Nonlinear Analysis 72 (2010) 4428–4437

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Positive solutions of the p(x)-Laplace equation with singularnonlinearityJingjing LiuDepartment of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China

a r t i c l e i n f o

Article history:Received 7 September 2009Accepted 10 February 2010

MSC:34B1535A1535G99

Keywords:p(x)-LaplacianSingular boundary value problemSub–supersolution method

a b s t r a c t

In this paper, using sub–supersolution method, we study the positive solution of the p(x)-Laplacian equations −∆p(x)u = f (x, u) with Dirichlet boundary condition on a boundeddomainΩ inRnwith C1,ω boundary for some0 < ω < 1 for a possibly singular nonlinearityf onΩ × (0,∞).

Crown Copyright© 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

LetΩ ⊆ Rn (n ≥ 1) be a bounded domain with C1,ω boundary for some 0 < ω < 1 and f : Ω × (0,∞) −→ [0,∞) bea given singular nonlinearity. In this paper we are concerned with the existence of weak solutions of the following singularboundary value problem:

−∆p(x)u = f (x, u) inΩ,u > 0 inΩ,u = 0 on ∂Ω

(1.1)

where p(x) ∈ C(Ω), 1 < p− = infx∈Ω p(x) ≤ p+ = supx∈Ω p(x) < +∞. Let p(x) satisfy the following condition:[P] The function p(x) is monotone when n = 1; there is a vector l ∈ Rn \ 0 such that for any x ∈ Ω , f (t) = p(x + tl) ismonotone for t ∈ Ix = t|x+ tl ∈ Ωwhen n > 1.The operator −∆p(x)u = −div(|∇u|p(x)−2∇u) is called p(x)-Laplacian, which becomes p-Laplacian when p(x) ≡

p (a constant). The p(x)-Laplacian possesses more complicated nonlinearities than the p-Laplacian. For example, it isinhomogeneous, and in general, it has no first eigenvalue, in other words, the infimum of the eigenvalues of p(x)-Laplacianequals 0 (see [1]). So some techniques used in the constant exponent case cannot be carried out for the variable exponentcase. Moreover, there are two cases about the infimum of the eigenvalues of p(x)-Laplacian, one is the infimum of theeigenvalue is an eigenvalue indeed, the other case is that the infimum of the eigenvalues is not an eigenvalue. In this paper,we consider the first case only.The aim of the present paper is to extend some results of [2] to the variable exponent case by using the sub–supersolution

method. The regularity results and the comparison principles are the bases of the sub–supersolution method. For the

E-mail address: jingjing830306@163.com.

0362-546X/$ – see front matter Crown Copyright© 2010 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2010.02.018

J. Liu / Nonlinear Analysis 72 (2010) 4428–4437 4429

regularity results in the variable exponent case, see [3–5], more precisely, for the L∞ and C0,αregularity, see [5]; for the localC1,α regularity of the minimizers of the corresponding integral functional, see [3]; for the global C1,α regularity, see [4].The present paper is organized as follows. In Section 2, we recall some basic facts about the variable exponent Lebesgue

and Sobolev spaces. In Section 3we provide basic facts and recall some lemmas that will be needed later.We also specify theconditions on the nonlinearity f that will be used throughout the paper. Section 4 contains the main result on the existenceof weak solutions of the problem (1.1). Several results that can be drawn from the main theorem are also given.

2. Preliminaries 1

Let Ω be a bounded domain of Rn(n ≥ 1) with a smooth boundary ∂Ω . Denote by S(Ω) the set of all measurable realfunctions defined on Ω . Note that two measurable functions are considered as the same element of S(Ω) when they areequal almost everywhere.

L∞+(Ω) = p ∈ L∞(Ω) : ess inf

x∈Ωp(x) > 1.

For p ∈ L∞+(Ω), denote

p− = p−(Ω) = ess infx∈Ωp(x), p+ = p+(Ω) = ess sup

x∈Ωp(x).

Define the variable exponent Lebesgue space Lp(x)(Ω) by

Lp(x)(Ω) =u ∈ S(Ω) :

∫Ω

|u(x)|p(x) dx <∞,

with the norm

‖u‖Lp(x)(Ω) = |u|p(x) = inf

λ > 0 :

∫Ω

∣∣∣∣u(x)λ∣∣∣∣p(x) dx ≤ 1

and the variable exponent Sobolev spaceW 1,p(x)(Ω) by

W 1,p(x)(Ω) =u ∈ Lp(x)(Ω) : |∇u| ∈ Lp(x)(Ω)

,

with the norm

‖u‖W1,p(x)(Ω) = |u|p(x) + |∇u|p(x) .

