Nonlinear Analysis 75 (2012) 3249–3261

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis

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

Boundary blow-up solutions to degenerate elliptic equations withnon-monotone inhomogeneous termsAhmed Mohammed a,∗, Seid Mohammed b

a Department of Mathematical Sciences, Ball State University, Muncie, IN 47306, USAb Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia

a r t i c l e i n f o

Article history:Received 27 October 2011Accepted 24 December 2011Communicated by S. Ahmad

We dedicate this paper to ProfessorGiovanni Porru on the occasion of his 70thbirthday

Keywords:Boundary blow-up solutionMinimality principleInfinity Laplacian

a b s t r a c t

Given a non-negative, and non-trivial continuous real-valued function h on Ω × [0,∞)such that h(x, 0) = 0 for all x ∈ Ω , we study the boundary value problem

1∞u = h(x, u) inΩu = ∞ on ∂Ω, (BVP)

where Ω ⊆ RN , N ≥ 2 is a bounded domain and 1∞ is the ∞-Laplacian, a degenerateelliptic operator. In this paper, we investigate conditions on the inhomogeneous termh(x, t) under which Problem (BVP) admits a solution or fails to admit a solution in C(Ω).Some notable features of this work are that h(x, t) is not required to have any specialstructure, and no monotonicity condition is imposed on h(x, t). Furthermore, h(x, t) maybe allowed to vanish in either of the variables.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

LetΩ be a bounded domain in RN for N ≥ 2, and h : Ω × [0,∞) → [0,∞) be a continuous function. In this work, wewish to study some general conditions on h in order for the following boundary value problem to (or not to) admit a solutionu ∈ C(Ω).

1∞u = h(x, u) inΩu = ∞ on ∂Ω. (1.1)

The boundary condition is understood in the sense that u(x) → ∞ as x → ∂Ω . The operator1∞ is the ∞-Laplacian, ahighly degenerate elliptic operator given by

1∞u := ⟨D2uDu,Du⟩ =

Ni,j=1

DiuDijuDju.

As a result of the high degeneracy of the∞-Laplacian, the associated Dirichlet problemsmay not have classical solutions,that is solutions in C2(Ω). Therefore solutions are understood in the viscosity sense, a concept to be defined in Section 2.For further insight on the ∞-Laplacian, we refer the interested reader to the papers [1,2].

Solutions of Problem (1.1), when they exist, are referred to as boundary blow-up solutions. The problem of existence,asymptotic boundary behavior and uniqueness of boundary blow-up solutions, in the context of the classical Laplace

∗ Corresponding author. Tel.: +1 765 285 8813; fax: +1 765 285 1721.E-mail addresses: amohammed@bsu.edu (A. Mohammed), mt@yahoo.com (S. Mohammed).

0362-546X/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2011.12.026

3250 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

operator, has been studied extensively and continues to draw the efforts of many researchers. Here we wish to mention theworks of Bandle and Marcus [3,4], Lazer and McKenna [5], Cîrstea and Rădulescu [6,7], García-Milián et al. [8–10], Lair [11],Marcus and Véron [12,13], Véron [14,15], Zhang [16] and Zhang et al. [17]. We refer the reader to the monograph [18] foran extensive list of references. We should also mention that similar investigations have been carried out when the Laplaceoperator is replaced by other elliptic operators. See [19–26].

In most of these investigations the inhomogeneous term h(x, t)was required to have some special structure which fallsinto one of three categories (see [9]): h(x, t) = h(t) is independent of x, or h(x, t) is separable in its variables, in the sensethat it is of the form h(x, t) = ϑ(x)h(t) or that the growth of h(x, t) in t is controlled by a function h(t), independent of x. Anexception to this is the paper [9], where inhomogeneous terms of the form h(x, t) = tp(x) are considered in Problem (1.1),with the ∞-Laplacian replaced by the classical Laplacian. See [9] for other papers that treat nonlinearities of similar type.

As far as the present authors are aware the question of existence of boundary blow-up problems in the context of the∞-Laplacian was first considered by Juutinen and Rossi in the paper [27]. In the paper [27], the authors study existence ofsolutions to Problem (1.1) for the case h(x, t) = tp with a constant p > 1, but with the ∞-Laplacian1∞ replaced by the socalled normalized ∞-Laplacian∆N

∞u := |Du|−21∞u.

In our recent work [28] we studied the existence of solutions to Problem (1.1) under the assumptions that h(x, t) > 0in Ω × (0,∞), h(x, 0) ≡ 0 in Ω , h(x, t) is non-decreasing in t for each x ∈ Ω and that an appropriate Keller–Ossermantype condition on h holds.We should point out that in the paper [28], none of the special structures traditionally required onh(x, t) is imposed. However, themonotonicity of the non-homogeneous term h in its second variable and the strict positivityof h were needed in [28] as these provide sufficient conditions for Comparison Principles to hold which are the main toolsused in [28] to study Problem (1.1). Recently, Dumont et al. [29] investigated the boundary value problem involving theclassical Laplace operator, to wit, 1u = f (u) in Ω with u = ∞ on ∂Ω , without any monotonicity or strict positivityassumption on the non-linearity f . In the paper [29], these authorswere able to circumvent the use of Comparison Principlesby using a Minimality Principle, See [29, Corollary 2.2].

Motivated by the work [29], in this paper we wish to study existence of solutions to Problem (1.1) without requiring anymonotonicity condition or strict positivity condition on the non-linearity h. Because h is no longer monotonic in the secondvariable, themethods of [27,28] are not applicable. Furthermore no special structure is imposed on the inhomogeneous termin Problem (1.1).

The paper is organized as follows. In Section 2 we will recall some preliminary concepts as well as useful results thatwill be needed in the paper. We will also fix some notations and provide some basic definitions. In Section 3 we begin bystudying aMinimality Principle which will replace the role played by the Comparison Principle. Thenwe proceed to identifysome sufficient conditions needed to prove existence of solutions to Problem (1.1). Section 4 is devoted to the study ofconditions on the inhomogeneous term h(x, t) that lead to existence of solutions of Problem (1.1). In Section 5, we considersome results on non-existence of solutions to Problem (1.1). In the event that the inhomogeneous term is separable, that is,it has the form h(x, t) = ϑ(x)f (t), then we find a necessary and sufficient condition on f in order for Problem (1.1) to admita solution u ∈ C(Ω, [0,∞)). Finally we have included an Appendix in which we provide some examples as well as a proofto a remark made in the main body of the paper. In this section, we also present an example to illustrate some of the mainresults of the paper.

