2 Mathematical Background
Let X be a Banach space and X* its topological dual. By 〈⋅,⋅〉 we denote 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, with
(
1
+
∥
u
n
∥
)
φ
′
(
u
n
)
→
0
in
X
*
as
n
→
∞
,
admits a strongly convergent subsequence.
This is a compactness-type condition on the functional φ which compensates for the fact the ambient space X need not be locally compact (usually X is infinite dimensional). It is more general than the usual Palais–Smale condition. Nevertheless, the C-condition leads to a deformation theorem from which one can derive the minimax theory of the critical values of φ. Prominent in this theory is the so-called “mountain pass theorem”, due to Ambrosetti and Rabinowitz [4]. Here we state it in a slightly more general form (see, for example, [13, p. 648]).
Theorem 2.1
Let X be a Banach space, let φ∈C1(X,R) satisfy the C-condition, let u0,u1∈X be such that
∥
u
1
-
u
0
∥
>
ρ
>
0
,
max
{
φ
(
u
0
)
,
φ
(
u
1
)
}
<
inf
{
φ
(
u
)
:
∥
u
-
u
0
∥
=
ρ
}
=
m
ρ
,
and let
c
=
inf
γ
∈
Γ
max
0
≤
t
≤
1
φ
(
γ
(
t
)
)
,
where
Γ
=
{
γ
∈
C
(
[
0
,
1
]
,
X
)
:
γ
(
0
)
=
u
0
,
γ
(
1
)
=
u
1
}
.
Then c≥mρ and c is a critical value of φ.
Let ϑ∈C1(0,∞) and assume that it satisfies the following growth conditions:
(2.1)
0
<
c
^
≤
t
ϑ
′
(
t
)
ϑ
(
t
)
≤
c
0
and
c
1
t
p
-
1
≤
ϑ
(
t
)
≤
c
2
(
1
+
t
p
-
1
)
for all t>0 and some c1,c2>0, 1<p<∞.
We introduce the precise conditions on the map y→a(y),y∈ℝN, involved in the definition of the differential operator.
We set a(y)=a0(|y|)y for all y∈ℝN with a0(t)>0 for all t>0, and assume the following:
a0∈C1(0,∞), t↦a0(t)t is strictly increasing on (0,∞),a0(t)t→0+ as t→0+ and
lim
t
→
0
+
a
0
′
(
t
)
t
a
0
(
t
)
>
-
1
.
For some c3>0 and all y∈ℝN∖{0},
|
∇
a
(
y
)
|
≤
c
3
ϑ
(
|
y
|
)
|
y
|
.
For all y∈ℝN∖{0} and all ξ∈ℝN,
ϑ
(
|
y
|
)
|
y
|
|
ξ
|
2
≤
(
∇
a
(
y
)
ξ
,
ξ
)
ℝ
N
.
If
G
0
(
t
)
=
∫
0
t
a
0
(
s
)
s
𝑑
s
for all
t
≥
0
,
then we have
-
c
~
≤
p
G
0
(
t
)
-
a
0
(
t
)
t
2
for all
t
≥
0
with
c
~
>
0
.
Remark 2.2
Hypotheses (Ha) (i)–(iii) are dictated by the nonlinear regularity theory of Lieberman [24] and the nonlinear maximum principle of Pucci and Serrin [36]. Hypothesis (Ha) (iv) corresponds to the particular features of our problem, but it is very mild and it is satisfied in all the major cases of interest as the examples below illustrate.
Set G(y)=G0(|y|) for all y∈ℝN. We have
∇
G
(
y
)
=
G
0
′
(
|
y
|
)
y
|
y
|
=
a
0
(
|
y
|
)
y
=
a
(
y
)
for all
y
∈
ℝ
N
∖
{
0
}
,
∇
G
(
0
)
=
0
.
So, G(⋅) is the primitive of a(⋅).
Hypotheses (Ha) imply that the functions G(⋅), G0(⋅) are both strictly convex and G0(⋅) is also strictly increasing. The convexity of G(⋅) and the fact that G(0)=0 imply
(2.2)
G
(
y
)
≤
(
a
(
y
)
,
y
)
ℝ
N
for all
y
∈
ℝ
N
.
The next lemma summarizes the main properties of the map a(⋅) and it is a straightforward consequence of hypotheses (Ha).
Lemma 2.3
If hypotheses (Ha) (i)–(iii) hold, then so do the following:
y
→
a
(
y
)
is continuous and strictly monotone, hence maximal monotone too,
|a(y)|≤c4(1+|y|p-1) for some c4>0 and all y∈ℝN,
(a(y),y)ℝN≥c1p-1|y|p for all y∈ℝN.
This lemma, together with (2.1) and (2.2), leads to the following growth estimates for the primitive G(⋅).
Corollary 2.4
If hypotheses (Ha) (i)–(iii) hold, then
c
1
p
-
1
|
y
|
p
≤
G
(
y
)
≤
c
5
(
1
+
|
y
|
p
)
for all
y
∈
ℝ
N
and some
c
5
>
0
.
Example 2.5
The following maps a(⋅) satisfy hypotheses (Ha):
a(y)=|y|p-2y with 1<p<∞. This map corresponds to the p-Laplace differential operator defined by
Δ
p
u
=
div
(
|
D
u
|
p
-
2
D
u
)
for all
u
∈
W
1
,
p
(
Ω
)
.
a(y)=|y|p-2y+|y|q-2y with 1<p<q<∞. This map corresponds to the (p,q)-differential operator defined by
Δ
p
u
+
Δ
q
u
for all
u
∈
W
1
,
p
(
Ω
)
.
Such differential operators arise in many physical applications, see [6] (quantum physics) and [8] (plasma physics). Recently there have been some existence and multiplicity results for such equations, see [5, 9, 15, 27, 34, 35, 39, 40].
a(y)=(1+|y|2)(p-2)/2y with 1<p<∞. This map corresponds to the generalized p-mean curvature differential operator defined by
div
(
(
1
+
|
D
u
|
2
)
(
p
-
2
)
/
2
D
u
)
for all
u
∈
W
1
,
p
(
Ω
)
.
a(y)=|y|p-2y(1+1/(1+|y|2p)1/2) with 1<p<∞.
Our hypothesis on the boundary weight function β(⋅) is the following:
β∈C1,α(∂Ω) with α∈(0,1),β≥0.
Remark 2.6
If β≡0, then we have the Neumann problem.
In the analysis of problem (1.4), in addition to the Sobolev space W1,p(Ω) we will also use the Banach space C1(Ω¯). This is an ordered Banach space with positive cone given by
C
+
=
{
u
∈
C
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
¯
}
.
On ∂Ω we use the (N-1)-dimensional surface (Hausdorff) measure σ(⋅) and using this measure we can define the Lebesgue spaces Lp(∂Ω) (1≤p≤∞). We know that there exists a unique continuous linear map γ0:W1,p(Ω)→Lp(∂Ω), known as the trace map, such that γ0(u)=u|∂Ω for all u∈W1,p(Ω)∩C(Ω¯). Recall that imγ0=W1/p′,p(∂Ω) (1p+1p′=1) and kerγ0=W01,p(Ω). Moreover, the trace map γ0 is compact in Lq(∂Ω) for all q∈[1,Np-pN-p). In the sequel for the sake of notational simplicity, we will drop the use of the trace map γ0. It is understood that all restrictions of Sobolev functions on the boundary ∂Ω are defined in the sense of traces.
In what follows, by ∥⋅∥ we denote the norm of the Sobolev space W1,p(Ω) defined by
∥
u
∥
=
[
∥
u
∥
p
p
+
∥
D
u
∥
p
p
]
1
/
p
for all
u
∈
W
1
,
p
(
Ω
)
.
For every x∈ℝ, we set x±=max{±x,0}. Then, for u∈W1,p(Ω), we define u±(⋅)=u(⋅)±. We have
u
=
u
+
-
u
-
,
|
u
|
=
u
+
+
u
-
and
u
+
,
u
-
∈
W
1
,
p
(
Ω
)
.
Also, by |⋅|N we denote the Lebesgue measure on ℝN and by Ap:W1,p(Ω)→W1,p(Ω)* the nonlinear map defined by
〈
A
p
(
u
)
,
h
〉
=
∫
Ω
|
D
u
|
p
-
2
(
D
u
,
D
h
)
ℝ
N
d
z
for all
u
,
h
∈
W
1
,
p
(
Ω
)
.
Let A:W1,p(Ω)→W1,p(Ω)* be the nonlinear map defined by
(2.3)
〈
A
(
u
)
,
h
〉
=
∫
Ω
(
a
(
D
u
)
,
D
h
)
ℝ
N
𝑑
z
for all
u
,
h
∈
W
1
,
p
(
Ω
)
.
The next proposition is a special case of a more general result of Gasinski and Papageorgiou [14, 16].
Proposition 2.7
The map A:W1,p(Ω)→W1,p(Ω)*, defined by (2.3), is bounded (that is, it maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type (S)+, that is, if
u
n
→
𝑤
u
in
W
1
,
p
(
Ω
)
𝑎𝑛𝑑
lim sup
n
→
+
∞
〈
A
(
u
n
)
,
u
n
-
u
〉
≤
0
,
then un→u in W1,p(Ω) as n→∞.
We consider the following nonlinear Robin problem:
(2.4)
{
-
div
a
(
D
u
(
z
)
)
=
f
0
(
z
,
u
(
z
)
)
in
Ω
,
∂
u
∂
n
a
+
β
(
z
)
|
u
|
p
-
2
u
=
0
on
∂
Ω
.
In this problem, f0:Ω×ℝ→ℝ is a Carathéodory function with critical growth in the x-variable, that is,
(2.5)
|
f
0
(
z
,
x
)
|
≤
a
0
(
z
)
(
1
+
|
x
|
p
*
-
1
)
for almost all
z
∈
Ω
and all
z
∈
ℝ
,
with a0∈L∞(Ω)+ and
p
*
=
{
N
p
N
-
p
if
p
<
N
,
+
∞
if
N
≤
p
.
By a weak solution of problem (2.4) we understand a function u∈W1,p(Ω) such that
∫
Ω
(
a
(
D
u
)
,
D
h
)
ℝ
N
𝑑
z
+
∫
∂
Ω
β
(
z
)
|
u
|
p
-
2
u
h
𝑑
σ
=
∫
Ω
f
0
(
z
,
u
)
h
𝑑
z
for all
h
∈
W
1
,
p
(
Ω
)
.
Next we establish the boundedness of weak solutions. Due to the critical growth of f(z,⋅), the Moser iteration technique used by Hu and Papageorgiou [18], and Winkert [43] does not work. Instead, we follow the approach of Garcia Azorero and Peral Alonso [11] (see also [41] for semilinear equations). An alternative method can be based on the work of Guedda and Véron [17].
Proposition 2.8
If hypotheses (Ha), (Hβ) hold and u∈W1,p(Ω) is a weak solution of (2.4), then u∈Lq(Ω) for all q∈[1,∞).
Proof.
Recalling that u=u+-u- and performing the argument on u+ and u- separately, we see that without any loss of generality, we may assume that u≥0.
For β>1 and λ>0, we introduce the following Lipschitz continuous functions:
H
(
t
)
=
{
t
β
if
0
≤
t
≤
λ
,
β
λ
β
-
1
(
t
-
λ
)
+
λ
β
if
λ
<
t
,
S
(
t
)
=
{
t
(
β
-
1
)
p
+
1
if
0
≤
t
≤
λ
,
β
(
(
β
-
1
)
p
+
1
)
λ
(
β
-
1
)
p
(
t
-
λ
)
+
λ
(
β
-
1
)
p
+
1
if
λ
<
t
.
It is easy to check that these two functions satisfy the following properties (see, e.g., [13, p. 194] or [11]):
S(t)≤tS′(t) for all t≥0,
c5H′(t)≤S′(t) for all t≥0 with c5>0 independent of λ>0,
tp-1S(t)≤c7H(t)p for all t≥0 with c7>0 independent of λ>0, and H(y),S(y)∈W1,p(Ω) for every y∈W1,p(Ω),
We fix β>1 such that βp<p*, and let ϑ∈Cc∞(ℝN,ℝ) with 0≤ϑ≤1 to be fixed precisely in the process of the proof. We use the test function
h
=
ϑ
p
S
(
u
)
∈
W
1
,
p
(
Ω
)
,
h
≥
0
.
