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

Authors: Li Yin; Yunrui Guo; Jing Yang; Bibo Lu; Qihu Zhang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

This paper investigates the followingp(x)-Laplacian equations with exponential nonlinearities: −Δp(x)u+ρ(x)ef(x,u)=0 in Ω, u(x)→+∞ as d(x,∂Ω)→0, where −Δp(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.


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## Body

## 1. Introduction

The study of differential equations and variational problems with nonstandardp(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
(P)-Δp(x)u+ρ(x)ef(x,u)=0,inΩ,u(x)⟶+∞,asd(x,∂Ω)⟶0,where-Δp(x)u=-div(|∇u|p(x)-2∇u) and  Ω=B(0,R)⊂ℝN is a bounded radial domain (B(0,R)={x∈ℝN∣|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 thatp(x), ρ(x), and f(x,u) satisfy the following.(H1) p(x)∈C1(Ω¯) is radial and satisfies (1.1)1<p-≤p+<+∞,wherep-=inf⁡Ωp(x),p+=sup⁡Ωp(x).(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(1.2)|f(x,t)|≤C1+C2|t|γ(x),∀(x,t)∈Ω×R,whereC1, 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 (1.3)ρ0(R-r)-β(r)≤ρ(r)≤ρ1(R-r)-β1(r)forr∈[σ,R)uniformly,whereρ0 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-Δp(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):(1.4)-Δpu+h(x)f(u)=0,inΩ,we refer to [19–26], and the following generalized Keller-Osserman condition is crucial(1.5)∫1∞1(F(t))1/pdt<+∞,whereF(t)=∫0tf(s)ds,but the typical form ofp(x)-Laplacian equation is(1.6)-Δp(x)u+|u|q(x)-2u=0,inΩ,and there are some differences between the results of (1.4) and (1.6) (see [16]).On the boundary blow-up solutions for the followingp-Laplacian equations with exponential nonlinearities (p is a constant):(1.7)-Δpu+eh(x)f(u)=0,inΩ,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: (1.8)-Δp(x)u+f(x,u)=0,inΩ,u(x)⟶+∞,asd(x,∂Ω)⟶0,on the condition thatf(x,·) satisfies polynomial growth condition.Ifp(x) is a function, the typical form of (P) is the following:(1.9)-Δp(x)u+ρ(x)e|u|q(x)-2u=0,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 ofp(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 withp(x)-Laplacian problems, we need some theories on the spaces Lp(x)(Ω), W1,p(x)(Ω) and properties of p(x)-Laplacian, which we will use later (see [6, 11]). Let(2.1)Lp(x)(Ω)={u∣uis a measurable real-valued function,∫Ω|u(x)|p(x)dx<∞}.We can introduce the norm onLp(x)(Ω) by(2.2)|u|p(x)=inf⁡{λ>0∣∫Ω|u(x)λ|p(x)dx≤1}.The space (Lp(x)(Ω), |·|p(x)) becomes a Banach space. We call it generalized Lebesgue space. The space (Lp(x)(Ω), |·|p(x)) is a separable, reflexive, and uniform convex Banach space (see [6, Theorems 1.10, 1.14] ).The spaceW1,p(x)(Ω) is defined by(2.3)W1,p(x)(Ω)={u∈Lp(x)(Ω)∣|∇u|∈Lp(x)(Ω)},and it can be equipped with the norm(2.4)‖u‖=|u|p(x)+|∇u|p(x),∀u∈W1,p(x)(Ω).W01,p(x)(Ω) is the closure of C0∞(Ω) in W1,p(x)(Ω). W1,p(x)(Ω) and W01,p(x)(Ω) are separable, reflexive, and uniform convex Banach spaces (see [6, Theorem 2.1]).Ifu∈Wloc1,p(x)(Ω)∩C(Ω), u is called a blow-up solution of (P) when it satisfies(2.5)∫Q|∇u|p(x)-2∇u∇qdx+∫Qρ(x)f(x,u)qdx=0,∀q∈W01,p(x)(Q),for any domainQ⋐Ω, and max(k-u,0)∈W01,p(x)(Ω) for every positive integer k.LetW0,loc1,p(x)(Ω)={u∣ there is an open domain Q⋐Ω  such that u∈W01,p(x)(Q)}, and define A:Wloc1,p(x)(Ω)∩C(Ω)→(W0,loc1,p(x)(Ω))* as(2.6)〈Au,φ〉=∫Ω(|∇u|p(x)-2∇u∇φ+ρ(x)ef(x,u)φ)dx,∀u∈Wloc⁡1,p(x)(Ω)∩C(Ω),∀φ∈W0,loc⁡1,p(x)(Ω).Lemma 2.1 (see [9, Theorem 3.1]).
