Existence and Asymptotic Behavior of Boundary Blow-Up Solutions for Weighted 𝑝(𝑥)-Laplacian Equations with Exponential Nonlinearities

Abstract

This paper investigates the following 𝑝(𝑥)-Laplacian equations with exponential nonlinearities: −Δ_𝑝(𝑥)𝑢+𝜌(𝑥)𝑒^{𝑓(𝑥,𝑢)}=0 in Ω, 𝑢(𝑥)→+∞ as 𝑑(𝑥,𝜕Ω)→0, where −Δ_𝑝(𝑥)𝑢=−div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is called 𝑝(𝑥)-Laplacian, 𝜌(𝑥)∈𝐶(Ω). The asymptotic behavior of boundary blow-up solutions is discussed, and the existence of boundary blow-up solutions is given.

1. Introduction

The study of differential equations and variational problems with nonstandard 𝑝(𝑥)-growth conditions is a new and interesting topic. On the background of this class of problems, we refer to [1–3]. Many results have been obtained on this kind of problems, for example, [4–18]. On the regularity of weak solutions for differential equations with nonstandard 𝑝(𝑥)-growth conditions, we refer to [4, 5, 8]. On the existence of solutions for 𝑝(𝑥)-Laplacian equation Dirichlet problems in bounded domain, we refer to [7, 9, 15, 18]. In this paper, we consider the following 𝑝(𝑥)-Laplacian equations with exponential nonlinearities 𝑢(−𝑥Δ)⟶_𝑝(𝑥)𝑢++𝜌∞(𝑥,)𝑒^{𝑓(𝑥,𝑢)}=0,inΩ,as𝑑(𝑥,𝜕Ω)⟶0,(P)

where −Δ_𝑝(𝑥)𝑢=−div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) and Ω=𝐵(0,𝑅)⊂ℝ^𝑁 is a bounded radial domain (𝐵(0,𝑅)={𝑥∈ℝ^𝑁∣|𝑥|<𝑅}). Our aim is to give the asymptotic behavior and the existence of boundary blow-up solutions for problem (P).

Throughout the paper, we assume that 𝑝(𝑥), 𝜌(𝑥), and 𝑓(𝑥,𝑢) satisfy the following.

(H₁) 𝑝(𝑥)∈𝐶¹(Ω) is radial and satisfies 1<𝑝⁻≤𝑝⁺<+∞,where𝑝⁻=inf_Ω𝑝(𝑥),𝑝⁺=sup_Ω𝑝(𝑥).(1.1)

(H₂) 𝑓(𝑥,𝑢) is radial with respect to 𝑥, 𝑓(𝑥,⋅) is increasing, and 𝑓(𝑥,0)=0 for any 𝑥∈Ω.

(H₃) 𝑓∶Ω×ℝ→ℝ is continuous and satisfies||𝑓(𝑥,𝑡)||≤𝐶₁+𝐶₂|𝑡|^𝛾(𝑥),∀(𝑥,𝑡)∈Ω×ℝ,(1.2)

where 𝐶₁, 𝐶₂ are positive constants and 0≤𝛾∈𝐶(Ω).

(H₄) 𝜌(𝑥)∈𝐶(Ω) is a radial nonnegative function, and there exists a constant 𝜎∈[𝑅/2,𝑅) such that 𝜌₀(𝑅−𝑟)^{−𝛽(𝑟)}≤𝜌(𝑟)≤𝜌₁(𝑅−𝑟)^{−𝛽1(𝑟)}for𝑟∈[𝜎,𝑅)uniformly,(1.3)

where 𝜌₀ and 𝜌₁ are positive constants and 𝛽(𝑟) and 𝛽₁(𝑟) are Lipschitz continuous on [𝜎,𝑅], which satisfy 𝛽(𝑟)≤𝛽₁(𝑟)<𝑝(𝑟) for any 𝑟∈[𝜎,𝑅].

The operator −Δ_𝑝(𝑥)𝑢=−div(|∇𝑢|^{𝑝(𝑥)−2}∇𝑢) is called 𝑝(𝑥)-Laplacian. Specifically, if 𝑝(𝑥)≡𝑝 (a constant), (P) is the well-known 𝑝-Laplacian problem. If 𝑓(𝑥,𝑢) can be represented as ℎ(𝑥)𝑓(𝑢), on the boundary blow-up solutions for the following 𝑝-Laplacian equations (𝑝 is a constant):−Δ_𝑝𝑢+ℎ(𝑥)𝑓(𝑢)=0,inΩ,(1.4)

we refer to [19–26], and the following generalized Keller-Osserman condition is crucial^∞11(𝐹(𝑡))^1/𝑝𝑑𝑡<+∞,where𝐹(𝑡)=^𝑡0𝑓(𝑠)𝑑𝑠,(1.5)

but the typical form of 𝑝(𝑥)-Laplacian equation is−Δ_𝑝(𝑥)𝑢+|𝑢|^{𝑞(𝑥)−2}𝑢=0,inΩ,(1.6)

and there are some differences between the results of (1.4) and (1.6) (see [16]).

On the boundary blow-up solutions for the following 𝑝-Laplacian equations with exponential nonlinearities (𝑝 is a constant):−Δ_𝑝𝑢+𝑒^{ℎ(𝑥)𝑓(𝑢)}=0,inΩ,(1.7)

we refer to [20–22], but the results on the boundary blow-up solutions for 𝑝(𝑥)-Laplacian equations are rare (see [16]).

In [16], the present author discussed the existence and asymptotic behavior of boundary blow-up solutions for the following 𝑝(𝑥)-Laplacian equations: 𝑢(𝑥−)Δ⟶_𝑝(𝑥)𝑢++∞𝑓,(𝑥,𝑢)=0,inΩ,as𝑑(𝑥,𝜕Ω)⟶0,(1.8)

on the condition that 𝑓(𝑥,⋅) satisfies polynomial growth condition.

If 𝑝(𝑥) is a function, the typical form of (P) is the following:−Δ_𝑝(𝑥)𝑢+𝜌(𝑥)𝑒^{|𝑢|𝑞(𝑥)−2𝑢}=0,(1.9)

and the method to construct subsolution and supersolution in [16] cannot give the exact asymptotic behavior of solutions for (P). Our results partially generalized the results of [20–22].

Because of the nonhomogeneity of 𝑝(𝑥)-Laplacian, 𝑝(𝑥)-Laplacian problems are more complicated than those of 𝑝-Laplacian ones (see [10]); another difficulty of this paper is that 𝑓(𝑥,𝑢) cannot be represented as ℎ(𝑥)𝑓(𝑢).

2. Preliminary

In order to deal with 𝑝(𝑥)-Laplacian problems, we need some theories on the spaces 𝐿^𝑝(𝑥)(Ω), 𝑊^1,𝑝(𝑥)(Ω) and properties of 𝑝(𝑥)-Laplacian, which we will use later (see [6, 11]). Let𝐿^𝑝(𝑥)(Ω)=𝑢∣𝑢isameasurablereal-valuedfunction,∫_Ω||𝑢(𝑥)||^𝑝(𝑥)𝑑𝑥<∞.(2.1)

We can introduce the norm on 𝐿^𝑝(𝑥)(Ω) by|𝑢|_𝑝(𝑥)=inf𝜆>0∣_Ω|||𝑢(𝑥)𝜆|||^𝑝(𝑥)𝑑𝑥≤1.(2.2)

The space (𝐿^𝑝(𝑥)(Ω), |⋅|_𝑝(𝑥)) becomes a Banach space. We call it generalized Lebesgue space. The space (𝐿^𝑝(𝑥)(Ω), |⋅|_𝑝(𝑥)) is a separable, reflexive, and uniform convex Banach space (see [6, Theorems 1.10, 1.14] ).