W 1,p(x)0 (Ω) denotes the closure of C∞0 (Ω) inW1,p(x)(Ω). |∇u|Lp(x)(Ω) is an equivalent norm onW

1,p(x)0 , we denote this norm

by ‖u‖. The spaces Lp(x)(Ω),W 1,p(x)(Ω) andW 1,p(x)0 (Ω) are all separable and reflexive Banach spaces. We refer to [6–8] forthe elementary properties of these spaces. Here, we only display some facts which will be used later.For u, v ∈ S(Ω), we write u ≤ v if u(x) ≤ v(x) for a.e. x ∈ Ω . Define u+(x) = maxu(x), 0 and u−(x) = max−u(x), 0.

Proposition 2.1 ([6]). The conjugate space of Lp(x) (Ω) is Lp0(x) (Ω), where 1p(x) +

1p0(x)= 1. For any u ∈ Lp(x) (Ω) and v ∈ Lp

0(x)

(Ω) ,∫Ω|uv| dx ≤ 2 |u|p(x) |v|p0(x).

Proposition 2.2 ([6]). Set ρ(u) =∫Ω|u(x)|p(x) dx. For u, uk ∈ Lp(x) (Ω), we have

(1) For u 6= 0, |u|p(x) = λ⇔ ρ( uλ) = 1;

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

(3) If |u|p(x) > 1, then |u|p−

p(x) ≤ ρ (u) ≤ |u|p+

p(x);

(4) If |u|p(x) < 1, then |u|p+

p(x) ≤ ρ (u) ≤ |u|p−

p(x);(5) limk→∞ |uk|p(x) = 0 ⇐⇒ limk→∞ ρ(uk) = 0;(6) |uk|p(x) →∞ ⇐⇒ ρ(uk)→∞.

Similar to Proposition 2.2, we have

Proposition 2.3. Set φ(u) =∫Ω|∇u|p(x)dx. For u, uk ∈ Lp(x) (Ω), we obtain

(1) For u 6= 0, ‖u‖ = λ⇔ φ( uλ) = 1;

(2) ‖u‖ < 1 (=1;> 1)⇔ φ(u) < 1 (=1;> 1);(3) If ‖u‖ > 1, then ‖u‖p

−

≤ φ (u) ≤ ‖u‖p+

;(4) If ‖u‖ < 1, then ‖u‖p

+

≤ φ (u) ≤ ‖u‖p−

;(5) limk→∞ ‖uk‖ = 0 ⇐⇒ limk→∞ φ(uk) = 0;(6) ‖uk‖ → ∞ ⇐⇒ φ(uk)→∞.

4430 J. Liu / Nonlinear Analysis 72 (2010) 4428–4437

3. Preliminaries 2

Throughout the paper the nonlinearity f : (0,∞) → [0,∞) will be assumed to be a Caratheodory function such thatf (·, t) is in Cθ (Ω) for some 0 < θ < 1. Unless specified otherwise we will also suppose thatΩ ∈ Rn is a bounded domainwith C1,ω boundary for some 0 < ω < 1. The positive constants that will be used in estimations are not necessarily thesame in every occurrence. In what follows, further conditions will be stipulated on f .Now, consider equation−∆p(x)u = f (x, u) inΩ,u = h on ∂Ω. (3.1)

Definition 3.1. (1) Given h ∈ W 1,p(x)(Ω), u ∈ W 1,p(x)(Ω) is called a (weak) solution of (3.1) if u− h ∈ W 1,p(x)0 (Ω) and∫Ω

|∇u|p−2∇u · ∇ϕ =∫Ω

f (x, u)ϕ (3.2)

for all ϕ ∈ W 1,p(x)0 (Ω).(2) u ∈ W 1,p(x)(Ω) is called a sub-solution (respectively a supersolution) of (3.1) if u ≤ (respectively≥) h on ∂Ω and for

all ϕ ∈ W 1,p(x)0 (Ω)with ϕ ≥ 0,∫Ω

|∇u|p−2∇u · ∇ϕ ≤ (respectively ≥)∫Ω

f (x, u)ϕ. (3.3)

Remark 3.2. (1) We say u ∈ W 1,p(x)0 (Ω) is a (weak) solution of (1.1) if u(x) > 0 in Ω and (3.2) holds. Obviously, (1.1) is aspecial case of (3.1).(2) Following usual practice, we shall write u ≤ h onΩ to mean (u− h)+ ∈ W 1,p(x)0 (Ω); u ≥ h onΩ to mean (u− h)−

∈ W 1,p(x)0 (Ω).

