Ahmed Mohammed, Vicenţiu D. Rădulescu ORCID logo and Antonio Vitolo

Blow-up solutions for fully nonlinear equations: Existence, asymptotic estimates and uniqueness

Open Access
De Gruyter | Published online: July 4, 2018

Abstract

The primary objective of the paper is to study the existence, asymptotic boundary estimates and uniqueness of large solutions to fully nonlinear equations H ( x , u , D u , D 2 u ) = f ( u ) + h ( x ) in bounded C 2 domains Ω n . Here H is a fully nonlinear uniformly elliptic differential operator, f is a non-decreasing function that satisfies appropriate growth conditions at infinity, and h is a continuous function on Ω that could be unbounded either from above or from below. The results contained herein provide substantial generalizations and improvements of results known in the literature.

MSC 2010: 35J60; 35J70

1 Introduction

Let Ω n be a bounded open set with C 2 boundary Ω . We consider the infinite boundary value problem

(1.1) { H [ u ] = f ( u ) + h ( x ) in  Ω u = on  Ω ,

where H [ u ] := H ( x , u , D u , D 2 u ) is a fully nonlinear uniformly elliptic operator. For u C 2 ( Ω ) , as usual, Du stands for the gradient of u while D 2 u denotes the Hessian matrix of u.

Let 𝒮 n be the set of n × n real symmetric matrices. Throughout this paper, we fix constants 0 < λ Λ and we set 𝒜 λ , Λ := { A 𝒮 n : λ I n A Λ I n } .

To specify our assumptions on H, we first recall the so-called Pucci extremal operators 𝒫 λ , Λ - : 𝒮 n and 𝒫 λ , Λ + : 𝒮 n (see [8]) defined by

𝒫 λ , Λ + ( X ) := sup A 𝒜 λ , Λ tr ( A X ) and 𝒫 λ , Λ - ( X ) := inf A 𝒜 λ , Λ tr ( A X ) for all  X 𝒮 n .

Here tr ( X ) stands for the trace of X 𝒮 n .

Given non-negative functions γ , χ C ( Ω ) let us set

(1.2) + ( x , t , p , X ) := 𝒫 λ , Λ + ( X ) + γ ( x ) | p | + χ ( x ) t - ,
(1.3) - ( x , t , p , X ) := 𝒫 λ , Λ - ( X ) - γ ( x ) | p | - χ ( x ) t +

for ( x , t , p , X ) Ω × × n × 𝒮 n , where t ± = max ( ± t , 0 ) .

The class of functions H : Ω × × n × 𝒮 n considered in this work will include

H ( x , t , p , X ) := 𝒫 λ , Λ ± ( X ) + 𝒦 ( x , p ) - χ ( x ) t ,

where 𝒦 : Ω × n is a continuous function such that

| 𝒦 ( x , q ) - 𝒦 ( x , p ) | γ ( x ) | q - p |

for some non-negative γ , χ C ( Ω ) .

In this paper, we will consider mappings H : Ω × × n × 𝒮 n such that for all x , y Ω , s , t , p , q n and X , Y 𝒮 n the following hold:

  1. (H-1)

    H is continuous, H ( x , 0 , 0 , O ) = 0 and

    - ( x , t - s ; q - p ; Y - X ) H ( x , t , q , Y ) - H ( x , s , p , X ) + ( x , t - s ; q - p ; Y - X ) .

  2. (H-2)

    | H ( x , t , p , X ) - H ( y , t , p , X ) | K X | x - y | + ω ( ( 1 + | p | ) | x - y | ) , where K 0 is a constant and ω : + + such that ω ( 0 + ) = 0 .

Here and throughout, + stands for the set of positive real numbers.

We now turn to the nonlinearity f in (1.1). Throughout this paper we will assume that f : is a continuous function that satisfies both of the following conditions:

  1. (f-1)

    f is non-decreasing, positive in + and f ( 0 ) = 0 .

  2. (f-2)

    f satisfies the Keller–Osserman condition; namely

    1 d t F ( t ) < , where  F ( t ) = 0 t f ( s ) 𝑑 s , t 0 .

Further conditions on f, as well as on h C ( Ω ) , that will be needed in this work will be explained later.

The study of large solutions has a long history. Perhaps a systematic study of large solutions started with the works of Keller [28] and Osserman [40]. Since then a huge amount of work has emerged focusing on existence and uniqueness of large solutions. An exhaustive list on large solutions is impossible and we only list [1, 5, 11, 10, 17, 18, 22, 24, 30, 31, 32, 33, 41, 42, 47, 21] and refer the interested reader to the references therein. We wish to single out the papers of García-Melián [22], López-Gómez and Luis Maire [30], and Marcus and Véron [32] on uniqueness of large solutions of Δ u = f ( u ) on smooth bounded domains under some general conditions on f. However, in order to put the problems we wish to consider in this paper in perspective, let us recall some works that are directly related to problem (1.1) with h ( x ) 0 on Ω. In [45], Verón studied the existence and uniqueness of solutions to problem (1.1) when h C ( Ω ) is non-positive, H [ u ] = tr ( A ( x ) D 2 u ) is uniformly elliptic, and f ( t ) = | t | κ - 1 t for some κ > 1 . Likewise, in [15], Diaz and Letelier investigated larges solutions of div ( | D u | p - 2 D u ) = f ( u ) + h ( x ) , p > 1 , in bounded C 2 domains when f is a non-decreasing function that satisfies a condition of Keller–Osserman type suited for the p-Laplace operator and h C ( Ω ) is non-positive. In [2], Alarcón and Quaas study existence, asymptotic boundary behavior and uniqueness of solutions to problem (1.1). In the paper [2], the authors consider the case when H [ u ] depends on D 2 u only, f satisfies the usual Keller–Osserman condition and h C ( Ω ) is non-positive. In a related work [46], one of the authors, Amendola and Galise show that H ( D u , D 2 u ) = c | u | p - 1 u + | u | q - 1 + h ( x ) has at most one positive large solution on a bounded domain Ω n with “local graph property” introduced by Marcus and Verón [31]. Here c , 0 < p < q , h 0 on Ω and H is a uniformly elliptic operator that is “homogeneous” of degree k [ p , q ] which satisfies appropriate structural conditions. We refer to [46] for more details on the results and conditions imposed.

In a recent paper, García-Melián studied existence and uniqueness of large solutions to Δ u = | u | p - 1 u + h ( x ) in bounded C 2 domains, where h C ( Ω ) is allowed to change sign. See also [47]. To the best of our knowledge, this more challenging case of a sign-changing inhomogeneous term h is investigated for the first time in the paper [23]. In [23], the author obtains existence of large solutions to the aforementioned equation for a large class of unbounded h C ( Ω ) and uniqueness result is proven under the restriction that h is bounded on Ω from above. Motivated by the work of [23], one of the authors and Porru [37] extended the work of [23] to large solutions of L u = f ( u ) + h , where L is a linear uniformly elliptic equations in non-divergence form with possibly unbounded lower-order terms. In [37], existence and uniqueness results are obtained when the inhomogeneous term h C ( Ω ) is allowed to be unbounded from above but with some restrictions, and with bounded coefficients for the first-order and zero-order terms.

The main objective of the present paper is to extend many of the above results to solutions of problem (1.1) by relaxing the conditions used in most of the aforementioned papers. In fact, the results contained herein are new for solutions of (1.1) when h is unbounded from above, even when H [ u ] = Δ u . Another feature of the current work is that we obtain existence of solutions not only when h is unbounded on Ω, but also when the coefficients of H are unbounded on Ω.

