1 Introduction
In this article, we study the existence and multiplicity of solutions for the differential inclusion problem with singular coefficients involving the p(x)-Laplacian of the form
-
div
(
|
∇
u
|
p
(
x
)
-
2
∇
u
)
∈
λ
a
1
(
x
)
∂
G
1
(
x
,
u
)
+
μ
a
2
(
x
)
∂
G
2
(
x
,
u
)
in
Ω
,
u
=
0
on
∂
Ω
,
where the following conditions are satisfied:
(P) Ω is a bounded open domain in ℝN, N ≥ 2, p
∈
C
(
Ω
̄
)
, 1 < p- := infΩ p(x) ≤ p+ := supΩ p(x) < +∞, λ, μ ∈ ℝ.
(A) For i = 1, 2,
a
i
∈
L
r
i
(
x
)
(
Ω
)
,
a
i
(
x
)
>
0
for x ∈ Ω, Gi(x, u) is measurable with respect to x (for every u ∈ ℝ) and locally Lipschitz with respect to u (for a.e. x ∈ Ω), ∂Gi : Ω × ℝ → ℝ is the Clarke sub-differential of Gi and |
ξ
i
|
≤
c
1
+
c
2
|
t
|
q
i
(
x
)
-
1
for x ∈ Ω, t ∈ ℝ and ξi ∈ ∂Gi, where ci is a positive constant,
r
i
,
q
i
∈
C
Ω
̄
,
r
i
-
>
1
,
q
i
-
>
1
, ri(x) > qi(x) for all x ∈ Ω, and
q
i
(
x
)
<
r
i
(
x
)
-
q
i
(
x
)
r
i
(
x
)
p
*
(
x
)
,
∀
x
∈
Ω
¯
,
here
p
*
(
x
)
=
N
p
(
x
)
N
-
p
(
x
)
if
p
(
x
)
<
N
,
∞
if
p
(
x
)
≥
N
.
(
A
1
)
q
1
+
<
p
-
.
(
A
2
)
q
2
-
>
p
+
.
A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardy exponents of the form
-
div
(
|
∇
u
|
p
(
x
)
-
2
∇
u
)
∈
λ
1
|
x
|
s
1
(
x
)
∂
G
1
(
x
,
u
)
+
μ
1
|
x
|
s
2
(
x
)
∂
G
2
(
x
,
u
)
in
Ω
,
u
=
0
on
∂
Ω
,
and in this case the assumption corresponding to (A) is the following
(
A
)
*
0
∈
Ω
̄
, for i = 1, 2, ∂Gi : Ω × ℝ → ℝ is the Clarke sub-differential of Gi and |
ξ
i
|
≤
c
1
+
c
2
|
t
|
q
i
(
x
)
-
1
for x ∈ Ω, t ∈ ℝ and ξi ∈ ∂Gi, where ci is a positive constant,
s
i
,
q
i
∈
C
Ω
̄
,
0
≤
s
i
-
≤
s
i
+
<
N
,
q
i
-
>
1
, and
q
i
(
x
)
<
N
-
s
i
(
x
)
q
i
(
x
)
N
p
*
(
x
)
,
∀
x
∈
Ω
̄
.
The operator -div(|∇u|p(x)-2 ∇u) is said to be the p(x)-Laplacian, and becomes p-Laplacian when p(x) ≡ p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous. The study of various mathematical problems with variable exponent growth condition has been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. One of the most studied models leading to problem of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electro-magnetic field 12. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal baro-tropic gas through a porous medium 34. Another field of application of equations with variable exponent growth conditions is image processing 5. The variable nonlinearity is used to outline the borders of the true image and to eliminate possible noise. We refer the reader to 67891011 for an overview of and references on this subject, and to 12131415161718192021 for the study of the p(x)-Laplacian equations and the corresponding variational problems.
Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions for Dirichlet boundary value problems with discontinuous nonlinearities has been widely investigated in recent years. Chang 22 extended the variational methods to a class of non-differentiable functionals, and directly applied the variational methods for non-differentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities. Later Kourogenis and Papageorgiou 23 obtained some nonsmooth critical point theories and applied these to nonlinear elliptic equations at resonance, involving the p-Laplacian with discontinuous nonlinearities. In the celebrated work 2425, Ricceri elaborated a Ricceri-type variational principle and a three critical points theorem for the Gâteaux differentiable functional, respectively. Later, Marano and Motreanu 2627 extended Ricceri's results to a large class of non-differentiable functionals and gave some applications to differential inclusion problems involving the p-Laplacian with discontinuous nonlinearities.
In 21, by means of the critical point theory, Fan obtain the existence and multiplicity of solutions for (1.1) under the condition of
G
i
(
x
,
⋅
)
∈
C
1
(
ℝ
)
and
g
i
=
G
i
′
satisfying the Carathéodory condition for i = 1, 2, x ∈ Ω. The aim of the present article is to generalize the main results of 21 to the case of the functional of problem (1.1) is nonsmooth.
This article is organized as follows: In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function; In Section 3, we give the variational principle which is needed in the sequel; In Section 4, using the critical point theory, we prove the existence and multiplicity results for problem (1.1).
2 Preliminaries
2.1 Variable exponent Sobolev spaces
Let Ω be a bounded open subset of ℝN, denote
L
+
∞
(
Ω
)
=
{
p
∈
L
∞
(
Ω
)
:
ess
inf
Ω
p
(
x
)
≥
1
}
.
For p
∈
L
+
∞
(
Ω
)
, denote
p
-
=
p
-
(
Ω
)
=
ess
inf
x
∈
Ω
p
(
x
)
,
p
+
=
p
+
(
Ω
)
=
ess
sup
x
∈
Ω
p
(
x
)
.
On the basic properties of the space W1,p(x)(Ω) we refer to 7282930. Here we display some facts which will be used later.
