
int
C
+
.
Proposition 3.4.
If Hypothesis 2.6 holds, then
for all
λ
∈
(
0
,
λ
+
*
]
, problem (
1.1
) has a smallest positive solution
u
¯
λ
∈
int
C
+
, that is,
u
¯
λ
≤
u
for all
u
∈
S
λ
+
,
for all
λ
∈
(
0
,
λ
-
*
]
, problem (
1.1
) has a biggest negative solution
v
¯
λ
∈
-
int
C
+
, that is,
v
≤
v
¯
λ
for all
v
∈
S
λ
-
.
Proof.
(a) First we show that
(3.42)
u
~
λ
≤
u
for all
u
∈
S
λ
+
(
0
<
λ
≤
λ
+
*
)
.
To this end, let
u
∈
S
λ
+
and consider the Caratheodory function
(3.43)
β
λ
(
z
,
x
)
=
{
λ
(
x
+
)
q
-
1
if
x
≤
u
(
z
)
,
λ
u
(
z
)
q
-
1
if
u
(
z
)
<
x
.
We set
B
λ
(
z
,
x
)
=
∫
0
x
β
λ
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
ψ
^
λ
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
ψ
^
λ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
B
λ
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From (3.43) it is clear that
ψ
^
λ
(
⋅
)
is coercive. Also, it is sequentially weakly lower semicontinuous.
So, we can find
u
~
λ
*
∈
W
0
1
,
p
(
Ω
)
such that
(3.44)
ψ
^
λ
(
u
~
λ
*
)
=
inf
[
ψ
^
λ
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
.
Let
y
∈
int
C
+
. Since
u
∈
S
λ
+
⊆
int
C
+
, using [17, Proposition 2.1], we can find
t
∈
(
0
,
1
]
small such that
t
y
≤
u
.
Then, from (3.43) and since
q
<
2
<
p
, we see that by choosing
t
∈
(
0
,
1
]
even smaller if necessary, we have
ψ
^
λ
(
t
y
)
<
0
, which implies
ψ
^
λ
(
u
~
λ
*
)
<
0
=
ψ
^
λ
(
0
)
(see (3.44)), and thus
u
~
λ
*
≠
0
.
From (3.44) we have
ψ
^
λ
′
(
u
~
λ
*
)
=
0
, hence
(3.45)
〈
A
p
(
u
~
λ
*
)
,
h
〉
+
〈
A
(
u
~
λ
*
)
,
h
〉
=
∫
Ω
β
λ
(
z
,
u
~
λ
*
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.45), we choose
h
=
-
(
u
~
λ
*
)
-
∈
W
0
1
,
p
(
Ω
)
.
Then
∥
D
(
u
~
λ
*
)
-
∥
p
p
+
∥
D
(
u
~
λ
*
)
-
∥
2
2
=
0
(see (3.43)), and therefore
u
~
λ
*
≥
0
and
u
~
λ
*
≠
0
.
Also, in (3.45), we choose
h
=
(
u
~
λ
*
-
u
)
+
∈
W
0
1
,
p
(
Ω
)
.
Then (see (3.12), (3.43), Hypothesis 2.6 (v) and note that
u
∈
S
λ
+
)
〈
A
p
(
u
~
λ
*
)
,
(
u
~
λ
*
-
u
)
+
〉
+
〈
A
(
u
~
λ
*
)
,
(
u
~
λ
*
-
u
)
+
〉
=
∫
Ω
λ
u
q
-
1
(
u
~
λ
*
-
u
)
+
𝑑
z
≤
∫
Ω
[
λ
u
q
-
1
+
f
(
z
,
u
)
]
(
u
~
λ
*
-
u
)
+
𝑑
z
=
〈
A
p
(
u
)
,
(
u
~
λ
*
-
u
)
+
〉
+
〈
A
(
u
)
,
(
u
~
λ
*
-
u
)
+
〉
,
hence
u
~
λ
*
≤
u
.
We have proved that
(3.46)
u
~
λ
*
∈
[
0
,
u
]
,
u
~
λ
*
≠
0
.
From (3.46), (3.45), (3.43) and Proposition 3.1, we conclude that
u
~
λ
*
=
u
~
λ
.
