Boundary Asymptotic and Uniqueness of Solution for a Problem with p(x)

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 609047, 14 pages doi:10.1155/2008/609047

Research Article Boundary Asymptotic and Uniqueness of Solution for a Problem with px-Laplacian Ionel Rovent¸a Department of Mathematics, University of Craiova, 200585 Craiova, Romania Correspondence should be addressed to Ionel Rovent¸a, [email protected] Received 14 April 2008; Revised 10 June 2008; Accepted 30 August 2008 Recommended by Alexander Domoshnitsky The goal of this paper is to prove the boundary asymptotic behavior of solutions for weighted px-Laplacian equations that take infinite value on a bounded domain. Copyright q 2008 Ionel Rovent¸a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction Let Ω be a bounded domain in Rn , and let f be a nonnegative, nondecreasing, and C1 function on R . We study here the asymptotic boundary behaviour of solutions for the problem Δpx u gxfu,

x ∈ Ω,

ux −→ ∞, as dist x, ∂Ω −→ 0,

1.1

where Ω ⊆ Rn is a bounded domain with C2 boundary. In additional conditions, the uniqueness of solutions is also discussed. For further details, see 1, 2 . Here, Δpx u : div|∇ux|px−2 ∇ux is the px-Laplacian, a function defined on Rn with 1 < px < ∞. First, Bieberbach in 3 , considered the problem Δu gxfu,

x ∈ Ω,

ux −→ ∞, as dist x, ∂Ω −→ 0.

1.2

He shows that 1.2 admits a unique solution if Ω is a smooth planar domain, f is the exponential function, and g is constant 1. Rademacher 4 extended the results to the threedimensional domains; Keller 5 and Osserman 6 gave a necessary and sufficient conditions to solve the problem 1.2 in the n-dimensional case if the domain satisfies inner and outer sphere conditions.

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Journal of Inequalities and Applications

Bandle and Marcus 7 found the boundary asymptotic of solutions for 1.2, when g is continuous and positive, and f satisfies fmt ≤ m1α ft for all 0 < m < 1 and all t ≥ t0 /m. Cˆırstea and R˘adulescu 8–10 prove the uniqueness and asymptotic behavior of solutions for problem 1.2, when f is regularly varying, and g ∈ C0,α Ω is a nonnegative function which is allowed to vanish on the boundary. In the study of p-Laplacian equations, the main difficulty arises from the lack of compactness, but here the px-Laplacian possesses more complicated inhomogeneous nonlinearities, so we need some special techniques. Problems of this type arise in many areas of applied physics including nuclear physics, field theory, solid waves, and problems of false vacuum. Zhang in 11 gives the existence and singularity of blowup solutions for the problem −Δpx u hx, u 0,

x ∈ Ω,

ux −→ ∞, as dist x, ∂Ω −→ 0,

1.3

where Ω B0, R ⊂ RN , px and fx, u satisfy the following. H1 px ∈ C1 Ω is a radial symmetric function and satisfies 1 < p− ≤ p < N,

where p− inf px, Ω

p sup px. Ω

1.4

H2 hx, u is radial with respect to x, hx, · is increasing, and hx, 0 0 for any x ∈ Ω. H3 h : Ω × R → R is a continuous function and satisfies |fx, t| ≤ C1 C2 |t|αx−1 ,

for every x, t ∈ Ω × R, C1 ≥ 0, C2 ≥ 0, α ∈ CΩ,

1 ≤ αx < p∗ x :

1.5

Npx . N − px

2. Preliminaries We recall in what follows some definitions and basic properties of the generalized Lebesgue1,px Sobolev spaces Lpx Ω, W 1,px Ω, and W0 Ω, where Ω is an open subset of Rn . In that context, we refer to the book of Musielak 12 and the papers of Kovácˇ ik and Rákosn´ık 13 and Fan et al. 14–16 . Set ∞ 2.1 L∞ Ω h; h ∈ L Ω, ess inf hx > 1 ∀x ∈ Ω . x∈Ω

For any h ∈ L∞ Ω, we define h ess sup hx, x∈Ω

h− ess inf hx.