Leth∈W1,p(x)(Ω)∩C(Ω), and X=h+W01,p(x)(Ω)∩C(Ω). Then, A:X→(W0,loc1,p(x)(Ω))* is strictly monotone.Lettingg∈(W0,loc1,p(x)(Ω))*, if 〈g,φ〉≥0,forallφ∈W0,loc1,p(x)(Ω) with φ≥0 a.e. in Ω, then denote g≥0 in (W0,loc1,p(x)(Ω))*; correspondingly, if -g≥0 in (W0,loc1,p(x)(Ω))*, then denote g≤0 in (W0,loc1,p(x)(Ω))*.Definition 2.2.
Letu∈Wloc1,p(x)(Ω)∩C(Ω). If Au≥0(Au≤0) in (W0,loc1,p(x)(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).
Letu,v∈Wloc1,p(x)(Ω)∩C(Ω) satisfy
(2.7)Au-Av≥0,in(W01,p(x)(Ω))*.
Let φ(x)=min{u(x)-v(x),0}. If φ(x)∈W01,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∈W1,p(x)(Ω) is a bounded weak solution of -Δp(x)u+ρ(x)ef(x,u)=0 in Ω, then u∈Cloc1,ϑ(Ω), where ϑ∈(0,1) is a constant.


## 3. Asymptotic Behavior of Boundary Blow-Up Solutions

Ifu is a radial solution for (P), then (P) can be transformed into(3.1)(rN-1|u′|p(r)-2u′)′=rN-1ρ(r)ef(r,u),r∈(0,R),u(0)=u0,u′(0)=0,u′(r)≥0,for0<r<R.It means thatu(r) is increasing.Theorem 3.1.
Iff(r,u) satisfies
(3.2)f(r,u)≥αus(asu⟶+∞)forr∈[σ,R)uniformly,
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 functiong(r,a,λ) on [0,Rλ) as
(3.3)g(r,a,λ)={(aln⁡⁡1(R-r)1-θ-λ)1/s+k,R0≤r<Rλ,k-∫rR0[a1/s(1-θ)(R-R0)-θs((R-R0)1-θ-λ)(ln⁡⁡1(R-R0)1-θ-λ)(1/s)-1](p(R0)-1)/(p(t)-1)×[(R0)N-1tN-1sin⁡ε(t-σ)]1/(p(t)-1)dt+(aln⁡⁡1(R-R0)1-θ-λ)1/s,σ<r<R0,k-∫σR0[a1/s(1-θ)(R-R0)-θs((R-R0)1-θ-λ)(ln⁡⁡1(R-R0)1-θ-λ)(1/s)-1](p(R0)-1)/(p(t)-1)×[(R0)N-1tN-1sin⁡ε(t-σ)]1/(p(t)-1)dt+(aln⁡⁡1(R-R0)1-θ-λ)1/s,r≤σ,
where θ<β(R)/p(R), a>(1/α)sup|x|≥R0p(x) are constants, R0∈(σ,R), R-R0 is small enough, parameter λ∈[0,(R-R0)1-θ/2], Rλ satisfies (R-Rλ)1-θ-λ=0, ε=π/2(R0-σ)(3.4)k=[2p+((1+s)/s+1/(1-θ))+|β|+/(1-θ)αln⁡⁡2(R-R0)(1-θ)]1/s+∫σR0[2a1/s(1-θ)s(R-R0)(ln⁡⁡2(R-R0)1-θ)(1/s)-1](p(R0)-1)/(p(t)-1)×[(R0)N-1tN-1sin⁡ε(t-σ)]1/(p(t)-1)dt.
Obviously, for any positive constanta, we have g(r,a,λ)∈C1[0,Rλ).
