2 Mathematical Background
Let X be a Banach space and X* its topological dual. By 〈⋅,⋅〉 we denote the duality brackets for the dual pair (X*,X). Also, let 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
+
∥
u
n
∥
)
φ
′
(
u
n
)
→
0
in
X
*
as
n
→
∞
,
admits a strongly convergent subsequence.
This compactness-type condition on the functional φ leads to a deformation theorem from which one derives the minimax theory of the critical values of φ. A basic result in this theory is the celebrated “mountain pass theorem” due to Ambrosetti and Rabinowitz [4]. Here, we state the result in a slightly more general form (see, for example, [16, p. 648]).
Theorem 2.1.
Let X be a Banach space, and assume that φ∈C1(X,R) satisfies the C-condition, u0,u1∈X, ∥u1-u0∥>ρ>0,
max
{
φ
(
u
0
)
,
φ
(
u
1
)
}
<
inf
{
φ
(
u
)
:
∥
u
-
u
0
∥
=
ρ
}
=
m
ρ
and
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 (that is, there exists u∈X such that φ′(u)=0, φ(u)=c).
Three Banach spaces will be central in our analysis of problem (1.1). We refer to the Dirichlet Sobolev spaces W01,p(Ω) and H01(Ω), and the Banach space C01(Ω¯)={u∈C1(Ω¯):u|∂Ω=0}.
By Poincaré’s inequality, the norm of W01,p(Ω) can be defined by
∥
u
∥
=
∥
D
u
∥
p
for all
u
∈
W
0
1
,
p
(
Ω
)
.
The space H01(Ω) is a Hilbert space and again the Poincaré inequality implies that we can choose as inner product
(
u
,
h
)
=
(
D
u
,
D
h
)
L
2
(
Ω
,
ℝ
N
)
for all
u
,
h
∈
H
0
1
(
Ω
)
.
The corresponding norm is
∥
u
∥
H
0
1
(
Ω
)
=
∥
D
u
∥
2
for all
u
∈
H
0
1
(
Ω
)
.
The Banach space C01(Ω¯) is an ordered Banach space with positive 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 is the usual normal derivative defined by ∂u∂n=(Du,n)ℝN, with n(⋅) being the outward unit normal on ∂Ω. Recall that C01(Ω¯) is dense in both W01,p(Ω) and H01(Ω).
Given x∈ℝ, we set x±=max{±x,0} and then define u±(⋅)=u(⋅)± for all u∈W01,p(Ω). We know that
u
±
∈
W
0
1
,
p
(
Ω
)
,
u
=
u
+
-
u
-
,
|
u
|
=
u
+
+
u
-
.
Also, we denote the Lebesgue measure on ℝN by |⋅|N, and if g:Ω×ℝ→ℝ is a measurable function (for example, a Carathéodory function), we define the Nemytskii map corresponding to g(⋅,⋅) by
N
g
(
u
)
(
⋅
)
=
g
(
⋅
,
u
(
⋅
)
)
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We will use the spectra of the operators (-Δp,W01,p(Ω)) and (-Δ,H01(Ω)). We start with the spectrum of (-Δp,W01,p(Ω)). So, consider the following nonlinear eigenvalue problem:
(2.1)
-
Δ
p
u
(
z
)
=
λ
^
|
u
(
z
)
|
p
-
2
u
(
z
)
in
Ω
(
1
<
p
<
∞
)
,
u
|
∂
Ω
=
0
.
We say that λ^∈ℝ is an eigenvalue of (-Δp,W01,p(Ω)) if problem (2.1) admits a nontrivial solution u^∈W01,p(Ω), known as the eigenfunction corresponding to λ^. We know that there exists the smallest eigenvalue λ^1(p)>0, which has the following properties:
λ^1(p) is isolated in the spectrum σ^(p) of (-Δp,W01,p(Ω)); in other words, there exists ϵ>0 such that (λ^1(p),λ^1(p)+ϵ)∩σ^(p)=∅.
λ^1(p) is simple; that is, if u^,u~∈W01,p(Ω) are eigenfunctions corresponding to λ^1(p), then u^=ξu~ with ξ∈ℝ∖{0}.
We have
(2.2)
λ
^
1
(
p
)
=
inf
{
∥
D
u
∥
p
p
∥
u
∥
p
p
:
u
∈
W
0
1
,
p
(
Ω
)
,
u
≠
0
}
.
In (2.2) the infimum is realized on the one-dimensional eigenspace corresponding to λ^1(p). The above properties imply that the elements of this eigenspace do not change sign. We point out that the nonlinear regularity theory (see, for example, [16, p. 737]) implies that all eigenfunctions of (-Δp,W01,p(Ω)) belong to C01(Ω¯). By u^1(p) we denote the positive Lp-normalized (that is, ∥u^1(p)∥p=1) eigenfunction corresponding to λ^1(p)>0. As we have already mentioned, u^1(p)∈C+∖{0} and, in fact, the nonlinear maximum principle (see, for example, [16, p. 738]) implies that u^1(p)∈intC+. An eigenfunction u^ which corresponds to an eigenvalue λ^≠λ^1(p) is nodal (sign changing). Since σ^(p) is closed and λ^1(p)>0 is isolated, the second eigenvalue λ^2(p) is well defined by
λ
^
2
(
p
)
=
min
{
λ
^
∈
σ
^
(
p
)
:
λ
^
>
λ
^
1
(
p
)
}
.
For additional eigenvalues, we employ the Ljusternik–Schnirelmann minimax scheme, which gives the entire nondecreasing sequence of eigenvalues {λ^k(p)}k⩾1 such that λ^k(p)→+∞. These eigenvalues are known as “variational eigenvalues” and, depending on the index used in the Ljusternik–Schnirelmann scheme, we can have various such sequences of variational eigenvalues, which all coincide in the first two elements λ^1(p) and λ^2(p), defined as described above. For the other elements we do not know if their sequences coincide. Here, we use the sequence constructed by using the Fadell–Rabinowitz [14] cohomological index (see [32]). Note that we do not know if the variational eigenvalues exhaust the spectrum σ^(p). We have full knowledge of the spectrum if N=1 (ordinary differential equations) and when p=2 (linear eigenvalue problem). In the latter case, we have σ^(2)={λ^k(2)}k⩾1 with 0<λ^1(2)<λ^2(2)<⋯<λ^k(2)→+∞ as k→∞. The corresponding eigenspaces, denoted by E(λ^k(2)), are linear spaces, and we have the orthogonal direct sum decomposition
H
0
1
(
Ω
¯
)
=
⊕
k
⩾
1
E
(
λ
^
k
(
2
)
)
¯
.
For all k∈ℕ, each E(λ^k(2)) is finite dimensional, E(λ^k(2))⊆C01(Ω¯), and has the so-called Unique Continuation Property (UCP for short), that is, if u∈E(λ^k(2)) vanishes on a set of positive measure in Ω, then u≡0. For every k∈ℕ, we define
H
¯
k
=
⊕
i
=
1
k
E
(
λ
^
i
(
2
)
)
and
H
^
k
+
1
=
⊕
i
⩾
k
+
1
E
(
λ
^
i
(
2
)
)
¯
=
H
¯
k
⊥
.
We have
H
0
1
(
Ω
)
=
H
¯
k
⊕
H
^
k
+
1
.
In this case all eigenvalues admit variational characterizations and we have
(2.3)
λ
^
1
(
2
)
=
inf
{
∥
D
u
∥
2
2
∥
u
∥
2
2
:
u
∈
H
0
1
(
Ω
)
,
u
≠
0
}
,
(2.4)
λ
^
k
(
2
)
=
sup
{
∥
D
u
∥
2
2
∥
u
∥
2
2
:
u
∈
H
¯
k
,
u
≠
0
}
=
inf
{
∥
D
u
∥
2
2
∥
u
∥
2
2
:
u
∈
H
^
k
,
u
≠
0
}
,
k
⩾
2
.
Again, the infimum in (2.3) is realized on the one-dimensional eigenspace E(λ^1(2)), while both the supremum and the infimum in (2.4) are realized on E(λ^k(2)).
As a consequence of the UCP, we have the following convenient inequalities.
Lemma 2.2.
If ϑ∈L∞(Ω) and, for k∈ℕ, ϑ(z)⩾λ^k(2) for almost all z∈Ω, with ϑ≢λ^k(2), then there exists a constant c0>0 such that
∥
D
u
∥
2
2
-
∫
Ω
ϑ
(
z
)
u
2
𝑑
z
⩽
-
c
0
∥
u
∥
2
for all
u
∈
H
¯
k
.
If ϑ∈L∞(Ω) and, for k∈ℕ, ϑ(z)⩽λ^k(2) for almost all z∈Ω, with ϑ≢λ^k(2), then there exists a constant c1>0 such that
∥
D
u
∥
2
2
-
∫
Ω
ϑ
(
z
)
u
2
𝑑
z
⩾
c
1
∥
u
∥
2
for all
u
∈
H
^
k
.
In what follows, let Ap:W01,p(Ω)→W-1,p′(Ω)=W01,p(Ω)* (1p+1p′=1, 1<p<∞) be the 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
0
1
,
p
(
Ω
)
.
By [23, p. 40], we have the following proposition.
Proposition 2.3.
The map Ap:W01,p(Ω)→W-1,p′(Ω) (1<p<∞) is bounded (that is, it 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
,
p
(
Ω
)
𝑎𝑛𝑑
lim sup
n
→
∞
〈
A
(
u
n
)
,
u
n
-
u
〉
⩽
0
,
then un→n in W01,p(Ω).
If p=2, then A2=A∈ℒ(H01(Ω),H-1(Ω)).
Consider a Carathéodory function f0:Ω×ℝ→ℝ such that
|
f
0
(
z
,
x
)
|
⩽
a
0
(
z
)
(
1
+
|
x
|
r
-
1
)
for almost all
z
∈
Ω
and all
x
∈
ℝ
,
with a0∈L∞(ℝ) and 1<r<p*, where the critical Sobolev exponent is defined by
p
*
=
{
N
p
N
-
p
if
p
<
N
,
+
∞
if
p
⩾
N
.