Therefore, (3.42) is true.
From [8], we know that
S
λ
+
is downward directed (that is, if
u
1
,
u
2
∈
S
λ
+
, then we can find
v
∈
S
λ
+
such that
u
≤
u
1
,
u
≤
u
2
).
Invoking [13, Lemma 3.10, p. 178], we can find a decreasing sequence
{
u
n
}
n
≥
1
⊆
S
λ
+
such that
inf
S
λ
+
=
inf
n
≥
1
u
n
.
Evidently,
{
u
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded (see Hypothesis 2.6 (i) and recall that
0
≤
u
n
≤
u
1
for all
n
∈
ℕ
).
So, we may assume that
(3.47)
u
n
→
𝑤
u
¯
λ
in
W
0
1
,
p
(
Ω
)
,
u
n
→
u
¯
λ
in
L
p
(
Ω
)
.
For every
n
∈
ℕ
, we have
(3.48)
〈
A
p
(
u
n
)
,
h
〉
+
〈
A
(
u
n
)
,
h
〉
=
∫
Ω
[
λ
u
n
q
-
1
+
f
(
z
,
u
n
)
]
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.48), we choose
h
=
u
n
-
u
¯
λ
∈
W
0
1
,
p
(
Ω
)
, pass to the limit as
n
→
∞
, use (3.47), and reasoning as in the proof of Proposition 3.3, via Proposition 2.2, we conclude that
(3.49)
u
n
→
u
¯
λ
in
W
0
1
,
p
(
Ω
)
.
So, in (3.48), we pass to the limit as
n
→
∞
and use (3.49).
Then
〈
A
p
(
u
¯
λ
)
,
h
〉
+
〈
A
(
u
¯
λ
)
,
h
〉
=
∫
Ω
[
λ
u
¯
λ
q
-
1
+
f
(
z
,
u
¯
λ
)
]
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
Also, from (3.42) and (3.49), we have
u
~
λ
≤
u
¯
λ
.
Therefore,
u
¯
λ
∈
S
λ
+
,
u
¯
λ
≤
u
for all
u
∈
S
λ
+
.
(b) Similarly for the set
S
λ
-
, we have
v
≤
v
~
λ
for all
v
∈
S
λ
-
.
Reasoning as in part (a), we produce a
v
¯
λ
∈
S
λ
-
such that
v
≤
v
¯
λ
for all
v
∈
S
λ
-
.
∎
Next we examine the maps
λ
↦
u
¯
λ
and
λ
↦
v
¯
λ
.
Proposition 3.5.
If Hypothesis 2.6 holds, then
the map
λ
↦
u
¯
λ
from
(
0
,
λ
+
*
]
into
C
+
is strictly increasing (that is, for
0
<
θ
<
λ
≤
λ
+
*
,
u
¯
λ
-
u
¯
θ
∈
int
C
+
) and left continuous;
the map
λ
↦
v
¯
λ
from
(
0
,
λ
-
*
]
into
-
C
+
is strictly decreasing (that is, for
0
<
θ
<
λ
≤
λ
-
*
,
u
¯
θ
-
u
¯
λ
∈
int
C
+
) and left continuous.
Proof.
(a) First we show that
λ
↦
u
¯
λ
is increasing.
We consider the Caratheodory function
(3.50)
m
θ
(
z
,
x
)
=
{
θ
(
x
+
)
q
-
1
+
f
(
z
,
x
+
)
if
x
≤
u
¯
λ
(
z
)
,
θ
u
¯
λ
(
z
)
q
-
1
+
f
(
z
,
u
¯
λ
(
z
)
)
if
u
¯
λ
(
z
)
<
x
.
We set
Γ
θ
(
z
,
x
)
=
∫
0
x
m
θ
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
σ
θ
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
σ
θ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
Γ
θ
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
This is coercive (see (3.50)) and sequentially weakly lower semicontinuous. Hence, we can find
y
θ
∈
W
0
1
,
p
(
Ω
)
such that
(3.51)
σ
θ
(
y
θ
)
=
inf
[
σ
θ
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
.