2.2

x∈Ω

For any px ∈ L∞ Ω, we define the variable exponent Lebesgue space Lpx Ω

u; u is a measurable real-valued function such that |ux|px dx < ∞ . Ω

2.3

Ionel Rovent¸a

3

We define a norm, the so-called Luxemburg norm, on this space by the formula ux px dx ≤ 1 . |u|px inf μ > 0; μ Ω

2.4

Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces 13, Theorem 2.5 , the Holder inequality holds 13, Theorem ¨ 2.1 , they are reflexive if and only if 1 < p− ≤ p < ∞ 13, Corollary 2.7 , and continuous functions are dense if p < ∞ 13, Theorem 2.11 . The inclusion between Lebesgue spaces also generalizes naturally 13, Theorem 2.8 . If 0 < |Ω| < ∞ and r1 , r2 are variable exponents so that r1 x ≤ r2 x almost everywhere in Ω, then there exists the continuous embedding Lr2 x Ω → Lr1 x Ω, whose norm does not exceed |Ω| 1. We denote by Lp x Ω the conjugate space of Lpx Ω, where 1/px 1/p x 1. type inequality For any u ∈ Lpx Ω and v ∈ Lp x Ω, the Holder ¨ 1 1 uv dx ≤ |u|px |v|p x p− p − Ω

2.5

holds true. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the Lpx Ω space, which is the mapping ρpx : Lpx Ω → R defined by ρpx u

Ω

|u|px dx.

2.6

If u ∈ Lpx Ω and p < ∞, then the following relations hold true: p−

p

p

p−

|u|px > 1 ⇒ |u|px ≤ ρpx u ≤ |u|px , |u|px < 1 ⇒ |u|px ≤ ρpx u ≤ |u|px ,

2.7

|un − u|px −→ 0 ⇐⇒ ρpx un − u −→ 0. Next, we define the variable Sobolev space: W 1,px Ω {u ∈ Lpx Ω; |∇u| ∈ Lpx Ω}.

2.8

On W 1,px Ω, we may consider one of the following equivalent norms: upx |u|px |∇u|px ,

2.9

∇ux px ux px |u| inf μ > 0; dx ≤ 1 . μ μ

2.10

or

Ω

1,px

We also define W0 W 1,px Ω and

Ω as the closure of C0∞ Ω in W 1,px Ω. Assuming p− > 1, the spaces

1,px W0 Ω

are separable and reflexive Banach spaces.

4

Journal of Inequalities and Applications Set Ipx u 1,px

For all u ∈ W0

Ω

|∇u|px |u|px dx.

2.11

Ω, the following relations hold true: −

−

u > 1 ⇒ up ≤ Ipx u ≤ up ,

2.12

u < 1 ⇒ up ≤ Ipx u ≤ up .

Finally, we remember some embedding results regarding variable exponent LebesgueSobolev spaces. For the continuous embedding between variable exponent Lebesgue-Sobolev spaces we refer to 15, Theorem 1.1 . If p : Ω → R is Lipschitz continuous and p < N, then for any q ∈ L∞ Ω with px ≤ qx ≤ Npx/N − px, there is a continuous embedding W 1,px Ω → Lqx Ω. In what concerns the compact embedding, we refer to 15, Theorem 1.3 . If Ω is a bounded domain in Rn , px ∈ CΩ, p > N, then for any qx ∈ L∞ Ω with ess infx∈Ω Npx/N − px − qx > 0 there is a compact embedding W 1,px Ω → Lqx Ω. Our aim is to find the asymptotic boundary behaviour to the problem Δpx u gxfu,

x ∈ Ω,

ux −→ ∞, as dist x, ∂Ω −→ 0.

2.13

# Suppose p ∈ L∞ and limdistx,∂Ω→0 px p . Consider g ∈ CΩ nonnegative, and f satisfies

f ∈ C1 0, ∞,

f0 0, ft > 0 , and f is nondecreasing on 0, ∞, t ∞ 1 dt < ∞, where Ft : fsds. 1/p# 1 Ft 0

2.14 2.15

We will refer to condition 2.15 as the generalized Keller-Osserman condition. Allow g to be unbounded on Ω or to vanish on ∂Ω. 3. Boundary asymptotic and uniqueness Let f satisfy the Keller-Osserman condition. Consider ψ as a function defined on 0, ∞ such that ∞ 1 ds, 3.1 ψt 1/p# # t q Fs where q# is the Holder conjugate of p# , that is, 1/q# 1/p# 1. The function ψ is decreasing so ψ has an inverse φ : 0, ψ0 → 0, ∞. Moreover, φ is a decreasing function with limt→0 φt ∞, and some computations give us 1/p#

φ t ψ −1 t −q# Fφt

,

|φ t|p

#

−2

φ t

q# fφt. p#

3.2

Ionel Rovent¸a

5

Now we will see that 1/q#

φ t p# q# Fφt − φ t q# fφt

3.3

.

