2 Mathematical Background and Hypotheses
Let X be a Banach space. We denote by X* the topological dual of X and by 〈⋅,⋅〉 the duality brackets for the pair (X*,X). Given φ∈C1(X,ℝ), we say that φ satisfies the “Cerami condition” (the “C-condition” for short) if the following property holds:
Every sequence {un}n⩾1⊆X such that {φ(un)}n⩾1⊆ℝ is bounded and (1+∥un∥)φ′(un)→0 in X* as n→∞, admits a strongly convergent subsequence.
This compactness-type condition on φ(⋅) is crucial in deriving 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 below.
Theorem 2.1.
Assume that φ∈C1(X,R) satisfies the C-condition, u0,u1∈X, ∥u1-u0∥>r,
max
{
φ
(
u
0
)
,
φ
(
u
1
)
}
<
inf
{
φ
(
u
)
:
∥
u
-
u
0
∥
=
r
}
=
m
r
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⩾mr and c is a critical value of φ (that is, there exists u∈X such that φ(u)=c and φ′(u)=0).
Recall that a Banach space X has the “Kadec–Klee property” if the following holds:
u
n
→
𝑤
u
in
X
,
∥
u
n
∥
→
∥
u
∥
⟹
u
n
→
u
in
X
.
It is an easy consequence of the parallelogram law that every Hilbert space has the Kadec–Klee property (see [5]).
In the study of problem ((${P_{\lambda}}$)), we will use the following three spaces:
H
1
(
Ω
)
,
C
1
(
Ω
¯
)
,
L
r
(
∂
Ω
)
(
1
⩽
r
⩽
∞
)
.
The Sobolev space H1(Ω) is a Hilbert space with inner product given by
(
u
,
h
)
=
∫
Ω
(
D
u
,
D
h
)
ℝ
N
𝑑
z
+
∫
Ω
u
h
𝑑
z
for all
u
,
h
∈
H
1
(
Ω
)
.
We denote by ∥⋅∥ the corresponding norm on H1(Ω). So, we have
∥
u
∥
=
[
∥
u
∥
2
2
+
∥
D
u
∥
2
2
]
1
/
2
for all
u
∈
H
1
(
Ω
)
.
The space C1(Ω¯) is an ordered Banach space with positive (order) cone
C
+
=
{
u
∈
C
1
(
Ω
¯
)
:
u
(
z
)
⩾
0
for all
z
∈
Ω
¯
}
.
This cone has a nonempty interior. Note that
D
+
=
{
u
∈
C
+
:
u
(
z
)
>
0
for all
z
∈
Ω
¯
}
⊆
int
C
+
.
In fact, D+ is the interior of C+ when the latter is furnished with the relative C(Ω¯)-norm topology.
On ∂Ω we consider the (N-1)-dimensional Hausdorff (surface) measure σ(⋅). Using this measure on ∂Ω, we can define in the usual way the “boundary” Lebesgue spaces Lr(∂Ω) (for 1⩽r⩽∞). From the theory of Sobolev spaces we know that there exists a unique continuous linear map γ0:H1(Ω)→L2(∂Ω) known as the “trace map” such that
γ
0
(
u
)
=
u
|
∂
Ω
for all
u
∈
H
1
(
Ω
)
∩
C
(
Ω
¯
)
.
So, the trace map assigns “boundary values” to every Sobolev function. The trace map is compact into Lp(∂Ω) for all 1⩽p<2(N-1)N-2 if N⩾3, and into Lp(∂Ω) for all 1⩽p⩽∞ if N=1,2. Also, we have
im
γ
0
=
H
1
2
,
2
(
∂
Ω
)
and
ker
γ
0
=
H
0
1
(
Ω
)
.
In what follows, for the sake of notational simplicity, we drop the use of the trace map γ0. All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces.
Next, we consider the following linear eigenvalue problem:
(2.1)
{
-
Δ
u
(
z
)
+
ξ
(
z
)
u
(
z
)
=
λ
^
u
(
z
)
in
Ω
,
∂
u
∂
n
+
β
(
z
)
(
u
)
=
0
on
∂
Ω
.
This problem was studied by D’Agui, Marano and Papageorgiou [2]. We impose the following conditions on the potential function ξ(⋅) and on the boundary coefficient β(⋅):
ξ
∈
L
s
(
Ω
)
with s>N.
β
∈
W
1
,
∞
(
∂
Ω
)
and β(z)⩾0 for all z∈∂Ω.
Remark.
The potential function ξ is both unbounded and sign-changing.
Remark.
If β≡0, then we recover the Neumann problem.
Let γ:H1(Ω)→ℝ be the C2-functional defined by
γ
(
u
)
=
∥
D
u
∥
2
2
+
∫
Ω
ξ
(
z
)
u
2
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
2
𝑑
σ
for all
u
∈
H
1
(
Ω
)
.
Problem (2.1) admits a smallest eigenvalue λ^1∈ℝ given by
(2.2)
λ
1
^
=
inf
{
γ
(
u
)
∥
u
∥
2
2
:
u
∈
H
1
(
Ω
)
,
u
≠
0
}
.
Moreover, there exists μ>0 such that
(2.3)
γ
(
u
)
+
μ
∥
u
∥
2
2
⩾
c
0
∥
u
∥
2
for some
c
0
>
0
and for all
u
∈
H
1
(
Ω
)
.
Using (2.3) and the special theorem for compact self-adjoint operators on Hilbert spaces, we produce the full spectrum of (2.2). This consists of a sequence {λ^k}k∈ℕ of distinct eigenvalues such that λ^k→+∞. Let E(λ^k) denote the eigenspace corresponding to the eigenvalue λ^k. By the regularity theory of Wang [15], we have
E
(
λ
^
k
)
⊆
C
1
(
Ω
¯
)
for all
k
∈
ℕ
.
Each eigenspace has the “Unique Continuation Property” (UCP for short). This means that if u∈E(λ^k) vanishes on a set of positive Lebesgue measure, then u≡0.
Let
H
¯
m
=
⊕
k
=
1
m
E
(
λ
^
k
)
and
H
^
m
=
H
¯
m
⊥
=
⊕
k
⩾
m
+
1
E
(
λ
^
k
)
¯
.
We have
H
1
(
Ω
)
=
H
¯
m
⊕
H
^
m
.
Moreover, for every m⩾2, we have variational characterizations for the eigenvalues λ^m analogue to that for λ^1 (see (2.2)):
(2.4)
λ
^
m
=
inf
{
γ
(
u
)
∥
u
∥
2
2
:
u
∈
H
^
m
-
1
,
u
≠
0
}
=
sup
{
γ
(
u
)
∥
u
∥
2
2
:
u
∈
H
¯
m
,
u
≠
0
}
,
m
⩾
2
.
In (2.2) the infimum is realized on E(λ^1), while in (2.4) both the infimum and the supremum are realized on E(λ^m). We know that dimE(λ^1)=1 (that is, the first eigenvalue λ^1 is simple). Hence the elements of E(λ^1) have constant sign. We denote by u^1∈C+∖{0} the positive L2-normalized eigenfunction (that is, ∥u^1∥2=1) corresponding to λ^1. By the strong maximum principle, we have u^1(z)>0 for all z∈Ω and if ξ+∈L∞(Ω) (that is, the potential function is bounded above), by the Hopf boundary point theorem we have u^1∈D+ (see [13, p. 120]).
Using (2.2), (2.4) and the above properties, we get the following useful inequalities.
Proposition 2.2.
If
ϑ
∈
L
∞
(
Ω
)
, ϑ(z)⩽λ^m for almost all z∈Ω, ϑ≢λ^m, m∈ℕ, then there exists c1>0 such that
c
1
∥
u
∥
2
⩽
γ
(
u
)
-
∫
Ω
ϑ
(
z
)
u
2
𝑑
z
for all
u
∈
H
^
m
-
1
.
If
ϑ
∈
L
∞
(
Ω
)
, ϑ(z)⩾λ^m for almost all z∈Ω, ϑ≢λ^m, m∈ℕ, then there exists c2>0 such that
γ
(
u
)
-
∫
Ω
ϑ
(
z
)
u
2
𝑑
z
⩽
-
c
2
∥
u
∥
2
for all
u
∈
H
¯
m
.