Exploiting as before the fact that
q
<
2
<
p
, we have
σ
θ
(
y
θ
)
<
0
=
σ
θ
(
0
)
.
Hence,
y
θ
≠
0
.
From (3.51) we have
σ
θ
′
(
y
θ
)
=
0
, therefore
(3.52)
〈
A
p
(
y
θ
)
,
h
〉
+
〈
A
(
y
θ
)
,
h
〉
=
∫
Ω
m
θ
(
z
,
y
θ
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (3.52), first let
h
=
-
y
θ
-
∈
W
0
1
,
p
(
Ω
)
.
Then, using (3.50), we have
y
θ
≥
0
,
y
θ
≠
0
.
Also, in (3.52), we choose
h
=
(
y
θ
-
u
¯
λ
)
+
∈
W
0
1
,
p
(
Ω
)
.
Then (see (3.50) and note that
θ
<
λ
)
〈
A
p
(
y
θ
)
,
(
y
θ
-
u
¯
λ
)
+
〉
+
〈
A
(
y
θ
)
,
(
y
θ
-
u
¯
λ
)
+
〉
=
∫
Ω
[
θ
u
¯
λ
q
-
1
+
f
(
z
,
u
¯
λ
)
]
(
y
θ
-
u
¯
λ
)
+
𝑑
z
≤
∫
Ω
[
λ
u
¯
λ
q
-
1
+
f
(
z
,
u
¯
λ
)
]
(
y
θ
-
u
¯
λ
)
+
𝑑
z
=
〈
A
p
(
u
¯
λ
)
,
(
y
θ
-
u
¯
λ
)
+
〉
+
〈
A
(
u
¯
λ
)
,
(
y
θ
-
u
¯
λ
)
+
〉
,
hence
y
θ
≤
u
¯
λ
.
We have proved that
(3.53)
y
θ
∈
[
0
,
u
¯
λ
]
,
y
θ
≠
0
,
which implies
y
θ
∈
S
θ
+
⊆
int
C
+
(see (3.50), (3.52)), and thus
u
¯
θ
≤
y
θ
≤
u
¯
λ
(see (3.53)).
Hence,
λ
↦
u
¯
λ
is increasing.
Now let
ρ
=
∥
u
¯
λ
∥
∞
and let
ξ
^
ρ
>
0
be as postulated by Hypothesis 2.6 (vi).
Then we have (see Hypothesis 2.6 (vi) and recall that
θ
<
λ
)
-
Δ
p
u
¯
θ
-
Δ
u
¯
θ
+
ξ
^
ρ
u
¯
θ
p
-
1
=
θ
u
¯
θ
q
-
1
+
f
(
z
,
u
¯
θ
)
+
ξ
^
ρ
u
¯
θ
p
-
1
=
λ
u
¯
θ
q
-
1
+
f
(
z
,
u
¯
θ
)
+
ξ
^
ρ
u
¯
θ
p
-
1
-
(
λ
-
θ
)
u
¯
θ
q
-
1
≤
λ
u
¯
θ
q
-
1
+
f
(
z
,
u
¯
θ
)
+
ξ
^
ρ
u
¯
θ
p
-
1
=
-
Δ
p
u
¯
λ
-
Δ
u
¯
λ
+
ξ
^
ρ
u
¯
λ
p
-
1
for a.a.
z
∈
Ω
.
Let
h
1
(
z
)
=
θ
u
¯
θ
(
z
)
q
-
1
+
f
(
z
,
u
¯
θ
(
z
)
)
+
ξ
^
ρ
u
¯
θ
(
z
)
p
-
1
,
h
2
(
z
)
=
λ
u
¯
λ
(
z
)
q
-
1
+
f
(
z
,
u
¯
λ
(
z
)
)
+
ξ
^
ρ
u
¯
λ
(
z
)
p
-
1
.
Evidently,
h
1
,
h
2
∈
L
∞
(
Ω
)
and
(
λ
-
θ
)
u
¯
θ
(
z
)
q
-
1
≤
h
2
(
z
)
-
h
1
(
z
)
for a.a.
z
∈
Ω
.
Since
u
¯
θ
∈
int
C
+
, it follows that
h
1
≺
h
2
and so, using Proposition 2.4, we conclude that
u
¯
λ
-
u
¯
θ
∈
int
C
+
.
