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Advances in Nonlinear Analysis

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


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Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

Ahmed Mohammed / Giovanni Porru
Published Online: 2017-06-08 | DOI: https://doi.org/10.1515/anona-2017-0065

Abstract

Motivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE Lu=f(u)+h(x) on bounded smooth domains Ωn, where L is a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation, f is a continuous non-decreasing function that satisfies the Keller–Osserman condition, while h is a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions on f and h that would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions on f and h for the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients of L to be bounded in Ω, but we still allow h to be unbounded on Ω.

Keywords: Large solutions; existence and uniqueness; semilinear elliptic equation

MSC 2010: 35J60; 35J70

1 Introduction

Let Ωn be a bounded domain with C2 boundary. We consider a uniformly elliptic nondivergence structure second order differential operator, namely,

Lu=aij(x)uxixj+bi(x)uxi+c(x)u.(1.1)

In (1.1) and throughout this paper, the summation convention over repeated indices from 1 to n is in effect. We will assume that [aij(x)] is an n×n symmetric matrix of continuous real-valued functions on Ω such that

|ξ|2aij(x)ξiξjΛ|ξ|2for all (x,ξ)Ω×N,(1.2)

for some constant Λ1.

In this paper we wish to study the question of existence and uniqueness of solutions to the following problem:

{Lu=f(u)+h(x),xΩ,u=,xΩ.(1.3)

Here h:Ω is a continuous function which may be unbounded on Ω, and f: is a continuous function with appropriate conditions to be specified later.

To study the existence of solutions to (1.3) we assume the following conditions. The coefficients bi(x) (1in) and c(x) in (1.1) are locally bounded Borel functions such that:

  • (DB)

    d(x)|bi(x)|=o(1) as d(x)0,

  • (DC)

    d2(x)|c(x)|η(d(x)) for all xΩ, with d(x)δ0 and c(x)0 for all xΩ.

In (DB) and (DC) we have used d(x) to denote the distance of xΩ from the boundary Ω, a notation we will continue to use throughout the paper. In (DC), δ0>0 is a positive constant and η:(0,δ0](0,η(δ0)] is an increasing function that satisfies the Dini condition

0δ0η(s)s𝑑s<.(1.4)

Following the pioneering works of Keller [15] and Osserman [19], problem (1.3) has been studied extensively by numerous authors when L is the Laplacian and h0. The reader is referred to the monograph [11] and the references therein for more discussion related to such problems. In [3], Bandle and Marcus investigated existence and asymptotic boundary behavior of solutions to (1.3) when h0c and biCα(Ω¯), i=1,,n, for some 0<α<1. Concerning problem (1.3) with h(x)0 on Ω, Véron in [24], and Díaz and Letelier in [6], established the existence and uniqueness of positive solutions to (1.3) for the nonlinearity f(t)=|t|p-1t, p>1, and for a non-positive unbounded inhomogeneous term hC(Ω) with appropriate growth condition on the boundary.

As far as we are aware, it was García-Melián who first studied problem (1.3) for a sign-changing and unbounded inhomogeneous term h in the recent paper [9]. He studied the existence of a solution to problem (1.3) when L is the Laplacian, f(t)=|t|p-1t for p>1 and h belonged to a large class of unbounded and sign-changing functions on Ω. He also obtained uniqueness of positive solutions of (1.3) for hC(Ω) with appropriate growth condition on the boundary, but bounded on Ω from above.

In this paper we wish to continue the aforementioned investigations with the objective of extending the results in several fronts. In all cases the class of inhomogeneous terms h we consider will include sign-changing and unbounded functions in Ω having appropriate growth conditions near the boundary. The necessity of some restriction on the growth of h near the boundary has already been noted in [9, Theorem 3]. As our first main result we will show the existence of solutions to (1.3), where the lower-order coefficients are allowed to be unbounded in Ω, and f comes from a wide class of nonlinearities. In addition, to the usual Keller–Osserman, we will require some mild conditions on f. As it turns out, if the inhomogeneous term h grows no faster than a suitable multiple of f(ϕ(d(x))) near the boundary of Ω, then problem (1.3) admits a solution. Here ϕ is a decreasing function on (0,a), for some a>0, that is associated with the nonlinearity f. For instance, ϕ(t)=t-2/(p-1) when f(t)=tp, p>1, for t>0. Our second main result concerns the asymptotic boundary estimates for solutions of (1.3) when the coefficients of L are bounded in Ω and, as a consequence, the uniqueness of solutions is obtained. These results are shown to hold for a large class of inhomogeneous terms h, which may change sign and be unbounded on Ω. In this regard, the asymptotic estimate and uniqueness results of this paper are new even when L is the Laplacian and f(t)=|t|p-1t for p>1, as we do not require h to be bounded from above.

We point out that problem (1.3) was also considered in [10, 25]. We direct the reader to [9] for a discussion of problem (1.3) in these papers. The reader is referred to the recent papers [2, 4, 5, 8, 21, 26], and references therein, on asymptotic behavior and uniqueness of singular solutions related to the content of this paper. In particular, we draw attention to the paper [5] in which the authors make a systematic use of Karamata variation theory to study uniqueness of boundary blow-up solutions. To the best of our knowledge, [8] is the first paper to investigate existence of boundary blow-up solutions of equations with nonmonotonic nonlinearity.