2. Preliminaries

Let us introduce some notations that we shall use throughout the remainder of the paper. We will use Ω to denote abounded domain in RN and diam(Ω) its diameter. Given r > 0, and x ∈ RN we will use B(x, r) to denote the ball in RN

of radius r and centered at x. There will also be occasions when we will use the notationB(x, r) to denote the compact setB(x, r) ∩ Ω . The set C(Ω) will stand for the space of continuous functions inΩ . Given a subset I of R, we write C(Ω, I) todenote the class of continuous functions inΩ that take values in I . We use the notation O ⊂⊂ Ω to denote an open subsetO ofΩ whose closure O is contained inΩ . Finally for δ > 0 we set

Ωδ := x ∈ Ω : dist (x, ∂Ω) < δ.

Given h ∈ C(Ω × R,R), let us consider the following partial differential equation, PDE, inΩ ⊆ RN .

1∞u = h(x, u) (x ∈ Ω). (2.1)

Definition 2.1. A function u ∈ C(Ω) is a viscosity sub-solution of the PDE (2.1) if for every ψ ∈ C2(Ω) with the propertythat u − ψ has a local maximum at some x0 ∈ Ω , there holds

∆∞ψ(x0) ≥ h(x0, u(x0)).

A function u ∈ C(Ω) is a viscosity super-solution of the PDE (2.1) if for everyψ ∈ C2(Ω), with the property that u −ψ hasa local minimum at some x0 ∈ Ω , then

∆∞ψ(x0) ≤ h(x0, u(x0)).

A function u ∈ C(Ω) is a viscosity solution of (2.1) if it is both a sub-solution and a super-solution of (2.1).

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3251

Let us consider the Dirichlet problem1∞u = h(x, u) inΩu = b on ∂Ω, (2.2)

where b ∈ C(∂Ω). We say u ∈ C(Ω) is a sub-solution (or a super-solution) of the above Dirichlet problem iff u is a sub-solution (or a super-solution) of (2.1) and u ≤ b (u ≥ b) on ∂Ω .

Finally, by a solution (sub-solution or super-solution) u of (1.1) we mean a solution (sub-solution or super-solution) u of(2.1) such that u = ∞ on ∂Ω .

Remark 2.2. If u ∈ C(Ω) is a sub-solution (or a super-solution) of (2.1) inΩ we indicate this by writing

1∞u ≥ h(x, u) inΩ (or1∞u ≤ h(x, u) inΩ).

Throughout the paper, solutions, sub-solutions and super-solutions of (2.1) are all to be understood in the viscosity sense,as given in Definition 2.1 above.

To proceed further we assume that the non-linearity h : Ω × R → R in Problem (2.1) satisfies the following condition.For each compact interval I ⊆ R

supΩ×I

|h(x, t)| < ∞. (2.3)

The following theorem holds. We refer the reader to the paper [30] for a proof.

Theorem 2.3. Let h ∈ C(Ω×R,R) satisfy condition (2.3), and b ∈ C(∂Ω). Suppose that u∗ ∈ C(Ω) is a sub-solution of (2.2) inΩ , and u∗

∈ C(Ω) is a super-solution of (2.2) in Ω such that u∗ ≤ u∗ in Ω . Then problem (2.2) admits a solution u ∈ C(Ω)such that u∗ ≤ u ≤ u∗ inΩ .

Remark 2.4. The solution u in Theorem 2.3 above is constructed by the so-called Perron’s method. More explicitly u isdefined as

u(x) = infα∈P

α(x), x ∈ Ω, (2.4)

where

P := α ∈ C(Ω) : 1∞α ≤ h(x, α) inΩ, α ≥ u∗ inΩ and α ≥ b on ∂Ω. (2.5)

As a consequence of Theorem 2.3, we have the following useful property of the solution defined by (2.4).

Theorem 2.5 (Minimality Principle). Under the assumptions given in Theorem 2.3 above, there is a unique solution u ∈ C(Ω)of (2.2) with the following properties.(i) u∗ ≤ u inΩ .(ii) Given any open set O ⊆ Ω and any super-solution β ∈ C(O) of (2.1) in O such that u∗ ≤ β in O and u ≤ β on ∂O, then

we have u ≤ β in O.Proof. Let u be the solution of (2.2) as given by (2.4), with u∗ ≤ u ≤ u∗ inΩ . Thus (i) holds trivially. Let O ⊆ Ω and let usfix β ∈ C(O) such that1∞β ≤ h(x, β) in O, u∗ ≤ β in O and u ≤ β on ∂O. Define v∗ as follows.

v∗(x) :=

minβ(x), u(x) x ∈ O

u(x) x ∈ Ω \ O.

Note that since u ≤ β on ∂O, we have v∗∈ C(Ω). Furthermore, we note that v∗

= b on ∂Ω . We claim that v∗ is asuper-solution of (2.2). To see this, let ϕ ∈ C2(Ω) such that v∗

− ϕ has a minimum at x0 ∈ Ω in a neighborhood N of x0.Case 1. Assume that x0 ∈ O. Suppose v∗(x0) = β(x0). Note that v∗

≤ β in O. Then

β(x0)− ϕ(x0) = v∗(x0)− ϕ(x0) ≤ v∗(x)− ϕ(x) ≤ β(x)− ϕ(x), x ∈ N ∩ O.

Therefore

1∞ϕ(x0) ≤ h(x0, β(x0)) = h(x0, v∗(x0)).

If v∗(x0) = u(x0), then a similar argument shows that

1∞ϕ(x0) ≤ h(x0, v∗(x0)).

Case 2. We now suppose that x0 ∈ Ω \ O. Then v∗(x0) = u(x0), and as in Case 1, we see that u(x0) − ϕ(x0) ≤ u(x) − ϕ(x)in N , and therefore1∞ϕ(x0) ≤ h(x0, v∗(x0)).In conclusion we see that v∗ is a super-solution of (2.2) such that u∗ ≤ v∗ inΩ , and v∗

= b on ∂Ω . Therefore v∗∈ P where

P is the class defined in (2.5). Thus, by the definition of u given in (2.4) we see that u ≤ v∗ inΩ . In particular, u ≤ v∗≤ β

in O, showing that (ii) also holds. The uniqueness is obvious.

3252 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

Remark 2.6. Note that the solution u ∈ C(Ω) defined by (2.4) depends on the sub-solution u∗ and the boundary datab ∈ C(∂Ω), and not on the choice of the super-solution u∗. Therefore, we will refer to the unique solution u ∈ C(Ω)of Problem (2.2) that satisfies properties (i) and (ii) of Theorem 2.5 as the minimal solution of (2.2) relative to the sub-solution u∗.