Let F0(z,x)=∫0xf0(z,s)𝑑s and consider the C1-functional φ0:W01,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 proposition is a special case of a more general result by Aizicovici, Papageorgiou and Staicu [2], see also [26, 29] for similar results in different spaces. All these results are consequences of the nonlinear regularity theory of Lieberman [20].
Proposition 2.4.
Let u0∈W01,p(Ω) be a local C01(Ω¯)-minimizer of φ0, that is, there exists ρ0>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 u0∈C01,α(Ω¯) for some α∈(0,1) and it is also a local W01,p(Ω)-minimizer of φ0, that is, there exists ρ1>0 such that
φ
0
(
u
0
)
⩽
φ
0
(
u
0
+
h
)
for all
h
∈
W
0
1
,
p
(
Ω
)
with
∥
h
∥
⩽
ρ
1
.
Finally, we recall some basic definitions and facts from Morse theory (critical groups), which we will use in the sequel.
So, let X be a Banach space, φ∈C1(X,ℝ) and c∈ℝ. We introduce the following sets:
K
φ
=
{
u
∈
X
:
φ
′
(
u
)
=
0
}
,
K
φ
c
=
{
u
∈
K
φ
:
φ
(
u
)
=
c
}
,
φ
c
=
{
u
∈
X
:
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 with integer coefficients for the pair (Y1,Y2). Given an isolated u∈Kφc, 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
,
where 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 independent of the particular choice of the neighborhood U.
Suppose that φ satisfies the C-condition and 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 (see, for example, [16, p. 628]) implies that this definition is independent of the choice of the level c<infφ(Kφ).
In the next section we prove an existence theorem under conditions of resonance both at ±∞ and at zero.
3 Existence of Nontrivial Solutions
The hypotheses on the reaction term f(z,x) are the following.
Hypotheses 3.1.
f:Ω×ℝ→ℝ is a Carathéodory function with the following properties:
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
.
There exists an integer m⩾1 such that
lim
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
-
2
x
=
λ
^
m
(
p
)
uniformly for almost all
z
∈
Ω
.
There exists τ∈(2,p) such that
0
<
β
0
⩽
lim inf
x
→
±
∞
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
|
x
|
τ
uniformly for almost all
z
∈
Ω
,
where F(z,x)=∫0xf(z,s)𝑑s.
There exist integer l⩾1, with dl≠m (dl=dimH¯l), δ>0 and η∈L∞(Ω) such that
λ
^
l
(
2
)
⩽
η
(
z
)
for almost all
z
∈
Ω
,
η
≢
λ
^
l
(
2
)
,
η
(
z
)
x
2
⩽
f
(
z
,
x
)
x
⩽
λ
^
l
+
1
x
2
for almost all
z
∈
Ω
and all
|
x
|
⩽
δ
,
and for every x≠0 the second inequality is strict on a subset of positive Lebesgue measure.
Remark 3.2.
Hypothesis 3.1 (ii) says that asymptotically as x→±∞, we have resonance with respect to some variational eigenvalue of (-Δp,W01,p(Ω)). Similarly, Hypothesis 3.1 (iv) permits resonance at zero with respect to the eigenvalue λ^l+1(2) of (-Δ,H01(Ω)). So, in a sense, we have a double resonance setting.
Let φ:H1(Ω)→ℝ be the energy (Euler) functional for problem (1.1) defined by
φ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Proposition 3.3.
If Hypotheses 3.1 (i), (ii), (iii) hold, then φ satisfies the C-condition.
Proof.
Let {un}n⩾1⊆W01,p(Ω) be a sequence such that
(3.1)
|
φ
(
u
n
)
|
⩽
M
1
for some
M
1
>
0
and all
n
∈
ℕ
,
(3.2)
(
1
+
∥
u
n
∥
)
φ
′
(
u
n
)
→
0
in
W
-
1
,
p
′
(
Ω
)
=
W
0
1
,
p
(
Ω
)
*
.
By (3.2), we have
(3.3)
|
〈
A
p
(
u
n
)
,
h
〉
+
〈
A
(
u
n
)
,
h
〉
-
∫
Ω
f
(
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 (3.3) we choose h=un∈W01,p(Ω) and obtain
(3.4)
-
∥
D
u
n
∥
p
p
-
∥
D
u
n
∥
2
2
+
∫
Ω
f
(
z
,
u
n
)
u
n
𝑑
z
⩽
ϵ
n
for all
n
∈
ℕ
.
On the other hand, from (3.1) we have
(3.5)
∥
D
u
n
∥
p
p
+
p
2
∥
D
u
n
∥
2
2
-
∫
Ω
p
F
(
z
,
u
n
)
𝑑
z
⩽
p
M
1
for all
n
∈
ℕ
.
We add (3.4) and (3.5) and obtain
(3.6)
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
⩽
M
2
+
(
1
-
p
2
)
∥
D
u
n
∥
2
2
for all
n
∈
ℕ
,
for some M2>0. Hypotheses 3.1 (i), (iii) imply that we can find β1∈(0,β0) and c2>0 such that
(3.7)
β
1
|
x
|
τ
-
c
2
⩽
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
for almost all
z
∈
Ω
and all
x
∈
ℝ
.
Returning to (3.6) and using (3.7), we have (recall that τ>2)
(3.8)
∥
u
n
∥
τ
τ
⩽
c
3
(
1
+
∥
D
u
n
∥
2
2
)
for all
n
∈
ℕ
,
for some c3>0.
Claim 1.
{un}n⩾1⊆W01,p(Ω) is bounded.
Arguing by contradiction, suppose that the claim is not true. By passing to a subsequence if necessary, we have
(3.9)
∥
u
n
∥
→
∞
.
Let yn=un∥un∥, n∈ℕ. Then ∥yn∥=1 for all n∈ℕ, and so we may assume that
(3.10)
y
n
→
𝑤
y
in
W
0
1
,
p
(
Ω
)
and
y
n
→
y
in
L
p
(
Ω
)
.
From (3.3) we have
(3.11)
|
〈
A
p
(
y
n
)
,
h
〉
+
1
∥
u
n
∥
p
-
2
〈
A
(
y
n
)
,
h
〉
-
∫
Ω
N
f
(
u
n
)
∥
u
n
∥
p
-
1
h
𝑑
z
|
⩽
ϵ
n
∥
h
∥
(
1
+
∥
u
n
∥
)
∥
u
n
∥
p
-
1
for all
n
∈
ℕ
.
Hypotheses 3.1 (i), (ii) imply that
(3.12)
|
f
(
z
,
x
)
|
⩽
c
4
(
1
+
|
x
|
p
-
1
)
for almost all
z
∈
Ω
and all
x
∈
ℝ
,
for some c4>0, and hence
(3.13)
{
N
f
(
u
n
)
∥
u
n
∥
p
-
1
}
n
⩾
1
⊆
L
p
′
(
Ω
)
is bounded.
In (3.11) we choose h=yn-y∈W01,p(Ω), pass to the limit as n→∞ and use (3.9), (3.10), (3.13) and the fact that p>2. Then limn→∞〈Ap(yn),yn-y〉=0, which implies (see Proposition 2.3)
(3.14)
y
n
→
y
in
W
0
1
,
p
(
Ω
)
⟹
∥
y
∥
=
1
.
From (3.8) we have
∥
y
n
∥
τ
τ
⩽
c
3
∥
u
n
∥
τ
+
c
3
∥
u
n
∥
τ
-
2
∥
D
y
n
∥
2
2
⩽
c
5
∥
u
n
∥
τ
-
2
for all
n
⩾
n
0
⩾
1
,
for some c5>0. This yields (see (3.9) and recall that τ>2)
y
n
→
0
in
L
τ
(
Ω
)
as
n
→
∞
.
Thus, y=0 (see (3.10)), a contradiction to (3.14). This proves the claim.
Because of Claim 1, we may assume that
(3.15)
u
n
→
𝑤
u
in
W
0
1
,
p
(
Ω
)
and
u
n
→
u
in
L
p
(
Ω
)
.
From (3.12) we see that
(3.16)
{
N
f
(
u
n
)
}
n
⩾
1
⊆
L
p
′
(
Ω
)
is bounded
.
So, if in (3.3) we choose h=yn-y∈W01,p(Ω), pass to the limit as n→∞ and use (3.15) and (3.16), then
lim
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
〉
+
〈
A
(
u
n
)
,
u
n
-
u
〉
]
=
0
,
and since A(⋅) is monotone, we have
lim sup
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
〉
+
〈
A
(
u
)
,
u
n
-
u
〉
]
⩽
0
.
From (3.15),
lim sup
n
→
∞
〈
A
p
(
u
n
)
,
u
n
-
u
〉
⩽
0
,
which implies (see Proposition 2.3)
u
n
→
u
in
W
0
1
,
p
(
Ω
)
.
Thus, φ satisfies the C-condition. ∎
We can have two approaches in the proof of the existence theorem. We present both because we believe that the particular tools used in each of them are of independent interest and can be used in different circumstances.
In the first approach we compute directly the critical groups at infinity of the energy functional φ. Note that Proposition 3.3 permits this computation.
Proposition 3.4.
If Hypotheses 3.1 (i), (ii), (iii) hold, then Cm(φ,∞)≠0.
Proof.
Let λ∈(λ^m(p),λ^m+1(p))∖σ^(p) and consider the C2-functional ψ:W01,p(Ω)→ℝ defined by
ψ
(
u
)
=
1
p
∥
D
u
∥
p
p
-
λ
p
∥
u
∥
p
p
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We also consider the homotopy h(t,u) defined by
h
(
t
,
u
)
=
(
1
-
t
)
φ
(
u
)
+
t
ψ
(
u
)
for all
(
t
,
u
)
∈
[
0
,
1
]
×
W
0
1
,
p
(
Ω
)
.
Claim 2.