The paper is organized as follows. In Section 2, we state the main conditions used on the nonlinearity f to study problem (1.3). In particular, we recall a lemma that will be useful in establishing the existence of solutions in the case when L has singular lower-order term coefficients. Section 3 is devoted to the study of existence of a solution to (1.3). In Section 4, we establish boundary asymptotic estimates of solutions to problem (1.3). Existence of positive solutions and uniqueness of solutions is investigated in Section 5. Finally, we have included an appendix where we prove some technical results that are used in the paper.

2 Preliminaries

Let Ωn be a bounded domain with C2 boundary. Throughout the paper, it will be convenient to use the following notations for a given δ>0:

Ωδ:={xΩ:d(x)<δ}andΩδ:={xΩ:d(x)>δ}.

Since Ω is a bounded C2 domain, we note that there exists μ>0 such that dC2(Ω¯μ) and |d(x)|=1 on Ωμ. See [12, Lemma 14.16] for a proof.

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-ψ)d+ψ instead of d, where ψCc2(Ω) is a cut-off function with 0ψ1 on Ω, ψ0 on Ωμ0 for some 0<μ0<μ, and ψ1 on Ωμ. Therefore, hereafter, we will always suppose that d is this modified distance function and that d is in C2(Ω¯) with |d|1 on Ωμ.

By a solution u of Lu=(,)H(x,u) we mean a twice weakly differentiable function on Ω such that

Lu(x)=(,)H(x,u(x))for almost every xΩ.

We start with the following extension of the classical maximum principle. We assume that tH(x,t) is non-decreasing on for each xΩ.

Lemma 2.1 (Comparison Principle).

Let u,wWloc2,n(Ω)C(Ω¯), and assume that LuH(x,u) in Ω and LwH(x,w) in Ω. If uw on Ω, then uw in Ω.

Proof.

Given ε>0,

L(w+ε)=Lw+εcLwH(x,w)H(x,w+ε)in Ω  and  u-(w+ε)<0on Ω.

Let δ>0 sufficiently small such that u-(w+ε)0 on Ωδ. Suppose that the open set

Ω0δ:={xΩδ:u(x)>w(x)+ε}

is non-empty. Since L(u-(w+ε))H(x,u)-H(x,w+ε)0 in Ω0δ and the coefficients of L are bounded on Ωδ, the maximum principle applies (see [12, Theorem 9.1]) and we conclude that u-(w+ε)0 on Ω0δ. This is an obvious contradiction to the assumption that Ω0δ is non-empty. Therefore, we must have uw+ε in Ωδ. Since δ>0 is arbitrary, we conclude that uw+ε in Ω. Since ε>0 is arbitrary, we find that uw on Ω, as desired. ∎

We consider the following conditions on the nonlinearity f in (1.3):

  • (A1)

    f: is a non-decreasing continuous function such that f(0)=0 with f(t)>0 for t>0.

  • (A2)

    f satisfies the Keller–Osserman condition, namely,

    1dtF(t)<,where F(t)=0tf(s)𝑑s.

  • (A3)

    We have

    lim inftF(t)tf(t)>0.

We make a few remarks about the above conditions.

Remark 2.2.

  • (i)

    Any regularly varying function at infinity with index 1<q< satisfies conditions (A2) and (A3). We recall that f is said to be regularly varying at infinity of index q if f is a measurable function defined on (a,) for some a>0 and

    limtf(ξt)f(t)=ξqfor all ξ>0.

  • (ii)

    f(t)=t satisfies (A3) but not (A2), while f(t)=et-1 satisfies (A2) but not (A3).

  • (iii)

    If f satisfies (A1), then it is clear that F(t)tf(t) for all t0. Moreover, if f satisfies both (A1) and (A2), then

    lim inftF(t)tf(t)12.

    We refer to Lemma A.1 in Appendix A for a proof of this assertion.

The reader may find more on regularly varying functions and some basic information on Karamata regular variation theory in [14, 23].

It is a well-known fact, see [13, 22], that if f satisfies (A1) and the Keller–Osserman condition (A2), then

limtF(t)f(t)=0andlimttf(t)=0.(2.1)

In fact, the following result holds for any f that satisfies (A1), (A2) and (A3).

Lemma 2.3.

Suppose that f satisfies (A1), (A2) and (A3). Then the following hold:

  • (i)

    lim suptF(t)f(t)tF(s)-1/2𝑑s<,

  • (ii)

    lim supttf(t)(tF(s)-1/2𝑑s)2<.

We refer the reader to [18] for a proof of the above lemma. We let ϕ to be the non-increasing function such that

ϕ(t)ds2F(s)=t,t>0.

It follows that

ϕ(t)=-2F(ϕ(t))andϕ′′(t)=f(ϕ(t)),t>0.

For later use, let us compute Lv, where v(x):=ϕ(ϱ(x)) for some ϱC2(Ω):

Lv=ϕ′′(ϱ)aijϱxiϱxj+ϕ(ϱ)(aijϱxixj+biϱxi)+cϕ(ϱ)=f(ϕ(ϱ))aijϱxiϱxj-2F(ϕ(ϱ))(aijϱxixj+biϱxi)+cϕ(ϱ)=f(ϕ(ϱ))(aijϱxiϱxj-2F(ϕ(ϱ))f(ϕ(ϱ))(aijϱxixj+biϱxi)+ϕ(ϱ)cf(ϕ(ϱ))).(2.2)

3 On existence of solutions to problem (1.3)

Throughout this section, we assume that the lower order coefficients bi(x) (1in) and c(x) of L satisfy conditions (DB) and (DC), respectively.