The paper is organized as follows. In Section 2, we state the main results of the paper. These results discuss existence, asymptotic boundary estimates and uniqueness of solutions to problem (1.1). Section 3 presents some basic facts that are consequences of the assumptions made in the Introduction. We also recall several useful results from the literature that will be used in our work. The ABP maximum principle will play a recurring role in our work. In Section 4 we will develop several existence results. Depending on the rate of growth of h near the boundary, we will either relax the conditions needed on f or require more restriction.

Asymptotic boundary estimates of solutions to problem (1.1) will be developed in Section 5. In the investigation of such estimates, a condition on f introduced by Martin Dindoš in [16] will have a prominent role.

Uniqueness of solution to problem (1.1) will be investigated in Section 6. In its most general form, the uniqueness result will use a condition on h that manifests through the growth of a solution ψ of - [ ψ ] = - h + . In particular, our uniqueness result allows h L n ( Ω ) C ( Ω ) in problem (1.1).

Finally, we have included an Appendix where some useful results on existence to boundary value problems involving H with unbounded coefficients are studied. These results are used in the main body of the paper and are of independent interest.

2 Main results

In this section we state the main results of the paper. To avoid use of technicalities, we have chosen to present these results in less general terms than given in the main body of the paper.

We begin by considering the non-increasing function ϕ : + + such that

(2.1) ϕ ( t ) d s 2 F ( s ) = t , t > 0 .

Our first existence result, as well as many others, will depend on the sizes of

(2.2) Θ f * ( h ± ) := lim sup d ( x ) 0 h ± ( x ) f ( ϕ ( d ( x ) ) ) ,

where d ( x ) is the distance of x Ω to the boundary Ω of Ω. We will denote (2.2) simply as Θ * ( h ) when there is no ambiguity concerning f.

Another important feature of our work is that we allow unbounded coefficients γ , χ C ( Ω ) subject to the conditions:

  1. (C-${\gamma}$)

    lim d ( x ) 0 γ ( x ) d ( x ) = 0 ,

  2. (C-${\chi}$)

    lim d ( x ) 0 χ ( x ) d 2 ( x ) = 0 .

To state our first existence result, we recall the following condition introduced by Dindoš in [16]. There is θ > 1 such that

  1. (f-theta)

    := lim inf t f ( θ t ) θ f ( t ) > 1 .

This condition, or a strengthened form thereof, will also appear in the study of boundary asymptotic estimates as well as in our uniqueness result. We remark that (f-theta), together with (f-1) implies (f-2). See Remark 3.6.

To obtain existence of solutions to problem (1.1) with the coefficients γ and χ allowed to be unbounded on Ω according to (C-γ) and (C-χ), we need control on the rate of growth of f at infinity and the following condition provides such control:

  1. (f-3)

    α := lim inf t F ( t ) t f ( t ) > 0 .

Perhaps a word on notational use is in order here. If we wish to use any condition (f-x) on a function g : , we will simply quote it as condition (g-x).

Referring to Section 3 for condition (f-4) we now state our first existence result.

Theorem 2.1.

Let Ω R n be a smooth bounded domain. Assume that (H-1), (H-2), (C-γ), (C-χ), (f-1), (f-2), (f-4) hold. Suppose that there is g : R R that satisfies (g-1), (g-3) and (g-θ) such that f g at infinity. Then there exists a constant Θ > 0 such that problem (1.1) admits a maximal solution whenever h C ( Ω ) satisfies h ( x ) = O ( f ( ϕ ( d ( x ) ) ) ) as d ( x ) 0 with Θ * ( h + ) < Θ . Here Θ = Θ ( λ , α , θ , ) , where θ , , α are the parameters in condition (g-3) and (g-θ).

We note that when f satisfies (f-3) and (f-theta), we may take f as the function g in Theorem 2.1, and if f satisfies (f-3), the choice g ( t ) = t p f ( t ) , where p > 0 will do in Theorem 2.1.

A complementary existence result can be obtained by prescribing an indirect control on the size of h + . This control is imposed on the growth, near the boundary, of a non-negative solution ψ of a PDE related to the Pucci maximal operator as follows:

  1. (D-h)

    The equation

    (2.3) + [ ψ ] - h +

    admits a non-negative solution ψ C ( Ω ) .

We refer to Remark 4.8 for a discussion on this condition.

Based on a result of Ancona [4], see also [34], we can relax condition (C-γ), while at the same time we need to strengthen condition (C-χ) to the following conditions, respectively:

  1. (B-${{\gamma}}$)

    sup x Ω γ ( x ) d ( x ) < ,

  2. (C-${{\chi_{\eta}}}$)

    sup x Ω d 2 ( x ) η ( d ( x ) ) χ ( x ) < ,

where η : ( 0 , R ] + is a non-decreasing function for some R diam ( Ω ) and satisfies the Dini condition

0 R η ( t ) t 𝑑 t < .

It will be convenient to refer to such a function as a Dini continuous function. Assuming that

h + ( x ) = O ( d ( x ) - 2 η ( d ( x ) ) ) as  d ( x ) 0 ,

one can use Lemma 4.3 to establish the existence of a maximal solution of (1.1). Moreover, this maximal solution is positive provided that the following is sufficiently small:

(2.4) h η := sup Ω h + ( x ) d - 2 ( x ) η ( d ( x ) ) < .

This allows for instance when h + ( x ) = O ( d - 2 + δ ( x ) ) as d ( x ) 0 for some δ > 0 . We wish to emphasize here that the only conditions on f needed are (f-1) and (f-2).

Theorem 2.2.

Let Ω R n be a bounded smooth domain. Assume (H-1), (H-2), (B-γ), (C- χ η ), (f-1), (f-2). If h C ( Ω ) is such that h η < for some Dini continuous function η, then problem (1.1) admits a maximal solution. Moreover, there exists a constant c > 0 such that the solution is positive whenever h η < c .

The proof relies on the existence of a positive solution ψ to problem (2.3). This approach based on condition (D-h) also proves to be useful in dealing with uniqueness for unbounded h, at least when the coefficients γ and χ are non-negative constants. The analysis on uniqueness will be carried out through boundary asymptotic estimates of solutions of (1.1).

To obtain boundary asymptotic estimates, we need Dindoš’ condition as well as control from below on h. In fact, we need to assume h - ( x ) = O ( f ( ϕ ( d ( x ) ) ) as d ( x ) 0 , or equivalently Θ f * ( h - ) < . However, we need the coefficients γ and χ be bounded, which without loss of generality, we take to be non-negative constants.

Theorem 2.3.

Let Ω R n be a bounded smooth domain. Assume (H-1) and (H-2) with γ and χ non-negative constants, (f-1) and (f-theta) for some θ > 1 . Let h C ( Ω ) be such that Θ * ( h - ) < and h η < for some Dini continuous function η. There exist constants 0 < A * A * < such that

A * lim inf d ( x ) 0 u ( x ) ϕ ( d ( x ) ) lim sup d ( x ) 0 u ( x ) ϕ ( d ( x ) ) A *

for all solutions u of (1.1).

We need further assumptions on H and on f in order to get uniqueness. These are the sub-homogeneity property: For all ( σ , x , t , p , X ) ( 1 , ) × Ω × × n × 𝒮 n

  1. (H-3)

    H ( x , σ t , σ p , σ X ) σ H ( x , t , p , X )

and the monotonicity condition

  1. (f-m)

    f ( t ) t is non-decreasing at infinity.

