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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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

Ahmed Mohammed / Vicenţiu D. RădulescuORCID iD: https://orcid.org/0000-0003-4615-5537
  • Corresponding author
  • Institute of Mathematics, Physics and Mechanics, 1000 Ljubljana, Slovenia; and Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland; and Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
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/ Antonio Vitolo
Published Online: 2018-07-04 | DOI: https://doi.org/10.1515/anona-2018-0134

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,Du,D2u)=f(u)+h(x) in bounded C2 domains Ωn. 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.

Keywords: Large solutions; existence and uniqueness; fully nonlinear elliptic equations

MSC 2010: 35J60; 35J70

1 Introduction

Let Ωn be a bounded open set with C2 boundary Ω. We consider the infinite boundary value problem

{H[u]=f(u)+h(x)in Ωu=on Ω,(1.1)

where H[u]:=H(x,u,Du,D2u) is a fully nonlinear uniformly elliptic operator. For uC2(Ω), as usual, Du stands for the gradient of u while D2u denotes the Hessian matrix of u.

Let 𝒮n be the set of n×n real symmetric matrices. Throughout this paper, we fix constants 0<λΛ and we set 𝒜λ,Λ:={A𝒮n:λInAΛIn}.

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

𝒫λ,Λ+(X):=supA𝒜λ,Λtr(AX)and𝒫λ,Λ-(X):=infA𝒜λ,Λtr(AX)for all X𝒮n.

Here tr(X) stands for the trace of X𝒮n.

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

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

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

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

H(x,t,p,X):=𝒫λ,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,χC(Ω).

In this paper, we will consider mappings H:Ω××n×𝒮n such that for all x,yΩ, s,t, p,qn and X,Y𝒮n the following hold:

  • (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).

  • (H-2)

    |H(x,t,p,X)-H(y,t,p,X)|KX|x-y|+ω((1+|p|)|x-y|), where K0 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:

  • (f-1)

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

  • (f-2)

    f satisfies the Keller–Osserman condition; namely

    1dtF(t)<,where F(t)=0tf(s)𝑑s,t0.

Further conditions on f, as well as on hC(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 Δu=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 hC(Ω) is non-positive, H[u]=tr(A(x)D2u) is uniformly elliptic, and f(t)=|t|κ-1t for some κ>1. Likewise, in [15], Diaz and Letelier investigated larges solutions of div(|Du|p-2Du)=f(u)+h(x), p>1, in bounded C2 domains when f is a non-decreasing function that satisfies a condition of Keller–Osserman type suited for the p-Laplace operator and hC(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 D2u only, f satisfies the usual Keller–Osserman condition and hC(Ω) is non-positive. In a related work [46], one of the authors, Amendola and Galise show that H(Du,D2u)=c|u|p-1u+|u|q-1+h(x) has at most one positive large solution on a bounded domain Ωn with “local graph property” introduced by Marcus and Verón [31]. Here c, 0<p<q, h0 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 Δu=|u|p-1u+h(x) in bounded C2 domains, where hC(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 hC(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 Lu=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 hC(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]=Δu. 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 hLn(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

ϕ(t)ds2F(s)=t,t>0.(2.1)

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

Θf*(h±):=lim supd(x)0h±(x)f(ϕ(d(x))),(2.2)

where d(x) is the distance of xΩ 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,χC(Ω) subject to the conditions:

  • (C-${\gamma}$)

    limd(x)0γ(x)d(x)=0,

  • (C-${\chi}$)

    limd(x)0χ(x)d2(x)=0.

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

  • (f-theta)

    :=lim inftf(θt)θf(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:

  • (f-3)

    α:=lim inftF(t)tf(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 ΩRn 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:RR that satisfies (g-1), (g-3) and (g-θ) such that fg at infinity. Then there exists a constant Θ>0 such that problem (1.1) admits a maximal solution whenever hC(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)=tpf(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:

  • (D-h)

    The equation

    +[ψ]-h+(2.3)

    admits a non-negative solution ψC(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:

  • (B-${{\gamma}}$)

    supxΩγ(x)d(x)<,

  • (C-${{\chi_{\eta}}}$)

    supxΩd2(x)η(d(x))χ(x)<,

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

0Rη(t)t𝑑t<.

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

h+(x)=O(d(x)-2η(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:

hη:=supΩh+(x)d-2(x)η(d(x))<.(2.4)

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 ΩRn be a bounded smooth domain. Assume (H-1), (H-2), (B-γ), (C-χη), (f-1), (f-2). If hC(Ω) is such that hη< 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 hη<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 Θf*(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 ΩRn 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 hC(Ω) be such that Θ*(h-)< and hη< for some Dini continuous function η. There exist constants 0<A*A*< such that

A*lim infd(x)0u(x)ϕ(d(x))lim supd(x)0u(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×𝒮n

  • (H-3)

    H(x,σt,σp,σX)σH(x,t,p,X)

and the monotonicity condition

  • (f-m)

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

The following uniqueness result holds.

Theorem 2.4.

Let ΩRn 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 hC(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 Ωn is a bounded open set with C2 boundary. In this work it will be convenient to use the following notations. Given δ>0,

Ωδ:={xΩ:d(x)<δ}

and

Ωδ:={xΩ:d(x)>δ},

where d(x) denotes the distance of xΩ to the boundary Ω. Since Ω is a bounded C2 domain, we note that there is μ>0 such that dC2(Ω¯μ) and |d(x)|=1 on Ωμ. 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 C2 function on Ω. For instance one can use (1-φ)d+φ instead of d, where φCc2(Ω) is a cut-off function with 0φ1 on Ω, φ0 on Ωμ0 for some 0<μ0<μ and φ1 on Ωμ. Therefore hereafter, we will always suppose that d is this modified distance function and that d is in C2(Ω¯) with |Dd|1 on Ωμ0.

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

𝒫λ,Λ+(X)=Λtr(X+)-λtr(X-)=Λei(X)>0ei(X)+λei(X)<0ei(X),𝒫λ,Λ-(X)=λtr(X+)-Λtr(X-)=λei(X)>0ei(X)+Λei(X)<0ei(X),

where X+ and X- are the positive and negative parts of X, respectively, and ei(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 ±:

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

for all c0 and (x,t,p,X),(x,s,q,Y)Ω××n×𝒮n.

Remark 3.1.

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

λtr(Y-X)H(x,t,p,Y)-H(x,t,p,X)Λtr(Y-X)whenever XY.

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

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

for st.