Note that if ξ≡0 and β≡0, then λ^1=0, while if ξ⩾0 and either ξ≢0 or β≢0, then λ^1>0. Also, the elements of E(λ^k) for k⩾2 are nodal (that is, sign-changing).
In addition to the eigenvalue problem (2.1), we can consider its weighted version. So, let m∈L∞(Ω), m(z)⩾0 for almost all z∈Ω, m≢0, and consider the following linear eigenvalue problem:
(2.5)
{
-
Δ
u
(
z
)
+
ξ
(
z
)
u
(
z
)
=
λ
~
m
(
z
)
u
(
z
)
in
Ω
,
∂
u
∂
n
+
β
(
z
)
u
=
0
on
∂
Ω
.
This eigenvalue problem exhibits the same properties as (2.1). So, the spectrum consists of a sequence {λ~k(m)}k∈ℕ of distinct eigenvalues such that λ~k(m)→+∞ as k→+∞. As for (2.1), the first eigenvalue λ~1(m) is simple and the elements of E(λ~1(m))⊆C1(Ω¯) have fixed sign, while the elements of E(λ~k(m))⊆C1(Ω¯) (for all k⩾2) are nodal. We have variational characterizations for all the eigenvalues as in (2.2) and (2.4) except that now the Rayleigh quotient is
γ
(
u
)
∫
Ω
m
(
z
)
u
2
𝑑
z
.
Moreover, the eigenspaces have the UCP property. These properties yield the following monotonicity property for the map m↦λ~k(m), k∈ℕ.
Proposition 2.3.
If m1,m2∈L∞(Ω), 0⩽m1(z)⩽m2(z) for almost all z∈Ω, m1≢0, m2≢m1, then
λ
~
k
(
m
2
)
<
λ
~
k
(
m
1
)
for all
k
∈
ℕ
.
Let f0:Ω×ℝ→ℝ be a Carathéodory function such that
|
f
0
(
z
,
x
)
⩽
a
0
(
z
)
[
1
+
|
x
|
r
-
1
]
|
for almost all
x
∈
ℝ
,
with a0∈L∞(Ω) and
1
<
r
⩽
2
*
=
{
2
N
N
-
2
if
N
⩾
3
,
+
∞
if
N
=
1
,
2
(the critical Sobolev exponent). Let F0(z,x)=∫0xf0(z,s)𝑑s and consider the C1-functional φ0:H1(Ω)→ℝ defined by
φ
0
(
u
)
=
1
2
γ
(
u
)
-
∫
Ω
F
0
(
z
,
u
)
𝑑
z
for all
u
∈
H
1
(
Ω
)
.
As in [10, Proposition 8], using the regularity theory of Wang [15], we obtain the following result.
Proposition 2.4.
Assume that u0∈H1(Ω) is a local C1(Ω¯)-minimizer of φ0(⋅), that is, there exists ρ1>0 such that
φ
0
(
u
0
)
⩽
φ
0
(
u
0
+
h
)
for all
h
∈
C
1
(
Ω
¯
)
,
∥
h
∥
C
1
(
Ω
¯
)
⩽
ρ
1
.
Then u0∈C1,α(Ω¯) with 0<α<1, and u0 is also a local H1(Ω)-minimizer of φ0, that is, there exists ρ2>0 such that
φ
0
(
u
0
)
⩽
φ
0
(
u
0
+
h
)
for all
h
∈
C
1
(
Ω
¯
)
,
∥
h
∥
⩽
ρ
2
.
Next, we recall some definitions and facts from Morse theory (critical groups). So, let X be a Banach space, let φ∈C1(X,ℝ) and let c∈ℝ. We introduce the following sets:
φ
c
=
{
u
∈
X
:
φ
(
u
)
⩽
c
}
,
K
φ
=
{
u
∈
X
:
φ
′
(
u
)
=
0
}
,
K
φ
c
=
{
u
∈
K
φ
:
φ
(
u
)
=
c
}
.
Given a topological pair (Y1,Y2) such that Y2⊆Y1⊆X, for every k∈ℕ0 we denote by Hk(Y1,Y2) the k-th-relative singular homology group for the pair (Y1,Y2) with integer coefficients. Suppose that u∈Kφc is isolated. 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 Kφ∩φc∩U={u}. The excision property of singular homology implies that the above definition of critical groups is independent of the choice of the neighborhood U. If u is a local minimizer of φ, then
C
k
(
φ
,
u
)
=
δ
k
,
0
ℤ
for all
k
∈
ℕ
0
.
Here, δk,m denotes the Kronecker symbol defined by
δ
k
,
m
=
{
1
if
k
=
m
,
0
if
k
≠
m
.
Next, let us fix our notation. If x∈ℝ, we set x±=max{±x,0}. For u∈W1,p(Ω) we define u±(⋅)=u(⋅)±. We know that
u
±
∈
W
1
,
p
(
Ω
)
,
u
=
u
+
-
u
-
,
|
u
|
=
u
+
+
u
-
.
Given a measurable function g:Ω×ℝ→ℝ (for example, a Carathéodory function), we denote by Ng(⋅) the Nemitsky (superposition) map defined by
N
g
(
u
)
(
⋅
)
=
g
(
⋅
,
u
(
⋅
)
)
for all
u
∈
W
1
,
p
(
Ω
)
.
Also, A∈ℒ(H1(Ω),H1(Ω)*) is defined by
〈
A
(
u
)
,
h
〉
=
∫
Ω
(
D
u
,
D
h
)
ℝ
N
𝑑
z
for all
u
,
h
∈
H
1
(
Ω
)
.
The hypotheses on the nonlinearity f(z,x) are the following:
f
:
Ω
×
ℝ
→
ℝ
is a Carathéodory function such that f(z,0)=0 for almost all z∈Ω and
For every ρ>0, there exists aρ∈L∞(Ω) such that
|
f
(
z
,
x
)
|
⩽
a
ρ
(
z
)
for almost all
z
∈
Ω
and for all
|
x
|
⩽
ρ
.
There exist functions η,η^∈L∞(Ω) and m∈ℕ, m⩾2, such that
λ
^
1
⩽
η
(
z
)
⩽
η
^
(
z
)
⩽
λ
^
m
for almost all
z
∈
Ω
,
η
≢
λ
^
1
,
η
^
≢
λ
^
m
,
η
^
(
z
)
⩽
lim inf
x
→
+
∞
f
(
z
,
x
)
x
⩽
lim sup
x
→
+
∞
f
(
z
,
x
)
x
⩽
η
^
(
z
)
uniformly for almost all
z
∈
Ω
,
and there exists η~>0 such that
-
η
^
⩽
lim inf
x
→
-
∞
f
(
z
,
x
)
x
⩽
lim sup
x
→
-
∞
f
(
z
,
x
)
x
⩽
λ
^
1
uniformly for almost all
z
∈
Ω
.
If F(z,x)=∫0xf(z,s)𝑑s, then
f
(
z
,
x
)
x
-
2
F
(
z
,
x
)
→
+
∞
uniformly for almost all
z
∈
Ω
as
x
→
-
∞
,
f
(
z
,
x
)
x
-
2
F
(
z
,
x
)
⩾
0
for almost all
z
∈
Ω
and for all
x
⩾
M
0
>
0
,
F
(
z
,
x
)
⩽
λ
^
m
2
x
2
for almost all
z
∈
Ω
and for all
x
∈
ℝ
.
There exist functions ϑ,ϑ^∈L∞(Ω) and l∈ℕ, l⩾m, such that
λ
^
l
⩽
ϑ
(
z
)
⩽
ϑ
^
(
z
)
⩽
λ
^
l
+
1
for almost all
z
∈
Ω
,
ϑ
≢
λ
^
l
,
ϑ
^
≠
λ
^
l
+
1
,
ϑ
(
z
)
⩽
lim inf
x
→
0
f
(
z
,
x
)
x
⩽
lim sup
x
→
0
f
(
z
,
x
)
x
⩽
ϑ
^
(
z
)
uniformly for almost all
z
∈
Ω
.