WhenR0<r<Rλ<R, we have
(3.5)g′=g′(r,a,λ)=a1/ss(ln⁡⁡1(R-r)1-θ-λ)(1/s)-1(1-θ)(R-r)-θ(R-r)1-θ-λ,|g′|p(r)-2g′=[(1-θ)a1/ss]p(r)-1(ln⁡⁡1(R-r)1-θ-λ)((1/s)-1)(p(r)-1)(R-r)-θ(p(r)-1)[(R-r)1-θ-λ]p(r)-1,(rN-1|g′|p(r)-2g′)′=rN-1[(1-θ)a1/ss]p(r)-1(ln⁡⁡1(R-r)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)],
where(3.6)Π(r)={rN-1[(1-θ)a1/s/s]p(r)-1}′(p(r)-1)rN-1[(1-θ)a1/s/s]p(r)-1(R-r)1-θ-λ(R-r)1-θ(R-r)+((1/s)-1)(1-θ)(ln⁡⁡(1/((R-r)1-θ-λ)))+(R-r)1-θ-λ(R-r)1-θ(R-r)((1/s)-1)p′(r)(p(r)-1)ln⁡⁡[ln⁡1(R-r)1-θ-λ]+θp′(r)(p(r)-1)(R-r)1-θ-λ(R-r)1-θ(R-r)ln⁡⁡1(R-r)+θ(R-r)1-θ-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].
If(R-R0) is small enough, it is easy to see that
(3.7)|Π(r)|≤ln⁡⁡1(R-r)1-θ-λ,forλ∈[0,(R-R0)1-θ2]uniformly,
and then(3.8)(rN-1|g′|p(r)-2g′)′≤rN-1[(1-θ)a1/ss]p(r)-1(ln⁡⁡1(R-r)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,Rλ).
Thus, when0<R-R0 is small enough, from (3.5) and (3.8), for λ∈[0,(R-R0)1-θ/2] uniformly, we have
(3.9)(rN-1|g′|p(r)-2g′)′≤2rN-1[(1-θ)a1/ss]p(r)-1(ln⁡⁡1(R-r)1-θ-λ)((1/s)-1)(p(r)-1)+1(p(r)-1)(R-r)-θp(r)[(R-r)1-θ-λ]p(r)≤rN-1ρ(r)(1(R-r)1-θ-λ)αa=rN-1ρ(r)eαgs≤rN-1ρ(r)ef(r,g),∀r∈(R0,Rλ).
Thus, when0<R-R0 is small enough, the following inequality is valid for λ∈[0,(R-R0)1-θ/2] uniformly:
(3.10)(rN-1|g′|p(r)-2g′)′≤rN-1ρ(r)f(r,g),∀r∈(R0,Rλ).
Obviously, ifR-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],
(3.11)(rN-1|g′|p(r)-2g′)′=ε(R0)N-1[a1/s(1-θ)(R-R0)-θs((R-R0)1-θ-λ)(ln⁡⁡1(R-R0)1-θ-λ)(1/s)-1](p(R0)-1)cos⁡(ε(r-σ))≤ε(R0)N-1[a1/s(1-θ)(R-R0)-θs(1/2)(R-R0)1-θ(ln⁡⁡2(R-R0)1-θ)(1/s)+1](p(R0)-1)≤ε(R0)N-1[2a1/s(1-θ)s(R-R0)(2(R-R0)1-θ)(1/s)+1](p(R0)-1)≤ε(R0)N-1[2a1/s(1-θ)s(2R-R0)((s+1)/s)(1-θ)+1]p+≤rN-1ρ(r)eαgs≤rN-1ρ(r)ef(r,g),σ<r<R0.
Thus,(3.12)(rN-1|g′|p(r)-2g′)′≤rN-1ρ(r)ef(r,g),σ<r<R0.
Obviously,(3.13)(rN-1|g′|p(r)-2g′)′=0≤rN-1ρ(r)ef(r,g),0≤r<σ.
Sinceg(x,a,λ)=g(|x|,a,λ) is a C1 function on B(0,Rλ), 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,Rλ), and then g(|x|,a,0) is a supersolution for (P).