Given kC(Ω×), a function uC2(Ω) is said to be a classical solution of equation H[u]=k(x,u) in Ω if and only if

H(x,u(x),Du(x),D2u(x))=k(x,u(x))for all xΩ.(3.5)

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

Let uUSC(Ω) (upper semicontinuous in Ω), resp. uLSC(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 xΩ and φC2(Ω) such that u-φ has a local maximum (resp. minimum) at x we have

H(x,u(x),Dφ(x),D2φ(x))k(x,u(x))(resp., H(x,u(x),Dφ(x),D2φ(x))k(x,u(x))).

A function uC(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 uC2(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):

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

for any function uC2(Ω), where ±[u]:=±(x,u,Du,D2u).

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 kC(Ω) and H[w]k(x) for some wC(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+]:=𝒫λ,Λ+(D2w+)+γ|Dw+|,

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 ORn be a bounded domain with diameter R. Suppose that H satisfies condition (H-1), assuming γ=γ+L(O)<. For kC(O)Ln(O), let wC(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 γR, such that

sup𝒪wsup𝒪w++CRk-Ln(𝒪).

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

H[w]0 in 𝒪,w0 on 𝒪w0 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 ORn 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,vC(O¯), then

+[u-v]a(u)-b(v)in 𝒪~:={x𝒪: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:

limtF(t)f(t)=0,limttf(t)=0,t>0.

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

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

limt0ϕ(t)=,

and

ϕ(t)=-2F(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 inftf(t)tq>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,

lim inftf(θjt)θjf(t)j(3.7)

as well as

lim suptf(θ-jt)θ-jf(t)-j.

Remark 3.8.

We remark that if f satisfies (f-1) and (f-3), then 0α12, 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:

lim suptF(t)f(t)tF(s)-12𝑑s<,(i)lim supttf(t)(tF(s)-12𝑑s)2<.(ii)

Remark 3.10.

Lemma 3.9 (ii) leads to the following observation when f satisfies conditions (f-1), (f-2) and (f-3). Suppose hC(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

limd0F(ϕ(d))f(ϕ(d))γ(x)=0.(3.8)

Assuming, in addition, (C-χ), we have

limd0ϕ(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))ds2F(s)|γ(x)|d(x).

Therefore, in light of Lemma 3.9 (i) and condition (C-γ), recalling that ϕ(δ) as δ0, 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)))d2(x)χ(x)d2(x)=ϕ(d(x))f(ϕ(d(x)))(ϕ(d(x))ds2F(s))2χ(x)d2(x)0as 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:

  • (i)

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

  • (ii)

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

We should point out that the constants cκ and δκ in Lemma 3.12 (i) depend on the parameter α in condition(f-3), while the constants cϱ and δϱ 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:

  • (f-4)

    limt-f(t)=-.

4 Existence

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

Lemma 4.1.

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

{H[v]f(v)+h(x)in B,v=on B.(4.1)

Proof.

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

Let us start with the case h0. We look for a solution of the form w=ϕ(ϱ)>0 with ϱ(x)=R2-|x|2. By (H-1), we have

H[w]+[w]𝒫λ,Λ+(D2w)+γ(x)|Dw|f(w)[4Λ|x|2+2F(ϕ(ϱ))f(ϕ(ϱ))(2nΛ+2γ(x)|x|)]f(w)[4ΛR2+2F(ϕ(ϱ))f(ϕ(ϱ))(2nΛ+2γ0R)],

where γ0=maxB¯γ(x). Taking a smaller concentric ball B, and using Remark 3.5, we take R sufficiently small so that 4ΛR2<12 and then

2F(ϕ(ϱ))f(ϕ(ϱ))(2nΛ+2γ0R)<12.

Thus w solves problem (4.1) with h0.

Next, we consider an arbitrary hC(Ω). Suppose h(x)-h0 and γ(x)γ0 in B, with h0,γ0+. Set v(x):=w(x)+C(R2-|x|2), with w(x) as considered above and C>0 to be suitably chosen. Then

H[v]+[w]+C+[R2-|x|2]f(w)-2C(λn-γ0R).

We now shrink R further, if necessary, to have λn-γ0Rλn2 and then we take C>0 large enough in order that Cλn>h0. 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 𝒪Ω. To accomplish this, we can apply, on noting that γ,χ,hC(𝒪¯), the existence theorem [13, Theorem 1.1].

Theorem 4.2.

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

Proof.

For j, let ujC(𝒪) be a solution of

{H[u]=f(u)+h(x)in 𝒪,u=jon 𝒪.(4.2)

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

+[w]f(uj)-f(uj+1)in 𝒪~:=𝒪{uj>uj+1}.

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

+[w]0in 𝒪~.

By the maximum principle, Proposition 3.3, we have ujuj+1 in 𝒪~ which is a contradiction. It follows that {uj} 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 ujv in B(z,R). Therefore

C1minB(z,R2)u1uj(x)maxB(z,R2)vC2in B(z,R2).

Hence {uj} 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,R2) (see [7, 44]). By Ascoli–Arzelà and stability results on viscosity solutions (see [12]), and taking into account the monotonicity of the sequence {uj}, we have that

u(x):=limjuj(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 hC(Ω). If the problem

{H[w]f(w)+h(x)in Ω,w=on Ω,(4.3)

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

Proof.

Let {Ωj} be an exhaustion of Ω by smooth domains so that Ω1Ω2Ω and jΩj=Ω. For each j we take the solution u=uj of problem (1.1) with Ωj instead of Ω (h is bounded in Ωj), provided by Theorem 4.2. On a fixed Ωj the solutions uk, with k>j, are bounded and by the maximum principle ukuj on Ωj. Moreover, {uk}kj is a non-increasing sequence on Ωj. Let us set

u(x):=limjuj(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 {uj} is uniformly bounded on each domain 𝒪Ω. Consequently, the sequence {uj} 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 ujw in Ωj for each j. Therefore uw in Ω and

lim infd(x)0u(x)lim infd(x)0w(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 uj as in the above shows that maximal solution u constructed above satisfies uw+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 ϕg, θg, g and Θg*(h±), respectively, the function ϕ, the numbers θ,>1 in (f-theta) and Remark 3.7, and the quantities Θg*(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:

*(g;θ*):=lim inftg(θ*t)θ*g(t).(4.4)

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:RR satisfying conditions (g-1), (g-3), and that there is θ*>0 such that *:=*(g;θ*)>1 and

ρ:=lim suptf(t)g(t)<λ*.(4.5)

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

Remark 4.5.