The space 𝑊^1,𝑝(𝑥)(Ω) is defined by𝑊^1,𝑝(𝑥)(Ω)=𝑢∈𝐿^𝑝(𝑥)(Ω)∣||∇𝑢||∈𝐿^𝑝(𝑥)(Ω),(2.3)

and it can be equipped with the norm ‖𝑢‖=|𝑢|_𝑝(𝑥)+||∇𝑢||_𝑝(𝑥),∀𝑢∈𝑊^1,𝑝(𝑥)(Ω).(2.4)

𝑊^{01,𝑝(𝑥)}(Ω) is the closure of 𝐶^∞0(Ω) in 𝑊^1,𝑝(𝑥)(Ω). 𝑊^1,𝑝(𝑥)(Ω) and 𝑊^{01,𝑝(𝑥)}(Ω) are separable, reflexive, and uniform convex Banach spaces (see [6, Theorem 2.1]).

If 𝑢∈𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶(Ω), 𝑢 is called a blow-up solution of (P) when it satisfies_𝑄||∇𝑢||^{𝑝(𝑥)−2}∇𝑢∇𝑞𝑑𝑥+_𝑄𝜌(𝑥)𝑓(𝑥,𝑢)𝑞𝑑𝑥=0,∀𝑞∈𝑊^{01,𝑝(𝑥)}(𝑄),(2.5)

for any domain 𝑄⋐Ω, and max(𝑘−𝑢,0)∈𝑊^{01,𝑝(𝑥)}(Ω) for every positive integer 𝑘.

Let 𝑊^{1,𝑝(𝑥)0,loc}(Ω)={𝑢∣ there is an open domain 𝑄⋐Ω such that 𝑢∈𝑊^{01,𝑝(𝑥)}(𝑄)}, and define 𝐴∶𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶(Ω)→(𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗ as⟨𝐴𝑢,𝜙⟩=_Ω||∇𝑢||^{𝑝(𝑥)−2}∇𝑢∇𝜙+𝜌(𝑥)𝑒^{𝑓(𝑥,𝑢)}𝜙𝑑𝑥,∀𝑢∈𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶(Ω),∀𝜙∈𝑊^{1,𝑝(𝑥)0,loc}(Ω).(2.6)

Lemma 2.1 (see [9, Theorem 3.1]). Let ℎ∈𝑊^1,𝑝(𝑥)(Ω)∩𝐶(Ω), and 𝑋=ℎ+𝑊^{01,𝑝(𝑥)}(Ω)∩𝐶(Ω). Then, 𝐴∶𝑋→(𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗ is strictly monotone.

Letting 𝑔∈(𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗, if ⟨𝑔,𝜙⟩≥0,forall𝜙∈𝑊^{1,𝑝(𝑥)0,loc}(Ω) with 𝜙≥0 a.e. in Ω, then denote 𝑔≥0 in (𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗; correspondingly, if −𝑔≥0 in (𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗, then denote 𝑔≤0 in (𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗.

Definition 2.2. Let 𝑢∈𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶(Ω). If 𝐴𝑢≥0(𝐴𝑢≤0) in (𝑊^{1,𝑝(𝑥)0,loc}(Ω))^∗, then 𝑢 is called a weak supersolution (weak subsolution) of (P).

Copying the proof of [14], we have the following.

Lemma 2.3 (comparison principle). Let 𝑢,𝑣∈𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶(Ω) satisfy 𝐴𝑢−𝐴𝑣≥0,in𝑊^{01,𝑝(𝑥)}(Ω)^∗.(2.7) Let 𝜙(𝑥)=min{𝑢(𝑥)−𝑣(𝑥),0}. If 𝜙(𝑥)∈𝑊^{01,𝑝(𝑥)}(Ω) (i.e., 𝑢≥𝑣 on 𝜕Ω), then 𝑢≥𝑣 a.e. in Ω.

Lemma 2.4 (see [8, Theorem 1.1]). Under the conditions (H1) and (H3), if 𝑢∈𝑊^1,𝑝(𝑥)(Ω) is a bounded weak solution of −Δ_𝑝(𝑥)𝑢+𝜌(𝑥)𝑒^{𝑓(𝑥,𝑢)}=0 in Ω, then 𝑢∈𝐶^1,𝜗loc(Ω), where 𝜗∈(0,1) is a constant.

3. Asymptotic Behavior of Boundary Blow-Up Solutions

If 𝑢 is a radial solution for (P), then (P) can be transformed into𝑟^𝑁−1||𝑢^||^{𝑝(𝑟)−2}𝑢^𝑢(0)=𝑢₀^=𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑢)},𝑟∈(0,𝑅),,𝑢^(0)=0,𝑢^(𝑟)≥0,for0<𝑟<𝑅.(3.1)

It means that 𝑢(𝑟) is increasing.

Theorem 3.1. If 𝑓(𝑟,𝑢) satisfies 𝑓(𝑟,𝑢)≥𝛼𝑢^𝑠(as𝑢⟶+∞)for𝑟∈[𝜎,𝑅)uniformly,(3.2) where 𝜎 is defined in (H4) and α and 𝑠 are positive constants, then there exists a supersolution Φ₁(𝑥) which satisfies Φ₁(𝑥)→+∞ (as 𝑑(𝑥,𝜕Ω)→0), such that for every solution 𝑢 of problem (P), one has 𝑢(𝑥)≤Φ₁(𝑥).