The following uniqueness result holds.

Theorem 2.4.

Let Ω R n be a bounded and smooth domain and assume (H-1)(H-3) with γ , χ non-negative constants. Assume that f satisfies conditions (f-1), (f-3), (f-theta) for all θ > 1 and (f-m). Suppose also that h C ( Ω ) satisfies Θ * ( h - ) < and h + ( x ) = O ( d - 2 ( x ) η ( d ( x ) ) ) as d ( x ) 0 for some Dini continuous function η. Then problem (1.1) admits at most one solution.

We should point out that the above asymptotic estimate and uniqueness results, which are stated here with the condition h + ( x ) = O ( d - 2 ( x ) η ( d ( x ) ) ) as d ( x ) 0 for some Dini continuous function η, have been established in this paper in a more general framework through control of the growth of the functions ψ given in condition (D-h). To the best of our knowledge, this approach appears here for the first time (see also [35]). The optimal growth on h + that this method leads to remains an open problem.

3 Preliminaries

Throughout the entire paper we suppose that Ω n is a bounded open set with C 2 boundary. In this work it will be convenient to use the following notations. Given δ > 0 ,

Ω δ := { x Ω : d ( x ) < δ }

and

Ω δ := { x Ω : d ( x ) > δ } ,

where d ( x ) denotes the distance of x Ω to the boundary Ω . Since Ω is a bounded C 2 domain, we note that there is μ > 0 such that d C 2 ( Ω ¯ μ ) and | d ( x ) | = 1 on Ω μ . See [25, Lemma 14.16] for a proof. In fact, by modifying the distance function d appropriately, we can suppose that d is a positive C 2 function on Ω. For instance one can use ( 1 - φ ) d + φ instead of d, where φ C c 2 ( Ω ) is a cut-off function with 0 φ 1 on Ω, φ 0 on Ω μ 0 for some 0 < μ 0 < μ and φ 1 on Ω μ . Therefore hereafter, we will always suppose that d is this modified distance function and that d is in C 2 ( Ω ¯ ) with | D d | 1 on Ω μ 0 .

It is helpful to keep in mind the following alternative description of the Pucci extremal operators:

𝒫 λ , Λ + ( X ) = Λ tr ( X + ) - λ tr ( X - ) = Λ e i ( X ) > 0 e i ( X ) + λ e i ( X ) < 0 e i ( X ) ,
𝒫 λ , Λ - ( X ) = λ tr ( X + ) - Λ tr ( X - ) = λ e i ( X ) > 0 e i ( X ) + Λ e i ( X ) < 0 e i ( X ) ,

where X + and X - are the positive and negative parts of X, respectively, and e i ( X ) , i = 1 , , n , are the eigenvalues of X, counted according multiplicity, in non-decreasing order.

The positive homogeneity, duality, sub-additive and super-additive properties of the Pucci extremal operators (see [8]) lead to the following useful properties of ± :

(3.1) ± ( x , c ( t , p , X ) ) = c ± ( x , t , p , X ) ,
(3.2) - ( x , t , p , X ) = - + ( x , - t , - p , - X ) ,
(3.3) + ( x , t , p , X ) + - ( x , s , q , Y ) + ( x , t + s , p + q , X + Y ) + ( x , t , p , X ) + + ( x , s , q , Y ) ,
(3.4) - ( x , t , p , X ) + - ( x , s , q , Y ) - ( x , t + s , p + q , X + Y ) + ( x , t , p , X ) + - ( x , s , q , Y )

for all c 0 and ( x , t , p , X ) , ( x , s , q , Y ) Ω × × n × 𝒮 n .

Remark 3.1.

From (H-1) it follows that H is uniformly elliptic, that is,

λ tr ( Y - X ) H ( x , t , p , Y ) - H ( x , t , p , X ) Λ tr ( Y - X ) whenever  X Y .

Moreover, (H-1) implies that H is non-increasing in t:

H ( x , t , p , X ) - H ( x , s , p , X ) 0

for s t .

Given k C ( Ω × ) , a function u C 2 ( Ω ) is said to be a classical solution of equation H [ u ] = k ( x , u ) in Ω if and only if

(3.5) H ( x , u ( x ) , D u ( x ) , D 2 u ( x ) ) = k ( x , u ( x ) ) for all  x Ω .

However, in this paper we consider functions u C ( Ω ) which are solutions in the viscosity sense, according to the following definition.

Let u USC ( Ω ) (upper semicontinuous in Ω), resp. u LSC ( Ω ) (lower semicontinuous in Ω). Then u is said to be a viscosity subsolution (resp., supersolution) in Ω of (3.5) if and only if for each x Ω and φ C 2 ( Ω ) such that u - φ has a local maximum (resp. minimum) at x we have

H ( x , u ( x ) , D φ ( x ) , D 2 φ ( x ) ) k ( x , u ( x ) ) ( resp.,  H ( x , u ( x ) , D φ ( x ) , D 2 φ ( x ) ) k ( x , u ( x ) ) ) .

A function u C ( Ω ) that is both a viscosity subsolution and viscosity supersolution in Ω of (3.5) is called a viscosity solution in Ω.

Remark 3.2.

It is well known that a function u C 2 ( Ω ) is a classical subsolution (supersolution) of (3.5) if and only if u is a viscosity subsolution (supersolution) of (3.5). The forward implication follows directly from the definition. For the reverse, we refer to [8, Corollary 2.6].

We note the following consequence of condition (H-1):

(3.6) - [ u ] H [ u ] + [ u ]

for any function u C 2 ( Ω ) , where ± [ u ] := ± ( x , u , D u , D 2 u ) .

In the sequel we will make an extensive use of a fundamental tool for pointwise estimates of viscosity solutions of elliptic equations, known as the Alexandroff–Bakelman–Pucci maximum principle (see, for instance, [6, 9, 3, 43]). For the convenience of the reader we recall below the version needed here.

For this, we first remark that if k C ( Ω ) and H [ w ] k ( x ) for some w C ( Ω ) , then by (3.6) it follows that + [ w ] k ( x ) . Note also that the latter implies that w + ( x ) = max ( w ( x ) , 0 ) satisfies + [ w + ] - k - ( x ) . Therefore, setting

γ + [ w + ] := 𝒫 λ , Λ + ( D 2 w + ) + γ | D w + | ,

we also have

γ + [ w + ] - k - ( x ) .

Consequently, the standard Alexandroff–Bakelman–Pucci maximum principle (see [6, Proposition 2.12]) leads to the following.

Proposition 3.3 (ABP estimate).

Let O R n be a bounded domain with diameter R. Suppose that H satisfies condition (H-1), assuming γ = γ + L ( O ) < . For k C ( O ) L n ( O ) , let w C ( O ¯ ) be a viscosity subsolution of equation H [ w ] = k ( x ) in O . There is a non-negative constant C, depending only on n , λ , Λ , and γ R , such that

sup 𝒪 w sup 𝒪 w + + C R k - L n ( 𝒪 ) .

In particular, under the assumptions of Proposition 3.3, the following sign propagation property holds:

H [ w ] 0  in  𝒪 , w 0  on  𝒪 w 0  in  𝒪 .

One then obtains a useful comparison principle by combining Proposition 3.3 and the following result which is based on [14, Proposition 2.1]. A justification for the reformulation presented below is sketched in [33, Lemma 2.5].

Lemma 3.4.

