Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 249, pp. 1–13.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
MULTIPLICITY OF SOLUTIONS FOR QUASILINEAR
EQUATIONS INVOLVING CRITICAL ORLICZ-SOBOLEV
NONLINEAR TERMS
JEFFERSON A. SANTOS
Abstract. In this work, we study the existence and multiplicity of solutions
for a class of problems involving the φ-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. Our main tool is the
variational method combined with the genus theory for even functionals.
1. Introduction
In this article, we consider the existence and multiplicity of solutions for the
quasilinear problem
− div φ(|∇u|)∇u = λφ∗ (|u|)u + f (x, u), in Ω
(1.1)
u = 0, on ∂Ω
where Ω ⊂ RN is a bounded domain in RN with smooth boundary, λ is a positive
parameter and φ : (0, +∞) → R is a continuous function satisfying
(φ(t)t)0 > 0
∀t > 0.
(1.2)
There exist l, m ∈ (1, N ) such that
l≤
φ(|t|)t2
≤m
Φ(t)
∀t 6= 0,
(1.3)
where
Z
|t|
mN
lN
, m∗ =
.
N
−
l
N
−m
0
Moreover, φ∗ (t)t is such that Sobolev conjugate function Φ∗ of Φ is its primitive;
R |t|
that is, Φ∗ (t) = 0 φ∗ (s)sds.
Related to function f : Ω × R → R, we assume the following:
(F1) f ∈ C(Ω × R, R) is odd with respect t and
f (x, t) = o φ(|t|)|t| , as |t| → 0 uniformly in x;
f (x, t) = o φ∗ (|t|)|t| , as |t| → +∞ uniformly in x;
Φ(t) =
φ(s)s ds,
l ≤ m < l∗ ,
l∗ =
2000 Mathematics Subject Classification. 35J62, 35B33, 35A15.
Key words and phrases. Quasilinear elliptic equations; critical growth; variational methods.
c
2013
Texas State University - San Marcos.
Submitted October 8, 2013. Published November 20, 2013.
Partially supported supported by PROCAD/CAPES-Brazil.
1
2
J. A. SANTOS
EJDE-2013/249
(F2) There is θ ∈ (m, l∗ ) such that F (x, t) ≤ θ1 f (x, t)t, for all t > 0 and a.e. in
Rt
Ω, where F (x, t) = 0 f (x, s)ds.
Problem (1.1) associated with nonhomogenous nonlinear Φ arises in various fields
of physics [16]:
(i) in nonlinear elasticity, Φ(t) = (1 + |t|2 )γ − 1 for γ ∈ (1, NN−2 ).
(ii) in plasticity,
Φ(t) = |t|p ln(1 + |t|) for 1 < p0 < p < N − 1 with p0 =
√
−1+ 1+4N
.
2
Rt
(iii) in generalized Newtonian fluids, Φ(t) = 0 s1−α (sinh−1 s)β ds, 0 ≤ α ≤ 1,
β > 0.
Our main result reads as follows.
Theorem 1.1. Assume that (1.2), (1.3), (F1) and (F2) are satisfied. Then, there
exist a sequence {λk } ⊂ (0, +∞) with λk+1 < λk , such that, for λ ∈ (λk+1 , λk ),
problem (1.1) has at least k pairs of nontrivial solutions.
The main difficulty to prove Theorem 1.1 is related to the fact that the nonlinearity f has a critical growth. In this case, it is not clear that functional energy
associated with (1.1) satisfies the well known (PS) condition, once that the embedding W 1,Φ (Ω) ,→ LΦ∗ (Ω) is not compact. To overcome this difficulty, we use
a version of the concentration compactness lemma due to Lions for Orlicz-Sobolev
space found in Fukagai, Ito and Narukawa [14]. We would like to mention that
Theorem 1.1 improves the main result found in [25].