We have proved that
λ
↦
u
¯
λ
is strictly increasing.
Next we show that the map
λ
↦
u
¯
λ
is left continuous. To this end, let
λ
n
→
λ
-
,
λ
≤
λ
+
*
. For each
n
∈
ℕ
, let
u
¯
n
=
u
¯
λ
n
∈
S
λ
n
+
⊆
int
C
+
be the minimal positive solution of problem (1.1) (see Proposition 3.3).
Then we have
(3.54)
〈
A
p
(
u
¯
n
)
,
h
〉
+
〈
A
(
u
¯
n
)
,
h
〉
=
∫
Ω
[
λ
n
u
¯
n
q
-
1
+
f
(
z
,
u
¯
n
)
]
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
, all
n
∈
ℕ
, and
(3.55)
u
¯
n
≤
u
¯
λ
for all
n
∈
ℕ
.
In (3.54), we choose
h
=
u
¯
n
∈
W
0
1
,
p
(
Ω
)
.
Then from Hypothesis 2.6 (i) and (3.55) we infer that
{
u
¯
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded.
Using [18, Corollary 8.6, p. 208], we can find
c
6
>
0
such that
∥
u
¯
n
∥
∞
≤
c
6
for all
n
∈
ℕ
.
Invoking [15, Theorem 1], we can find
α
∈
(
0
,
1
)
and
c
7
>
0
such that
u
¯
n
∈
C
0
1
,
α
(
Ω
¯
)
,
∥
u
¯
∥
C
0
1
,
α
(
Ω
¯
)
≤
c
7
for all
n
∈
ℕ
.
Since
C
0
1
,
α
(
Ω
¯
)
↪
C
0
1
(
Ω
¯
)
compactly, we may assume that
(3.56)
u
¯
n
→
u
^
λ
in
C
0
1
(
Ω
¯
)
.
Passing to the limit as
n
→
∞
in (3.54) and using (3.56), we obtain
u
^
λ
∈
S
λ
+
. We claim that
u
¯
λ
=
u
^
λ
.
If this is not true, then we can find
z
0
∈
Ω
such that
u
¯
λ
(
z
0
)
<
u
^
λ
(
z
0
)
, hence (see (3.56))
u
¯
λ
(
z
0
)
<
u
¯
n
(
z
0
)
for all
n
≥
n
0
.
This contradicts the fact that
λ
↦
u
¯
λ
is increasing.
Hence,
u
^
λ
=
u
¯
λ
and so we conclude the left continuity of the map
λ
↦
u
¯
λ
.
∎
Summarizing the results obtained in this section, we can state the following theorem.
Theorem 3.6.
If Hypothesis 2.6 holds, then
there exists
λ
+
*
>
0
such that for
λ
∈
(
0
,
λ
+
*
]
, we have
problem (
1.1
) has at least two positive solutions
u
0
,
u
^
∈
int
C
+
,
u
0
≠
u
^
,
u
0
-
u
~
λ
,
u
^
-
u
~
λ
∈
int
C
+
,
problem (
1.1
) has a smallest positive solution
u
¯
λ
∈
int
C
+
and
λ
↦
u
¯
λ
is strictly increasing and left continuous;
there exists
λ
-
*
>
0
such that for
λ
∈
(
0
,
λ
-
*
]
, we have
problem (
1.1
) has at least two negative solutions
v
0
,
v
^
∈
-
int
C
+
,
v
0
≠
v
^
,
v
~
λ
-
v
0
,
v
~
λ
-
v
^
∈
int
C
+
,
problem (
1.1
) has a biggest negative solution
v
¯
λ
∈
-
int
C
+
and
λ
↦
v
¯
λ
is strictly decreasing and left continuous.
4 Nodal solutions
In this section we focus on nodal (sign changing) solutions for problem (1.1). So, let
λ
*
=
min
{
λ
+
*
,
λ
-
*
}
(see Theorem 3.6). Let
λ
∈
(
0
,
λ
*
]
and consider the
C
1
-functional
τ
^
λ
:
H
0
1
(
Ω
)
→
ℝ
defined by
τ
^
λ
(
u
)
=
1
2
∥
D
u
∥
2
2
-
λ
q
∥
u
∥
q
q
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
H
0
1
(
Ω
)
.