There exist η∈ℝ and δ^>0 such that
h
(
t
,
u
)
⩽
η
⟹
(
1
+
∥
u
∥
)
∥
h
u
′
(
t
,
u
)
∥
*
⩾
δ
^
for all
t
∈
[
0
,
1
]
.
We argue indirectly. So, suppose that the claim is not true. Since h(⋅,⋅) maps bounded sets to bounded sets, we can find {tn}n⩾1⊆[0,1] and {un}n⩾1⊆W01,p(Ω) such that
(3.17)
t
n
→
t
,
∥
u
n
∥
→
∞
,
h
(
t
n
,
u
n
)
→
-
∞
and
(
1
+
∥
u
n
∥
)
h
u
′
(
t
n
,
u
n
)
→
0
in
W
-
1
,
p
′
(
Ω
)
.
From the last convergence in (3.17), we have
(3.18)
|
〈
A
p
(
u
n
)
,
h
〉
+
(
1
-
t
n
)
〈
A
(
u
n
)
,
h
〉
-
(
1
-
t
n
)
∫
Ω
f
(
z
,
u
n
)
h
𝑑
z
-
t
n
∫
Ω
λ
|
u
n
|
p
-
2
u
n
h
𝑑
z
|
⩽
ϵ
n
∥
h
∥
1
+
∥
u
n
∥
for all h∈W01,p(Ω) with ϵn→0+.
Let yn=un∥un∥, n∈ℕ. Then ∥yn∥=1 for all n∈ℕ, and so we may assume that
(3.19)
y
n
→
𝑤
y
in
W
0
1
,
p
(
Ω
)
and
y
n
→
y
in
L
p
(
Ω
)
.
From (3.18) we have
(3.20)
|
〈
A
p
(
y
n
)
,
h
〉
+
1
-
t
n
∥
u
n
∥
p
-
2
〈
A
(
y
n
)
,
h
〉
-
(
1
-
t
n
)
∫
Ω
N
f
(
u
n
)
∥
u
n
∥
p
-
1
h
𝑑
z
-
t
n
∫
Ω
λ
|
y
n
|
p
-
2
y
n
h
𝑑
z
|
⩽
ϵ
n
∥
h
∥
(
1
+
∥
u
n
∥
)
∥
u
n
∥
p
-
1
for all n∈ℕ.
From (3.12) and (3.19), we see that
{
N
f
(
u
n
)
∥
u
n
∥
p
-
1
}
n
⩾
1
⊆
L
p
′
(
Ω
)
is bounded
.
Hence, by passing to a subsequence if necessary and using Hypothesis 3.1 (ii), we obtain (see [15])
(3.21)
N
f
(
u
n
)
∥
u
n
∥
p
-
1
→
𝑤
λ
^
m
(
p
)
|
y
|
p
-
2
y
in
L
p
′
(
Ω
)
.
In (3.20) we choose h=yn-y∈W01,p(Ω), pass to the limit as n→∞ and use (3.17), (3.19), (3.21) and the fact that 2<p. Then limn→∞〈Ap(yn),yn-y〉=0, which implies (see Proposition 2.3)
(3.22)
y
n
→
y
in
W
0
1
,
p
(
Ω
)
⟹
∥
y
∥
=
1
.
We return to (3.20), pass to the limit as n→∞ and use (3.21) and (3.22). We obtain
〈
A
p
(
y
)
,
h
〉
=
∫
Ω
λ
t
|
y
|
p
-
2
y
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
, with
λ
t
=
(
1
-
t
)
λ
^
m
(
p
)
+
t
λ
,
hence
(3.23)
-
Δ
p
y
(
z
)
=
λ
t
|
y
(
z
)
|
p
-
2
y
(
z
)
for almost all
z
∈
Ω
,
y
|
∂
Ω
=
0
.
If λt∉σ^(p), then from (3.23) it follows that y=0, a contradiction (see (3.22)).
If λt∈σ^(p), then for E={z∈Ω:y(z)≠0}, we have |E|N>0. Hence,
|
u
n
(
z
)
|
→
+
∞
for almost all
z
∈
Ω
,
and thus
(3.24)
lim inf
n
→
∞
f
(
z
,
u
n
(
z
)
)
u
n
(
z
)
-
p
F
(
z
,
u
n
(
z
)
)
|
u
n
(
z
)
|
τ
⩾
β
0
>
0
for almost all
z
∈
E
.
From (3.24), Hypothesis 3.1 (iii) and Fatou’s lemma, we have
(3.25)
lim inf
n
→
∞
1
∥
u
n
∥
τ
∫
E
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
>
0
.
Note that Hypothesis 3.1 (iii) implies that we can find M3>0 such that
(3.26)
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
⩾
0
for almost all
z
∈
Ω
and all
|
x
|
⩾
M
3
.
Then, in view of (3.26) and Hypothesis 3.1 (i), we have
1
∥
u
n
∥
τ
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
=
1
∥
u
n
∥
τ
∫
Ω
∩
{
|
u
n
|
⩾
M
3
}
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
+
1
∥
u
n
∥
τ
∫
Ω
∩
{
|
u
n
|
<
M
3
}
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
⩾
1
∥
u
n
∥
τ
∫
E
∩
{
|
u
n
|
⩾
M
3
}
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
-
c
6
∥
u
n
∥
τ
⩾
1
∥
u
n
∥
τ
∫
E
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
-
c
7
∥
u
n
∥
τ
for all
n
∈
ℕ
,
for some c6,c7>0. Hence, by (3.25),
(3.27)
lim inf
n
→
∞
1
∥
u
n
∥
τ
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
>
0
.
From the third convergence in (3.17), we see that we can find n0∈ℕ such that
(3.28)
∥
D
u
n
∥
p
p
+
(
1
-
t
n
)
p
2
∥
D
u
n
∥
2
2
-
(
1
-
t
n
)
∫
Ω
p
F
(
z
,
u
n
)
𝑑
z
-
t
n
∫
Ω
λ
|
u
n
|
p
𝑑
z
⩽
-
1
for all
n
⩾
n
0
.
In (3.18) we choose h=un∈W01,p(Ω). Then
(3.29)
-
∥
D
u
n
∥
p
p
-
(
1
-
t
n
)
∥
D
u
n
∥
2
2
+
(
1
-
t
n
)
∫
Ω
f
(
z
,
u
n
)
u
n
𝑑
z
+
t
n
∫
Ω
λ
|
u
n
|
p
𝑑
z
⩽
ϵ
m
for all
n
∈
ℕ
.
Since ϵn→0+, by choosing n0∈ℕ even bigger if necessary, we can get
(3.30)
ϵ
n
∈
(
0
,
1
)
for all
n
⩾
n
0
.
By adding (3.28) and (3.29), and using (3.30), we have
(
1
-
t
n
)
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
⩽
(
1
-
t
n
)
(
1
-
p
2
)
∥
D
u
n
∥
2
2
.
We may assume that tn≠1 for all n∈ℕ. Otherwise, t=1, and so λt=λ∉σ(p), hence y=0, a contradiction to (3.22). Then
1
∥
u
n
∥
τ
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
⩽
(
1
-
p
2
)
1
∥
u
n
∥
τ
-
2
∥
D
y
n
∥
2
2
for all
n
∈
ℕ
.
Since p>τ>2, it follows from (3.17) and (3.22) that
lim sup
n
→
∞
∫
Ω
[
f
(
z
,
u
n
)
u
n
-
p
F
(
z
,
u
n
)
]
𝑑
z
⩽
0
,
which contradicts (3.27). This proves the claim.
In fact, the above argument with minor changes shows that for every t∈[0,1], h(t,⋅) satisfies the C-condition. So, [9, Theorem 5.1.12] (see also [19, Proposition 3.2]) implies that
C
k
(
h
(
0
,
⋅
)
,
∞
)
=
C
k
(
h
(
1
,
⋅
)
,
∞
)
for all
k
∈
ℕ
0
,
and therefore
C
k
(
φ
,
∞
)
=
C
k
(
ψ
,
∞
)
for all
k
∈
ℕ
0
.
Since λ∉σ^(p), we have Kψ={0}, and so Ck(ψ,∞)=Ck(ψ,0) for all k∈ℕ0. Hence,
C
k
(
φ
,
∞
)
=
C
k
(
ψ
,
0
)
for all
k
∈
ℕ
0
.
But by [32, Proposition 1.1], we have Cm(ψ,0)≠0. So, Cm(φ,∞)≠0. ∎
In the second approach, we avoid the computation of the critical groups of φ at infinity. Instead we use the following result which is essentially due to Perera [32, Lemma 4.1], here adapted to our setting.
Proposition 3.5.
If Hypotheses 3.1 (i), (ii), (iii) hold, then there exist r>0 and φ0∈C1(W01,p(Ω)) such that
φ
0
(
u
)
=
{
φ
(
u
)
if
∥
u
∥
⩽
r
,
ψ
(
u
)
if
∥
u
∥
⩾
2
1
/
p
r
,
K
φ
0
=
K
φ
and Cm(φ0,∞)≠0.
Proof.
Let ψ∈C2(W01,p(Ω)) be as in the proof of Proposition 3.4. Also let τ:W01,p(Ω)→ℝ be the C1-functional defined by
τ
(
u
)
=
∫
Ω
F
(
z
,
u
)
𝑑
z
-
λ
p
∥
u
∥
p
p
-
1
2
∥
D
u
∥
2
2
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Evidently, we have
(3.31)
φ
(
u
)
=
ψ
(
u
)
-
τ
(
u
)
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Since λ∉σ^(p), the functional ψ satisfies the C-condition, and so
μ
=
inf
{
∥
ψ
′
(
u
)
∥
*
:
u
∈
W
0
1
,
p
(
Ω
)
,
∥
u
∥
=
1
}
>
0
.
We have
ψ
′
(
u
)
=
A
p
(
u
)
-
λ
|
u
|
p
-
2
u
,
hence the (p-1)-homogeneity of ψ′(⋅) implies that
(3.32)
inf
{
∥
ψ
′
(
u
)
∥
*
:
u
∈
W
0
1
,
p
(
Ω
)
,
∥
u
∥
=
r
}
=
r
p
-
1
μ
>
0
(
r
>
0
)
.