We cite the papers of Alves and Barreiro [3], Alves, Gonçalves and Santos
[4], Bonano, Bisci and Radulescu [5], Cerny [7], Clément, Garcia-Huidobro and
Manásevich [9], Donaldson [10], Fuchs and Li [12], Fuchs and Osmolovski [13], Fukagai, Ito and Narukawa [14, 15], Gossez [17], Mihailescu and Raduslescu [19, 20],
Mihailescu and Repovs [21], Pohozaev [22] and references therein, where quasilinear
problems like (1.1) have been considered in bounded and unbounded domains of
RN . In some those papers, the authors have mentioned that this class of problem
arises in applications, such as, nonlinear elasticity, plasticity and non-Newtonian
fluids.
This paper is organized as follows: In Section 2, we collect some preliminaries on
Orlicz-Sobolev spaces that will be used throughout the paper, which can be found
in [1], [2], [11] and [23]. In Section 3, we recall an abstract theorem involving genus
theory that will use in the proof of Theorem 1.1 and prove some technical lemmas,
and then we prove Theorem 1.1.
2. Preliminaries on Orlicz-Sobolev spaces
First of all, we recall that a continuous function A : R → [0, +∞) is a N -function
if:
(A1) A is convex.
(A2) A(t) = 0 if and only if t = 0.
(A3) A(t)
t → 0 as t → 0, and A(t)/t → ∞ as t → +∞.
(A4) A is an even function.
In what follows, we say that a N -function A satisfies the ∆2 -condition if, there
exists t0 ≥ 0 and k > 0 such that
A(2t) ≤ kA(t) ∀t ≥ t0 .
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MULTIPLICITY OF SOLUTIONS
3
This condition can be rewritten of the following way: For each s > 0, there exists
Ms > 0 and t0 ≥ 0 such that
A(st) ≤ Ms A(t),
∀t ≥ t0 .
(2.1)
Fixed an open set Ω ⊂ RN and a N-function A satisfying ∆2 -condition, the space
LA (Ω) is the vectorial space of the measurable functions u : Ω → R such that
Z
A(u) < ∞.
Ω
The space LA (Ω) endowed with Luxemburg norm,
Z
n
o
u
|u|A = inf α > 0 :
A
≤1 ,
α
Ω
e
is a Banach space. The complement function of A, denoted by A(s),
is given by
the Legendre transformation,
e = max{st − A(t)}
A(s)
t≥0
for s ≥ 0.
e are complementary each other. Moreover, we have the
The functions A and A
Young’s inequality
e
st ≤ A(t) + A(s),
∀t, s ≥ 0.
(2.2)
Using this inequality, it is possible to prove the Hölder type inequality
Z
uv ≤ 2|u|A |v|Ae, ∀u ∈ LA (Ω) and v ∈ LAe(Ω).
(2.3)
Ω
Another important function related to function A is the Sobolev’s conjugate function A∗ of A, defined by
Z t
A−1 (s)
ds, for t > 0.
A−1
(t)
=
∗
(N +1)/N
0 s
∗
∗
When A(t) = |t|p for 1 < p < N , we have A∗ (t) = p∗ p |t|p , where p∗ =
pN
N −p .
Hereafter, we denote by W01,A (Ω) the Orlicz-Sobolev space obtained by the completion of C0∞ (Ω) with respect to norm
kuk = |∇u|A + |u|A .
e satisfy the ∆2 -condition, then the
An important property is that: If A and A
spaces LA (Ω) and W 1,A (Ω) are reflexive and separable. Moreover, the ∆2 -condition
also implies that
Z
un → u in LA (Ω) ⇐⇒
A(|un − u|) → 0
(2.4)
Ω
and
un → u in W
1,A
Z
(Ω) ⇐⇒
Z
A(|un − u|) → 0 and
Ω
A(|∇un − ∇u|) → 0. (2.5)
Ω
Another important inequality was proved by Donaldson and Trudinger [10],
which establishes that for all open Ω ⊂ RN and there is a constant SN = S(N ) > 0
such that
|u|A∗ ≤ SN |∇u|A , u ∈ W01,A (Ω).
(2.6)
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J. A. SANTOS
EJDE-2013/249
Moreover, exist C0 > 0 such that
Z
Z
A(u) ≤ C0
A(|∇u|), u ∈ W01,A (Ω).