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

ρ:=lim suptf(t)g(t)<.(4.6)

In fact, suppose that (4.6) holds. In Remark 3.7 we have seen that the function g satisfies (3.7) with θ=θg and =g for all j. We choose j large enough to have ρ<λgj, obtaining (4.5) with *=gj. We also point out that if λ1 and fg 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 ϕg and Θg*(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*-1ϕg(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 Ωμ, where |Dd|=1, yields

H[w]-[w]𝒫λ,Λ-(D2w)-γ(x)|Dw|-χ(x)w+θ*-1{𝒫λ,Λ-(ϕ′′(d)DdDd)+𝒫λ,Λ-(ϕ(d)D2d)-γ(x)ϕ(d)|Dd|-χ(x)ϕ(d)}θ*-1g(ϕ(d)){λ-2G(ϕ(d))g(ϕ(d))(𝒫λ,Λ+(D2d)+γ(x))-ϕ(d)g(ϕ(d))χ(x)}.(4.7)

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

2G(ϕ(d))g(ϕ(d))|𝒫λ,Λ+(D2d)|<λε3,(4.8)2G(ϕ(d))g(ϕ(d))γ(x)<λε3,(4.9)ϕ(d)g(ϕ(d))χ(x)<λε3.(4.10)

In (4.8) we used Remark 3.5 and the fact that dC2(Ω¯). In (4.9) and (4.10) we employed Corollary 3.11.

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

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

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*-1ϕ(d))+h(x),xΩμ.

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

To finish the proof, we will choose A>0 large enough so that w is a subsolution also in Ωμ2. For this, let

m=minΩμ2H[θ*-1ϕ(d(x))],Mh=maxΩμ2h(x).

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

f(θ*-1ϕ(d)-A)<m-Mh

so that in Ωμ2, once again by (H-1),

H[w]mf(θ*-1ϕ(d)-A)+Mhf(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 ρ1, 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 λgj>ρ. Therefore with *=gj 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(tp) at infinity for some p>1, then for any hC(Ω) such that h+(x)=O(d-2p/(p-1)) as d(x)0, problem (1.1) admits a maximal solution. In particular, if f(t)=o(tp) 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-1t. 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*-1λ>Θ*(h+). Since in (4.5) we have ρ=0, and Θg*(h+)<Θ=θ*-1λ, 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 2pp-1>Q. ∎

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

{H[u]=ulog3(|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 ψ0 in Ω satisfies (2.3) if and only if the following holds:

γ+[ψ]:=𝒫λ,Λ+(D2ψ)+γ(x)|Dψ|=+[ψ]-h+(x)in Ω.(4.12)

Remark 4.8.

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

γ+[v]=-h+(x),xΩ,

admits a solution vC(Ω) that is bounded in Ω from below. In fact, setting c=infΩv in this situation, then ψ:=v-c is a non-negative solution of (4.12). If, in addition, γLq(Ω) for some q>n and h+Lp(Ω) for some p>p0, then the equation γ+[v]=-h+ has a solution vC(Ω¯), in which case condition (D-h) holds. Here, p0=p0(n,λ,Λ)(n2,n) is the Escauriaza exponent [19] (see also Crandall and Świȩch, [27]) such that for any hLp(Ω) with p>p0, solutions of 𝒫λ,Λ+(D2u)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-χη) as described in the Introduction.

We now have the following existence result.

Theorem 4.9.

Assume (H-1), (H-2), (B-γ), (C-χη), (f-1), (f-2). If hC(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 ψ0 of (2.3) which is bounded above on Ω. Then w:=v-ψ satisfies

-[w]-[v]-+[ψ]=f(v)+h+f(w)+hin Ω.

Consequently,

H[w]f(w)+hin Ω.

Since ψ(x) is bounded above and v(x), we note that w(x) as xΩ. 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-χη), (f-1), (f-2). If hC(Ω)Lp(Ω) for some p>p0, where p0 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 hη 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 ψ1C(Ω)C(Ω¯) such that

+[ψ1]-η(d(x))d2(x),xΩ.(4.13)

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 v0 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 inequality1 (see [38]) we note that actually v>0 in Ω. Let w:=v-cψ1, where c>0 is chosen such that cmaxΩψ1<minΩv. It follows that w>0 in Ω and if hη<c, then (4.13) implies that

+[cψ1]-η(d(x))d2(x)c-η(d(x))d2(x)hη-h+(x),xΩ.

Consequently, we have

H[w]=H[v-cψ1]-[v-cψ1]-[v]-c+[ψ1]f(v)+h+f(w)+h.

We now invoke Lemma 4.3 to conclude that (1.1) has a solution u such that wu 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 supd(x)0ψ(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).

  • (i)

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

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

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

    lim supd(x)0u(x)ϕ(d(x))A*.(5.2)

  • (ii)

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

    lim suptf(A*t)A*f(t)<λ𝑎𝑛𝑑Ξ*(ψ)<A*,(5.3)

    then for any supersolution u of ( 1.1 ) we have

    0<A*:=A*-Ξ*(ψ)lim infd(x)0u(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:

Ωρ-:={xΩ:ρ<d(x)<μ},Ωρ+:={xΩ: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),xΩρ-,

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

2F(ϕ(d-ρ))f(ϕ(d-ρ))(|𝒫λ,Λ-(D2d)|+γ)Λε.

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

+[w*]𝒫λ,Λ-(D2w*)+γ|Dw*|A*f(ϕ(d-ρ))[Λ+2F(ϕ(d-ρ))f(ϕ(d-ρ))(|𝒫λ,Λ-(D2d)|+γ)]A*f(ϕ(d-ρ))Λ(1+ε).(5.4)

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))>0in Ωμ.

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

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

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

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

Using (5.6) in (5.5), we find

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

Estimating (5.4) with (5.7), we get

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

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

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

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

By the comparison principle we conclude that uw*+B* in Ωρ-. Therefore

u(x)ϕ(d(x)-ρ)-B*ϕ(d(x)-ρ)A*for xΩρ-.

On letting ρ0+, we see that the following holds on Ωμ:

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),xΩρ+

and we wish to show that w* is a subsolution of equation H[w*]=f(w*) in Ωρ+ provided μ is sufficiently small. For ε(0,12) to be chosen small enough, using Remark 3.5, we take a sufficiently small μ>0 in order that

2F(ϕ(d+ρ))f(ϕ(d+ρ))(|𝒫λ,Λ+(D2d)|+γ)+ϕ(d+ρ)f(ϕ(d+ρ))λε.

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

H[w*]-[w*]𝒫λ,Λ-(D2w*)-γ|Dw*|-χw*A*f(ϕ(d+ρ))[λ-2F(ϕ(d+ρ))f(ϕ(d+ρ))(|𝒫λ,Λ+(D2d)|+γ)-ϕ(d+ρ)f(ϕ(d+ρ))χ]A*λ(1-ε)f(ϕ(d+ρ)).(5.9)

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 Ωρ+ 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 Ωρ+. On the other hand, considering the function ψ(x) provided by condition (D-d), we also have in Ωρ+

H[u+ψ]H[u]++[ψ]f(u)+h(x)-h+(x)f(u).