Denote by S(Ω) the set of all measurable real functions defined on Ω. Two functions in S(Ω) are considered as the same element of S(Ω) when they are equal almost everywhere. For p
∈
L
+
∞
(
Ω
)
, define the spaces Lp(x) (Ω) and W1,p(x) (Ω) by
L
p
(
x
)
Ω
=
u
∈
S
(
Ω
)
:
∫
Ω
|
u
(
x
)
|
p
(
x
)
d
x
<
∞
with the norm
|
u
|
L
p
(
x
)
Ω
=
|
u
|
p
(
x
)
=
inf
λ
>
0
:
∫
Ω
u
(
x
)
λ
p
(
x
)
d
x
≤
1
,
and
W
1
,
p
x
Ω
=
u
∈
L
p
x
Ω
:
∇
u
∈
L
p
x
Ω
with the norm
|
|
u
|
|
W
1
,
p
(
x
)
(
Ω
)
=
|
u
|
L
p
(
x
)
(
Ω
)
+
|
∇
u
|
L
p
(
x
)
(
Ω
)
.
Denote by
W
0
1
,
p
(
x
)
(
Ω
)
the closure of
C
0
∞
(
Ω
)
in W1,p(x) (Ω) . Hereafter, we always assume that p- > 1.
Proposition 2.1. 731 The spaces Lp(x) (Ω) , W1,p(x) (Ω) and
W
0
1
,
p
(
x
)
(
Ω
)
are separable and reflexive Banach spaces.
Proposition 2.2. 731 The conjugate space of Lp(x) (Ω) is
L
p
0
(
x
)
(
Ω
)
, where
1
p
(
x
)
+
1
p
0
(
x
)
=
1
. For any u ∈ Lp(x) (Ω) and v ∈
L
p
0
(
x
)
(
Ω
)
,
∫
Ω
|
u
v
|
d
x
≤
2
|
u
|
p
(
x
)
|
v
|
p
0
(
x
)
.
Proposition 2.3. 731 In
W
0
1
,
p
(
x
)
(
Ω
)
the Poincaré inequality holds, that is, there exists a positive constant c such that
|
u
|
L
p
(
x
)
(
Ω
)
≤
c
|
∇
u
|
L
p
(
x
)
(
Ω
)
,
∀
u
∈
W
0
1
,
p
x
Ω
.
So |
∇
u
|
L
p
(
x
)
(
Ω
)
is an equivalent norm in
W
0
1
,
p
(
x
)
(
Ω
)
.
Proposition 2.4. 7282931 Assume that the boundary of Ω possesses the cone property and p
∈
C
(
Ω
̄
)
. If q
∈
C
(
Ω
̄
)
and 1
≤
q
(
x
)
<
p
*
(
x
)
f
o
r
x
∈
Ω
̄
, then there is a compact embedding W1,p(x)(Ω) → Lq(x) (Ω).
Let us now consider the weighted variable exponent Lebesgue space.
Let a ∈ S(Ω) and a(x) > 0 for x ∈ Ω. Define
L
a
(
x
)
p
(
x
)
Ω
=
u
∈
S
(
Ω
)
:
∫
Ω
a
(
x
)
u
(
x
)
p
(
x
)
d
x
<
∞
with the norm
u
L
a
(
x
)
p
(
x
)
Ω
=
u
(
p
(
x
)
,
a
(
x
)
)
=
inf
λ
>
0
:
∫
Ω
a
(
x
)
u
(
x
)
λ
p
(
x
)
d
x
≤
1
,
then
L
a
(
x
)
p
(
x
)
(
Ω
)
is a Banach space. The following proposition follows easily from the definition of |
u
|
L
a
(
x
)
p
(
x
)
(
Ω
)
.
Proposition 2.5. (see 731) Set ρ(u) = ∫Ω a(x)|u(x)|p(x) dx. For u
,
u
k
∈
L
a
(
x
)
p
(
x
)
(
Ω
)
, we have
(1)
F
o
r
u
≠
0
,
|
u
|
(
p
(
x
)
,
a
(
x
)
)
=
λ
⇔
ρ
u
λ
=
1
.
(2)
|
u
|
(
p
(
x
)
,
a
(
x
)
)
<
1
(
=
1
;
>
1
)
⇔
ρ
(
u
)
<
1
(
=
1
;
>
1
)
.
(3)
I
f
|
u
|
(
p
(
x
)
,
a
(
x
)
)
>
1
,
t
h
e
n
|
u
|
(
p
(
x
)
,
a
(
x
)
)
p
-
≤
ρ
u
≤
|
u
|
(
p
(
x
)
,
a
(
x
)
)
p
+
.
(4)
I
f
|
u
|
(
p
(
x
)
,
a
(
x
)
)
<
1
,
t
h
e
n
|
u
|
(
p
(
x
)
,
a
(
x
)
p
+
≤
ρ
(
u
)
≤
|
u
|
(
p
(
x
)
,
a
(
x
)
)
p
−
.
(5)
lim
k
→
∞
|
u
k
|
(
p
(
x
)
,
a
(
x
)
)
=
0
⇔
lim
k
→
∞
ρ
(
u
k
)
=
0
.
(6)
|
u
k
|
(
p
(
x
)
,
a
(
x
)
)
→
∞
⇔
ρ
(
u
k
)
→
∞
.
Proposition 2.6. (see 21) Assume that the boundary of Ω possesses the cone property and p
∈
C
(
Ω
̄
)
. Suppose that a ∈ Lr(x)(Ω), a(x) > 0 for x ∈ Ω, r
∈
C
(
Ω
̄
)
and r- > 1. If
q
∈
C
(
Ω
̄
)
and
1
≤
q
(
x
)
<
r
(
x
)
-
1
r
(
x
)
p
*
(
x
)
:
=
p
a
(
x
)
*
(
x
)
,
∀
x
∈
Ω
̄
,
then there is a compact embedding
W
1
,
p
(
x
)
(
Ω
)
→
L
a
(
x
)
q
(
x
)
(
Ω
)
.