The next result, a consequence of these conditions and Lemma 2.3, will prove useful in establishing the existence of solutions to problem (1.3).

Corollary 3.1.

Suppose that f satisfies conditions (A1), (A2) and (A3). Then

limd(x)0|c(x)|ϕ(d(x))f(ϕ(d(x)))=0𝑎𝑛𝑑limd(x)0F(ϕ(d(x)))f(ϕ(d(x)))i|bi(x)|=0.

Proof.

We prove the first limit and omit the second as the proof is similar. We have

|c(x)|ϕ(d(x))f(ϕ(d(x)))=d2(x)|c(x)|ϕ(d(x))d2(x)f(ϕ(d(x)))=|c(x)|d2(x)ϕ(d(x))f(ϕ(d(x)))(ϕ(d(x))F(s)-1/2𝑑s)2η(d(x))ϕ(d(x))f(ϕ(d(x)))(ϕ(d(x))F(s)-1/2𝑑s)2.

On noting that η(0+)=0, the claim follows from Lemma 2.3 (ii).∎

Remark 3.2.

It is clear that (DB) and (DC) hold when the coefficients 𝐛 and c of L are bounded on Ω. Therefore, Corollary 3.1 holds when the coefficients of L are bounded. In fact, in this case, condition (A3) is not needed. One only needs to recall (2.1).

Let Ξ*:=infξ>0g*(ξ) and Ξ*:=supξ>0g*(ξ), where

g*(ξ):=ξ-lim suptf(ξt)f(t)andg*(ξ):=Λξ-lim inftf(ξt)f(t).

Here, Λ is the ellipticity constant of L as noted in condition (1.2).

As an example, we observe that for any regularly varying function (at infinity) f of index 1<q<, it can be easily seen that

Ξ*=-andΞ*=(q-1)(1q)q/(q-1).(3.1)

On the other hand, for ϖ>2, we note that f(t)=tlogϖ(|t|+1) is regularly varying at infinity of index q=1 and satisfies (A1), (A2) and (A3). Computation shows that Ξ*=0=Ξ*.

Remark 3.3.

On noting that g*(1)=0, we see that Ξ*0. Moreover, we also have Ξ*0 for any f that satisfies (A1) and (A3). We refer the reader to Lemma A.3 in Appendix A.

We need some conditions on f and h in order to prove the existence of a solution to (1.3). We require the following assumption on f:

  • (f4)

    limt-f(t)=-.

We should note that if f:[0,)[0,) satisfies (A1) and (A2), then the odd extension of f to satisfies (f4).

In addition to condition (f4), we will also require some growth restrictions, near the boundary Ω, on the inhomogeneous term h in (1.3). We state one of these conditions on h as follows:

  • (h1)

    Θ*(h):=lim supd(x)0h(x)f(ϕ(d(x)))<Ξ*.

The main result of this section gives the existence of a solution to problem (1.3). We employ the sub-solution and super-solution technique to establish the result. In preparation for this, let us consider a function H:Ω× which is non-decreasing in in the second variable for each xΩ, and H(,t)C(Ω¯) for each t.

From [18] we recall the following result on the solvability of a class of Dirichlet problems with continuous boundary data.

Lemma 3.4.

Given gC(Ω), the following Dirichlet problem admits a solution uWloc2,p(Ω)C(Ω¯) for each 1p<:

{Lu=H(x,u)in Ω,u=gon Ω.(3.2)

Remark 3.5.

Suppose that all coefficients of L are bounded and belong to Cα(Ω) for some 0<α<1. If, in addition to the hypotheses on H, we suppose HCα(Ω×), then problem (3.2) admits a solution uC2(Ω)C(Ω¯). This is a consequence of the elliptic regularity theory, see [12, Theorem 9.19] with k=1.

We will use Lemma 3.4 to study the following infinite boundary value problem.

{Lu=H(x,u)in Ω,u=on Ω.(3.3)

As a simple consequence of Lemma 3.4 we obtain the following.

Lemma 3.6.

Let u*,u*Wloc2,p(Ω)C(Ω) for 1<p< be such that

Lu*H(x,u*)in Ω,Lu*H(x,u*)in Ω  𝑎𝑛𝑑  u*=on Ω.

Assume that u*L(Ω) or u*= on Ω. If u*u* in Ω, then there exists a solution uWloc2,p(Ω)C(Ω) of (3.3) with u*uu* a.e. in Ω.

Proof.

Let us first consider the case when u*L(Ω). For each positive integer jsupΩu*, by Lemma 3.4, we let ujWloc2,pC(Ω¯) be a solution of

{Luj=H(x,uj)in Ω,u=jon Ω.

By the Comparison Principle, Lemma 2.1, we have

u*ujuj+1u*in Ω for all j.

By proceeding as in the proof of [18, Lemma 3.2], we conclude that {uj} converges locally uniformly to a solution uC2(Ω) of problem (3.3) on Ω with u*uu*.