Moreover, since ψ0 we have w*-B*u+ψ on Ωρ+. Therefore, by the comparison principle (see Proposition 3.3 and Lemma 3.4) we find

w*-B*u+ψin Ωρ+,

and therefore we have

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

On letting ρ0+, we see that the following holds on Ωμ:

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 infd(x)0u(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*1A* 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 hC(Ω) satisfies Θ*(h-)<. Then there exist constants 0<A*A*< such that if (D-h) holds with ψC(Ω) and Ξ*(ψ)<A*, then

A*lim infd(x)0u(x)ϕ(d(x))lim supd(x)0u(x)ϕ(d(x))A*(5.10)

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-)<θj(j-Λ),θ-j1and-j<λ.

Then, recalling Remark 3.7, we see that inequality (5.1) holds with the choice A*=θj. 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-χη), 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-j4. If Ξ*(ψ)<Ξ, then the assumption of Theorem 5.1 are satisfied and

A*=A*-Ξ*(ψ)A*-Ξ=34θ-j,

so that

lim infd(x)0v(x)ϕ(d(x))A*>0.

As a consequence, we have

lim infd(x)0w(x)ϕ(d(x))lim infd(x)0v(x)ϕ(d(x))-lim supd(x)0ψ(x)ϕ(d(x))A*-Ξ*(ψ)34θ-j-Ξ=θ-j2>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, γLq(Ω) for some q>n and h+Lp(Ω) for some p>p0, 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 ψC(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 Ξ*(ψ), where ψ is as in condition (D-h), is sufficiently small as required for the conclusion of Theorem 5.3 to hold. Since all other assumptions of Theorem 5.3 hold, this would establish the desired result. Nevertheless, the assumption concerning the smallness of Ξ*(ψ) is satisfied since for some Dini continuous function η we have h+(x)=O(η(d(x))d-2(x)) as d(x)0. Due to the existence result of Ancona already used in Theorem 2.2 and discussed in (A.2) of the Appendix, there is a non-negative ψC2(Ω)C(Ω¯) such that

+(ψ)-η(d(x))d2(x).

Therefore, on noting that Ξ*(ψ)=0, the proof is complete. ∎

6 Uniqueness

In this section we discuss uniqueness of solutions of (1.1) under some additional conditions on H and f. The asymptotic boundary behavior of solutions results of Section 5 will be crucial in developing the uniqueness result. We will assume H satisfies the sub-homogeneity condition (H-3) and we will also suppose that f satisfies the Dindoš’ condition (f-theta) for all θ>1.

Remark 6.1.

Suppose that (f-theta) holds for all θ>1. Then θθ is non-decreasing on (1,). In particular, given θ¯>1 and 1<τ<θ¯, there is t¯:=t(θ¯,τ) such that

f(θt)τf(t)for all (θ,t)(θ¯,)×(t¯,).

The next result shows that any two solutions of (1.1) have the same rate of growth near the boundary. This result is based on condition (D-h) and will be proved under the technical assumption that the limit supremum Ξ*(ψ) is actually a limit which will be denoted by Ξ(ψ):

Ξ*(ψ):=limd(x)0ψ(x)ϕ(d(x))Ξ(ψ).

Theorem 6.2.

Let (H-1)(H-3) hold with γ,χ non-negative constants. Assume that f satisfies conditions (f-1), (f-3) and (f-theta) for all θ>1. Suppose that hC(Ω) is such that Θ*(h-)< and that (D-h) holds for some ψC(Ω). There is a constant Ξ>0 such that if Ξ*(ψ)=Ξ(ψ)<Ξ, then we have

limd(x)0u(x)v(x)=1(6.1)

for any solutions u and v of (1.1).

Proof.

Let u and v be two solutions of (1.1) in Ω, and let Ξ>0 be the constant in Corollary 5.2. If Ξ(ψ)<Ξ, then the estimates (5.10) lead to

θ:=lim supd(x)0u(x)v(x)<.

To prove the theorem, we show that θ1, for then reversing the roles of u and v we will have

1lim infd(x)0v(x)u(x)lim supd(x)0v(x)u(x)=θ1.

By way of contradiction, let us suppose θ>1. Let Ψ(x):=ψ(x)-Ξ(ψ)ϕ(d(x)). Note that

θ:=lim supd(x)0u(x)v(x)+Ψ(x).

We fix ε0>0 small enough such that θ-ε0>1. Let w:=v+Ψ. Given ε(0,ε0), there is δ:=δ(ε)(0,μ) such that

{u(x)w(x)θ+εif d(x)δ,u(xε)w(xε)>θ-εfor some xε with d(xε)<23δ.(6.2)

According to Corollary 5.2, we observe that

lim infd(x)0w(x)ϕ(d(x))A*

for some positive constant A*. Therefore w(x) as d(x)0. For the remainder of the proof we will suppose μ>0 is sufficiently small such that all of the following hold for xΩμ:

  • (i)

    Since (f-theta) holds for all θ>1 and recalling Remark 6.1 for ε(0,ε0), we have

    f((θ-ε)v(x))-(θ-ε)f(v(x))mf(v(x))>0,(6.3)

    with m:=θ-ε0-1>0 (set θ¯=θ-ε0 and apply Remark 6.1 with τ:=12(1+θ¯)).

  • (ii)

    Use Corollary 5.2 to obtain

    12A*ϕ(d(x))w(x),v(x)2A*ϕ(d(x)).(6.4)

  • (iii)

    By Lemma 3.12, (ii) there is a positive constant cϕ such that

    ϕ(3r)cϕϕ(r)for 0<r<μ.(6.5)

Let 0<ε<ε0 be fixed, but arbitrary and let δ=δ(ε) be the corresponding positive number such that (6.2) holds.

Let us now consider

𝒪:={xΩ:u(x)>(θ-ε)w(x)}B(xε,r),where r:=12d(xε)<13δ,

so that

rd(x)3rfor all x𝒪Ωδ.(6.6)

Let us note that, for sufficiently small μ>0, see (4.7),

-[ϕ(d(x))]>0,xΩμ.