Now we give a comparison principle as follows.

Lemma 3.3. Let g(x, t) : Ω × R→ R be measurable and nondecreasing in t. Let u, v ∈ W 1.p(x)(Ω) satisfy

−∆p(x)u+ g(x, u) ≤ −∆p(x)v + g(x, v) (x ∈ Ω).

If u ≤ v on ∂Ω , then u ≤ v onΩ .

Proof. It is an obvious result according to Proposition 2.3. in [9] by using reduction to absurdity.

Lemma 3.4 (See [1]). If n = 1, then the infimum of the eigenvalues of p(x)-Laplacian > 0 if and only if the function p(x) ismonotone.

Lemma 3.5 (See [1]). Let n > 1, if there is a vector l ∈ Rn \ 0 such that for any x ∈ Ω , f (t) = p(x + tl) is monotone fort ∈ Ix = t|x+ tl ∈ Ω, then he infimum of the eigenvalues of p(x)-Laplacian> 0.

In this paper we shall use the notation q(x) for the Höder conjugate of p(x) > 1, that is q(x) := p(x)p(x)−1 . Also we let p

∗(x)be the Sobolev conjugate exponent of p(x), namely,

p∗(x) =

Np(x)N − p(x)

if p(x) < N,

∞ if p(x) ≥ N.

Beforewe state conditions on f thatwill be needed in the paper, let us identify a classG of functions g : (0,∞)→ (0,∞)and a class B of non-negative functions b in Lq(x)(Ω) such that the following conditions hold:

[G1] There is ρ ≥ 1 and a positive constant C such that g(t) ≤ C for all t ≥ ρ.[G2] One of the conditions (1), (2), or (3) below holds, where h(t) = tg(t), t > 0:

(1) h(t) ≤ C for all 0 < t < δ and some positive constants C and δ.(2) h is nondecreasing.(3) (i) g is nonincreasing.

(ii) For each ∈ (0, 1) there is a constant Cθ ≥ 1 such that g(θ t) ≤ Cθg(t) for all t > 0.

(iii) There is ω ∈ C10 (Ω)with ω > 0 onΩ such that bg(ω) ∈ Lp∗p∗−1 (Ω).

J. Liu / Nonlinear Analysis 72 (2010) 4428–4437 4431

We are now ready to list the conditions on f that will be used in this paper. In stating the conditions, we use the notation

γs(x) = supf (x, t) : t ≥ s (x ∈ Ω)

for any s > 0.

[F1] γs(x) ∈ Lq(x)(Ω) for each s > 0.[F2] There is (g, b) ∈ G × B that satisfies conditions [G1]–[G2], a measurable function a for which x ∈ Ω : 0 < a(x) ≤ 1

has a positive measure, and 0 ≤ a ≤ b such that(1) f (x, s) ≥ a(x) for (x, s) ∈ Ω × (0,∞),(2) f (x, s) ≤ b(x)g(s) for (x, s) ∈ Ω × [ρ,∞),(3) f (x, s)g(t) ≤ Cf (x, t)g(s) for x ∈ Ω , 0 < s < t and some constant C > 0.

We start with the following lemma.

Lemma 3.6. Let f satisfy [F1] and h ∈ W 1,p(x)(Ω)with h > 0 onΩ . Let ψsub be a sub-solution andψsup a supersolution of (3.1)onΩ . If 0 < infΩ ψsub ≤ ψsub ≤ ψsup onΩ then there is a solution u ∈ W 1,p(x)(Ω) of (3.1) such that ψsub ≤ u ≤ ψsup a.e. onΩ .

Proof. The lemma is a consequence of [10, Theorem 4.14]. For completeness we include the short proof.Take 0 < l = infΩ ψsub and let F : Ω × R→ R be defined by

F(x, t) =f (x, l) if t < l,f (x, t) if t ≥ l.

Then F(x, ·) is Hölder continuous on R for each x ∈ Ω and |F(x, t)| ≤ γl(x) onΩ . Consider now the Dirichlet problem−∆p(x)u = F(x, u) inΩ,u = h on ∂Ω.

Note that ψsub is a sub-solution, and ψsup is a supersolution of this problem. Since γl ∈ Lq(x)(Ω), it follows by [10, Theorem4.14] that this problem admits a solution u ∈ W 1,p(x)(Ω) such that ψsub ≤ u ≤ ψsup. Finally we note that u is a solution of(3.1) as claimed because of u ≥ l.