Now assume that u*= on Ω. Let 𝒪j:={xΩ:d(x)>1/j} for j=1,2,, and let ujWloc2,p(𝒪j)C(𝒪¯j) be a solution of

{Luj=H(x,uj)in 𝒪j,uj=u*on 𝒪j.

By the Comparison Principle, we see that u*uju* on 𝒪¯j for each j1. Consequently,1 we also note that ujuj+1 in 𝒪j. Thus, we have

u*ujuj+1u*on 𝒪j.

Again, by proceeding as in the proof of [18, Lemma 3.2], we conclude that {uj} converges locally uniformly to a solution uWloc2,p(Ω)C(Ω) of problem (3.3) with u*uu* on Ω. ∎

For the rest of the paper, we will assume that f satisfies both conditions (A1) and (A2).

We are now ready to state the following theorem on the existence of a solution to (1.3).

Theorem 3.7.

Suppose that f satisfies (A3) and (f4). We also assume that hC(Ω) satisfies (h1). Then problem (1.3) has a solution uWloc2,p(Ω)C(Ω) for 1p<. Moreover, u is the maximal solution.

Proof.

There is no loss of generality in supposing that pn. Let {𝒪j} be the sequence of open subsets of Ω introduced in the proof of Lemma 3.6 above. Let us first show that the following has a solution:

{Lw=f(w)+hin 𝒪j,w=on 𝒪j.(3.4)

To this end, let vj be a solution of Lvj=f(vj) in 𝒪j and vj= on 𝒪j. If h0 in 𝒪j, then wj:=vj is a super-solution of (3.4). Otherwise, let zj be a solution of Lzj=min𝒪¯jh in 𝒪j with zj=0 on 𝒪j. Note that zj>0. Then wj:=vj+zj satisfies

Lwj=L(vj+zj)=f(vj)+min𝒪¯jhf(vj+zj)+h(x)=f(wj)+h(x),x𝒪j,

and

wj=on 𝒪j.

Thus, in any case wj is a super-solution of (3.4). We now proceed to construct a sub-solution of (3.4). To this end, we claim that there are positive constants A and α such that

w=Aϕ(d(x))-α

is a sub-solution of (1.3) in Ω. To see this, let ϱ:=d in (2.2), and we estimate (2.2) in Ωμ as follows:

Lwf(ϕ(d))[A-A2F(ϕ(d))f(ϕ(d))(|L0d|+i|bi(x)|)-A|c(x)|ϕ(d)f(ϕ(d))]-αcf(ϕ(d))[A-A2F(ϕ(d))f(ϕ(d))(|L0d|+i|bi(x)|)-A|c(x)|ϕ(d)f(ϕ(d))].(3.5)

In the last inequality, we have used the fact that -αc0 for any α>0. By (h1), since Ξ*>Θ*, we let A such that g*(A)>Θ*. Let us choose ε>0 sufficiently small so that

Θ*+2ε<g*(A)=A-lim supd(x)0f(Aϕ(d(x)))f(ϕ(d(x))).

Therefore, there exists 0<δ<μ such that for xΩδ, we have

A-ε>Θ*+ε2+f(Aϕ(d(x)))f(ϕ(d(x))).

Recalling (2.1) and Corollary 3.1, we can take δ>0 sufficiently small so that for all xΩδ,

A2F(ϕ(d))f(ϕ(d))(|L0d|+i|bi(x)|)+A|c(x)|ϕ(d)f(ϕ(d))<ε.(3.6)

Likewise, by the definition of Θ*:=Θ*(h), we have (by shrinking δ further, if necessary)

h(x)f(ϕ(d(x)))<Θ*+ε2for all xΩδ.(3.7)

Putting (3.5), (3.6) and (3.7) together, we find that w(x)=Aϕ(d(x))-α satisfies the following for all xΩδ:

Lwf(ϕ(d(x)))(A-ε)f(ϕ(d(x)))[h(x)f(ϕ(d(x)))+f(Aϕ(d(x)))f(ϕ(d(x)))]=h(x)+f(Aϕ(d(x)))f(w)+h(x)for any α>0.

Next we choose α so that w is a sub-solution of Lw=f(w)+h(x) on Ωδ. For this, let

η:=min{ALϕ(d):xΩδ}.

We should point out that η is independent of α. The hypothesis on f shows that

f(Aϕ(δ)-α)+max{h(x):xΩδ}-as α.

Thus, we can choose α>0 sufficiently large so that

f(Aϕ(δ)-α)+max{h(x):xΩδ}η.

Having fixed such α, we see that in Ωδ the following holds:

Lw=ALϕ(d(x))-αcηf(Aϕ(δ)-α)+max{h(x):xΩδ}f(Aϕ(d(x))-α)+h(x)=f(w)+h(x).

Thus, we have shown that w is a sub-solution of (1.3) in Ω. Moreover, for each positive integer j, the Comparison Principle shows that wwj in 𝒪j. Therefore, by Lemma 3.6, problem (3.4) has a solution uj such that wujwj in 𝒪j. Since uj is bounded on 𝒪j-1, the Comparison Principle shows that ujuj-1 on 𝒪j-1, that is, {uk}k=j-1 is a decreasing sequence in 𝒪j-1. Let us set

u(x):=limjuj(x),xΩ.