Definition 3.1. A positive measurable function defined on a, ∞, for some a > 0 is said to be regularly varying at infinity of index q ∈ R, written f ∈ RVq , provided that ftz tq z→∞ fz

∀t > 0.

lim

3.4

Remark 3.2. Let f ∈ RVσ1 σ 2 > p be a nondecreasing and continuous function. Then, F ∈ RVσ2 and therefore F −1/p ∈ RV−σ−2/p . Since −σ − 2/p < −1, the observation made in 1 allows us to conclude that F −1/p ∈ L1 1, ∞ and hence f satisfies the generalized Keller-Osserman condition. Remark 3.3. Let f ∈ RVσ1 . By the change of variable s tz, we infer that z

Fz

fsds

1

3.5

zftzdt.

0

0

Because f ∈ RVσ1 is continuous, there exists an > 0 such that for every z > we have ftz/fz ≤ tσ1 1. Hence, using Lebesgue’s dominated convergence theorem, the limit shifts with the integral and so Fz z→∞ zfz

1

ftz dt z→∞ fz

1

tσ1 dt

lim

lim

0

0

1 . σ2

3.6

Lemma 3.4. Consider f satisfies the Keller-Osserman condition, then one has #

Fs1/q 0. s→∞ fs

3.7

lim

Proof. We sketch the proof only in the context of regularly varying functions. For further details regarding general case, see 17 . Let f ∈ RVσ1 σ > p# − 2. From fz/Fz 1/zzfz/Fz and using limz→∞ Fz/zfz 1/σ 2, then there exists a constant C > 0 such that limz→∞ Fz/zσ2 C. Finally, limz→∞ fz/zσ1 limz→∞ Fz/zσ2 zfz/Fz Cσ 2. It remains to consider the limit 1/q#

#

Fz1/q Fz/zσ2 lim lim z→∞ fz z→∞ fz/zσ1

z−1/p

#

σ21

0,

3.8

because −1/p# σ 2 1 < 0. Another way to prove the lemma is to consider the derivative #

Fs1/p #

# 1 fzFz−1/q > 0. p#

3.9 #

If Fs1/p is bounded then there exists M > 0 such that 0 < Fs1/p < # M, ∀t > 1. By integrating from 1 to t, it follows that 0 < Fs1/p < Mt and also

6

Journal of Inequalities and Applications #

1/Mt < 1/Fs1/p < ∞, ∀t > 1. If we integrate again from 1 to ∞, we obtain a contradiction with the Keller-Osserman condition. # In conclusion, the derivative of Fs1/q is unbounded. Using L’Hospitals rule, we should obtain much more #

Fz1/q ∞. z→∞ z

3.10

lim

The following lemma will be useful later; see 1 . Lemma 3.5. Let q# ∈ R be the Holder conjugate of p# > 1. If f ∈ RVσ1 σ > p# − 2, then

∞ # # limz→∞ Fz1/q /fz z Fs−1/p ds σ 2 − p# /p# 2 σ. Proof. By applying L’Hospitals rule, we obtain # σ 2 − p# zFz−1/p 1 zfz lim − 1 lim ∞ . # # z→∞ p# Fs−1/p ds z→∞ p Fz z

3.11

In conclusion, σ 2 − p# Fz Fz1/q zFz−1/p lim . lim

z→∞ fz ∞ Fs−1/p# ds z→∞ ∞ Fs−1/p# ds zfz p# 2 σ z z #

#

3.12

Corollary 3.6. Let f ∈ RVσ1 σ > p# be a continuous function. Then, q# σ p# − 2 |φ t|p −2 φ t − # . lim t→0 tfφt σ2 p #