Let O R n be a bounded domain, and let a ( t ) , b ( t ) be continuous functions on R . Suppose that H satisfies (H-1) and (H-2). If H [ u ] a ( u ) and H [ v ] b ( v ) for some u , v C ( O ¯ ) , then

+ [ u - v ] a ( u ) - b ( v ) in  𝒪 ~ := { x 𝒪 : u ( x ) > v ( x ) } .

As mentioned in the Introduction, we will assume throughout the paper that f : satisfies conditions (f-1) and (f-2). We recall some useful consequences of these assumptions.

Remark 3.5.

It is well known that if f satisfies (f-1) and (f-2), then both the following limits hold:

lim t F ( t ) f ( t ) = 0 , lim t t f ( t ) = 0 , t > 0 .

The reader is referred to [24, 26] for a proof.

The non-increasing function ϕ : + + defined in (2.1) satisfies

lim t 0 ϕ ( t ) = ,

and

ϕ ( t ) = - 2 F ( ϕ ( t ) ) , ϕ ′′ ( t ) = f ( ϕ ( t ) ) .

Here we mention some easy, but useful consequences of the Dindoš’ condition (f-theta).

Remark 3.6.

First we point out that assuming (f-1), condition (f-theta) with θ > 1 implies (f-2). More precisely, we have

lim inf t f ( t ) t q > 0

for some q > 1 . For a proof we refer to [35, Lemma 2.2].

Remark 3.7.

Note that, by iterating (f-theta), we also have, for all j ,

(3.7) lim inf t f ( θ j t ) θ j f ( t ) j

as well as

lim sup t f ( θ - j t ) θ - j f ( t ) - j .

Remark 3.8.

We remark that if f satisfies (f-1) and (f-3), then 0 α 1 2 , where α is the infimum in condition (f-3). We refer to [36, Lemma 6.1] for a proof.

We also recall the following two lemmas from [36], and [39], respectively.

Lemma 3.9.

Suppose that f satisfies (f-1), (f-2) and (f-3). Then:

(i) lim sup t F ( t ) f ( t ) t F ( s ) - 1 2 𝑑 s < ,
(ii) lim sup t t f ( t ) ( t F ( s ) - 1 2 𝑑 s ) 2 < .

Remark 3.10.

Lemma 3.9 (ii) leads to the following observation when f satisfies conditions (f-1), (f-2) and (f-3). Suppose h C ( Ω ) such that h + ( x ) = O ( d - 2 ( x ) ϕ ( d ( x ) ) ) as d ( x ) 0 . Then Θ * ( h + ) < .

The next result, a consequence of Lemma 3.9, will prove useful in establishing the existence of solutions to problem (1.1).

Corollary 3.11.

Suppose that condition (f-1), (f-2), (f-3) are satisfied. Assuming, in addition, (C-γ), we have

(3.8) lim d 0 F ( ϕ ( d ) ) f ( ϕ ( d ) ) γ ( x ) = 0 .

Assuming, in addition, (C-χ), we have

lim d 0 ϕ ( d ) f ( ϕ ( d ) ) χ ( x ) = 0 .

Proof.

To show (3.8), observe that

F ( ϕ ( d ( x ) ) ) f ( ϕ ( d ( x ) ) ) | γ ( x ) | = F ( ϕ ( d ( x ) ) ) d ( x ) f ( ϕ ( d ( x ) ) ) | γ ( x ) | d ( x ) = F ( ϕ ( d ( x ) ) ) f ( ϕ ( d ( x ) ) ) ϕ ( d ( x ) ) d s 2 F ( s ) | γ ( x ) | d ( x ) .

Therefore, in light of Lemma 3.9 (i) and condition (C-γ), recalling that ϕ ( δ ) as δ 0 , the right-hand side tends to zero as d ( x ) 0 . In a similar way, using Lemma 3.9 (ii) and condition (C-χ), we get

ϕ ( d ( x ) ) f ( ϕ ( d ( x ) ) ) χ ( x ) = ϕ ( d ( x ) ) f ( ϕ ( d ( x ) ) ) d 2 ( x ) χ ( x ) d 2 ( x ) = ϕ ( d ( x ) ) f ( ϕ ( d ( x ) ) ) ( ϕ ( d ( x ) ) d s 2 F ( s ) ) 2 χ ( x ) d 2 ( x ) 0 as  d ( x ) 0 .

The next lemma will be useful in the proof of Theorem 6.2, and hence Theorem 2.4.

Lemma 3.12.

Let f satisfy conditions (f-1) and (f-3). Then:

  1. (i)

    Given any κ > 0 , there are positive constants t κ and c κ such that f ( κ t ) c κ f ( t ) for all t > t κ .

  2. (ii)

    If, in addition, (f-theta) holds, then given ϱ > 1 , there are constants δ ϱ > 0 and c ϱ > 0 such that ϕ ( ϱ t ) c ϱ ϕ ( t ) for all 0 < t < δ ϱ .

We should point out that the constants c κ and δ κ in Lemma 3.12 (i) depend on the parameter α in condition(f-3), while the constants c ϱ and δ ϱ Lemma 3.12 (ii) also depend on θ and θ in condition (f-theta). See [35, Lemmas 2.12, 2.13 and 2.15].

The following condition which holds for any odd function f that satisfies (f-1) will be needed in one of our existence results:

  1. (f-4)

    lim t - f ( t ) = - .

4 Existence

We start this section with a result that shows the existence of supersolutions of (1.1) in balls B Ω of suitably small radii.

Lemma 4.1.

Suppose that assumptions (H-1), (f-1), (f-2) are satisfied. Suppose also h C ( Ω ) . If B Ω is a ball of sufficiently small radius, then there exists a supersolution v C ( B ) :

(4.1) { H [ v ] f ( v ) + h ( x ) in  B , v = on  B .

Proof.

We may suppose that B is centered at the origin, that is, B := { x n : | x | < R } with R < dist ( 0 , Ω ) .

Let us start with the case h 0 . We look for a solution of the form w = ϕ ( ϱ ) > 0 with ϱ ( x ) = R 2 - | x | 2 . By (H-1), we have

H [ w ] + [ w ]
𝒫 λ , Λ + ( D 2 w ) + γ ( x ) | D w |
f ( w ) [ 4 Λ | x | 2 + 2 F ( ϕ ( ϱ ) ) f ( ϕ ( ϱ ) ) ( 2 n Λ + 2 γ ( x ) | x | ) ]
f ( w ) [ 4 Λ R 2 + 2 F ( ϕ ( ϱ ) ) f ( ϕ ( ϱ ) ) ( 2 n Λ + 2 γ 0 R ) ] ,

where γ 0 = max B ¯ γ ( x ) . Taking a smaller concentric ball B, and using Remark 3.5, we take R sufficiently small so that 4 Λ R 2 < 1 2 and then

2 F ( ϕ ( ϱ ) ) f ( ϕ ( ϱ ) ) ( 2 n Λ + 2 γ 0 R ) < 1 2 .

Thus w solves problem (4.1) with h 0 .

Next, we consider an arbitrary h C ( Ω ) . Suppose h ( x ) - h 0 and γ ( x ) γ 0 in B, with h 0 , γ 0 + . Set v ( x ) := w ( x ) + C ( R 2 - | x | 2 ) , with w ( x ) as considered above and C > 0 to be suitably chosen. Then

H [ v ] + [ w ] + C + [ R 2 - | x | 2 ] f ( w ) - 2 C ( λ n - γ 0 R ) .