Since {uj} is locally uniformly bounded, by standard Schauder estimates, we see that uC2(Ω) satisfies Lu=f(u)+h in Ω. From this and the inequality wu in Ω, we see that u is a solution of (1.3), as desired.

If v is any solution of (1.3), then Comparison Principle shows that vuj on 𝒪j for all j1. Consequently, we have vu in Ω, and therefore u is a maximal solution of (1.3), as claimed. ∎

Remark 3.8.

If the coefficients of L are bounded in Ω, we note that condition (A3) is not needed in the above theorem, see Remark 3.2.

4 On the boundary asymptotic estimates of solutions to problem (1.3)

In this section we will assume that the coefficients bi(x)(1in) and c(x) are bounded on Ω. It should be recalled that we always assume conditions (A1) and (A2) on f.

To discuss asymptotic boundary estimates of solutions to (1.3), we need an additional condition on h which we now make explicit:

  • (h2)

    Θ*(h):=lim infd(x)0h(x)f(ϕ(d(x)))>Ξ*.

Remark 4.1.

If h is bounded from above on Ω, then Θ*Θ*0.

We have the following lemma on asymptotic boundary estimates of solutions to (1.3).

Lemma 4.2.

Let hC(Ω).

  • (i)

    If h satisfies (h2) , then there exists a positive constant A* such that for any solution uWloc2,n(Ω) of ( 1.3 ), we have

    lim supd(x)0u(x)ϕ(d(x))A*.(4.1)

  • (ii)

    If h satisfies (h1) and is bounded from above, then there exists a positive constant A* such that for any solution uWloc2,n(Ω) of ( 1.3 ), we have

    A*lim infd(x)0u(x)ϕ(d(x)).(4.2)

Proof.

For any 0<ρ<μ, let us consider the sets

Ωρ-:={xΩ:ρ<d(x)<μ}andΩρ+:={xΩ:0<d(x)<μ-ρ}.

Proof of (4.1). For an appropriate choice of a positive constant A*, we will show that

w*(x):=A*ϕ(d(x)-ρ),xΩρ-,

is a super-solution of (1.3) on Ωρ- for all 0<ρ<μ and sufficiently small μ. To this end, we estimate (2.2) with ϱ(x):=d(x)-ρ as follows. Recalling that c0 in Ω, we estimate

Lw*f(ϕ(d(x)-ρ))[A*Λ+A*2F(ϕ(d-ρ))f(ϕ(d-ρ))(|L0d|+𝐛L)].(4.3)

Let ε>0, to be specified later. By (2.1), we can take μ>0 sufficiently small so that for all xΩρ-,

A*2F(ϕ(d-ρ))f(ϕ(d-ρ))(|L0d|+𝐛L)<ε.(4.4)

Therefore, from (4.3) and (4.4), we conclude that for xΩμ-,

Lw*f(ϕ(d-ρ)))(A*Λ+ε).

Now, let ε be chosen so that Θ*-2ε>Ξ*. We pick a number A*:=A*(Λ,f,Θ*(h))>0 such that

Θ*-2ε>g*(A*)=A*Λ-lim inf0<d(x)-ρ0f(A*ϕ(d(x)-ρ))f(ϕ(d(x)-ρ)).

That is,

A*Λ+ε<Θ*-ε+lim inf0<d(x)-ρ0f(A*ϕ(d(x)-ρ))f(ϕ(d(x)-ρ)).

We also suppose that μ is sufficiently small, so that for (x,ρ)Ωρ-×(0,μ),

A*Λ+ε<Θ*-ε2+f(A*ϕ(d(x)-ρ))f(ϕ(d(x)-ρ)).

Let us now suppose that Θ*(h)0. In this case, for (x,ρ)Ωρ-×(0,μ), we have

Lw*f(ϕ(d(x)-ρ))(A*Λ+ε)f(ϕ(d(x)-ρ))[Θ*-ε2+f(A*ϕ(d(x)-ρ))f(ϕ(d(x)-ρ))]=(Θ*-ε2)f(ϕ(d(x)-ρ))+f(A*ϕ(d(x)-ρ))(Θ*-ε2)f(ϕ(d(x)))+f(A*ϕ(d(x)-ρ))h(x)+f(w*),

from the definition of Θ*. Let B*:=max{u(x):d(x)μ}. Then uw*+B* on Ωρ- and

Lu=f(u)+h,L(w*+B*)Lw*f(w*)+hf(w*+B*)+hon Ωρ-.

By the Comparison Principle, Lemma 2.1, 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*.

Now, let d(x)0 to obtain (4.1) when Θ*(h)0.

We now consider the case Θ*(h)>0. Then h is non-negative near Ω. Let {𝒪j} be the sequence of open subsets of Ω defined as in the proof of Lemma 3.6 and m:=min{h(x):xΩ¯}. We note that m>-. Let vj be a solution of (3.4) with hm. If u is any solution of (1.3), then Lu=f(u)+hf(u)+m. By the Comparison Principle, uvj in 𝒪j for all j. Proceeding as in the proof of Theorem 3.7, one can show that v=limjvj is a solution of (1.3) on Ω with hm on Ω. Clearly, uv in Ω, and since v is a large solution of Lv=f(v)+m on Ω and Θ*(m)=0, the case considered above applied to v shows that

lim supd(x)0u(x)ϕ(d(x))lim supd(x)0v(x)ϕ(d(x))A*,

where A*=A*(Λ,f,Θ*(h)), and therefore (4.1) holds.