We have
(2.6)
∫
Ω
(
a
(
D
u
)
,
D
h
)
ℝ
N
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
p
-
1
h
𝑑
σ
=
∫
Ω
f
0
(
z
,
u
)
h
𝑑
z
.
Note that
D
h
=
p
ϑ
p
-
1
S
(
u
)
D
ϑ
+
ϑ
p
G
′
(
u
)
D
u
,
and so
(2.7)
∫
Ω
(
a
(
D
u
)
,
D
h
)
ℝ
N
𝑑
z
=
p
∫
Ω
ϑ
p
-
1
S
(
u
)
(
a
(
D
u
)
,
D
ϑ
)
ℝ
N
𝑑
z
+
∫
Ω
ϑ
p
G
′
(
u
)
(
a
(
D
u
)
,
D
u
)
ℝ
N
𝑑
z
.
Using (2.7) in (2.6), we have
(2.8)
∫
Ω
ϑ
p
G
′
(
u
)
(
a
(
D
u
)
,
D
u
)
ℝ
N
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
p
-
1
h
𝑑
σ
≤
∫
Ω
f
0
(
z
,
u
)
h
𝑑
z
-
p
∫
Ω
ϑ
p
-
1
S
(
u
)
(
a
(
D
u
)
,
D
ϑ
)
ℝ
N
𝑑
z
.
From Lemma 2.3 and since ϑpS′(u)≥0 (see P1), we have
(2.9)
c
1
p
-
1
∫
Ω
ϑ
p
S
′
(
u
)
|
D
u
|
p
𝑑
z
≤
∫
Ω
ϑ
p
S
′
(
u
)
(
a
(
D
u
)
,
D
u
)
ℝ
N
𝑑
z
.
Also, using (P1) and Young’s inequality with ϵ>0, we have
|
p
∫
Ω
ϑ
p
-
1
S
(
u
)
(
a
(
D
u
)
,
D
ϑ
)
ℝ
N
𝑑
z
|
≤
p
∫
Ω
ϑ
p
-
1
S
(
u
)
|
a
(
D
u
)
|
|
D
ϑ
|
𝑑
z
=
p
∫
Ω
ϑ
p
-
1
S
(
u
)
1
/
p
S
(
u
)
(
p
-
1
)
/
p
|
a
(
D
u
)
|
|
D
ϑ
|
𝑑
z
≤
p
∫
Ω
ϑ
p
-
1
|
a
(
D
u
)
|
S
(
u
)
1
/
p
(
u
S
′
(
u
)
)
(
p
-
1
)
/
p
|
D
ϑ
|
𝑑
z
≤
ϵ
∫
Ω
ϑ
p
|
a
(
D
u
)
|
p
/
(
p
-
1
)
S
′
(
u
)
𝑑
z
+
c
ϵ
∫
Ω
u
p
-
1
S
(
u
)
|
D
ϑ
|
𝑑
z
(2.10)
≤
ϵ
∫
Ω
c
8
(
1
+
|
D
u
|
p
)
S
′
(
u
)
ϑ
p
𝑑
z
+
c
ϵ
∫
Ω
u
p
-
1
S
(
u
)
|
D
ϑ
|
p
𝑑
z
for some c8>0, with cϵ>0 (see Lemma 2.3).
We return to (2.8) and use (2.9) and (2.10). So, choosing ϵ∈(0,1) small and since ∫∂Ωβ(z)up-1h𝑑σ≥0, we have (see (2.5) and P3)
∫
Ω
ϑ
p
S
′
(
u
)
|
D
u
|
p
𝑑
z
≤
ϵ
c
9
∫
Ω
ϑ
p
S
′
(
u
)
𝑑
z
+
c
9
c
ϵ
∫
Ω
u
p
-
1
S
(
u
)
|
D
ϑ
|
p
𝑑
z
+
c
9
∫
Ω
f
0
(
z
,
u
)
ϑ
p
S
(
u
)
𝑑
z
≤
ϵ
c
9
∫
Ω
ϑ
p
S
′
(
u
)
𝑑
z
+
c
9
c
ϵ
∫
Ω
H
(
u
)
p
|
D
ϑ
|
p
𝑑
z
(2.11)
+
c
9
∥
a
0
∥
∞
∫
Ω
ϑ
p
S
(
u
)
𝑑
z
+
c
9
∥
a
0
∥
∞
∫
Ω
u
p
*
-
1
ϑ
p
S
(
u
)
𝑑
z
for some c9>0. Using (P2), we obtain
(2.12)
c
10
∫
Ω
(
ϑ
H
′
(
u
)
)
p
|
D
u
|
p
𝑑
z
≤
∫
Ω
ϑ
p
S
′
(
u
)
|
D
u
|
p
𝑑
z
for some
c
10
>
0
.
Then, on account of (P3), (2.11) and (2.12), we have the following estimate:
∫
Ω
|
D
(
ϑ
H
(
u
)
)
|
p
d
z
=
∫
Ω
|
ϑ
H
′
(
u
)
D
u
+
H
(
u
)
D
ϑ
|
p
d
z
≤
c
11
[
∫
Ω
|
ϑ
H
′
(
u
)
D
u
|
p
d
z
+
∫
Ω
H
(
u
)
p
|
D
ϑ
|
p
d
z
]
≤
c
12
[
∫
Ω
ϑ
p
S
′
(
u
)
|
D
u
|
p
𝑑
z
+
∫
Ω
H
(
u
)
p
|
D
ϑ
|
p
𝑑
z
]
≤
c
13
[
∫
Ω
H
(
u
)
p
|
D
ϑ
|
p
d
z
+
∫
Ω
u
p
*
-
p
(
ϑ
H
(
u
)
)
p
d
z
+
∫
Ω
ϑ
p
S
(
u
)
d
z
]
(2.13)
≤
c
14
[
∫
Ω
H
(
u
)
p
|
D
ϑ
|
p
𝑑
z
+
∫
Ω
u
p
*
-
p
(
ϑ
H
(
u
)
)
p
𝑑
z
+
1
]
for some c1i>0, i=1,2,3,4, since 0≤ϑ≤1 and |Dϑ(⋅)| is bounded.
We choose ρ>0 such that for any ball Bρ of radius ρ>0 with Bρ∩Ω≠∅, we have
(2.14)
∥
u
∥
L
p
*
(
B
ρ
∩
Ω
)
p
*
-
p
≤
1
η
c
14
with
η
>
0
(recall that W1,p(Ω)↪Lp*(Ω)).
Given z0∈Ω choose ϑ∈Cc∞(ℝN) with 0≤ϑ≤1, suppϑ=B¯ρ(z0) and ϑ≡1 on B¯ρ/2(z0). Using Hölder’s inequality and (2.14), we have
∫
Ω
u
p
*
-
p
ϑ
p
H
(
u
)
p
𝑑
z
=
∫
B
ρ
(
z
0
)
∩
Ω
u
p
*
-
p
ϑ
p
H
(
u
)
p
𝑑
z
≤
(
∫
Ω
ϑ
p
*
H
(
u
)
p
*
𝑑
z
)
p
/
p
*
(
∫
B
ρ
(
z
0
)
∩
Ω
u
p
*
𝑑
z
)
(
p
*
-
p
)
/
p
*
(2.15)
≤
1
η
c
14
(
∫
Ω
ϑ
p
*
H
(
u
)
p
*
𝑑
z
)
p
/
p
*
.
Note that for δ>0, u→δ∥u∥p*+∥Du∥p is an equivalent norm on the Sobolev space W1,p(Ω) (see, for example, [13, p. 227]). So, by choosing δ>0 small, we can find c15>0 such that
(2.16)
(
∫
Ω
ϑ
p
*
H
(
u
)
p
*
d
z
)
p
/
p
*
≤
c
15
∫
Ω
|
D
(
ϑ
H
(
u
)
)
|
p
d
z
.
Using (2.16) in (2.15), we obtain
(2.17)
∫
Ω
u
p
*
-
p
ϑ
p
H
(
u
)
p
d
z
≤
c
15
η
c
14
∫
Ω
|
D
(
ϑ
H
(
u
)
)
|
p
d
z
.
Returning to (2.13) and using (2.17) with η>c15c14, we have
∫
Ω
|
D
(
ϑ
H
(
u
)
)
|
p
d
z
≤
c
16
[
∫
Ω
H
(
u
)
p
d
z
+
1
]
for some c16>0, and hence, by (2.16),
(2.18)
(
∫
Ω
ϑ
p
*
H
(
u
)
p
*
𝑑
z
)
p
/
p
*
≤
c
17
[
∫
Ω
H
(
u
)
p
𝑑
z
+
1
]
for some c17>0. Letting λ→+∞ in (2.18) (see the definition of H(⋅) in the beginning of the proof) yields
(2.19)
(
∫
B
ρ
/
2
(
z
0
)
∩
Ω
u
β
p
*
𝑑
z
)
p
/
p
*
≤
c
17
[
∫
Ω
u
β
p
𝑑
z
+
1
]
.
Since βp<p* and u∈W1,p(Ω), we have that u∈Lβp*(Bρ/2(z0)∩Ω). Then, from (2.19) and since Ω is totally bounded, we infer that u∈Lβp*(Ω). Fix ϵ0>0 such that β-ϵ0>1. Then, by repeating the above argument, we can generate a sequence {βn}n≥1 such that β1p*<βp*,βn≥(β-ϵ0)n and u∈Lβnp*(Ω) for all n∈ℕ. Since (β-ϵ0)n→+∞, we conclude that u∈Lq(Ω) for all q∈[1,∞). ∎
Next we will establish the essential boundedness of u and produce a useful bound for its L∞-norm. We start with a lemma, which is essentially [20, Lemma 5.1, p. 71]. For completeness in our argument we include it here.
Lemma 2.9
If u∈W1,p(Ω),0≤u,q∈(1,p*),k0>1 and c¯>0 are such that
(2.20)
∫
E
k
|
D
u
|
p
d
z
≤
c
¯
k
p
|
E
k
|
N
p
/
q
for all
k
≥
k
0
,
where Ek={z∈Ω:u(z)≥k}, then there exists M1=M1(Ω,c¯,q,k0)>0 such that ∥u∥∞≤M.
Proof.
From [20, p. 45], we know that
(2.21)
(
∫
E
k
(
u
-
k
)
p
d
z
)
1
/
p
≤
c
18
(
∫
E
k
|
D
u
|
p
d
z
)
1
/
p
|
E
k
|
N
1
/
p
-
1
/
p
*
for some c18>0. Using (2.20), (2.21) and Hölder’s inequality, we have
∫
E
k
(
u
-
k
)
d
z
≤
(
∫
E
k
(
u
-
k
)
p
d
z
)
1
/
p
|
E
k
|
N
1
-
1
/
p
≤
c
18
(
∫
E
k
|
D
u
|
p
d
z
)
1
/
p
|
E
k
|
N
1
/
p
-
1
/
p
*
|
E
k
|
1
-
1
/
p
(2.22)
≤
c
19
k
|
E
k
|
1
+
1
/
q
-
1
/
p
*
for all
k
≥
k
0
.
Let ϑ=1q-1p*>0 (recall that ϑ∈(1,p*)). Then from (2.22) we have
(2.23)
∫
E
k
(
u
-
k
)
𝑑
z
≤
c
19
k
|
E
k
|
N
1
+
ϑ
for all
k
≥
k
0
.
We set (see Ziemer [44, p. 19])
ξ
(
k
)
=
∫
E
k
(
u
-
k
)
d
z
=
∫
k
∞
|
E
s
|
N
d
s
,
and have
(2.24)
-
ξ
′
(
k
)
=
|
E
k
|
N
.
From (2.23) we have
ξ
(
k
)
-
1
/
(
1
+
ϑ
)
≥
(
c
19
k
)
-
1
/
(
1
+
ϑ
)
|
E
k
|
N
-
1
,
and using (2.24) this yields
(2.25)
-
ξ
′
(
k
)
ξ
(
k
)
-
1
/
(
1
+
ϑ
)
≥
(
c
19
k
)
-
1
/
(
1
+
ϑ
)
.
