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Existence of solutions for a differential inclusion problem with singular

coefficients involving the $p(x)$-Laplacian

Boundary Value Problems 2012, 2012:11 doi:10.1186/1687-2770-2012-11

Guowei Dai (daiguowei@nwnu.edu.cn)Ruyun Ma (mary@nwnu.edu.cn)Qiaozhen Ma (maqzh@nwnu.edu.cn)

ISSN 1687-2770

Article type Research

Submission date 5 November 2011

Acceptance date 9 February 2012

Publication date 9 February 2012

Article URL http://www.boundaryvalueproblems.com/content/2012/1/11

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Existence of solutions for a differential inclusion problem with singular coefficients involving the

p(x)-Laplacian

Guowei Dai∗, Ruyun Ma and Qiaozhen Ma

Department of Mathematics, Northwest Normal University, Lanzhou 730070, P.R China

∗Corresponding author: daiguowei@nwnu.edu.cn

Email addresses:

RM: mary@nwnu.edu.cnQM: maqzh@nwnu.edu.cn

AbstractUsing the non-smooth critical point theory we investigate the existence andmultiplicity of solutions for a differential inclusion problem with singular coefficients

involving the p(x)-Laplacian.

Keywords: p(x)-Laplacian; differential inclusion; singularity.

Mathematics Subject Classification 2000: 35D05; 35J20; 35J60; 35J70

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

where the following conditions are satisfied:

(P) Ω is a bounded open domain in RN , N ≥ 2, p ∈ C(Ω), 1 < p − := infΩp(x) ≤

p+:= supΩp(x) < +∞, λ, µ ∈ R.

(A) For i = 1, 2, a i ∈ L r i (x) (Ω), a i (x) > 0 for x ∈ Ω, G i (x, u) is measurable with spect to x (for every u ∈ R) and locally Lipschitz with respect to u (for a.e x ∈ Ω),

re-∂G i : Ω × R → R is the Clarke sub-differential of G i and |ξ i | ≤ c1+ c2|t| q i (x)−1 for x ∈ Ω,

t ∈ R and ξ i ∈ ∂G i , where c i is a positive constant, r i , q i ∈ C(Ω), r −

A typical example of (1.1) is the following problem involving subcritical Sobolev-Hardyexponents of the form

and in this case the assumption corresponding to (A) is the following

(A)∗ 0 ∈ Ω, for i = 1, 2, ∂G i : Ω × R → R is the Clarke sub-differential of G i and

|ξ i | ≤ c1 + c2|t| q i (x)−1 for x ∈ Ω, t ∈ R and ξ i ∈ ∂G i , where c i 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)

∗ (x), ∀x ∈ Ω. (1.5)

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)-p-Laplacian possesses more complicated nonlinearities than the p-Laplacian; for example, it is inhomogeneous The study of var-

ious mathematical problems with variable exponent growth condition has been receivedconsiderable attention in recent years These problems are interesting in applications andraise many difficult mathematical problems One of the most studied models leading toproblem of this type is the model of motion of electro-rheological fluids, which are charac-terized by their ability to drastically change the mechanical properties under the influence

of an exterior electro-magnetic field [1, 2] Problems with variable exponent growth tions also appear in the mathematical modeling of stationary thermo-rheological viscousflows of non-Newtonian fluids and in the mathematical description of the processes fil-tration of an ideal baro-tropic gas through a porous medium [3, 4] Another field of

condi-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 toeliminate possible noise We refer the reader to [6–11] for an overview of and references

on this subject, and to [12–21] 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 partialdifferential equations with discontinuous nonlinearities, the existence of multiple solu-tions for Dirichlet boundary value problems with discontinuous nonlinearities has beenwidely investigated in recent years Chang [22] extended the variational methods to aclass of non-differentiable functionals, and directly applied the variational methods fornon-differentiable functionals to prove some existence theorems for PDE with discontin-uous nonlinearities Later Kourogenis and Papageorgiou [23] obtained some nonsmoothcritical point theories and applied these to nonlinear elliptic equations at resonance, in-

volving the p-Laplacian with discontinuous nonlinearities In the celebrated work [24, 25],