Hypothesis 2.6 (v) and Proposition 2.1 of [14] imply the following result.
Proposition 4.1.
If Hypothesis 2.6 holds and
λ
>
0
, then
C
k
(
τ
^
λ
,
0
)
=
0
for all
k
∈
N
0
.
Let
τ
λ
=
τ
^
λ
|
W
0
1
,
p
(
Ω
)
.
Since
W
0
1
,
p
(
Ω
)
↪
H
0
1
(
Ω
)
densely, we can apply [19, Theorem 16] (see also [4, p. 14]) and have
C
k
(
τ
λ
,
0
)
=
C
k
(
τ
^
λ
,
0
)
for all
k
∈
ℕ
0
, thus (see [14])
(4.1)
C
k
(
τ
λ
,
0
)
=
0
for all
k
∈
ℕ
0
.
Let
φ
λ
:
W
0
1
,
p
(
Ω
)
→
ℝ
be the energy functional for problem (1.1) defined by
φ
λ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
λ
q
∥
u
∥
q
q
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Then we have
|
φ
λ
(
u
)
-
τ
λ
(
u
)
|
=
1
p
∥
D
u
∥
p
p
=
1
p
∥
u
∥
p
,
|
〈
φ
λ
′
(
u
)
-
τ
λ
′
(
u
)
,
h
〉
|
=
|
〈
A
p
(
u
)
,
h
〉
|
≤
∥
D
u
∥
p
p
-
1
∥
h
∥
p
for all
h
∈
W
0
1
,
p
(
Ω
)
.
Therefore,
∥
φ
λ
′
(
u
)
-
τ
λ
′
(
u
)
∥
*
≤
∥
D
u
∥
p
p
-
1
=
∥
u
∥
p
-
1
.
Then the
C
1
-continuity property of critical groups (see [11, Theorem 5.126, p. 836]) implies that
C
k
(
φ
λ
,
0
)
=
C
k
(
τ
λ
,
0
)
for all
k
∈
ℕ
0
,
hence (see (4.1))
(4.2)
C
k
(
φ
λ
,
0
)
=
0
for all
k
∈
ℕ
0
.
Let
λ
∈
(
0
,
λ
*
]
and let
u
¯
λ
∈
int
C
+
,
v
¯
λ
∈
-
int
C
+
be two extremal constant sign solutions for problem (1.1) (see Theorem 3.6).
We introduce the following trunction of the reaction in problem (1.1):
(4.3)
g
λ
(
z
,
x
)
=
{
λ
|
v
¯
λ
(
z
)
|
q
-
2
v
¯
λ
(
z
)
+
f
(
z
,
v
¯
λ
(
z
)
)
if
x
≤
v
¯
λ
(
z
)
,
λ
|
x
|
q
-
1
x
+
f
(
z
,
x
)
if
v
¯
λ
(
z
)
≤
x
≤
u
¯
λ
(
z
)
λ
u
¯
λ
(
z
)
q
-
1
+
f
(
z
,
u
¯
λ
(
z
)
)
if
u
¯
λ
(
z
)
<
x
.
This is a Caratheodory function.
Set
G
λ
(
z
,
x
)
=
∫
0
x
g
λ
(
z
,
s
)
𝑑
s
and consider the
C
1
-functional
φ
~
λ
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
~
λ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
G
λ
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We know that
φ
~
λ
∈
C
1
(
W
0
1
,
p
(
Ω
)
)
. Using (4.3), we can easily show that
(4.4)
K
φ
~
λ
⊆
[
v
¯
λ
,
u
¯
λ
]
∩
C
0
1
(
Ω
¯
)
.
Therefore, the nontrivial critical points of
φ
~
λ
, distinct from
v
¯
λ
and
u
¯
λ
, are nodal solutions of (1.1).
So, we assume that
K
φ
~
λ
is finite.
Otherwise, we already have an infinity of nodal solutions for problem (1.1).