The following proposition plays an important role in the present article.
Proposition 2.7. Assume that the boundary of Ω possesses the cone property and p
∈
C
(
Ω
̄
)
. Suppose that a ∈ Lr(x)(Ω), a(x) > 0 for x ∈ Ω,
r
∈
C
(
Ω
̄
)
and r(x) > q(x) for all x ∈ Ω. If
q
∈
C
(
Ω
̄
)
and
1
≤
q
(
x
)
<
r
(
x
)
-
q
(
x
)
r
(
x
)
p
*
(
x
)
,
∀
x
∈
Ω
̄
,
then there is a compact embedding
W
1
,
p
(
x
)
(
Ω
)
→
L
(
a
(
x
)
)
q
(
x
)
q
(
x
)
(
Ω
)
.
Proof. Set
r
1
(
x
)
=
r
(
x
)
q
(
x
)
, then
r
1
-
>
1
and
(
a
(
x
)
)
q
(
x
)
∈
L
r
1
(
x
)
(
Ω
)
. Moreover, from (2.2) we can get
1
≤
q
(
x
)
<
r
1
(
x
)
-
1
r
1
(
x
)
p
*
(
x
)
,
∀
x
∈
Ω
̄
.
Using Proposition 2.6, we see that the embedding
W
1
,
p
(
x
)
(
Ω
)
→
L
(
a
(
x
)
)
q
(
x
)
q
(
x
)
(
Ω
)
is compact.
■
2.2 Generalized gradient of the locally Lipschitz function
Let (X, || · ||) be a real Banach space and X* be its topological dual. A function f : X → ℝ is called locally Lipschitz if each point u ∈ X possesses a neighborhood Ωu such that |f(u1) - f(u2)| ≤ L||u1 - u2|| for all u1, u2 ∈ Ωu, for a constant L > 0 depending on Ωu. The generalized directional derivative of f at the point u ∈ X in the direction v ∈ X is
f
0
(
u
,
v
)
=
lim
sup
w
→
u
,
t
→
0
1
t
(
f
(
w
+
t
v
)
-
f
(
w
)
)
.
The generalized gradient of f at u ∈ X is defined by
∂
f
(
u
)
=
{
u
*
∈
X
*
:
⟨
u
*
,
φ
⟩
≤
f
0
(
u
;
φ
)
for all
φ
∈
X
}
,
which is a non-empty, convex and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u ∈ X is a critical point of f if 0 ∈ ∂f(u). For further details, we refer the reader to Chang 22.
We list some fundamental properties of the generalized directional derivative and gradient that will be used throughout the article.
Proposition 2.8. (see 2232) (1) Let j : X → ℝ be a continuously differentiable function. Then ∂j(u) = {j'(u)}, j0(u; z) coincides with 〈j' (u), z〉X and (f + j)0(u, z) = f0(u; z) + 〈j' (u), z〉X for all u, z ∈ X.
(2) The set-valued mapping u → ∂f(u) is upper semi-continuous in the sense that for each u0 ∈ X, ε > 0, v ∈ X, there is a δ > 0, such that for each w ∈ ∂f (u) with ||w - u0|| < δ, there is w0 ∈ ∂f (u0)
⟨
w
-
w
0
,
v
⟩
<
ε
.
(3) (Lebourg's mean value theorem) Let u and v be two points in X. Then there exists a point w in the open segment joining u and v and
x
w
*
∈
∂
f
(
w
)
such that
f
(
u
)
-
f
(
v
)
=
⟨
x
w
*
,
u
-
v
⟩
X
.
(4) The function
m
(
u
)
=
min
w
∈
∂
f
(
u
)
w
X
*
exists, and is lower semi continuous; i.e.,
lim
inf
u
→
u
0
m
(
u
)
≥
m
(
u
0
)
.
In the following we need the nonsmooth version of Palais-Smale condition.
Definition 2.1. We say that φ satisfies the (PS)c-condition if any sequence {un} ⊂ X such that φ(un) → c and m(un) → 0, as n → +∞, has a strongly convergent subsequence, where m(un) = inf{||u*||X* : u* ∈ ∂φ (un)}.
In what follows we write the (PS)c-condition as simply the PS-condition if it holds for every level c ∈ ℝ for the Palais-Smale condition at level c.
3 Variational principle
In this section we assume that Ω and p(x) satisfy the assumption (P). For simplicity we write X
=
W
0
1
,
p
(
x
)
(
Ω
)
and ||u|| = |∇u|p(x) for u ∈ X. Denote by un ⇀ u and un → u the weak convergence and strong convergence of sequence {un} in X, respectively, denote by c and ci the generic positive constants, Bρ = {u ∈ X : ||u|| < ρ}, Sρ = {u ∈ X : ||u|| = ρ}.
Set
F
(
x
,
t
)
=
λ
a
1
(
x
)
G
1
(
x
,
t
)
+
μ
a
2
(
x
)
G
2
(
x
,
t
)
,
where ai and Gi (i = 1, 2) are as in (A).
Define the integral functional
φ
(
u
)
=
∫
Ω
1
p
(
x
)
|
∇
u
|
p
(
x
)
d
x
-
∫
Ω
F
(
x
,
u
)
d
x
,
∀
u
∈
X
.
We write
J
(
u
)
=
∫
Ω
1
p
(
x
)
|
∇
u
|
p
(
x
)
d
x
,
Ψ
(
u
)
=
∫
Ω
F
(
x
,
u
)
d
x
,
then it is easy to see that J ∈ C1(X, ℝ) and φ = J - Ψ.
Below we give several propositions that will be used later.