Since λ>λ^m(p) and p>2, it follows that
(3.33)
lim sup
∥
u
∥
→
∞
τ
′
(
u
)
∥
u
∥
p
-
1
⩽
0
.
From (3.31) we have
φ
′
(
u
)
=
ψ
′
(
u
)
-
τ
′
(
u
)
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Hence, using (3.32) and (3.33),
(3.34)
φ
′
(
u
)
>
0
and
φ
′
(
u
)
+
τ
′
(
u
)
>
0
for all
∥
u
∥
>
r
.
Let ξ:ℝ+→[0,1] be a C1-function such that |ξ′(t)|⩽1 for all t⩾0 and
(3.35)
ξ
(
t
)
=
{
0
if
t
∈
[
0
,
1
]
,
1
if
t
⩾
2
.
We define
φ
0
(
u
)
=
φ
(
u
)
+
ξ
(
∥
u
∥
p
r
p
)
τ
(
u
)
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Evidently, φ0∈C1(W01,p(Ω)) and from (3.34) and (3.35), it follows that
(3.36)
φ
0
(
u
)
=
{
φ
(
u
)
if
∥
u
∥
⩽
r
,
ψ
(
u
)
if
∥
u
∥
⩾
2
1
/
p
r
,
K
φ
0
=
K
φ
⊆
B
¯
r
.
Moreover, by (3.36), it is clear that
C
k
(
φ
0
,
∞
)
=
C
k
(
ψ
,
∞
)
for all
k
∈
ℕ
0
,
and, since Kψ={0} and λ∉σ^(p), we have
C
k
(
φ
0
,
∞
)
=
C
k
(
ψ
,
0
)
for all
k
∈
ℕ
.
Thus, Cm(φ0,∞)≠0, see [32, Proposition 1.1]. ∎
Next, we turn our attention to the critical groups of φ at the origin. To compute them we only need a subcritical growth on f(z,⋅) and the behavior of f(z,⋅) near zero. So, we introduce the following weaker set of hypotheses on f(z,x).
Hypotheses 3.6.
f:Ω×ℝ→ℝ is a Carathéodory function with the following properties:
|f(z,x)|⩽a(z)(1+|x|r-1) for almost all z∈Ω and all x∈ℝ, with a∈L∞(Ω)+, p⩽r<p*.
There exist l∈ℕ,δ>0 and η∈L∞(Ω) such that
λ
^
l
(
2
)
⩽
η
(
z
)
for almost all
z
∈
Ω
,
η
≢
λ
^
l
(
2
)
,
η
(
z
)
x
2
⩽
f
(
z
,
x
)
x
⩽
λ
^
l
+
1
(
2
)
x
2
for almost all
z
∈
Ω
and all
|
x
|
⩽
δ
,
and for every x≠0, the second inequality is strict on a set of positive Lebesgue measure.
Proposition 3.7.
If Hypotheses 3.6 hold and the functional φ satisfies the C-condition, then Ck(φ,0)=δk,dlZ for all k∈N0 with dl=dimH¯l.
Proof.
We consider the C2-functional ψ^:H01(Ω)→ℝ defined by
ψ
^
(
u
)
=
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
(
z
,
u
)
𝑑
z
for all
u
∈
H
0
1
(
Ω
)
.
We set ψ=ψ^|W01,p(Ω) (recall that p>2).
Claim 3.
Ck(ψ,0)=δk,dlℤ for all k∈ℕ0.
To prove this claim, let ϑ∈(λ^l(2),λ^l+1(2)) and consider the C2-functional τ:H01(Ω)→ℝ defined by
τ
(
u
)
=
1
2
∥
D
u
∥
2
2
-
ϑ
2
∥
u
∥
2
2
for all
u
∈
H
0
1
(
Ω
)
.
We also consider the homotopy h(t,u) defined by
h
(
t
,
u
)
=
(
1
-
t
)
ψ
^
(
u
)
+
t
τ
(
u
)
for all
(
t
,
u
)
∈
[
0
,
1
]
×
H
0
1
(
Ω
)
.
First consider t∈(0,1]. Let u∈C01(Ω¯) with ∥u∥C01(Ω¯)⩽δ, where δ>0 is as in Hypothesis 3.6 (ii). Let 〈⋅,⋅〉0 denote the duality brackets for the pair (H-1(Ω),H01(Ω)). Then we have
(3.37)
〈
h
u
′
(
t
,
u
)
,
v
〉
=
(
1
-
t
)
〈
ψ
^
′
(
u
)
,
v
〉
0
+
t
〈
τ
′
(
u
)
,
v
〉
0
for all
v
∈
H
0
1
(
Ω
)
.
Recall that
H
¯
l
=
⊕
k
=
1
l
E
(
λ
^
k
(
2
)
)
,
H
^
l
+
1
=
H
¯
l
⊥
=
⊕
k
⩾
l
+
1
E
(
λ
^
k
(
2
)
)
and consider the orthogonal direct sum decomposition
H
0
1
(
Ω
)
=
H
¯
l
⊕
H
^
l
+
1
.
So, every u∈H01(Ω) admits a unique sum decomposition
u
=
u
¯
+
u
^
,
with
u
¯
∈
H
¯
l
,
u
^
∈
H
^
l
+
1
.
In (3.37) we choose v=u^-u¯. Exploiting the orthogonality of the component spaces, we have
(3.38)
〈
ψ
^
(
u
)
,
u
^
-
u
¯
〉
0
=
∥
D
u
^
∥
2
2
-
∥
D
u
¯
∥
2
2
-
∫
Ω
f
(
z
,
u
)
(
u
^
-
u
¯
)
𝑑
z
.
Hypothesis 3.6 (ii) implies that
(3.39)
η
(
z
)
⩽
f
(
z
,
x
)
x
⩽
λ
^
l
+
1
(
2
)
for almost all
z
∈
Ω
and all
0
<
|
x
|
⩽
δ
,
and the second inequality is, for every x≠0, strict on a set of positive Lebesgue measure. Set y=u^-u¯. Then, using (3.39), we have
f
(
z
,
u
)
(
u
^
-
u
¯
)
=
f
(
z
,
u
)
y
=
f
(
z
,
u
)
u
u
y
⩽
{
λ
^
l
+
1
(
2
)
(
u
^
2
-
u
¯
2
)
if
u
y
⩾
0
,
η
(
z
)
(
u
^
2
-
u
¯
2
)
if
u
y
<
0
(3.40)
⩽
λ
^
l
+
1
(
2
)
u
^
2
-
η
(
z
)
u
¯
2
for almost all
z
∈
Ω
.
Returning to (3.38) and using (3.40), we obtain (see Hypothesis 3.6 (ii) and (2.4))
(3.41)
〈
ψ
^
′
(
u
)
,
u
^
-
u
¯
〉
0
⩾
∥
D
u
^
∥
2
2
-
λ
^
l
+
1
(
2
)
∥
u
^
∥
2
2
-
[
∥
D
u
¯
∥
2
2
-
λ
^
l
(
2
)
∥
u
¯
∥
2
2
]
⩾
0
.
Also, using Lemma 2.2, for some c9>0, we have
(3.42)
〈
τ
′
(
u
)
,
u
^
-
u
¯
〉
0
=
∥
D
u
^
∥
2
2
-
ϑ
∥
u
^
∥
2
2
-
[
∥
D
u
¯
∥
2
2
-
ϑ
∥
u
¯
∥
2
2
]
⩾
c
9
∥
u
∥
2
.
So, if we use (3.41) and (3.42) in (3.37), then
〈
h
u
′
(
t
,
u
)
,
u
^
-
u
¯
〉
⩾
t
c
9
∥
u
∥
2
>
0
for all
t
∈
(
0
,
1
]
.
Standard regularity theory implies that
K
h
(
t
,
⋅
)
⊆
C
0
1
(
Ω
¯
)
for all
t
∈
[
0
,
1
]
.
Therefore, we infer that for all t∈(0,1], u=0 is isolated in Kh(t,⋅).
We have h(0,⋅)=ψ^(⋅). Next, we show that 0∈Kψ^ s isolated. Arguing by contradiction, suppose that we could find {un}n⩾1⊆H01(Ω) such that
(3.43)
u
n
→
0
in
H
1
(
Ω
)
and
ψ
^
′
(
u
n
)
=
0
for all
n
∈
ℕ
0
.
From the equation in (3.43), we have
(3.44)
-
Δ
u
n
(
z
)
=
f
(
z
,
u
n
(
z
)
)
for almost all
z
∈
Ω
,
u
n
|
∂
Ω
=
0
,
n
∈
ℕ
.
From (3.44) and standard regularity theory (see, for example, [16, pp. 737–738]), we can find α∈(0,1) and c10>0 such that
(3.45)
u
n
∈
C
0
1
,
α
(
Ω
¯
)
and
∥
u
n
∥
C
0
1
,
α
(
Ω
¯
)
⩽
c
10
for all
n
∈
ℕ
.
Exploiting the compact embedding of C01,α(Ω¯) into C1(Ω¯) and using (3.45) and (3.43), we obtain
u
n
→
0
in
C
0
1
(
Ω
¯
)
.
Therefore, we can find n0∈ℕ such that
|
u
n
(
z
)
|
⩽
δ
for all
n
⩾
n
0
and all
z
∈
Ω
¯
,
hence (see Hypothesis 3.6 (ii))
η
(
z
)
u
n
(
z
)
2
⩽
f
(
z
,
u
n
(
z
)
)
u
n
(
z
)
⩽
λ
^
l
+
1
(
2
)
u
n
(
z
)
2
for almost all
z
∈
Ω
and all
n
⩾
n
0
.