3.13

Proof. See 1, Corollary 3.2 . 1,px

We say that u ∈ Wloc Ω is a continuous local weak solution to the equation Δpx u gxfu on the domain Ω if and only if 1,px |∇u|px−2 ∇u∇ϕdx − gxfuxϕdx, ϕ ∈ W0 D, 3.14 D

D

for every subdomain D Ω. 1,px

∗

Ω as Define L : W 1,px Ω → W0 Lu, ϕ |∇u|px−2 ∇u∇ϕdx gxfuxϕdx, Ω

3.15

Ω

1,px

for every u ∈ W 1,px Ω, v ∈ W0

Ω. 1,px

Lemma 3.7 Comparison principle. Let u, v ∈ W 1,px Ω satisfy Lu − Lv ≥ 0 in W0 1,px

and consider ϕx min{ux − vx, 0}. If ϕx ∈ W0 in Ω. Proof. See 18 .

∗

Ω

Ω (i.e., u ≥ v on ∂Ω), then u ≥ v a.e.

Ionel Rovent¸a

7

Remark 3.8. L defined below satisfy the condition from 18 . Here, we use Lu ≥ Lv in the form px−2 px−2 |∇v| ∇v∇ϕdx gxfvxϕdx ≤ |∇u| ∇u∇ϕdx gxfuxϕdx. Ω

Ω

Ω

Ω

3.16

Given a real number b, let Λb be the class of all positive monotonic functions k ∈ L1 0, ν ∩ C1 0, ν that satisfy t lim t→0

0

ks ds b. kt

3.17

We will let Λ stand for the union of all Λb as b ranges over 0, ∞. For k ∈ Λ, we use the

t notation λt 0 ksds.

t Remark 3.9. Observe that limt→0 0 ks/ktds 0 for any k ∈ Λ. Moreover, in 3.17, 0 ≤ b ≤ 1 if k is nondecreasing and b ≥ 1 if k is nonincreasing. This is true because

t

t ks/ktds 1/kt 0 ksds tkξ/kt, for an ξ ∈ 0, t. 0 Let z be a C2 function on a domain Ω in Rn , f satisfies the Keller-Osserman condition and v φz. A direct computation show that Δpx v px − 1|φ z|px−2 φ z|∇z|px |φ z|px−2 φ zΔpx z |φ z|px−1 ln|φ z||∇z|px−2 ∇px∇zx.

3.18

Theorem 3.10. Suppose f ∈ RVσ1 , σ ≥ p# − 2 satisfies 2.14. Let Ω ⊆ Rn be a bounded C2 domain and let g ∈ CΩ be a nonnegative function such that lim

gx

dx→0 kdx px

A

3.19

for some positive constant A and some k ∈ Λb . Then, any local weak solution u of 2.13 satisfies # 1/2σ−p# p b2 σ − p# ux lim . dx→0 φλdx A2 σ

3.20

Proof. Consider ρ > 0 and Ωρ : {x ∈ Ω : 0 < dx < ρ}.

3.21

We know that Ω is a C2 bounded domain, so there exists positive constant μ, depending only on Ω, such that d ∈ C2 Ωμ ,

|∇d| ≡ 1 on Ωμ ,

3.22

where dx distx, ∂Ω. The existence of such a constant is given by the smoothness of the domain Ω. For ρ ∈ Γ : 0, μ/2, let Ω−ρ : Ωμ \ Ωρ ,

Ωρ : Ωμ−ρ .

3.23

8


The proof concerns two cases which discuss the monotonicity of the function k. In the first case, we take k ∈ Λb for some b and k nonincreasing on 0, ν for some ν > 0. Without loss of generality, we can consider ν > μ. Let z± x : λdx ± λρ,

for x ∈ Ω±ρ .