Remark.
Hypothesis H(f) (ii) implies that f(z,⋅) has asymmetric behavior as x→±∞ (jumping nonlinearity). Moreover, as x→-∞ we can have resonance with respect to the principal eigenvalue λ^1. Hypothesis H(f) (iii) implies that this resonance is from the left of λ^1 in the sense that
λ
^
1
x
2
-
2
F
(
z
,
x
)
→
+
∞
uniformly for almost all
z
∈
Ω
as
x
→
-
∞
.
Note that hypotheses H(f) (i), (ii) and (iv) imply that
(2.6)
|
f
(
z
,
x
)
|
⩽
c
3
|
x
|
for almost all
z
∈
Ω
for all
x
∈
ℝ
and for some
c
3
>
0
.
For every λ>0, let φλ:H1(Ω)→ℝ be the energy functional for problem ((${P_{\lambda}}$)) defined by
φ
λ
(
u
)
=
1
2
γ
(
u
)
+
λ
q
∥
u
∥
q
q
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
H
1
(
Ω
)
.
Evidently, φλ∈C1(H1(Ω),ℝ).
Let μ>0 be as in (2.3). We introduce the following truncations-perturbations of the reaction in problem ((${P_{\lambda}}$)):
(2.7)
{
k
λ
+
(
z
,
x
)
=
{
0
if
x
⩽
0
,
f
(
z
,
x
)
-
λ
x
q
-
1
+
μ
x
if
x
>
0
,
k
λ
-
(
z
,
x
)
=
{
f
(
z
,
x
)
-
λ
|
x
|
q
-
2
x
+
μ
x
if
x
<
0
,
0
if
x
⩾
0
.
Both are Carathéodory functions. We set
K
λ
±
(
z
,
x
)
=
∫
0
x
k
λ
±
(
z
,
s
)
𝑑
s
and consider the C1-functionals φ^λ±:H1(Ω)→ℝ defined by
φ
^
λ
±
(
u
)
=
1
2
γ
(
u
)
+
μ
2
∥
u
∥
2
2
-
∫
Ω
K
λ
±
(
z
,
u
)
𝑑
z
for all
u
∈
H
1
(
Ω
)
.
3 Compactness Conditions for the Functionals
We consider the functionals φ^λ± and φλ and we show that they satisfy the compactness-type condition.
Proposition 3.1.
If hypotheses H(ξ), H(β) and H(f) hold, then for every λ>0 the functional λ^λ+ satisfies the C-condition.
Proof.
We consider a sequence {un}n⩾1⊆H1(Ω) such that
|
φ
^
λ
+
(
u
n
)
|
⩽
M
1
for some
M
1
>
0
and for all
n
∈
ℕ
,
(3.1)
(
1
+
∥
u
n
∥
)
(
φ
^
λ
+
)
′
(
u
n
)
→
0
in
H
1
(
Ω
)
*
as
n
→
+
∞
.
From (3.1) we have
(3.2)
|
〈
A
(
u
n
)
,
h
〉
+
∫
Ω
[
ξ
(
z
)
+
μ
]
u
n
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
n
h
𝑑
σ
-
∫
Ω
k
λ
+
(
z
,
u
n
)
h
𝑑
z
|
⩽
ϵ
n
∥
h
∥
1
+
∥
u
n
∥
for all h∈H1(Ω), with ϵn→0+. In (3.2) we choose h=-un-∈H1(Ω). Then
γ
(
u
n
-
)
+
μ
∥
u
n
-
∥
2
2
⩽
ϵ
n
for all
n
∈
ℕ
(see (2.7))
,
⟹
c
0
∥
u
n
-
∥
2
⩽
ϵ
n
for all
n
∈
ℕ
(see (2.3))
,
(3.3)
⟹
u
n
-
→
0
in
H
1
(
Ω
)
as
n
→
∞
.
From (3.2) and (3.3) we have
(3.4)
|
〈
A
(
u
n
+
)
,
h
〉
+
∫
Ω
ξ
(
z
)
u
n
+
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
n
+
h
𝑑
σ
-
∫
Ω
[
f
(
z
,
u
n
+
)
-
λ
(
u
n
+
)
q
-
1
]
h
𝑑
z
|
⩽
ϵ
n
′
∥
h
∥
for all h∈H1(Ω), with ϵn′→0+ (see (2.7)).
We show that {un+}n⩾1⊆H1(Ω) is bounded. Arguing by contradiction, suppose that
(3.5)
∥
u
n
+
∥
→
∞
as
n
→
∞
.
Let
y
n
=
u
n
+
∥
u
n
+
∥
,
n
∈
ℕ
.
Then ∥yn∥=1 and yn⩾0 for all n∈ℕ. So, we may assume that
(3.6)
y
n
→
𝑤
y
in
H
1
(
Ω
)
and
y
n
→
y
in
L
2
(
Ω
)
and in
L
2
(
∂
Ω
)
,
y
⩾
0
.
Using (3.4), we obtain
|
〈
A
(
y
n
)
,
h
〉
+
∫
Ω
ξ
(
z
)
y
n
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
y
n
h
𝑑
σ
+
λ
∥
u
n
+
∥
2
-
q
∫
Ω
y
n
q
-
1
h
𝑑
z
-
∫
Ω
N
f
(
u
n
+
)
∥
u
n
+
∥
h
𝑑
z
|
(3.7)
⩽
ϵ
′
∥
h
∥
∥
u
n
+
∥
for all
n
∈
ℕ
.
From (2.6) we see that
(3.8)
{
N
f
(
u
n
+
)
∥
u
n
+
∥
}
n
⩾
1
⊆
L
2
(
Ω
)
is bounded
.
So, by passing to a subsequence if necessary and using hypothesis H(f) (ii), we have (see [1, Proof of Proposition 16])
(3.9)
N
f
(
u
n
+
)
∥
u
n
+
∥
→
𝑤
ν
(
z
)
y
in
L
2
(
Ω
)
,
η
(
z
)
⩽
ν
(
z
)
⩽
η
^
(
z
)
for almost all
z
∈
Ω
.
If in (3.7) we choose h=yn-y∈H1(Ω), passing to the limit as n→∞ and using (3.5), (3.6), (3.8) and the fact that q<2, we obtain
lim
n
→
∞
〈
A
(
y
n
)
,
y
n
-
y
〉
=
0
,
⟹
∥
D
y
n
∥
2
→
∥
D
y
∥
2
,
(3.10)
⟹
y
n
→
y
in
H
1
(
Ω
)
(by the Kadec–Klee property), and hence
∥
y
∥
=
1
.
In (3.7) we pass to the limit as n→∞ and use (3.9). We obtain
〈
A
(
y
)
,
h
〉
+
∫
Ω
ξ
(
z
)
y
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
y
h
𝑑
σ
=
∫
Ω
ν
(
z
)
y
h
𝑑
z
for all
h
∈
H
1
(
Ω
)
,
which implies
-
Δ
y
(
z
)
+
ξ
(
z
)
y
(
z
)
=
ν
(
z
)
y
(
z
)
for almost all
z
∈
Ω
,
(3.11)
∂
y
∂
n
+
β
(
z
)
y
=
0
on
∂
Ω
(see [9]).
From (3.9) and Proposition 2.3 we have
(3.12)
λ
~
1
(
ν
)
<
λ
~
1
(
λ
^
1
)
=
1
.
Then (3.11), (3.12) and the fact that ∥y∥=1 (see (3.10)) imply that y(⋅) must be nodal. But this contradicts (3.6). Therefore,
{
u
n
+
}
n
⩾
1
⊆
H
1
(
Ω
)
is bounded
,
⟹
{
u
n
}
n
⩾
1
⊆
H
1
(
Ω
)
is bounded (see (3.3))
.
We may assume that
(3.13)
u
n
→
𝑤
u
in
H
1
(
Ω
)
and
u
n
→
u
in
L
2
(
Ω
)
and in
L
2
(
∂
Ω
)
.
In (3.2) we choose h=un-u∈H1(Ω), pass to the limit as n→∞ and use (3.13) and (2.6). Then
lim
n
→
∞
〈
A
(
u
n
)
,
u
n
-
u
〉
=
0
,
⟹
u
n
→
u
in
H
1
(
Ω
)
(again by the Kadec–Klee property)
,
⟹
φ
^
λ
+
satisfies the C-condition
.