Let k*=esssupΩu and integrate (2.25) from k0 to k*. Then
(
k
*
)
ϑ
/
(
1
+
ϑ
)
≤
k
0
ϑ
/
(
1
+
ϑ
)
+
c
19
ξ
(
k
0
)
ϑ
/
(
1
+
ϑ
)
=
M
1
(
1
+
ϑ
)
/
ϑ
.
∎
Now we are ready to establish the essential boundedness of the weak solutions of problem (2.4) and provide a useful description of their bound.
Proposition 2.10
If hypotheses (Ha), (Hβ) hold and u∈W1,p(Ω) is a weak solution of problem (2.4), then there exists M2=M2(p,N,∥u∥p*,Ω)>0 such that ∥u∥∞≤M2.
Proof.
As in the proof of Proposition 2.8, without any loss of generality, we may assume that u≥0.
Let uk=(u-k)+∈W1,p(Ω) and Ek=suppukk∈ℕ. Since u∈W1,p(Ω) is a weak solution of the Robin problem (2.4), we have
(2.26)
∫
Ω
(
a
(
D
u
)
,
D
h
)
ℝ
N
𝑑
z
+
∫
∂
Ω
β
(
z
)
u
p
-
1
h
𝑑
σ
=
∫
Ω
f
0
(
z
,
u
)
h
𝑑
z
for all
h
∈
W
1
,
p
(
Ω
)
.
In (2.26), we choose h=uk∈W1,p(Ω) and, by hypothesis (Hβ), we obtain
∫
E
k
(
a
(
D
u
)
,
D
u
)
ℝ
N
𝑑
z
≤
∫
E
k
f
0
(
z
,
u
)
u
𝑑
z
,
which implies (see Lemma 2.3)
(2.27)
c
1
p
-
1
∫
E
k
|
D
u
|
p
𝑑
z
≤
∫
E
k
f
0
(
z
,
u
)
u
𝑑
z
.
Note that, using (2.5), we have
|
∫
E
k
f
0
(
z
,
u
)
u
d
z
|
≤
∫
E
k
|
f
0
(
z
,
u
)
|
|
u
|
d
z
≤
c
20
∫
E
k
(
1
+
u
p
*
-
1
)
u
𝑑
z
≤
2
c
20
∫
E
k
u
p
*
-
1
u
𝑑
z
(since
k
∈
ℕ
)
(2.28)
=
2
c
20
∫
E
k
u
p
u
p
*
-
p
𝑑
z
for some c20>0,
We choose q∈(p,p*). Using Proposition 2.8, we have up∈Lq/p(Ω) and up*-p∈Lq/(q-p)(Ω). Note that pq+q-pq=1. So, using Hölder’s inequality in (2.28), and in view of Proposition 2.8, (2.19) and [20, p. 45], we have
|
∫
E
k
f
0
(
z
,
u
)
u
d
z
|
≤
2
c
20
(
∫
E
k
u
q
d
z
)
p
/
q
(
∫
E
k
u
(
p
*
-
p
)
q
/
(
p
-
q
)
d
z
)
(
q
-
p
)
/
q
≤
c
21
(
∫
E
k
u
q
𝑑
z
)
p
/
q
=
c
21
(
∫
E
k
(
u
-
k
+
k
)
q
𝑑
z
)
p
/
q
≤
c
22
(
∫
E
k
(
u
-
k
)
q
𝑑
z
)
p
/
q
+
c
22
k
p
|
E
k
|
N
p
/
q
≤
c
23
|
E
k
|
N
1
/
q
-
1
/
p
*
∫
E
k
|
D
u
|
p
d
z
+
c
22
k
p
|
E
k
|
N
p
/
q
for some c21=c21(∥u∥p*)>0, c22>0 and c23>0.
Returning to (2.27) and choosing k∈ℕ big so that |Ek|N is small, we have
(2.29)
∫
E
k
|
D
u
|
p
d
z
≤
c
24
k
p
|
E
k
|
N
p
/
q
for some c24>0 (note that all the above estimation constants depend only on (p,N,∥u∥p*,Ω)). Then, from (2.29) and Lemma 2.9, we see that we can find M2=M2(p,N,∥u∥p*,Ω)>0 such that
u
∈
L
∞
(
Ω
)
with
∥
u
∥
∞
≤
M
2
.
∎
Remark 2.11
As we already said, an alternative approach can be based on the work of Guedda and Véron, see [17]. Indeed, let
K
(
z
)
=
sign
(
u
)
f
0
(
z
,
u
(
z
)
)
1
+
|
u
(
z
)
|
p
-
1
.
Then from (2.5) we have
|
K
(
z
)
|
≤
c
25
(
1
+
|
u
(
z
)
|
p
*
-
1
)
1
+
|
u
(
z
)
|
p
-
1
≤
c
26
(
1
+
|
u
(
z
)
|
p
*
-
p
)
for almost all
z
∈
Ω
,
for some c25,c26>0. Note that p*-p=p2N-p for p<N and recall that u∈Lp*(Ω). Hence, K∈LN/p(Ω). We have
-
div
a
(
D
u
(
z
)
)
=
K
(
z
)
|
u
(
z
)
|
p
-
2
u
(
z
)
+
sign
(
u
)
K
(
z
)
=
K
(
z
)
(
|
u
(
z
)
|
p
-
2
u
(
z
)
+
sign
(
u
)
)
=
f
0
(
z
,
u
(
z
)
)
1
+
|
u
(
z
)
|
p
-
1
(
1
+
|
u
(
z
)
|
p
-
1
)
=
f
0
(
z
,
u
(
z
)
)
.
So, keeping in mind that for every ϵ>0, u↦ϵ∥u∥p*+∥Du∥p is an equivalent norm on W1,p(Ω), we can follow the proof of [17, Proposition 2.1] (with suitable modifications to accommodate the more general differential operator and the boundary term), to prove that u∈Lq(Ω) for all q∈[1,+∞). Then we can continue with Lemma 2.9 and Proposition 2.10 to reach the desired conclusion.
We can use Proposition 2.10 to prove a result comparing Sobolev and Hölder local minimizers of certain C1-functionals. Such a result was first proved by Brezis and Nirenberg [7] for functionals defined on H01(Ω) and it was extended to functionals defined on W01,p(Ω) by Garcia Azorero, Peral Alonso and Manfredi [12] and to functionals defined on W1,p(Ω) by Motreanu and Papageorgiou [30], and Papageorgiou and Rădulescu [33]. All these works involve perturbation terms with subcritical growth. Our result here is more general, since the functional is more general and the perturbation has critical growth.
So, let F0(z,x)=∫0xf0(z,s)𝑑s and consider the C1-functional φ0:W1,p(Ω)→ℝ defined by
φ
0
(
u
)
=
∫
Ω
G
(
D
u
)
𝑑
z
+
1
p
∫
∂
Ω
β
(
z
)
|
u
|
p
𝑑
σ
-
∫
Ω
F
0
(
z
,
u
)
𝑑
z
for all
u
∈
W
1
,
p
(
Ω
)
.
Proposition 2.12
If u0∈W1,p(Ω) is a local C1(Ω¯) minimizer of φ0, that is, we can find ρ0>0 such that
φ
0
(
u
0
)
≤
φ
0
(
u
0
+
h
)
for all
h
∈
C
1
(
Ω
¯
)
with
∥
h
∥
C
1
(
Ω
¯
)
≤
ρ
0
,
then u0∈C1,η(Ω¯) for some η∈(0,1) and u0 is also a local W1,p(Ω)-minimizer of φ0, that is, we can find ρ1>0 such that
φ
0
(
u
0
)
≤
φ
0
(
u
0
+
h
)
for all
h
∈
W
1
,
p
(
Ω
)
with
∥
h
∥
≤
ρ
1
.
Proof.
Since by hypothesis u0 is a local C1(Ω¯)-minimizer of φ0, for every h∈C1(Ω¯) and for t>0 small, we have φ0(u0)≤φ0(u0+th), and hence
(2.30)
0
≤
〈
φ
0
′
(
u
0
)
,
h
〉
for all
h
∈
C
1
(
Ω
¯
)
.
Recalling that C1(Ω¯) is dense in W1,p(Ω), from (2.30) we infer that φ0′(u0)=0, and therefore
(2.31)
〈
A
(
u
0
)
,
h
〉
+
∫
∂
Ω
β
(
z
)
|
u
0
|
p
-
2
u
0
h
𝑑
σ
=
∫
Ω
f
0
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
1
,
p
(
Ω
)
.
From the nonlinear Green’s identity, we have
(2.32)
〈
A
(
u
0
)
,
h
〉
=
〈
-
div
a
(
D
u
0
)
,
h
〉
+
〈
∂
u
0
∂
n
a
,
h
〉
∂
Ω
for all
h
∈
W
1
,
p
(
Ω
)
,
where by 〈⋅,⋅〉∂Ω we denote the duality brackets for the pair (W-1/p′,p′(∂Ω),W1/p′,p(∂Ω)). Note that
div
a
(
D
u
0
)
∈
W
-
1
,
p
′
(
Ω
)
=
W
0
1
,
p
(
Ω
)
*
.
So, if by 〈⋅,⋅〉0 we denote the duality brackets for the pair (W-1,p′(Ω),W01,p(Ω)), from (2.32), we have
〈
-
div
a
(
D
u
0
)
,
h
〉
0
=
〈
A
(
u
0
)
,
h
〉
0
=
〈
A
(
u
0
)
,
h
〉
for all
h
∈
W
0
1
,
p
(
Ω
)
⊆
W
1
,
p
(
Ω
)
.
Hence, by (2.31),
〈
-
div
a
(
D
u
0
)
,
h
〉
0
=
∫
Ω
f
0
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
,
and therefore
(2.33)
-
div
a
(
D
u
0
(
z
)
)
=
f
0
(
z
,
u
0
(
z
)
)
for almost all
z
∈
Ω
.
From (2.31), (2.32) and (2.33), we obtain
(2.34)
〈
∂
u
0
∂
n
a
+
β
(
z
)
|
u
0
|
p
-
2
u
0
,
h
〉
∂
Ω
=
0
for all
h
∈
W
1
,
p
(
Ω
)
.
Recall that, if γ0 is the trace map, then imγ0=W1/p′,p(∂Ω). So, from (2.34) it follows that
∂
u
0
∂
n
a
+
β
(
z
)
|
u
0
|
p
-
2
u
0
=
0
in
W
-
1
/
p
′
,
p
′
(
∂
Ω
)
.
From Proposition 2.10 we have that u0∈L∞(Ω). So, the nonlinear regularity result of Lieberman [24, p. 320] implies that
u
0
∈
C
1
,
η
(
Ω
¯
)
for some
η
∈
(
0
,
1
)
.
Next we show that u0 is also a local W1,p(Ω)-minimizer of φ0. We argue indirectly. So, we assume that u0 is not a local W1,p(Ω)-minimizer of φ0. Given ϵ>0, we consider the set
B
¯
ϵ
*
=
{
h
∈
W
1
,
p
(
Ω
)
:
∥
h
∥
p
*
≤
ϵ
}
,
and define
(2.35)
m
ϵ
*
=
inf
{
φ
0
(
u
0
+
h
)
:
h
∈
B
¯
ϵ
*
}
.
By our contradiction hypothesis, we have
(2.36)
m
ϵ
*
<
φ
0
(
u
0
)
.
Let {hn}n≥1⊆B¯ϵ* be a minimizing sequence for (2.35). Then, since u→∥u∥p*+∥Du∥ is an equivalent norm on the Sobolev space W1,p(Ω), we see that {hn}n≥1⊆W1,p(Ω) is bounded and so we may assume that
h
n
→
𝑤
h
^
ϵ
in
W
1
,
p
(
Ω
)
and in
L
p
*
(
Ω
)
,
(2.37)
h
n
(
z
)
→
h
^
ϵ
(
z
)
for almost all
z
∈
Ω
.
Using the extended Fatou’s lemma, we see that φ0 is sequentially weakly lower semicontinuous. So, we have
φ
0
(
u
0
+
h
^
ϵ
)
≤
lim inf
n
→
∞
φ
0
(
u
0
+
h
n
)
.