3.24

Some computations give us |∇z± |px kpx d|∇d|px , Δpx z± px − 1kpx−2 dk d|∇d|px kpx−1 dΔpx d

3.25

kpx−1 d lnkd|∇d|px−2 ∇px∇dx. Using these computations in 3.18 with v± φz± , we find that Δpx v± px − 1|φ z± |px−2 φ z± |∇z± |px |φ z± |px−2 φ z± Δpx z± |φ z± |px−1 ln|φ z± ||∇z± |px−2 ∇px∇z± x px − 1kpx d|φ z± |px−2 φ z± |∇d|px px − 1kpx−2 dk d|φ z± |px−2 φ z± |∇d|px

3.26

kpx−1 d|φ z± |px−2 φ z± Δpx d kpx−1 d lnkd|∇d|px−2 ∇px∇dx|φ z± |px−2 φ z± kpx−1 d|∇d|px−2 ∇px∇dx|φ z± |px−1 ln|φ z± |. Given 0 < < A/2, we define two numbers ϑ± by ϑ± :

p# b2 σ − p# A ± 2 2 σ

1/2σ−p# ,

3.27

where A is the limit in 3.19. Let w± x ϑ± φz± x, x ∈ Ω±ρ . For simplicity, let Lz : −Δpx z gxfz. In order to apply the comparison principle, we have to prove that for ρ ∈ Γ and μ ≥ 0 sufficiently small, we have Lw− ≥ 0 on Ω−ρ ,

Lw ≤ 0 on Ωρ .

3.28

From 3.26, recalling that px − 1 px/qx and |∇d| 1 on Ωμ , we find Lw± gxfw± − Δpx w± Lw± ϑ± px−1 kpx dfφz± gxfϑ± φz± ± ± ± ± x D x D x D x , × D 2 3 1 4 ϑ± px−1 kpx dfφz±

3.29

Ionel Rovent¸a

9

where D1± x −

|φ z± |px−2 φ z± Δpx d, kdfφz±

D2± x −

px k d |φ z± |px−2 φ z± |∇d|px , qx k2 d fφz±

px |φ z± |px−2 φ z± , qx fφz±

|φ z± |px−2 φ z± ln kdϑ± φ z± ln φ z± ± D4 x − |∇d|px−2 ∇px∇dx.

kdf φ z± 3.30

D3± x −

Recalling that φ < 0 and z± x ≤ 2λdx, we infer |D1± x| −

|φ z± |px−2 φ z± |Δpx d| kdfφz±

3.31

px−2 ± φ z λdx |φ z± | ≤ −2 |Δpx d|. kdx z± xfφz±

Since d ∈ C2 Ωμ , limdistx,∂Ω→0 px p# , by Lemma 3.5 and Remark 3.9 from below, we obtain lim

Ω±ρ ×Γdx,ρ→0 ,0

D1± x 0.

3.32

Using limdistx,∂Ω→0 px p# and Corollary 3.2, we have q# σ 2 − p# |φ z± |px−2 φ z± − . p# σ 2 Ω±ρ ×Γdx,ρ→0 ,0 z± xfφz± lim

3.33

With the same argument, we obtain |D4± x| ≤ 2

px−2 ± φ z λdx kdϑ± |φ z± | ln ± . kdx φ z z± xfφz±

3.34

Next, we prove lim

Ω±ρ ×Γdx,ρ→0

λdx kdϑ± ln ± 0. |φ z | ,0 kdx

3.35

Recall that 1/p#

φ z −q# Fφz #

where Az Fφz/φzσ2 1/p .

−q#

1/p#

Azφzσ2/p , #

3.36

10

Journal of Inequalities and Applications If we integrate from 0 to t, we obtain φt−α1 1/p# −q# −α 1

t

3.37

Asds, 0

where α σ 2/p# > 1. Here, when x → 0, we have dx → ρ and z± x → 0. Dividing by t and taking to the limit, we have φt−α1 C, t→0 t

3.38

lim

where C is a suitable positive constant. From 3.36, it follows that limt→0 φt/tβ −C1 , where β 1/−α 1σ 2/p# < 0, and C1 is a positive suitable constant. Now, it will be easy to infer that kdϑ± λdx kdϑ± λdx ln lim ln |φ z± | Ω±ρ ×Γdx,ρ→0 ,0 kdx |λβ z± | Ω±ρ ×Γdx,ρ→0 ,0 kdx lim

lnkt/λβ t lim 0. t→0 kt/λt

3.39

Finally, limΩ±ρ ×Γdx,ρ→0 ,0 D4± x 0. Since k is nonincreasing, we have k dz− ≥ k dλd and k dz ≤ k dλd. Therefore, D2− D3− −

px k d |φ z± |px−2 φ z± px |φ z− |px−2 φ z− − qx k2 d fφz± qx fφz−

px k dλd |φ z± |px−2 φ z± px |φ z− |px−2 φ z− − − D2 D3− , ≥− qx k2 d z− fφz± qx fφz−

3.40

D . and similarly D2 D3 ≤ D 2 3 From 3.19, we see that corresponding to , there is 0 < δ < A/2 such that A − kpx dx ≤ gx ≤ A k p dx,

0 < dx < δ .