We now shrink R further, if necessary, to have λ n - γ 0 R λ n 2 and then we take C > 0 large enough in order that C λ n > h 0 . Since f ( w ) f ( v ) , we see that v solves (4.1). ∎

Next, we show the existence of solutions of problem (1.1) when Ω is replaced by any 𝒪 Ω . To accomplish this, we can apply, on noting that γ , χ , h C ( 𝒪 ¯ ) , the existence theorem [13, Theorem 1.1].

Theorem 4.2.

Assume (H-1), (H-2), (f-1), (f-2). If O Ω , then problem (1.1) with O instead of Ω has a solution.

Proof.

For j , let u j C ( 𝒪 ) be a solution of

(4.2) { H [ u ] = f ( u ) + h ( x ) in  𝒪 , u = j on  𝒪 .

(See [13, Theorem 1.1].) By Lemma 3.4, the difference w := u j - u j + 1 satisfies the differential inequality

+ [ w ] f ( u j ) - f ( u j + 1 ) in  𝒪 ~ := 𝒪 { u j > u j + 1 } .

Therefore, assuming that 𝒪 ~ is non-empty, we would have

+ [ w ] 0 in  𝒪 ~ .

By the maximum principle, Proposition 3.3, we have u j u j + 1 in 𝒪 ~ which is a contradiction. It follows that { u j } is a non-decreasing sequence.

Let B ( z , R ) 𝒪 be a ball of sufficiently small radius R, and let v be the corresponding solution of (4.1) provided by Lemma 4.1. A further application of the maximum principle shows that u j v in B ( z , R ) . Therefore

C 1 min B ( z , R 2 ) u 1 u j ( x ) max B ( z , R 2 ) v C 2 in  B ( z , R 2 ) .

Hence { u j } is a sequence of locally uniformly bounded solutions of (4.2), and so via the Harnack inequality also locally equi-Hölder continuous in B ( z , R 2 ) (see [7, 44]). By Ascoli–Arzelà and stability results on viscosity solutions (see [12]), and taking into account the monotonicity of the sequence { u j } , we have that

u ( x ) := lim j u j ( x )

is a continuous viscosity solution of problem (1.1). See, for instance, [20, 46]. ∎

The following lemma shows that a maximal solution can be found for problem (1.1) if a subsolution exists.

Lemma 4.3.

Assume (H-1), (H-2), (f-1), (f-2) and h C ( Ω ) . If the problem

(4.3) { H [ w ] f ( w ) + h ( x ) in  Ω , w = on  Ω ,

has a solution w, then problem (1.1) has a maximal solution u such that u w in Ω. If h + 0 , then the solution is non-negative in Ω.

Proof.

Let { Ω j } be an exhaustion of Ω by smooth domains so that Ω 1 Ω 2 Ω and j Ω j = Ω . For each j we take the solution u = u j of problem (1.1) with Ω j instead of Ω (h is bounded in Ω j ), provided by Theorem 4.2. On a fixed Ω j the solutions u k , with k > j , are bounded and by the maximum principle u k u j on Ω j . Moreover, { u k } k j is a non-increasing sequence on Ω j . Let us set

u ( x ) := lim j u j ( x ) in  Ω .

Using the subsolution w in Ω of the hypothesis and supersolutions on balls of sufficiently small radius as provided by Lemma 4.1, we can show that the sequence { u j } is uniformly bounded on each domain 𝒪 Ω . Consequently, the sequence { u j } is equi-Hölder continuous. Therefore u ( x ) is a continuous viscosity solution of equation H [ u ] = f ( u ) + h ( x ) in Ω (see the proof of Theorem 4.2). It is clear from the maximum principle that u j w in Ω j for each j. Therefore u w in Ω and

lim inf d ( x ) 0 u ( x ) lim inf d ( x ) 0 w ( x ) = .

To prove the second assertion in the lemma, suppose h + 0 in Ω. Then by conditions (H-1) and (f-1) we note that w + is also a solution of (4.3). Therefore, comparison with each u j as in the above shows that maximal solution u constructed above satisfies u w + 0 in Ω, which was to be shown. ∎

We are ready to prove our first existence theorem for problem (1.1), where an auxiliary function g satisfying Dindoš’ condition will be employed. As mentioned in the Introduction, we will refer to conditions (f-1)(f-4), (f-theta), respectively, as conditions (g-1)–(g-4), (g-θ) when we use g instead of f. Similarly, we denote by ϕ g , θ g , g and Θ g * ( h ± ) , respectively, the function ϕ, the numbers θ , > 1 in (f-theta) and Remark 3.7, and the quantities Θ g * ( h ± ) in (2.2) when we consider g instead of f.

In the statement of the theorem it will be convenient to use the following notation for any positive constant θ * > 0 :

(4.4) * ( g ; θ * ) := lim inf t g ( θ * t ) θ * g ( t ) .

We recall that for g that satisfies (g-1) and Dindoš’ condition there is θ * > 1 such that * ( g ; θ * ) > 1 .

Theorem 4.4.

Assume (H-1), (H-2), (f-1), (f-2), (f-4), (C-γ), (C-χ). Suppose there exists a function g : R R satisfying conditions (g-1), (g-3), and that there is θ * > 0 such that * := * ( g ; θ * ) > 1 and

(4.5) ρ := lim sup t f ( t ) g ( t ) < λ * .

Then there exists a positive constant Θ = Θ ( λ , ρ , θ * , * ) such that problem (1.1) admits a maximal solution provided h C ( Ω ) satisfies h + ( x ) = O ( g ( ϕ g ( d ( x ) ) ) ) as d ( x ) 0 with Θ g * ( h + ) < Θ . In fact, we may choose Θ = θ * - 1 ( λ - ρ * - 1 ) .

Remark 4.5.

If g satisfies Dindoš’ condition with θ g > 1 and g > 1 , then (4.5) is equivalent to

(4.6) ρ := lim sup t f ( t ) g ( t ) < .

In fact, suppose that (4.6) holds. In Remark 3.7 we have seen that the function g satisfies (3.7) with θ = θ g and = g for all j . We choose j large enough to have ρ < λ g j , obtaining (4.5) with * = g j . We also point out that if λ 1 and f g holds at infinity, then we can take θ * = * = 1 in the theorem.

Proof.

Let us first observe that (f-2) and (4.5) show that g satisfies the Keller–Osserman condition (g-2). For notational simplicity, let us denote ϕ g and Θ g * ( h + ) by ϕ and Θ * , respectively. According to Lemma 4.3 it is enough to show that problem (4.3) admits a solution. To this end, we search for a solution in the form

w ( x ) = θ * - 1 ϕ g ( d ( x ) ) - A

with the constant A > 0 to be suitably chosen.

Denoting by G the antiderivative of g vanishing at the origin, direct computation in Ω μ , where | D d | = 1 , yields

H [ w ] - [ w ]
𝒫 λ , Λ - ( D 2 w ) - γ ( x ) | D w | - χ ( x ) w +
θ * - 1 { 𝒫 λ , Λ - ( ϕ ′′ ( d ) D d D d ) + 𝒫 λ , Λ - ( ϕ ( d ) D 2 d ) - γ ( x ) ϕ ( d ) | D d | - χ ( x ) ϕ ( d ) }
(4.7) θ * - 1 g ( ϕ ( d ) ) { λ - 2 G ( ϕ ( d ) ) g ( ϕ ( d ) ) ( 𝒫 λ , Λ + ( D 2 d ) + γ ( x ) ) - ϕ ( d ) g ( ϕ ( d ) ) χ ( x ) } .