The following lemma will be useful in the sequel. We refer the reader to the papers [31,28] for a proof.

Lemma 2.7. Let h ∈ C(Ω × R, [0,∞)) and uk be a sequence of functions in C(Ω) that satisfy the Eq. (2.1) inΩ . Suppose thatfor any O ⊂⊂ Ω there are constants mO and MO such that mO ≤ uk ≤ MO in O for all k. Then there is a subsequence of uk

that converges locally uniformly inΩ to a solution u ∈ C(Ω) of (2.1).

The Minimality Principle, Theorem 2.5, and the above lemma can now be used to establish the following useful result onexistence of blow-up solution to Problem (1.1).

Lemma 2.8. Let Ω ⊆ RN be a bounded domain, and h ∈ C(Ω × R, [0,∞)) satisfy condition (2.3). Let u∗ ∈ C(Ω) be a sub-solution of (2.1) inΩ and u∗

∈ C(Ω) be a super-solution of (1.1) inΩ such that u∗ ≤ u∗ inΩ . Then there is a solution u ∈ C(Ω)of (1.1) such that u∗ ≤ u ≤ u∗ inΩ .

Proof. Let ℓ := supx∈Ω u∗(x), and for each positive integer j, we consider the boundary value problem1∞u = h(x, u) inΩu = ℓ+ j on ∂Ω. (Dj)

For each positive integer j, we note that u∗ ≤ wj in Ω where wj := ℓ + j and that wj is a super-solution of (Dj). Bythe minimality principle, Theorem 2.5, we pick the minimal solution uj ∈ C(Ω) of Problem (Dj) relative to u∗, so thatu∗ ≤ uj ≤ ℓ + j in Ω . Since u∗ is a super-solution of (1.1), using property (ii) of Theorem 2.5 on Ωδ for arbitrarily smallδ > 0, we conclude that in fact u∗ ≤ uj ≤ u∗ inΩ for any j. Let us now observe that uj+1 is a super-solution of (Dj) such thatu∗ ≤ uj+1 in Ω , and uj ≤ uj+1 on ∂Ω . Since uj is the minimal solution of Problem (Dj) relative to u∗, again using property(ii) of Theorem 2.5, we conclude that uj ≤ uj+1 inΩ . Thus we have constructed a non-decreasing sequence uj of solutionsof Problem (2.1) in C(Ω) such that u∗ ≤ uj ≤ u∗ inΩ for all j = 1, 2, . . .. By Lemma 2.7, we conclude that uj contains asubsequence that converges locally uniformly to a solution u ∈ C(Ω). Consequently we conclude that u∗ ≤ u ≤ u∗ in Ω ,and that u = ∞ on ∂Ω . Thus u ∈ C(Ω) is a solution of Problem (1.1) such that u∗ ≤ u ≤ u∗ inΩ , as asserted.

Remark 2.9. In the sequel, Lemma 2.8will be usedwhen the inhomogeneous term h(x, t) satisfies the condition h(x, 0) ≡ 0inΩ . In this casewe take u∗ ≡ 0 as a sub-solution of (2.1) and according to Lemma 2.8, one needs only to find a non-negativesuper-solution u∗ of Problem (1.1) to deduce existence of a non-negative solution to (1.1).

3. Existence of boundary blow-up solutions

Let us now consider g ∈ C([0,∞), [0,∞)) such that g(0) = 0 and g(c) > 0 for some c > 0. We define Φ : (0,∞) →

(0,∞] by

Φ(t) :=1

√2

∞

t

14√G(s)− G(t)

ds, where G(t) :=

t

0g(ζ ) dζ . (3.1)

By convention we takeΦ(t) = ∞ when the integral diverges or s ∈ [t,∞) : G(s) = G(t) is a set of positive measure.We will use the following Keller–Osserman condition, hereafter referred to as the KO condition, on g . See [29].

Φ(t) < ∞ for some t > 0. (3.2)

An argument as in [29] shows that the KO condition (3.2) is equivalent to the following condition.

lim inft→∞

Φ(t) = 0. (3.3)

Remark 3.1. We remark that if g(t) > 0 for t > 0, and g is non-decreasing in (0,∞), then Condition (3.2) is equivalent tothe condition that

∞

t

14√G(s)

ds < ∞ for some t > 0.

This follows from the easily verifiable estimate∞

2t

ds4√G(s)

≤

∞

2t

ds4√G(s)− G(t)

≤4√2

∞

2t

ds4√G(s)

, t > 0. (3.4)

See the Appendix in [28] for details.

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3253

Let us now consider the following boundary blow-up problem in balls. Here B stands for a ball in RN .1∞u = g(u) in Bu = ∞ on ∂B. (3.5)

We begin with the following lemma.

Lemma 3.2. Suppose g satisfies the KO condition (3.2). There is R > 0 such that for any x0 ∈ RN , Problem (3.5) admits anon-negative solution u ∈ C(B) in B = B(x0, R).

Proof. By the KO condition (3.2), we fix a > 0 such thatΦ(a) < ∞. Let G be defined as in (3.1), and set

ψ(t) :=1

√2

t

a

ds(G(s)− G(a))1/4

, t > a.

Since Φ(a) < ∞, on replacing ψ by ψ − ψ(a+), if necessary, and setting R := ψ(∞) = Φ(a) < ∞, we note thatψ : [a,∞) → [0, R) is a well-defined, continuous increasing function. Let φ : [0, R) → [a,∞) be the inverse of ψ so thatφ(0) = a and φ(r) → ∞ as r → R−. Observing that

r =1

√2

φ(r)

a

ds(G(s)− G(a))1/4

, for 0 ≤ r < R,

direct computation shows that, for 0 < r < R

φ′(r) = [4G(φ(r))− G(a)]1/4 andφ′′(r) = g(φ(r))[4G(φ(r))− G(a)]−1/2.

Therefore φ ∈ C2((0, R)) ∩ C1([0, R)), and φ is a solution of the following initial value problem(φ′(r))2φ′′(r) = g(φ(r)), r > 0φ(0) = a, φ′(0) = 0. (3.6)

Now, wemake the simple observation that v(x) := φ(|x− x0|) is a viscosity super-solution of (3.5) in B. To see this, let usfirst note that v ∈ C2(B \ x0)∩ C1(B). Therefore by virtue of the differential equation in (3.6) it is clear that v is a viscositysolution of the PDE in (3.5) in B \ x0. On the other hand, suppose ϕ ∈ C2(B) such that v−ϕ has a minimum at x0 ∈ B. Notethat Dϕ(x0) = Dv(x0) = 0. Therefore1∞ϕ(x0) = 0 ≤ g(v(x0)), showing that v is indeed a non-negative super-solution of(3.5) inΩ , as claimed.