Proposition 3.1. (see 19) The functional J : X → ℝ is convex. The mapping J' : X → X* is a strictly monotone, bounded homeomorphism, and is of (S+) type, namely
u
n
⇀
u
a
n
d
lim
¯
n
→
∞
J
′
(
u
n
)
(
u
n
-
u
)
≤
0
i
m
p
l
i
e
s
u
n
→
u
.
Proposition 3.2. Ψ is weakly-strongly continuous, i.e., un ⇀ u implies Ψ(un) → Ψ(u).
Proof. Define ϒ1 = ∫Ω G1(x, u) dx and ϒ2 = ∫Ω G2(x, u) dx. In order to prove Ψ is weakly-strongly continuous, we only need to prove ϒ1 and ϒ2 are weakly-strongly continuous. Since the proofs of ϒ1 and ϒ2 are identical, we will just prove ϒ1.
We assume un ⇀ u in X. Then by Proposition 2.8.3, we have
ϒ
1
(
u
n
)
-
ϒ
1
(
u
)
=
∫
Ω
(
G
1
(
x
,
u
n
)
-
G
1
(
x
,
u
)
)
d
x
=
∫
Ω
ξ
n
(
x
)
(
u
n
-
u
)
d
x
,
where ξn ∈ ∂G1(,τn(x)) for some τn(x) in the open segment joining u and un. From Chang 22 we know that
ξ
n
∈
L
q
1
0
(
x
)
(
Ω
)
. So using Proposition 2.5, we have
ϒ
1
(
u
n
)
-
ϒ
1
(
u
)
→
0
.
■
Proposition 3.3. Assume (A) holds and F satisfies the following condition:
(B) F
(
x
,
u
)
≤
θ
λ
a
1
(
x
)
⟨
ξ
1
,
u
⟩
+
θ
μ
a
2
(
x
)
⟨
ξ
2
,
u
⟩
+
b
(
x
)
+
∑
i
=
1
m
d
i
(
x
)
|
u
|
k
i
(
x
)
for a.e.x ∈ Ω, all u ∈ X and ξ1 ∈ ∂G1, ξ2 ∈ ∂G2, where θ is a constant, θ
<
1
p
+
,
b
∈
L
1
(
Ω
)
,
d
i
∈
L
h
i
(
x
)
(
Ω
)
,
h
i
,
k
i
∈
C
(
Ω
̄
)
,
k
i
(
x
)
<
h
i
(
x
)
-
1
h
i
(
x
)
p
*
(
x
)
f
o
r
x
∈
Ω
¯
,
k
i
+
<
p
-
.
Then φ satisfies the nonsmooth (PS) condition on X.
Proof. Let {un} be a nonsmooth (PS) sequence, then by (B) we have
c
+
1
+
u
n
≥
φ
(
u
n
)
-
θ
⟨
ω
,
u
n
⟩
=
∫
Ω
1
p
(
x
)
-
θ
∇
u
n
p
(
x
)
d
x
-
∫
Ω
(
F
(
x
,
u
n
)
-
θ
λ
a
1
(
x
)
⟨
ξ
1
,
u
n
⟩
-
θ
μ
a
2
(
x
)
⟨
ξ
2
,
u
n
⟩
)
d
x
≥
1
p
+
-
θ
|
|
u
n
|
|
p
-
-
c
1
-
∫
Ω
b
(
x
)
+
∑
i
=
1
m
d
i
(
x
)
|
u
n
|
k
i
(
x
)
d
x
≥
1
p
+
-
θ
|
|
u
n
|
|
p
-
-
c
2
-
∑
i
=
1
m
|
u
n
|
(
k
i
(
x
)
,
d
i
(
x
)
)
k
i
+
≥
1
p
+
-
θ
|
|
u
n
|
|
p
-
-
c
2
-
c
3
∑
i
=
1
m
|
|
u
n
|
|
k
i
+
,
and consequently {un} is bounded.
Thus by passing to a subsequence if necessary, we may assume that un ⇀ u in X as n → ∞. We have
⟨
J
′
(
u
n
)
,
u
n
-
u
⟩
-
∫
Ω
λ
ξ
1
n
(
x
)
a
1
(
x
)
(
u
n
-
u
)
-
∫
Ω
μ
ξ
2
n
(
x
)
a
2
(
x
)
(
u
n
-
u
)
d
x
≤
ε
n
|
|
u
n
-
u
|
|
with εn ↓ 0, where ξin(x) ∈ ∂Gi(x, un) for a.e. x ∈ Ω, i = 1, 2. From Chang 22 or Theorem 1.3.10 of 33, we know that
ξ
i
n
(
x
)
∈
L
q
1
0
(
x
)
,
i
=
1
,
2
. Since X is embedded compactly in
L
(
a
i
(
x
)
)
q
i
(
x
)
q
i
(
x
)
(
Ω
)
, we have that un → u as n → ∞ in
L
(
a
i
(
x
)
)
q
i
(
x
)
q
i
(
x
)
(
Ω
)
,
i
=
1
,
2
. So using Proposition 2.2, we have
∫
Ω
ξ
i
n
(
x
)
a
i
(
x
)
(
u
n
-
u
)
d
x
→
0
as
n
→
∞
,
i
=
1
,
2
.
Therefore we obtain
lim
sup
n
→
∞
⟨
J
′
(
u
n
)
,
u
n
-
u
⟩
≤
0
. But we know that J' is a mapping of type (S+). Thus we have
u
n
→
u
in
X
.
Remark 3.1. Note that our condition (1.2) is stronger than (1.2) of 21. Because Ψ' is weakly-strongly continuous in 21, to verify that φ satisfies (PS) condition on X, it is enough to verify that any (PS) sequence is bounded. However, in this paper we do not know whether ξ(u) is weakly-strongly continuous, where ξ(u) ∈ ⇀Ψ. Therefore, it will be very useful to consider this problem.
Below we denote
F
1
(
x
,
t
)
=
λ
a
1
(
x
)
G
1
(
x
,
t
)
,
F
2
(
x
,
t
)
=
μ
a
2
(
x
)
G
2
(
x
,
t
)
.