We also consider the positive and negative truncations of
φ
~
λ
, namely, we consider the
C
1
-functionals
φ
~
λ
±
:
W
0
1
,
p
(
Ω
)
→
ℝ
defined by
φ
~
λ
±
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
G
λ
(
z
,
±
u
±
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We can easily show that
K
φ
~
λ
+
⊆
[
0
,
u
¯
λ
]
∩
C
0
1
(
Ω
¯
)
,
K
φ
~
λ
-
⊆
[
v
¯
λ
,
0
]
∩
C
0
1
(
Ω
¯
)
.
The extremality of
u
¯
λ
and
v
¯
λ
implies that
(4.5)
K
φ
~
λ
+
=
{
0
,
u
¯
λ
}
,
K
φ
~
λ
-
=
{
0
,
v
¯
λ
}
.
We compute the critical groups of
φ
~
λ
.
Proposition 4.2.
If Hypothesis 2.6 holds and
λ
∈
(
0
,
λ
*
]
, then
C
k
(
φ
λ
,
0
)
=
C
k
(
φ
~
λ
,
0
)
for all
k
∈
N
0
.
Proof.
We consider the homotopy
h
^
(
t
,
u
)
defined by
h
^
(
t
,
u
)
=
(
1
-
t
)
φ
λ
(
u
)
+
t
φ
~
λ
(
u
)
for all
t
∈
[
0
,
1
]
,
u
∈
W
0
1
,
p
(
Ω
)
.
Suppose we can find
{
t
n
}
n
≥
1
⊆
[
0
,
1
]
and
{
u
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
such that
(4.6)
t
n
→
t
,
u
n
→
0
in
W
0
1
,
p
(
Ω
)
,
h
u
′
(
t
n
,
u
n
)
=
0
for all
n
∈
ℕ
.
Then we have
〈
A
p
(
u
n
)
,
h
〉
+
〈
A
(
u
n
)
,
h
〉
=
(
1
-
t
n
)
∫
Ω
[
λ
|
u
n
|
q
-
2
u
n
+
f
(
z
,
u
n
)
]
h
𝑑
z
+
t
n
∫
Ω
g
λ
(
z
,
u
n
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
and all
n
∈
ℕ
,
which implies
(4.7)
{
-
Δ
p
u
n
(
z
)
-
Δ
u
n
(
z
)
=
(
1
-
t
n
)
[
λ
|
u
n
(
z
)
|
q
-
2
u
n
(
z
)
+
f
(
z
,
u
n
(
z
)
)
]
+
t
n
g
λ
(
z
,
u
n
(
z
)
)
for a.a.
z
∈
Ω
,
u
n
|
Ω
=
0
.
Evidently,
{
u
n
}
n
≥
1
⊆
W
0
1
,
p
(
Ω
)
is bounded (see (4.6), (4.7)).
Then as before using (4.7) and [15, Theorem 1], we can find
α
∈
(
0
,
1
)
and
c
8
>
0
such that
(4.8)
u
n
∈
C
0
1
,
α
(
Ω
¯
)
,
∥
u
n
∥
C
0
1
,
α
(
Ω
¯
)
≤
c
8
for all
n
∈
ℕ
.
From (4.6), (4.8) and the compact embedding of
C
0
1
,
α
(
Ω
¯
)
into
C
0
1
(
Ω
¯
)
, we have
u
n
→
0
in
C
0
1
(
Ω
¯
)
,
hence
u
n
∈
[
v
¯
λ
,
u
¯
λ
]
for all
n
≥
n
0
,
and therefore
{
u
n
}
n
≥
n
0
⊆
K
φ
~
λ
(see (4.3)),
a contradiction to our assumption that
K
φ
~
λ
is finite. Therefore, (4.6) can not occur, and from homotopy invariance of critical groups (see [11, Theorem 5.125, p. 836]), we have
C
k
(
φ
λ
,
0
)
=
C
k
(
φ
~
λ
,
0
)
for all
k
∈
ℕ
0
.
∎
From Proposition 4.2 and (4.2), we infer that
(4.9)
C
k
(
φ
~
λ
,
0
)
=
0
for all
k
∈
ℕ
0
.
Proposition 4.3.