Ω
(2.7)
Ω
This inequality shows the following embedding is continuous
W01,A (Ω) ,→ LA∗ (Ω).
If Ω is a bounded domain and the two limits hold
B(t)
B(t)
< +∞, lim sup
= 0,
lim sup
A(t)
A
t→0
∗ (t)
|t|→+∞
(2.8)
then the embedding
W01,A (Ω) ,→ LB (Ω)
(2.9)
is compact.
e and Φ∗ and theirs proofs can
The next four lemmas involving the functions Φ, Φ
e Φ∗
be found in [14]. Hereafter, Φ is the N -function given in the introduction and Φ,
are the complement and conjugate functions of Φ respectively.
Lemma 2.1. Assume (1.2) and (1.3). Then
Z |t|
Φ(t) =
sφ(s)ds,
0
e ∈ ∆2 . Hence, LΦ (Ω), W 1,Φ (Ω) and W 1,Φ (Ω) are reflexive
is a N -function with Φ, Φ
0
and separable spaces.
e and Φ
e ∗ satisfy the inequality
Lemma 2.2. The functions Φ, Φ∗ , Φ
e
e ∗ (φ∗ (|t|)t) ≤ Φ∗ (2t), ∀t ≥ 0.
Φ(φ(|t|)t)
≤ Φ(2t), Φ
(2.10)
l
m
Lemma 2.3. Assume that (1.2) and (1.3) hold and let ξ0 (t) = min{t , t }, ξ1 (t) =
max{tl , tm }, for all t ≥ 0. Then
ξ0 (ρ)Φ(t) ≤ Φ(ρt) ≤ ξ1 (ρ)Φ(t) for ρ, t ≥ 0,
Z
ξ0 (|u|Φ ) ≤
Φ(u) ≤ ξ1 (|u|Φ ) for u ∈ LΦ (Ω).
Ω
Lemma 2.4. The function Φ∗ satisfies the inequality
Φ0 (t)t
≤ m∗ for t > 0.
l∗ ≤ ∗
Φ∗ (t)
As an immediate consequence of the Lemma 2.4, we have the following result
∗
∗
Lemma 2.5. Assume that (1.2) and (1.3) hold and let ξ2 (t) = min{tl , tm },
∗
∗
ξ3 (t) = max{tl , tm } for all t ≥ 0. Then
ξ2 (ρ)Φ∗ (t) ≤ Φ∗ (ρt) ≤ ξ3 (ρ)Φ∗ (t) for ρ, t ≥ 0,
Z
ξ2 (|u|Φ∗ ) ≤
Φ∗ (u)dx ≤ ξ3 (|u|Φ∗ ) for u ∈ LA∗ (Ω).
Ω
e be the complement of Φ and put
Lemma 2.6. Let Φ
l
m
ξ4 (s) = min{s l−1 , s m−1 },
l
m
ξ5 (s) = max{s l−1 , s m−1 },
Then the following inequalities hold
e
e
e
ξ4 (r)Φ(s)
≤ Φ(rs)
≤ ξ5 (r)Φ(s),
r, s ≥ 0;
s ≥ 0.
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MULTIPLICITY OF SOLUTIONS
5
Z
ξ4 (|u|Φ
e) ≤
Ω
e
Φ(u)dx
≤ ξ5 (|u|Φ
e ), u ∈ LΦ
e (Ω).
3. An abstract theorem and technical lemmas
In this section we recall an important abstract theorem involving genus theory,
which will use in the proof of Theorem 1.1. After, we prove some technical lemmas
that will use to show that the energy functional associated with problem (1.1)
satisfies the hypotheses of the abstract theorem.
3.1. An abstract theorem. Let E be a real Banach space and Σ the family of
sets Y ⊂ E\{0} such that Y is closed in E and symmetric with respect to 0; that
is,
Σ = {Y ⊂ E\{0}; Y is closed in E and Y = −Y }.