Then from (3.45) and the previous argument, we have
(3.46)
f
(
z
,
u
n
(
z
)
)
(
u
^
n
-
u
¯
n
)
(
z
)
⩽
λ
^
l
+
1
(
2
)
u
^
n
(
z
)
2
-
η
(
z
)
u
¯
n
(
z
)
2
for almost all
z
∈
Ω
and all
n
⩾
n
0
.
From (3.44) we have
〈
A
(
u
n
)
,
v
〉
=
∫
Ω
f
(
z
,
u
n
)
v
𝑑
z
for all
v
∈
H
0
1
(
Ω
)
.
Choosing v=u^n-u¯n∈H01(Ω) and using the orthogonality of the component spaces and (3.46), we obtain
∫
Ω
(
D
u
n
,
D
u
^
n
-
D
u
¯
n
)
ℝ
N
𝑑
z
=
∥
D
u
^
n
∥
2
2
-
∥
D
u
¯
n
∥
2
2
=
∫
Ω
f
(
z
,
u
n
)
(
u
^
n
-
u
¯
n
)
𝑑
z
⩽
∫
Ω
[
λ
^
l
+
1
(
2
)
u
^
n
2
-
η
(
z
)
u
¯
n
2
]
𝑑
z
.
Hence, by (2.4) and Lemma 2.2 (a),
0
⩽
∥
D
u
^
n
∥
2
2
-
λ
^
l
+
1
(
2
)
∥
u
^
n
∥
2
2
⩽
∥
D
u
^
n
∥
2
2
-
∫
Ω
η
(
z
)
u
¯
n
2
𝑑
z
⩽
-
c
11
∥
u
¯
n
∥
2
for all
n
⩾
n
0
,
for some c11>0. Therefore,
u
¯
n
=
0
and
u
^
n
∈
E
(
λ
^
l
+
1
(
2
)
)
for all
n
∈
ℕ
.
Then un=u^n for all n⩾n0, and the UCP implies that
(3.47)
u
n
(
z
)
≠
0
for almost all
z
∈
Ω
and all
n
⩾
n
0
.
From (3.44) and (3.47) we have (see Hypothesis 3.6 (ii))
λ
^
l
+
1
(
2
)
∥
u
n
∥
2
2
=
∫
Ω
f
(
z
,
u
n
)
u
n
𝑑
z
<
λ
^
l
+
1
(
2
)
∥
u
n
∥
2
2
for all
n
⩾
n
0
,
a contradiction. Therefore, 0∈Kψ^ is isolated and we can conclude that 0∈Kh(t,⋅) is isolated for all t∈[0,1].
So, [12, Theorem 5.2] implies that
C
k
(
ψ
^
,
0
)
=
C
k
(
τ
,
0
)
for all
k
∈
ℕ
0
,
and thus (see [23, Theorem 6.51])
C
k
(
ψ
^
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
Since W01,p(Ω) is dense in H01(Ω), it follows that (see [25] and [8, p. 14])
C
k
(
ψ
^
,
0
)
=
C
k
(
ψ
,
0
)
for all
k
∈
ℕ
0
,
and thus
(3.48)
C
k
(
ψ
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
We have
(3.49)
|
φ
(
u
)
-
ψ
(
u
)
|
⩽
1
p
∥
u
∥
p
and
〈
φ
′
(
u
)
-
ψ
′
(
u
)
,
v
〉
|
⩽
c
12
∥
u
∥
p
-
1
∥
v
∥
for all
v
∈
H
0
1
(
Ω
)
,
for some c12>0, which implies
(3.50)
∥
φ
′
(
u
)
-
ψ
′
(
u
)
∥
*
⩽
c
12
∥
u
∥
p
-
1
.
Then (3.49), (3.50) and the C1-continuity of the critical groups (see [12, Theorem 5.1]), imply that
C
k
(
φ
,
0
)
=
C
k
(
ψ
,
0
)
for all
k
∈
ℕ
0
,
and hence (see (3.48))
C
k
(
φ
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
This completes the proof. ∎
Now we are ready for the existence theorem.
Theorem 3.8.
If Hypotheses 3.1 hold, then problem (1.1) admits a nontrivial solution u0∈C01(Ω¯).
Proof.
As we have already mentioned, we can use two approaches.
In the first, we use Proposition 3.4 and have that Cm(φ,∞)≠0. So, there exists u0∈W01,p(Ω) such that
(3.51)
u
0
∈
K
φ
and
C
m
(
φ
,
u
0
)
≠
0
.
On the other hand, from Proposition 3.7, we have
(3.52)
C
k
(
φ
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
Recalling that dl≠m (see Hypothesis 3.1 (iv)) and comparing (3.51) and (3.52), we see that u0≠0.
In the second approach, we use Proposition 3.5. According to that result, we have Cm(φ0,∞)≠0. So, we can find u0∈W01,p(Ω) such that
(3.53)
u
0
∈
K
φ
0
and
C
m
(
φ
0
,
u
0
)
≠
0
.
Note that φ0|B¯r=φ|B¯r (see Proposition 3.5). So,
C
k
(
φ
0
,
0
)
=
C
k
(
φ
,
0
)
for all
k
∈
ℕ
0
,
and thus (see Proposition 3.7)
(3.54)
C
k
(
φ
0
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
Again, since dl≠m, from (3.53) and (3.54), it follows that (see Proposition 3.5)
u
0
≠
0
and
u
0
∈
K
φ
.
So, with both approaches we produced a nontrivial critical point u0 of the functional φ. Then u0 is a nontrivial solution of (1.1). Invoking [18, Theorem 7.1], we have u0∈L∞(Ω). So, we apply [20, Theorem 1] and conclude that u0∈C1(Ω¯). ∎
4 Multiple Nontrivial Solutions
In this section we strengthen the conclusions on the reaction term f(z,x) and prove a multiplicity theorem. More precisely, the new conditions on f(z,x) are the following.
Hypotheses 4.1.
f:Ω×ℝ→ℝ is a Carathéodory function with the following properties:
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
.
There exists an integer m⩾1 such that
lim
x
→
±
∞
f
(
z
,
x
)
|
x
|
p
-
2
x
=
λ
^
m
(
p
)
uniformly for almost all
z
∈
Ω
.
There exists τ∈(2,p) such that
0
<
β
0
⩽
lim inf
x
→
±
∞
f
(
z
,
x
)
x
-
p
F
(
z
,
x
)
|
x
|
τ
uniformly for almost all
z
∈
Ω
,
where F(z,x)=∫0xf(z,s)𝑑s.
There exist functions w±∈W1,p(Ω)∩C(Ω¯) and constants c±∈ℝ such that
w
-
(
z
)
⩽
c
-
<
0
<
c
+
⩽
w
+
(
z
)
for all
z
∈
Ω
¯
,
f
(
z
,
w
+
(
z
)
)
⩽
0
⩽
f
(
z
,
w
-
(
z
)
)
for almost all
z
∈
Ω
,
A
p
(
w
-
)
+
A
(
w
-
)
⩽
0
⩽
A
p
(
w
+
)
+
A
(
w
+
)
in
W
-
1
,
p
′
(
Ω
)
=
W
0
1
,
p
(
Ω
)
*
.
There exist an integer l⩾1 with dl≠m (dl=dimH¯l), δ>0 and η∈L∞(Ω) such that
λ
^
l
(
2
)
⩽
η
(
z
)
for almost all
z
∈
Ω
,
η
≢
λ
^
l
(
2
)
,
η
(
z
)
x
2
⩽
f
(
z
,
x
)
x
⩽
λ
^
l
+
1
(
2
)
x
2
for almost all
z
∈
Ω
and all
|
x
|
⩽
δ
,
and for x≠0, the second inequality is strict on a set of positive Lebesgue measure.
For every ρ>0, there exists ξ^ρ>0 such that for almost all z∈Ω, the function z↦f(z,x)+ξ^ρ|x|p-2x is nondecreasing on [-ρ,ρ].
Remark 4.2.
We see that in comparison to the Hypotheses 3.1, we have added Hypotheses 4.1 (iv), (vi). So, the problem remains resonant at both ±∞ and at zero. Hypothesis 3.1 (iv) is satisfied if, for example, we can find c-<0<c+ such that
f
(
z
,
c
+
)
⩽
0
⩽
f
(
z
,
c
-
)
for almost all
z
∈
Ω
.
Therefore, this hypothesis implies that near zero f(z,⋅) exhibits an oscillatory behavior.
First, we produce two constant sign solutions.
Proposition 4.3.
If Hypotheses 4.1 (i), (iv), (v), (vi) hold, then problem (1.1) admits two nontrivial smooth solutions of constant sign:
u
0
∈
int
C
+
,
with
u
0
(
z
)
<
w
+
(
z
)
for all
z
∈
Ω
¯
,
v
0
∈
-
int
C
+
,
with
w
-
(
z
)
<
v
0
(
z
)
for all
z
∈
Ω
¯
.
Proof.
First, we produce the positive solution.
We introduce the following Carathéodory function:
(4.1)
f
^
+
(
z
,
x
)
=
{
0
if
x
<
0
,
f
(
z
,
x
)
if
0
⩽
x
⩽
w
+
(
z
)
,
f
(
z
,
w
+
(
z
)
)
if
w
+
(
z
)
<
x
.
We set F^+(z,x)=∫0xf^+(z,s)𝑑s and consider the C1-functional φ^+:W01,p(Ω)→ℝ defined by
φ
^
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
^
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From (4.1) it is clear that φ^+ is coercive. Also, using the Sobolev embedding theorem, we see that φ^+ is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find u0∈W01,p(Ω) such that
(4.2)
φ
^
+
(
u
0
)
=
inf
{
φ
^
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
}
.
From (4.2) we have φ^+′(u0)=0, and hence
(4.3)
〈
A
p
(
u
0
)
,
h
〉
+
〈
A
(
u
0
)
,
h
〉
=
∫
Ω
f
^
+
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
.