If Hypothesis 2.6 holds and
0
<
λ
≤
λ
*
, then problem (1.1) admits two nodal solutions
y
0
,
y
^
∈
C
0
1
(
Ω
¯
)
,
y
0
∈
inf
C
0
1
(
Ω
¯
)
[
v
¯
λ
,
u
¯
λ
]
.
Proof.
From (4.4) and (4.5) we know that
(4.10)
K
φ
~
λ
⊆
[
v
¯
λ
,
u
¯
λ
]
∩
C
0
1
(
Ω
¯
)
,
K
φ
~
λ
+
=
{
0
,
u
¯
λ
}
,
K
φ
~
λ
-
=
{
0
,
v
¯
λ
}
.
Claim:
u
¯
λ
∈
int
C
+
and
v
¯
λ
∈
-
int
C
+
are local minimizers of
φ
~
λ
.
From (4.3) it is clear that
φ
~
λ
+
is coercive. Also, it is sequentially weakly lower semicontinuous.
So, we can find
u
¯
λ
*
∈
W
0
1
,
p
(
Ω
)
such that
(4.11)
φ
~
λ
+
(
u
¯
λ
*
)
=
inf
[
φ
~
λ
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
]
.
As before, since
q
<
2
<
p
, we have
φ
~
λ
+
(
u
¯
λ
*
)
<
0
=
φ
~
λ
+
(
0
)
,
hence
u
¯
λ
*
≠
0
and
u
¯
λ
*
∈
K
φ
~
λ
+
(see (4.11)), and thus
u
¯
λ
*
=
u
¯
λ
(see (4.10)).
It is clear from (4.3) that
φ
~
λ
|
C
+
=
φ
~
λ
+
|
C
+
.
Since
u
¯
λ
∈
int
C
+
, it follows that
u
¯
λ
is a local
C
0
1
(
Ω
¯
)
-minimizer of
φ
~
λ
, hence
u
¯
λ
is a local
W
0
1
,
p
(
Ω
)
-minimizer of
φ
~
λ
(see Proposition 2.3).
We can argue in a similar manner for
v
¯
λ
∈
-
int
C
+
, using this time the functional
φ
~
λ
-
.
This proves the claim.
Without loss of generality, we may assume that
(4.12)
φ
~
λ
(
v
¯
λ
)
≤
φ
~
λ
(
u
¯
λ
)
.
Recall that
K
φ
~
λ
is finite.
So, on account of the claim and (4.12), we can find
ρ
∈
(
0
,
1
)
small such that
(4.13)
φ
~
λ
(
v
¯
λ
)
≤
φ
~
λ
(
u
¯
λ
)
<
inf
[
φ
~
λ
(
u
)
:
∥
u
-
u
¯
λ
∥
=
ρ
]
=
m
~
λ
,
∥
v
¯
λ
-
u
¯
λ
∥
>
ρ
.
We know that
φ
~
λ
is coercive (see (4.3)).
Therefore, we have that
(4.14)
φ
~
λ
satisfies the
C
-condition
(see [17, Proposition 2.2]). From (4.13), (4.14) we see that we can use Theorem 2.1 (the mountain pass theorem).
Therefore, we can find
y
0
∈
W
0
1
,
p
(
Ω
)
such that
y
0
∈
K
φ
~
λ
⊆
[
v
¯
λ
,
u
¯
λ
]
∩
C
0
1
(
Ω
¯
)
(see (4.10)) and
m
~
λ
≤
φ
~
λ
(
y
0
)
(see (4.13)), thus
y
0
∉
{
v
¯
λ
,
u
¯
λ
}
(see (4.13)).
Since
y
0
is a critical point of mountain pass type for
φ
~
λ
, we have
(4.15)
C
1
(
φ
~
λ
,
y
0
)
≠
0
(see [18, Proposition 6.100, p. 176]).
From (4.9) and (4.15), we infer that
y
0
≠
0
, hence
y
0
∈
C
0
1
(
Ω
¯
)
is a nodal solution of (1.1).
Let
a
(
y
)
=
|
y
|
p
-
2
y
+
y
for all
y
∈
ℝ
N
.