Hereafter, let us denote by γ(Y ) the genus of Y ∈ Σ (see [24, pp. 45]). Moreover,
we set
Kc = {u ∈ E; I(u) = c and I 0 (u) = 0},
Ac = {u ∈ E; I(u) ≤ c}.
Next, we recall a version of the Mountain Pass Theorem for even functionals,
whose proof can be found in [24].
Theorem 3.1. Let E be an infinite dimensional Banach space with
E = V ⊕ X, where V is finite dimensional and let I ∈ C 1 (E, R) be a even function
with I(0) = 0, and satisfying:
(I1) there are constants β, ρ > 0 such that I(u) ≥ β > 0, for each u ∈ ∂Bρ ∩ X;
(I2) there is Υ > 0 such that I satisfies the (P S)c condition, for 0 < c < Υ;
e ⊂ E, there is R = R(E)
e > 0 such
(I3) for each finite dimensional subspace E
e R (0).
that I(u) ≤ 0 for all u ∈ E\B
Suppose V is k dimensional and V = span{e1 , . . . , ek }. For m ≥ k, inductively
choose em+1 6∈ Em := span{e1 , . . . , em }. Let Rm = R(Em ) and Dm = BRm ∩ Em .
Define
Gm := h ∈ C(Dm , E); h is odd and h(u) = u, ∀u ∈ ∂BRm ∩ Em },
(3.1)
(3.2)
Γj := h(Dm \Y ); h ∈ Gm , m ≥ j, Y ∈ Σ, and γ(Y ) ≤ m − j .
For each j ∈ N, let
cj = inf max I(u).
(3.3)
K∈Γj u∈K
Then, 0 < β ≤ cj ≤ cj+1 for j > k, and if j > k, cj < Υ and cj is critical value of
I. Moreover, if cj = cj+1 = · · · = cj+l = c < Υ for j > k, then γ(Kc ) ≥ l + 1.
3.2. Technical lemmas. Associated with problem (1.1), we have the energy functional Jλ : W01,Φ (Ω) → R defined by
Z
Z
Z
Jλ (u) =
Φ(|∇u|) − λ
Φ∗ (u) −
F (x, u).
Ω
Ω
1
Ω
W01,Φ (Ω), R
with
By conditions (F1) and (F2), Jλ ∈ C
Z
Z
Z
0
Jλ (u) · v =
φ(|∇u|)∇u∇v − λ
φ∗ (|u|)uv −
f (x, u)v,
Ω
Ω
Ω
6
J. A. SANTOS
EJDE-2013/249
for any u, v ∈ W01,Φ (Ω). Thus, critical points of Jλ are weak solutions of problem (1.1).
Lemma 3.2. Under the conditions (F1) and (F2), the functional Jλ satisfies (I1).
Proof. On the one hand, from (F1) and (F2), for a given > 0, there exists C > 0
such that
|F (x, t)| ≤ Φ(t) + C Φ∗ (t),
∀(x, t) ∈ Ω̄ × R.
(3.4)
Combining (2.7) with (3.4),
Z
Jλ (u) ≥ (1 − C0 )
Z
Φ(|∇u|) − (1 + C )
Φ∗ (u).
Ω
Ω
For is small enough and kuk = ρ ' 0, from (2.6) and Lemma 2.4, it follows that
∗
∗
l
l
Jλ (u) ≥ C1 |∇u|m
Φ − C2 SN |∇u|Φ
for some positive constants C1 and C2 . For m < l∗ , if ρ is small enough, there is
β > 0 such that
Jλ (u) ≥ β > 0 ∀u ∈ ∂Bρ (0),
which completes the proof.
Lemma 3.3. Under conditions (F1) and (F2), the functional Jλ satisfies (I3).
Proof. Suppose (I3) does not hold. Then, there is a finite dimensional subspace
e ⊂ W 1,Φ (Ω) and a sequence (un ) ⊂ E\B
e n (0) satisfying
E
0
Jλ (un ) > 0,
∀n ∈ N.
(3.5)
A direct computation shows that given > 0, there is a constant M > 0 such that
− M − Φ∗ (t) ≤ F (x, t) ≤ M + Φ∗ (t),
∀(x, t) ∈ Ω × R.