Defining the functiongm(|x|,a-ϵ)=g(r,a-ϵ,1/m) on [0,R1/m), where a-ϵ>(1/α)sup|x|≥R0p(x), then gm(|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 gm(|x|,a-ϵ)≥u(x) for any x∈B(0,R1/m). For any x∈B(0,R)∖B(0,R0), we have gm(|x|,a-ϵ)≥gm+1(|x|,a-ϵ), when m is large enough. Thus
(3.14)u(x)≤lim⁡m⟶+∞gm(|x|,a-ϵ),∀x∈B(0,R)∖B(0,R0).
Whend(x,∂Ω)>0 is small enough, we have
(3.15)lim⁡m⟶+∞gm(|x|,a-ϵ)<(aln⁡⁡1(R-r)1-θ)1/s+k≤g(|x|,a,0).
According to the comparison principle, we get thatg(|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.
Iff(r,u) satisfies
(3.16)f(r,u)⟶-∞(asu⟶-∞)forr∈[σ,R)uniformly,f(r,u)≤δus(asu⟶+∞)forr∈[σ,R)uniformly,
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).
Letz1 be a radial solution of
(3.17)-Δp(x)z1(x)=-μ,inΩ1=B(0,σ),z1=0,on∂Ω1,
whereμ>2(maxr∈[0,R0]ρ(r)+1)2(p+-1)/(p--1) is a positive constant. We denote z1=z1(r)=z1(|x|). Then, z1 satisfies
(3.18)-(rN-1|z1′|p(r)-2z1′)′=-rN-1μ,z1(σ)=0,z1′(0)=0,z1′=|rμN|1/(p(r)-1),z1=-∫rσ|rμN|1/(p(r)-1)dr.
Denotehb(r,λ) on [σ,R0] as
(3.19)hb(r,λ)=∫rR0{(R0)N-1tN-1t-σR0-σ[b(1-θ)(R-R0)-θs((R-R0)1-θ+λ)(bln⁡⁡1(R-R0)1-θ+λ)(1/s)-1]p(R0)-1+(σ)N-1tN-1R0-tR0-σ|σμN|}1/(p(t)-1)dt.
It is easy to see that(3.20)-hb′(σ,λ)=z1′(σ)=|σμN|1/(p(σ)-1),-hb′(R0,λ)=b(1-θ)(R-R0)-θs((R-R0)1-θ+λ)(bln⁡⁡1(R-R0)1-θ+λ)(1/s)-1.
Define the function v(r,b,λ) on [0,R) as
(3.21)v(r,b,λ)={(bln⁡⁡1(R-r)1-θ+λ)1/s-k*,R0≤r<R,(bln⁡⁡1(R-R0)1-θ+λ)1/s-k*-hb(r,λ),σ<r<R0,-∫rσ|rμN|1/(p(r)-1)dr+(bln⁡⁡1(R-R0)1-θ+λ)1/s-k*-hb(σ,λ),r≤σ,where θ∈(β1(R)/p(R),1), b∈(0,(1/δ)inf|x|≥R0p(x)) are constants, R0∈(σ,R),R-R0 is small enough, parameter λ∈[0,(R-R0)1-θ/2], and
(3.22)k*=M+(bln⁡⁡1(R-R0)1-θ)1/s,where M satisfies
(3.23)(σ)N-11R0-σ≥rN-1ρ(r)ef(r,y),∀y≤-M,∀r∈[0,R0].
Obviously, for any positive constantb, v(r,b,λ)∈C1[0,R).
By computation, whenr∈(R0,R), we have
(3.24)v′=v′(r,b,λ)=b1/ss(ln⁡⁡1(R-r)1-θ+λ)1/s-1(1-θ)(R-r)-θ(R-r)1-θ+λ,|v′|p(r)-2v′=[(1-θ)b1/ss]p(r)-1(ln⁡⁡1(R-r)1-θ+λ)(1/s-1)(p(r)-1)(R-r)-θ(p(r)-1)[(R-r)1-θ+λ]p(r)-1,(rN-1|v′|p(r)-2v′)′=rN-1[(1-θ)b1/ss]p(r)-1(ln⁡⁡1(R-r)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)),
where(3.25)Λ(r)={rN-1[(1-θ)b1/s/s]p(r)-1}′(p(r)-1)rN-1[(1-θ)b1/s/s]p(r)-1(R-r)+(1/s-1)(1-θ)(ln⁡⁡(1/((R-r)1-θ+λ)))[(R-r)1-θ+λ]×(R-r)1-θ+(1/s-1)p′(r)(p(r)-1)(R-r)ln⁡⁡[ln⁡1(R-r)1-θ+λ]+θp′(r)(p(r)-1)(R-r)ln⁡⁡1(R-r)+(1-θ)[(R-r)1-θ+λ](R-r)1-θ+-p′(r)p(r)-1(R-r)ln⁡⁡[(R-r)1-θ+λ].