Proof. Define the function 𝑔(𝑟,𝑎,𝜆) on [0,𝑅_𝜆) as 𝑔(𝑟,𝑎,𝜆)=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩𝑎ln1(𝑅−𝑟)^1−𝜃−𝜆^1/𝑠𝑘−_𝑅0𝑟+𝑘,𝑅₀≤𝑟<𝑅_𝜆,⎡⎢⎢⎣𝑎^1/𝑠(1−𝜃)𝑅−𝑅₀^−𝜃𝑠𝑅−𝑅₀^1−𝜃−𝜆ln𝑅−𝑅₀1^1−𝜃−𝜆^(1/𝑠)−1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)/(𝑝(𝑡)−1)}×𝑅⁰^𝑁−1𝑡^𝑁−1+𝑎ln1sin𝜖(𝑡−𝜎)^{1/(𝑝(𝑡)−1)}𝑑𝑡𝑅−𝑅₀^1−𝜃−𝜆^1/𝑠𝑘−_𝑅0𝜎,𝜎<𝑟<𝑅₀,⎡⎢⎢⎣𝑎^1/𝑠(1−𝜃)𝑅−𝑅₀^−𝜃𝑠𝑅−𝑅₀^1−𝜃−𝜆ln𝑅−𝑅₀1^1−𝜃−𝜆^(1/𝑠)−1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)/(𝑝(𝑡)−1)}×𝑅⁰^𝑁−1𝑡^𝑁−1+𝑎ln1sin𝜖(𝑡−𝜎)^{1/(𝑝(𝑡)−1)}𝑑𝑡𝑅−𝑅₀^1−𝜃−𝜆^1/𝑠,𝑟≤𝜎,(3.3) where 𝜃<𝛽(𝑅)/𝑝(𝑅), 𝑎>(1/𝛼)sup_{|𝑥|≥𝑅0}𝑝(𝑥) are constants, 𝑅₀∈(𝜎,𝑅), 𝑅−𝑅₀ is small enough, parameter 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2], 𝑅_𝜆 satisfies (𝑅−𝑅_𝜆)^1−𝜃−𝜆=0, 𝜖=𝜋/2(𝑅₀−𝜎)𝑘=2𝑝⁺((1+𝑠)/𝑠+1/(1−𝜃))+||𝛽||⁺/(1−𝜃)𝛼ln2𝑅−𝑅₀^(1−𝜃)^1/𝑠+^𝑅0_𝜎⎡⎢⎢⎣2𝑎^1/𝑠(1−𝜃)𝑠𝑅−𝑅₀ln𝑅−𝑅2₀^1−𝜃^(1/𝑠)−1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)/(𝑝(𝑡)−1)}×𝑅⁰^𝑁−1𝑡^𝑁−1sin𝜖(𝑡−𝜎)^{1/(𝑝(𝑡)−1)}𝑑𝑡.(3.4)
Obviously, for any positive constant 𝑎, we have 𝑔(𝑟,𝑎,𝜆)∈𝐶¹[0,𝑅_𝜆).
When 𝑅₀<𝑟<𝑅_𝜆<𝑅, we have 𝑔^=𝑔^(𝑟,𝑎,𝜆)=𝑎^1/𝑠𝑠ln(𝑅−𝑟)1^1−𝜃−𝜆^(1/𝑠)−1(1−𝜃)(𝑅−𝑟)^−𝜃||𝑔(𝑅−𝑟)^1−𝜃−𝜆_,||^{𝑝(𝑟)−2}𝑔^=(1−𝜃)𝑎^1/𝑠𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃−𝜆^{((1/𝑠)−1)(𝑝(𝑟)−1)}(𝑅−𝑟)^{−𝜃(𝑝(𝑟)−1)}(𝑅−𝑟)^1−𝜃−𝜆^{𝑝(𝑟)−1}𝑟_𝑁−1,||𝑔^||^{𝑝(𝑟)−2}𝑔^^=𝑟^𝑁−1(1−𝜃𝑠)𝑎^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃−𝜆^{((1/𝑠)−1)(𝑝(𝑟)−1)}×(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃𝑝(𝑟)}(𝑅−𝑟)^1−𝜃−𝜆^𝑝(𝑟)[(1−𝜃)+Π(𝑟)],(3.5)
where Π(𝑟)=𝑟^𝑁−1(1−𝜃)𝑎^1/𝑠/𝑠^{𝑝(𝑟)−1}^(𝑝(𝑟)−1)𝑟^𝑁−1(1−𝜃)𝑎^1/𝑠/𝑠^{𝑝(𝑟)−1}(𝑅−𝑟)^1−𝜃−𝜆(𝑅−𝑟)^1−𝜃(𝑅−𝑟)+((1/𝑠)−1)(1−𝜃)+(𝑅−𝑟)_1−𝜃ln1/(𝑅−𝑟)^1−𝜃−𝜆−𝜆(𝑅−𝑟)^1−𝜃(𝑅−𝑟)((1/𝑠)−1)𝑝^(𝑟)+𝜃𝑝_(𝑝(𝑟)−1)lnln(𝑅−𝑟)1^1−𝜃−𝜆(𝑟)(𝑝(𝑟)−1)(𝑅−𝑟)^1−𝜃−𝜆(𝑅−𝑟)^1−𝜃(𝑅−𝑟)ln1+𝜃(𝑅−𝑟)_1−𝜃(𝑅−𝑟)−𝜆(𝑅−𝑟)^1−𝜃+−𝑝^(𝑟)𝑝(𝑟)−1(𝑅−𝑟)(𝑅−𝑟)^1−𝜃−𝜆(𝑅−𝑟)^1−𝜃ln(𝑅−𝑟)^1−𝜃−𝜆.(3.6)
If (𝑅−𝑅₀) is small enough, it is easy to see that ||Π(𝑟)||≤ln1(𝑅−𝑟)^1−𝜃−𝜆,for𝜆∈0,𝑅−𝑅⁰^1−𝜃2uniformly,(3.7)
and then 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^≤𝑟^𝑁−1(1−𝜃𝑠)𝑎^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃−𝜆^{((1/𝑠)−1)(𝑝(𝑟)−1)+1}×(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃𝑝(𝑟)}(𝑅−𝑟)^1−𝜃−𝜆^𝑝(𝑟),∀𝑟∈𝑅₀,𝑅_𝜆.(3.8)
Thus, when 0<𝑅−𝑅₀ is small enough, from (3.5) and (3.8), for 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2] uniformly, we have 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^≤2𝑟^𝑁−1(1−𝜃𝑠)𝑎^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃−𝜆^{((1/𝑠)−1)(𝑝(𝑟)−1)+1}(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃𝑝(𝑟)}(𝑅−𝑟)^1−𝜃−𝜆^𝑝(𝑟)≤𝑟^𝑁−1𝜌(𝑟)1(𝑅−𝑟)^1−𝜃−𝜆^𝛼𝑎=𝑟^𝑁−1𝜌(𝑟)𝑒^𝛼𝑔𝑠≤𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑔)},∀𝑟∈𝑅₀,𝑅_𝜆.(3.9)
Thus, when 0<𝑅−𝑅₀ is small enough, the following inequality is valid for 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2] uniformly: 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^≤𝑟^𝑁−1𝜌(𝑟)𝑓(𝑟,𝑔),∀𝑟∈𝑅₀,𝑅_𝜆.(3.10)
Obviously, if 𝑅−𝑅₀ is small enough, then 𝑔≥[((2𝑝⁺((𝑠+1)/𝑠+1/(1−𝜃))+|𝛽|⁺/(1−𝜃))/𝛼)ln(2/(𝑅−𝑅₀)^1−𝜃)]^1/𝑠 is large enough. Since 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2], 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^=𝜖𝑅₀^𝑁−1⎡⎢⎢⎣𝑎^1/𝑠(1−𝜃)𝑅−𝑅₀^−𝜃𝑠𝑅−𝑅₀^1−𝜃−𝜆ln𝑅−𝑅₀1^1−𝜃−𝜆^(1/𝑠)−1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)}≤𝜖𝑅₀cos(𝜖(𝑟−𝜎))^𝑁−1⎡⎢⎢⎣𝑎^1/𝑠(1−𝜃)𝑅−𝑅₀^−𝜃𝑠(1/2)𝑅−𝑅₀^1−𝜃lnR−𝑅2₀^1−𝜃^(1/𝑠)+1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)}≤𝜖𝑅₀^𝑁−1⎡⎢⎢⎣2𝑎^1/𝑠(1−𝜃)𝑠𝑅−𝑅₀𝑅−𝑅2₀^1−𝜃^(1/𝑠)+1⎤⎥⎥⎦^{(𝑝(𝑅0)−1)}≤𝜖𝑅₀^𝑁−12𝑎^1/𝑠(1−𝜃)𝑠𝑅−2𝑅₀^{((𝑠+1)/𝑠)(1−𝜃)+1}^𝑝+≤𝑟^𝑁−1𝜌(𝑟)𝑒^𝛼𝑔𝑠≤𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑔)},𝜎<𝑟<𝑅₀.(3.11)
Thus, 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^≤𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑔)},𝜎<𝑟<𝑅₀.(3.12)
Obviously, 𝑟^𝑁−1||𝑔^||^{𝑝(𝑟)−2}𝑔^^=0≤𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑔)},0≤𝑟<𝜎.(3.13)
Since 𝑔(𝑥,𝑎,𝜆)=𝑔(|𝑥|,𝑎,𝜆) is a 𝐶¹ function on 𝐵(0,𝑅_𝜆), if 0<𝑅−𝑅₀ is small enough (𝑅₀ depends on 𝑅, 𝑝, 𝑠, α), from (3.10), (3.12), and (3.13), for any 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2], we can see that 𝑔(|𝑥|,𝑎,𝜆) is a supersolution for (P) on 𝐵(0,𝑅_𝜆), and then 𝑔(|𝑥|,𝑎,0) is a supersolution for (P).
Defining the function 𝑔_𝑚(|𝑥|,𝑎−𝜀)=𝑔(𝑟,𝑎−𝜀,1/𝑚) on [0,𝑅_1/𝑚), where 𝑎−𝜀>(1/𝛼)sup_{|𝑥|≥𝑅0}𝑝(𝑥), then 𝑔_𝑚(|𝑥|,𝑎−𝜀) is a supersolution for (P) on 𝐵(0,𝑅−(1/𝑚)). If 𝑢 is a solution for (P), according to the comparison principle, we get that 𝑔_𝑚(|𝑥|,𝑎−𝜀)≥𝑢(𝑥) for any 𝑥∈𝐵(0,𝑅_1/𝑚). For any 𝑥∈𝐵(0,𝑅)⧵𝐵(0,𝑅₀), we have 𝑔_𝑚(|𝑥|,𝑎−𝜀)≥𝑔_𝑚+1(|𝑥|,𝑎−𝜀), when 𝑚 is large enough. Thus 𝑢(𝑥)≤lim_𝑚⟶+∞𝑔_𝑚(|𝑥|,𝑎−𝜀),∀𝑥∈𝐵(0,𝑅)⧵𝐵0,𝑅₀.(3.14)
When 𝑑(𝑥,𝜕Ω)>0 is small enough, we have lim_𝑚⟶+∞𝑔_𝑚(|𝑥|,𝑎−𝜀)<𝑎ln1(𝑅−𝑟)^1−𝜃^1/𝑠+𝑘≤𝑔(|𝑥|,𝑎,0).(3.15)
According to the comparison principle, we get that 𝑔(|𝑥|,𝑎,0)≥𝑢(𝑥),forall𝑥∈𝐵(0,𝑅); then Φ₁(𝑥)=Φ₁(|𝑥|)=𝑔(|𝑥|,𝑎,0) is a radial upper control function of all of the solutions for (P), and Φ₁(𝑥)=Φ₁(|𝑥|) is a radial supersolution for (P). The proof is completed.