Therefore, by Lemma 2.8 we conclude that Problem (3.5) has a non-negative solution u ∈ C(B) in B := B(x0, R).

Next we show that Problem (3.5) admits solutions in balls of arbitrarily small radii. The argument is similar to that of theabove lemma.

Lemma 3.3. Suppose g satisfies the KO condition (3.2). Problem (3.5) admits a non-negative solution in balls of arbitrarily smallradius.

Proof. Let us fix x0 ∈ RN , and consider the set

S := R > 0 : Problem (3.5) has a non-negative solution in B(x0, R).

Lemma 3.2 shows that this set is non-empty. Let R0 = inf S, and suppose R0 > 0. By the KO condition (3.3) we can choosea > 0, large enough, such thatΦ(a) < R0. Let φ be a solution of

(φ′(r))2φ′′(r) = g(φ(r)) in (0, R) and φ(0) = a, φ′(0) = 0. (3.7)

Here (0, R) is the maximal interval of existence. We observe that φ′ > 0 in (0, R) and direct computation shows that

r =1

√2

φ(r)

a

14√G(t)− G(a)

ds ≤ Φ(a), for 0 < r < R.

Therefore R < ∞. Moreover, we note that φ(r) → ∞ as r → R−. In particular, we have

R = Φ(a) < R0.

But an argument similar to the one in the proof of Lemma 3.2 above, shows that v(x) := φ(|x− x0|) is a non-negative super-solution of (3.5) in B := B(x0, R). Therefore, by Lemma 2.8, we conclude that Problem (3.5) has a non-negative solutionu ∈ C(B). But then this leads to a contradiction as R < R0. Therefore we must have R0 = 0, and this concludes the proof ofthe lemma.

3254 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

Remark 3.4. From here on we will assume, without further mention, that the inhomogeneous term h(x, t) in Problem (1.1)satisfies the following conditions.

(1): h : Ω × [0,∞) → [0,∞) is continuous and non-trivial.(2): h(x, 0) ≡ 0 inΩ .

Next we identify a fairly general condition on h that would ensure the existence of a non-negative solution to Problem(1.1). Toward this end, for each (x, r) ∈ Ω × (0,∞), we define h∗(t; x, r) : [0,∞) → [0,∞) by

h∗(t; x, r) := minh(z, t) : z ∈B(x, r),where we recallB(x, r) = B(x, r) ∩Ω . We make the following observations.

(i) h∗(t; x, r) > 0 for some (x, t, r) ∈ Ω × (0,∞)× (0,∞).(ii) h∗(0; x, r) = 0 for all (x, r) ∈ Ω × (0,∞).(iii) h∗(t; x, r) is continuous in t ∈ R.

To introduce some definitions, we need a few notations. For (x, r) ∈ Ω × (0,∞), we set

H∗(t; x, r) :=

t

0h∗(ζ ; x, r) dζ ∀t > 0.

In analogywith (3.1)we use the following notation. Given x ∈ Ω such that h∗(τ ; x, r) > 0 for some (τ , r) ∈ (0,∞)×(0,∞)we set

Φ∗(t; x, r) :=1

√2

∞

t

14√H∗(s; x, r)− H∗(t; x, r)

ds, ∀t > 0.

Remark 3.5. We point out that, since h∗(t; x, r) is non-increasing in r , it follows thatΦ∗(t; x, r) is non-decreasing in r . Wedirect the reader to the Appendix for a proof.

We now give the following definition.

Definition 3.6. We say that h(x, t) satisfies a KO∗ condition at x0 ∈ Ω if there is a pair (α, r) ∈ (0,∞)× (0,∞) such that

Φ∗(α; x0, r) < ∞. (3.8)

If h satisfies a KO∗ condition at x0, then in view of Remark 3.5 we note that Φ∗(α; x0, r) < ∞ for some α > 0 and allsufficiently small r > 0.

As an example, we consider h(x, t) = ϑ(x)tp(x)(1 + cos λt), where p ∈ C(Ω) is positive, ϑ ∈ C(Ω) is a non-negativeand non-trivial function, and λ is a constant. We point out that when ϑ(x) > 0 in Ω and λ = 0, in which case h(x, t) isnon-decreasing in t for each x ∈ Ω , Problem (1.1) was considered in the paper [28]. For the general case, let x0 ∈ Ω suchthat ϑ(x0) > 0, and p(x0) > 3. Then h satisfies a KO∗ condition at x0. We direct the reader to the Appendix for a detaileddiscussion of this example.

Remark 3.7. Suppose h(x, t) > 0 for all (x, t) ∈ Ω × (0,∞), and h(x, t) is non-decreasing in t for each x ∈ Ω . Therequirement that h satisfies a KO∗ condition at x0 ∈ Ω is equivalent to the condition that for some r > 0 and some α > 0,

∞

α

14√H∗(s; x0, r)

ds < ∞.

See Remark 3.1.

Theorem 3.8. Suppose O ⊂⊂ Ω , and assume h satisfies a KO∗ condition at each x ∈ ∂O. Then there is a constant C > 0 suchthat

0 ≤ ub ≤ C in O

for any minimal solution ub ∈ C(Ω) of (2.2) relative to the sub-solution u0 ≡ 0 and any boundary data b ∈ C(∂Ω, [0,∞)).

Proof. Let b ∈ C(∂Ω) such that b ≥ 0 on ∂Ω , and suppose ub is the minimal solution of (2.2) relative to u0 ≡ 0. ByLemma 3.3, given z ∈ ∂O there is a ball B(z, rz) ⊆ Ω with sufficiently small radius rz > 0 such that the following problemadmits a non-negative solution vz ∈ C(B(z, rz)).

1∞w = h∗(w; z, rz) in B(z, rz)w = ∞ on ∂B(z, rz).

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3255

Since 1∞vz = h∗(vz(x); z, rz) ≤ h(x, vz(x)) for x ∈ B(z, rz), by the Minimality Principle, Theorem 2.5 we see that ub ≤ vzon B(z, rz). In particular, ub ≤ Mz in the ball B(z, rz/2), where

Mz := maxvz(x) : x ∈ B(z, rz/2).

From the open cover U := B(z, rz/2) : z ∈ ∂O of ∂O we pick a finite subcover

B(zj, rzj/2) : j = 1, 2, . . . ,m.

Therefore

ub ≤ C on ∂O,

where C := maxMzj : j = 1, . . . ,m. Since C > 0 is a super-solution of1∞u = h(x, u) such that ub ≤ C on ∂O, again bythe Minimality Principle we see that u0 ≤ ub ≤ C in O, as claimed.