The proof is now complete. ∎
Proposition 3.2.
If hypotheses H(ξ), H(β) and H(f) hold, then for every λ>0 the functional φ^λ- is coercive.
Proof.
According to hypothesis H(f) (iii), given any ρ>0, we can find M2=M2(ρ)>0 such that
(3.14)
ρ
⩽
f
(
z
,
x
)
x
-
2
F
(
z
,
x
)
for almost all
z
∈
Ω
and for all
x
⩽
-
M
2
.
We have
d
d
x
(
F
(
z
,
x
)
x
2
)
=
f
(
z
,
x
)
x
2
-
2
x
F
(
z
,
x
)
x
4
=
f
(
z
,
x
)
x
-
2
F
(
z
,
x
)
|
x
|
2
x
⩽
ρ
|
x
|
2
x
for almost all
z
∈
Ω
and all
x
⩽
-
M
2
(see (3.14))
,
which implies
(3.15)
F
(
z
,
v
)
v
2
-
F
(
z
,
y
)
y
2
⩾
ρ
2
(
1
y
2
-
1
v
2
)
for almost all
z
∈
Ω
and for all
v
⩽
y
⩽
-
M
2
.
From hypothesis H(f) (ii) we have
(3.16)
-
η
~
⩽
lim inf
x
→
-
∞
2
F
(
z
,
x
)
x
2
⩽
lim sup
x
→
-
∞
2
F
(
z
,
x
)
x
2
⩽
λ
^
1
uniformly for almost all
z
∈
Ω
.
If in (3.15) we let v→-∞ and use (3.16), then
λ
^
1
y
2
-
2
F
(
z
,
y
)
⩾
ρ
for almost all
z
∈
Ω
and for all
y
⩽
-
M
2
,
(3.17)
⟹
λ
^
1
y
2
-
2
F
(
z
,
y
)
→
+
∞
uniformly for almost all
z
∈
Ω
as
y
→
-
∞
.
Suppose to the contrary that λ^λ- is not coercive. This means that we can find {un}n⩾1⊆H1(Ω) such that
(3.18)
∥
u
n
∥
→
∞
as
n
→
∞
and
φ
^
λ
-
(
u
n
)
⩽
M
3
for some
M
3
>
0
and for all
n
∈
ℕ
.
Let
v
n
=
u
n
∥
u
n
∥
,
n
∈
ℕ
.
Then ∥vn∥=1 for all n∈ℕ, and so we may assume that
(3.19)
v
n
→
𝑤
v
in
H
1
(
Ω
)
and
v
n
→
v
in
L
2
(
Ω
)
and in
L
2
(
∂
Ω
)
.
From (3.18) we have
1
2
γ
(
u
n
)
+
μ
2
∥
u
n
∥
2
2
-
∫
Ω
K
λ
-
(
z
,
u
n
)
𝑑
z
⩽
M
3
for all
n
∈
ℕ
,
(3.20)
⟹
1
2
γ
(
v
n
)
+
μ
2
∥
v
n
∥
2
2
-
∫
Ω
K
λ
-
(
z
,
u
n
)
∥
u
n
∥
2
𝑑
z
⩽
M
3
∥
u
n
∥
2
for all
n
∈
ℕ
.
From (2.6) we obtain
|
F
(
z
,
x
)
|
⩽
c
3
2
x
2
for almost all
z
∈
Ω
and for all
x
∈
ℝ
,
⟹
{
K
λ
-
(
⋅
,
u
n
(
⋅
)
)
∥
u
n
∥
2
}
n
⩾
1
⊆
L
1
(
Ω
)
is uniformly integrable (see (2.7) and (3.19))
.
Hence, by the Dunford–Pettis theorem and hypothesis H(f) (ii) we have
(3.21)
K
λ
-
(
⋅
,
u
n
(
⋅
)
)
∥
u
n
∥
2
→
𝑤
1
2
[
e
~
(
z
)
+
μ
]
(
v
-
)
2
in
L
1
(
Ω
)
as
n
→
∞
with -η~⩽e~(z)⩽λ^1 for almost all z∈Ω (see [1]).
We return to (3.20) and pass to the limit as n→∞ in (3.18), (3.19) and (3.21). Since γ(⋅) is sequentially weakly lower semicontinuous on H1(Ω), we obtain (see (2.3))
1
2
γ
(
v
)
+
μ
2
∥
v
∥
2
2
⩽
1
2
∫
Ω
[
e
~
(
z
)
+
μ
]
(
v
-
)
2
𝑑
z
(3.22)
⟹
γ
(
v
-
)
⩽
∫
Ω
e
~
(
z
)
(
v
-
)
2
𝑑
z
.
First, we assume that e~≢λ^1 (see (3.21)). Then by (3.22) and Proposition 2.2 we have c1∥v-∥2⩽0, which implies
(3.23)
v
⩾
0
.
Then on account of (3.19) and (3.23) we have
(3.24)
v
n
-
→
𝑤
0
in
H
1
(
Ω
)
and
v
n
-
→
0
in
L
2
(
Ω
)
and in
L
2
(
∂
Ω
)
.
In (3.20) we pass to the limit as n→∞ and use (3.24), (3.22) and the sequential weak lower semicontinuity of γ(⋅). We obtain
γ
(
v
+
)
+
μ
∥
v
+
∥
2
2
⩽
0
,
⟹
c
0
∥
v
+
∥
2
⩽
0
(see (2.3))
,
⟹
v
=
0
(see (3.23))
.
From (3.20) we obtain ∥Dvn∥2→0, which implies vn→0 in H1(Ω) (see (3.19)), which contradicts the fact that ∥vn∥=1 for all n∈ℕ.
Next, we assume that e~(z)=λ^1 for almost all z∈Ω. From (3.22) and (2.2) we have γ(v-)=λ^1∥v-∥22, which implies
(3.25)
v
-
=
τ
u
^
1
for some
τ
⩾
0
.
If τ=0, then v⩾0 and, arguing as above (see the part of the proof after (3.23)), we obtain v=0, contradicting the fact that ∥vn∥=1 for all n∈ℕ. If τ>0, then from (3.25) we have
v
(
z
)
<
0
for all
z
∈
Ω
.
This means that
u
n
-
(
z
)
→
-
∞
for almost all
z
∈
Ω
as
n
→
∞
,
⟹
λ
^
1
u
n
-
(
z
)
2
-
2
F
(
z
,
u
n
-
(
z
)
)
→
+
∞
for almost all
z
∈
Ω
as
n
→
∞
(see (3.17))
,
⟹
∫
Ω
[
λ
^
1
(
u
n
-
)
2
-
2
F
(
z
,
u
n
-
)
]
𝑑
z
→
+
∞
as
n
→
∞
(by Fatou’s lemma, see (3.17))
,
⟹
γ
(
u
n
-
)
-
2
∫
Ω
F
(
z
,
-
u
n
-
)
𝑑
z
→
+
∞
as
n
→
∞
(see (2.2))
,
⟹
2
φ
^
λ
-
(
u
n
-
)
→
+
∞
as
n
→
∞
(see (2.7))
.
But this contradicts (3.18). We conclude that φ^λ- is coercive. ∎
This proposition leads to the following corollary (see [6, Proposition 2.2]).
Corollary 3.3.
If hypotheses H(ξ), H(β) and H(f) hold, then for every λ>0 the functional φ^λ- satisfies the C-condition.
Next, we turn our attention to the energy functional φλ, λ>0.
Proposition 3.4.
If hypotheses H(ξ), H(β) and H(f) hold, then for every λ>0 the functional φλ satisfies the C-condition.
Proof.
We consider a sequence {un}n⩾1⊆H1(Ω) such that
(3.26)
|
φ
λ
(
u
n
)
|
⩽
M
4
for some
M
4
>
0
and for all
n
∈
ℕ
,
(3.27)
(
1
+
∥
u
n
∥
)
φ
λ
′
(
u
n
)
→
0
in
H
1
(
Ω
)
*
as
n
→
∞
.