In (4.3) we first choose h=-u0-∈W01,p(Ω). Then ∥Du0-∥pp+∥Du0-∥22=0 (see (4.1)), and thus u0⩾0. Also, in (4.3) we choose h=(u0-w+)+∈W01,p(Ω). Then, by (4.1) and Hypothesis 4.1 (iv),
∫
Ω
|
D
u
0
|
p
-
2
(
D
u
0
,
D
(
u
0
-
w
+
)
+
)
ℝ
N
d
z
+
∫
Ω
(
D
u
0
,
D
(
u
0
-
w
+
)
+
)
ℝ
N
d
z
=
∫
Ω
f
(
z
,
w
+
)
(
u
0
-
w
+
)
+
𝑑
z
⩽
∫
Ω
|
D
w
+
|
p
-
2
(
D
w
+
,
D
(
u
0
-
w
+
)
+
)
ℝ
N
d
z
+
∫
Ω
(
D
w
+
,
D
(
u
0
-
w
+
)
+
)
ℝ
N
d
z
.
Thus,
∫
Ω
(
|
D
u
0
|
p
-
2
D
u
0
-
|
D
w
+
|
p
-
2
D
w
+
,
D
(
u
0
-
w
+
)
+
)
ℝ
N
𝑑
z
+
∥
D
(
u
0
-
w
+
)
+
∥
2
2
⩽
0
,
and hence u0⩽w+.
So, we have proved that
u
0
∈
[
0
,
w
+
]
=
{
y
∈
W
0
1
,
p
(
Ω
)
:
0
⩽
y
(
z
)
⩽
w
+
(
z
)
for almost all
z
∈
Ω
}
.
Then, on account of (4.1), equation (4.3) becomes
〈
A
p
(
u
0
)
,
h
〉
+
〈
A
(
u
0
)
,
h
〉
=
∫
Ω
f
(
z
,
u
0
)
h
𝑑
z
for all
h
∈
W
0
1
,
p
(
Ω
)
,
which implies
(4.4)
-
Δ
p
u
0
(
z
)
-
Δ
u
0
(
z
)
=
f
(
z
,
u
0
(
z
)
)
for almost all
z
∈
Ω
,
u
0
|
∂
Ω
=
0
,
and hence u0∈C+ (by the nonlinear regularity theory, see [20]).
Since p>2, given ϵ>0, we can find δ0∈(0,min{δ,C+}) (δ>0 as in Hypothesis 4.1 (v)) such that
(4.5)
1
p
|
y
|
p
⩽
ϵ
2
|
y
|
2
for all
y
∈
ℝ
N
with
|
y
|
⩽
δ
0
.
Recall that u^1(2)∈intC+. So, we can find small t∈(0,1) such that
∥
t
u
^
1
(
2
)
∥
C
0
1
(
Ω
¯
)
⩽
δ
0
.
By (4.5), (4.1) and Hypothesis 4.1 (v), since δ0⩽δ, we have
φ
^
+
(
t
u
^
1
(
2
)
)
⩽
ϵ
+
1
2
t
2
∥
D
u
^
1
(
2
)
∥
2
2
-
1
2
t
2
∫
Ω
η
(
z
)
u
^
1
(
2
)
2
𝑑
z
=
t
2
[
ϵ
2
λ
^
1
(
2
)
∥
u
^
1
(
2
)
∥
2
2
-
1
2
∫
Ω
(
η
(
z
)
-
λ
^
1
(
2
)
)
u
^
1
(
2
)
2
d
z
)
]
<
0
,
by choosing ϵ>0 small enough (see Lemma 2.2 (b)). Then φ^+(u0)<0=φ^+(0) (see (4.2)), and hence u0≠0.
Let ρ=∥u0∥∞ and let ξ^ρ>0 be as postulated by Hypothesis 4.1 (vi). Then, by (4.4), we have
(4.6)
Δ
p
u
0
(
z
)
+
Δ
u
0
(
z
)
⩽
ξ
^
ρ
u
0
(
z
)
p
-
1
for almost all
z
∈
Ω
.
Let V(y)=|y|p-2y+y for all y∈ℝN. Evidently,
div
(
V
(
D
u
)
)
=
Δ
p
u
+
Δ
u
for all
u
∈
W
0
1
,
p
(
Ω
)
.
We have V∈C1(ℝN,ℝN) and
∇
V
(
y
)
=
|
y
|
p
-
2
[
I
+
(
p
-
2
)
y
⊕
y
|
y
|
2
]
+
I
,
which implies
(4.7)
(
∇
V
(
y
)
ξ
,
ξ
)
ℝ
N
⩾
|
ξ
|
2
for all
y
∈
ℝ
N
and all
ξ
∈
ℝ
N
.
Then (4.7), (4.6) and the tangency principle of Pucci and Serrin [33, Theorem 2.5.2] imply that
0
<
u
0
(
z
)
for all
z
∈
Ω
.
Next, using the boundary point lemma (see [33, Theorem 5.5.1]), we obtain
(4.8)
u
0
∈
int
C
+
.
Also, Hypothesis 4.1 (iv) implies
(4.9)
A
p
(
u
0
)
+
A
(
u
0
)
-
N
f
(
u
0
)
=
0
⩽
A
p
(
w
+
)
+
A
(
w
+
)
-
N
f
(
w
+
)
in
W
-
1
,
p
′
(
Ω
)
.
So, once more (4.7), (4.9) and the tangency principle of Pucci and Serrin [33, Theorem 2.5.2], imply that
u
0
(
z
)
<
w
+
(
z
)
for all
z
∈
Ω
,
and, by Hypothesis 4.1 (iv), we have
u
0
(
z
)
<
w
+
(
z
)
for all
z
∈
Ω
¯
.
Similarly, to produce the negative solution, we introduce the Carathéodory function
f
^
-
(
z
,
x
)
=
{
f
(
z
,
w
-
(
z
)
)
if
x
<
w
-
(
z
)
,
f
(
z
,
x
)
if
w
-
(
z
)
⩽
x
⩽
,
0
0
if
0
<
x
.
We set F^-(z,x)=∫0xf^-(z,s)𝑑s and consider the C1-functional φ^-:W01,p(Ω)→ℝ defined by
φ
^
-
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
^
-
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Working with φ^- and using (4.8), we produce a solution of (1.1), v0∈W01,p(Ω), such that
v
0
∈
-
int
C
+
,
w
-
(
z
)
<
v
0
(
z
)
for all
z
∈
Ω
¯
.
This completes the proof. ∎
In fact, we can show that we have extremal constant sign solutions in the order intervals [0,w+] and [w-,0], that is, we can show that there is a smallest positive solution u*∈intC+ in [0,w+] and a biggest negative solution v*∈-intC+ in [w-,0].
Proposition 4.4.
If Hypotheses 4.1 (i), (iv), (v), (vi) hold, then problem (1.1) admits a smallest positive solution u*∈intC+ in [0,w+] and a biggest negative solution v*∈-intC+ in [w-,0].
Proof.
First we produce the smallest positive solution in [0,w+]. Let S^+ be the set of positive solutions of problem (1.1) in the order interval [0,w+]. From Proposition 4.3 and its proof, we have
S
^
+
≠
∅
and
S
^
+
⊆
[
0
,
w
+
]
∩
int
C
+
.
Invoking [17, Lemma 3.10, p. 178], we infer that we can find {un}n⩾1⊆S^+ such that infS^+=infn⩾1un.
We have
(4.10)
A
p
(
u
n
)
+
A
(
u
n
)
=
N
f
(
u
n
)
,
0
⩽
u
n
⩽
w
+
for all
n
∈
ℕ
,
which implies that {un}n⩾1⊆W01,p(Ω) is bounded. So, we may assume that
(4.11)
u
n
→
𝑤
u
*
in
W
0
1
,
p
(
Ω
)
and
u
n
→
u
*
in
L
p
(
Ω
)
as
n
→
∞
.
On (4.10) we act with un-u*∈W01,p(Ω), pass to the limit as n→∞ and use (4.11). Then
lim
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
*
〉
+
〈
A
(
u
n
)
,
u
n
-
u
*
〉
]
=
0
,
and since A is monotone, we have
lim sup
n
→
∞
[
〈
A
p
(
u
n
)
,
u
n
-
u
*
〉
+
〈
A
(
u
*
)
,
u
n
-
u
*
〉
]
⩽
0
.
Thus, by (4.11), lim supn→∞〈Ap(un),un-u*〉⩽0, which implies (see Proposition 2.3)
(4.12)
u
n
→
u
*
in
W
0
1
,
p
(
Ω
)
.
Passing to the limit as n→∞ in (4.10) and using (4.12), we obtain
A
p
(
u
*
)
+
A
(
u
*
)
=
N
f
(
u
*
)
,
0
⩽
u
*
⩽
w
+
.
Hence,
-
Δ
p
u
*
(
z
)
-
Δ
u
*
(
z
)
=
f
(
z
,
u
*
(
z
)
)
for almost all
z
∈
Ω
,
u
*
|
∂
Ω
=
0
,
0
⩽
u
*
⩽
w
+
.
Then u*∈C+ (by the nonlinear regularity theory, see [20]) is a nonnegative solution of (1.1). If we can show that u*≠0, then u*∈S^+ and u*=infS^+.
To this end, we proceed as follows. Hypotheses 4.1 (i), (v) imply that we can find c13>0 such that
(4.13)
f
(
z
,
x
)
⩾
η
(
z
)
x
-
c
13
x
p
-
1
for almost all
z
∈
Ω
and all
0
⩽
x
⩽
w
+
(
z
)
.
Let g:Ω×ℝ→ℝ be the Carathéodory function defined by
(4.14)
g
(
z
,
x
)
=
{
0
if
x
<
0
,
η
(
z
)
x
-
c
13
x
p
-
1
if
0
⩽
x
⩽
w
+
(
z
)
,
η
(
z
)
w
+
(
z
)
-
c
13
w
+
(
z
)
p
-
1
if
w
+
(
z
)
<
x
.
We consider the auxiliary Dirichlet problem
(4.15)
-
Δ
p
u
(
z
)
-
Δ
u
(
z
)
=
g
(
z
,
u
(
z
)
)
in
Ω
,
u
|
∂
Ω
=
0
.