In preparation for stating our main theorem, we start with the following definition.

Definition 3.9. We say that h satisfies an annular KO∗ condition at x0 ∈ Ω if and only if there is an open set O withx0 ∈ O ⊂⊂ Ω such that h satisfies a KO∗ condition at every x ∈ ∂O.

We are now ready to state and prove our main result on existence of solutions to Problem (1.1).

Theorem 3.10. Suppose that h satisfies an annular KO∗ condition at every point of Ωδ for some δ > 0. Then Problem (1.1) admitsa non-negative solution u ∈ C(Ω).

Proof. For each positive integer k, consider the Dirichlet problem1∞u = h(x, u) inΩu = k on ∂Ω. (3.9)

For each k, note thatw ≡ 0 is a sub-solution and v ≡ k is a super-solution of (3.9). Let uk be the minimal solution to (3.9)with respect to w, so that 0 ≤ uk ≤ k for all k. By the Minimality Principle, Theorem 2.5, we see that uk ≤ uk+1 onΩ . Wewish to show that the sequence uk is locally uniformly convergent inΩ . For this it suffices to show that the sequence uk

is bounded in

Dε = x ∈ Ω : dist(x, ∂Ω) > ε

for each 0 < ε < δ. To this end, first we show that given z ∈ ∂Dε , there are positive constants rz > 0 andMz > 0 such that

0 ≤ uk ≤ Mz in B(z, rz) for all k = 1, 2, . . . .

Obviously we have ∂Dε ⊆ Ωδ . Let z ∈ ∂Dε . By hypothesis there is an open set O ⊂⊂ Ω containing z such that h satisfies aKO∗ condition at each x ∈ ∂O. But then, by Theorem 3.8 there is a constantMz, ε , independent of k, such that 0 ≤ uk ≤ Mz, εin O. In particular, there is a ball B(z, rz) ⊆ Ω such that 0 ≤ uk ≤ Mz, ε in B(z, rz) for all k, as claimed.

Therefore we have shown that given z ∈ ∂Dε there is a ball B(z, rz) ⊆ Ω and a positive constant Mz,ε , independent of k,such that

0 ≤ uk ≤ Mz,ε in B(z, rz) for all k = 1, 2, . . . .

By compactness it follows that there is a constantMε > 0, independent of k, such that

0 ≤ uk ≤ Mε, on ∂Dε for all k = 1, 2, . . . .

By the Minimality Principle, Theorem 2.5, we see that

0 ≤ uk ≤ Mε, in Dε for all k = 1, 2, . . . .

By Lemma 2.7 it follows that the sequence uk has a subsequence that converges locally uniformly to u ∈ C(Ω) such that1∞u = h(x, u) inΩ , in the viscosity sense. Since uk ≤ u inΩ , it follows that u(x) → ∞ as x → ∂Ω , as desired.

The next result shows that under appropriate conditions on the inhomogeneous term h, Problem (1.1) admits infinitelymany non-negative solutions. For this, we assume that there isϖ > 0 such that

h(x, t) ≤ h(x, t +ϖ) ∀(x, t) ∈ Ω × R. (3.10)

Proposition 3.11. Suppose h(x, j) ≡ 0 inΩ for all j ∈ J, where J is some unbounded set of positive integers, and assume thatCondition (3.10) holds for someϖ > 0. If Problem (1.1) has a non-negative solution in C(Ω), then Problem (1.1) admits infinitelymany non-negative solutions.

3256 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

Proof. Without loss of generality we assume that ϖ ≥ 1, and let us fix j1 ∈ J and x0 ∈ Ω . Suppose v ∈ C(Ω) is a non-negative blow-up solution of (1.1). Letw1 := j1 and v1 := v+ j1ϖ . We note thatw1 is a sub-solution of (2.1) and, in view of(3.10), v1 is a super-solution of (1.1) such thatw1 ≤ v1 inΩ . By Lemma 2.8, we find a solution u1 ∈ C(Ω) of (1.1) such thatw1 ≤ u1 ≤ v1 inΩ . Let j2 > u1(x0) and setw2 := j2 and v2 := v+ j2ϖ . Thenw2 is a sub-solution of (2.1) and v2 is a super-solution of (1.1) such that w2 ≤ v2. Again, by Lemma 2.8, we find a solution u2 ∈ C(Ω) of (1.1) with w2 ≤ u2 ≤ v2, andwe note that u1(x0) < u2(x0). We inductively continue in this manner to produce an infinite number of blow-up solutionsuj ∈ C(Ω) of Problem (1.1) such that 0 < uℓ(x0) < uℓ+1(x0) for ℓ = 1, 2, . . . .

Next we provide an example. Let p ∈ C(Ω) be positive, and ϑ ∈ C(Ω) be a non-negative, and non-trivial function.Suppose p(x) > 3 and ϑ(x) > 0 for all x ∈ ∂Ω . Then, given any λ ∈ R, the following problem has a non-negative solutionu ∈ C(Ω).

1∞u = ϑ(x)up(x)(1 + cos(λu(x))) inΩu = ∞ on ∂Ω. (3.11)

This follows from Theorem 3.10, as h(x, t) = ϑ(x)tp(x)(1+ cos λt) satisfies the annular KO∗ condition in a neighborhood,relative to Ω , of ∂Ω . See the Appendix for details. Moreover, Proposition 3.11 shows that Problem (3.11) admits infinitelymany non-negative solutions provided λ = 0.

4. Non-existence of boundary blow-up solutions

In this section, we study some conditions on h that would ensure the non-existence of solutions to Problem (1.1). For thiswe consider a modified form of the KO∗ condition.

For each (x, r) ∈ Ω × (0,∞), define h∗(t; x, r) : [0,∞) → [0,∞) by

h∗(t; x, r) := maxh(z, s) : (z, s) ∈B(x, r)× [0, t].

Again, it follows easily from Remark 3.4 that h∗(t; x, r) > 0 in (0,∞) for some (x, r) ∈ Ω × (0,∞), h∗(0; x, r) = 0 andh∗(t; x, r) is non-decreasing in t > 0, as well as in r > 0. One can also see that h∗(t; x, r) is continuous in t . For t > 0, weset

H∗(t; x, r) :=

t

0h∗(ζ ; x, r) dζ .

In analogy with (3.1) we let

Φ∗(t; x, r) :=1

√2

∞

t

14√H∗(s; x, r)− H∗(t; x, r)

ds.

Remark 4.1. Since h∗(t; x, r) is non-decreasing in r , it follows thatΦ∗(t; x, r) is non-increasing in r .