From (3.27) we have
|
〈
A
(
u
n
)
,
h
〉
+
∫
Ω
ξ
(
z
)
u
n
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
n
h
𝑑
σ
+
λ
∫
Ω
|
u
n
|
q
-
2
u
n
h
𝑑
σ
-
∫
Ω
f
(
z
,
u
n
)
h
𝑑
z
|
(3.28)
⩽
ϵ
n
∥
h
∥
1
+
∥
u
n
∥
for all
h
∈
H
1
(
Ω
)
,
with
ϵ
n
→
0
+
.
In (3.28) we choose h=un∈H1(Ω). Then
(3.29)
-
γ
(
u
n
)
-
λ
∥
u
n
∥
q
q
+
∫
Ω
f
(
z
,
u
n
)
u
n
𝑑
z
⩽
ϵ
n
for all
n
∈
ℕ
.
On the other hand, from (3.26) we have
(3.30)
γ
(
u
n
)
+
2
λ
q
∥
u
n
∥
q
q
-
∫
Ω
2
F
(
z
,
u
n
)
𝑑
z
⩽
2
M
4
for all
n
∈
ℕ
.
We add (3.29) and (3.30). Recalling that q<2, we obtain
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
2
F
(
z
,
u
n
)
]
𝑑
z
⩽
M
5
for all
n
∈
ℕ
.
Using hypothesis H(f) (iii), we see that
(3.31)
∫
Ω
[
f
(
z
,
-
u
n
-
)
(
-
u
n
-
)
-
2
F
(
z
,
-
u
n
-
)
]
𝑑
z
⩽
M
5
for all
n
∈
ℕ
.
We use (3.31) to show that {un-}n⩾1⊆H1(Ω) is bounded. Arguing by contradiction, we may assume that
(3.32)
∥
u
n
-
∥
→
∞
as
n
→
∞
.
Let
y
n
=
u
n
-
∥
u
n
-
∥
,
n
∈
ℕ
.
Then ∥yn∥=1 and yn⩾0 for all n∈ℕ. We may assume that
(3.33)
y
n
→
𝑤
y
in
H
1
(
Ω
)
and
y
n
→
y
in
L
2
(
Ω
)
and in
L
2
(
∂
Ω
)
,
y
⩾
0
.
In (3.28) we choose h=-un-∈H1(Ω). Then
γ
(
u
n
-
)
+
λ
∥
u
n
-
∥
q
q
-
∫
Ω
f
(
z
,
-
u
n
-
)
(
-
u
n
-
)
𝑑
z
⩽
ϵ
n
for all
n
∈
ℕ
,
(3.34)
⟹
γ
(
y
n
)
+
λ
∥
u
n
-
∥
2
-
q
∥
y
n
∥
q
q
-
∫
Ω
N
f
(
-
u
n
-
)
∥
u
n
-
∥
y
n
𝑑
z
⩽
ϵ
n
∥
u
n
-
∥
2
for all
n
∈
ℕ
.
From (2.6) we see that
{
N
f
(
-
u
n
-
)
∥
u
n
-
∥
}
n
⩾
1
⊆
L
2
(
Ω
)
is bounded
.
So, by passing to a subsequence if necessary and using hypothesis H(f) (ii), we have
(3.35)
N
f
(
-
u
n
-
)
∥
u
n
-
∥
→
𝑤
e
~
(
z
)
y
in
L
2
(
Ω
)
as
n
→
∞
with -η~⩽e~(z)⩽λ^1 for almost all z∈Ω.
Returning to (3.34), passing to the limit as n→∞ and using (3.32) (recall that q<2), (3.33), (3.35) and the sequential weak lower semicontinuity of γ(⋅), we obtain
(3.36)
γ
(
y
)
⩽
∫
Ω
e
~
(
z
)
y
2
𝑑
z
.
First, we assume that e~≢λ^1 (see (3.35)). Then from (3.36) and Proposition 2.2 we get c1∥y∥2⩽0, which implies y=0. From this and (3.34) we infer that ∥Dyn∥2→0, which implies yn→0 in H1(Ω), which contradicts the fact that ∥yn∥=1 for all n∈ℕ.
We now assume that e~(z)=λ^1 for almost all z∈Ω. Then from (3.36) and (2.2) we have
y
=
τ
u
^
1
with
τ
⩾
0
.
If τ=0, then y=0 and, as above, we have
y
n
→
0
in
H
1
(
Ω
)
,
a contradiction since ∥yn∥=1 for all n∈ℕ. If τ>0, then y(z)>0 for all z∈Ω, and so
u
n
-
(
z
)
→
+
∞
for almost all
z
∈
Ω
,
⟹
f
(
z
,
-
u
n
-
(
z
)
)
(
-
u
n
-
)
(
z
)
-
2
F
(
z
,
-
u
n
-
(
z
)
)
→
+
∞
for almost all
z
∈
Ω
(see hypothesis
H
(
f
)
(iii)),
⟹
∫
Ω
[
f
(
z
,
-
u
n
-
)
(
-
u
n
-
)
-
2
F
(
z
,
-
u
n
-
)
]
𝑑
z
→
+
∞
(by Fatou’s lemma)
.
This contradicts (3.31). Therefore,
(3.37)
{
u
n
-
}
n
⩾
1
⊆
H
1
(
Ω
)
is bounded
.
Next, we show that {un+}n⩾1⊆H1(Ω) is bounded. From (3.28) and (3.37) we have
|
〈
A
(
u
n
+
)
,
h
〉
+
∫
Ω
ξ
(
z
)
u
n
+
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
n
+
h
𝑑
σ
+
λ
∫
Ω
(
u
n
+
)
q
-
1
h
𝑑
z
-
∫
Ω
f
(
z
,
u
n
+
)
h
𝑑
z
|
⩽
M
6
for some M6>0 and all n∈ℕ. Using this bound and a contradiction argument as in the proof of Proposition 3.1, we show that
{
u
n
+
}
n
⩾
1
⊆
H
1
(
Ω
)
is bounded
,
⟹
{
u
n
}
n
⩾
1
⊆
H
1
(
Ω
)
is bounded (see (3.37))
.
From this, as before (see the proof of Proposition 3.1), via the Kadec–Klee property, we conclude that φλ satisfies the C-condition. ∎
4 Multiplicity Theorems
In this section, using variational methods, truncation and perturbation techniques and Morse theory, we prove two multiplicity theorems for problem ((${P_{\lambda}}$)) when λ>0 is small. In the first result, we produce four nontrivial smooth solutions, while in the second theorem, under stronger conditions on f(z,⋅), we establish the existence of five nontrivial smooth solutions.
We start with a result which allows us to satisfy the mountain pass geometry (see Theorem 2.1) and also to distinguish the solutions we produce from the trivial one.
Proposition 4.1.
If hypotheses H(ξ), H(β) and H(f) hold, then u=0 is a local minimizer of φλ and of φ^λ± for every λ>0.
Proof.
We give the proof for the functional φλ. The proofs for the φ^λ± are similar.
Recall that
(4.1)
|
F
(
z
,
x
)
|
⩽
c
3
2
|
x
|
2
for almost all
z
∈
Ω
and for all
x
∈
ℝ
(see (2.6))
.
Then for u∈C1(Ω¯)∖{0} we have
φ
λ
(
u
)
⩾
λ
q
∥
u
∥
q
q
-
[
c
8
2
+
∥
ξ
∥
∞
]
∥
u
∥
2
2
(see (4.1) and hypotheses
H
(
ξ
)
,
H
(
β
)
)
.
⩾
λ
q
∥
u
∥
q
q
-
c
4
∥
u
∥
∞
2
-
q
∥
u
∥
q
q
(with
c
4
=
[
c
1
2
+
∥
ξ
∥
∞
]
>
0
)
=
[
λ
q
-
c
4
∥
u
∥
∞
2
-
q
]
∥
u
∥
q
q
.