Theorem 3.2. If 𝑓(𝑟,𝑢) satisfies 𝑓(𝑟,𝑢)⟶−∞(as𝑢⟶−∞)for𝑟∈[𝜎,𝑅)uniformly,𝑓(𝑟,𝑢)≤𝛿𝑢^𝑠(as𝑢⟶+∞)for𝑟∈[𝜎,𝑅)uniformly,(3.16) where 𝜎 is defined in (H₄) and 𝛿 and 𝑠 are positive constants, then there exists a subsolution Φ₂(𝑥) which satisfies Φ₂(𝑥)→+∞ (as 𝑑(𝑥,𝜕Ω)→0), such that for every solution 𝑢(𝑥) for problem (P), one has 𝑢(𝑥)≥Φ₂(𝑥).

Proof. We will prove this theorem in the following two cases.(i)𝛽₁(𝑅)>0. (ii)𝛽₁(𝑅)≤0. Case 1 (𝛽₁(𝑅)>0). Let 𝑧₁ be a radial solution of −Δ_𝑝(𝑥)𝑧₁(𝑥)=−𝜇,inΩ₁=𝐵(0,𝜎),𝑧₁=0,on𝜕Ω₁,(3.17)
where 𝜇>2(max_{𝑟∈[0,𝑅0]}𝜌(𝑟)+1)^{2(𝑝+−1)/(𝑝−−1)} is a positive constant. We denote 𝑧₁=𝑧₁(𝑟)=𝑧₁(|𝑥|). Then, 𝑧₁ satisfies −𝑟^𝑁−1||𝑧^1||𝑧^{𝑝(𝑟)−2}𝑧^1^=−𝑟^𝑁−1𝜇,𝑧₁(𝜎)=0,𝑧^1(0)=0,_1=|||𝑟𝜇𝑁|||^{1/(𝑝(𝑟)−1)},𝑧₁=−^𝜎𝑟|||𝑟𝜇𝑁|||^{1/(𝑝(𝑟)−1)}𝑑𝑟.(3.18)
Denote ℎ_𝑏(𝑟,𝜆) on [𝜎,𝑅₀] as ℎ_𝑏(𝑟,𝜆)=^𝑅0_𝑟𝑅⁰^𝑁−1𝑡^𝑁−1𝑡−𝜎𝑅₀−𝜎⎡⎢⎢⎣𝑏(1−𝜃)𝑅−𝑅⁰^−𝜃𝑠𝑅−𝑅₀^1−𝜃+𝜆𝑏ln𝑅−𝑅₀1^1−𝜃+𝜆^(1/𝑠)−1⎤⎥⎥⎦^{𝑝(𝑅0)−1}+(𝜎)^𝑁−1𝑡^𝑁−1𝑅₀−𝑡𝑅₀−𝜎|||𝜎𝑁𝜇|||^{1/(𝑝(𝑡)−1)}𝑑𝑡.(3.19)
It is easy to see that −ℎ^𝑏(𝜎,𝜆)=𝑧^1(𝜎)=|||𝜎𝜇𝑁|||^{1/(𝑝(𝜎)−1)}−ℎ_𝑏,𝑅₀,𝜆=𝑏(1−𝜃)𝑅−𝑅⁰^−𝜃𝑠𝑅−𝑅₀^1−𝜃+𝜆𝑏ln𝑅−𝑅₀1^1−𝜃+𝜆^(1/𝑠)−1.(3.20) Define the function 𝑣(𝑟,𝑏,𝜆) on [0,𝑅) as 𝑣(𝑟,𝑏,𝜆)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑏ln1(𝑅−𝑟)^1−𝜃+𝜆^1/𝑠𝑏ln1−𝑘^∗,𝑅₀≤𝑟<𝑅,𝑅−𝑅₀^1−𝜃+𝜆^1/𝑠−_𝜎𝑟−𝑘^∗−ℎ_𝑏(𝑟,𝜆),𝜎<𝑟<𝑅₀,|||𝑟𝜇𝑁|||^{1/(𝑝(𝑟)−1)}𝑑𝑟+𝑏ln1𝑅−𝑅₀^1−𝜃+𝜆^1/𝑠−𝑘^∗−ℎ_𝑏(𝜎,𝜆),𝑟≤𝜎,(3.21)where 𝜃∈(𝛽₁(𝑅)/𝑝(𝑅),1), 𝑏∈(0,(1/𝛿)inf_{|𝑥|≥𝑅0}𝑝(𝑥)) are constants, 𝑅₀∈(𝜎,𝑅),𝑅−𝑅₀ is small enough, parameter 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2], and 𝑘^∗=𝑀+𝑏ln1𝑅−𝑅₀^1−𝜃^1/𝑠,(3.22)where 𝑀 satisfies (𝜎)^𝑁−11𝑅₀−𝜎≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑦)},∀𝑦≤−𝑀,∀𝑟∈0,𝑅₀.(3.23)
Obviously, for any positive constant 𝑏, 𝑣(𝑟,𝑏,𝜆)∈𝐶¹[0,𝑅).
By computation, when 𝑟∈(𝑅₀,𝑅), we have 𝑣^=𝑣^(𝑟,𝑏,𝜆)=𝑏^1/𝑠𝑠ln(𝑅−𝑟)1^1−𝜃+𝜆^1/𝑠−1(1−𝜃)(𝑅−𝑟)^−𝜃||𝑣(𝑅−𝑟)^1−𝜃+𝜆_,||^{𝑝(𝑟)−2}𝑣^=(1−𝜃)𝑏^1/𝑠𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃+𝜆^{(1/𝑠−1)(𝑝(𝑟)−1)}(𝑅−𝑟)^{−𝜃(𝑝(𝑟)−1)}(𝑅−𝑟)^1−𝜃+𝜆^{𝑝(𝑟)−1}𝑟_𝑁−1,||𝑣^||^{𝑝(𝑟)−2}𝑣^^=𝑟^𝑁−1(1−𝑠𝜃)𝑏^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃+𝜆^{(1/𝑠−1)(𝑝(𝑟)−1)}×(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃(𝑝(𝑟)−1)−1}(𝑅−𝑟)^1−𝜃+𝜆^{𝑝(𝑟)−1}(𝜃+Λ(𝑟)),(3.24)
where Λ(𝑟)=𝑟^𝑁−1(1−𝜃)𝑏^1/𝑠/𝑠^{𝑝(𝑟)−1}^(𝑝(𝑟)−1)𝑟^𝑁−1(1−𝜃)𝑏^1/𝑠/𝑠^{𝑝(𝑟)−1}(𝑅−𝑟)+(1/𝑠−1)(1−𝜃)×(𝑅−𝑟)_1−𝜃ln1/(𝑅−𝑟)^1−𝜃+𝜆(𝑅−𝑟)^1−𝜃+𝜆+(1/𝑠−1)𝑝^(𝑟)(𝑝(𝑟)−1)(𝑅−𝑟)lnln(𝑅−𝑟)1^1−𝜃+𝜆+𝜃𝑝^(𝑟)+(1−𝜃)(𝑝(𝑟)−1)(𝑅−𝑟)ln(𝑅1−𝑟)(𝑅−𝑟)^1−𝜃+𝜆(𝑅−𝑟)^1−𝜃+−𝑝^(𝑟)𝑝(𝑟)−1(𝑅−𝑟)ln(𝑅−𝑟)^1−𝜃+𝜆.(3.25)
By computation, when 𝑅−𝑅₀ is small enough, for 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2] uniformly, we have 𝑟^𝑁−1||𝑣^||^{𝑝(𝑟)−2}𝑣^^≥𝑟^𝑁−1(1−𝑠𝜃)𝑏^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃+𝜆^{(1/𝑠−1)(𝑝(𝑟)−1)}×(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃(𝑝(𝑟)−1)−1}(𝑅−𝑟)^1−𝜃+𝜆^{𝑝(𝑟)−1}𝜃1−1≥𝜃22𝑟^𝑁−1(1−𝑠𝜃)𝑏^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃+𝜆^{(1/𝑠−1)(𝑝(𝑟)−1)}×(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃(𝑝(𝑟)−1)−1}(𝑅−𝑟)^1−𝜃+𝜆^𝑝(𝑟)(𝑅−𝑟)^1−𝜃≥𝜃2𝑟^𝑁−1(1−𝑠𝜃)𝑏^1/𝑠^{𝑝(𝑟)−1}ln(𝑅−𝑟)1^1−𝜃+𝜆^{(1/𝑠−1)(𝑝(𝑟)−1)}(𝑝(𝑟)−1)(𝑅−𝑟)^{−𝜃𝑝(𝑟)}(𝑅−𝑟)^1−𝜃+𝜆^𝑝(𝑟)≥𝑟^𝑁−1𝜌₁(𝑅−𝑟)^{−𝛽1(𝑟)}𝑒^𝛿𝑣𝑠≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑣)},∀𝑟∈𝑅₀,𝑅.(3.26)
Then, for 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2] uniformly, we have 𝑟^𝑁−1||𝑣^||^{𝑝(𝑟)−2}𝑣^^≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑣)},∀𝑟∈𝑅₀,𝑅.(3.27)