We now give the following definition.

Definition 4.2. We say that h(x, t) satisfies a KO∗ condition at x0 ∈ Ω if there is a pair (r, α) ∈ (0,∞)× (0,∞) such that

Φ∗(α; x0, r) < ∞. (4.1)

If h satisfies a KO∗ condition at x0, then in view of Remark 4.1 we note that Φ∗(α; x0, r) < ∞ for some α > 0 and allsufficiently big r > 0.

It is clear that Φ∗(t; x, r) ≤ Φ∗(t; x, r) for all (t, x, r) ∈ (0,∞) × Ω × (0,∞), and that equality holds if h(x, t) isindependent of x, and is non-decreasing in t . In the latter case we see that the KO∗ and the KO∗ conditions are equivalent tothe KO condition.

To state the next result, we will assume that the following holds.

lim supt→∞

h(x0, t) > 0 for some x0 ∈ Ω. (4.2)

The following holds.

Theorem 4.3. Suppose h satisfies condition (4.2) at some x0 ∈ Ω . If Problem (1.1) admits a solution u ∈ C(Ω), then h satisfiesa KO∗ condition at x0.

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3257

Proof. Let u ∈ C(Ω) be a solution of (1.1), and let r0 := diam(Ω). Suppose that h satisfies Condition (4.2) at x0 ∈ Ω . Byvirtue of (4.2), we fix a > u(x0) such that h∗(a; x0, r0) > 0. SinceΩ ⊆ B(x0, r0), we note that u is a super-solution of

1∞v = h∗(v; x0, r0), inΩv = ∞ on ∂Ω.

Let φ be a solution of(φ′(r))2φ′′(r) = h∗(φ(r); x0, r0), 0 < r < Rφ(0) = a, φ′(0) = 0 (4.3)

in (0, R) for some R > 0. Then φ ∈ C2((0, R)) ∩ C1([0, R)). We assume that (0, R) is the maximal interval of existence of φ.We claim that, in the ball B := B(x0, R), the functionw(x) := φ(|x − x0|) satisfies

1∞w ≥ h∗(w; x0, r0). (4.4)

To see this, let ϕ ∈ C2(B) such thatw − ϕ has a maximum at z ∈ B. That is

φ(|x − x0|)− φ(|z − x0|) ≤ ϕ(x)− ϕ(z), x ∈ B.

Since w ∈ C∞(B \ x0), Eq. (4.3) shows that 1∞ϕ(z) ≥ h∗(w(z); x0, r0) provided that z = x0. Therefore in what followswe assume that z = x0. Note that in this case Dϕ(x0) = Dw(x0) = φ′(0) = 0. Then as x → x0, we have

φ(|x − x0|)− φ(0) ≤ ϕ(x)− ϕ(x0) =12⟨D2ϕ(x0)(x − x0), x − x0⟩ + o(|x − x0|2). (4.5)

By continuity there is δ > 0 such that 0 < m := mina≤t≤a+δ h∗(t; x0, r0). Therefore there is δ∗ > 0 such that for 0 < r < δ∗

we conclude, from (4.3), that

√2 r =

φ(r)

a

14√H∗(t; x0, r0)− H∗(a; x0, r0)

dt

≤1

4√m

φ(r)

a(t − a)−1/4 dt

=4(φ(r)− a)3/4

3 4√m

.

Thus, for a positive constant C independent of 0 < r < δ∗, we obtain the inequality

φ(r)− a ≥ Cr4/3, 0 < r < δ∗.

Now, in (4.5), let us take x = x0 + tewhere e is a unit vector and t > 0 is sufficiently small. Then (4.5) implies that

Ct4/3 ≤ φ(t)− a ≤12t2⟨D2ϕ(x0)e, e⟩ + t2o(1) as t → 0+.

But this contradicts the fact that ϕ is twice differentiable at x0. Therefore, we conclude that there is no ϕ ∈ C2(B) such thatw − ϕ has a maximum at x0. This shows that w is indeed a solution of (4.4) in the ball B := B(x0, R). Now suppose thatR > r0. By the Comparison Principle, [28, Lemma 2.5], applied in Ω , we would have w ≤ u in Ω . But this contradicts thechoice of a = w(x0) as a > u(x0). Therefore R ≤ r0. In particular R is finite, and thus we must have φ(r) → ∞ as r → R−.Note that φ′ > 0. For 0 < r < R, computation shows that

1√2

φ(r)

a

14√H∗(t; x0, r0)− H∗(a; x0, r0)

dt = r.

In particularΦ∗(a; x0, R) < ∞. Therefore, indeed h satisfies an KO∗ condition at x0.

Let us now consider the following condition.∞

α

13√h∗(t; x, r)

dt = ∞. (4.6)

The following result holds.

Theorem 4.4. Suppose that h satisfies condition (4.6) at some (x0, r) ∈ ∂Ω × (0,∞) and that h(x0, τ ) > 0 for some τ > 0.Then Problem (1.1) has no solution in C(Ω, [0,∞)).

Proof. Since h∗(t; x0, r) is continuous and non-decreasing in t , the proof is similar to that of [28, Theorem 3.4] and isomitted.

3258 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

5. The case h(x, t) = ϑ(x)f (t)

Following Lair in [11], we make the following definition.

Definition 5.1. A function ϑ : Ω → [0,∞) is said to be CΩ-positive if and only if given x0 ∈ Ω with ϑ(x0) = 0 there isO ⊂⊂ Ω such that x0 ∈ O and ϑ(x) > 0 for x ∈ ∂O.

Let us now consider the following Dirichlet problem.1∞u = ϑ(x)f (u) inΩu = ∞ on ∂Ω. (5.1)

Suppose that f (0) = 0 and f (c) > 0 for some c > 0. A simple computation shows that if x0 ∈ Ω such that ϑ(x0) > 0, thenfor a sufficiently small r > 0,

Φ∗(t; x0, r) =1

4

minB(x0,r)ϑ

Φ(t). (5.2)

If, in addition, f is non-decreasing in (0,∞), then

Φ∗(t; x0, r) =1

4

maxB(x0,r)ϑ

Φ(t). (5.3)

In both cases (5.2) and (5.3),Φ is defined as in (3.1) but with g replaced by f .Let us mention that when f is non-decreasing the notions KO∗ and KO∗ coincide.The next result is a consequence of Theorem 3.10 and Theorem 4.3. The analogue of the following theorem, in the context

of the classical Laplacian operator, was proved in the paper [11] under the additional assumptions that f (t) > 0 for t > 0.