(3.6)
Consequently,
Z
Jλ (un ) ≤
Z
Φ(|∇un |)dx − λ
Ω
Z
Φ∗ (un ) + M |Ω|.
Φ∗ (un ) + Ω
Ω
Fixing = λ/2, and using Lemma 2.5, we obtain
Z
λ
Jλ (un ) ≤
Φ(|∇un |) − ξ3 (|un |Φ∗ ) + M |Ω|.
2
Ω
(3.7)
Using that dim Ẽ < ∞, we know that any two norms are equivalent in Ẽ. Then,
using that kun k → ∞, we can assume that |un |Φ∗ > 1. Thereby, from Lemmas 2.3
and 2.5,
λ
l∗
Jλ (un ) ≤ |∇un |m
Φ − |un |Φ∗ + M |Ω|.
2
Using again the equivalence of the norms in Ẽ, there is C > 0 such that
∗
λ
Jλ (un ) ≤ kun km − Ckun kl + M |Ω|.
2
Recalling that m < l∗ , the above inequality implies that there is n0 ∈ N such that
Jλ (un ) < 0,
which contradicts (3.5).
∀n ≥ n0 ,
Lemma 3.4. Under conditions (F1)–(F2), any (PS) sequence for Jλ is bounded in
W01,Φ (Ω).
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MULTIPLICITY OF SOLUTIONS
7
Proof. Let {un } be a (P S)d sequence of Jλ . Then,
Jλ (un ) → d,
Jλ0 (un ) → 0
as n → +∞.
We claim that {un } is bounded. Indeed, note that
Z
Z
1 0
1
Jλ (un ) − Jλ (un )un =
Φ(|∇un |) −
φ(|∇un |)|∇un |2
θ
θ Ω
Ω
Z
Z
λ
−λ
φ∗ (|un |)u2n
Φ∗ (un ) +
θ Ω
Ω
Z
Z
1
f (x, un )un .
−
F (x, un ) +
θ Ω
Ω
Consequently,
Z
Z
1
1
λ
Φ(|∇un |)
φ∗ (|un |)u2n − Φ∗ (un ) = Jλ (un ) − Jλ0 (un )un −
θ
Ω θ
Ω
Z
1
+
φ(|∇un |)|∇un |2
θ Ω
Z
1
+
F (x, un ) − f (x, un )un .
θ
Ω
Then, by (1.3), (F2) and Lemma 2.4, for n sufficiently large,
Z
Z
l∗
m
λ( − 1)
Φ∗ (un ) ≤ C + 1 + kun k + ( − 1)
Φ(|∇un |),
θ
θ
Ω
Ω
which implies that
[λ(
l∗
− 1)]
θ
Z
Φ∗ (un ) ≤ C + kun k,
Ω
where C is a positive constant, and so
Z
Φ∗ (un )dx ≤ C(1 + kun k).
(3.8)
Ω
By (3.6) and (3.8),
Z
Z
Z
Φ(|∇un |) ≤ Jλ (un ) + λ
Φ∗ (un ) +
F (x, un )dx
Ω
Ω
Ω
Z
≤ C + on (1) + (λ + )
Φ∗ (un )
Ω
≤ C(1 + kun k) + on (1).
Therefore, for n sufficiently large,
Z
Φ(|∇un |) ≤ C (1 + kun k) .
Ω
If kun k > 1, from Lemma 2.5, it follows that
kun kl ≤ C(1 + kun k).