When 𝑅−𝑅₀ is small enough, forall𝑟∈(𝜎,𝑅₀), since 𝑣≤−𝑀, it is easy to see that 𝑟^𝑁−1||𝑣^||^{𝑝(𝑟)−2}𝑣^^≥𝑟^𝑁−1||ℎ^||^{𝑝(𝑟)−2}ℎ^^=𝑅₀^𝑁−11𝑅₀−𝜎⎡⎢⎢⎣𝑏(1−𝜃)𝑅−𝑅⁰^−𝜃𝑠𝑅−𝑅₀^1−𝜃+𝜆𝑏ln𝑅−𝑅₀1^1−𝜃+𝜆^1/𝑠−1⎤⎥⎥⎦^{𝑝(𝑅0)−1}−(𝜎)^𝑁−11≥(𝜎)_𝑁−1𝑅₀−𝜎|||𝜎𝑁𝜇|||1≥𝑟_𝑁−1𝑅₀−𝜎𝜌(𝑟)𝑒^{𝑓(𝑟,𝑣)},(3.28)
Then, 𝑟^𝑁−1||𝑣^||^{𝑝(𝑟)−2}𝑣^^≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑣)},∀𝑟∈𝜎,𝑅₀.(3.29)
Obviously, 𝑟^𝑁−1||𝑣^||^{𝑝(𝑟)−2}𝑣^^=𝑟^𝑁−1𝜇≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑣)},∀𝑟∈(0,𝜎).(3.30)
Combining (3.27), (3.29), and (3.30), when 𝑅−𝑅₀ is large enough, for any 𝜆∈[0,(𝑅−𝑅₀)^1−𝜃/2], one can see that 𝑣(𝑟,𝑎,𝜆) is a subsolution for (P).
Define the function 𝑣_𝑚(𝑟,𝑏+𝜀) on 𝐵(0,𝑅) as 𝑣_𝑚(𝑟,𝑏+𝜀)=𝑣_𝑚𝑟,𝑏+𝜀,1𝑚,(3.31)
where 𝜀 is a small enough positive constant such that (𝑏+𝜀)<(1/𝛿)inf_{|𝑥|≥𝑅0}𝑝(𝑥).
For any 𝑚=1,2,…, we can see that 𝑣_𝑚(𝑟,𝑏+𝜀)∈𝐶¹([0,𝑅)) is a subsolution for (P) on 𝐵(𝑅₀,𝑅). According to the comparison principle, we get that 𝑣_𝑚(𝑟,𝑏+𝜀)≤𝑢(𝑥) for any 𝑥∈𝐵(0,𝑅). For any 𝑥∈𝐵(0,𝑅)⧵𝐵(0,𝑅₀), we have 𝑣_𝑚(|𝑥|,𝑏+𝜀)≤𝑣_𝑚+1(|𝑥|,𝑏+𝜀). Thus 𝑢(𝑥)≥lim_𝑚⟶+∞𝑣_𝑚(|𝑥|,𝑏+𝜀),∀𝑥∈𝐵(0,𝑅)⧵𝐵0,𝑅₀.(3.32)
When 𝑑(𝑥,𝜕Ω) is small enough, we havelim_𝑚→+∞𝑣_𝑚(|𝑥|,𝑏+𝜀)>𝑣(|𝑥|,𝑏,0).
According to the comparison principle, we get that 𝑣(|𝑥|,𝑏,0)≤𝑢(𝑥),∀𝑥∈𝐵(0,𝑅); then Φ₂(𝑥)=Φ₂(|𝑥|)=𝑣(|𝑥|,𝑏,0) is a radial lower control function of all of the solutions for (P), and Φ₂(𝑥) is a radial subsolution for (P).Case 2 (𝛽₁(𝑅)≤0). Let 𝜇>2(max_{𝑟∈[0,𝑅0]}𝜌(𝑟)+1)^{2(𝑝+−1)/(𝑝−−1)} be a positive constant. Denote 𝜛_𝑏(𝑟,𝜆) on [𝜎,𝑅₀] as 𝜛_𝑏(𝑟,𝜆)=^𝑅0_𝑟𝑅⁰^𝑁−1𝑡^𝑁−1𝑡−𝜎𝑅₀−𝜎𝑠𝑅+𝜆𝑏−𝑅₀𝑏ln𝑅+𝜆−𝑅⁰⁻¹^1/𝑠−1^{𝑝(𝑅0)−1}+(𝜎)^𝑁−1𝑡^𝑁−1𝑅₀−𝑡𝑅₀−𝜎|||𝜎𝑁𝜇|||^{1/(𝑝(𝑡)−1)}𝑑𝑡.(3.33)
It is easy to see that −𝜛^𝑏(𝜎,𝜆)=𝑧^1(𝜎)=|||𝜎𝜇𝑁|||^{1/(𝑝(𝜎)−1)},−𝜛^𝑏𝑅₀,𝜆=𝑏𝑠𝑅+𝜆−𝑅₀𝑏ln𝑅+𝜆−𝑅⁰⁻¹^1/𝑠−1.(3.34)
Define the function 𝜂(𝑟,𝑏,𝜆) on 𝐵(0,𝑅) as 𝜂(𝑟,𝑏,𝜆)=⎧⎪⎪⎨⎪⎪⎩𝑏ln(𝑅+𝜆−𝑟)⁻¹𝑏ln𝑅+𝜆−𝑅₀^1/𝑠−𝑘^∗,𝑅₀≤𝑟<𝑅,−_𝜎𝑟⁻¹^1/𝑠−𝑘^∗−𝜛_𝑏(𝑟,𝜆),𝜎<𝑟<𝑅₀,|||𝑟𝜇𝑁|||^{1/(𝑝(𝑟)−1)}𝑑𝑟+𝑏ln𝑅+𝜆−𝑅₀⁻¹^1/𝑠−𝑘^∗−𝜛_𝑏(𝜎,𝜆),𝑟≤𝜎,(3.35)where 𝑏∈(0,(1/𝛿)inf_{|𝑥|≥𝑅0}[𝑝(𝑥)−𝛽₁(𝑥)]) is a constant, 𝑅₀∈(𝜎,𝑅),𝑅−𝑅₀ is small enough, parameter 𝜆∈[0,(𝑅−𝑅₀)/2], and 𝑘^∗=𝑀+𝑏ln1𝑅−𝑅₀^1/𝑠,(3.36)where 𝑀 is defined in (3.23).
Obviously, for any positive constant 𝑏, 𝜂(𝑟,𝑏,𝜆)∈𝐶¹[0,𝑅).
Similar to the proof of Case 1, when 𝑅−𝑅₀ is small enough, we have 𝑟^𝑁−1||𝜂^||^{𝑝(𝑟)−2}𝜂^^≥𝑟^𝑁−1𝑏𝑠^1/𝑠^{𝑝(𝑟)−1}(𝑝(𝑟)−1)(𝑅+𝜆−𝑟)^{−𝑝(𝑟)}ln(𝑅+𝜆−𝑟)⁻¹^{(1/𝑠−1)(𝑝(𝑟)−1)}1−1≥𝑟_𝑁−12𝜌(𝑟)𝑒^{𝑓(𝑟,𝜂)},∀𝑟∈𝑅₀,𝑅.(3.37)
When 𝑅−𝑅₀ is small enough, forall𝑟∈(𝜎,𝑅₀), from the definition of 𝑘^∗, it is easy to see that 𝑟^𝑁−1||𝜂^||^{𝑝(𝑟)−2}𝜂^^≥(𝜎)^𝑁−11𝑅₀−𝜎≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝜂)}.(3.38)
Obviously 𝑟^𝑁−1||𝜂^||^{𝑝(𝑟)−2}𝜂^^=𝑟^𝑁−1𝜇≥𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝜂)},∀𝑟∈(0,𝜎).(3.39)
Combining (3.37), (3.38), and (3.39), when 𝑅−𝑅₀ is large enough, for any 𝜆∈[0,(𝑅−𝑅₀)/2], one can see that 𝜂(𝑟,𝑎,𝜆) is a subsolution for (P).
Define the function 𝜂_𝑚(𝑟,𝑏+𝜖) on 𝐵(0,𝑅) as 𝜂_𝑚(𝑟,𝑏+𝜖)=𝜂𝑟,𝑏+𝜖,1𝑚,(3.40)
where 𝜖 is a small enough positive constant such that (𝑏+𝜖)<(1/𝛿)inf_{|𝑥|≥𝑅0}𝑝(𝑥).
We can see that 𝜂_𝑚(𝑟,𝑏+𝜖)∈𝐶¹[0,𝑅) is a subsolution for (P) for any 𝑚=1,2…. According to the comparison principle, we get that 𝜂_𝑚(𝑟,𝑏+𝜖)≤𝑢(𝑥) for any 𝑥∈𝐵(0,𝑅). For any 𝑥∈𝐵(0,𝑅)⧵𝐵(0,𝑅₀), we have 𝜂_𝑚(|𝑥|,𝑏+𝜖)≤𝜂_𝑚+1(|𝑥|,𝑏+𝜖). Then, 𝑢(𝑥)≥lim_𝑚→+∞𝜂_𝑚(|𝑥|,𝑏+𝜖),∀𝑥∈𝐵(0,𝑅)⧵𝐵0,𝑅₀.(3.41)
When 𝑑(𝑥,𝜕Ω) is small enough, we have lim_𝑚⟶+∞𝜂_𝑚(|𝑥|,𝑏+𝜖)>𝜂(|𝑥|,𝑏,0).(3.42)
According to the comparison principle, we get that 𝜂(|𝑥|,𝑏,0)≤𝑢(𝑥),∀𝑥∈𝐵(0,𝑅); then Φ₂(𝑥)=Φ₂(|𝑥|)=𝜂(|𝑥|,𝑏,0) is a radial lower control function of all of the solutions for (P), and Φ₂(𝑥)=Φ₂(|𝑥|) is a radial subsolution for (P).