Theorem 5.2. Suppose ϑ ∈ C(Ω, [0,∞)) is a CΩ-positive function, and f : [0,∞) → [0,∞) a non-trivial and continuousfunction such that f (0) = 0.

(i) If f satisfies the KO condition (3.2), then Problem (5.1) admits a non-negative solution u ∈ C(Ω).(ii) Suppose f is non-decreasing. If Problem (5.1) admits a non-negative solution u in C(Ω), then f satisfies the KO condition (3.2).

Proof. (i) Suppose first that f satisfies the KO condition (3.2). We note that if ϑ(x0) > 0 at some x0 ∈ Ω , then by virtue of(5.2) we see that h satisfies a KO∗ condition at x0. Therefore, it suffices to note that the CΩ-positivity of ϑ together withthe assumption that f satisfies the KO condition (3.2) imply that h(x, t) := ϑ(x)f (t) satisfies the annular KO∗ conditionat every point ofΩδ for any δ > 0. Therefore statement (i) of the theorem follows from Theorem 3.10.

(ii) For the converse, let us assume that u ∈ C(Ω) is a non-negative solution of Problem (5.1). Note that Condition (4.2)holds at any x0 ∈ Ω for which ϑ(x0) > 0. By Theorem 4.3, h(x, t) = ϑ(x)f (t) satisfies a KO∗ condition at such pointx0 ∈ Ω . That is, Φ∗(a; x0, r) < ∞ for some a > 0 and r > 0. But then, according to (5.3), we find that Φ(a) < ∞,whereΦ is as in (3.1) with G replaced by F(t) =

t0 f (s) ds. That is f satisfies the KO condition (3.2). This concludes the

proof of the theorem.

Acknowledgments

The authors would like to express their gratitude to the referees for carefully reading the manuscript and for their usefulsuggestions that improved the paper.

Appendix

A.1. An example

As an example to illustrate some of our main results, we take the nonlinearity

h(x, t) = ϑ(x)tp(x)(1 + cos λt), (A.1)

where p ∈ C(Ω) is positive, ϑ ∈ C(Ω) is a non-negative and non-trivial function, and λ is a constant.

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3259

A.1.1. On existenceLet x0 ∈ Ω such that ϑ(x0) > 0 and p(x0) > 3. Let us set

ϑ∗(x0, r) := minϑ(x) : x ∈B(x0, r), and p∗(x0, r) := minp(x) : x ∈B(x0, r). (A.2)

For t ≥ 1 we see that

h∗(t; x0, r) ≥ ϑ∗(x0, r)tp∗(x0,r)(1 + cos λt).

We show below that h satisfies a KO∗ condition at x0.To see this let r > 0 be sufficiently small such that p∗(x0, r) > 3. Let β be a fixed positive constant such that cosβ = 1, andα := β/λ > 1. Then for t > α and sufficiently close to α, we find that

H(t; x0, r)− H(α; x0, r) =

t

α

h∗(s; x0, r) ds

≥ ϑ∗(x0, r) t

α

sp∗(x0,r)(1 + cos(λs)) ds

≥ ϑ∗(x0, r) t

α

sp∗(x0,r) ds

≥ ϑ∗(x0, r)αp∗(x0,r)(t − α). (A.3)

Therefore for a sufficiently small t1 with α < t1 we have t1

α

14√H(s; x0, r)− H(α; x0, r)

ds < ∞.

On the other hand, using t

α

sk cos(λs) ds =tk

λsin(λt)−

kλ

t

α

sk−1 sin(λs) ds,

we find that

limt→∞

1(t − α)k+1

t

α

sk(1 + cos(λs)) ds =1

k + 1.

Therefore we can find t2 > t1 sufficiently large such that t

α

sp∗(x0,r)(1 + cos(λs)) ds ≥ C(t − α)p∗(x0,r)+1 for t ≥ t2.

Therefore, the above inequality and the first inequality in (A.3) show that

H∗(s; x0, r)− H∗(α; x0, r) ≥ C(s − α)p∗(x0,r)+1, for s ≥ t2.

Consequently, since p∗(x0, r) > 3, we have∞

t2

14√H∗(s; x0, r)− H∗(α; x0, r)

ds < ∞.

Thus, in this case, we see thatΦ∗(α; x0, r) < ∞, as claimed.Therefore, if p(x) > 3 onΩδ for some δ > 0 then Theorem 3.10 shows that Problem (1.1) admits a non-negative solution

u ∈ C(Ω) with h(x, t) given as in (A.1). Now let λ = 0. Then h(x, t) satisfies Condition (3.10) with ϖ = 2π/|λ|. Sinceh(x, tj) ≡ 0 in Ω for all j, where tj := (2j + 1)π/|λ|, Proposition 3.11 shows that Problem (1.1) actually admits infinitelymany non-negative solutions in C(Ω).

A.1.2. On non-existenceLet h be as in (A.1). Then there is a positive constant C such that for t ≥ 1

h∗(t; x, r) = maxϑ(z)tp(z)(1 + cos(λs)) : z ∈B(x, r), 0 ≤ s ≤ t ≤ CtmaxB(x,r)p(z). (A.4)

Let us suppose that either of the following conditions holds.

(i) p(x) < 3 whenever ϑ(x) > 0 for all x ∈ Ω .(ii) p(x) < 3 and ϑ(x) > 0 for some x ∈ ∂Ω .

3260 A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261

We claim that Problem (1.1) has no non-negative solution u ∈ C(Ω) under either of the above conditions. To see this, let usfix x0 ∈ Ω such that p(x0) < 3 and ϑ(x0) > 0. We first observe that, by continuity, 0 < p(x) ≤ γ < 3 for all x ∈ B(x0, r)and a sufficiently small r > 0. Now let us distinguish two cases, viz., x0 ∈ Ω , and x0 ∈ ∂Ω . First we suppose that x0 ∈ Ω .Note that in this case h satisfies Condition (4.2). Furthermore, in view of (A.4) we have

H∗(s; x0, r)− H∗(t; x0, r) =

s

th∗(τ ; x0, r) dτ ≤ C(sγ+1

− tγ+1), s ≥ t ≥ 1,

for some C > 0. Thus Φ∗(t; x0, r) = ∞ for all t > 0. Therefore, Theorem 4.3 shows that Problem (1.1) has no solutionin C(Ω, [0,∞)) if ϑ(x) > 0 implies that p(x) < 3 for all x ∈ Ω . That is, (1.1) has no non-negative solution in C(Ω) ifCondition (1) above holds.