By (H-1) and (H-3) we have the following on Ωμ:

H[(θ-ε)w](θ-ε)H[v]+(θ-ε)+[Ψ](θ-ε)H[v]+(θ-ε)+[ψ]-(θ-ε)Ξ(ψ)-[ϕ(d(x))]=(θ-ε)f(v)-(θ-ε)h-.(6.7)

By appealing to Lemma 3.12 (i) and Lemma 3.9, respectively, and by shrinking μ if necessary, we can suppose that both of the following hold for 0<r<μ, and some positive constant cf, not necessarily the same:

f(12A*cϕϕ(r))cff(ϕ(r))(6.8)

and

f(ϕ(r))=12r2(ϕ(r)dsF(s))2f(ϕ(r))cfϕ(r)2r2.(6.9)

From the monotonicity of the functions f, and ϕ (non-increasing), we find that for all x𝒪Ωμ,

f(v(x))f(12A*ϕ(d(x)))(by (6.4))f(12A*ϕ(3r))f(12A*cϕϕ(r))(by (6.5))cff(ϕ(r))(by (6.8)).(6.10)

On noting (6.7) and the fact that H[u]f(u)+h, we invoke Lemma 3.4 to find that the following hold on 𝒪:

γ+[u-(θ-ε)w]=+[u-(θ-ε)w](f(u)-(θ-ε)f(v))+h++(θ-ε-1)h-f((θ-ε)v)-(θ-ε)f(v)mf(v)(by (6.3)).(6.11)

Using (6.10) in (6.11) and recalling (6.6), we find that

γ+[u-(θ-ε)w]mcff(ϕ(r)),x𝒪.(6.12)

We now observe that for all yB¯(xε,r) we have

f(ϕ(r))cf2ϕ(r)r2(by (6.9))cf2ϕ(d(y))r2(by (6.6))cf4A*w(y)r2(by (6.4)).(6.13)

Therefore, from (6.12) and (6.13) we conclude

γ+[u-(θ-ε)w]Cr2w(y)on 𝒪 and for all yB¯(xε,r).(6.14)

In (6.14) we have set C:=14A*mcf2 which, we should note, is independent of ε.

For arbitrary, but fixed yB¯(xε,r) we consider the following auxiliary function:

z(x):=aCw(y)(1-|x-xε|2r2).

Since z(x) is concave and smooth, choosing 0<a(2(Λn+bμ))-1 we have

γ-[z]=-ΛΔz-b|Dz|-ΛΔzL(Ω¯)-bDzL(Ω¯)-2aC[Λn+bμ]w(y)r2-Cr2w(y).(6.15)

From (6.14), (6.15) and (3.3) the following inequality holds on 𝒪:

γ+[u-(θ-ε)w+z]Mγ+[u-(θ-ε)w]+Mγ-[z]0for all yB¯(xε,r).

By the Alexandroff–Bakelman–Pucci maximum principle (see Proposition 3.3), we find that there is yε𝒪 such that

u(xε)-(θ-ε)w(xε)+z(xε)[u(yε)-(θ-ε)w(yε)+z(yε)]+.(6.16)

We infer that yεB(xε,r). In fact, supposing the contrary yεB(xε,r), then u(yε)=(θ-ε)w(yε), and hence we would have a contradiction:

z(xε)<u(xε)-(θ-ε)w(xε)+z(xε)z(yε).

Therefore indeed yεB(xε,r). Consequently, from (6.16) we obtain

aCw(y)=z(xε)u(yε)-(θ-ε)w(yε)for all yB¯(xε,r).(6.17)

Therefore, since d(yε)<δ, we use (6.2) to estimate (6.17) as

aCw(y)(θ+ε)w(yε)-(θ-ε)w(yε)=2εw(yε).(6.18)

Setting y=yε in (6.18) and rearranging, we find

aCw(yε)2εw(yε).

Taking, ε=min{12ε0,14aC} leads to a contradiction, thus completing the proof of the theorem. ∎

Our next result is a uniqueness theorem, and this requires the monotonicity condition (f-m) on f stated in the Introduction.

Theorem 6.3.

Suppose that the assumptions of Theorem 6.2 on H and f are satisfied and that (f-m) holds as well. Suppose also that hC(Ω) is such that Θ*(h-)< and (D-h) holds with ψ. There is a constant Ξ>0 such that if Ξ*(ψ)=Ξ(ψ)<Ξ, then problem (1.1) admits at most one solution.

Proof.

Let ψC(Ω) be as in (D-h). We suppose that Ξ(ψ)<Ξ, where Ξ is the positive constant in Theorem 6.2. Note that Theorem 6.2 applies. Let u be a positive solution of (1.1). According to Theorem 5.3, and by taking Ξ smaller if needed, problem (1.1) admits a maximal solution v and therefore uv in Ω. For the purpose of obtaining a contradiction, we suppose that u(x0)<v(x0) for some x0Ω. Let ε>0 be arbitrary. Consider wε:=(1+ε)u+εψ. Then the set

𝒪ε:={xΩ:wε(x)<v(x)}

contains x0 for all sufficiently small ε>0. Given zΩ, it is clear that there is ε such that z𝒪ε. If this is not the case, then (1+ε)u(z)+εψ(z)v(z) for all sufficiently small ε>0. But then this implies u(z)=v(z). On recalling that uv in Ω and hence by Lemma 3.4, we have γ+[v-u]f(v)-f(u)0, the Strong Maximum Principle implies that u=v in Ω, contradicting u(x0)<v(x0). As a consequence of (6.1) we see that 𝒪εΩ for each ε>0.

By (f-m), we let t0>0 such that f(t)t is non-decreasing for tt0. Let 0<ϱ<μ such that u(x)t0 for all xΩϱ. We note that there is ε0>0 such that

𝒪ε,ϱ:=𝒪εΩϱ,0<ε<ε0,

is non-empty. Otherwise there is a sequence {εj} that converges to zero and

𝒪εjΩϱ=.

Then (1+εj)u(x)+εjψv(x) on Ωϱ, for all j. This implies uv on Ωϱ, and hence u=v on Ωϱ. But then γ+[v-u]f(v)-f(u)0 in Ω, and v=u on Ωϱ implies, by the Alexandroff–Bakelman–Pucci maximum principle, that uv on Ωϱ and therefore u=v in Ω, which contradicts the assumption that u(x0)<v(x0).

By (H-1) we see that

H[wε]H[(1+ε)u]+ε+[ψ].(6.19)

Therefore, by (H-3) and (6.19) we have the following in 𝒪ε,ϱ:

H[wε](1+ε)H[u]+ε+[ψ](1+ε)f(u)+(1+ε)h-εh+f((1+ε)u)+h-εh-f(wε)+h(by (f-1)).

On 𝒪ε,ϱ, 0<ε<ε0, we see that

γ+[v-wε]f(v)-f(wε)(by Lemma 3.4)0.