Proof of (4.2).

We consider the function

w*(x):=A*ϕ(d(x)+ρ),xΩρ+.

For an appropriate choice of A*, we show that w* is a sub-solution in Ωρ+ for all ρ<μ, assuming that μ is sufficiently small. We assume first that Ξ*>0. Let us fix ε>0 such that 3ε<Ξ*. Then, by definition, there exists a positive real number A* such that Ξ*/2+3ε/2<g*(A*). That is,

12Ξ*+lim supd(x)+ρ0f(A*ϕ(d(x)+ρ))f(ϕ(d(x)+ρ))<A*-32ε.

Therefore, assuming that μ is sufficiently small, the following holds in Ωρ+ for all 0<ρ<μ:

12Ξ*+f(A*ϕ(d(x)+ρ))f(ϕ(d(x)+ρ))<A*λ-ε.

By shrinking μ if necessary, we can invoke (2.1) to estimate

A*[2F(ϕ(d(x)+ρ))f(ϕ(d(x)+ρ))(|L0d|+𝐛L)+ϕ(d(x)+ρ)f(ϕ(d(x)+ρ))cL]<εfor all xΩρ+ and 0<ρ<μ.

Thus, for any 0<ρ<μ, the following chain of inequalities hold on Ωρ+:

Lw*f(ϕ(d(x)+ρ))[A*-A*2F(ϕ(d(x)+ρ))f(ϕ(d(x)+ρ))(|L0d|+𝐛L)-A*cLϕ(d(x)+ρ)f(ϕ(d(x)+ρ))]f(ϕ(d(x)+ρ))(A*-ε)f(ϕ(d(x)+ρ))(12Ξ*+f(A*ϕ(d(x)+ρ))f(ϕ(d(x)+ρ)))12Ξ*f(ϕ(d(x)+ρ))+f(A*ϕ(d(x)+ρ)).(4.5)

Recall that Ξ*>0. Since h is bounded from above on Ω, we can assume μ is small enough so that

12Ξ*f(ϕ(d(x)+ρ))h(x),xΩρ+.

Thus, for Ξ*>0 and sufficiently small μ>0, we have shown that

Lw*f(w*)+h(x),xΩρ+, for all 0<ρ<μ.

Now let us suppose that Ξ*=0. Then there exists A*>0 such that for ε>0 and d+ρ small, we have

A*-f(A*ϕ(d+ρ))f(ϕ(d+ρ))>-ε,

which can be rewritten as

f(A*ϕ(d+ρ))f(ϕ(d+ρ))-2ε<A*-ε.

Using this in estimate (4.5), we find

Lw*f(A*ϕ(d+ρ))-2εf(ϕ(d+ρ)).(4.6)

By (h1), we recall that -Θ*<Ξ*=0. There is no loss in generality if we assume that Θ*>-. At this point we use condition (h1), that is,

lim supd(x)0h(x)f(ϕ(d))=Θ*<0.(4.7)

On noting that f(ϕ(d+ρ))f(ϕ(d)), and using (4.7), we find

h(x)f(ϕ(d+ρ))12Θ*.

Choosing ε=-Θ*/4 in the above inequality, we find

h(x)-2εf(ϕ(d+ρ)).

Inserting the latter estimate into (4.6), for sufficiently small ρ>0, yields

Lw*f(w*)+h(x),xΩρ+.

In conclusion, we have shown that in either of the cases, the following holds:

Lw*f(w*)+h(x),xΩρ+.

Let B*:=A*ϕ(μ). Note that w*-B*u on Ωρ+ and

L(w*-B*)Lw*f(w*)+hf(w*-B*)+handL(u)=f(u)+hin Ωρ+.

Therefore, w*-B*u in Ωρ+ and

A*u(x)ϕ(d(x)+ρ)+B*ϕ(d(x)+ρ)for xΩρ+.

On letting ρ0+, we see that

A*u(x)ϕ(d(x))+B*ϕ(d(x))on Ωμ.

On recalling that ϕ(d(x)) as d(x)0, we get

A*lim infd(x)0u(x)ϕ(d(x)),

and the proof is complete. ∎

In Lemma 4.2, we required h in (1.3) to be bounded from above in order to get the asymptotic estimate (4.2). Next, we wish to remove this restriction to allow h to be unbounded on Ω. However, the growth of h near the boundary of Ω needs to be constrained in such a way that a Dirichlet problem with zero boundary data is solvable. We use the notation h+(x):=max{h(x),0}.

  • (h3)

    We assume that d2(x)h+(x) is bounded in Ω and that the following Dirichlet problem admits a solution wWloc2,n(Ω)C(Ω¯):

    {Lw=h+(x)in Ω,w=0on Ω.(4.8)

Suppose that hC(Ω) satisfies (h3). Then we note that there exists a positive constant C such that h+(x)Cd-2(x) in Ω. Therefore, we have

0h+(x)f(ϕ(d(x)))Cd2(x)f(ϕ(d(x))),xΩ.(4.9)

By Lemma 2.3 (ii), we note that

limt1(tF(s)-1/2𝑑s)2f(t)=0.(4.10)

Thus, from (4.9) and (4.10), we conclude that

Θ*(h+)=Θ*(h+)=limd(x)0h+(x)f(ϕ(d(x)))=0.