Theorem 3.3. If 𝑓(𝑟,𝑢) satisfies lim_𝑢⟶+∞𝑓(𝑟,𝑢)𝑢^𝑠=𝛿(as𝑢⟶+∞)for𝑟∈[𝜎,𝑅)uniformly,(3.43) where 𝜎 is defined in (H₄), 𝛿 and 𝑠 are positive constants, 𝜌(𝑟)=𝜌₀(𝑅−𝑟)^{−𝛽(𝑟)}, where 𝛽(𝑅)<𝑝(𝑅), then each solution 𝑢(𝑥) for (P) satisfies lim_{|𝑥|⟶𝑅}𝑢(𝑥)(𝑝(𝑅)/𝛿)ln1/(𝑅−|𝑥|)^1−𝜃^1/𝑠=1,where𝜃=𝛽(𝑅)𝑝(𝑅).(3.44)

Proof. It is easy to be seen from Theorems 3.1 and 3.2

4. The Existence of Boundary Blow-Up Solutions

Theorem 4.1. If inf_𝑥∈Ω𝑝(𝑥)>𝑁 and 𝑓(𝑟,𝑢) satisfies 𝑓(𝑟,𝑢)≥𝑎𝑢^𝑠(as𝑢→+∞)for𝑟∈[𝜎,𝑅)uniformly,(4.1) where 𝜎 is defined in (H₄), 𝑎 and 𝑠 are positive constants, then (P) possesses a boundary blow-up solution.