In (4.7), given ε > 0 , we may pass to a smaller μ = μ ( ε ) > 0 , if necessary, to ensure that

(4.8) 2 G ( ϕ ( d ) ) g ( ϕ ( d ) ) | 𝒫 λ , Λ + ( D 2 d ) | < λ ε 3 ,
(4.9) 2 G ( ϕ ( d ) ) g ( ϕ ( d ) ) γ ( x ) < λ ε 3 ,
(4.10) ϕ ( d ) g ( ϕ ( d ) ) χ ( x ) < λ ε 3 .

In (4.8) we used Remark 3.5 and the fact that d C 2 ( Ω ¯ ) . In (4.9) and (4.10) we employed Corollary 3.11.

For μ > 0 small enough, the following chain of inequalities holds in Ω μ :

H [ w ] λ θ * - 1 g ( ϕ ( d ) ) ( 1 - ε ) from (4.7)–(4.10)
= ( f ( θ * - 1 ϕ ( d ) ) + h + ( x ) ) λ ( 1 - ε ) f ( θ * - 1 ϕ ( d ) ) θ * - 1 g ( ϕ ( d ) ) + h + ( x ) θ * - 1 g ( ϕ ( d ( x ) ) )
= ( f ( θ * - 1 ϕ ( d ) ) + h + ( x ) ) λ ( 1 - ε ) f ( θ * - 1 ϕ ( d ) ) g ( θ * - 1 ϕ ( d ) ) g ( θ * - 1 ϕ ( d ) ) θ * - 1 g ( ϕ ( d ) ) + h + ( x ) θ * - 1 g ( ϕ ( d ( x ) ) )
(4.11) ( f ( θ * - 1 ϕ ( d ) ) + h + ( x ) ) λ ( 1 - ε ) ( ( ρ + ε ) * - 1 + θ * ( Θ * + ε ) ) ( 1 + ε ) .

To get (4.11), we used (2.2), (4.4), and (4.5). Now, if Θ * < θ * - 1 ( λ - ρ * - 1 ) , taking ε ( 0 , 1 ) small enough, by (4.5) we have

H [ w ] f ( θ * - 1 ϕ ( d ) ) + h ( x ) , x Ω μ .

Since f ( θ * - 1 ϕ ( d ) ) f ( θ * - 1 ϕ ( d ) - A ) = f ( w ) , it follows that w is a subsolution in Ω μ .

To finish the proof, we will choose A > 0 large enough so that w is a subsolution also in Ω μ 2 . For this, let

m = min Ω μ 2 H [ θ * - 1 ϕ ( d ( x ) ) ] , M h = max Ω μ 2 h ( x ) .

By (f-4) we can choose A such that

f ( θ * - 1 ϕ ( d ) - A ) < m - M h

so that in Ω μ 2 , once again by (H-1),

H [ w ] m f ( θ * - 1 ϕ ( d ) - A ) + M h f ( w ) + h ( x ) .

This concludes the proof. ∎

Theorem 2.1 now follows as a direct consequence of Theorem 4.4 as we show below.

Proof of Theorem 2.1.

By assumption we note that ρ 1 , where ρ is the constant in (4.5). Since g satisfies condition (g-θ), we use Remark 3.7 to choose j large enough such that λ g j > ρ . Therefore with * = g j we observe that (4.5) holds. Now we take Θ as in Theorem 4.4 to obtain the conclusion of Theorem 2.1. ∎

We also make note of the following special case.

Corollary 4.6.

Assume (H-1), (H-2), (f-1), (f-2), (f-4), (C-γ), (C-χ). If f ( t ) = o ( t p ) at infinity for some p > 1 , then for any h C ( Ω ) such that h + ( x ) = O ( d - 2 p / ( p - 1 ) ) as d ( x ) 0 , problem (1.1) admits a maximal solution. In particular, if f ( t ) = o ( t p ) at infinity for any p > 1 , and h + ( x ) = O ( d - Q ) for some constant Q > 0 as d ( x ) 0 , then problem (1.1) has a maximal solution.

Proof.

Let g ( t ) = | t | p - 1 t . Note that in this case (4.4) holds for any θ * > 0 . In particular, the hypothesis on h allows us to choose θ * > 0 small enough such that θ * - 1 λ > Θ * ( h + ) . Since in (4.5) we have ρ = 0 , and Θ g * ( h + ) < Θ = θ * - 1 λ , Theorem 4.4 shows that problem (1.1) admits a maximal solution. To prove the second assertion, it suffices to observe that given a constant Q > 0 we pick p > 1 such that 2 p p - 1 > Q . ∎

Corollary 4.6 shows, for instance, that the following problem admits a solution for any h C ( Ω ) such that d ( x ) q h + ( x ) is bounded in Ω for some q > 0 :

{ H [ u ] = u log 3 ( | u | + 1 ) + h ( x ) in  Ω , u = on  Ω .

Remark 4.7.

If f satisfies (f-3) and (4.4) such that λ * > 1 , then one can use f for g in (4.5).

An alternative existence result can be obtained by imposing indirect control on h through the solvability of (2.3). We readily note that ψ 0 in Ω satisfies (2.3) if and only if the following holds:

(4.12) γ + [ ψ ] := 𝒫 λ , Λ + ( D 2 ψ ) + γ ( x ) | D ψ | = + [ ψ ] - h + ( x ) in  Ω .

Remark 4.8.

Condition (D-h) is satisfied, for instance, if

γ + [ v ] = - h + ( x ) , x Ω ,

admits a solution v C ( Ω ) that is bounded in Ω from below. In fact, setting c = inf Ω v in this situation, then ψ := v - c is a non-negative solution of (4.12). If, in addition, γ L q ( Ω ) for some q > n and h + L p ( Ω ) for some p > p 0 , then the equation γ + [ v ] = - h + has a solution v C ( Ω ¯ ) , in which case condition (D-h) holds. Here, p 0 = p 0 ( n , λ , Λ ) ( n 2 , n ) is the Escauriaza exponent [19] (see also Crandall and Świȩch, [27]) such that for any h L p ( Ω ) with p > p 0 , solutions of 𝒫 λ , Λ + ( D 2 u ) h ( x ) satisfy the maximum principle. We refer the reader to [29, Theorem 7.1] for details.

In the present approach for existence of solutions to (1.1), we can relax condition (C-γ) by requiring the weaker condition (B-γ), while we need to strengthen condition (C-χ) to condition (C- χ η ) as described in the Introduction.

We now have the following existence result.

Theorem 4.9.

Assume (H-1), (H-2), (B-γ), (C- χ η ), (f-1), (f-2). If h C ( Ω ) and (D-h) holds with ψ bounded above, then problem (1.1) admits a maximal solution.

Proof.

Let v be a solution of

{ - [ v ] = f ( v ) in  Ω , v = on  Ω .

We direct the reader to the Appendix, Lemma A.4, for justification of the existence of such a solution.

We now use condition (D-h) to find a solution ψ 0 of (2.3) which is bounded above on Ω. Then w := v - ψ satisfies

- [ w ] - [ v ] - + [ ψ ] = f ( v ) + h + f ( w ) + h in  Ω .

Consequently,

H [ w ] f ( w ) + h in  Ω .

Since ψ ( x ) is bounded above and v ( x ) , we note that w ( x ) as x Ω . Therefore w is a large solution of (4.3). We invoke Lemma 4.3 to conclude the proof. ∎

The following gives a generalization of the existence results of Alarcón and Quaas [2].

Corollary 4.10.