4. Existence of weak solutions

We now show the existence of a solution of (1.1) inW 1,p(x)0 (Ω).

Theorem 4.1. Suppose that f satisfies [F1]–[F2]. Then problem (1.1) admits a solution in W1,p(x)0 (Ω).

Proof. By replacing g and b by C−1g and Cb respectively, if necessary, we can assume that g(t) ≤ 1 for all t ≥ ρ in condition[G1]. For k = 1, 2, . . ., let ψk be the solution of

−∆p(x)u(x) = ak(x) inΩ,u(x) = k−1 on ∂Ω (4.1)

where for k = 1, 2, . . .,

ak(x) = mina(x),

k+ 1k

.

Likewise, let ψ∞ be the solution of (4.1) with k = ∞. Note that since ak(x) ∈ L∞(Ω) the problem (4.1) does indeed havea solution. By the comparison lemma, we have 0 ≤ ψ∞ ≤ ψk ≤ ψ1 for all k = 1, 2, . . . and ψk ≥ k−1 on Ω for allk = 1, 2, . . .. By the Strong Maximum Principle [11], we note that ψ∞ > 0 on Ω . So, we can get −∆p(x)ψk ≤ a in Ω , andψk = k−1 on ∂Ω .For each k ∈ Z+ let us now consider the Dirichlet problem−∆p(x)u(x) = f (x, u(x)) inΩ,u(x) = k−1 on ∂Ω. (BVPk)

By (1) of [F2]we observe that

−∆p(x)ψk − f (x, ψk) ≤ −∆p(x)ψk − a(x) ≤ 0

and thus ψk is a sub-solution of (BVPk) for all k.

4432 J. Liu / Nonlinear Analysis 72 (2010) 4428–4437

Now, let ψ be a solution of−∆p(x)u(x) = b(x) inΩ,u(x) = ρ on ∂Ω. (4.2)

Note that (4.2) has a solution since b ∈ Lq(x)(Ω). By comparison principle, we note that ψk ≤ ψ for all k = 1, 2, . . . andψ ≥ ρ onΩ . Then by (2) of [F2]we have

−∆p(x)ψ − f (x, ψ) ≥ −∆p(x)ψ − b(x) = 0.

Thus ψ is a supersolution of (BVPk) on Ω for any k = 1, 2, . . .. Let us fix a positive integer m. By Lemma 3.6 let um bea solution of (BVPm) such that ψm ≤ um ≤ ψ on Ω . Note that um is a supersolution of (BVPm+1) and therefore, again byLemma3.6, there is a solution um+1 of (BVPm+1) such thatψm+1 ≤ um+1 ≤ um onΩ . Continuing in thismanner, we constructa sequence uk of solutions of (BVPk) such that for all k ≥ m

ψ∞ ≤ uk+1 ≤ uk ≤ · · · ≤ um ≤ ψ inΩ.

Let us also note that uk ≥ k−1 onΩ . We define

u(x) = limk→∞

uk(x) (x ∈ Ω). (4.3)

We use ϕk := uk − 1/k as a test function for the solution uk onΩ (by condition [F1]). We obtain∫Ω

|∇uk|p(x) =∫Ω

|∇uk|p(x)−2∇uk · ∇(uk − ε/k) =∫Ω

f (x, uk)(uk − ε/k) ≤∫Ω

f (x, uk)uk. (4.4)

We now use condition [F2] to show that there is a positive constant C , independent of k, such that∫Ω

f (x, uk)uk ≤ C . (4.5)

First we note that conditions (2) and (3) of [F2] imply that

f (x, uk)uk =f (x, uk)ukg(uk)

g(uk)uk ≤ Cf (x, ψ)g(ψ)

g(uk)uk.

If (1) of [G2] holds, then this together with [G1] implies

f (x, ψ)g(ψ)

g(uk)uk ≤ Mbmaxρ, uk ≤ Mbψ

for some positive constantM .If (2) of [G2] holds, then recalling g(ψ) ≤ 1, it follows from (2) of [F2] that

f (x, ψ)g(ψ)

g(uk)uk ≤ f (x, ψ)ψ ≤ bψ.