By computation, whenR-R0 is small enough, for λ∈[0,(R-R0)1-θ/2] uniformly, we have
(3.26)(rN-1|v′|p(r)-2v′)′≥rN-1[(1-θ)b1/ss]p(r)-1(ln⁡⁡1(R-r)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)-1θ(1-12)≥θ2rN-1[(1-θ)b1/ss]p(r)-1(ln⁡⁡1(R-r)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≥θ2rN-1[(1-θ)b1/ss]p(r)-1(ln⁡⁡1(R-r)1-θ+λ)(1/s-1)(p(r)-1)(p(r)-1)(R-r)-θp(r)[(R-r)1-θ+λ]p(r)≥rN-1ρ1(R-r)-β1(r)eδvs≥rN-1ρ(r)ef(r,v),∀r∈(R0,R).
Then, forλ∈[0,(R-R0)1-θ/2] uniformly, we have
(3.27)(rN-1|v′|p(r)-2v′)′≥rN-1ρ(r)ef(r,v),∀r∈(R0,R).
WhenR-R0 is small enough, forallr∈(σ,R0), since v≤-M, it is easy to see that
(3.28)(rN-1|v′|p(r)-2v′)′≥(rN-1|h′|p(r)-2h′)′=(R0)N-11R0-σ[b(1-θ)(R-R0)-θs((R-R0)1-θ+λ)(bln⁡⁡1(R-R0)1-θ+λ)1/s-1]p(R0)-1-(σ)N-11R0-σ|σμN|≥(σ)N-11R0-σ≥rN-1ρ(r)ef(r,v),
Then,(3.29)(rN-1|v′|p(r)-2v′)′≥rN-1ρ(r)ef(r,v),∀r∈(σ,R0).
Obviously,(3.30)(rN-1|v′|p(r)-2v′)′=rN-1μ≥rN-1ρ(r)ef(r,v),∀r∈(0,σ).
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 functionvm(r,b+ϵ) on B(0,R) as
(3.31)vm(r,b+ϵ)=vm(r,b+ϵ,1m),
whereϵ is a small enough positive constant such that (b+ϵ)<(1/δ)inf|x|≥R0p(x).
For anym=1,2,…, we can see that vm(r,b+ϵ)∈C1([0,R)) is a subsolution for (P) on B(R0,R). According to the comparison principle, we get that vm(r,b+ϵ)≤u(x) for any x∈B(0,R). For any x∈B(0,R)∖B(0,R0), we have vm(|x|,b+ϵ)≤vm+1(|x|,b+ϵ). Thus
(3.32)u(x)≥lim⁡m⟶+∞vm(|x|,b+ϵ),∀x∈B(0,R)∖B(0,R0).
Whend(x,∂Ω) is small enough, we havelimm→+∞vm(|x|,b+ϵ)>v(|x|,b,0).
According to the comparison principle, we get thatv(|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).
Letμ>2(maxr∈[0,R0]ρ(r)+1)2(p+-1)/(p--1) be a positive constant. Denote ϖb(r,λ) on [σ,R0] as
(3.33)ϖb(r,λ)=∫rR0{(R0)N-1tN-1t-σR0-σ[bs(R+λ-R0)(bln⁡⁡(R+λ-R0)-1)1/s-1]p(R0)-1+(σ)N-1tN-1R0-tR0-σ|σμN|}1/(p(t)-1)dt.
It is easy to see that(3.34)-ϖb′(σ,λ)=z1′(σ)=|σμN|1/(p(σ)-1),-ϖb′(R0,λ)=bs(R+λ-R0)(bln⁡⁡(R+λ-R0)-1)1/s-1.