Ricceri elaborated a Ricceri-type variational principle and a three critical points rem for the Gˆateaux differentiable functional, respectively Later, Marano and Motreanu[26, 27] extended Ricceri’s results to a large class of non-differentiable functionals and

theo-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, ·) ∈ C1(R) and g i = G 0

i satisfying the

Carath´eodory 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 prelimi-

nary knowledge on variable exponent Sobolev spaces and the generalized gradient of thelocally 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 andmultiplicity results for problem (1.1)

2 Preliminaries

Let Ω be a bounded open subset of RN , denote L ∞

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

|u| L p(x)(Ω)= |u| p(x) = inf

W 1,p(x)(Ω) =©u ∈ L p(x) (Ω) : |∇u| ∈ L p(x)(Ω)ª

with the norm

kuk W 1,p(x)(Ω)= |u| L p(x)(Ω)+ |∇u| L p(x)(Ω).

Denote by W01,p(x) (Ω) the closure of C ∞

0 (Ω) in W 1,p(x) (Ω) Hereafter, we always assume that p − > 1.

Proposition 2.1 [7, 31] The spaces L p(x) (Ω) , W 1,p(x) (Ω) and W01,p(x) (Ω) are

sepa-rable and reflexive Banach spaces.

Proposition 2.2 [7, 31] The conjugate space of L p(x) (Ω) is L p0(x) (Ω) , where 1

p(x) +

1

p0(x) = 1 For any u ∈ L p(x) (Ω) and v ∈ L p0(x) (Ω) , RΩ|uv| dx ≤ 2 |u| p(x) |v| p0(x)

Proposition 2.3 [7, 31] In W01,p(x) (Ω) the Poincar´e inequality holds, that is, there

exists a positive constant c such that

|u| L p(x)(Ω) ≤ c |∇u| L p(x)(Ω) , ∀u ∈ W01,p(x) (Ω)

So |∇u| L p(x)(Ω) is an equivalent norm in W01,p(x) (Ω)

Proposition 2.4 [7, 28, 29, 31] Assume that the boundary of Ω possesses the cone

property and p ∈ C(Ω) If q ∈ C(Ω) and 1 ≤ q(x) < p ∗ (x) for x ∈ Ω, then there is a

compact embedding W 1,p(x) (Ω) → L q(x) (Ω).

Let us now consider the weighted variable exponent Lebesgue space

Let a ∈ S(Ω) and a(x) > 0 for x ∈ Ω Define

with the norm

(2) |u| (p(x),a(x)) < 1 (= 1; > 1) ⇔ ρ(u) < 1 (= 1; > 1).

(3) If |u| (p(x),a(x)) > 1, then |u| p (p(x),a(x)) − ≤ ρ (u) ≤ |u| p (p(x),a(x))+ .

(4) If |u| (p(x),a(x)) < 1, then |u| p (p(x),a(x)+ ≤ ρ (u) ≤ |u| p (p(x),a(x)) −

(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

prop-erty and p ∈ C(Ω) Suppose that a ∈ L r(x) (Ω), a(x) > 0 for x ∈ Ω, r ∈ C(Ω) and r − > 1.

then there is a compact embedding W 1,p(x) (Ω) → L q(x) a(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 ∈ L r(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, (2.2)

then there is a compact embedding W 1,p(x) (Ω) → L q(x) (a(x)) q(x) (Ω).

Proof Set r1(x) = r(x) q(x) , then r −

1 > 1 and (a(x)) q(x) ∈ L r1(x)(Ω) Moreover, from(2.2) we can get

1 ≤ q(x) < r1(x) − 1

r1(x) p

∗ (x), ∀x ∈ Ω.

Using Proposition 2.6, we see that the embedding W 1,p(x) (Ω) → L q(x) (a(x)) q(x)(Ω) is compact

Let (X, k · k) 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 Ω u

such that |f (u1)−f (u2)| ≤ Lku1−u2k for all u1, u2 ∈ Ω u , for a constant L > 0 depending

on Ωu The generalized directional derivative of f at the point u ∈ X in the direction

The generalized gradient of f at u ∈ X is defined by

∂f (u) = {u ∗ ∈ X ∗ : hu ∗ , ϕi ≤ f0(u; ϕ) for all ϕ ∈ X}, which is a non-empty, convex and w ∗ -compact subset of X, where h·, ·i 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 dient that will be used throughout the article

gra-Proposition 2.8 (see [22, 32]) (1) Let j : X → R be a continuously differentiable

function Then ∂j(u) = {j 0 (u)}, j0(u; z) coincides with hj 0 (u), zi X and (f + j)0(u, z) =

f0(u; z) + hj 0 (u), zi 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 k w − u0k < δ, there is w0 ∈ ∂f (u0)

|hw − w0, vi| < 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 ∗

u→u0

m(u) ≥ m(u0).