Since ∥hϵ∥p*≤ϵ (see (2.37)), it follows that mϵ*=φ(u0+h^ϵ), hence, by (2.36), h^ϵ≠0. By the Lagrange multiplier rule (see, for example, [32, p. 35]), we can find λϵ≤0 such that
〈
φ
0
′
(
u
0
+
h
^
ϵ
)
,
v
〉
=
λ
ϵ
∫
Ω
|
h
^
ϵ
|
p
*
-
2
h
^
ϵ
v
d
z
for all
v
∈
W
1
,
p
(
Ω
)
,
which implies
(2.38)
〈
A
(
u
0
,
h
^
ϵ
)
,
v
〉
+
∫
∂
Ω
β
(
z
)
|
u
0
+
h
^
ϵ
|
p
-
2
(
u
0
+
h
^
ϵ
)
v
d
σ
=
∫
Ω
f
0
(
z
,
u
0
+
h
^
ϵ
)
v
d
z
+
λ
ϵ
∫
Ω
|
h
^
ϵ
|
p
*
-
2
h
^
ϵ
v
d
z
for all v∈W1,p(Ω). From (2.38), as above using the nonlinear Green’s identity, we obtain
(2.39)
{
-
div
a
(
D
(
u
0
+
h
^
ϵ
)
(
z
)
)
=
f
0
(
z
,
(
u
0
+
h
^
ϵ
)
(
z
)
)
+
λ
ϵ
|
h
^
ϵ
(
z
)
|
p
*
-
2
h
^
ϵ
(
z
)
for almost all
z
∈
Ω
,
∂
(
u
0
+
h
^
ϵ
)
∂
n
a
+
β
(
z
)
|
u
0
+
h
^
ϵ
|
p
-
2
(
u
0
+
h
^
ϵ
)
=
0
on
∂
Ω
.
First assume that λϵ∈[-1,0] for all ϵ∈(0,1]. Then, from (2.39) and Proposition 2.10, we can find M3>0 such that
(2.40)
∥
u
0
+
h
^
ϵ
∥
∞
≤
M
3
for all
ϵ
∈
(
0
,
1
]
.
Invoking the regularity result of Lieberman [24], we can find η∈(0,1) and M4>0 such that
(2.41)
u
0
+
h
^
ϵ
∈
C
1
,
η
(
Ω
¯
)
,
∥
u
0
+
h
^
ϵ
∥
C
1
(
Ω
¯
)
≤
M
4
for all
ϵ
∈
(
0
,
1
]
.
Next suppose that there exists ϵn↓0 such that λn=λϵn<-1 for all n∈ℕ. From (2.39) with h^n=h^ϵn, we have
(2.42)
-
1
|
λ
n
|
div
a
(
D
(
u
0
+
h
^
n
)
(
z
)
)
=
1
|
λ
n
|
f
0
(
z
,
(
u
0
+
h
^
n
)
(
z
)
)
+
|
h
^
n
(
z
)
|
p
*
-
2
h
^
n
(
z
)
for almost all z∈Ω. Also, from the first part of the proof, we have
(2.43)
-
1
|
λ
n
|
div
a
(
D
u
0
(
z
)
)
=
1
|
λ
n
|
f
0
(
z
,
u
0
(
z
)
)
for almost all
z
∈
Ω
.
Let μ>1 and consider the function |h^n|μh^nn∈ℕ. We have
D
(
|
h
^
n
|
μ
h
^
n
)
=
|
h
^
n
|
μ
D
h
^
n
+
μ
h
^
n
h
^
n
|
h
^
n
|
|
h
^
n
|
μ
-
1
D
h
^
n
=
(
μ
+
1
)
|
h
^
n
|
μ
D
h
^
n
,
which, by (2.41) and the fact that u0∈C1,η(Ω¯), implies
|
h
^
n
|
μ
h
^
n
∈
W
1
,
p
(
Ω
)
.
Using this as test function, from (2.5), (2.40), (2.42) and (2.43), we have
0
≤
〈
A
(
u
0
+
h
n
)
-
A
(
u
0
)
,
|
h
^
n
|
μ
h
^
n
〉
+
∫
∂
Ω
β
(
z
)
[
|
u
0
+
h
^
n
|
p
-
2
(
u
0
+
h
^
n
)
-
|
u
0
|
p
-
2
u
0
]
𝑑
σ
=
∫
Ω
[
f
0
(
z
,
u
0
+
h
^
n
)
-
f
0
(
z
,
u
0
)
]
|
h
^
n
|
μ
h
^
n
d
z
+
λ
n
∫
Ω
|
h
^
n
|
p
*
+
μ
d
z
≤
M
5
∫
Ω
|
h
^
n
|
μ
+
1
d
z
+
λ
n
∫
Ω
|
h
^
n
|
p
*
+
μ
d
z
≤
M
5
|
Ω
|
N
(
p
*
-
1
)
/
(
p
*
+
μ
)
∥
h
^
n
∥
p
*
+
μ
μ
+
1
+
λ
n
∥
h
^
n
∥
p
*
+
μ
p
*
+
μ
for some M5>0 and all n∈ℕ, where we have used Hölder’s inequality with the exponents p*+μμ+1,p*+μp*-1. Thus,
|
λ
n
|
∥
h
^
n
∥
p
*
+
μ
p
*
-
1
≤
M
5
|
Ω
|
N
(
p
*
-
1
)
/
(
p
*
+
μ
)
,
and hence
∥
h
^
n
∥
p
*
+
μ
p
*
-
1
≤
M
6
for some M6>0 (independent of μ>1) and all n∈ℕ (recall that |λn|>1). Since μ>1 is arbitrary, we let μ→∞ and obtain that
∥
h
^
n
∥
∞
≤
M
7
for some M7>0 and all n∈ℕ. So, the nonlinear regularity theory of Lieberman [24] implies that for some η∈(0,1) and some M8>0, we have (see (2.42) and recall that u0∈C1,η(Ω¯))
h
^
n
∈
C
1
,
η
(
Ω
¯
)
,
∥
h
^
n
∥
C
1
(
Ω
¯
)
≤
M
8
for all
n
∈
ℕ
.
Therefore, in both cases (case 1: λϵ∈[-1,0] for all ϵ∈(0,1] and case 2: λϵn<-1 for some ϵn↓0), we reach the same uniform C1,η(Ω¯) bounds for the sequence {h^n}n≥1⊆W1,p(Ω) such that (see (2.36))
φ
0
(
u
0
+
h
n
)
<
φ
0
(
u
0
)
for all
n
∈
ℕ
.
Recalling that ∥h^n∥p*≤ϵn for all n∈ℕ and exploiting the compact embedding of C1,η(Ω¯) into C1(Ω¯), we have
h
^
n
→
0
in
C
1
(
Ω
¯
)
,
hence
u
0
+
h
^
n
→
u
0
in
C
1
(
Ω
¯
)
,
and therefore
φ
0
(
u
0
)
≤
φ
0
(
u
0
+
h
^
n
)
for all
n
≥
n
0
.
But recall that
φ
0
(
u
0
+
h
^
n
)
<
φ
0
(
u
0
)
for all
n
∈
ℕ
,
a contradiction. This proves that u0∈C1,η(Ω¯) is also a local W1,p(Ω)-minimizer of φ0. ∎
Remark 2.13
A careful reading of the proof of Proposition 2.8 reveals that the result remains valid if instead we use the more general nonlinear boundary condition
∂
u
∂
n
a
=
ξ
(
z
,
u
)
on
∂
Ω
with ξ∈C0,η(∂Ω×ℝ), 0<η<1, such that
|
ξ
(
z
,
x
)
|
≤
c
25
|
x
|
τ
for all
(
z
,
x
)
∈
∂
Ω
×
ℝ
,
with c25>0 and τ∈(1,p]. For simplicity in our presentation, we have used in problem (2.4) the Robin boundary condition from problem (1.1), simplifying this way a little the necessary estimates.
As we already mentioned in the introduction, we will also use tools from Morse theory (critical groups). So, let us recall some basic definitions and facts from that theory.
Given a Banach space X, a function φ∈C1(X,ℝ) and 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
}
.
Let (Y1,Y2) be a topological pair such that Y2⊆Y1⊆X and k∈ℕ0. By Hk(Y1,Y2) we denote the kth-relative singular homology group for the topological pair (Y1,Y2) with integer coefficients. The critical groups of φ at an isolated u∈Kφc are defined by
C
k
(
φ
,
u
)
=
H
k
(
φ
c
∩
U
,
φ
c
∩
U
∖
{
u
}
)
for all
k
∈
ℕ
0
.
Here U is 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 dependent of the choice of the neighborhood U of u.
Suppose that φ satisfies the C-condition and that infφ(Kφ)>-∞. Let c<infφ(Kφ). The critical groups of φ at infinity are defined by
C
k
(
φ
,
∞
)
=
H
k
(
X
,
φ
c
)
for all
k
∈
ℕ
0
.
The second deformation theorem implies that this definition is independent of the choice of the level c<infφ(Kφ).
Suppose that φ∈C1(X,ℝ) satisfies the C-condition and that Kφ is finite. We define
M
(
t
,
u
)
=
∑
k
∈
ℕ
0
rank
C
k
(
φ
,
u
)
t
k
for all
t
∈
ℝ
and all
u
∈
K
φ
,
P
(
t
,
∞
)
=
∑
k
∈
ℕ
0
rank
C
k
(
φ
,
∞
)
t
k
for all
t
∈
ℝ
.
The Morse relation says that
(2.44)
∑
u
∈
K
φ
M
(
t
,
u
)
=
P
(
t
,
∞
)
+
(
1
+
t
)
Q
(
t
)
for all
t
∈
ℝ
,
where Q(t)=∑k∈ℕ0βktk is a formal series in t∈ℝ with nonnegative integer coefficients βk.
Finally, from [33] we recall that the nonlinear eigenvalue problem
{
-
Δ
p
u
(
z
)
=
λ
^
|
u
(
z
)
|
p
-
2
u
(
z
)
in
Ω
,
∂
u
∂
n
p
+
β
(
z
)
|
u
|
p
-
2
u
=
0
on
∂
Ω
has a smallest eigenvalue λ^1(p,β)≥0. If β≢0, then λ^1(p,β)>0, while if β≡0 then λ^1(p,0)=λ^1(p)=0 (Neumann problem). The eigenfunctions corresponding to this eigenvalue have constant sign and
λ
^
1
(
p
,
β
)
=
inf
{
∥
D
u
∥
p
p
+
∫
∂
Ω
β
(
z
)
|
u
|
p
𝑑
σ
∥
u
∥
p
p
:
u
∈
W
1
,
p
(
Ω
)
,
u
≠
0
}
.
By u^1(p,β) we denote the Lp-normalized (that is, ∥u^1(p,β)∥p=1) positive eigenfunction corresponding to λ^1(p,β). We have
λ
^
1
(
p
,
β
)
=
∥
D
u
^
1
(
p
,
β
)
∥
p
p
+
∫
∂
Ω
β
(
z
)
|
u
^
1
(
p
,
β
)
|
p
𝑑
σ
,
and from the nonlinear regularity theory and the nonlinear maximum principle, we have u^1(p,β)∈intC+.
3 Three Solutions Theorem
The hypotheses on the reaction f(z,x) are as follows:
f:Ω×ℝ→ℝ is a Carathéodory function with the following properties:
We have
lim
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
*
-
2
x
=
0
uniformly 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 all
|
x
|
≤
ρ
.
If F(z,x)=∫0xf(z,s)𝑑s, then
lim
x
→
±
∞
F
(
z
,
x
)
|
x
|
p
=
+
∞
uniformly for almost all
z
∈
Ω
.
If ξ(z,x)=f(z,x)x-pF(z,x), then there exists η∈L1(Ω)+ such that
ξ
(
z
,
x
)
≤
ξ
(
z
,
y
)
+
η
(
z
)
for almost all
z
∈
Ω
and all
0
≤
x
≤
y
or
y
≤
x
≤
0
.
There exist δ>0 and γδ>0 such that
-
γ
δ
|
x
|
p
≤
f
(
z
,
x
)
x
for almost all
z
∈
Ω
and all
|
x
|
≤
δ
.
If β≢0, then there exists ℑ∈L∞(Ω)+ such that ℑ(z)≤λ^1(p,β^) for almost all z∈Ω, ℑ≢λ^1(p,β^), with β^=p-1c1β and
lim sup
x
→
0
f
(
z
,
x
)
|
x
|
p
-
2
x
≤
ℑ
(
x
)
uniformly for almost all
z
∈
Ω
.