Using that l > 1, the above inequality gives that {un } is bounded in W01,Φ (Ω). As a consequence of the above result, if {un } is a (PS) sequence for Jλ , we can
extract a subsequence of {un }, still denoted by {un } and u ∈ W01,Φ (Ω), such that
8
J. A. SANTOS
•
•
•
•
EJDE-2013/249
un * u in W01,Φ (Ω);
un * u in LΦ∗ (Ω);
un → u in LΦ (Ω);
un (x) → u(x) a.e. in Ω;
From the concentration compactness lemma of Lions in Orlicz-Sobolev space
found in [14], there exist two nonnegative measures µ, ν ∈ M(RN ), a countable set
J , points {xj }j∈J in Ω and sequences {µj }j∈J , {νj }j∈J ⊂ [0, +∞), such that
X
Φ(|∇un |) → µ ≥ Φ(|∇u|) +
µj δxj in M(RN )
(3.9)
j∈J
Φ∗ (un ) → ν = Φ∗ (u) +
X
in M(RN )
νj δxj
(3.10)
j∈J
∗
l∗
m∗
∗
∗
l∗
∗
m∗
m
l
m
l
µj l , SN
µjm , SN
µjm },
νj ≤ max{SN
µjl , SN
(3.11)
where SN satisfies (2.6).
Next, we will show an important estimate for {νi }, from below. Firs, we prove
a technical lemma.
Lemma 3.5. Under the conditions of Lemma 3.4. If {un } is a (P S) sequence for
Jλ and {νj } as above, then for each j ∈ J ,
β
− α
l β−1
SN β−1
∗
λm
νj ≥
∗
∗
∗
or
νj = 0,
∗
l
for some α ∈ {l∗ , m∗ } and β ∈ { ll , ml , m
,m
m }.
Proof. Let ψ ∈ C0∞ (RN ) such that
ψ(x) = 1 in B1/2 (0),
supp ψ ⊂ B1 (0),
0 ≤ ψ(x) ≤ 1 ∀x ∈ RN .
For each j ∈ Γ and > 0, let us define
ψ (x) = ψ
x − xj ,
∀x ∈ RN .
Then {ψ un } is bounded in W01,Φ (Ω). Since Jλ0 (un ) → 0, we have
Jλ0 (un )(ψ un ) = on (1),
or equivalently,
Z
Z
Z
2
φ(|∇un |)∇un ∇ (un ψ ) = on (1) + λ
φ∗ (|un |)un ψ +
f (x, un )un ψ
Ω
Ω
Z
ZΩ
(3.12)
≤ on (1) + λm∗
Φ∗ (un )ψ +
f (x, un )un ψ .
Ω
Ω
Using that
lim
t→+∞
f (x, t)tψ (x)
= 0,
Φ∗ (t)
uniformly in x ∈ Ω
and that limn→+∞ f (x, un )un ψ = 0 a.e. on Ω, we have by compactness Lemma of
Strauss [8] (note that this result is still true when we replace RN by Ω)
Z
Z
lim
f (x, un )un ψ =
f (x, u)uψ .
(3.13)
n→∞
Ω
Ω
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MULTIPLICITY OF SOLUTIONS
9
On the other hand, by (1.3)
Z
Z
Z
2
φ(|∇un |)∇un ∇(un ψ ) =
φ(|∇un |)|∇un | ψ +
φ(|∇un |)(∇un ∇ψ )un
Ω
Ω
ZΩ
Z
≥l
Φ(|∇u|)ψ +
φ(|∇un |)(∇un ∇ψ )un .
Ω
Ω
(3.14)
By Lemmas 2.2 and 2.6, the sequence {|φ(|∇un |)∇un |Φ
e } is bounded. Thus, there
is a subsequence {un } such that
φ(|∇un |)∇un * w
e1
N
weakly in LΦ
e (Ω, R ),
N
for some w
e1 ∈ LΦ
e (Ω, R ). Since un → u in LΦ (Ω),
Z
Z
φ(|∇un |)(∇un ∇ψ )un → (w
e1 ∇ψ )u.
Ω
Ω
Thus, combining (3.12), (3.13), (3.14), and letting n → ∞, we have
Z
Z
Z
Z
l
ψ dµ + (w
e1 ∇ψ )u ≤ λm∗
ψ dν +
f (x, u)uψ .
Ω
Ω
Ω
(3.15)
Ω
Now we show that the second term of the left-hand side converges 0 as → 0.