Now let us assume that x0 ∈ ∂Ω , and we recall that p(x0) > 0. Since ϑ(x0) > 0, we see that h(x0, τ ) > 0 for some τ > 0.According to (A.4) there is a sufficiently small r > 0 and positive constants C and γ < 3 such that h∗(t; x0, r) ≤ Ctγ for allt ≥ 1. Consequently, h satisfies Condition (4.6) and thus, by Theorem 4.4, we conclude that Problem (1.1) has no solution inC(Ω, [0,∞)).

A.2. Monotonicity ofΦ∗(t; x, r) andΦ∗(t; x, r) in r

We show thatΦ∗(t; x, r) is non-decreasing in r . Suppose that r1 < r2. Then form the definition of h∗, we see that

h∗(ζ ; x, r1) ≥ h∗(ζ ; x, r2) for ζ > 0.

Therefore for s ≥ t > 0 we have

H∗(s; x, r1)− H∗(t; x, r1) =

s

th∗(ζ ; x, r1) dζ

≥

s

th∗(ζ ; x, r2) dζ

≥ H∗(s; x, r2)− H∗(t; x, r2).

Consequently, we haveΦ∗(t; x, r1) ≤ Φ∗(t; x, r2).ThatΦ∗(t; x, r) is non-increasing in r can be shown in a similar manner.

References

[1] M.G. Crandall, A visit with the ∞-Laplace equation, in: Calculus of Variations and Nonlinear Partial Differential Equations, in: Lecture Notes in Math.,vol. 1927, Springer, Berlin, 2008, pp. 75–122.

[2] R.R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal. 123 (1) (1993) 51–74.[3] C. Bandle, M. Marcus, Large solutions of semilinear elliptic equations: existence uniqueness, and asymptotic behavior, J. Anal. Math. 58 (1992) 9–24.[4] C. Bandle, M. Marcus, Asymptotic behavior of solutions and their derivatives for semilinear elliptic problems with blowup on the boundary, Ann. Inst.

Henri Poincaré 12 (1995) 155–171.[5] A.C. Lazer, P.J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations 7 (3–4) (1994) 1001–1019.[6] F. Cîrstea, V. Rădulescu, Existence anduniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp.Math. 4 (3) (2002) 559–586.[7] F. Cîrstea, V. Rădulescu, Boundary blow-up in nonlinear elliptic equations of Bieberbach–Rademacher type, Trans. Amer. Math. Soc. 359 (7) (2007)

3275–3286.[8] J. García-Melián, R. Letelier, J.C. Sabina de Lis, Uniqueness and asymptotic behviour for functions of semilinear problemswith boundary blow-up, Proc.

Amer. Math. Soc. 129 (2001) 3593–3602.[9] J. García-Melián, J.D. Rossi, J.C. Sabina de Lis, Large solutions for the Laplacian with a power nonlinearity given by a variable exponent, Ann. Inst.

H. Poincar Anal. Non Linaire 26 (3) (2009) 889–902.[10] J. García-Melián, J.C. Sabina de Lis, J. Rossi, Existence, asymptotic behavior and uniqueness for large solutions to 1u = eq(x)u , Adv. Nonlinear Stud. 9

(2) (2009) 395–424.[11] A.V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl. 240 (1) (1999)

205–218.[12] M. Marcus, L. Véron, Uniqueness of solutions with blowup at the boundary of a class of nonlinear elliptic equations, C.R. Acad. Sci. Paris Sér. I Math.

317 (6) (1993) 559–563.[13] M. Marcus, L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blowup for a class of nonlinear elliptic equations, Ann. Inst.

H. Poincaré Anal. Non Lineairé 14 (2) (1997) 237–274.[14] L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math. 59 (1992) 231–250.[15] L. Véron, Large solutions of elliptic equations with strong absorption, in: Elliptic and Parabolic Problems, in: Progr. Nonlinear Differential Equations

Appl., Birkhäuser, Basel, 2005, pp. 453–464.[16] Z. Zhang, The asymptotic behaviour of solutions with boundary blow-up for semilinear elliptic equations with nonlinear gradient terms, Nonlinear

Anal. 62 (6) (2005) 1137–1148.[17] Z. Zhang, Y. Ma, L. Mi, X. Li, Blow-up rates of large solutions for elliptic equations, J. Differential Equations 249 (1) (2010) 180–199.[18] M. Ghergu, V. Radulescu, Singular Elliptic Equations. Bifurcation and Asymptotic Analysis, in: Oxford Lecture Series in Mathematics and Its

Applications, vol. 37, Oxford University Press, 2008, 320 pp.[19] A. Colesanti, P. Salani, E. Francini, Convexity and asymptotic estimates for large solutions of Hessian equations, Differential Integral Equations 13

(2000) 1459–1472.[20] G. Diaz, R. Letelier, Explosive solutions of quasilinear elliptic eqiuations: existence and uniqueness, Nonlinear Anal. 20 (1993) 97–125.[21] Y. Du, Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89 (2003) 277–302.[22] F. Gladiali, G. Porru, Estimates for explosive solutions to p-Laplace equations, in: Progress in Partial Differential Equations (Pont-á-Mousson 1997),

in: Pitman Res. Notes Math. Series, vol. 1, Longman, 1998, pp. 117–127. 383.

A. Mohammed, S. Mohammed / Nonlinear Analysis 75 (2012) 3249–3261 3261

[23] B. Guan, H.-Y. Jian, The Monge–Ampre equation with infinite boundary value, Pacific J. Math. 216 (2004) 77–94.[24] J. Matero, The Bieberbach–Rademacher problem for the Monge–Ampére operator, Manuscripta Math. 91 (1996) 379–391.[25] J. Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math. 69 (1996) 229–247.[26] P. Salani, Boundary blow-up problems for Hessian equations, Manuscripta Math. 96 (1998) 281–294.[27] P. Juutinen, J.D. Rossi, Large solutions for the infinity Laplacian, Adv. Calc. Var. 1 (3) (2008) 271–289.[28] A. Mohammed, S. Mohammed, On boundary blow-up solutions to equations involving the ∞-Laplacian, Nonlinear Anal. 74 (2011) 5238–5252.[29] S. Dumont, L. Dupaigne, O. Goubet, V. Rădulescu, Back to the KellerOsserman condition for boundary blow-up solutions, Adv. Nonlinear Stud. 7 (2)

(2007) 271–298.[30] T. Bhattacharya, A. Mohammed, Inhomogeneous Dirichlet problems involving the infinity-Laplacian, Adv. Differential Equations 17 (3–4) (2012)

225–266.[31] T. Bhattacharya, A. Mohammed, On solutions to Dirichlet problems involving the infinity-Laplacian, Adv. Calc. Var. 4 (2011) 445–487.