We claim that this problem has a unique solution u¯∈intC+. First, we show the existence of a nontrivial solution. So, let ψ+:W01,p(Ω)→ℝ be the energy (Euler) functional for problem (4.15) defined by
ψ
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
G
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
,
where G(z,x)=∫0xg(z,s)𝑑s. Evidently, ψ+ is coercive (see (4.14)) and sequentially weakly lower semicontinuous. So, we can find u¯∈W01,p(Ω) such that
ψ
+
(
u
¯
)
=
inf
{
ψ
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
}
.
As in the proof of Proposition 4.3, using Hypothesis 4.1 (v), we have (see (4.14))
ψ
+
(
u
¯
)
<
0
=
ψ
+
(
0
)
and
u
¯
∈
[
0
,
w
+
]
,
hence u¯∈Kψ+⊆[0,w+]∩intC+.
Next, we show that this solution is unique. For this purpose, we consider the integral functional j:L1(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
.
By [7, Lemma 4] and [13, Lemma 1], we have that j(⋅) is convex. Suppose that y¯∈W01,p(Ω) is another nontrivial solution of (4.15). Then again we have y¯∈[0,w+]∩intC+. Let domj={u∈L1(Ω):j(u)<+∞} (the effective domain of j). For every h∈C01(Ω¯), we have
u
¯
2
+
t
h
∈
dom
j
and
y
¯
2
+
t
h
∈
dom
j
for
|
t
|
⩽
1
small
.
Then we can easily see that j(⋅) is Gâteaux differentiable at u¯2 and at y¯2 in the direction h. Moreover, using the nonlinear Green’s identity (see, for example, [16, p. 211]), we have
j
′
(
u
¯
2
)
(
h
)
=
1
2
∫
Ω
-
Δ
p
u
¯
-
Δ
u
¯
u
¯
h
𝑑
z
=
1
2
∫
Ω
[
η
(
z
)
-
c
13
u
¯
p
-
2
]
h
𝑑
z
,
j
′
(
y
¯
2
)
(
h
)
=
1
2
∫
Ω
-
Δ
p
y
¯
-
Δ
y
¯
y
¯
h
𝑑
z
=
1
2
∫
Ω
[
η
(
z
)
-
c
13
y
¯
p
-
2
]
h
𝑑
z
,
see (4.15) and (4.14). The convexity of j(⋅) implies the monotonicity of j′(⋅). Hence,
0
⩽
∫
Ω
[
y
¯
p
-
2
-
u
¯
p
-
2
]
(
u
¯
2
-
y
¯
2
)
𝑑
z
⟹
u
¯
=
y
¯
.
This proves the uniqueness of the nontrivial solution u¯∈[0,w+]∩intC+ of the auxiliary problem (4.15).
Claim 4.
u¯⩽u for all u∈S^+.
Let u∈S^+ and consider the Carathéodory function k:Ω×ℝ→ℝ defined by
(4.16)
k
(
z
,
x
)
=
{
0
if
x
<
0
,
η
(
z
)
x
-
c
13
x
p
-
1
if
0
⩽
x
⩽
u
(
z
)
,
η
(
z
)
u
(
z
)
-
c
13
u
(
z
)
p
-
1
if
u
(
z
)
<
x
.
We set K(z,x)=∫0xk(z,s)𝑑s and consider the C1-functional ψ^+:W01,p(Ω)→ℝ defined by
ψ
^
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
K
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Again, ψ^+ is coercive (see (4.16)) and sequentially weakly lower semicontinuous. So, we can find u~∈W01,p(Ω) such that
(4.17)
ψ
^
+
(
u
~
)
=
inf
{
ψ
^
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
}
.
Let t∈(0,1) be small such that tu^1(2)⩽u (see [22, Proposition 2.1] and recall that u∈intC+). Then, by taking t∈(0,1) even smaller if necessary and using Hypothesis 4.1 (v), we have ψ^+(tu^1(2))<0, which implies ψ^+(u~)<ψ^+(0)=0, hence u~≠0. Using (4.13) and the fact that u∈S^+, we can show that Kψ^+⊆[0,u]. From (4.17) we have u~∈Kψ^+∖{0}⊆[0,u]∖{0}, which implies (see (4.16) and recall that u¯ is the unique solution of (4.15)) u~=u¯. Thus,
u
¯
⩽
u
for all
u
∈
S
^
+
.
This proves the claim.
On account of Claim 4, we have u¯⩽u*, and so
u
*
∈
S
^
+
,
u
*
=
inf
S
^
+
.
Similarly, if S^- is the set of negative solutions of (1.1) in [w-,0], then
S
^
-
≠
∅
and
S
^
-
⊆
[
w
-
,
0
]
∩
(
-
int
C
+
)
(see Proposition 4.3 and its proof). Reasoning as above, we can show that there exists v*∈[w-,0]∩(-intC+) which is the biggest negative solution of (1.1) in [w-,0]. ∎
Using these extremal constant sign solutions of (1.1), we can generate a nodal (that is, sign changing) solution. To do this, we need a slightly stronger condition on f(z,⋅) near zero (see Hypothesis 4.1 (v)). The new hypotheses on the reaction f(z,x) are the following.
Hypotheses 4.5.
The conditions on the Carathéodory function f:Ω×ℝ→ℝ are the same as in Hypotheses 4.1, the only difference being that here we have l⩾2.
Proposition 4.6.
If Hypotheses 4.5 (i), (iv), (v), (vi) hold, then problem (1.1) admits a nodal solution y0 in [v*,u*]∩C01(Ω¯).
Proof.
Let u*∈intC+ and v*∈-intC+ be the two extremal constant sign solutions of (1.1) produced in Proposition 4.4. Let e:Ω×ℝ→ℝ be the Carathéodory function defined by
(4.18)
e
(
z
,
x
)
=
{
f
(
z
,
v
*
(
z
)
)
if
x
<
v
*
(
z
)
,
f
(
z
,
x
)
if
v
*
(
z
)
⩽
x
⩽
u
*
(
z
)
,
f
(
z
,
u
*
(
z
)
)
if
u
*
(
z
)
<
x
.
We set E(z,x)=∫0xe(z,s)𝑑s, and consider the C1-functional τ:W01,p(Ω)→ℝ defined by
τ
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
E
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Also, we consider the positive and negative truncations of e(z,⋅), namely the Carathéodory functions
e
±
(
z
,
x
)
=
e
(
z
,
±
x
±
)
.
We set E±(z,x)=∫0xe±(z,s)𝑑s and consider the C1-functionals τ±:W01,p(Ω)→ℝ defined by
τ
±
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
E
±
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
As before (see the proof of Proposition 4.3), using (4.18), we can show that
K
τ
⊆
[
v
*
,
u
*
]
,
K
τ
+
⊆
[
0
,
u
*
]
,
K
τ
-
⊆
[
v
*
,
0
]
.
The extremality of u*∈intC+ and v*∈-intC+ implies that
(4.19)
K
τ
⊆
[
v
*
,
u
*
]
,
K
τ
+
=
{
0
,
u
*
}
,
K
τ
-
=
{
0
,
v
*
}
.
Claim 5.
u*∈intC+ and v*∈-intC+ are local minimizers of τ.
The functional τ+ is coercive (see (4.18)) and sequentially weakly lower semicontinuous. So, we can find u^*∈W01,p(Ω) such that
τ
+
(
u
^
*
)
=
inf
{
τ
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
}
.
As in the proof of Proposition 4.4 (see the part of the proof after (4.17)), we have τ+(u^*)<0=τ+(0), hence u^*≠0. Since u^*∈Kτ+={0,u*}, it follows that u^*=u*∈intC+ (see (4.19)). Note that τ|C+=τ+|C+, which implies that u*∈intC+ is a local C01(Ω¯)-minimizer of τ. Hence, by Proposition 2.4, u*∈intC+ is a local W01,p(Ω)-minimizer of τ. Similar arguments apply for v*∈-intC+, using this time the functional τ-. This proves Claim 5.
We may assume that
τ
(
v
*
)
⩽
τ
(
u
*
)
.
The reasoning is similar if the opposite inequality holds. Also, we may assume that Kτ is finite. Indeed, if Kτ is infinite, then on account of (4.19), we see that we already have an infinity of nodal solutions, which belong to C01(Ω¯) (nonlinear regularity theory). Then Claim 5 implies that we can find ρ∈(0,1) small such that
(4.20)
τ
(
v
*
)
⩽
τ
(
u
*
)
<
inf
{
τ
(
u
)
:
∥
u
-
u
*
∥
=
ρ
}
=
m
ρ
,
∥
v
*
-
u
*
∥
>
ρ
(see the proof of [1, Proposition 29]). The functional τ(⋅) is coercive (see (4.18)) and so τ(⋅) satisfies the C-condition (see [31]). Therefore, from (4.20), we see that we can apply Theorem 2.1 (the mountain pass theorem). So, we can find y0∈W01,p(Ω) such that
(4.21)
y
0
∈
K
τ
and
m
ρ
⩽
τ
(
y
0
)
.
From (4.19), (4.20), (4.21) and the nonlinear regularity theory (see [20]), we infer that
y
0
∈
[
v
*
,
u
*
]
∩
C
0
1
(
Ω
¯
)
,
y
0
∉
{
v
*
,
u
*
}
.
Also, from [23, Corollary 6.81], we have
(4.22)
C
1
(
τ
,
y
0
)
≠
0
.
Let f^:Ω×ℝ→ℝ be the Carathéodory function defined by
f
^
(
z
,
x
)
=
{
f
(
z
,
w
-
(
z
)
)
if
x
<
w
-
(
z
)
,
f
(
z
,
x
)
if
w
-
(
z
)
⩽
x
⩽
w
+
(
z
)
,
f
(
z
,
w
+
(
z
)
)
if
w
+
(
z
)
<
x
.