Consequently, we see that

Θ*(h)=Θ*(-h-)andΘ*(h)=Θ*(-h-).(4.11)

The following result complements Lemma 4.2.

Theorem 4.3.

Suppose that hC(Ω) satisfies (h1), (h2) and (h3). Then there exist constants 0<A*A* such that both estimates (4.1) and (4.2) hold for any solution uWloc2,n(Ω) of (1.3).

Proof.

Estimate (4.1) is proved in Lemma 4.2. Thus, it only remains to show (4.2). For j=1,2,, let vj be the solution of

Lvj=f(vj)-min{j,h-}in Ω,vj=jon Ω.

The Comparison Principle shows that {vj} is an increasing sequence. Let w be the unique solution of (4.8). Since w<0 on Ω, we find that

L(vj+w)=f(vj)-min{j,h-}+h+f(vj+w)+hin Ω,vj+w=jon Ω.

Now, if u is any solution of (1.3) in Ω, then again by the Comparison Principle, we find

vj+w(x)u(x)for all j.

Letting j, we obtain

v+wuin Ω,

where

v(x):=limjvj(x),xΩ.

Then v is a large solution of Lv=f(v)-h- on Ω.

Since h satisfies (h1) and (h3), we see that Θ*(-h-)=Θ*(h)<Ξ*. Since, in addition, -h- is bounded from above we invoke Lemma 4.2 (ii) to infer that there exists a constant A*>0 such that v satisfies the asymptotic estimate (4.2). Since w is bounded on Ω, we have

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

and this concludes the proof of (4.2). ∎

The following corollary is noteworthy.

Corollary 4.4.

Let hC(Ω) and suppose that |h| satisfies (h3). If Ξ*<0<Ξ*, then there exist constants 0<A*A* such that for any solution u of (1.3),

A*lim infd(x)0u(x)ϕ(d(x))lim supd(x)0u(x)ϕ(d(x))A*.(4.12)

Proof.

Since |h| satisfies (h3), we recall from (4.11) that Θ*(h)=Θ*(h)=0. Therefore, by hypothesis, h satisfies (h1) and (h2). Thus, the conclusion of the corollary follows from Theorem 4.3. ∎

Remark 4.5.

Suppose that hC(Ω) satisfies d2(x)|h(x)|η(d(x)) for all xΩ, with d(x)δ0, for some δ0, where η:(0,δ0](0,η(δ0)] is an increasing function that satisfies (1.4). According to [1, Theorem 4] or [16, Theorem 5.2], we note that |h| satisfies condition (h3). Therefore, if f satisfies (A1) and is regularly varying at infinity of order q>1, then (3.1), together with Corollary 4.4, shows that any solution u of (1.3) satisfies (4.12) for some constants 0<A*A*.

Remark 4.6.

Suppose that hC(Ω) satisfies (h1), (h2) and (h3). If u¯ is a maximal solution of (1.3), then there exist constants A1 and δ>0 such that for any solution of u of (1.3),

u¯Auin Ωδ.

This follows from Theorem 4.3 with A:=A*/A*1, and it was obtained in [9] for Δu=|u|q-1u+h, q>1, under the assumption that h is bounded from above in Ω.

5 On uniqueness and existence of positive solutions to problem (1.3)

In this section, we continue to assume that all the coefficients of L are bounded in Ω.

Our next result shows that problem (1.3) admits at most one non-negative solution even when h is allowed to be unbounded in Ω.

Theorem 5.1.

Assume that f satisfies (f4), and hC(Ω) satisfies (h1), (h2) and (h3). If f is convex on [0,), then problem (1.3) has at most one non-negative solution in Wloc2,n(Ω).

The proof is omitted since it is similar to that of [18, Theorem 4.4], see also [7, 8, 9, 17]. In particular, we mention [8] in which the authors prove uniqueness to the boundary blow-up problem associated with Δu=f(u) in balls, where f is non-decreasing in (0,) and convex at infinity.

For our next result, we suppose that fC1(). If uWloc2,n(Ω), then f(u)W1,n(Ω). This follows from [12, Theorem 7.8]. Furthermore, we suppose that the coefficients of L are Hölder continuous in Ω. If uWloc2,n(Ω) is a solution of (1.3) with h0, then, since f(u)Wloc1,n(Ω), from the elliptic regularity theory, we conclude that uC2(Ω), see [12, Theorem 9.19].

In Theorem 3.7 we showed that problem (1.3) has a solution. This solution may change sign in Ω. In the next theorem, we consider the existence of a positive solution to (1.3). We use an adaptation of the argument in [9].

Theorem 5.2.

Let 0<β<1. There exists a constant C>0, depending on β and Ω, such that for every hC(Ω) satisfying

supΩd2-β(x)h+(x)C,(5.1)

problem (1.3) admits a positive solution in Wloc2,n(Ω).

Proof.