Proof. In order to deal with the existence of boundary blow-up solutions, let us consider the problem −Δ𝑢_𝑝(𝑥)(𝑢𝑥+)=𝜌(𝑐𝑟,)𝑒^{𝑓(𝑥,𝑢)}=0,inΩ₀,for𝑥∈𝜕Ω₀,(4.2) where 𝑐 is a positive constant and Ω₀⋐Ω is a radial subdomain of Ω. Since inf_𝑥∈Ω𝑝(𝑥)>𝑁, then W^1,𝑝(𝑥)(Ω₀)↪𝐶^𝛼(Ω₀), where 𝛼∈(0,1). The relative functional of (4.2) is 𝜙=_Ω01𝑝(𝑥)||∇𝑢(𝑥)||^𝑝(𝑥)𝑑𝑥+_Ω0𝐹(𝑥,𝑢)𝑑𝑥,(4.3) where 𝐹(𝑥,𝑢)=∫^𝑢0𝑒^{𝑓(𝑥,𝑡)}𝑑𝑡. Since 𝜙 is coercive in 𝑋∶=𝑐+𝑊^{01,𝑝(𝑥)}(Ω₀), then 𝜙 possesses a nontrivial minimum point 𝑢. So, problem (4.2) possesses a weak solution 𝑢.
Since 𝑎𝑢^𝑠≤𝑓(𝑟,𝑢)≤𝐶₁+𝐶₂|𝑢|^𝛾(𝑥), from Theorems 3.1 and 3.2, we get that (P) possesses a supersolution 𝑔^∗(𝑥) and a subsolution 𝑔_∗(𝑥), which satisfy 𝑔^∗(𝑥)≥𝑔_∗(𝑥), when 𝑑(𝑥,𝜕Ω) (the distance from 𝑥 to 𝜕Ω) is small enough. According to the comparison principle, we get that 𝑔^∗(𝑥)≥𝑔_∗(𝑥) for any 𝑥∈Ω.
Denote 𝐷_𝑗={𝑥∣|𝑥|<1−1/(𝑗+1)𝑅} (𝑗=1,2,…). Let us consider the problem −Δ𝑢_{𝑝(𝑥)𝑗}𝑢_𝑗+𝜌(𝑥)𝑒^{𝑓(𝑥,𝑢𝑗)}=0,in𝐷_𝑗,(𝑥)=𝑔_∗(𝑥),for𝑥∈𝜕𝐷_𝑗,(4.4)
and the relative functional is 𝜙=_𝐷𝑗1𝑝(𝑥)||∇𝑢^𝑗(𝑥)||^𝑝(𝑥)𝑑𝑥+_𝐷𝑗𝜌(𝑥)𝐹𝑥,𝑢_𝑗𝑑𝑥.(4.5)
Let 𝑔_∗𝑗(𝑥)=𝑔_∗(𝑥)∣_𝐷𝑗. Since the functional 𝜙 is coercive in 𝑋_𝑗=𝑔_∗𝑗(𝑥)+𝑊^{01,𝑝(𝑥)}(𝐷_𝑗), then 𝜙 has a nontrivial minimum point 𝑢_𝑗. Therefore, problem (4.4) has a weak solution 𝑢_𝑗.
According to the comparison principle, we get that 𝑔_∗(𝑥)≤𝑢_𝑗(𝑥) for any 𝑥∈𝐷_𝑗 (𝑗=1,2,… ). Since 𝑢_𝑗(𝑥)=𝑔_∗(𝑥) for any 𝑥∈𝜕𝐷_𝑗, then 𝑢_𝑗(𝑥)≤𝑢_𝑗+1(𝑥) for any 𝑥∈𝜕𝐷_𝑗 (𝑗=1,2,…). According to the comparison principle, we get that 𝑢_𝑗(𝑥)≤𝑢_𝑗+1(𝑥) for any 𝑥∈𝐷_𝑗 (𝑗=1,2,…).
Since 𝑔^∗(𝑥) is a supersolution and 𝑔^∗(𝑥)≥𝑔_∗(𝑥) for any 𝑥∈Ω, so we have 𝑢_𝑗(𝑥)=𝑔_∗(𝑥)≤𝑔^∗(𝑥) for any 𝑥∈𝜕𝐷_𝑗 (𝑗=1,2,…). According to the comparison principle, we get that 𝑢_𝑗(𝑥)≤𝑔^∗(𝑥) for any 𝑥∈𝐷_𝑗 (𝑗=1,2,…).
Since 𝑔^∗(𝑥) and 𝑔_∗(𝑥) are locally bounded, from Lemma 2.4, each weak solution of (4.4) is a 𝐶^1,𝛼loc function. The 𝐶^1,𝛼 interior regularity result implies that the sequences {𝑢_𝑗} and {∇𝑢_𝑗} are equicontinuous in 𝐷₂, and hence we can choose a subsequence, which we denoted by {𝑢^1𝑗}, such that 𝑢^1𝑗→𝑤₁ and ∇𝑢^1𝑗→𝜛₁ uniformly on 𝐷₁ for some 𝑤₁∈𝐶(𝐷₁) and 𝜛₁∈(𝐶(𝐷₁))^𝑁. In fact, 𝜛₁=∇𝑤₁ on 𝐷₁, and from the interior 𝐶^1,𝛼 estimate, we conclude that ∇𝑤₁∈(𝐶^𝛼(𝐷₁))^𝑁 for some 0<𝛼<1. Thus, 𝑤₁∈𝑊^1,𝑝(𝑥)(𝐷₁)∩𝐶^1,𝛼(𝐷₁). From the 𝐶^1,𝛼 interior regularity result, we see that |∇𝑢_𝑗|^𝑝−1|∇𝜙|≤𝐶|∇𝜙| on 𝐷₁, and since the function 𝜉→|𝜉|^𝑝−2𝜉 is continuous on ℝ^𝑁, it follows that |∇𝑢^1𝑗(𝑥)|^𝑝−2∇𝑢^1𝑗(𝑥)⋅∇𝜙(𝑥)→|∇𝑤₁(𝑥)|^𝑝−2∇𝑤₁(𝑥)⋅∇𝜙(𝑥) for 𝑥∈𝐷₁. Thus, by the dominated convergence theorem, we have _𝐷1||∇𝑢^1𝑗(𝑥)||^𝑝−2∇𝑢^1𝑗(𝑥)⋅∇𝜙(𝑥)𝑑𝑥⟶_𝐷1||∇𝑤₁(𝑥)||^𝑝−2∇𝑤₁(𝑥)⋅∇𝜙(𝑥)𝑑𝑥,∀𝜙∈𝑊^{01,𝑝(𝑥)}𝐷₁.(4.6)
Furthermore, since 0≤𝑓(𝑢^1𝑗)≤𝑓(𝑢^1𝑗+1) and 𝑓(𝑢^1𝑗(𝑥))→𝑓(𝑤₁(𝑥)) for each 𝑥∈𝐷₁, by the monotone convergence theorem, we obtain _𝐷1𝜌𝑒^{𝑓(𝑢1𝑗)}𝑞𝑑𝑥⟶_𝐷1𝜌𝑒^𝑓(𝑤1)𝑞𝑑𝑥,∀𝑞∈𝑊^{01,𝑝(𝑥)}𝐷₁.(4.7)
Therefore, it follows that _𝐷1||∇𝑤₁(𝑥)||^𝑝−2∇𝑤₁(𝑥)⋅∇𝑞(𝑥)𝑑𝑥+_𝐷1𝜌𝑒^𝑓(𝑤1)𝑞𝑑𝑥=0,∀𝑞∈𝑊^{01,𝑝(𝑥)}𝐷₁,(4.8)
and hence 𝑤₁ is a weak solution for −Δ_𝑝(𝑥)𝑤₁+𝜌𝑒^𝑓(𝑤1)=0 on 𝐷₁.
Thus, there exists a subsequence of {𝑢_𝑗} which we denote it by{𝑢^1𝑗}, such that 𝑢^1𝑗→𝑤₁ in 𝐷₁ (as 𝑗→∞), where 𝑤₁∈𝑊^1,𝑝(𝑥)(𝐷₁)∩𝐶^1,𝛼1(𝐷₁) and satisfies _𝐷1||∇𝑤₁||^{𝑝(𝑥)−2}∇𝑤₁∇𝑞𝑑𝑥+_𝐷1𝜌(𝑥)𝑒^{𝑓(𝑥,𝑤1)}𝑞𝑑𝑥=0,∀𝑞∈𝑊^{01,𝑝(𝑥)}𝐷₁.(4.9)
Similarly, we can prove that there exists a subsequence of {𝑢^1𝑗} which we denote by{𝑢^2𝑗}, such that 𝑢^2𝑗→𝑤₂ in 𝐷₂ (as 𝑗→∞), where 𝑤₂∈𝑊^1,𝑝(𝑥)(𝐷₂)∩𝐶^1,𝛼2(𝐷₂) satisfies 𝑤₁=𝑤₂∣_𝐷1 and _𝐷2||∇𝑤₂||^{𝑝(𝑥)−2}∇𝑤₂∇𝑞𝑑𝑥+_𝐷2𝜌(𝑥)𝑒^{𝑓(𝑥,𝑤2)}𝑞𝑑𝑥=0,∀𝑞∈𝑊^{01,𝑝(𝑥)}𝐷₂.(4.10)
Repeating the above steps, we can get a subsequence of {𝑢^𝑖𝑗∣𝑗=1,2,…} which we denote by {𝑢^𝑗𝑖+1∣𝑗=1,2,…} (𝑖=1,2,…) and satisfies the following.(1⁰) For any fixed 𝑖, {𝑢^𝑗𝑖+1} is a subsequence of{𝑢^𝑖𝑗}.(2⁰) For any fixed 𝑖, 𝑢^𝑗𝑖+1→𝑤_𝑖+1 in 𝐷_𝑖+1 (as 𝑗→∞), where 𝑤_𝑖+1∈𝑊^1,𝑝(𝑥)(𝐷_𝑖+1)∩𝐶^1,𝛼𝑖+1(𝐷_𝑖+1) satisfies 𝑤_𝑖=𝑤_𝑖+1∣_𝐷𝑖.(3⁰)For any fixed 𝑖, 𝑤_𝑖 satisfies _𝐷𝑖||∇𝑤_𝑖||^{𝑝(𝑥)−2}∇𝑤_𝑖∇𝑞𝑑𝑥+_𝐷𝑖𝜌(𝑥)𝑒^{𝑓(𝑥,𝑤𝑖)}𝑞𝑑𝑥=0,∀𝑞∈𝑊^{01,𝑝(𝑥)}𝐷_𝑖.(4.11)
Thus, we can conclude that
(i){𝑢^𝑗𝑗}is a subsequence of{𝑢_𝑗},(ii) there exists a function 𝑤∈𝑊^{1,𝑝(𝑥)loc}(Ω)∩𝐶^1,𝛼loc(Ω) such that 𝑤_𝑖=𝑤∣_𝐷𝑖, and for any 𝑥∈Ω, there exists a constant 𝑗_𝑥 such that when 𝑗≥𝑗_𝑥, 𝑢^𝑗𝑗(𝑥) is defined at 𝑥, and lim_𝑗→∞𝑢^𝑗𝑗(𝑥)=𝑤(𝑥),(iii)_Ω||∇𝑤||^{𝑝(𝑥)−2}∇𝑤∇𝑞𝑑𝑥+_Ω𝜌(𝑥)𝑒^{𝑓(𝑥,𝑤)}𝑞𝑑𝑥=0,∀𝑞∈𝑊^{1,𝑝(𝑥)0,loc}(Ω).(4.12)
Obviously, 𝑤 is a boundary blow-up solution for (P).
This completes the proof.