So, if
∥
u
∥
∞
⩽
∥
u
∥
C
1
(
Ω
¯
)
<
(
λ
q
c
4
)
1
2
-
q
,
then φλ(u)>0=φλ(0). Hence,
u
=
0
is a local
C
1
(
Ω
¯
)
-minimizer of
φ
λ
(
⋅
)
,
⟹
u
=
0
is a local
H
1
(
Ω
)
-minimizer of
φ
λ
(
⋅
)
(see Proposition 2.4)
.
The proofs for the functionals φ^λ± are similar. ∎
With the next proposition we guarantee that for small λ>0 the functional φ^λ+(⋅) satisfies the mountain pass geometry (see Theorem 2.1).
Proposition 4.2.
If hypotheses H(ξ), H(β) and H(f) hold, then we can find λ*>0 such that for all λ∈(0,λ*) there is t0=t0(λ)>0 for which we have φ^λ+(t0u^1)<0.
Proof.
Let r>2. From hypothesis H(f) (iv) and (4.1), we see that given ϵ>0 we can find c5=c5(ϵ,r)>0 such that
(4.2)
F
(
z
,
x
)
⩾
1
2
[
ϑ
(
z
)
-
ϵ
]
x
2
-
c
5
x
r
for almost all
z
∈
Ω
and for all
x
⩾
0
.
Then for all t>0 we have
φ
^
λ
+
(
t
u
^
1
)
=
t
2
2
γ
(
u
^
1
)
+
λ
t
q
q
∥
u
^
1
∥
q
q
-
∫
Ω
F
(
z
,
t
u
^
1
)
𝑑
z
(see (2.7))
⩽
t
2
2
[
γ
(
u
^
1
)
-
∫
Ω
ϑ
(
z
)
u
^
1
2
𝑑
z
+
ϵ
]
+
λ
t
q
q
∥
u
^
1
∥
q
q
+
c
5
t
r
∥
u
^
1
∥
r
r
(
see (4.2) and recall that
∥
u
^
1
∥
2
=
1
)
(4.3)
=
t
2
2
[
∫
Ω
(
λ
^
1
-
ϑ
(
z
)
)
u
^
1
2
𝑑
z
+
ϵ
]
+
λ
t
q
q
∥
u
^
1
∥
q
q
+
c
5
t
r
∥
u
^
1
∥
r
r
.
Note that
k
*
=
∫
Ω
(
ϑ
(
z
)
-
λ
^
1
)
u
^
1
2
𝑑
z
>
0
(see hypothesis
H
(
f
)
(iv))
.
Choosing ϵ∈(0,k*), we see from (4.3) that
(4.4)
φ
^
λ
+
(
t
u
^
1
)
⩽
-
c
6
t
2
+
λ
c
7
t
q
+
c
8
t
r
=
[
-
c
6
+
λ
c
7
t
q
-
2
+
c
8
t
r
-
2
]
t
2
for some
c
6
,
c
7
,
c
8
>
0
.
Consider the function
𝒥
λ
(
t
)
=
λ
c
7
t
q
-
2
+
c
8
t
r
-
2
for all
t
>
0
.
Evidently, 𝒥λ∈C1(0,+∞), and since 1<q<2<r, we see that
𝒥
λ
(
t
)
→
+
∞
as
t
→
0
+
and as
t
→
+
∞
.
So, we can find t0∈(0,+∞) such that
𝒥
λ
(
t
0
)
=
min
{
𝒥
(
t
)
:
0
<
t
<
+
∞
}
,
⟹
𝒥
λ
′
(
t
0
)
=
0
,
⟹
λ
c
7
(
2
-
q
)
t
0
q
-
3
=
c
8
(
r
-
2
)
t
0
r
-
3
,
⟹
t
0
=
t
0
(
λ
)
=
[
λ
c
7
(
2
-
q
)
c
8
(
r
-
2
)
]
1
r
-
q
.
Then
𝒥
λ
(
t
0
)
=
λ
c
7
[
c
8
(
r
-
2
)
]
2
-
q
r
-
q
[
λ
c
2
(
2
-
q
)
]
2
-
q
r
-
q
+
c
8
[
λ
c
2
(
2
-
q
)
]
r
-
2
2
-
q
[
c
8
(
r
-
2
)
]
r
-
2
2
-
q
.
Since 2-qr-q<1, we see that
𝒥
λ
(
t
0
)
→
0
+
as
λ
→
0
+
.
So, we can find λ*>0 such that
𝒥
λ
(
t
0
)
<
c
6
for all
λ
∈
(
0
,
λ
*
)
.
Then it follows from (4.4) that
φ
^
λ
+
(
t
0
u
^
1
)
<
0
for all
λ
∈
(
0
,
λ
*
)
.
This completes the proof of Proposition 4.2. ∎
Remark.
In fact, a careful reading of the above proof reveals that
(4.5)
φ
^
λ
-
(
-
t
0
u
^
1
)
<
0
for all
λ
∈
(
0
,
λ
*
)
.
Proposition 4.3.
If hypotheses H(ξ), H(β) and H(f) hold and λ∈(0,λ*), then there exists u0∈C1(Ω¯) with u0(z)<0 for all z∈Ω and
φ
^
λ
-
(
u
0
)
=
inf
{
φ
^
λ
-
(
u
)
:
u
∈
H
1
(
Ω
)
}
<
0
.
Proof.
From Proposition 3.2 we know that φ^λ- is coercive. Also, the Sobolev embedding theorem and the compactness of the trace map imply that φ^λ- is sequentially weakly lower semicontinuous. Hence, by the Weierstrass–Tonelli theorem, we can find u0∈H1(Ω) such that
(4.6)
φ
^
λ
-
(
u
0
)
=
inf
{
φ
^
λ
-
(
u
)
:
u
∈
W
1
,
p
(
Ω
)
}
.
From (4.5) we see that φ^λ-(u0)<0=λ^λ-(0), which implies u0≠0.
From (4.6) we have (φ^λ-)′(u0)=0, which implies
(4.7)
〈
A
(
u
0
)
,
h
〉
+
∫
Ω
[
ξ
(
z
)
+
μ
]
u
0
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
0
h
𝑑
σ
=
∫
Ω
k
λ
-
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
H
1
(
Ω
)
.
In (4.7) we choose h=u0+∈H1(Ω). Then
γ
(
u
0
+
)
+
μ
∥
u
0
+
∥
2
2
=
0
(see (2.7))
,
⟹
c
0
∥
u
0
+
∥
2
⩽
0
(see (2.3))
,
⟹
u
0
⩽
0
,
u
0
≠
0
.
From (4.7) and (2.7) it follows that
〈
A
(
u
0
)
,
h
〉
+
∫
Ω
ξ
(
z
)
u
0
h
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
0
h
𝑑
σ
=
∫
Ω
[
f
(
z
,
u
0
)
-
λ
|
u
0
|
q
-
2
u
0
]
h
𝑑
z
for all
h
∈
H
1
(
Ω
)
,
which implies
-
Δ
u
0
(
z
)
+
ξ
(
z
)
u
0
(
z
)
=
f
(
z
,
u
0
(
z
)
)
-
λ
|
u
0
(
z
)
|
q
-
2
u
0
(
z
)
for almost all
z
∈
Ω
,
(4.8)
∂
u
0
∂
n
+
β
(
z
)
u
0
=
0
on
∂
Ω
(see [9]).
Let
τ
λ
(
z
,
x
)
=
f
(
z
,
x
)
-
λ
|
x
|
q
-
2
x
and
k
^
λ
(
z
)
=
τ
λ
(
z
,
u
0
(
z
)
)
1
+
|
u
0
(
z
)
|
for
λ
>
0
.
Hypotheses H(f) (i) and (ii) imply that
|
τ
λ
(
z
,
x
)
|
⩽
c
9
[
1
+
|
x
|
]
for almost all
z
∈
Ω
and all
x
∈
ℝ
,
with
c
9
=
c
9
(
λ
)
>
0
,
⟹
|
k
^
λ
(
z
)
|
=
|
τ
λ
(
z
,
u
0
(
z
)
)
|
1
+
|
u
0
(
z
)
|
⩽
c
9
for almost all
z
∈
Ω
,
⟹
k
^
λ
∈
L
∞
(
Ω
)
.