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 {u n } ⊂ X

such that ϕ(u n ) → c and m(u n ) → 0, as n → +∞, has a strongly convergent

subse-quence, where m(u n ) = inf{ku ∗ k X ∗ : u ∗ ∈ ∂ϕ(u n )}.

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 = W01,p(x) (Ω) and kuk = |∇u| p(x) for u ∈ X Denote by u n * u and u n → u the

weak convergence and strong convergence of sequence {u n } in X, respectively, denote by

c and c i the generic positive constants, B ρ = {u ∈ X : kuk < ρ}, S ρ = {u ∈ X : kuk = ρ}.

Set

F (x, t) = λa1(x)G1(x, t) + µa2(x)G2(x, t), (3.1)

where a i and G i (i = 1, 2) are as in (A).

Define the integral functional

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 0 : X → X ∗ is a strictly monotone, bounded homeomorphism, and is of (S+) type,

namely

u n * u and lim n→∞ J 0 (u n )(u n − u) ≤ 0 implies u n → u.

Proposition 3.2 Ψ is weakly–strongly continuous, i.e., u n * u implies Ψ(u n ) → Ψ(u).

Proof Define Υ1 = RΩG1(x, u) dx and Υ2 = RΩG2(x, u) dx In order to prove Ψ is

weakly–strongly continuous, we only need to prove Υ1 and Υ2 are weakly–strongly tinuous Since the proofs of Υ1 and Υ2 are identical, we will just prove Υ1

con-We assume u n * u in X Then by Proposition 2.8.3, we have

Ω

ξ n (x)(u n − u) dx,

where ξ n ∈ ∂G1(, τ n (x)) for some τ n (x) in the open segment joining u and u n From

Chang [22] we know that ξ n ∈ L q0(x)(Ω) So using Proposition 2.5, we have

Υ1(u n ) − Υ1(u) → 0.

Proposition 3.3 Assume (A) holds and F satisfies the following condition:

(B) F (x, u) ≤ θλa1(x)hξ1, ui+θµa2(x)hξ2, ui+b(x)+Pm i=1 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 ∈ L1(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) for x ∈ Ω, k+

i < p − Then ϕ satisfies the nonsmooth (PS) condition on X.

Proof Let {u n } be a nonsmooth (PS) sequence, then by (B) we have

c + 1 + ku n k ≥ ϕ(u n ) − θhω, u n i

=Z

Ω

µ1

and consequently {u n } is bounded.

Thus by passing to a subsequence if necessary, we may assume that u n * u in X as

(a i (x)) qi(x) (Ω), we have that u n → u as n → ∞ in L q i (x)

Therefore we obtain lim sup

n→∞ hJ 0 (u n ), u n − ui ≤ 0 But we know that J 0 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 Ψ0

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

F1(x, t) = λa1(x)G1(x, t), F2(x, t) = µa2(x)G2(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) ≤ θhu, ξ2i 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.

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,

G2(x, u) ≤ θhu, ξ2i + c3, and F2(x, u) ≤ θµa2(x)hu, ξ2i + c3µa2(x),

so we have

F (x, u) − θλa1(x)hξ1, ui − θµa2(x)hξ2, ui = (F1(x, u) − θλa1(x)hξ1, ui)

+ (F2(x, u) − θµa2(x)hξ2, ui)

≤ c1a1(x) + c2a1(x) |u| q1(x) + c3µa2(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

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 , kuk≤γ

|Ψ (u)|

Then β k → 0 as k → ∞.

Proposition 3.6 (Nonsmooth Mountain pass theorem, see [23, 33]) 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 ke1k > r, max{ϕ(0), ϕ(e1)} ≤ inf{ϕ(u) : kuk = 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∈Γmax

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 nach space, ϕ : X → R be an invariant locally Lipschitz functional, the subspaces X k , Y k and Z k are defined by (3.3).

Ba-If, for every k ∈ N, there exist ρ k > r k > 0 such that

then ϕ has an unbounded sequence of critical values.

Proposition 3.8 (Nonsmooth dual Fountain theorem, see [37]) Assume (F1) is satisfied