Evidently,
a
∈
C
(
ℝ
N
,
ℝ
N
)
(recall that
2
<
p
) and
div
a
(
D
u
)
=
Δ
p
u
+
Δ
u
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We have
∇
a
(
y
)
=
|
y
|
p
-
2
[
id
+
y
⊗
y
|
y
|
2
]
+
id
for all
y
∈
ℝ
N
,
hence
(4.16)
(
∇
a
(
y
)
ξ
,
ξ
)
ℝ
N
≥
|
ξ
|
2
for all
y
,
ξ
∈
ℝ
N
.
We know that
v
¯
λ
≤
y
0
≤
u
¯
λ
,
with
y
0
≠
v
¯
λ
,
y
0
≠
u
¯
λ
.
Using (4.16) and the tangency principle of Pucci and Serrin [26, p. 35], we have
(4.17)
v
¯
λ
(
z
)
<
y
0
(
z
)
<
u
¯
λ
(
z
)
for all
z
∈
Ω
.
Let
ρ
=
max
{
∥
u
¯
λ
∥
∞
,
∥
v
¯
λ
∥
∞
}
and let
ξ
^
ρ
>
0
be as postulated by Hypothesis 2.6 (vi).
Then (see (4.17) and Hypothesis 2.6 (vi))
-
Δ
p
y
0
-
Δ
y
0
+
ξ
^
ρ
|
y
0
|
p
-
2
y
0
=
λ
|
y
0
|
p
-
2
y
0
+
f
(
z
,
y
0
)
+
ξ
^
ρ
|
y
0
|
p
-
2
y
0
≤
λ
u
¯
λ
q
-
1
+
f
(
z
,
u
¯
λ
)
+
ξ
^
ρ
u
¯
λ
p
-
1
(4.18)
=
-
Δ
p
u
¯
λ
-
Δ
u
¯
λ
+
ξ
^
ρ
u
¯
λ
p
-
1
for a.a.
z
∈
Ω
.
Let
h
1
(
z
)
=
λ
|
y
0
(
z
)
|
q
-
2
y
0
(
z
)
+
f
(
z
,
y
0
(
z
)
)
+
ξ
^
ρ
|
y
0
(
z
)
|
p
-
2
y
0
(
z
)
,
h
2
(
z
)
=
λ
u
¯
λ
(
z
)
q
-
1
+
f
(
z
,
u
¯
λ
(
z
)
)
+
ξ
^
ρ
u
¯
λ
(
z
)
p
-
1
.
Evidently,
h
1
,
h
2
∈
L
∞
(
Ω
)
and
h
2
(
z
)
-
h
1
(
z
)
≥
λ
[
u
λ
(
z
)
q
-
1
-
|
y
0
(
z
)
|
q
-
2
y
0
(
z
)
]
,
which implies
h
1
≺
h
2
(see Hypothesis 2.6 (vi), (4.17) and recall
u
¯
λ
∈
int
C
+
,
y
0
∈
C
0
1
(
Ω
¯
)
).
Then from (4.18) and Proposition 2.4, we infer that
u
¯
λ
-
y
0
∈
int
C
+
.
Similarly, we show that
y
0
-
v
¯
λ
∈
int
C
+
.
Therefore,
(4.19)
y
0
∈
int
C
0
1
(
Ω
¯
)
[
v
¯
λ
,
u
¯
λ
]
.
On the other hand, from [12, Proposition 17] (see also [20]), we have a nodal solution of (1.1) such that
(4.20)
y
^
∉
int
C
0
1
(
Ω
¯
)
[
v
¯
λ
,
u
¯
λ
]
.
From (4.19) and (4.20) it follows that
y
0
≠
y
^
.
∎
We can state the following multiplicity result for problem (1.1).
Theorem 4.4.
If Hypothesis 2.6 holds, then there exists
λ
*
>
0
such that for all
λ
∈
(
0
,
λ
*
]
, problem (1.1) has at least six nontrivial smooth solutions
u
0
,
u
^
∈
int
C
+
,
u
0
≠
u
^
,
v
0
,
v
^
∈
-
int
C
+
,
v
0
≠
v
^
,
y
0
,
y
^
∈
C
0
1
(
Ω
¯
)
both nodal
.