Define the functionη(r,b,λ) on B(0,R) as
(3.35)η(r,b,λ)={(bln⁡⁡(R+λ-r)-1)1/s-k*,R0≤r<R,(bln⁡⁡(R+λ-R0)-1)1/s-k*-ϖb(r,λ),σ<r<R0,-∫rσ|rμN|1/(p(r)-1)dr+(bln⁡⁡(R+λ-R0)-1)1/s-k*-ϖb(σ,λ),r≤σ,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
(3.36)k*=M+(bln⁡⁡1R-R0)1/s,where M is defined in (3.23).
Obviously, for any positive constantb, η(r,b,λ)∈C1[0,R).
Similar to the proof of Case1, when R-R0 is small enough, we have
(3.37)(rN-1|η′|p(r)-2η′)′≥rN-1(b1/ss)p(r)-1(p(r)-1)(R+λ-r)-p(r)(ln⁡⁡(R+λ-r)-1)(1/s-1)(p(r)-1)(1-12)≥rN-1ρ(r)ef(r,η),∀r∈(R0,R).
WhenR-R0 is small enough, forallr∈(σ,R0), from the definition of k*, it is easy to see that
(3.38)(rN-1|η′|p(r)-2η′)′≥(σ)N-11R0-σ≥rN-1ρ(r)ef(r,η).
Obviously(3.39)(rN-1|η′|p(r)-2η′)′=rN-1μ≥rN-1ρ(r)ef(r,η),∀r∈(0,σ).
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 functionηm(r,b+ε) on B(0,R) as
(3.40)ηm(r,b+ε)=η(r,b+ε,1m),
whereε is a small enough positive constant such that (b+ε)<(1/δ)inf|x|≥R0p(x).
We can see thatηm(r,b+ε)∈C1[0,R) is a subsolution for (P) for any m=1,2…. According to the comparison principle, we get that ηm(r,b+ε)≤u(x) for any x∈B(0,R). For any x∈B(0,R)∖B(0,R0), we have ηm(|x|,b+ε)≤ηm+1(|x|,b+ε). Then,
(3.41)u(x)≥lim⁡m→+∞ηm(|x|,b+ε),∀x∈B(0,R)∖B(0,R0).
Whend(x,∂Ω) is small enough, we have
(3.42)lim⁡m⟶+∞ηm(|x|,b+ε)>η(|x|,b,0).
According to the comparison principle, we get thatη(|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.
Iff(r,u) satisfies
(3.43)lim⁡u⟶+∞f(r,u)us=δ(asu⟶+∞)forr∈[σ,R)uniformly,
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
(3.44)lim⁡|x|⟶Ru(x)((p(R)/δ)(ln⁡⁡1/(R-|x|)1-θ))1/s=1,whereθ=β(R)p(R).Proof.
It is easy to be seen from Theorems3.1 and 3.2


## 4. The Existence of Boundary Blow-Up Solutions

Theorem 4.1.
Ifinfx∈Ωp(x)>N and f(r,u) satisfies
(4.1)f(r,u)≥aus(asu→+∞)forr∈[σ,R)uniformly,
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(4.2)-Δp(x)u+ρ(r)ef(x,u)=0,inΩ0,u(x)=c,forx∈∂Ω0,
where c is a positive constant and Ω0⋐Ω is a radial subdomain of Ω. Since infx∈Ωp(x)>N, then W1,p(x)(Ω0)↪Cα(Ω0¯), where α∈(0,1). The relative functional of (4.2) is
(4.3)φ=∫Ω01p(x)|∇u(x)|p(x)dx+∫Ω0F(x,u)dx,
where F(x,u)=∫0uef(x,t)dt. Since φ is coercive in X:=c+W01,p(x)(Ω0), then φ possesses a nontrivial minimum point u. So, problem (4.2) possesses a weak solution u.
Sinceaus≤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.
DenoteDj={x∣|x|<1-1/(j+1)R} (j=1,2,…). Let us consider the problem
(4.4)-Δp(x)uj+ρ(x)ef(x,uj)=0,inDj,uj(x)=g*(x),forx∈∂Dj,
and the relative functional is(4.5)φ=∫Dj1p(x)|∇uj(x)|p(x)dx+∫Djρ(x)F(x,uj)dx.