Assume (H-1), (H-2), (B-γ), (C- χ η ), (f-1), (f-2). If h C ( Ω ) L p ( Ω ) for some p > p 0 , where p 0 is the Escauriaza exponent, then problem (1.1) admits a maximal solution.

Proof.

Note that by Remark 4.8, condition (D-h) holds. Therefore Theorem 4.9 shows that problem (1.1) admits a maximal solution. ∎

As pointed out in the Introduction condition (2.4) allows us to show existence of solutions to (1.1), and the problem admits a positive solution provided h η is sufficiently small. This is the content of Theorem 2.2, which we now prove.

Proof of Theorem 2.2.

Let η be the Dini continuous function as provided in the hypothesis. Then according to [4], there is a positive function ψ 1 C ( Ω ) C ( Ω ¯ ) such that

(4.13) + [ ψ 1 ] - η ( d ( x ) ) d 2 ( x ) , x Ω .

We refer to the Appendix for how this assertion follows from the work of Ancona in [4]. Therefore (2.4) together with (4.13) shows that condition (D-h) holds. Thus Theorem 4.9 shows that problem (1.1) admits a solution.

Let v 0 be a large solution of - [ v ] = f ( v ) in Ω. We refer to the Appendix for the existence of such a large solution. By the Harnack inequality[1] (see [38]) we note that actually v > 0 in Ω. Let w := v - c ψ 1 , where c > 0 is chosen such that c max Ω ψ 1 < min Ω v . It follows that w > 0 in Ω and if h η < c , then (4.13) implies that

+ [ c ψ 1 ] - η ( d ( x ) ) d 2 ( x ) c - η ( d ( x ) ) d 2 ( x ) h η - h + ( x ) , x Ω .

Consequently, we have

H [ w ] = H [ v - c ψ 1 ] - [ v - c ψ 1 ] - [ v ] - c + [ ψ 1 ] f ( v ) + h + f ( w ) + h .

We now invoke Lemma 4.3 to conclude that (1.1) has a solution u such that w u in Ω, and thus completing the proof of the theorem. ∎

5 Boundary asymptotic estimates

In this and the subsequent section, except in Theorem 5.3 and in the Appendix, we will assume that γ and χ in (1.2) and (1.3) are non-negative constants.

Boundary asymptotic estimates of solutions to (1.1) can be derived provided condition (D-h) holds with ψ ( x ) = O ( ϕ ( d ( x ) ) as d ( x ) 0 . The size of the following quantity will play a critical role in this derivation:

Ξ * ( ψ ) := lim sup d ( x ) 0 ψ ( x ) ϕ ( d ( x ) ) .

We have the following theorem on asymptotic boundary estimates of solutions to (1.1).

Theorem 5.1.

Assume (H-1), (H-2) with γ , χ non-negative constants, (f-1), (f-2).

  1. (i)

    Suppose that h C ( Ω ) satisfies Θ * ( h - ) < . If there exists a positive constant A * 1 such that

    (5.1) Θ * ( h - ) A * < lim inf t f ( A * t ) A * f ( t ) - Λ ,

    then for any continuous subsolution u of ( 1.1 ) we have

    (5.2) lim sup d ( x ) 0 u ( x ) ϕ ( d ( x ) ) A * .

  2. (ii)

    Suppose that h C ( Ω ) satisfies (D-h) . If there exists a positive constant A * 1 such that

    (5.3) lim sup t f ( A * t ) A * f ( t ) < λ 𝑎𝑛𝑑 Ξ * ( ψ ) < A * ,

    then for any supersolution u of ( 1.1 ) we have

    0 < A * := A * - Ξ * ( ψ ) lim inf d ( x ) 0 u ( x ) ϕ ( d ( x ) ) .

Proof.

In the proof that follows it will be convenient to write Θ * for Θ * ( h - ) . For any 0 < ρ < μ let us consider the following subsets of Ω:

Ω ρ - := { x Ω : ρ < d ( x ) < μ } , Ω ρ + := { x Ω : 0 < d ( x ) < μ - ρ } .

We start with the proof of (i), which will be carried out by showing that

w * ( x ) := A * ϕ ( d ( x ) - ρ ) , x Ω ρ - ,

is a supersolution of the PDE in (1.1) on Ω ρ - for all 0 < ρ < μ and sufficiently small μ > 0 . Let u C ( Ω ) be a supersolution of (1.1) which we may suppose u > 0 in Ω ρ - . Given ε > 0 , we use Remark 3.5 to obtain, for a sufficiently small μ > 0 ,

2 F ( ϕ ( d - ρ ) ) f ( ϕ ( d - ρ ) ) ( | 𝒫 λ , Λ - ( D 2 d ) | + γ ) Λ ε .

Then, recalling (3.6) and the expression of ϕ , ϕ ′′ , computation shows that

+ [ w * ] 𝒫 λ , Λ - ( D 2 w * ) + γ | D w * |
A * f ( ϕ ( d - ρ ) ) [ Λ + 2 F ( ϕ ( d - ρ ) ) f ( ϕ ( d - ρ ) ) ( | 𝒫 λ , Λ - ( D 2 d ) | + γ ) ]
(5.4) A * f ( ϕ ( d - ρ ) ) Λ ( 1 + ε ) .

On using the assumption that Θ * ( h - ) < , we obtain the following estimates. Let us note that, according to (5.1), we can take μ sufficiently small that

f ( A * ϕ ( d - ρ ) ) A * f ( ϕ ( d - ρ ) ) - h - ( x ) A * f ( ϕ ( d - ρ ) ) > 0 in  Ω μ .

By shrinking μ > 0 further, if necessary, the following hold in Ω μ :

A * f ( ϕ ( d - ρ ) ) = f ( A * ϕ ( d - ρ ) ) - h - ( x ) f ( A * ϕ ( d - ρ ) ) A * f ( ϕ ( d - ρ ) ) - h - ( x ) A * f ( ϕ ( d - ρ ) )
f ( A * ϕ ( d - ρ ) ) - h - ( x ) f ( A * ϕ ( d - ρ ) ) A * f ( ϕ ( d - ρ ) ) - h - ( x ) A * f ( ϕ ( d ( x ) ) )
(5.5) f ( A * ϕ ( d - ρ ) ) - h - ( x ) f ( A * ϕ ( d - ρ ) ) A * f ( ϕ ( d - ρ ) ) - ( 1 + ε ) Θ * + ε A * .

By (5.1), we can choose μ > 0 sufficiently small so that

(5.6) f ( A * ϕ ( d - ρ ) ) A * f ( ϕ ( d - ρ ) ) ( 1 + ε ) ( Λ + Θ * + ε A * ) .

Using (5.6) in (5.5), we find

(5.7) A * f ( ϕ ( d - ρ ) ) f ( A * ϕ ( d - ρ ) ) - h - ( x ) Λ ( 1 + ε ) .

Estimating (5.4) with (5.7), we get

(5.8) + [ w * ] f ( A * ϕ ( d - ρ ) ) - h - ( x ) f ( A * ϕ ( d - ρ ) ) + h ( x ) ,

which, on recalling (3.6), shows that w * is a supersolution in Ω ρ - of the PDE in (1.1).

Next, let u C ( Ω ) be a subsolution of (1.1) and set B * := max { u ( x ) : d ( x ) μ } . Then u w * + B * on Ω ρ - and we note that the following inequalities hold on Ω ρ - :

H [ w * + B * ] + [ w * ]
f ( w * ) + h from (5.8)
f ( w * + B * ) + h .