Recalling 𝒪ε,ϱΩ, we see that v-wεC(𝒪¯ε,ϱ), and hence by the ABP maximum principle, Proposition 3.3, we have

v-wεmax𝒪ε,ϱ(v-wε).(6.20)

Let us note that 𝒪ε,ϱ=(𝒪εΩϱ)(𝒪εΩϱ). We also observe that the maximum on the right of (6.20) cannot occur on 𝒪εΩρ. Therefore for 0<εε0 the maximum on the right of (6.20) is achieved on 𝒪εΩϱ. Since 𝒪εΩ=, this means

v-wεmax𝒪ε{x:d(x)=ϱ}(v-wε).

We let ε0+ to obtain

v-umaxd(x)=ϱ(v-u):=κin Ωϱ.

On noting that

{γ+[v-(u+κ)]f(v)-f(u)0in Ω,vu+κon Ωρ,

we conclude, by the maximum principle, that vu+κ on Ωρ. Consequently, vu+κ on Ω. Let us also note that κv(x0)-u(x0)>0. Since γ+[v-(u+κ)]0 in Ω, by the Strong Maximum Principle (see [3] for instance) we conclude v=u+κ in Ω. Now we find that

f(u)+h(x)=H(x,u,Du,D2u)=H(x,u,D(u+κ),D2(u+κ))H(x,u+κ,D(u+κ),D2(u+κ))  (by (H-1), see Remark 3.1)=H(x,v,Dv,D2v)=f(v)+h(x).

But uv in Ω implies that f(u)f(v). Therefore we have f(u)=f(v)=f(u+κ) in Ω. If x*Ωϱ, then according to (f-4) we have

f(u(x*))u(x*)f(u(x*)+κ)u(x*)+κ.

Therefore, since f(u(x*))=f(u(x*)+κ), we have u(x*)u(x*)+κ. Of course, this is not possible. This proves the uniqueness theorem. ∎

Remark 6.4.

Suppose that the assumptions of Theorem 6.3 on H and f hold. Assume that h+C(Ω)Lp(Ω) for some p>p0, where p0 is the Escauriaza exponent. Then problem (1.1) admits at most one solution. Thus Theorem 6.3 improves the uniqueness result of [2].

Finally, we give the proof of Theorem 2.4.

Proof of Theorem 2.4.

Note that all the assumptions of Theorem 6.3, except for the existence of ψ that satisfies (D-h) with Ξ*(ψ)=Ξ(ψ) small enough, are stated explicitly in the theorem to be proved. As already observed in the proof of Theorem 2.3, this missing assumption is satisfied with Ξ*(ψ)=Ξ(ψ)=0 by the hypothesis made on h+. Thus the proof is completed upon invoking Theorem 6.3. ∎

A Appendix

Given a bounded C2 domain Ωn, in this appendix, we wish to study the existence of non-negative solution to

{H[u]=f(u)in Ω,u=on Ω,

under the assumption that H satisfies (H-1), (H-2) and f satisfies (f-1), (f-2). The coefficients γ,χC(Ω) are allowed to be unbounded with their growth near the boundary controlled in accordance with conditions (C-γ) and (C-χη).

Our analysis is based on a result due to Ancona [4, Proposition 11, Remark 6.1]. Let η be a Dini continuous function. According to the result in [4] cited above, there is a positive function ψC(Ω)C(Ω¯) such that

{Lψ-η(d(x))d2(x)in Ω,ψ=0on Ω,(A.1)

for any uniformly elliptic differential operator Lw:=tr(A(x)D2w)+b(x)Dw with fixed ellipticity constants 0<λΛ. Here A(x):=[aij(x)] with A(x)𝒜λ,Λ, and |b(x)| is continuous on Ω such that |b(x)|d(x) sufficiently small near the boundary Ω (see [4, Remark 6.1]).

Since 𝒫λ,Λ+(D2w):=sup{tr(AD2w):λInAΛIn}, it follows that (A.1) is valid with γ+ taking the place of L. Therefore we have

{H[ψ]+[ψ]=γ+[ψ]-η(d(x))d2(x)in Ω,ψ=0on Ω.(A.2)

As a consequence of (A.2) we may proceed as in [4, Theorem 4] to prove the following existence result. While we only need a special case of the next lemma for our purpose, we present it in the generality stated as it may be of independent interest. Due to the nonlinearity of the underlying operator the proof requires careful consideration.

Lemma A.1.

Let ϑC(Ω) with d2(x)|ϑ(x)|η(d(x)) near the boundary Ω for some Dini continuous function η. For any constant κ0 the Dirichlet problem

{H[w]=ϑ(x)in Ω,w=κon Ω,(A.3)

has a viscosity solution wC(Ω¯).

Proof.

We present the proof in three steps. Throughout the proof we fix an exhaustion {𝒪j} of Ω by smooth subdomains 𝒪jΩ.

Step 1. We first show the existence of a solution wC(Ω¯) to

{H[w]=-ϑ-in Ω,w=κon Ω.(A.4)

By hypothesis, there is a positive constant c=c(ϑ) such that

ϑ-(x)cη(d(x))d2(x)in Ω.

Then, taking (A.2) into consideration, we see that +[cψ]-ϑ- in Ω. Let z:=cψ+κ. Then +[z]-ϑ- in Ω and z=κ on Ω. For each j, let wjC(𝒪¯j) be the solution of

{H[wj]=-ϑ-in 𝒪j,wj=κon 𝒪j.(A.5)

For the existence of a solution to problem (A.5) we refer, for instance, to [13, Theorem 1.1]. Note that +[κ-wj]ϑ-, and therefore by Proposition 3.3 we see that wjκ in 𝒪j. Similarly, we have wjwj+1 in 𝒪j for all j. Furthermore, we have

+[wj-z]0in 𝒪j.

Since zκ in Ω, again by Proposition 3.3, we have wjz in 𝒪j. Let

w:=limjwjin Ω.

Then we note that w is a viscosity solution of H[w]=-ϑ- in Ω, and since κwz, we conclude that w is a solution of (A.4) as desired.

Step 2. Here we show the existence of a solution vC(Ω¯) to

{H[v]=ϑ+in Ω,v=κon Ω.(A.6)

On recalling the assumptions on ϑ and χ, we note that there is a positive constant C:=C(ϑ,κ,χ) such that

ϑ++κχCη(d(x))d2(x)in Ω.

By (A.2) we conclude that +[Cψ]-ϑ+-κχ. Let z:=-Cψ+κ. Then -[z]-[-Cw]-κχ=ϑ+ in Ω and z=κ on Ω. Thus H[z]ϑ+ in Ω and z=κ on Ω. For each positive integer j, let vjC(O¯j) such that

{H[vj]=ϑ+in 𝒪j,vj=κon 𝒪j.

Then +[z-vj]0 in 𝒪j and since zκ in Ω, we see from the maximum principle that hence zvj in 𝒪j. Similarly, by the maximum principle vjκ in 𝒪j. As before, vjvj+1 in 𝒪j. Consequently, the limit v solves (A.6) and vκ in Ω.