If β≡0, then f(z,x)x≤0 for almost all z∈Ω and all |x|≤δ.
Remark 3.1
Hypothesis (Hf) (i) is more general than the usual polynomial subcritical growth condition which says that
(3.1)
|
f
(
z
,
x
)
|
≤
c
26
(
1
+
|
x
|
r
-
1
)
for almost all
z
∈
Ω
and all
x
∈
ℝ
,
with c26>0 and 1<r<p*. For example the function (for the sake of simplicity we drop the z-dependence)
f
(
x
)
=
|
x
|
p
*
-
2
x
ln
(
(
1
+
|
x
|
p
)
)
-
p
p
*
|
x
|
p
*
|
x
|
p
-
2
x
ln
(
1
+
|
x
|
p
)
2
(
1
+
|
x
|
p
)
,
with primitive
F
(
x
)
=
1
p
*
|
x
|
p
*
ln
(
1
+
|
x
|
p
)
,
satisfies hypothesis (Hf) (i) but fails to satisfy the subcritical polynomial growth (3.1). The lack of compactness in the embedding of W1,p(Ω) into Lp*(Ω) is a source of difficulties which we have to overcome. We do this without any appeal to the concentration-compactness principle (see Ambrosetti and Malchiodi [3, p. 252]). It is not clear how hypothesis (Hf) (i) can lead to concentration phenomena and for this reason our approach avoids the use of the concentration-compactness method of Lions. Instead we show that despite the almost critical growth of the reaction term f(z,⋅) (see hypothesis (Hf) (i)), the compactness condition is still valid for the energy functional of the problem and so we can proceed with the usual variational methods of critical point theory. Hypothesis (Hf) (iv) implies that
f
(
z
,
0
)
=
0
for almost all
z
∈
Ω
.
Then hypothesis (Hf) (iii) implies
ξ
(
z
,
0
)
=
0
≤
ξ
(
z
,
x
)
+
η
(
z
)
for almost all
z
∈
Ω
and all
x
∈
ℝ
,
hence
p
F
(
z
,
x
)
≤
f
(
z
,
x
)
x
+
η
(
z
)
for almost all
z
∈
Ω
,
and therefore, from hypothesis (Hf) (ii), we obtain
lim
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
-
2
x
=
+
∞
uniformly for almost all
z
∈
Ω
.
Hypotheses (Hf) (ii)–(iii) replace the AR-condition and allow in our framework superlinear reactions with “slower” growth near ±∞ which fail to satisfy the AR-condition (see the examples below). Hypothesis (Hf) (iii) is a quasimonotonicity condition on ξ(z,⋅) and it is satisfied if, for example, we can find M9>0 such that for almost all z∈Ω,
x
→
f
(
z
,
x
)
x
p
-
1
is nondecreasing on
[
M
9
,
+
∞
)
and
x
→
f
(
z
,
x
)
|
x
|
p
-
2
x
is nonincreasing on
(
-
∞
,
-
M
9
]
.
More restrictive versions of hypothesis (Hf) (iii) were used by Li and Yang [22], Liu [26], Miyagaki and Souto [28], and Sun [38]. We should mention that all these conditions originate from the important work of Jeanjean [19] (see also Struwe [37]), who was the first to employ an alternative to the AR-condition. So, Jeanjean [19] assumed (for p=2) that there exists ϑ≥1 such that
ξ
(
z
,
s
x
)
≤
ϑ
ξ
(
z
,
x
)
for almost all
z
∈
Ω
, all
x
∈
ℝ
and
s
∈
[
0
,
1
]
.
The disadvantage of this condition is that it is global. In contrast, the previous remarks show that condition (Hf) (iii) avoids this global character and so it is a quite generic condition. For a further discussion and comparison of these extensions of the AR-condition, we refer to the paper by Li and Yang [22].
Example 3.2
The following primitive functions satisfy hypotheses (Hf) (for the sake of simplicity we drop the z-dependence):
F
1
(
x
)
=
1
q
|
x
|
q
-
1
p
|
x
|
p
,
F
2
(
x
)
=
1
p
*
|
x
|
p
*
ln
(
1
+
|
x
|
p
*
)
+
{
-
1
p
|
x
|
p
if
|
x
|
≤
1
,
1
p
|
x
|
p
ln
|
x
|
-
1
p
if
1
<
|
x
|
,
with 1<p<q<p*. Note that f2(x)=ddxF2(x) fails to satisfy (3.1) and the AR-condition.
We introduce the following truncations-perturbations of the reaction term f(z,⋅):
(3.2)
f
^
+
(
z
,
x
)
=
{
0
if
x
≤
0
,
f
(
z
,
x
)
+
x
p
-
1
if
x
>
0
,
(3.3)
f
^
-
(
z
,
x
)
=
{
f
(
z
,
x
)
+
|
x
|
p
-
2
x
if
x
<
0
,
0
if
x
≥
0
.
Both are Carathéodory functions. We set
F
^
±
(
z
,
x
)
=
∫
0
x
f
^
±
(
z
,
s
)
𝑑
s
,
and consider the C1-functionals φ^±:W1,p(Ω)→ℝ defined by
φ
^
±
(
u
)
=
∫
Ω
G
(
D
u
)
𝑑
z
+
1
p
∥
u
∥
p
p
±
1
p
∫
∂
Ω
β
(
z
)
(
u
±
)
p
𝑑
σ
-
∫
Ω
F
^
±
(
z
,
u
)
𝑑
z
for all
u
∈
W
1
,
p
(
Ω
)
.
Also, let φ:W1,p(Ω)→ℝ be the energy functional for problem (1.4) defined by
φ
(
u
)
=
∫
Ω
G
(
D
u
)
𝑑
z
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
W
1
,
p
(
Ω
)
.
Evidently, φ∈C1(W1,p(Ω)).
Proposition 3.3
If hypotheses (Ha), (Hβ) and (Hf) hold, then the functionals φ^± satisfy the C-condition.
Proof.
We give the proof (similarly, in two other occurrences) for the functional φ^+; the proof for φ^- is similar.
Consider a sequence {un}n≥1⊆W1,p(Ω) such that
(3.4)
|
φ
^
+
(
u
n
)
|
≤
M
10
for some
M
10
>
0
and all
n
∈
ℕ
,
(3.5)
(
1
+
∥
u
n
∥
)
φ
^
+
′
(
u
n
)
→
0
in
W
1
,
p
(
Ω
)
*
as
n
→
∞
.
From (3.5) we have
(3.6)
|
〈
A
(
u
n
)
,
h
〉
+
∫
Ω
|
u
n
|
p
-
2
u
n
h
d
z
+
∫
∂
Ω
β
(
z
)
(
u
n
+
)
p
-
1
h
d
σ
-
∫
Ω
f
^
+
(
z
,
u
n
)
h
d
z
|
≤
ϵ
n
∥
h
∥
1
+
∥
u
n
∥
for all h∈W1,p(Ω) with ϵn→0+. In (3.6) we choose h=-un-∈W1,p(Ω). Then, by Lemma 2.3 and (3.2),
c
1
p
-
1
∥
D
u
n
-
∥
p
p
+
∥
u
n
-
∥
p
p
≤
ϵ
n
for all
n
∈
ℕ
,
hence
(3.7)
u
n
-
→
0
in
W
1
,
p
(
Ω
)
.
We use (3.7) in (3.4). Then, because of Corollary 2.4 and (3.2), we have
(3.8)
|
∫
Ω
p
G
(
D
u
n
+
)
𝑑
z
+
∫
∂
Ω
β
(
z
)
(
u
n
+
)
p
𝑑
σ
-
∫
Ω
p
F
(
z
,
u
n
+
)
𝑑
z
|
≤
M
11
for all
n
∈
ℕ
,
for some M11>0. In (3.6) we choose h=un+∈W1,p(Ω). Then
(3.9)
-
∫
Ω
(
a
(
D
u
n
+
)
,
D
u
n
+
)
ℝ
N
𝑑
z
-
∫
∂
Ω
β
(
z
)
(
u
n
+
)
p
𝑑
σ
+
∫
Ω
f
(
z
,
u
n
+
)
u
n
+
𝑑
z
≤
ϵ
n
for all
n
∈
ℕ
.
We add (3.8) and (3.9) and use hypothesis (Ha) (iv) to obtain
(3.10)
∫
Ω
ξ
(
z
,
u
n
+
)
𝑑
z
≤
M
12
for all
n
∈
ℕ
,
for some M12>0.
Claim 1
{un+}n≥1⊆W1,p(Ω) is bounded.
We argue indirectly. So, suppose that Claim 1 is not true. By passing to a subsequence if necessary, we may assume that ∥un+∥→∞. Let yn=un+∥un+∥, n∈ℕ. Then ∥yn∥=1,yn≥0 for all n∈ℕ, and so we may assume that
(3.11)
y
n
→
𝑤
y
in
W
1
,
p
(
Ω
)
and
y
n
→
y
in
L
p
(
Ω
)
and in
L
p
(
∂
Ω
)
,
y
≥
0
.
Suppose that y≠0 and let Ω+(y)={y>0}. Then |Ω+(y)|N>0 and we have
u
n
+
(
z
)
→
+
∞
for almost all
z
∈
Ω
+
(
y
)
.
Hypothesis (Hf) (ii) implies that
F
(
z
,
u
n
+
(
z
)
)
∥
u
n
+
∥
p
=
F
(
z
,
u
n
+
(
z
)
)
u
n
+
(
z
)
p
y
n
(
z
)
p
→
+
∞
for almost all
z
∈
Ω
+
(
y
)
.
From this fact and Fatou’s lemma (see also hypothesis (Hf) (ii) and (3.11)), we have
(3.12)
∫
Ω
F
(
z
,
u
n
+
)
∥
u
n
+
∥
p
𝑑
z
→
+
∞
.
From (3.8) and in view of Corollary 2.4, hypothesis (Hβ) and (3.11) (recall also that p>1), we have
∫
Ω
F
(
z
,
u
n
+
)
∥
u
n
+
∥
p
≤
M
11
+
1
∥
u
n
+
∥
p
∫
Ω
G
(
D
u
n
+
)
𝑑
z
+
∫
∂
Ω
β
(
z
)
y
n
p
𝑑
σ
≤
c
27
(
1
+
∥
y
n
∥
p
)
(3.13)
≤
c
28
for all
n
∈
ℕ
,
for some c27,c28>0. Comparing (3.12) and (3.13), we reach a contradiction.
So, we assume that y≡0. Let k>0 and set vn=(kp)1/pyn for all n∈ℕ. From (3.11) we have
(3.14)
v
n
→
𝑤
0
in
W
1
,
p
(
Ω
)
and
v
n
→
0
in
L
p
(
Ω
)
and in
L
p
(
∂
Ω
)
.
Let c29=supn≥1∥vn∥p*p*<+∞ (see (3.14)). Hypothesis (Hf) (i) implies that given ϵ>0, we can find cϵ>0 such that
(3.15)
|
F
(
z
,
x
)
|
≤
ϵ
2
c
29
|
x
|
p
*
+
c
ϵ
for almost all
z
∈
Ω
and all
x
∈
ℝ
.
From (3.15), for every measurable set E⊆Ω with |EN|≤ϵ2cϵ, we have
|
∫
E
F
(
z
,
v
n
)
d
z
|
≤
∫
E
|
F
(
z
,
v
n
)
|
d
z
≤
ϵ
2
c
29
∥
v
n
∥
p
*
p
*
+
c
ϵ
|
E
|
N
≤
ϵ
2
+
ϵ
2
=
ϵ
for all
n
∈
ℕ
,
hence {F(⋅,vn(⋅))}n≥1⊆L1(Ω) is uniformly integrable. Since F(z,vn(z))→0 for almost all z∈Ω, from the extended dominated convergence theorem (Vitali’s theorem), we have
(3.16)
∫
Ω
F
(
z
,
v
n
)
𝑑
z
→
0
as
n
→
∞
.
Recall that we have assumed that ∥un+∥→∞. So, we can find n0∈ℕ such that
(3.17)
0
<
(
k
p
)
1
/
p
1
∥
u
n
+
∥
≤
1
for all
n
≥
n
0
.