In Theorem 4.1, when inf_𝑥∈Ω𝑝(𝑥)>𝑁, the existence of solutions for (P) is given. In the following, we will consider the existence of solutions for (P) in the general case 1<inf_𝑥∈Ω𝑝(𝑥)≤sup_𝑥∈Ω𝑝(𝑥)<∞. We need to do some preparation. Let us consider 𝑟^𝑁−1||𝑢^||^{𝑝(𝑟)−2}𝑢^𝑢_^=𝑟^𝑁−1𝜌(𝑟)𝑒^{𝑓(𝑟,𝑢)},𝑟∈0,𝑅_𝜆,(0)=0,𝑢𝑅_𝜆=𝑑,(I)

where 𝑅_𝜆∈(0,𝑅) and 𝑑 is a constant.

Lemma 4.2. If Φ₂(𝑅_𝜆)≤𝑑≤Φ₁(𝑅_𝜆), where Φ₁ and Φ₂ are defined in Theorems 3.13.2, respectively, then (4.13) has a solution 𝑢 satisfying Φ₂(𝑟)≤𝑢(𝑟)≤Φ₁(𝑟),∀𝑟∈0,𝑅_𝜆.(4.13)

Proof. Denote ℎ(𝑟,𝑢)=⎧⎪⎨⎪⎩𝑒^{𝑓(𝑟,Φ1(𝑟))}𝑒+arctan𝑢(𝑟)−Φ₁(𝑟),𝑢(𝑟)>Φ₁(𝑟),_{𝑓(𝑟,𝑢)}𝑒,Φ₂(𝑟)≤𝑢(𝑟)≤Φ₁(𝑟),_{𝑓(𝑟,Φ2(𝑟))}+arctan𝑢(𝑟)−Φ₂(𝑟),𝑢(𝑟)<Φ₂(𝑟).(4.14)
Let 𝜌_𝐸(𝑡)=𝜌(|𝑡|),andℎ_𝐸(𝑡,𝑢)=ℎ(|𝑡|,𝑢),forall𝑡∈[−𝑅_𝜆,𝑅_𝜆]. Let us consider the even solutions of the following |𝑡|^𝑁−1||𝑢^||^{𝑝(|𝑡|)−2}𝑢^𝑢^−=𝑅|_𝜆𝑡|^𝑁−1𝜌_𝐸(𝑡)ℎ_𝐸(𝑡,𝑢),𝑡∈−𝑅_𝜆,𝑅_𝜆,=𝑑,𝑢𝑅_𝜆=𝑑.(II)
It is easy to see that 𝑢 is an even solution for (4.15) if and only if 𝑢 is even and 𝑢=𝑑−^𝑅𝜆_𝑟|𝑡|^1−𝑁^𝑡0|𝑠|^𝑁−1𝜌(𝑠)ℎ(𝑠,𝑢(𝑠))𝑑𝑠^{1/(𝑝(𝑡)−1)}𝑑𝑡,∀𝑟∈0,𝑅_𝜆.(4.15)
Denote Ψ(𝑢,𝜇)=𝜇𝑑−𝜇∫^𝑅𝜆_𝑟[|𝑡|^1−𝑁∫^𝑡0|𝑠|^𝑁−1𝜌(𝑠)ℎ(𝑠,𝑢(𝑠))𝑑𝑠]^{1/(𝑝(𝑡)−1)}𝑑𝑡. Similar to the proof of Lemma 2.3 of [18], for any 𝜇∈[0,1], it is easy to see that Ψ(𝑢,𝜇) is compact continuous and bounded from 𝐶^1𝐸[0,𝑅_𝜆] to 𝐶^1𝐸[0,𝑅_𝜆], where 𝐶^1𝐸[0,𝑅_𝜆]={𝑢∈𝐶¹[0,𝑅_𝜆]∣𝑢iseven}. Thus, 𝑢=Ψ(𝑢,1) has a solution 𝑢 in 𝐶^1𝐸[0,𝑅_𝜆] and satisfies 𝑢^(0)=lim_𝑟→0⁺𝑢^(𝑟)=0. Then, 𝑢(|𝑡|) is an even solution for (4.15).
Denote Φ_1,𝐸(𝑡)=Φ₁(|𝑡|),Φ_2,𝐸(𝑡)=Φ₂(|𝑡|). From the definitions of Φ₁and Φ₂, we can see that Φ^1(0)=0=Φ^2(0); therefore, Φ_1,𝐸(𝑡) and Φ_2,𝐸(𝑡) are supersolution and subsolution for (4.15), respectively.
Since Φ₂(𝑅_𝜆)≤𝑢(𝑅_𝜆)≤Φ₁(𝑅_𝜆) and ℎ_𝐸(𝑡,⋅) is increasing, from the comparison principle, we have Φ_2,𝐸(𝑡)≤𝑢(𝑡)≤Φ_1,𝐸(𝑡),∀𝑡∈−𝑅_𝜆,𝑅_𝜆.(4.16)
It means that 𝑢 is a solution for (4.13) and 𝑢 satisfies Φ₂(𝑟)≤𝑢(𝑟)≤Φ₁(𝑟),∀𝑟∈0,𝑅_𝜆.(4.17)
Thus 𝑢 is a radial solution for (P). This completes the proof.

Theorem 4.3. If 𝑓(𝑟,𝑢) satisfies 𝑓(𝑟,𝑢)≥𝑎𝑢^𝑠(as𝑢⟶+∞)for𝑟∈[𝜎,𝑅)uniformly,(4.18) where 𝜎 is defined in (H₄) and 𝑎 and 𝑠 are positive constants, then (P) possesses a boundary blow-up solution.

Proof. From Lemma 4.2, we have that (4.4) has a weak solution 𝑢_𝑗(𝑥)=𝑢_𝑗(|𝑥|)=𝑢_𝑗(𝑟). Similar to the proof of Theorem 4.1, we can obtain the existence of solutions for (P).

Acknowledgments

This paper is partly supported by the National Science Foundation of China (10701066& 10926075 & 10971087) China Postdoctoral Science Foundation-funded project (20090460969), and the Natural Science Foundation of Henan Education Committee (2008-755-65).