Step 3. We now show the existence of a solution to (A.3). Let w and v be solutions of (A.4) and (A.6), respectively. We recall that vκw in Ω. Let ujC(𝒪¯j) be a solution of

{H[uj]=ϑin 𝒪j,uj=von 𝒪j.

Then

+[v-uj]ϑ-0in 𝒪j.

By the maximum principle we have vuj in Ω. Similarly, +[uj-w]ϑ+0 in 𝒪j. Moreover, uj=vw on 𝒪j. Therefore ujw in 𝒪j. We also observe that ujuj+1 in 𝒪j. Therefore the limit u:=limjuj satisfies vuw, and hence u satisfies (A.3) as was to be shown. ∎

One can use Lemma A.1 to establish solvability of the Dirichlet problem where the right-hand side depends on the unknown. To be specific, we consider the following Dirichlet problem:

{H[u]=f(u)in Ω,u=κon Ω,(DH)

for any non-negative constant κ.

For this we need the following comparison lemma, which is a direct consequence of Proposition 3.3 and Lemma 3.4. For completeness we include the short proof.

Lemma A.2.

Suppose w,vC(Ω¯) such that H[v]f(v) and H[w]f(w) in Ω. If vw on Ω, then vw on Ω.

Proof.

Given ε>0, we note that v<w+ε in Ωδ for any sufficiently small δ>0. Moreover, it follows from condition (H-1) that

H[w+ε]H(x,w+ε,Dw,D2w)H[w]f(w)f(w+ε).

Suppose that 𝒪:={w+ε>v}Ωδ is non-empty. By Lemma 3.4, we have +[w+ε-v]f(w+ε)-f(v)0 in 𝒪:={w+ε>v}Ωδ. Therefore, by ABP, vw+ε in 𝒪, which is a contradiction. It follows that 𝒪 is empty and vw+ε. Since ε and δ are arbitrary, we obtain the desired conclusion. ∎

Now we can state the following result on the solvability of problem (DH).

Lemma A.3.

Suppose that f:RR satisfies (f-1). Given a constant κ0, problem (DH) has a solution uC(Ω¯).

Proof.

We recall from (3.6) that H[κ]+[κ]=0. Let u0C(Ω¯) such that

{H[u0]=f(κ)in Ω,u0=κon Ω.

The existence of such solutions follows from Lemma A.1. Since +[u0-κ]0 in Ω, by Lemma A.2 we find that u0κ in Ω.

For j1, let ujC(𝒪¯j) be a solution of

{H[uj]=f(uj)in 𝒪j,uj=u0on 𝒪j.(A.7)

Existence of a solution to problem (A.7) follows, for instance, from [13, Theorem 1.1]. Since +[uj-κ]0 in Ω, we proceed as in the above to conclude that ujκ in Ω. Consequently,

+[u0-uj]f(κ)-f(uj)0in 𝒪j.

Therefore, again by the comparison principle we find that u0uj in 𝒪j for all j. Similarly, it follows that ujuj+1 in 𝒪j. Thus in summary, we have shown that

u0ujuj+1κin 𝒪j for all j=1,2,.

By Ascoli–Arzelá and the stability of viscosity solutions we conclude that u:=limuj is a solution of (DH). ∎

Lemma A.4.

Let f satisfy conditions (f-1) and (f-2). The following problem has a solution:

{H[u]=f(u)in Ω,u=on Ω.(A.8)

Proof.

Now that we know problem (DH) is solvable for any constant κ0, we can proceed as in the proof of Theorem 4.2 to show that problem (A.8) has a solution uC(Ω). Since f(0)=0, we note that u0 in Ω. ∎

References

  • [1]

    N. Abatangelo, Very large solutions for the fractional Laplacian: Towards a fractional Keller–Osserman condition, Adv. Nonlinear Anal. 6 (2017), no. 4, 383–405.  Web of ScienceGoogle Scholar

  • [2]

    S. Alarcón and A. Quaas, Large viscosity solutions for some fully nonlinear equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), no. 4, 1453–1472.  CrossrefWeb of ScienceGoogle Scholar

  • [3]

    M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second-order elliptic equations in unbounded domains, Abstr. Appl. Anal. 2008 (2008), Article ID 178534.  Web of ScienceGoogle Scholar

  • [4]

    A. Ancona, On strong barriers and an inequality of Hardy for domains in 𝐑n, J. Lond. Math. Soc. (2) 34 (1986), no. 2, 274–290.  Google Scholar

  • [5]

    C. Bandle and M. Marcus, “Large” solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992), 9–24.  CrossrefGoogle Scholar

  • [6]

    L. Caffarelli, M. G. Crandall, M. Kocan and A. Świȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), no. 4, 365–397.  CrossrefGoogle Scholar

  • [7]

    L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213.  CrossrefGoogle Scholar

  • [8]

    L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, American Mathematical Society, Providence, 1995.  Google Scholar

  • [9]

    I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo, The Alexandrov–Bakelman–Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Comm. Partial Differential Equations 30 (2005), no. 10–12, 1863–1881.  CrossrefGoogle Scholar

  • [10]

    O. Costin and L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, J. Differential Equations 249 (2010), no. 4, 931–964.  Web of ScienceCrossrefGoogle Scholar

  • [11]

    O. Costin, L. Dupaigne and O. Goubet, Uniqueness of large solutions, J. Math. Anal. Appl. 395 (2012), no. 2, 806–812.  CrossrefGoogle Scholar

  • [12]

    M. G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67.  CrossrefGoogle Scholar

  • [13]

    M. G. Crandall, M. Kocan, P. L. Lions and A. Świȩch, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations 1999 (1999), Paper No. 24.  Google Scholar

  • [14]

    F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 317–330.  Google Scholar

  • [15]

    G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: Existence and uniqueness, Nonlinear Anal. 20 (1993), no. 2, 97–125.  CrossrefGoogle Scholar

  • [16]

    M. Dindoš, Large solutions for Yamabe and similar problems on domains in Riemannian manifolds, Trans. Amer. Math. Soc. 363 (2011), no. 10, 5131–5178.  CrossrefGoogle Scholar

  • [17]

    S. Dumont, L. Dupaigne, O. Goubet and V. Rădulescu, Back to the Keller–Osserman condition for boundary blow-up solutions, Adv. Nonlinear Stud. 7 (2007), no. 2, 271–298.  Google Scholar

  • [18]

    L. Dupaigne, M. Ghergu and V. Rădulescu, Lane–Emden–Fowler equations with convection and singular potential, J. Math. Pures Appl. (9) 87 (2007), no. 6, 563–581.  CrossrefWeb of ScienceGoogle Scholar