We shall use the following conditions.
(B1) ∃ c0 > 0 such that G2(x, t) ≥ - c0 for x ∈ Ω and t ∈ ℝ.
(B2) ∃
θ
∈
0
,
1
p
+
and M > 0 such that 0 < G2(x, u) ≤ θ 〈u, ξ2〉 for x∈ Ω, u ∈ X and |u| ≥ M, ξ2 ∈ ⇀G2.
Corollary 3.1. Assume (P), (A) and (A1) hold. Then φ satisfies nonsmooth (PS) condition on X provided either one of the following conditions is satisfied.
(1). λ ∈ ℝ and μ = 0.
(2). λ ∈ ℝ, μ = 0 and (B1) holds.
(3). λ ∈ ℝ, μ ∈ ℝ and (B2) holds.
Proof. In case (1) or (2), we have, for x ∈ Ω and t ∈ ℝ,
F
(
x
,
t
)
≤
F
1
(
x
,
t
)
+
|
μ
|
c
0
a
2
(
x
)
≤
(
c
1
a
1
(
x
)
+
|
μ
|
c
0
a
2
(
x
)
)
+
c
2
a
1
(
x
)
|
t
|
q
1
(
x
)
,
which shows that the condition (B) with θ = 0 is satisfied.
In case (3), noting that (B2) and (A) imply (B1), by the conclusion (1) and (2) we know φ satisfies (PS) condition if μ ≤ 0. Below assume μ > 0. The conditions (B2) and (A) imply that, for x ∈ Ω and u ∈ X,
G
2
(
x
,
u
)
≤
θ
⟨
u
,
ξ
2
⟩
+
c
3
,
and
F
2
(
x
,
u
)
≤
θ
μ
a
2
(
x
)
⟨
u
,
ξ
2
⟩
+
c
3
μ
a
2
(
x
)
,
so we have
F
(
x
,
u
)
-
θ
λ
a
1
(
x
)
⟨
ξ
1
,
u
⟩
-
θ
μ
a
2
(
x
)
⟨
ξ
2
,
u
⟩
=
(
F
1
(
x
,
u
)
-
θ
λ
a
1
(
x
)
⟨
ξ
1
,
u
⟩
)
+
(
F
2
(
x
,
u
)
-
θ
μ
a
2
(
x
)
⟨
ξ
2
,
u
⟩
)
≤
c
1
a
1
(
x
)
+
c
2
a
1
(
x
)
|
u
|
q
1
(
x
)
+
c
3
μ
a
2
(
x
)
,
which shows (B) holds. The proof is complete. ■
As X is a separable and reflexive Banach space, there exist (see [34, Section 17])
e
n
n
=
1
∞
⊂
X
and
f
n
n
=
1
∞
⊂
X
*
such that
f
n
e
m
=
δ
n
,
m
=
1
if
n
=
m
0
if
n
≠
m
,
X
=
span
¯
{
e
n
:
n
=
1
,
2
,
…
,
}
,
X
*
=
span
¯
W
*
{
f
n
:
n
=
1
,
2
,
…
,
}
.
For k = 1, 2, . . . , denote
X
k
=
span{
e
k
}
,
Y
k
=
⊕
j
=
1
k
X
j
,
Z
k
=
⊕
j
=
k
∞
X
j
¯
.
Proposition 3.5. 35 Assume that Ψ : X → ℝ is weakly-strongly continuous and Ψ (0) = 0. Let γ > 0 be given. Set
β
k
=
β
k
(
γ
)
=
sup
u
∈
Z
k
,
u
≤
γ
|
Ψ
(
u
)
|
.
Then βk → 0 as k → ∞.
Proposition 3.6. (Nonsmooth Mountain pass theorem, see 2333) If X is a reflexive Banach space, φ : X → ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)c-condition, and for some r > 0 and e1 ∈ X with ||e1|| > r, max{φ(0), φ(e1)} ≤·inf{φ(u) : ||u|| = r}. Then φ has a nontrivial critical u ∈ X such that the critical value c = φ(u) is characterized by the following minimax principle
c
=
inf
γ
∈
Γ
max
t
∈
[
0
,
1
]
φ
(
γ
(
t
)
where Γ = {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e1}.
Proposition 3.7. (Nonsmooth Fountain theorem, see 36) Assume (F1) X is a Banach space, φ : X → ℝ be an invariant locally Lipschitz functional, the subspaces Xk, Yk and Zk are defined by (3.3).
If, for every k ∈ ℕ, there exist ρk > rk > 0 such that
(F2)
a
k
:
=
inf
u
∈
Z
k
‖
u
‖
=
r
k
φ
(
u
)
→
∞
,
k
→
∞
,
(F3)
b
k
:
=
max
u
∈
Y
k
‖
u
‖
=
ρ
k
φ
(
u
)
≤
0
,
(F4) φ satisfies the nonsmooth (PS)c condition for every c > 0, then φ has an unbounded sequence of critical values.
Proposition 3.8. (Nonsmooth dual Fountain theorem, see 37) Assume (F1) is satisfied and there is a k0 > 0 such that, for each k ≥ k0, there exists ρk > γk > 0 such that
(D1)
a
k
:
=
inf
u
∈
Z
k
‖
u
‖
=
ρ
k
φ
(
u
)
≥
0
,
(D2)
b
k
:
=
max
u
∈
Y
k
‖
u
‖
=
r
k
φ
(
u
)
<
0
,
(D3)
d
k
:
=
inf
u
%
Z
k
|
|
u
|
|
≤
ρ
k
φ
(
u
)
→
0
,
k
→
∞
,
(D4) φ satisfies the nonsmooth
(
PS
)
c
*
condition for every c
∈
[
d
k
0
,
0
)
, then φ has a sequence of negative critical values converging to 0.