From (4.8) we have
-
Δ
u
0
(
z
)
=
[
ξ
(
z
)
-
k
^
λ
(
z
)
]
u
0
(
z
)
+
k
^
λ
(
z
)
for almost all
z
∈
Ω
,
∂
u
0
∂
n
+
β
(
z
)
u
0
=
0
on
∂
Ω
(recall that u0⩽0). Since (ξ-k^λ)(⋅)∈Ls(Ω) (for s>N), we deduce by [15, Lemma 5.1] that
u
0
∈
L
∞
(
Ω
)
.
Then the Calderon–Zygmund estimates (see [15, Lemma 5.2]) imply that
u
0
∈
(
-
C
+
)
∖
{
0
}
.
Moreover, the Harnack inequality (see [13, p. 163, Theorem 7.2.1]) implies that
u
0
(
z
)
<
0
for all
z
∈
Ω
.
This completes the proof. ∎
Remark.
The negative sign of the concave term does not allow us to conclude that u0∈-D+ when ξ+∈L∞(Ω) (by Hopf’s boundary point theorem, see [13, p. 120]).
Now we can state and prove our first multiplicity theorem.
Theorem 4.4.
Assume that hypotheses H(ξ), H(β) and H(f) hold. Then there exists λ^>0 such that for all λ∈(0,λ^) problem ((${P_{\lambda}}$)) has at least four nontrivial solutions
u
0
,
u
^
∈
(
-
C
+
)
∖
{
0
}
,
u
0
(
z
)
,
u
^
(
z
)
<
0
for all
z
∈
Ω
,
v
0
∈
C
+
∖
{
0
}
,
v
0
(
z
)
>
0
for all
z
∈
Ω
,
y
0
∈
C
1
(
Ω
¯
)
∖
{
0
}
.
Proof.
From Proposition 4.3 and its proof (see (4.8)) we already have one solution
u
0
∈
(
-
C
+
)
∖
{
0
}
,
u
0
(
z
)
<
0
for all
z
∈
Ω
,
when
λ
∈
(
0
,
λ
*
)
.
This solution is a global minimizer of the functional φ^λ-.
Claim.
The solution u0 is a local minimizer of the energy functional φλ.
We first show that u0 is a local C1(Ω¯)-minimizer of φλ. Arguing by contradiction, suppose that we could find a sequence {un}n⩾1⊆C1(Ω¯) such that
(4.9)
u
n
→
u
0
in
C
1
(
Ω
¯
)
as
n
→
∞
and
φ
λ
(
u
n
)
<
φ
λ
(
u
0
)
for all
n
∈
ℕ
.
Then for all n∈ℕ, we have
0
>
φ
λ
(
u
n
)
-
φ
λ
(
u
0
)
=
φ
λ
(
u
n
)
-
φ
^
λ
-
(
u
0
)
(since
φ
λ
|
(
-
C
+
)
=
φ
^
λ
-
|
(
-
C
+
)
,
see (2.7)
)
⩾
φ
λ
(
u
n
)
-
φ
^
λ
-
(
u
n
)
(recall that
u
0
is a global minimizer of
φ
^
λ
-
)
=
1
2
γ
(
u
n
)
+
λ
q
∥
u
n
∥
q
q
-
∫
Ω
F
(
z
,
u
n
)
𝑑
z
-
1
2
γ
(
u
n
)
-
μ
2
∥
u
n
+
∥
2
2
-
λ
q
∥
u
n
-
∥
q
q
+
∫
Ω
F
(
z
,
-
u
n
-
)
𝑑
z
(see (2.7))
=
λ
q
∥
u
n
+
∥
q
q
-
μ
2
∥
u
n
+
∥
2
2
-
∫
Ω
F
(
z
,
u
n
+
)
𝑑
z
⩾
λ
q
∥
u
n
+
∥
q
q
-
(
μ
+
c
3
2
)
∥
u
n
+
∥
2
2
(see (4.1)
)
⩾
λ
q
∥
u
n
+
∥
q
q
-
c
10
∥
u
n
+
∥
∞
2
-
q
∥
u
n
+
∥
q
q
(4.10)
=
[
λ
q
-
c
10
∥
u
n
+
∥
∞
2
-
q
]
∥
u
n
+
∥
q
q
,
where
c
10
=
μ
+
c
3
2
>
0
.
From (4.9) we have
u
n
+
→
0
in
C
1
(
Ω
¯
)
(recall that
u
0
|
Ω
<
0
)
.
Therefore, we can find n0∈ℕ such that
λ
q
>
c
10
∥
u
n
+
∥
∞
2
-
q
for all
n
⩾
n
0
,
⟹
0
>
φ
λ
(
u
n
)
-
φ
(
u
0
)
>
0
for all
n
⩾
n
0
(see (4.10))
,
a contradiction. Hence we have that
u
0
is a local
C
1
(
Ω
¯
)
-minimizer of
φ
λ
,
⟹
u
0
is a local
H
1
(
Ω
)
-minimizer of
φ
λ
(see Proposition 2.4)
.
This proves the claim.
Using (2.7) and the regularity theory of Wang [15], we can see that
(4.11)
K
λ
^
λ
-
⊆
(
-
C
+
)
and
K
φ
^
λ
+
⊆
C
+
for all
λ
>
0
.
On account of (4.11), we see that we may assume that both critical sets Kφ^λ- and Kφ^λ+ are finite or, otherwise, we already have an infinity of nontrivial smooth solutions of constant sign and so we are done.
From Proposition 4.1 we know that u=0 is a local minimizer of φ^λ- for all λ>0. Since Kφ^λ- is finite, we can find ρ∈(0,∥u0∥) small such that (see [1, Proof of Proposition 29])
(4.12)
φ
^
λ
-
(
u
0
)
<
0
=
φ
^
λ
-
(
0
)
<
inf
{
φ
^
λ
-
(
u
)
:
∥
u
∥
=
ρ
}
=
m
^
ρ
-
.
From Corollary 3.3 we know that
(4.13)
φ
^
λ
-
satisfies the C-condition
.
Then (4.12) and (4.13) permit the use of Theorem 2.1 (the mountain pass theorem). So, we can find u^∈H1(Ω) such that
u
^
∈
K
φ
^
λ
-
⊆
(
-
C
+
)
(see (4.11))
and
φ
^
λ
-
(
u
0
)
<
0
=
φ
^
λ
-
(
0
)
<
m
^
ρ
-
⩽
φ
^
λ
-
(
u
^
)
.
It follows that
u
^
∈
(
-
C
+
)
∖
{
0
,
u
0
}
is a solution of (Pl) (see (2.7))
.
As before, Harnack’s inequality implies that
u
^
(
z
)
<
0
for all
z
∈
Ω
.
Now we use once more Proposition 4.1 to find ρ0∈(0,t0) small enough such that
(4.14)
0
=
φ
^
λ
+
(
0
)
<
inf
{
φ
^
λ
+
(
u
)
:
∥
u
∥
=
ρ
0
}
=
m
^
ρ
0
+
,
λ
>
0
.
Proposition 4.2 implies that we can find λ*>0 such that
(4.15)
φ
^
λ
+
(
t
0
u
^
1
)
<
0
for all
λ
∈
(
0
,
λ
*
)
with
t
0
=
t
0
(
λ
)
>
0
.
Moreover, Proposition 2.5 implies that
(4.16)
φ
^
λ
+
satisfies the C-condition for all
λ
>
0
.
Then, on account of (4.14)–(4.16), we can apply Theorem 2.1 (the mountain pass theorem) and produce v0∈H1(Ω) such that
v
0
∈
K
φ
^
λ
+
⊆
C
+
(see (4.11))
and
0
=
φ
^
λ
+
(
0
)
<
m
^
ρ
+
⩽
φ
^
λ
+
(
v
0
)
,
⟹
v
0
∈
C
+
∖
{
0
}
is a solution of (Pl)
,
λ
∈
(
0
,
λ
*
)
(see (2.7))
.
Once again, Harnack’s inequality guarantees that
v
0
(
z
)
>
0
for all
z
∈
Ω
.
Let l∈ℕ be as in hypothesis H(f) (iv) and set
H
¯
l
=
⊕
k
=
1
l
E
(
λ
^
k
)
,
H
^
l
=
H
¯
l
⊥
=
⊕
k
⩾
l
+
1
E
(
λ
^
k
)
¯
.