Therefore, since ψ ∈ Lp(x)(Ω) and b ∈ Lq(x)(Ω), the estimate (4.5) holds if either (1) or (2) of [G2] holds.Suppose now that (3) of [G2] holds. By [11] we note that ψ∞ > 0 on Ω , and ∂ψ∞/∂ν > 0 on ∂Ω , where ν is the unit

inner normal. Thus, we have infΩ(ψ∞/ω) > 0, where ω is as in condition 3(iii) of [G2]. That is θω ≤ ψ∞ on Ω for some0 < θ ≤ 1. Then on using 3(i) and 3(ii) of [G2], we estimate

f (x, ψ)g(ψ)

g(uk)uk ≤f (x, ψ)g(ψ)

g(θω)ψ ≤ Cθbg(ω)ψ.

Since ψ ∈ W 1,p(x)(Ω) ⊆ Lp∗

(Ω), the assumption in 3(iii) of [G2] shows that estimate (4.5) holds in this case as well.Thus estimates (4.4) and (4.5) show that the sequenceψk is bounded inW

1,p(x)0 (Ω). We pick a subsequence, still denoted

by ψk, such that it converges weakly inW1,p(x)0 (Ω), strongly in Lp(x)(Ω) and pointwise a.e. onΩ . Note that ψk converges

to u ∈ W 1,p(x)0 (Ω), and that uk converges weakly to u inW 1,p(x)(Ω), strongly in Lp(x)(Ω) and a.e. onΩ . Now letΩ0 b Ω .OnΩ0 we have

|(f (x, uk)− f (x, uj))(uk − uj)| ≤ 4ψl(x)ψ,

where l = minΩ ψ∞, and hence proceeding as in [12] one can show that∫Ω0

(|∇uk|p(x)−2∇uk − |∇uj|p(x)−2∇uj) · ∇(uk − uj)→ 0

J. Liu / Nonlinear Analysis 72 (2010) 4428–4437 4433

as k, j→∞. Then we can get∫Ω0

|∇uk −∇uj|p(x) → 0 k, j→∞ (4.6)

by using the inequality (see [13])

[(|ξ |p−2ξ − |η|p−2η)(ξ − η)] · (|ξ |p + |η|p)(2−p)/p ≥ (p− 1)|ξ − η|p, 1 < p < 2;

(|ξ |p−2ξ − |η|p−2η)(ξ − η) ≥

(12

)p|ξ − η|p, p ≥ 2.

Since uk converges strongly to u in Lp(x)(Ω0), the limits (4.6) shows that the sequence uk is Cauchy inW 1,p(x)(Ω0), andhence converges to u inW 1,p(x)(Ω0). In conclusion, given any compact setΩ0 ⊂ Ω , we can find a subsequence of uk thatconverges to u strongly inW 1,p(x)(Ω0). Let us take note of the following estimates. If p(x) ≥ 2, then by the Hölder inequalityPropositions 2.1 and 2.3, we have∫

Ω0

|∇uk −∇u|(|∇uk| + |∇u|)p(x)−2dx ≤ 2‖∇uk −∇u‖Lp(x)(Ω0 )‖(|∇uk| + |∇u|)p(x)−2‖Lq(x)(Ω0)

= 2‖∇uk −∇u‖Lp(x)(Ω0)(‖(|∇uk| + |∇u|)p(x)−2‖q−

Lq(x)(Ω0))1q−

≤ 2‖uk − u‖W1,p(x)(Ω0)

(∫Ω0

(|∇uk| + |∇u|)(p(x)−2)p(x)p(x)−1 dx

) 1q−

.

Here we consider ‖(|∇uk| + |∇u|)p(x)−2‖Lq(x)(Ω0) ≥ 1 and we can just change q− to q+ if ‖(|∇uk| + |∇u|)p(x)−2‖Lq(x)(Ω0) ≤ 1.

Then we will give an estimate about (∫Ω0(|∇uk| + |∇u|)

(p(x)−2)p(x)p(x)−1 dx)

1q− . Let m(x) = (p(x)−2)p(x)

p(x)−1 , n(x) = p(x)−1p(x)−2 then we can

get ∫Ω0

(|∇uk| + |∇u|)(p(x)−2)p(x)p(x)−1 dx =

∫Ω0

(|∇uk| + |∇u|)m(x)dx

≤ C‖(|∇uk| + |∇u|)m(x)‖Ln(x)(Ω0)