By the comparison principle we conclude that u w * + B * in Ω ρ - . Therefore

u ( x ) ϕ ( d ( x ) - ρ ) - B * ϕ ( d ( x ) - ρ ) A * for  x Ω ρ - .

On letting ρ 0 + , we see that the following holds on Ω μ :

u ( x ) ϕ ( d ( x ) ) - B * ϕ ( d ( x ) ) A * ,

from which we get (5.2) upon letting d ( x ) 0 .

Now we turn to the proof of (ii). For this consider the function

w * ( x ) := A * ϕ ( d ( x ) + ρ ) , x Ω ρ +

and we wish to show that w * is a subsolution of equation H [ w * ] = f ( w * ) in Ω ρ + provided μ is sufficiently small. For ε ( 0 , 1 2 ) to be chosen small enough, using Remark 3.5, we take a sufficiently small μ > 0 in order that

2 F ( ϕ ( d + ρ ) ) f ( ϕ ( d + ρ ) ) ( | 𝒫 λ , Λ + ( D 2 d ) | + γ ) + ϕ ( d + ρ ) f ( ϕ ( d + ρ ) ) λ ε .

Then, on recalling (3.6) and the expression for ϕ , ϕ ′′ , direct computation in Ω ρ + shows that

H [ w * ] - [ w * ] 𝒫 λ , Λ - ( D 2 w * ) - γ | D w * | - χ w *
A * f ( ϕ ( d + ρ ) ) [ λ - 2 F ( ϕ ( d + ρ ) ) f ( ϕ ( d + ρ ) ) ( | 𝒫 λ , Λ + ( D 2 d ) | + γ ) - ϕ ( d + ρ ) f ( ϕ ( d + ρ ) ) χ ]
(5.9) A * λ ( 1 - ε ) f ( ϕ ( d + ρ ) ) .

In the above, provided μ > 0 is small enough, by (5.3) we can make

f ( A * ϕ ( d + ρ ) ) A * f ( ϕ ( d + ρ ) ) λ ( 1 - ε ) .

Inserting this in (5.9), we get

H [ w * ] f ( A * ϕ ( d + ρ ) ) f ( w * ) ,

and this shows that w * is a subsolution in Ω ρ + as claimed.

Set B * := A * ϕ ( μ ) > 0 . By the structure conditions,

H [ w * - B * ] H [ w * ] - + [ B * ] f ( w * ) f ( w * - B * )

and w * - B * u on Ω ρ + . On the other hand, considering the function ψ ( x ) provided by condition (D-d), we also have in Ω ρ +

H [ u + ψ ] H [ u ] + + [ ψ ] f ( u ) + h ( x ) - h + ( x ) f ( u ) .

Moreover, since ψ 0 we have w * - B * u + ψ on Ω ρ + . Therefore, by the comparison principle (see Proposition 3.3 and Lemma 3.4) we find

w * - B * u + ψ in  Ω ρ + ,

and therefore we have

A * u ( x ) ϕ ( d ( x ) + ρ ) + ψ ( x ) + B * ϕ ( d ( x ) + ρ ) for  x Ω ρ + .

On letting ρ 0 + , we see that the following holds on Ω μ :

A * u ( x ) ϕ ( d ( x ) ) + ψ ( x ) + B * ϕ ( d ( x ) ) .

On recalling that ϕ ( d ( x ) ) , as d ( x ) 0 , and using condition (5.3), we get

A * lim inf d ( x ) 0 u ( x ) ϕ ( d ( x ) ) + Ξ * ( ψ ) ,

and this concludes the proof of the second part of the theorem with A * := A * - Ξ * ( ψ ) . ∎

If f satisfies Dindoš’ condition (f-theta) for some θ > 1 , then we can easily choose constants 0 < A * 1 A * such that (5.1) and (5.3) both hold. This leads to the following corollary.

Corollary 5.2.

Assume (H-1), (H-2) with γ , χ non-negative constants, (f-1), (f-theta) and assume that h C ( Ω ) satisfies Θ * ( h - ) < . Then there exist constants 0 < A * A * < such that if (D-h) holds with ψ C ( Ω ) and Ξ * ( ψ ) < A * , then

(5.10) A * lim inf d ( x ) 0 u ( x ) ϕ ( d ( x ) ) lim sup d ( x ) 0 u ( x ) ϕ ( d ( x ) ) A *

for all solutions u of (1.1).

Proof.

By hypothesis, f satisfies condition (f-theta) for some θ > 1 . We now take the smallest j , depending on , λ and Λ, such that

Θ * ( h - ) < θ j ( j - Λ ) , θ - j 1 and - j < λ .

Then, recalling Remark 3.7, we see that inequality (5.1) holds with the choice A * = θ j . If we now also require Ξ * ( ψ ) < θ - j , then both inequalities in (5.3) hold with the choice A * = θ - j . We invoke Theorem 5.1 (i) and (ii) to complete the proof. ∎

In addition to requiring condition (D-h) in the statement of Theorem 4.9, we also needed the solution ψ of (2.3) to be bounded on Ω from above. Thanks to Theorem 5.1 (i), we can now relax this restriction as we now show.

Theorem 5.3.

Assume (H-1), (H-2) with γ , χ satisfying (C-γ) and (C- χ η ), respectively, (f-1) and (f-theta). There exists a positive constant Ξ = Ξ ( θ , , λ ) such that if condition (D-h) holds with a solution ψ such that Ξ * ( ψ ) < Ξ , then problem (1.1) admits a maximal solution.

Proof.

By Lemma A.4 of the Appendix let v be a large solution of - [ v ] = f ( v ) in Ω. Consider the function w = v - ψ with ψ such that Ξ * ( ψ ) < Ξ . Here Ξ > 0 is a constant to be suitably chosen soon. As in the proof of Theorem 4.9, we can show that - [ w ] f ( w ) + h .

It remains to prove that w ( x ) as d ( x ) 0 , and then invoke Lemma 4.1 in order to complete the proof. To this end, we use (f-theta) to find j large enough (see Remark 3.7) in order that - j < λ , set A * = θ - j and Ξ = θ - j 4 . If Ξ * ( ψ ) < Ξ , then the assumption of Theorem 5.1 are satisfied and

A * = A * - Ξ * ( ψ ) A * - Ξ = 3 4 θ - j ,

so that

lim inf d ( x ) 0 v ( x ) ϕ ( d ( x ) ) A * > 0 .

As a consequence, we have

lim inf d ( x ) 0 w ( x ) ϕ ( d ( x ) ) lim inf d ( x ) 0 v ( x ) ϕ ( d ( x ) ) - lim sup d ( x ) 0 ψ ( x ) ϕ ( d ( x ) )
A * - Ξ * ( ψ ) 3 4 θ - j - Ξ = θ - j 2 > 0 ,

and w ( x ) when d ( x ) 0 , as we wanted to show. ∎

The following is an immediate consequence of Theorem 5.3.

Corollary 5.4.

Assume that H, f and γ , χ satisfy the hypotheses of Theorem 5.3. If, in addition, γ L q ( Ω ) for some q > n and h + L p ( Ω ) for some p > p 0 , then problem (1.1) has a maximal solution.

Proof.

It suffices to note, by Remark 4.8, that condition (D-h) holds with non-negative ψ C ( Ω ¯ ) , and hence Ξ * ( ψ ) = 0 . Therefore the conclusion follows from Theorem 5.3. ∎

We conclude this section with the proof of Theorem 2.3.

Proof of Theorem 2.3.

We wish to show that