Remark 3.2. We say φ that satisfies the nonsmooth
(
PS
)
c
*
condition at level c ∈ ℝ (with respect to (Yn)) if any sequence {un} ⊂ X such that
n
j
→
∞
,
u
n
j
∈
Y
n
j
,
φ
(
u
n
j
)
→
c
,
m
|
Y
n
j
(
u
n
)
→
0
contains a subsequence converging to a critical point of φ.
4 Existence and multiplicity of solutions
In this section, using the critical point theory, we give the existence and multiplicity results for problem (1.1). We shall use the following assumptions:
(
O
1
)
∃
δ
1
>
0
,
c
3
>
0
and
q
3
∈
C
(
Ω
̄
)
with
q
3
(
x
)
<
p
a
1
(
x
)
*
(
x
)
for
x
∈
Ω
̄
and
q
3
+
<
p
-
,
such that
G
1
(
x
,
t
)
≥
c
3
t
q
3
(
x
)
,
∀
x
∈
Ω
,
∀
t
∈
0
,
δ
1
.
(
O
2
)
∃
δ
2
>
0
,
c
4
>
0
and
q
4
∈
C
(
Ω
̄
)
with
q
4
(
x
)
<
p
a
2
(
x
)
*
(
x
)
for
x
∈
Ω
̄
and
q
4
-
>
p
+
,
such that
|
G
2
(
x
,
t
)
|
≤
c
4
|
t
|
q
4
(
x
)
,
∀
x
∈
Ω
,
∀
|
t
|
≤
δ
2
.
(S) For i = 1, 2, Gi(x, -t) = Gi(x, t), ∀x ∈ Ω, ∀t ∈ ℝ.
Remark 4.1.
(1) It follows from (A), (A2) and (O2) that
G
2
(
x
,
t
)
≤
c
4
|
t
|
q
4
(
x
)
+
c
5
|
t
|
q
2
(
x
)
,
∀
x
∈
Ω
,
∀
t
∈
ℝ
.
(2)It follows from (A) and (B2) that (see [33, p. 298])
G
(
x
,
t
)
≥
c
6
|
t
|
1
/
θ
-
c
7
,
∀
x
∈
Ω
,
∀
t
∈
ℝ
.
The following is the main result of this article.
Theorem 4.1. Assume (P), (A), (A1) hold.
(1) If (B1) holds, then for every λ ∈ ℝ and μ ≤ 0, problem (1.1) has a solution which is a minimizer of the corresponding functional φ.
(2) If (B1), (A2), (O1), (O2) hold, then for every λ > 0 and μ ≤ 0, problem (1.1) has a nontrivial solution v1 such that v1 is a minimizer of φ and φ(v1) < 0.
(3) If (A2), (B2), (O2) hold, then for every μ > 0, there exists λ0(μ) > 0 such that when |λ| ≤ λ0(μ), problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0.
(4) If (A2), (B2), (O1), (O2) holds, then for every μ > 0, there exists λ0(μ) > 0 such that when 0 < λ ≤ λ0(μ), problem (1.1) has two nontrivial solutions u1 and v1 such that φ(u1) > 0 and φ(v1) < 0.
(5) If (A2), (B2), (O1), (O2) and (S) holds, then for every μ > 0 and λ ∈ ℝ, problem (1.1) has a sequence of solutions {±uk} such that φ(±uk) → ∞ as k → ∞.
(6) If (A2), (B2), (O1), (O2) and (S) holds, then for every λ > 0 and μ ∈ ℝ, problem (1.1) has a sequence of solutions {±vk} such that φ(±vk) < 0 and φ(±vk) → 0 as k → ∞.
Proof. We will use c, c' and ci as a generic positive constant. By Corollary 3.1, under the assumptions of Theorem 4.1, φ satisfies nonsmooth (PS) condition. We write
Ψ
1
(
u
)
=
λ
∫
Ω
a
1
(
x
)
G
1
(
x
,
u
)
d
x
,
Ψ
2
(
u
)
=
μ
∫
Ω
a
2
(
x
)
G
2
(
x
,
u
)
d
x
,
then Ψ = Ψ1 + Ψ2, φ(u) = J(u) - Ψ (u) = J(u) - Ψ1(u) - Ψ2(u). Firstly, we use
Ψ
i
^
to denote its extension to
L
q
i
(
x
)
(
Ω
)
, where i = 1, 2. From (A) and Theorem 1.3.10 of 33 (or Chang 22), we see that
Ψ
i
^
(u) is locally Lipschitz on
L
q
i
(
x
)
(
Ω
)
and ∂
Ψ
i
^
(
u
)
⊆
{
ξ
i
(
x
)
∈
L
q
i
0
(
Ω
)
:
ξ
i
(
u
)
∈
∂
G
i
(
x
,
u
)
}
for a.e. x ∈ Ω and i = 1, 2. In view of Proposition 2.4 and Theorem 2.2 of 22, we have that
Ψ
i
=
Ψ
i
^
|
X
is also locally Lipschitz, and ∂Ψ1(u) ⊆ λ ∫Ω a1(x) ∂G1(x, u) dx, ∂Ψ2(u) ⊆ μ ∫Ω a2(x) ∂G1(x, u) dx, (see 38), where
Ψ
i
^
|
X
stands for the restriction of
Ψ
i
^
to X for i = 1, 2. Therefore, φ is a locally Lipschitz functional on X.
(1) Let λ ∈ ℝ and μ ≤ 0. By (A),
|
Ψ
1
(
u
)
|
≤
c
1
∫
Ω
a
1
(
x
)
|
u
|
q
1
(
x
)
d
x
+
c
2
≤
c
1
(
|
u
|
(
q
1
(
x
)
,
a
1
(
x
)
)
q
1
+
+
c
3
≤
c
4
|
|
u
|
|
q
1
+
+
c
3
.