We have
H
1
(
Ω
)
=
H
¯
l
⊕
H
^
l
and
dim
H
¯
l
<
+
∞
.
Consider u∈H¯l. We have
φ
λ
(
u
)
=
1
2
γ
(
u
)
+
λ
q
∥
u
∥
q
q
-
∫
Ω
F
(
z
,
u
)
𝑑
z
(4.17)
⩽
1
2
[
γ
(
u
)
-
∫
Ω
ϑ
(
z
)
u
2
𝑑
z
+
ϵ
∥
u
∥
2
]
+
c
11
[
λ
∥
u
∥
q
+
∥
u
∥
r
]
(4.18)
⩽
1
2
[
-
c
2
+
ϵ
]
∥
u
∥
2
+
c
11
[
λ
∥
u
∥
q
+
∥
u
∥
r
]
,
where (4.17) holds for some c11>0 and follows from (4.2) and the fact that all norms on H¯l are equivalent, and (4.18) follows from Proposition 2.2. Choosing ϵ∈(0,c2), we have
φ
λ
(
u
)
⩽
[
-
c
12
+
λ
c
11
∥
u
∥
q
-
2
+
c
11
∥
u
∥
r
-
2
]
∥
u
∥
2
for some
c
12
>
0
.
Reasoning as in the proof of Proposition 4.3, we can find λ^∈(0,λ*] such that for all λ∈(0,λ^] there exists ρλ>0 for which we have
(4.19)
φ
λ
(
u
)
<
0
for all
u
∈
H
¯
l
,
∥
u
∥
=
ρ
λ
.
For u∈H^l we have
φ
λ
(
u
)
⩾
1
2
γ
(
u
)
+
λ
q
∥
u
∥
q
q
-
λ
^
m
2
∥
u
∥
2
2
(see hypothesis
H
(
f
)
(iii))
⩾
1
2
[
γ
(
u
)
-
λ
^
l
∥
u
∥
2
2
]
+
λ
q
∥
u
∥
q
q
(since
l
⩾
m
)
(4.20)
⩾
0
.
Finally, consider the half-space
H
+
=
{
t
u
^
1
+
u
~
:
t
⩾
0
,
u
~
∈
H
^
l
}
.
Exploiting the orthogonality of H^l and H¯l, for every u∈H+ we have
φ
λ
(
u
)
⩾
1
2
[
t
2
γ
(
u
^
1
)
+
γ
(
u
~
)
]
-
λ
^
m
2
[
t
2
∥
u
^
1
∥
2
2
+
∥
u
~
∥
2
2
]
(see hypothesis
H
(
f
)
(iii))
(4.21)
⩾
0
(since
u
~
∈
H
^
l
,
l
⩾
m
).
Then (4.19)–(4.21) permit the use of [12, Theorem 3.1]. So, we can find y0∈H1(Ω) such that
y
0
∈
K
φ
λ
⊆
C
1
(
Ω
¯
)
(by the regularity theory of Wang [15])
,
(4.22)
φ
λ
(
y
0
)
<
0
=
φ
λ
(
0
)
and
C
d
l
-
1
(
φ
λ
,
y
0
)
≠
0
(
d
l
=
dim
H
¯
l
)
.
From (4.22) it is clear that y0≠0. Recall that
0
<
φ
λ
(
u
^
)
,
φ
λ
(
v
0
)
(since
φ
λ
=
φ
^
λ
-
|
(
-
C
+
)
=
φ
^
λ
+
|
C
+
)
.
Therefore, it follows from (4.22) that
y
0
∉
{
u
^
,
v
0
,
0
}
.
Also, by the claim we have that u0 is a local minimizer of φλ. Hence,
(4.23)
C
k
(
φ
λ
,
u
0
)
=
δ
k
,
0
ℤ
for all
k
∈
ℕ
0
.
Note that dl⩾2 (since l⩾m⩾2). Therefore,
d
l
-
1
⩾
1
,
and so from (4.22) and (4.23) we infer that
y
0
≠
u
0
.
So, we conclude that y0∈C1(Ω¯)∖{0} is a fourth nontrivial solution of ((${P_{\lambda}}$)) (for all λ∈(0,λ^)) distinct from u0, u^ and v0. ∎
If we strengthen the hypotheses on f(z,⋅), we can improve the above multiplicity theorem and produce a fifth nontrivial smooth solution.
The new conditions on the nonlinearity f(z,x) are the following:
f
:
Ω
×
ℝ
→
ℝ
is a measurable function such that for almost all z∈Ω, f(z,0)=0, f(z,⋅)∈C1(ℝ), hypotheses H(f)′ (i), (ii) and (iii) are the same as the corresponding hypotheses H(f) (i), (ii) and (iii), and, furthermore,
(iv) there exist l∈ℕ, l⩾m such that
f
x
′
(
z
,
0
)
=
lim
x
→
0
f
(
z
,
x
)
x
uniformly for almost all
z
∈
Ω
,
f
x
′
(
z
,
0
)
∈
[
λ
^
l
,
λ
^
l
+
1
]
for almost all
z
∈
Ω
,
f
x
′
(
⋅
,
0
)
≢
λ
^
l
,
f
x
′
(
⋅
,
0
)
≢
λ
^
l
+
1
.
Theorem 4.5.
If hypotheses H(ξ), H(β) and H(f)′ hold, then there exists λ^>0 such that for all λ∈(0,λ^) problem ((${P_{\lambda}}$)) has at least five nontrivial solutions
u
0
,
u
^
∈
(
-
C
+
)
,
u
0
(
z
)
<
0
for all
z
∈
Ω
,
v
0
∈
C
+
,
v
0
(
z
)
>
0
for all
z
∈
Ω
,
y
0
,
y
^
∈
C
1
(
Ω
¯
)
∖
{
0
}
.
Proof.
Now we have φλ∈C2(H1(Ω)∖{0},ℝ). Similarly, φ^λ±∈C2(H1(Ω)∖{0},ℝ).
The solutions u0,u^,v0,y0 are a consequence of Theorem 4.4. From Proposition 4.1 and (4.23) we have
(4.24)
C
k
(
φ
λ
,
u
0
)
=
C
k
(
φ
λ
,
0
)
=
δ
k
,
0
ℤ
for all
k
∈
ℕ
0
,
λ
∈
(
0
,
λ
^
)
.
Also, from the proof of Theorem 4.4 we know that u^ is a critical point of φ^λ- of mountain pass type, and v0 is a critical point of φ^λ+ of mountain pass type.
Invoking [7, Corollary 6.102], we have
(4.25)
C
k
(
φ
λ
-
,
u
^
)
=
C
k
(
φ
^
λ
+
,
v
0
)
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
.
The continuity in the C1-norm of the critical groups (see [5, p. 836, Theorem 5.126]) implies that
(4.26)
C
k
(
φ
^
λ
-
,
u
^
)
=
C
k
(
φ
λ
,
u
^
)
for all
k
∈
ℕ
0
,
(4.27)
C
k
(
φ
^
λ
+
,
v
0
)
=
C
k
(
φ
λ
,
v
0
)
for all
k
∈
ℕ
0
.
From (4.25)–(4.27) it follows that
(4.28)
C
k
(
φ
λ
,
u
^
)
=
C
k
(
φ
λ
,
v
0
)
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
.
The fourth nontrivial solution y0∈C1(Ω¯) was produced by using [12, Theorem 3.1]. According to that theorem, we can also find another function y^∈H1(Ω), y^≠y0, such that
(4.29)
y
^
∈
K
φ
λ
⊆
C
1
(
Ω
¯
)
and
C
d
l
(
φ
λ
,
y
^
)
≠
0
(
d
l
⩾
2
)
.
From (4.24)–(4.29) we conclude that
y
^
∈
C
1
(
Ω
¯
)
∖
{
u
0
,
u
^
,
v
0
,
y
0
,
0
}
is the fifth nontrivial solution of problem ((${P_{\lambda}}$)) for all λ∈(0,λ^). ∎
und
Dušan D. Repovš