Claim 1: {f (x, un )} is bounded in LΦ
e ∗ (Ω). In fact, by (F1) and Lemma 2.2 we
have
Z
Z
Z
e ∗ (f (x, un )) ≤ c1
e ∗ (φ∗ (|un |)un ) + c2
e ∗ (φ(|un |)un )
Φ
Φ
Φ
Ω
Ω
[|un |>1]
Z
e ∗ (φ(|un |)un )
+ c3
Φ
[|un |≤1]
Z
Z
e ∗ (φ(|un |)un ) + c3 |Ω|.
≤ c1
Φ∗ (un ) + c2
Φ
[|un |>1]
Ω
∗
Hence, by (1.3), Lemma 2.3 and m < l ,
Z
Z
Z
e
e
e ∗ (|un |m−1 ) + C3 |Ω|
Φ∗ (f (x, un )) ≤ C1
Φ∗ (φ∗ (|un |)un ) + C2
Φ
Ω
Ω
[|un |>1]
Z
Z
e ∗ (|un |l∗ −1 ) + C3 |Ω|.
≤ C1
Φ∗ (un ) + C2
Φ
[|un |>1]
Ω
Now, by Lemmas 2.4 and 2.2,
Z
Z
e ∗ (f (x, un )) ≤ K1
Φ
Φ∗ (un ) + K2 |Ω| < +∞.
Ω
Ω
From Claim 1, there is a subsequence {un } such that
φ∗ (|un |)un + f (x, un ) * w
e2
weakly in LΦ
e ∗ (Ω),
for some w
e2 ∈ LΦ
e ∗ (Ω). Since
Z
Z
0
Jλ (un )v =
φ(|∇un |)∇un ∇v −
(φ∗ (|un |)un + f (x, un )) v → 0,
Ω
as n → ∞ for any v ∈
Ω
W01,Φ (Ω),
Z
Ω
(w
e1 ∇v − w
e2 v) = 0,
10
J. A. SANTOS
EJDE-2013/249
for any v ∈ W01,Φ (Ω). Substituting v = uψ we have
Z
(w
e1 ∇(uψ ) − w
e2 uψ ) = 0.
Ω
Namely,
Z
Z
Ω
(w
e1 ∇ψ )u = −
Ω
(w
e1 ∇u − w
e2 u)ψ .
Noting w
e1 ∇u − w
e2 u ∈ L1 (Ω), we see that right-hand side tends to 0 as → 0.
Hence we have
Z
(w
e1 ∇ψ )u → 0,
Ω
as → 0.
Letting → 0 in (3.15), we obtain lµj ≤ λm∗ νj . Hence,
lλ β β
−α
SN
νj ≤ µβj ≤
νj ,
m∗
∗
∗
∗
∗
l
,m
for some α ∈ {l∗ , m∗ }, β ∈ { ll , ml , m
m }, and so
β
− α
l
β−1
or νj = 0.
νj ≥
SN β−1
∗
λm
Lemma 3.6. Assume that (F1)–(F2). Then, Jλ satisfies (P S)d for d ∈ (0, dλ )
where
n l∗ − θ
β
l
l∗ m∗ l∗ m∗ o
dλ = min
( ∗ ) β−1 ; α ∈ {l∗ , m∗ }, β ∈ { ,
, ,
} .
α
1
l l m m
θSNβ−1 λ β−1 m
Proof. Using that Jλ (un ) = d + on (1) and Jλ0 (un ) = on (1), we have
1
d = lim I(un ) = lim Jλ (un ) − Jλ0 (un )un
n→∞
n→∞
θ
Z
Z
h
l∗
m
Φ(|∇un |) + λ( − 1)
Φ∗ (un )
≥ lim (1 − )
n→∞
θ Ω
θ
Ω
Z
i
1
−
F (x, un ) − f (x, un )un
θ
Ω
Z
l∗
Φ∗ (un ).
≥ λ( − 1)
θ
Ω
Recalling that
Z
hZ
X i
lim
Φ∗ (un )dx =
Φ∗ (u) +
νj ≥ νj ,
n→∞
Ω
Ω
j∈J
we derive that
α
β
l β−1
− β−1
l∗
−1
S
N
θ
λm∗
α
β
1
− β−1
l∗ − θ l β−1
=
SN
λ 1−β ,
∗
θ
m
∗
∗
l ∗ m∗
for some α ∈ {l∗ , m∗ }, β ∈ { ll , ml , m
, m }, which is an absurd. From this, we
must have νj = 0 for any j ∈ J , leading to
Z
Z
Φ∗ (un ) →
Φ∗ (u).