Consider the C1-functional ψ^+:W1,p(Ω)→ℝ defined by
ψ
^
+
(
u
)
=
c
1
p
(
p
-
1
)
∥
D
u
∥
p
p
+
1
p
∥
u
∥
p
p
+
1
p
∫
∂
Ω
β
(
z
)
(
u
+
)
p
𝑑
σ
-
∫
Ω
F
^
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
1
,
p
(
Ω
)
.
Let tn∈[0,1] be such that
ψ
^
+
(
t
n
u
n
+
)
=
max
0
≤
t
≤
1
ψ
^
+
(
t
u
n
+
)
for all
n
∈
ℕ
.
From (3.16) we see that we can find n1∈ℕ, n1≥n0 such that
(3.18)
∫
Ω
F
(
z
,
v
n
)
𝑑
z
≤
c
1
2
(
p
-
1
)
k
for all
n
≥
n
1
.
Using (3.17), (3.18) and hypothesis (Hβ), we have
(3.19)
ψ
^
+
(
t
n
u
n
+
)
≥
ψ
^
+
(
v
n
)
≥
c
1
k
p
-
1
-
c
1
k
2
(
p
-
1
)
-
∥
v
n
∥
p
p
=
c
1
k
3
(
p
-
1
)
for all
n
≥
n
1
.
Recall that k>0 is arbitrary. So, from (3.19) it follows that
(3.20)
ψ
^
+
(
t
n
u
n
+
)
→
+
∞
as
n
→
∞
.
From (3.4) and (3.7) and since ψ^+≤φ^+ (see Corollary 2.4), we see that
(3.21)
{
ψ
^
+
(
u
n
+
)
}
n
≥
1
⊆
ℝ
is bounded
.
Also, we have
(3.22)
ψ
^
+
(
0
)
=
0
.
From (3.20)–(3.22), it follows that we can find n2∈ℕ such that
(3.23)
t
n
∈
(
0
,
1
)
for all
n
≥
n
2
.
Then, for n≥n2, we have
d
d
t
ψ
^
+
(
t
u
n
+
)
|
t
=
t
n
=
0
,
which, by the chain rule, yields
c
1
p
-
1
〈
A
p
(
t
n
u
n
+
)
,
u
n
+
〉
+
∫
∂
Ω
β
(
z
)
(
t
n
u
n
+
)
p
-
1
u
n
+
𝑑
σ
=
∫
Ω
f
(
z
,
t
n
(
u
n
+
)
u
n
+
)
𝑑
z
,
and therefore
(3.24)
c
1
p
-
1
∥
D
(
t
n
u
n
+
)
∥
p
p
+
∫
∂
Ω
β
(
z
)
(
t
n
u
n
+
)
p
𝑑
σ
=
∫
Ω
f
(
z
,
t
n
u
n
+
)
(
t
n
u
n
+
)
𝑑
z
for all
n
≥
n
2
.
From hypothesis (Hf) (iii) and (3.23), we have
(3.25)
∫
Ω
f
(
z
,
t
n
u
n
+
)
(
t
n
u
n
+
)
𝑑
z
≤
∫
Ω
ξ
(
z
,
u
n
+
)
𝑑
z
+
∫
Ω
p
F
(
z
,
t
n
u
n
+
)
𝑑
z
+
∥
η
∥
1
for all
n
≥
n
2
.
Using (3.25) in (3.24), from (3.10) we obtain
(3.26)
ψ
^
+
(
t
n
u
n
+
)
≤
M
12
+
∥
η
∥
1
=
M
13
for all
n
≥
n
2
.
Comparing (3.20) and (3.26), we reach a contradiction. This proves Claim 1. ∎From (3.7) and Claim 1, it follows that {un}n≥1⊆W1,p(Ω) is bounded. So, we way assume that
(3.27)
u
n
→
𝑤
u
in
W
1
,
p
(
Ω
)
and
u
n
→
u
in
L
p
(
Ω
)
and in
L
p
(
∂
Ω
)
.
Let c30=supn≥1∥un∥p*p*<+∞ (see (3.27)). Hypothesis (Hf) (i) implies that given ϵ>0, we can find c^ϵ>0 such that
|
f
(
z
,
x
)
|
≤
ϵ
2
c
30
|
x
|
p
*
-
1
+
c
^
ϵ
for almost all
z
∈
Ω
and all
x
∈
ℝ
.
For E⊆Ω measurable, we have
|
∫
E
f
(
z
,
u
n
)
(
u
n
-
u
)
d
z
|
≤
∫
E
|
f
(
z
,
u
n
)
∥
u
n
-
u
|
d
z
(3.28)
≤
ϵ
2
c
30
∫
E
|
u
n
|
p
*
-
1
|
u
n
-
u
|
d
z
+
c
^
ϵ
∫
E
|
u
n
-
u
|
d
z
.
Using Hölder’s inequality, we have (recall that 1p*+1(p*)′=1)
(3.29)
c
^
ϵ
∫
E
|
u
n
-
u
|
𝑑
z
≤
c
^
ϵ
|
E
|
N
1
/
(
p
*
)
′
∥
u
n
-
u
∥
p
*
≤
2
c
^
ϵ
|
E
|
N
1
/
(
p
*
)
′
c
30
1
/
p
*
.
Thus,
(3.30)
ϵ
2
c
30
∫
E
|
u
n
|
p
*
-
1
|
u
n
-
u
|
d
z
≤
ϵ
2
c
30
∥
u
n
∥
p
*
p
*
-
1
∥
u
n
-
u
∥
p
*
≤
ϵ
2
for all
n
∈
ℕ
.
Choose E⊆Ω measurable with
|
E
|
N
≤
ϵ
2
(
2
c
^
ϵ
)
(
p
*
)
′
c
30
1
/
p
*
-
1
.
Then from (3.29) we have
(3.31)
c
^
ϵ
∫
E
|
u
n
-
u
|
d
z
≤
ϵ
2
for all
n
∈
ℕ
.
From (3.28), (3.30) and (3.31), it follows that
sup
n
≥
1
∫
E
|
f
(
z
,
u
n
)
|
|
u
n
-
u
|
d
z
≤
ϵ
,
hence {f(⋅,un(⋅))(un-u)(⋅)}n≥1⊆L1(Ω) is uniformly integrable. From (3.27) we have (at least for a subsequence) that
f
(
z
,
u
n
(
z
)
)
(
u
n
-
u
)
(
z
)
→
0
for almost all
z
∈
Ω
.
So, employing the extended dominated convergence theorem (Vitali’s theorem), we have
(3.32)
∫
Ω
f
(
z
,
u
n
)
(
u
n
-
u
)
𝑑
z
→
0
as
n
→
∞
.
In (3.6), we choose h=un-u∈W1,p(Ω), pass to the limit as n→∞, and use (3.27), (3.32) and hypothesis (Hβ). Then
lim
n
→
∞
〈
A
(
u
n
)
,
u
n
-
u
〉
=
0
,
and by Proposition 2.7,
u
n
→
u
in
W
1
,
p
(
Ω
)
,
which implies that φ^+ satisfies the C-condition. Similarly for φ^- using (3.3). ∎
A careful reading of the above proof shows with minor and straightforward changes, we can have the same result for the energy functional φ. Therefore, we can state the following proposition.
Proposition 3.4
If hypotheses (Ha), (Hβ) and (Hf) hold, then the energy functional φ satisfies the C-condition.
Hypothesis (Hf) (ii) leads easily to the following result.
Proposition 3.5
If hypotheses (Ha), (Hβ) and (Hf) hold and u∈intC+, then φ^±(tu)→-∞ as t→+∞.
The next result establishes the mountain pass geometry (see Theorem 2.1) for the functionals φ^±. Also, this result will be useful in generating a third nontrivial solution for problem (1.4), since it identifies the nature of u=0∈Kφ.
Proposition 3.6
If hypotheses (Ha), (Hβ) and (Hf) hold, then u=0 is a local minimizer of the functional φ^± and φ.
Proof.
We do the proof for the functional φ^+; the proofs for φ^- and φ are similar.
First suppose β≢0. Hypothesis (Hf) (iv) implies that given ϵ>0, we can find δ1=δ1(ϵ)>0 such that
(3.33)
F
(
z
,
x
)
≤
1
p
(
ℑ
(
z
)
+
ϵ
)
|
x
|
p
for almost all
z
∈
Ω
and all
|
x
|
≤
δ
1
.
Let u∈C1(Ω¯) with ∥u∥C1(Ω¯)≤δ1. Then, in view of (3.2), (3.33), [33] and (Hf) (iv), we have
φ
^
+
(
u
)
=
∫
Ω
G
(
D
u
)
𝑑
z
+
1
p
∥
u
-
∥
p
p
+
1
p
∫
∂
Ω
β
(
z
)
(
u
+
)
p
𝑑
σ
-
∫
Ω
F
(
z
,
u
+
)
𝑑
z
≥
c
1
p
(
p
-
1
)
[
∥
D
u
+
∥
p
p
+
∫
∂
Ω
β
^
(
z
)
(
u
+
)
p
𝑑
σ
-
∫
Ω
ℑ
(
z
)
(
u
+
)
p
𝑑
z
-
ϵ
∥
u
+
∥
p
]
+
1
p
[
c
1
p
-
1
∥
D
u
-
∥
p
p
+
∥
u
-
∥
p
p
]
≥
(
c
31
-
ϵ
)
∥
u
+
∥
p
+
c
32
∥
u
-
∥
p
for some c31,c32>0, with β^=p-1c1β. Choosing ϵ∈(0,c31), from (3.33) we infer that
φ
^
+
(
u
)
≥
c
33
∥
u
∥
p
for all
u
∈
C
1
(
Ω
¯
)
with
∥
u
∥
C
1
(
Ω
¯
)
≤
δ
1
,
hence u=0 is a local C1(Ω¯)-minimizer of φ^+, and therefore, by Proposition 2.12, u=0 is a local W1,p(Ω)-minimizer of φ^+.
Next suppose that β≡0. Let δ>0 be as postulated by hypothesis (Hf) (iv) and let u∈C1(Ω¯) with ∥u∥C1(Ω¯)≤δ. Then hypothesis (Hf) (iv) implies
-
∫
Ω
F
(
z
,
u
)
𝑑
z
≥
0
.
So, we have
φ
^
+
(
u
)
≥
0
=
φ
^
+
(
0
)
for all
u
∈
C
1
(
Ω
¯
)
with
∥
u
∥
C
1
(
Ω
¯
)
≤
δ
,
and again by Proposition 2.12, u=0 is a local W1,p(Ω)-minimizer of φ^+.
Similarly for the functionals φ^- and φ. ∎
Proposition 3.7
If hypotheses (Ha), (Hβ) and (Hf) hold, then Kφ^+⊆C+ and Kφ^-⊆C+.
Proof.
Let u∈Kφ^+. Then φ^+′(u)=0 and (3.2) imply
(3.34)
〈
A
(
u
)
,
h
〉
+
∫
Ω
|
u
|
p
-
2
u
h
d
z
+
∫
∂
Ω
β
(
z
)
(
u
+
)
p
-
1
h
d
σ
=
∫
Ω
[
f
(
z
,
u
+
)
+
(
u
+
)
p
-
1
]
h
d
z
for all
h
∈
W
1
,
p
(
Ω
)
.
In (3.34) we choose h=-u-∈W1,p(Ω). Then, by Lemma 2.3,
c
1
p
-
1
∥
D
u
-
∥
p
p
+
∥
u
-
∥
p
p
≤
0
,
hence u≥0. From Proposition 2.10 we have that u∈L∞(Ω). So, we can use the regularity theory of Lieberman [24, p. 320] and have that u∈C+. Therefore,
K
φ
^
+
⊆
C
+
.
Similarly, for the functional φ^-, using this time (3.3), we show that Kφ^-⊆-C+. ∎
Now we are ready to produce two constant sign solutions for problem (1.4).
Proposition 3.8
If hypotheses (Ha), (Hβ) and (Hf) hold, then problem (1.4) has at least two constant sign solutions u0∈intC+ and v0∈-intC+.
Proof.
Proposition 3.7 together with (3.2) and (3.3) indicate that we may assume that Kφ^+ and Kφ^- are infinite or, otherwise, we already have a whole sequence of distinct solutions of constant sign.