  • [19]

    L. Escauriaza, W2,n a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J. 42 (1993), no. 2, 413–423.  Google Scholar

  • [20]

    M. J. Esteban, P. L. Felmer and A. Quaas, Superlinear elliptic equation for fully nonlinear operators without growth restrictions for the data, Proc. Edinb. Math. Soc. (2) 53 (2010), no. 1, 125–141.  CrossrefWeb of ScienceGoogle Scholar

  • [21]

    G. Galise and A. Vitolo, Viscosity solutions of uniformly elliptic equations without boundary and growth conditions at infinity, Int. J. Differ. Equ. 2011 (2011), Article ID 453727.  Google Scholar

  • [22]

    J. García-Melián, Uniqueness of positive solutions for a boundary blow-up problem, J. Math. Anal. Appl. 360 (2009), no. 2, 530–536.  CrossrefWeb of ScienceGoogle Scholar

  • [23]

    J. García-Melián, Large solutions for an elliptic equation with a nonhomogeneous term, J. Math. Anal. Appl. 434 (2016), no. 1, 872–881.  CrossrefWeb of ScienceGoogle Scholar

  • [24]

    M. Ghergu and V. D. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Ser. Math. Appl. 37, Clarendon Press, Oxford, 2008.  Google Scholar

  • [25]

    D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977.  Google Scholar

  • [26]

    F. Gladiali and G. Porru, Estimates for explosive solutions to p-Laplace equations, Progress in Partial Differential Equations. Vol. 1 (Pont-à-Mousson 1997), Pitman Res. Notes Math. Ser. 383, Longman, Harlow (1998), 117–127.  Google Scholar

  • [27]

    J. A. Goldstein, An appreciation of my teacher, M. M. Rao, Stochastic Processes and Functional Analysis, Lecture Notes Pure Appl. Math. 238, Dekker, New York (2004), 3–5.  Google Scholar

  • [28]

    J. B. Keller, On solutions of Δu=f(u), Comm. Pure Appl. Math. 10 (1957), 503–510.  Google Scholar

  • [29]

    S. Koike and A. Świȩch, Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients, J. Math. Soc. Japan 61 (2009), no. 3, 723–755.  CrossrefWeb of ScienceGoogle Scholar

  • [30]

    J. López-Gómez and L. Maire, Uniqueness of large positive solutions, Z. Angew. Math. Phys. 68 (2017), no. 4, Article ID 86.  Web of ScienceGoogle Scholar

  • [31]

    M. Marcus and L. Véron, Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 2, 237–274.  CrossrefGoogle Scholar

  • [32]

    M. Marcus and L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal. 144 (1998), no. 3, 201–231.  CrossrefGoogle Scholar

  • [33]

    M. Marcus and L. Véron, The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption, Comm. Pure Appl. Math. 56 (2003), no. 6, 689–731.  CrossrefGoogle Scholar

  • [34]

    J. H. Michael, A general theory for linear elliptic partial differential equations, J. Differential Equations 23 (1977), no. 1, 1–29.  CrossrefGoogle Scholar

  • [35]

    A. Mohammed and G. Porru, Harnack inequality for non-divergence structure semilinear elliptic equations, Adv. Nonlinear Anal. (2016), 10.1515/anona-2016-0050.  Google Scholar

  • [36]

    A. Mohammed and G. Porru, Large solutions for non-divergence structure equations with singular lower order terms, Nonlinear Anal. Real World Appl. 35 (2017), 470–482.  CrossrefWeb of ScienceGoogle Scholar

  • [37]

    A. Mohammed and G. Porru, Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term, Adv. Nonlinear Anal. (2017), 10.1515/anona-2017-0065.  Web of ScienceGoogle Scholar

  • [38]

    A. Mohammed, G. Porru and A. Vitolo, Harnack inequality for nonlinear elliptic equations with strong absorption, J. Differential Equations 263 (2017), no. 10, 6821–6843.  CrossrefGoogle Scholar

  • [39]

    A. Mohammed and A. Vitolo, Large solutions of fully nonlinear equations: Existence and uniqueness, preprint.  

  • [40]

    R. Osserman, On the inequality Δuf(u), Pacific J. Math. 7 (1957), 1641–1647.  Google Scholar

  • [41]

    V. D. Rădulescu, Singular phenomena in nonlinear elliptic problems: From blow-up boundary solutions to equations with singular nonlinearities, Handbook of Differential Equations: Stationary Partial Differential Equations. Vol. IV, Handb. Differ. Equ., Elsevier, Amsterdam (2007), 485–593.  Google Scholar

  • [42]

    D. Repovš, Asymptotics for singular solutions of quasilinear elliptic equations with an absorption term, J. Math. Anal. Appl. 395 (2012), no. 1, 78–85.  Web of ScienceCrossrefGoogle Scholar

  • [43]

    A. Świȩch, W1,p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations 2 (1997), no. 6, 1005–1027.  Google Scholar

  • [44]

    N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of nonlinear elliptic equations, Rev. Mat. Iberoam. 4 (1988), no. 3–4, 453–468.  Google Scholar

  • [45]

    L. Véron, Semilinear elliptic equations with uniform blow-up on the boundary, J. Anal. Math. 59 (1992), 231–250.  CrossrefGoogle Scholar

  • [46]

    A. Vitolo, M. E. Amendola and G. Galise, On the uniqueness of blow-up solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst. 2013 (2013), 771–780.  Google Scholar

  • [47]

    Z. Zhang, A boundary blow-up elliptic problem with an inhomogeneous term, Nonlinear Anal. 68 (2008), no. 11, 3428–3438.  CrossrefWeb of ScienceGoogle Scholar

Footnotes

  • 1

    Let ρ>0 such that v>0 in Ωρ. For a fixed p>1, let g(t)=f(t) for tmaxΩρv and g(t)=f(maxΩρv)tp/(maxΩρv)p for t>maxΩρv. Then v is a solution of -[v]=g(v) in Ωρ. Since g satisfies all the assumptions of [38, Theorem 2.8], we see that v>0 in Ωρ by the Harnack inequality. 

About the article

Received: 2018-06-13

Accepted: 2018-06-13

Published Online: 2018-07-04

Published in Print: 2019-03-01


Funding Source: Javna Agencija za Raziskovalno Dejavnost RS

Award identifier / Grant number: P1-0292

Award identifier / Grant number: J1-8131

Award identifier / Grant number: J1-7025

Award identifier / Grant number: N1-0064

Award identifier / Grant number: N1-0083

Vicenţiu D. Rădulescu was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. He also acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 39–64, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0134.

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