(3.16)
d≥λ
Ω
Ω
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MULTIPLICITY OF SOLUTIONS
11
Combining the last limit with Brézis and Lieb [6], we obtain
Z
Φ∗ (un − u) → 0 as n → ∞,
Ω
from where it follows by Lemma 2.5
un → u in LΦ∗ (Ω).
Now, as
Jλ0 (un )un
= on (1), the last limit gives
Z
Z
φ(|∇un |)|un |2 = λ
φ∗ (|un |)u2n +
f (x, un )un + on (1).
Z
Ω
Ω
Ω
In what follows, let us denote by {Pn } the sequence
Pn (x) = hφ(|∇un (x)|)∇un (x) − φ(|∇u(x)|)∇u(x), ∇un (x) − ∇u(x)i.
Since Φ is convex in R and Φ(|.|) is C 1 class in RN , has Pn (x) ≥ 0. From definition
of {Pn },
Z
Z
Z
Z
2
Pn =
φ(|∇un |)|∇un | −
φ(|∇un |)∇un ∇u −
φ(|∇u|)∇u∇(un − u).
Ω
Ω
Ω
Ω
W01,Φ (Ω),
Recalling that un * u in
we have
Z
φ(|∇u|)∇u∇(un − u) → 0
as n → ∞,
(3.17)
Ω
which implies that
Z
Z
Z
Pn =
φ(|∇un |)|∇un |2 −
φ(|∇un |)∇un ∇u + on (1).
Ω
Ω
Ω
= on (1) and Jλ0 (un )u = on (1), we derive
On the other hand, from
Z
Z
Z
2
0≤
Pn = λ
φ∗ (|un |)|un | − λ
φ∗ (|un |)un u
Ω
Ω
ZΩ
Z
+
f (x, un )un −
f (x, un )u + on (1).
Jλ0 (un )un
Ω
Ω
Combining (3.16) with the compactness Lemma of Strauss [8], we deduce that
Z
Pn → 0 as n → ∞.
Ω
Using that Φ is convex, from a result due to Dal Maso and Murat [18], it follows
that
∇un (x) → ∇u(x) a.e. Ω.
Now, using Lebesgue’s Theorem,
Z
Φ(|∇un − ∇u|)dx → 0,
Ω
which shows that
un → u in W01,Φ (Ω).
(3.18)
The next lemma is similar to [25, Lemma 5] and its proof will be omitted.
12
J. A. SANTOS
EJDE-2013/249
Lemma 3.7. Under conditions (F1)–(F2), there is sequence {Mm } ⊂ (0, +∞)
independent of λ with Mm ≤ Mm+1 , such that for any λ > 0,
cλm = inf max Jλ (u) < Mm .
K∈Γm u∈K
(3.19)
Proof of Theorem 1.1. For each k ∈ N, choose λk such that Mk < dλk . Thus, for
λ ∈ (λk+1 , λk ),
0 < cλ1 ≤ cλ2 ≤ · · · ≤ cλk < Mk ≤ dλ .
By Theorem 3.1, the levels cλ1 ≤ cλ2 ≤ · · · ≤ cλk are critical values of Jλ . Thus, if
cλ1 < cλ2 < · · · < cλk ,
the functional Jλ has at least k critical points. Now, if cλj = cλj+1 for some j =
1, 2, . . . , k, it follows from Theorem 3.1 that Kcλj is an infinite set [24, Cap. 7].
Then, in this case, problem (1.1) has infinitely many solutions.
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Jefferson A. Santos
Universidade Federal de Campina Grande, Unidade Acadêmica de Matemática e Estatı́stica, CEP: 58429-900, Campina Grande - PB, Brazil
E-mail address: jefferson@dme.ufcg.edu.br