= C(‖(|∇uk| + |∇u|)m(x)‖n′

Ln(x)(Ω0))1n′

≤ C(∫

Ω0

(|∇uk| + |∇u|)m(x)·n(x)dx) 1n′

= C(∫

Ω0

(|∇uk| + |∇u|)p(x)dx) 1n′

≤ 2p++1C

(∫Ω0

(|∇uk|p(x) + |∇u|p(x))dx) 1n′

≤ C(‖∇uk‖

p+

Lp(x)(Ω0)+ ‖∇uk‖

p−

Lp(x)(Ω0)+ ‖∇u‖p

+

Lp(x)(Ω0)+ ‖∇u‖p

−

Lp(x)(Ω0)

) 1n′

≤ C

where

(‖(|∇uk| + |∇u|)m(x)‖n′

Ln(x)(Ω0))1n′ = min

(‖(|∇uk| + |∇u|)m(x)‖n

±

Ln(x)(Ω0))1n±

and in the last inequality, we have used the boundedness of uk inW 1,p(x)(Ω0). So we can get that∫Ω0

|∇uk −∇u|(|∇uk| + |∇u|)p(x)−2dx ≤ C‖uk − u‖W1,p(x)(Ω0). (4.7)

Similarly, let d(x) = p(x)p(x)−1 we can get∫

Ω0

|∇uk −∇u|p(x)−1dx ≤ C‖|∇uk −∇u|p(x)−1‖Ld(x)(Ω0)

= C(‖|∇uk −∇u|p(x)−1‖d

′

Ld(x)(Ω0)

) 1d′

4434 J. Liu / Nonlinear Analysis 72 (2010) 4428–4437

≤ C(∫

Ω0

|∇uk −∇u|(p(x)−1)d(x)dx) 1d′

= C(∫

Ω0

|∇uk −∇u|p(x)dx) 1d′

≤ C(‖∇uk −∇u‖

p+

Lp(x)(Ω0)+ ‖∇uk −∇u‖

p−

Lp(x)(Ω0)

) 1d′

≤ C(‖∇uk −∇u‖

p+

W1,p(x)(Ω0)+ ‖∇uk −∇u‖

p−

W1,p(x)(Ω0)

) 1d′, (4.8)

where(‖|∇uk −∇u|p(x)−1‖d

′

Ld(x)(Ω0)

) 1d′= min

(‖|∇uk −∇u|p(x)−1‖d

±

Ld(x)(Ω0)

) 1d±.

We note that the inequality above is (4.8). We now recall some useful inequalities (see [14]) that hold for all ξ , η ∈ Rn:∣∣|ξ |p−2ξ − |η|p−2η∣∣ ≤ C |ξ − η|(|ξ | + |η|)p−2 if p ≥ 2,C |ξ − η|p−1 if 1 < p ≤ 2

(4.9)

where C is a positive constant independent of ξ and η. The estimates (4.7) and (4.8) togetherwith the inequalities (4.9) showthat

limk→∞

∫Ω0

∣∣|∇uk|p−2∇uk − |∇u|p−2∇u∣∣ = 0. (4.10)

Now let ϕ ∈ C∞0 (Ω)with suppϕ ⊆ Ω0 b Ω . It then follows from the limit (4.10) that∫Ω

|∇uk|p(x)−2∇uk · ∇ϕ→∫Ω

|∇u|p(x)−2∇u · ∇ϕ. (4.11)

Since |f (x, uk)ϕ| ≤ Cγl(x) onΩ0 and γl(x) ∈ L1(Ω), it follows that∫Ω

f (x, uk)ϕ→∫Ω

f (x, u)ϕ. (4.12)

Therefore (4.11) and (4.12) show that the identity (3.2) holds for all ϕ ∈ C∞0 (Ω).We proceed to show that it also holds for any ϕ ∈ W 1,p(x)0 (Ω). We choose a sequence ηk of non-negative functions in

C∞0 (Ω) such that ηk → |ω| inW1,p(x)0 (Ω). By going to a subsequence if necessary, we can assume that ηk → |ω| a.e. onΩ .

Then, by the Fatou lemma and the Höder inequality and Proposition 2.3 we see that∣∣∣∣∫Ω

f (x, u)ω∣∣∣∣ ≤ ∫

Ω

f (x, u)|ω|

≤ lim infk→∞

∫Ω

f (x, u)ηk

= limk→∞

∫Ω

|∇u|p(x)−2∇u · ∇ηk

≤ (‖u‖p+

W1,p(x)0 (Ω)+ ‖u‖p

−

W1,p(x)0 (Ω))M limk→∞‖ηk‖W1,p(x)0 (Ω)

= (‖u‖p+

W1,p(x)0 (Ω)+ ‖u‖p

−

W1,p(x)0 (Ω))M‖ω‖W1,p(x)0 (Ω)

whereM is an appropriate positive constant according to the computation. Now if ϕ ∈ W 1,p(x)0 (Ω), and ϕk → ϕ, then takingω := ϕk − ϕ in the above inequality shows that

limk→∞

∫Ω

f (x, uk)ϕ =∫Ω

f (x, u)ϕ.