By (B1), Ψ2(u) ≤ - μc0 ∫Ω a2(x) dx = c5. Hence φ
(
u
)
≥
1
p
+
|
|
u
|
|
p
-
-
c
4
|
|
u
|
|
q
1
+
-
c
6
. By (A1),
q
1
+
<
p
-
, so φ is coercive, that is, φ(u) → ∞ as ||u|| → ∞. Thus φ has a minimize which is a solution of (1.1).
(2) Let λ > 0, μ ≤ 0 and the assumptions of (2) hold. By the above conclusion (1), φ has a minimize v1. Take
v
0
∈
C
0
∞
(
Ω
)
such that 0 ≤ v0(x) ≤ min{δ1, δ2},
∫
Ω
a
1
(
x
)
v
0
(
x
)
q
3
(
x
)
d
x
=
d
1
>
0
and
∫
Ω
a
2
(
x
)
v
0
(
x
)
q
4
(
x
)
d
x
=
d
2
>
0
. By (O1) and (O2) we have, for t ∈ (0, 1) small enough,
φ
(
t
v
0
)
=
∫
Ω
1
p
(
x
)
|
t
∇
v
0
|
p
(
x
)
d
x
-
λ
∫
Ω
a
1
(
x
)
G
1
(
x
,
t
v
0
(
x
)
)
d
x
-
μ
∫
Ω
a
2
(
x
)
G
2
(
x
,
t
v
0
(
x
)
)
d
x
≤
t
p
-
∫
Ω
1
p
(
x
)
|
∇
v
0
|
p
(
x
)
d
x
-
λ
∫
Ω
a
1
(
x
)
c
3
(
t
v
0
(
x
)
)
q
3
(
x
)
d
x
-
μ
∫
Ω
a
2
(
x
)
c
4
(
t
v
0
(
x
)
)
q
4
(
x
)
d
x
≤
t
p
-
∫
Ω
1
p
(
x
)
|
∇
v
0
|
p
(
x
)
d
x
-
t
q
3
+
λ
c
3
d
1
-
t
q
4
-
μ
c
4
d
2
.
Since
q
3
+
<
p
-
<
q
4
-
, we can find t0 ∈ (0, 1) such that φ(t0v0) < 0, and this shows φ(v1) = infu∈X φ(u) < 0. So v1 ≠ 0 because φ(0) = 0. The conclusion (2) is proved.
(3) Let μ > 0 and the assumptions of (3) hold. By Remark 4.1.(1), for sufficiently small ||u||
Ψ
2
(
u
)
≤
μ
∫
Ω
a
2
(
x
)
c
4
|
u
|
q
4
(
x
)
+
c
5
|
u
|
q
2
(
x
)
d
x
≤
μ
c
4
|
u
|
(
q
4
(
x
)
,
a
2
(
x
)
)
q
4
-
+
μ
c
5
|
u
|
(
q
2
(
x
)
,
a
2
(
x
)
)
q
2
-
≤
μ
c
8
|
|
u
|
|
q
4
-
+
|
|
u
|
|
q
2
-
.
Since
p
+
<
q
2
-
and
p
+
<
q
4
-
, there exists γ > 0 and α > 0 such that J(u) - Ψ2(u) ≥ α for u ∈ Sγ. We can find λ0(μ) > 0 such that when |λ| ≤ λ0(μ), Ψ1(u) ≤ α/2 for u ∈ Sγ. So when |λ| ≤ λ0(μ), φ(u) ≥ α/2 > 0 for u ∈ Sγ. By Remark 4.1.(2), noting that 1
θ
>
p
+
>
q
1
+
, we can find a u0 ∈ X such that ||u0|| > γ and φ(u0) < 0. By Proposition 3.6 problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0.
(4) Let μ > 0 and the assumptions of (4) hold. By the conclusion (3), we know that, there exists λ0(μ) > 0 such that when 0 < λ ≤ λ0(μ), problem (1.1) has a nontrivial solution u1 such that φ(u1) > 0. Let γ and α be as in the proof of (3), that is, φ(u) ≥ α/2 > 0 for u ∈ Sγ. By (O1), (O2) and the proof of (2), there exists w ∈ X such that ||w|| < γ and φ(w) < 0. It is clear that there is v1 ∈ Bγ, a minimizer of φ on Bγ. Thus v1 is a nontrivial solution of (1.1) and φ(v1) < 0.
(5) Let μ > 0, λ ∈ ℝ and the assumptions of (5) hold. By (S), we can use the nonsmooth version Fountain theorem with the antipodal action of ℤ2 to prove (5) (see Proposition 3.7). Denote
Ψ
(
u
)
=
∫
Ω
F
(
x
,
u
)
d
x
=
λ
∫
Ω
a
1
(
x
)
G
1
(
x
,
u
)
d
x
+
μ
∫
Ω
a
2
(
x
)
G
2
(
x
,
u
)
d
x
.
Let βk(γ) be as in Proposition 3.5. By Proposition 3.5, for each positive integer n, there exists a positive integer k0(n) such that βk(n) ≤ 1 for all k ≥ k0(n). We may assume k0(n) < k0(n + 1) for each n. We define {γk : k = 1, 2, . . . , } by
γ
k
=
n
if
k
0
(
n
)
≤
k
<
k
0
(
n
+
1
)
1
if
1
≤
k
<
k
0
(
1
)
.
Note that γk → ∞ as k → ∞. Then for u ∈ Zk with ||u|| = γk we have
φ
(
u
)
=
∫
Ω
1
p
(
x
)
|
∇
u
|
p
(
x
)
d
x
-
Ψ
(
u
)
≥
1
p
+
(
γ
k
)
p
-
-
1
and consequently
inf
u
∈
Z
k
,
u
=
γ
k
φ
u
→
∞
as
k
→
∞
,
i.e., the condition (F2) of Proposition 3.7 is satisfied.