From Proposition 3.6 we know that u=0 is a local minimizer of φ^+. So, we can find ρ∈(0,1) small such that (see the proof of [1, Proposition 29])
(3.35)
φ
^
+
(
0
)
=
0
<
inf
{
φ
^
+
(
u
)
:
∥
u
∥
=
ρ
}
=
m
^
ρ
+
.
Combining (3.35) with Propositions 3.3 and 3.5, we see that we can apply Theorem 2.1 (the mountain pass theorem). So, by Proposition 3.7, we can find u0∈W1,p(Ω) such that
(3.36)
u
0
∈
K
φ
^
+
⊆
C
+
and
m
^
ρ
+
≤
φ
^
+
(
u
0
)
.
From (3.35) and (3.36), we have that u0≠0. Also, since u0≥0 (see (3.36)), by (3.2), we have
〈
A
(
u
0
)
,
h
〉
+
∫
∂
Ω
β
(
z
)
u
0
p
-
1
h
𝑑
σ
=
∫
Ω
f
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
1
,
p
(
Ω
)
.
Thus,
(3.37)
{
-
div
a
(
D
u
0
(
z
)
)
=
f
(
z
,
u
0
(
z
)
)
for almost all
z
∈
Ω
,
∂
u
0
∂
n
a
+
β
(
z
)
u
0
p
-
1
=
0
on
∂
Ω
.
Hypothesis (Hf) (iv) implies that given ρ>0, we can find ξ^ρ>0 such that
(3.38)
f
(
z
,
x
)
x
+
ξ
^
ρ
|
x
|
p
≥
0
for almost all
z
∈
Ω
and all
|
x
|
≤
ρ
.
Let ρ=∥u0∥∞ (recall that u0∈C+∖{0}) and let ξ^ρ>0 as in (3.38). Then from (3.37) we have
(3.39)
div
a
(
D
u
0
(
z
)
)
≤
ξ
^
ρ
u
0
(
z
)
p
-
1
for almost all
z
∈
Ω
.
Let γ(t)=a0(t)t for t>0. Then (1.2) and hypothesis (Ha) (ii) ensure that
γ
′
(
t
)
t
=
a
0
′
(
t
)
t
2
+
a
0
(
t
)
t
≥
c
1
t
p
-
1
.
By integration, we obtain
(3.40)
∫
0
t
γ
′
(
s
)
s
𝑑
s
=
γ
(
t
)
t
-
∫
0
t
γ
(
s
)
𝑑
s
=
a
0
(
t
)
t
2
-
G
0
(
t
)
≥
c
1
p
t
p
for all
t
>
0
.
Let
d
^
(
t
)
=
a
0
(
t
)
t
2
-
G
0
(
t
)
and
d
^
0
(
t
)
=
c
1
p
t
p
for all
t
>
0
.
Let s>0 and consider the following two sets:
C
1
=
{
t
∈
(
0
,
1
)
:
d
^
(
t
)
≥
s
}
,
C
2
=
{
t
∈
(
0
,
1
)
:
d
^
0
(
t
)
≥
s
}
.
From (3.40) we see that C2⊆C1 and so infC1≤infC2. Therefore, d^-1(s)≤d^0-1(s) (see, e.g., [21, p. 6]). Then for δ>0 we have
∫
0
δ
1
d
^
-
1
(
ξ
^
ρ
p
s
p
)
𝑑
s
≥
∫
0
δ
1
d
^
0
-
1
(
ξ
^
ρ
p
s
p
)
𝑑
s
=
ξ
^
ρ
p
∫
0
δ
d
s
s
=
+
∞
.
Hence, because of (3.39), we can apply the nonlinear strong maximum principle of Pucci and Serrin [36, p. 111] and have that
u
0
(
z
)
>
0
for all
z
∈
Ω
.
Then the boundary point theorem of Pucci and Serrin [36, p. 120] implies that u0∈intC+.
Similarly, working with the functional φ^-, we produce a second constant sign solution v0∈-intC+. ∎
To produce a third nontrivial solution, we will use Morse theoretical tools (critical groups). To this end we compute the critical groups of φ at infinity.
Proposition 3.9
If hypotheses (Ha), (Hβ) and (Hf) hold and infφ(Kφ)>-∞, then Ck(φ,∞)=0 for all k∈N0.
Proof.
From hypotheses (Hf) (i)–(ii) we see that given γ>0, we can find c34=c34(γ)>0 such that
(3.41)
F
(
z
,
x
)
≥
γ
|
x
|
p
-
c
34
for almost all
z
∈
Ω
and all
x
∈
ℝ
.
Let u∈∂B1={u∈W1,p(Ω):∥u∥=1} and t>0. On account of Corollary 2.4, (3.41) and hypothesis (Hβ), we have
(3.42)
φ
(
t
u
)
≤
t
p
[
c
35
∥
D
u
∥
p
p
+
c
36
∥
u
∥
L
p
(
∂
Ω
)
p
-
γ
∥
u
∥
p
p
]
+
c
37
for some c35,c36,c37>0. Because γ>0 is arbitrary, from (3.42) we see that
(3.43)
φ
(
t
u
)
→
-
∞
as
t
→
-
∞
.
Also, using the chain rule, and hypotheses (Ha) (iv) and (Hf) (iii), we have
d
d
t
φ
(
t
u
)
=
〈
φ
′
(
t
u
)
,
u
〉
=
1
t
〈
φ
′
(
t
u
)
,
t
u
〉
=
1
t
[
∫
Ω
(
a
(
t
D
u
)
,
t
D
u
)
ℝ
N
𝑑
z
+
∫
∂
Ω
β
(
z
)
|
t
u
|
p
𝑑
σ
-
∫
Ω
f
(
z
,
t
u
)
t
u
𝑑
z
]
≤
1
t
[
∫
Ω
p
G
(
t
D
u
)
𝑑
z
+
∫
∂
Ω
β
(
z
)
|
t
u
|
p
𝑑
σ
-
∫
Ω
p
F
(
z
,
t
u
)
𝑑
z
+
c
38
]
=
1
t
[
p
φ
(
t
u
)
+
c
38
]
for some c38>0. Then (3.43) implies that for large t>0 we have φ(tu)≤ℑ0<-c38, and thus
d
d
t
φ
(
t
u
)
<
0
for large
t
>
0
.
Therefore, we can find a unique r(u)>0 such that φ(r(u)u)=ℑ0. The implicit function theorem implies that r∈C(∂B1). We extend r(⋅) to all of W1,p(Ω)∖{0} by
r
0
(
u
)
=
1
∥
u
∥
r
(
u
∥
u
∥
)
for all
u
∈
W
1
,
p
(
Ω
)
∖
{
0
}
.
Then r0∈C(W1,p(Ω)∖{0}) and φ(r0(u)u)=ℑ0. Also, if φ(u)=ℑ0, then r0(u)=1. So, we set
(3.44)
r
^
0
(
u
)
=
{
1
if
φ
(
u
)
≤
ℑ
0
,
r
0
(
u
)
if
ℑ
0
<
φ
(
u
)
.
Evidently, r^0∈C(W1,p(Ω)∖{0}). Consider the deformation h(t,u) defined by
h
(
t
,
u
)
=
(
1
-
t
)
u
+
t
r
^
0
(
u
)
u
for all
(
t
,
u
)
∈
[
0
,
1
]
×
(
W
1
,
p
(
Ω
)
∖
{
0
}
)
.
We have
h
(
0
,
u
)
=
u
,
h
(
1
,
u
)
=
r
^
0
(
u
)
u
∈
φ
ℑ
0
and (see (3.44))
h
(
t
,
⋅
)
|
φ
ℑ
0
=
id
|
φ
ℑ
0
for all
t
∈
[
0
,
1
]
.
It follows that
(3.45)
φ
ℑ
0
is a strong deformation retract of
W
1
,
p
(
Ω
)
∖
{
0
}
.
We consider the radial retraction r~:W1,p(Ω)∖{0}→ℝ defined by
r
~
(
u
)
=
u
∥
u
∥
for all
u
∈
W
1
,
p
(
Ω
)
∖
{
0
}
.
This map is continuous and r~|∂B1=id|∂B1. Therefore, ∂B1 is a retract of W1,p(Ω)∖{0}. We consider the deformation h~(t,u) defined by
h
~
(
t
,
u
)
=
(
1
-
t
)
u
+
t
r
~
(
u
)
for all
(
t
,
u
)
∈
[
0
,
1
]
×
(
W
1
,
p
(
Ω
)
∖
{
0
}
)
.
Then
h
~
(
0
,
u
)
=
u
,
h
~
(
1
,
u
)
=
r
~
(
u
)
∈
∂
B
1
and
h
~
(
1
,
⋅
)
|
∂
B
1
=
id
|
∂
B
1
.
Hence, we infer that
(3.46)
∂
B
1
is a deformation retract of
W
1
,
p
(
Ω
)
∖
{
0
}
.
From (3.45) and (3.46), it follows that φℑ0 and ∂B1 are homotopy equivalent, hence
H
k
(
W
1
,
p
(
Ω
)
,
φ
ℑ
0
)
=
H
k
(
W
1
,
p
(
Ω
)
,
∂
B
1
)
for all
k
∈
ℕ
0
,
and therefore, by choosing ℑ0<0 even more negative if necessary, we have
(3.47)
C
k
(
φ
,
∞
)
=
H
k
(
W
1
,
p
(
Ω
)
,
∂
B
1
)
for all
k
∈
ℕ
0
.
The space W1,p(Ω) is infinite dimensional and so ∂B1 is contractible. Hence, from [29, p. 147], we have
H
k
(
W
1
,
p
(
Ω
)
,
∂
B
1
)
=
0
for all
k
∈
ℕ
0
,
and therefore, by (3.47),
C
k
(
φ
,
∞
)
=
0
for all
k
∈
ℕ
0
.
∎
With suitable changes in the above proof, we can also compute the critical groups at infinity for the functionals φ^±. So, we have the following proposition.
Proposition 3.10
Assume that hypotheses (Ha), (Hβ) and (Hf) hold and also that infφ^±(Kφ^±)>-∞. Then Ck(φ^±,∞)=0 for all k∈N0.
Proof.
We do the proof for φ^+ the proof for the functional φ^- being similar.
Let ∂B1+={u∈∂B1:u+≠0}. Consider the deformation h+:[0,1]×∂B1+→∂B1+ defined by
h
+
(
t
,
u
)
=
(
1
-
t
)
u
+
t
u
^
1
(
p
,
β
)
∥
(
1
-
t
)
u
+
t
u
^
1
(
p
,
β
)
∥
for all
(
t
,
u
)
∈
[
0
,
1
]
×
∂
B
1
+
.
We have
h
+
(
1
,
u
)
=
u
^
1
(
p
,
β
)
∥
u
^
1
(
p
,
β
)
∥
∈
∂
B
1
+
,
hence ∂B1+ is contractible. Hypotheses (Hf) (ii)–(iii) imply that for every u∈∂B1+, we have
(3.48)
φ
^
+
(
t
u
)
→
-
∞
as
t
→
+
∞
.
For u∈∂B1+ and t>0, using the chain rule, (3.2), and hypotheses (Ha) (iv) and(Hf) (iii), we have
d
d
t
φ
^
+
(
t
u
)
=
〈
φ
^
+
′
(
t
u
)
,
u
〉
=
1
t
〈
φ
^
+
′
(
t
u
)
,
t
u
〉
=
1
t
[
∫
Ω
(
a
(
t
D
u
)
,
t
D
u
)
ℝ
N
𝑑
z
+
∥
t
u
-
∥
p
p
+
∫
∂
Ω
β
(
z
)
(
t
u
+
)
p
𝑑
σ
-
∫
Ω
f
(
z
,
t
u
+
)
t
u
+
𝑑
z
]
≤
1
t
[
p
G
(
t
D
u
)
d
z
+
∥
t
u
-
∥
p
p
+
∫
∂
Ω
β
(
z
)
(
t
u
+
)
p
𝑑
z
-
∫
Ω
p
F
(
z
,
t
u
+
)
𝑑
z
+
c
39
]
(3.49)
=
1
t
[
p
φ
^
+
(
t
u
)
+
c
39
]
.