3.41

Suppose that the constant μ in 3.22 is chosen such that μ < δ . Using 3.40 in the left side of the inequality 3.29, we get Lw− ≥ ϑ−

px−1 px

k

dfφz−

A − fϑ− φz− ϑ− px−1 fφz−

− − − x D x D x . D1− x D 3 4 2 3.42

Using the same method, we obtain

px−1 px

Lw ≤ ϑ

k

dfφz

A fϑ φz ϑ px−1 fφz

D1 x

D2 x D3 x D4 x . 3.43

Ionel Rovent¸a

11

From 3.2, we infer lim


−

px |φ z± |px−2 φ z± −1. qx fφz±

3.44

By Lemma 3.5, we have lim

Ω±ρ ×Γdx,ρ→0

p# b2 σ − p# ± ± D . x D x − 3 2 2σ ,0

3.45

Thus, as Ω±ρ × Γ dx, ρ → 0 , 0 , the expression in the bracket of 3.42 converges to #

A − ϑ− 2σ−p −

# p b2 σ − p# p# b2 σ − p# A− −1 > 0, 2σ A − 2 2σ

3.46

and the expression in the bracket of 3.43 converges to #

A ϑ 2σ−p −

# p b2 σ − p# p# b2 σ − p# A −1 < 0. 2σ A 2 2σ

3.47

Therefore, for ρ ∈ Γ and μ ≥ 0 sufficiently small, we conclude that Lw− ≥ 0 on Ω−ρ ,

Lw ≤ 0 on Ωρ .

3.48

Thus, −Δp w− ≥ −gxfw−

on Ω−ρ ,

−Δp w ≤ −gxfw

on Ωρ .

3.49

Suppose now that k is nondecreasing. Given 0 < < A/2, from 3.19 we deduce A − kpx dx − ρ ≤ A − kpx dx ≤ gx ≤ A k px dx ≤ A kpx dx ρ.

3.50

Consider w± x ϑ± φλdx ± ρ ϑ± φy±

for x ∈ Ω±ρ .

3.51

Here, y± x : λdx ± ρ. For simplicity, we will use d± for dx ± ρ. Similarly, we obtain Lw± ϑ± px−1 kpx d±fφy± gxfϑ± φy± ± ± ± ± x T x T x T x , × T 2 3 1 4 ϑ± px−1 kpx d±fφy±

3.52

12


where T1± x −

|φ y± |px−2 φ y± Δpx d, kd±fφy±

T2± x −

px k d± |φ y± |px−2 φ y± |∇d|px , qx k2 d± fφy±

T3± x −

px |φ y± |px−2 φ y± , qx fφy±

T4± x −

|φ y± |px−2 φ y± lnkd±ϑ± |φ y± | ln|φ y± | |∇d|px−2 ∇px∇dx. kd±fφy± 3.53

By Lemma 3.5 and Remark 3.9, we obtain a similar inequality by replacing z± with y± : lim

T1± x 0,

lim

T4± x 0,



lim


T2± x T3± x −

3.54

p# b2 σ − p# . 2σ

Now, suppose u is a nonnegative solution of 2.13. We note that u ≤ w− Cu μ

on ∂Ω−ρ ,

3.55

where Cu μ : max{ux | dx ≥ μ}. Since φ and k are nonincreasing follows w x ≤ ϑ φλμ for dx μ − ρ. Therefore, w ≤ u Cw μ on ∂Ωρ ,

3.56

where Cw μ : ϑ φλμ. Furthermore, u and w satisfy the hypothesis from comparison principle on Ωρ for Gx, t −gxft. Moreover, since f is nondecreasing, w− Cu μ and u Cw μ satisfy the hypothesis from comparison principle on Ω−ρ and Ωρ , respectively, for Gx, t −gxft. Therefore by comparison principle, we infer ux ≤ w− x Cu μ x ∈ ∂Ω−ρ ,

w x ≤ u Cw xμ x ∈ ∂Ωρ .