By (A), (A1), (B2) and Remark 4.1.(2), we have
φ
(
u
)
≤
1
p
-
|
|
u
|
|
p
+
+
c
1
|
λ
|
(
|
u
|
(
q
1
(
x
)
,
a
1
(
x
)
)
)
q
1
+
-
c
6
μ
|
u
|
(
1
/
θ
,
a
2
(
x
)
)
1
/
θ
+
c
9
.
Noting that 1
θ
>
p
+
>
q
1
+
and all norms on a finite dimensional vector space are equivalent each other, we can see that, for each Yk, φ(u) → - ∞ as u ∈ Yk and ||u|| → ∞. Thus for each k there exists ρk > γk such that φ(u) < 0 for u ∈ Yk ∩ Sρk, so the condition (F3) of Proposition 3.7 is satisfied. As was noted earlier, φ satisfies nonsmooth (PS) condition. By Proposition 3.7 the conclusion (5) is true.
(6) Let λ > 0, μ ∈ ℝ and the assumptions of (5) hold. Let us verify the conditions of the Nonsmooth dual Fountain theorem (see Proposition 3.8). By (S), φ is invariant on the antipodal action of ℤ2. For Ψ(u) = ∫Ω F(x, u)dx = Ψ1(u)+ Ψ2(u) let βk(1) be as in Proposition 3.5, that is
β
k
(
1
)
=
sup
u
∈
Z
k
,
|
|
u
|
|
≤
1
|
Ψ
(
u
)
|
.
By Proposition 3.5, there exists a positive integer k0 such that
β
k
(
1
)
≤
1
2
p
+
for all k ≥ k0. Setting ρk = 1, then for k ≥ k0 and u ∈ Zk ∩ S1, we have
φ
(
u
)
≥
1
p
+
-
1
2
p
+
=
1
2
p
+
>
0
,
which shows that the condition (D1) of Proposition 3.8 is satisfied.
Since X
=
W
0
1
,
p
(
x
)
is the closure of
C
0
∞
(
Ω
)
in
W
1
,
p
(
x
)
(
Ω
)
, we may choose {Yk : k = 1, 2, . . . , }, a sequence of finite dimensional vector subspaces of X defined by (3.5), such that
Y
k
⊂
C
0
∞
(
Ω
)
for all k. For each Yk, because all norms on Yk are equivalent each other, there is ε ∈ (0, 1) such that for every u
∈
Y
k
∩
B
ε
,
|
u
|
∞
≤
min
{
δ
1
,
δ
2
}
,
|
u
|
(
q
3
(
x
)
,
a
1
(
x
)
)
≤
1
and |
u
|
(
q
4
(
x
)
,
a
2
(
x
)
)
≤
1
By (O1) and (O2), for u ∈ Yk ∩ Bε we have
φ
(
u
)
≤
1
p
-
|
|
u
|
|
p
-
-
λ
c
3
∫
Ω
a
1
(
x
)
|
u
|
q
3
(
x
)
d
x
+
|
μ
|
c
4
∫
Ω
a
2
(
x
)
|
u
|
q
4
(
x
)
d
x
≤
1
p
-
|
|
u
|
|
p
-
-
λ
c
3
|
u
|
(
q
3
(
x
)
,
a
1
(
x
)
)
q
3
+
+
|
μ
|
c
4
|
u
|
(
q
4
(
x
)
,
a
2
(
x
)
)
q
4
-
.
Because
q
3
+
<
p
-
<
q
4
-
there exists γk ∈ (0, ε) such that
b
k
:
=
max
u
∈
Y
k
,
|
|
u
|
|
=
γ
k
φ
u
<
0
,
thus the condition (D2) of Proposition 3.8 is satisfied.
Because Yk ∩ Zk ≠ ∅ and γk < ρk, we have
d
k
:
=
inf
u
∈
Z
k
,
|
|
u
|
|
≤
ρ
k
φ
u
≤
b
k
:
=
max
u
∈
Y
k
,
|
|
u
|
|
=
r
k
φ
u
<
0
.
On the other hand, for any u ∈ Zk with ||u|| ≤ 1 = ρk, we have φ(u) = J(u) - Ψ(u) ≥ -Ψ(u) ≥ -βk(1). Noting that βk → 0 as k → ∞, we obtain dk → 0, i.e., (D3) of Proposition 3.8 is satisfied.
Finally let us prove that φ satisfies nonsmooth
(
PS
)
c
*
condition for every c ∈ R. Suppose
{
u
n
j
}
⊂
X
,
n
j
→
∞
,
u
n
j
∈
Y
n
j
,
φ
u
n
j
→
c
and
m
|
Y
n
j
u
n
j
→
0
. Similar to the process of verifying the (PS) condition in the proof of Proposition 3.3, we can get
u
n
j
→
u
in X. Let us prove 0 ∈ ∂φ(u) below. Notice that
0
≤
m
(
u
)
=
m
(
u
)
-
m
(
u
n
j
)
+
m
(
u
n
j
)
=
m
(
u
)
-
m
(
u
n
j
)
+
m
|
Y
n
j
(
u
n
j
)
.
Using Proposition 2.8.4, Going to limit in the right side of above equation, we have
m
(
u
)
≤
0
,
so m(u) ≡ 0, i.e., 0 ∈ ∂φ(u), this shows that φ satisfies the nonsmooth
(
PS
)
c
*
condition for every c ∈ ℝ. So all conditions of Proposition 3.8 are satisfied and the conclusion (6) follows from Proposition 3.8. The proof of Theorem 4.1 is complete. ■
Remark 4.2
Theorem 4.1 includes several important special cases. In particular, in the case of the problem (1.4), i.e., in the case that
a
1
(
x
)
=
1
|
x
|
s
1
(
x
)
,
a
2
(
x
)
=
1
|
x
|
s
2
(
x
)
,
all conditions of Theorem 4.1 are satisfied provided (P), (A*), (A1), and (A2) hold.