We also have

lim∫Ω

|∇u|p(x)−2∇u · ∇ϕk =∫Ω

|∇u|p(x)−2∇u · ∇ϕ.

J. Liu / Nonlinear Analysis 72 (2010) 4428–4437 4435

It follows that (3.2) holds for all ϕ ∈ W 1,p(x)0 (Ω). Thus u ∈ W 1,p(x)0 (Ω) is a solution of (1.1) such that ψ∞ ≤ u ≤ ψ0 onΩ .

As an immediate corollary we have the following, which reduces to Corollary 3.2 of [2] when p(x) ≡ p.

Corollary 4.2. Let g : (0,∞)→ (0,∞) be nonincreasing, and assume that g ∈ L1(0, δ) for some δ > 0. If f (x, t) = b(x)g(t)for some non-trivial and non-negative b ∈ Lq(x)(Ω), then (1.1) has a solution.

Proof. It is enough to realize that if g satisfies the hypothesis, then tg(t) ≤ C for all 0 < t < δ and some positive constantC . Thus, g satisfies both the conditions [G1] and (1) of [G2]. Therefore f satisfies both the conditions [F1]− [F2]. The corollarythen follows from Theorem 4.1.

For the next result, we will need the following condition on g : (0,∞)→ (0,∞), where g is Cα , 0 < α < 1.

[G3] limt→0+ g(t) = ∞.

Consider

G(t) :=∫ d

tg(s)ds, (4.13)

where 0 < d ≤ ∞ is chosen such that the integral (4.13) is finite for 0 < t < d. We defineψ : [0, d] → [0, ψ(d)] to be theincreasing function

ψ(t) =∫ t

0

1G(s)1/p−

ds. (4.14)

Let c = ψ(d) and ϕ : [0, c] → [0, d] be the inverse of ψ . We note some properties of ϕ in the following remark.

Remark 4.3. (1) ϕ satisfies−p−|ϕ′|p

−−2ϕ′′ = g(ϕ) in (0, c),

ϕ(t) > 0 in (0, c],ϕ(0) = 0.

(2) From (1) we see that ϕ′ is decreasing on (0, c) and therefore ϕ′(0+) ∈ (0,∞]. Furthermore ϕ is increasing on (0, c].(3) If, in addition g is nonincreasing, then on invoking Lemma 2.1 of [15] we see that

limt→0+

G(t)g(t)= 0.

Therefore, in this case

−ϕ′(t)ϕ′′(t)

=G(ϕ(t))g(ϕ(t))

·1

ϕ′(t)

is bounded on (0, c].

Remark 4.4. (1) Let ψ(d− δ) = c − ε where δ, ε > 0. Then we can get

ϕ′(c − ε) =1

ψ ′(d− δ)=

1G(d− δ)1/p−

=1∫ d

d−δ g(s)ds.

It follows that ϕ′(c − ε)→∞when δ→ 0. So there is a δ0 > 0 such that ϕ′(c − ε0) > 1, where c − ε0 = ψ(d− δ0).(2) By Lemmas 3.4 and 3.5 and p(x) satisfies the condition [P], we can define the first eigenvalue λ1 = the infimum of theeigenvalues>0 from [1].

By Remark 4.4(2), let z be an positive eigenfunction of −∆p(x) on Ω corresponding to the first eigenvalue λ1. Letz1(x) = µz(x) so that 0 < z1(x) < c−ε0 for all x ∈ Ω whereµ is a positive constant. It is known that z(x), z1(x) ∈ C

1,α0 (Ω)

for some 0 < α < 1 and that |∇z1| 6= 0 on ∂Ω by the regularity results of eigenfunction.

Theorem 4.5. Let f (x, t) = b(x)g(t) where b ∈ L∞(Ω) with infΩ b > 0 and g satisfies 3(i), 3(ii) of [G2] and [G3]. Thenproblem (1.1) has a solution if and only if ϕ(z1)g(ϕ(z1)) ∈ L1(Ω).