Let UC2(Ω) be the maximal solution of (1.3) with h0 on Ω (see the remark right after Theorem 5.1). By the tangency principle (see [20, Theorem 2.1.3]), we note that minΩU>0. Let wWloc2,n(Ω)C(Ω¯) be the unique solution of Lw=h+ in Ω with w=0 on Ω. The existence of such a solution is justified by [1, Theorem 4]. Moreover, w<0 in Ω and there exists a positive constant Cβ>0 such that (see [1, Remark 6.2])

supΩ|d-β(x)w(x)|CβsupΩ(d2-β(x)h+(x)).

Therefore, we have

|w(x)|=dβ(x)|d-β(x)w(x)|Cβ(diamΩ)βsupΩd2-β(x)h+(x)CCβ(diamΩ)βfor all xΩ.

We choose C>0 in (5.1) so that

CCβ(diamΩ)β<minΩU(x).

Thus, U+w>0 in Ω, and

L(U+w)=f(U)+h+f(U+w)+hin Ω.

Let {uj} be the sequence of solutions of (3.4) constructed in the proof of Theorem 3.7. Then, by the Comparison Principle, we note that U+wuj in 𝒪j for all j. Therefore, U+wu in Ω, showing that u is a positive solution of (1.3). ∎

A Appendix

We use this appendix to prove two lemmas and make some remarks, which may be of independent interest.

Lemma A.1.

Suppose that f satisfies (A1). If f satisfies the Keller–Osserman condition (A2), then

lim inftF(t)tf(t)12.(A.1)

Proof.

Let

α:=lim inftF(t)tf(t).

Given τ0>0, we see that

ln(F(t)F(τ0))=τ0tf(s)F(s)𝑑s,t>τ0.

That is,

F(t)=F(τ0)exp(τ0tf(s)F(s)𝑑s),t>τ0.(A.2)

Now assume, contrary to (A.1), that α>1/2. We choose τ0>0 so that

F(t)tf(t)12for all tτ0.(A.3)

Using (A.3) in (A.2), we obtain

F(t)F(τ0)exp(2τ0tdss)=F(τ0)τ0-2t2for all tτ0.

This shows that (A2) fails. ∎

Remark A.2.

(i)  We remark that if tt-1f(t) is non-decreasing at infinity, then

lim suptF(t)tf(t)12.

To see this, suppose tt-1f(t) is non-decreasing on (τ0,) for some τ0>0. Then

F(t)=F(τ0)+τ0tf(s)𝑑sF(τ0)+f(t)t0ts𝑑s=F(τ0)+12tf(t),t>τ0.

Therefore, we have

F(t)tf(t)F(τ0)tf(t)+12,t>τ0,

from which the claim follows.

(ii)  The converse of Lemma A.1 is false, as the example f(t)=tlog2(t+1) shows.

Lemma A.3.

Suppose that f satisfies (A1) and (A3). Then Ξ*0.

Proof.

Let α be as in the proof of Lemma A.1. By (A3), we have α>0. Let us fix ρ so that 0<ρ<α. Then there exists τ0:=τ(ρ)>0 such that

F(t)tf(t)ρfor all tτ0.(A.4)

If 0<κ<1, then, using (A.4) in (A.2), we find

F(κt)F(t)=exp(-κttf(s)F(s)𝑑s)κ1/ρfor all tτ0κ.(A.5)

As a consequence of (A.5), for 0<κ<1, we find

κ1/ρF(κt)F(t)2κf(κt)f(t/2),t>τ0κ.(A.6)

Therefore, for 0<κ<1, we have shown that

f(2κt)f(t)12κ(1-ρ)/ρ,tτ02κ.(A.7)

For an arbitrary, but fixed 0<ξ<2, we take κ:=ξ/2 in (A.7) to find

f(ξt)f(t)12(ξ2)(1-ρ)/ρ,tτ0ξ.

Therefore,

lim inftf(ξt)f(t)12(ξ2)(1-ρ)/ρ,0<ξ<2.

Consequently, we get

g*(ξ)Λξ-12(ξ2)(1-ρ)/ρ,0<ξ<2.

Therefore, since 0<ρ<1, we note that the right-hand side tends to 0 as ξ0. This shows that

Ξ*=infξ>0g*(ξ)0,

and the proof is complete. ∎

We complement the above lemma with the following remark.

Remark A.4.

Suppose that f satisfies (A1) and

lim suptF(t)tf(t)<12.

Then Ξ*=-. To see this let

lim suptF(t)tf(t)<σ<12.(A.8)

Let κ>1. Again, using (A.8) in (A.2), we can find some τ0>0 such that

F(κt)F(t)κ1/σ,t>τ0.

Proceeding as in (A.6) we find

f(2κt)f(t)12κ(1-σ)/σ,tτ0.

That is,

f(ξt)f(t)2-1/σξ(1-σ)/σ,t>τ0.

Consequently, we have

g*(ξ)Λξ-2-1/σξ(1-σ)/σ.

Recalling that 1/σ>2, the assertion follows on letting ξ.

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Footnotes

  • 1

    Note that uj+1u* on 𝒪j+1 and hence uj=u*uj+1 on 𝒪j. Therefore, the Comparison Principle applies. 

About the article

Received: 2017-03-17

Revised: 2017-04-05

Accepted: 2017-04-06

Published Online: 2017-06-08


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 517–532, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0065.

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