We set F^(t,x)=∫0xf^(z,s)𝑑s and consider the C1-functional φ^:W01,p(Ω)→ℝ defined by
φ
^
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
F
^
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From Proposition 3.7, we know that (recall that dl=dimH¯l)
(4.23)
C
k
(
φ
^
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
Claim 6.
Ck(τ,0)=δk,dlℤ for all k∈ℕ0.
We consider the homotopy h(t,u) defined by
h
(
t
,
u
)
=
(
1
-
t
)
φ
^
(
u
)
+
t
τ
(
u
)
for all
(
t
,
u
)
∈
[
0
,
1
]
×
W
0
1
,
p
(
Ω
)
.
Suppose we can find {tn}n⩾1⊆[0,1] and {un}n⩾1⊆W01,p(Ω) such that
(4.24)
t
n
→
t
,
u
n
→
0
in
W
0
1
,
p
(
Ω
)
,
h
u
′
(
t
n
,
u
n
)
=
0
for all
n
∈
ℕ
.
From the equality in (4.24) we have
A
p
(
u
n
)
+
A
(
u
n
)
=
(
1
-
t
n
)
N
f
^
(
u
n
)
+
t
n
N
τ
(
u
n
)
for all
n
∈
ℕ
,
which implies
(4.25)
-
Δ
p
u
n
(
z
)
-
Δ
u
n
(
z
)
=
(
1
-
t
n
)
f
^
(
z
,
u
n
(
z
)
)
+
t
n
e
(
z
,
u
n
(
z
)
)
for almost all
z
∈
Ω
,
u
n
|
∂
Ω
=
0
.
By (4.24), (4.25) and [18, Theorem 7.1] (see also [23, Corollary 8.7]), we can find c14>0 such that
(4.26)
∥
u
n
∥
∞
⩽
c
14
for all
n
∈
ℕ
.
Then, from (4.26) and [20, Theorem 1], we infer that there exist α∈(0,1) and c15>0 such that
u
n
∈
C
0
1
,
α
(
Ω
¯
)
,
∥
u
n
∥
C
0
1
,
α
(
Ω
¯
)
⩽
c
15
for all
n
∈
ℕ
.
Since C01,α(Ω¯) is embedding compactly in C01(Ω¯), it follows that (see (4.24))
u
n
→
0
in
C
0
1
(
Ω
¯
)
,
which implies
u
n
∈
[
v
*
,
u
*
]
for all
n
⩾
n
0
,
and thus (see (4.19)) {un}n⩾n0⊆Kτ, a contradiction to our hypothesis that Kτ is finite.
So, (4.24) cannot happen and this shows that 0∈Kh(t,⋅) is isolated uniformly in t∈[0,1]. Hence, the homotopy invariance of critical groups, [12, Theorem 5.2], implies that
C
k
(
h
(
0
,
⋅
)
,
0
)
=
C
k
(
h
(
1
,
⋅
)
,
0
)
for all
k
∈
ℕ
0
,
and thus
C
k
(
φ
^
,
0
)
=
C
k
(
τ
,
0
)
for all
k
∈
ℕ
0
.
Therefore, by (4.23),
C
k
(
τ
,
0
)
=
δ
k
,
d
l
ℤ
for all
k
∈
ℕ
0
.
This proves Claim 6.
Since l⩾2 (see Hypotheses 4.5), we have dl⩾2. So, from Claim 6 and (4.22), it follows that y0≠0. Therefore, y0∈[v*,u*]∩C01(Ω¯)∖{0} is nodal. ∎
So far we have not used the asymptotic conditions at ±∞ (that is, Hypotheses 4.5 (ii), (iii)). Next, by using them, we will generate two more nontrivial smooth solutions of constant sign, for a total of five nontrivial smooth solutions all with sign information and ordered.
Theorem 4.7.
If Hypotheses 4.5 hold, then problem (1.1) admits the following five nontrivial smooth solutions:
u
0
,
u
^
∈
int
C
+
,
u
^
-
u
0
∈
C
+
∖
{
0
}
,
v
0
,
v
^
∈
-
int
C
+
,
v
0
-
v
^
∈
C
+
∖
{
0
}
,
y
0
∈
[
v
0
,
u
0
]
∩
C
0
1
(
Ω
¯
)
(nodal).
Proof.
Propositions 4.3 and 4.6 provide the following three nontrivial smooth solutions:
u
0
∈
[
0
,
w
+
]
∩
int
C
+
,
with
(
w
+
-
u
0
)
(
z
)
>
0
for all
z
∈
Ω
¯
,
v
0
∈
[
w
-
,
0
]
∩
(
-
int
C
+
)
,
with
(
u
0
-
w
-
)
(
z
)
>
0
for all
z
∈
Ω
¯
,
y
0
∈
[
v
0
,
u
0
]
∩
C
0
1
(
Ω
¯
)
(nodal).
On account of Proposition 4.4, we may assume that u0 and v0 are extremal constant sign solutions (that is, u0=u* and v0=v*).
We consider the Carathéodory function γ+:Ω×ℝ→ℝ defined by
(4.27)
γ
+
(
z
,
x
)
=
{
f
(
z
,
u
0
(
z
)
)
if
x
⩽
u
0
(
z
)
,
f
(
z
,
x
)
if
u
0
(
z
)
<
x
,
and set Γ+(z,x)=∫0xγ+(z,s)𝑑s. We consider the C1-functional σ+:W01,p(Ω)→ℝ defined by
σ
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
Γ
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
Using (4.27), we can easily show that
(4.28)
K
σ
+
⊆
[
u
0
]
=
{
u
∈
W
0
1
,
p
(
Ω
)
:
u
0
(
z
)
⩽
u
(
z
)
for almost all
z
∈
Ω
}
.
Note that u0∈Kσ+. We may assume that
(4.29)
K
σ
+
∩
[
u
0
,
w
+
]
=
{
u
0
}
.
Otherwise, we already have a second positive solution u^⩾u0,u^≠u0,u^∈C01(Ω¯). Consider the following Carathéodory function:
(4.30)
γ
^
+
(
z
,
x
)
=
{
γ
+
(
z
,
x
)
if
x
⩽
w
+
(
z
)
,
γ
+
(
z
,
w
+
(
z
)
)
if
w
+
(
z
)
<
x
.
We set Γ^+(z,x)=∫0xγ^+(z,s)𝑑s and consider the C1-functional σ^+:W01,p(Ω)→ℝ defined by
σ
^
+
(
u
)
=
1
p
∥
D
u
∥
p
p
+
1
2
∥
D
u
∥
2
2
-
∫
Ω
Γ
^
+
(
z
,
u
)
𝑑
z
for all
u
∈
W
0
1
,
p
(
Ω
)
.
From (4.30) it is clear that σ^+ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u~0∈W01,p(Ω) such that
(4.31)
σ
^
+
(
u
~
0
)
=
inf
{
σ
^
+
(
u
)
:
u
∈
W
0
1
,
p
(
Ω
)
}
.
Using (4.30), we can show that (see also (4.28))
(4.32)
K
σ
^
+
⊆
[
u
0
,
w
+
]
.
Then (4.29), (4.31) and (4.32) imply that
(4.33)
u
~
0
=
u
0
∈
[
0
,
w
+
]
,
(
w
+
-
u
0
)
(
z
)
>
0
for all
z
∈
Ω
¯
.
From (4.30) we see that σ+|[0,w+]=σ^+|[0,w+], which, in view of (4.33), implies that u0 is a local C01(Ω¯)-minimizer of σ+. Hence, u0 is a local W01,p(Ω)-minimizer of σ+ (see Proposition 2.4).
Because of (4.28), we see that we may assume that Kσ+ is finite or otherwise we already have infinite positive and smooth (by the nonlinear regularity theory) solutions of (1.1), all bigger than u0. Hence, we can find small ρ∈(0,1) such that
(4.34)
σ
+
(
u
0
)
<
inf
{
σ
+
(
u
)
:
∥
u
-
u
0
∥
=
ρ
}
=
m
ρ
+
.
Reasoning as in the proof of Proposition 3.3, we can establish that
(4.35)
σ
+
satisfies the C-condition
.
Note that in this case, due to (4.27), for any Cerami sequence {un}n⩾1⊆W01,p(Ω), we have automatically that {un-}n⩾1⊆W01,p(Ω) is bounded.
Hypotheses 4.5 (i), (ii) imply that we can find ϑ>λ^m(p) and c16>0 such that
(4.36)
F
(
z
,
x
)
⩽
ϑ
p
x
p
+
c
16
for almost all
z
∈
Ω
and all
x
⩾
0
.
Since u^1(p)∈intC+, we can find t⩾1 big such that tu^1(p)⩾u0 (see [22, Proposition 2.1]). Then (see (4.36) and recall that ∥u^1(p)∥p=1)
(4.37)
σ
(
t
u
^
1
(
p
)
)
⩽
t
p
p
λ
^
1
(
p
)
+
t
2
2
∥
D
u
^
1
(
p
)
∥
2
2
-
t
p
p
ϑ
+
c
17
=
t
p
p
[
λ
^
1
(
p
)
-
ϑ
]
+
t
2
2
∥
D
u
^
1
(
p
)
∥
2
2
+
c
17
,
for some c17>0. Since ϑ>λ^1(p) and p>2, from (4.37), it follows that
(4.38)
σ
(
t
u
^
1
(
p
)
)
→
-
∞
as
t
→
+
∞
.
Then (4.34), (4.35) and (4.38) permit the use of Theorem 2.1 (the mountain pass theorem). So, we can find u^∈W01,p(Ω) such that
(4.39)
u
^
∈
K
σ
+
and
m
ρ
+
⩽
σ
+
(
u
^
)
.
From (4.27), (4.28), (4.34) and (4.39), it follows that u0⩽u^, u^≠u0, and u^∈intC+ is a solution of (1.1).
Similarly, by working with v0∈[w-,0]∩(-intC+) on the negative semiaxis as above, we produce v^∈-intC+, v^⩽v0, v^≠v0, a second negative solution for problem (1.1). ∎
and
Dušan D. Repovš