Letg*j(x)=g*(x)∣Dj. Since the functional φ is coercive in Xj=g*j(x)+W01,p(x)(Dj), then φ has a nontrivial minimum point uj. Therefore, problem (4.4) has a weak solution uj.
According to the comparison principle, we get thatg*(x)≤uj(x) for any x∈Dj (j=1,2,… ). Since uj(x)=g*(x) for any x∈∂Dj, then uj(x)≤uj+1(x) for any x∈∂Dj (j=1,2,…). According to the comparison principle, we get that uj(x)≤uj+1(x) for any x∈Dj (j=1,2,…).
Sinceg*(x) is a supersolution and g*(x)≥g*(x) for any x∈Ω, so we have uj(x)=g*(x)≤g*(x) for any x∈∂Dj (j=1,2,…). According to the comparison principle, we get that uj(x)≤g*(x) for any x∈Dj (j=1,2,…).
Sinceg*(x) and g*(x) are locally bounded, from Lemma 2.4, each weak solution of (4.4) is a Cloc1,α function. The C1,α interior regularity result implies that the sequences {uj} and {∇uj} are equicontinuous in D2, and hence we can choose a subsequence, which we denoted by {uj1}, such that uj1→w1 and ∇uj1→ϖ1 uniformly on D1 for some w1∈C(D1) and ϖ1∈(C(D1))N. In fact, ϖ1=∇w1 on D1, and from the interior C1,α estimate, we conclude that ∇w1∈(Cα(D1))N for some 0<α<1. Thus, w1∈W1,p(x)(D1)∩C1,α(D1). From the C1,α interior regularity result, we see that |∇uj|p-1|∇φ|≤C|∇φ| on D1, and since the function ξ→|ξ|p-2ξ is continuous on ℝN, it follows that |∇uj1(x)|p-2∇uj1(x)·∇φ(x)→|∇w1(x)|p-2∇w1(x)·∇φ(x) for x∈D1. Thus, by the dominated convergence theorem, we have
(4.6)∫D1|∇uj1(x)|p-2∇uj1(x)⋅∇φ(x)dx⟶∫D1|∇w1(x)|p-2∇w1(x)⋅∇φ(x)dx,∀φ∈W01,p(x)(D1).
Furthermore, since0≤f(uj1)≤f(uj+11) and f(uj1(x))→f(w1(x)) for each x∈D1, by the monotone convergence theorem, we obtain
(4.7)∫D1ρef(uj1)qdx⟶∫D1ρef(w1)qdx,∀q∈W01,p(x)(D1).
Therefore, it follows that(4.8)∫D1|∇w1(x)|p-2∇w1(x)⋅∇q(x)dx+∫D1ρef(w1)qdx=0,∀q∈W01,p(x)(D1),
and hencew1 is a weak solution for -Δp(x)w1+ρef(w1)=0 on D1.
Thus, there exists a subsequence of{uj} which we denote it by{uj1}, such that uj1→w1 in D1 (as j→∞), where w1∈W1,p(x)(D1)∩C1,α1(D1) and satisfies
(4.9)∫D1|∇w1|p(x)-2∇w1∇qdx+∫D1ρ(x)ef(x,w1)qdx=0,∀q∈W01,p(x)(D1).
Similarly, we can prove that there exists a subsequence of{uj1} which we denote by{uj2}, such that uj2→w2 in D2 (as j→∞), where w2∈W1,p(x)(D2)∩C1,α2(D2) satisfies w1=w2∣D1 and
(4.10)∫D2|∇w2|p(x)-2∇w2∇qdx+∫D2ρ(x)ef(x,w2)qdx=0,∀q∈W01,p(x)(D2).
Repeating the above steps, we can get a subsequence of{uji∣j=1,2,…} which we denote by {uji+1∣j=1,2,…} (i=1,2,…) and satisfies the following.(10)
For any fixedi, {uji+1} is a subsequence of{uji}.(20)
For any fixedi, uji+1→wi+1 in Di+1 (as j→∞), where wi+1∈W1,p(x)(Di+1)∩C1,αi+1(Di+1) satisfies wi=wi+1∣Di.(30)
For any fixedi, wi satisfies
(4.11)∫Di|∇wi|p(x)-2∇wi∇qdx+∫Diρ(x)ef(x,wi)qdx=0,∀q∈W01,p(x)(Di).