3.57

Hence for x ∈ Ωρ ∩ Ω−ρ , we have ϑ −

Cw μ Cu μ ux ≤ ≤ ϑ− , φλdx ± λρ φλdx ± λρ φλdx ± λρ Cw μ Cu μ ux ≤ ≤ ϑ− . ϑ − φλdx ± ρ φλdx ± ρ φλdx ± ρ

3.58

Ionel Rovent¸a

13

Letting ρ → 0, we see that ϑ −

Cw μ Cu μ ux ≤ ≤ ϑ− , φλdx φλdx φλdx

3.59

for all x ∈ Ωμ . On recalling that φt → ∞ as t → 0, we obtain ϑ ≤ lim inf dx→0

ux ux ≤ lim sup ≤ ϑ− . φλdx φλdx dx→0

3.60

The claimed result follows if we take → 0 . If we want to prove the uniqueness of solutions for 2.13, we need additional condition on f; see 1 . The proof is similar to the proof from 1 . Acknowledgment The author would like to express his gratitude to the referee and to the editor for their detailed and helpful comments. The author has been partially supported by CNCSIS Grant no. 589/2007. References 1 A. Mohammed, “Boundary asymptotic and uniqueness of solutions to the p-Laplacian with infinite boundary values,” Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 480–489, 2007. 2 Q. Zhang, “Boundary blow-up solutions to px-Laplacian equations with exponential nonlinearities,” Journal of Inequalities and Applications, vol. 2008, Article ID 279306, 8 pages, 2008. 3 L. Bieberbach, “Δu eu und die automorphen Funktionen,” Mathematische Annalen, vol. 77, no. 2, pp. 173–212, 1916. 4 H. Rademacher, “Einige besondere probleme der partiellen Differentialgleichungen,” in Die Differential und Integralgleichungen der Mechanik und Physik I, P. Frank and R. von Mises, Eds., pp. 838–845, Rosenborg, New York, NY, USA, 2nd edition, 1943. 5 J. B. Keller, “On solution of Δu fu,” Communications on Pure and Applied Mathematics, vol. 10, no. 4, pp. 503–510, 1957. 6 R. Osserman, “On the inequality Δu ≥ fu,” Pacific Journal of Mathematics, vol. 7, pp. 1641–1647, 1957. 7 C. Bandle and M. Marcus, “‘Large’ solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour,” Journal d’Analyse Mathématique, vol. 58, no. 1, pp. 9–24, 1992. 8 F.-C. Cˆırstea and V. R˘adulescu, “Uniqueness of the blow-up boundary solution of logistic equations with absorbtion,” Comptes Rendus Mathématique. Académie des Sciences. Paris, vol. 335, no. 5, pp. 447– 452, 2002. 9 F.-C. Cˆırstea and V. R˘adulescu, “Nonlinear problems with boundary blow-up: a Karamata regular variation theory approach,” Asymptotic Analysis, vol. 46, no. 3-4, pp. 275–298, 2006. 10 F.-C. Cˆırstea and V. R˘adulescu, “Boundary blow-up in nonlinear elliptic equations of Bieberbach— Rademacher type,” Transactions of the American Mathematical Society, vol. 359, no. 7, pp. 3275–3286, 2007. 11 Q. Zhang, “Existence and asymptotic behavior of positive solutions to px-Laplacian equations with singular nonlinearities,” Journal of Inequalities and Applications, vol. 2007, Article ID 19349, 9 pages, 2007. 12 J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983. 13 O. Kovácˇ ik and J. Rákosn´ık, “On spaces Lpx and W 1,px ,” Czechoslovak Mathematical Journal, vol. 41116, no. 4, pp. 592–618, 1991. 14 X. Fan and X. Han, “Existence and multiplicity of solutions for px-Laplacian equations in RN ,” Nonlinear Analysis: Theory, Methods & Applications, vol. 59, no. 1-2, pp. 173–188, 2004. 15 X. Fan, J. Shen, and D. Zhao, “Sobolev embedding theorems for spaces W k,px Ω,” Journal of Mathematical Analysis and Applications, vol. 262, no. 2, pp. 749–760, 2001.

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