From (3.48) and (3.49), it follows that
(3.50)
d
d
t
φ
^
+
(
t
u
)
<
-
c
39
p
<
0
for large
t
>
0
.
Choose
ξ
0
<
min
{
-
c
39
p
,
inf
B
¯
1
φ
^
+
}
(recall that B¯1={u∈W1,p(Ω):∥u∥≤1}). Given u∈∂B1+, because of (3.50) we see that there is unique s0(u)≥1 such that
(3.51)
{
φ
^
+
(
t
u
)
>
ξ
0
if
t
∈
[
0
,
s
0
(
u
)
)
,
φ
^
+
(
t
u
)
=
ξ
0
if
t
=
s
0
(
u
)
,
φ
^
+
(
t
u
)
<
ξ
0
if
s
0
(
u
)
<
t
.
The implicit function theorem implies that s0∈C(∂B1+). Note that (see (3.51))
φ
^
+
ξ
0
=
{
t
u
:
u
∈
∂
B
1
+
,
t
≥
γ
0
(
u
)
}
.
We define E+={tu:u∈∂B1+,t≥1}. We have φ^+ξ0⊆E+. We consider the deformation h^+(r,tu) defined by
h
^
+
(
r
,
t
u
)
=
{
(
1
-
r
)
t
u
+
r
s
0
(
u
)
u
if
t
∈
[
0
,
s
0
(
u
)
]
,
t
u
if
s
0
(
u
)
<
t
,
for all
(
r
,
t
u
)
∈
[
0
,
1
]
×
E
+
.
We have (see (3.51))
h
^
+
(
0
,
t
u
)
=
t
u
,
h
^
+
(
1
,
t
u
)
∈
φ
^
+
ξ
0
and
h
^
+
(
r
,
⋅
)
|
φ
^
+
ξ
0
=
id
|
φ
^
+
ξ
0
for all
r
∈
[
0
,
1
]
.
Therefore, φ^+ξ0 is a strong deformation retract of E+. Hence,
H
k
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
0
)
=
H
k
(
W
1
,
p
(
Ω
)
,
E
+
)
for all
k
∈
ℕ
0
,
and thus (by choosing ξ0<0 even more negative if necessary)
(3.52)
C
k
(
φ
^
+
,
∞
)
=
H
k
(
W
1
,
p
(
Ω
)
,
E
+
)
for all
k
∈
ℕ
0
.
Consider the deformation
h
+
*
(
r
,
t
u
)
=
(
1
-
r
)
t
u
+
r
t
u
∥
t
u
∥
for all
(
r
,
t
u
)
∈
[
0
,
1
]
×
E
+
.
We see that
h
+
*
(
0
,
t
u
)
=
t
u
,
h
+
*
(
1
,
t
u
)
∈
∂
B
1
+
and
h
+
*
(
1
,
⋅
)
|
∂
B
1
+
=
id
|
∂
B
1
+
.
Therefore, ∂B1+ is a deformation retract of E+. Hence,
H
k
(
W
1
,
p
(
Ω
)
,
∂
B
1
+
)
=
H
k
(
W
1
,
p
(
Ω
)
,
E
+
)
for all
k
∈
ℕ
0
,
which implies (recall that ∂B1+ is contractible)
H
k
(
W
1
,
p
(
Ω
)
,
E
+
)
=
0
for all
k
∈
ℕ
0
.
Thus, by (3.52),
C
k
(
φ
^
+
,
∞
)
=
0
for all
k
∈
ℕ
0
.
Similarly for the functional φ^-. ∎
Using Propositions 3.9 and 3.10, we can compute precisely the critical groups of the energy functional φ at the two constant sign solutions u0∈intC+ and v0∈-intC+ produced in Proposition 3.8.
First, we relate the critical groups of φ with those of φ^±. In what follows we assume that the critical sets Kφ and Kφ^± are finite. Otherwise, we already have a whole sequence of distinct solutions of (1.4) (see Proposition 3.7, (3.2) and (3.3)).
Proposition 3.11
If hypotheses (Ha), (Hβ) and (Hf) hold, then
C
k
(
φ
,
u
0
)
=
C
k
(
φ
^
+
,
u
0
)
𝑎𝑛𝑑
C
k
(
φ
,
u
0
)
=
C
k
(
φ
^
-
,
v
0
)
for all
k
∈
ℕ
0
.
Proof.
We do the proof for the triple (φ,φ^+,u0), the proof for the other triple (φ,φ^-,v0) being similar.
We consider the homotopy
h
(
t
,
u
)
=
(
1
-
t
)
φ
(
u
)
+
t
φ
^
+
(
u
)
for all
(
t
,
u
)
∈
[
0
,
1
]
×
W
1
,
p
(
Ω
)
.
Suppose we can find {tn}n≥1⊆[0,1] and {un}n≥1⊆W1,p(Ω) such that
(3.53)
t
n
→
t
,
u
n
→
u
0
in
W
1
,
p
(
Ω
)
and
h
u
′
(
t
n
,
u
n
)
=
0
for all
n
∈
ℕ
.
Then, from the equation in (3.53) and (3.2), we have
〈
A
(
u
n
)
,
v
〉
+
∫
∂
Ω
β
(
z
)
(
u
n
+
)
p
-
1
v
𝑑
σ
-
t
n
∫
∂
Ω
β
(
z
)
(
u
n
-
)
p
-
1
v
𝑑
σ
-
t
n
∫
Ω
(
u
n
-
)
p
-
1
v
𝑑
z
=
∫
Ω
f
(
z
,
u
n
+
)
v
𝑑
z
+
(
1
-
t
n
)
∫
Ω
f
(
z
,
-
u
n
-
)
𝑑
z
for all
v
∈
W
1
,
p
(
Ω
)
,
which implies
{
-
div
a
(
D
u
n
(
z
)
)
-
t
n
u
n
-
(
z
)
p
-
1
=
f
(
z
,
u
n
+
(
z
)
)
+
(
1
-
t
n
)
f
(
z
,
-
u
n
-
(
z
)
)
for almost all
z
∈
Ω
,
∂
u
n
∂
n
a
+
β
(
z
)
(
(
u
n
+
)
p
-
1
-
t
n
(
u
n
-
)
p
-
1
)
=
0
on
∂
Ω
.
From Proposition 2.10 we know that there exists M14>0 such that
∥
u
n
∥
∞
≤
M
14
for all
n
∈
ℕ
.
So, from Lieberman [24] we know that there exist α∈(0,1) and M15>0 such that
(3.54)
u
n
∈
C
1
,
α
(
Ω
¯
)
and
∥
u
n
∥
C
1
,
α
(
Ω
¯
)
≤
M
15
for all
n
∈
ℕ
.
Because of (3.53) and since C1,α(Ω¯) is embedded compactly into C1(Ω¯), from (3.54) we have
u
n
→
u
0
in
C
1
(
Ω
¯
)
.
Recall that u0∈intC+ (see Proposition 3.8). So, we can find n0∈ℕ such that
u
n
∈
int
C
+
for all
n
≥
n
0
,
hence {un}n≥n0 are distinct (positive) solutions of (1.4) (see (3.53)), a contradiction (recall that we have assumed Kφ^+ is finite). Therefore (3.53) can not happen. Then, invoking [10, Theorem 5.2] (the homotopy invariance of critical groups), we have
C
k
(
φ
,
u
0
)
=
C
k
(
φ
^
+
,
u
0
)
for all
k
∈
ℕ
0
.
In a similar fashion we show that
C
k
(
φ
,
v
0
)
=
C
k
(
φ
^
-
,
v
0
)
for all
k
∈
ℕ
0
.
∎
Proposition 3.12
If hypotheses (Ha), (Hβ) and (Hf) hold, then Ck(φ^+,u0)=Ck(φ^-,v0)=δk,1Z for all k∈N0.
Proof.
We do the proof for the pair (φ^+,u0), the proof for the pair (φ^-,v0) being similar.
From Proposition 3.7 we know that Kφ^+⊆C+. So, we may assume that
(3.55)
K
φ
^
+
=
{
0
,
u
0
}
or, otherwise, we already have a third nontrivial solution for problem (1.4) which in fact is positive. From the proof of Proposition 3.8 (see (3.35) and (3.36)) we have
0
=
φ
^
+
(
0
)
<
m
ρ
+
≤
φ
^
+
(
u
0
)
.
Let ξ-<0<ξ+<mρ+, and consider the triple of sets
φ
^
+
ξ
-
⊆
φ
^
+
ξ
+
⊆
W
1
,
p
(
Ω
)
.
For this triple of sets, we consider the following corresponding long exact sequence of singular homology groups (see [29, p. 143]):
(3.56)
⋯
→
H
k
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
-
)
→
i
*
H
k
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
+
)
→
∂
*
H
k
-
1
(
φ
^
+
ξ
+
,
φ
^
+
ξ
-
)
→
⋯
,
with i* being the homomorphism induced by the inclusion i:(W1,p(Ω),φ^+ξ-)→(W1,p(Ω),φ^+ξ+) and ∂* is the boundary homomorphism. From (3.55) and since ξ-<0=φ^+(0), we have (see Proposition 3.10)
(3.57)
H
k
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
-
)
=
C
k
(
φ
^
+
,
∞
)
=
0
for all
k
∈
ℕ
0
.
Also, we have
0
=
φ
^
+
(
0
)
<
ξ
+
<
φ
^
+
(
u
0
)
.
Then from (3.55) we have
(3.58)
H
k
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
+
)
=
C
k
(
φ
^
+
,
u
0
)
for all
k
∈
ℕ
0
.
Similarly, we have (see Proposition 3.6)
(3.59)
H
k
-
1
(
φ
^
+
ξ
+
,
φ
^
+
ξ
-
)
=
C
k
-
1
(
φ
^
+
,
0
)
=
δ
k
-
1
,
0
ℤ
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
.
From (3.57)–(3.59) and the exactness of (3.56), we see that only the tail of that chain (that is, k=1) is nontrivial. From the rank theorem, the exactness of (3.56), and using (3.57) and (3.59), we have
(3.60)
rank
H
1
(
W
1
,
p
(
Ω
)
,
φ
^
+
ξ
+
)
=
rank
ker
∂
*
+
rank
im
∂
*
=
rank
im
i
*
+
rank
im
∂
*
≤
1
.
From the proof of Proposition 3.8 we know that u0 is a critical point of φ^+ of mountain pass type. Therefore,
(3.61)
C
1
(
φ
^
+
,
u
0
)
≠
0
.
From (3.58), (3.60), (3.61) and recalling that only for k=1 the chain (3.56) is nontrivial, we conclude that
C
k
(
φ
^
+
,
u
0
)
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
.
Similarly, for the pair (φ^-,v0). ∎
From Propositions 3.11 and 3.12, we infer the following corollary.
Corollary 3.13
If hypotheses (Ha), (Hβ) and (Hf) hold, then Ck(φ,u0)=Ck(φ,v0)=δk,1Z for all k∈N0.
Now we ready for the “three solutions theorem” for problem (1.4).
Theorem 3.14
If hypotheses (Ha), (Hβ) and (Hf) hold, then problem (1.4) has at least three nontrivial solutions u0∈intC+, v0∈-intC+ and y0∈C1(Ω¯).
Proof.
From Proposition 3.8 we already have two constant sign solutions u0∈intC+ and v0∈-intC+. Suppose that these are the only nontrivial solutions of problem (1.4) (that is, Kφ={0,u0,v0}). From Corollary 3.13 we have
(3.62)
C
k
(
φ
,
u
0
)
=
C
k
(
φ
,
v
0
)
=
δ
k
,
1
ℤ
for all
k
∈
ℕ
0
.
From Proposition 3.6 we have
(3.63)
C
k
(
φ
,
0
)
=
δ
k
,
0
ℤ
for all
k
∈
ℕ
0
.
Finally, Proposition 3.9 implies that
(3.64)
C
k
(
φ
,
∞
)
=
0
for all
k
∈
ℕ
0
.
From (3.62)–(3.64) and the Morse relation with t=-1 (see (2.44)), we have 2(-1)1+(-1)0=0, which implies (-1)1=0, a contradiction. So, we can find y0∈Kφ,y0∉{0,u0,v0}. This is the third nontrivial solution of problem (1.4) and, as before, the nonlinear regularity theory implies y0∈C1(Ω¯). ∎