Thus, we can conclude that
(i)
{ujj}is a subsequence of{uj},(ii)
there exists a functionw∈Wloc1,p(x)(Ω)∩Cloc1,α(Ω) such that wi=w∣Di, and for any x∈Ω, there exists a constant jx such that when j≥jx, ujj(x) is defined at x, and limj→∞ujj(x)=w(x),(iii)
(4.12)∫Ω|∇w|p(x)-2∇w∇qdx+∫Ωρ(x)ef(x,w)qdx=0,∀q∈W0,loc⁡1,p(x)(Ω).
Obviously,w is a boundary blow-up solution for (P).
This completes the proof.In Theorem4.1, when infx∈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<infx∈Ωp(x)≤supx∈Ωp(x)<∞. We need to do some preparation. Let us consider
(I)(rN-1|u′|p(r)-2u′)′=rN-1ρ(r)ef(r,u),r∈(0,Rλ),u′(0)=0,u(Rλ)=d,whereRλ∈(0,R) and d is a constant.Lemma 4.2.
IfΦ2(Rλ)≤d≤Φ1(Rλ), where Φ1 and Φ2 are defined in Theorems 3.13.2, respectively, then (4.13) has a solution u satisfying
(4.13)Φ2(r)≤u(r)≤Φ1(r),∀r∈[0,Rλ].Proof.
Denote(4.14)h(r,u)={ef(r,Φ1(r))+arctan(u(r)-Φ1(r)),u(r)>Φ1(r),ef(r,u),Φ2(r)≤u(r)≤Φ1(r),ef(r,Φ2(r))+arctan(u(r)-Φ2(r)),u(r)<Φ2(r).
LetρE(t)=ρ(|t|),andhE(t,u)=h(|t|,u),forallt∈[-Rλ,Rλ]. Let us consider the even solutions of the following
(II)(|t|N-1|u′|p(|t|)-2u′)′=|t|N-1ρE(t)hE(t,u),t∈(-Rλ,Rλ),u(-Rλ)=d,u(Rλ)=d.
It is easy to see thatu is an even solution for (4.15) if and only if u is even and
(4.15)u=d-∫rRλ[|t|1-N∫0t|s|N-1ρ(s)h(s,u(s))ds]1/(p(t)-1)dt,∀r∈[0,Rλ].
DenoteΨ(u,μ)=μd-μ∫rRλ[|t|1-N∫0t|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 CE1[0,Rλ] to CE1[0,Rλ], where CE1[0,Rλ]={u∈C1[0,Rλ]∣uiseven}. Thus, u=Ψ(u,1) has a solution u in CE1[0,Rλ] and satisfies u′(0)=limr→0+u′(r)=0. Then, u(|t|) is an even solution for (4.15).
DenoteΦ1,E(t)=Φ1(|t|),Φ2,E(t)=Φ2(|t|). From the definitions of Φ1and Φ2, we can see that Φ1′(0)=0=Φ2′(0); therefore, Φ1,E(t) and Φ2,E(t) are supersolution and subsolution for (4.15), respectively.
SinceΦ2(Rλ)≤u(Rλ)≤Φ1(Rλ) and hE(t,·) is increasing, from the comparison principle, we have
(4.16)Φ2,E(t)≤u(t)≤Φ1,E(t),∀t∈[-Rλ,Rλ].
It means thatu is a solution for (4.13) and u satisfies
(4.17)Φ2(r)≤u(r)≤Φ1(r),∀r∈[0,Rλ].
Thusu is a radial solution for (P). This completes the proof.Theorem 4.3.
Iff(r,u) satisfies
(4.18)f(r,u)≥aus(asu⟶+∞)forr∈[σ,R)uniformly,
where σ is defined in (H4) and a and s are positive constants, then (P) possesses a boundary blow-up solution.Proof.
From Lemma4.2, we have that (4.4) has a weak solution uj(x)=uj(|x|)=uj(r). Similar to the proof of Theorem 4.1, we can obtain the existence of solutions for (P).



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*Source: 971268-2010-12-09.xml*