4436 J. Liu / Nonlinear Analysis 72 (2010) 4428–4437

Proof. Suppose ϕ(z1)g(ϕ(z1)) ∈ L1(Ω). Since ω := ϕ(z1) ∈ C10 (Ω) it follows that conditions [G1] and 3(iii) of [G2] hold.Therefore the hypotheses of Theorem 4.1 are satisfied by f , and thus problem (1.1) has a solution. For the converse, letω = βϕ(z1)where β ≥ 1 to be determined shortly. Then

|∇ω|p(x)−2∇ω = βp(x)−1(ϕ′(z1))p(x)−1|∇z|p(x)−2∇z · µp(x)−1.

Let η ∈ C∞0 (Ω) and η ≥ 0. By Remarks 4.3(2) and 4.4(1) we can know that ϕ′(z1) ≥ 1. By Remark 4.3(1), (3), and recalling

that z is an eigenfunction of the p(x)-Laplacian, direct computation shows that∫Ω

|∇ω|p(x)−2∇ω · ∇η =

∫Ω

(βp(x)−1|∇z|p(x)−2∇z · µp(x)−1 · ∇((ϕ′(z1))p(x)−1η)

−βp(x)−1µp(x)−2(p(x)− 1) · |∇z|p(x)ϕ′(z1)p(x)−2ϕ′′(z1)η)dx

≥ M1

∫Ω

|∇z|p(x)−2∇z · ∇((ϕ′(z1))p(x)−1η)dx

−M2

∫Ω

(p(x)− 1)|∇z|p(x)ϕ′(z1)p−−2ϕ′′(z1)ηdx

= M1

∫Ω

|∇z|p(x)−2∇z · ∇((ϕ′(z1))p(x)−1η)dx+M2

∫Ω

p(x)− 1p−

|∇z|p(x)g(ϕ(z1))ηdx

= λ1M1

∫Ω

zp(x)−1(ϕ′(z1))p(x)−1ηdx+M2

∫Ω

p(x)− 1p−

|∇z|p(x)g(ϕ(z1))ηdx

where M1, M2 are positive constants determined by our computation and M1, M2 →∞ when β →∞. Consequently weget ∫

Ω

|∇ω|p(x)−2∇ω · ∇η −

∫Ω

bg(ω)η ≥∫Ω

|∇ω|p(x)−2∇ω · ∇η − bg(ϕ)ηdx

=

∫Ω

ηg(ϕ(z))[λ1M1zp(x)−1

(ϕ′(z1))p(x)−1

g(ϕ(z))+ M2

p(x)− 1p−

|∇z|p(x) − b]dx

≥

∫Ω

ηg(ϕ(z))

[M

(zp(x)−1

(ϕ′(z1))p−−1

g(ϕ(z))+p(x)− 1p−

|∇z|p(x))− b

]dx

=

∫Ω

ηg(ϕ(z))[M(zp(x)−1 ·

1p−−ϕ′(z1)ϕ′′(z1)

+p(x)− 1p−

|∇z|p(x))− b

]dx

whereM = minλ1M1, M2.Note that, since |∇z| 6= 0 on ∂Ω , we have

infΩ

[zp(x)−1 ·

1p−−ϕ′(z1)ϕ′′(z1)

+p(x)− 1p−

|∇z|p(x)]> 0

and thus, by choosing β ≥ 1 sufficiently big, we see that∫Ω

|∇ω|p(x)−2∇ω · ∇η ≥

∫Ω

bg(ϕ(z))η ≥∫Ω

bg(ω)η

for all η ∈ C∞0 (Ω)with η ≥ 0. A straightforward application of the Fatou lemma shows that the above inequality holds forall non-negative η ∈ W 1,p(x)0 (Ω). Thus, if u is a solution of (1.1) by the comparison principle, Lemma 3.3, we have

u(x) ≤ βϕ(z(x)) (x ∈ Ω).

Using ϕ := u in (3.2) we note that

(infΩ)

∫Ω

g(u)u ≤∫Ω

bg(u)u =∫Ω

|∇u|p(x)−2 <∞.

Therefore since, by condition 3(ii) of [G2],

Cβϕ(z)g(ϕ(z)) ≤ βϕ(z)g(βϕ(z)) ≤ ug(u) onΩ

for some positive constant Cβ , we have∫Ω

ϕ(z)g(ϕ(z)) <∞.

The proof is complete.

J. Liu / Nonlinear Analysis 72 (2010) 4428–4437 4437

Acknowledgements

The authors would like to thank professor Xianling Fan for clear suggestions and the reviewer(s) for clear valuablecomments and suggestions.

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