A Nonlocal Operator Breaking the Keller–Osserman Condition

Raúl Ferreira; Mayte Pérez-Llanos

doi:10.1515/ans-2016-6011

Article Open Access

A Nonlocal Operator Breaking the Keller–Osserman Condition

Raúl Ferreira and Mayte Pérez-Llanos

Published/Copyright: January 10, 2017

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From the journal Advanced Nonlinear Studies Volume 17 Issue 4

Abstract

This work is concerned about the existence of solutions to the nonlocal semilinear problem

{ - ∫ ℝ N J ⁢ ( x - y ) ⁢ ( u ⁢ ( y ) - u ⁢ ( x ) ) ⁢ 𝑑 y + h ⁢ ( u ⁢ ( x ) ) = f ⁢ ( x ) , x ∈ Ω , u = g , x ∈ ℝ N ∖ Ω ,

verifying that limx→∂⁡Ω,x∈Ω⁡u⁢(x)=+∞, known in the literature as large solutions. We find out that the relation between the diffusion and the absorption term is not enough to ensure such existence, not even assuming that the boundary datum g blows up close to ∂⁡Ω. On the contrary, the role to obtain large solutions is played only by the interior source f, which gives rise to large solutions even without the presence of the absorption. We determine necessary and sufficient conditions on f providing large solutions and compute the blow-up rates of such solutions in terms of h and f. Finally, we also study the uniqueness of large solutions.

Keywords: Nonlocal Diffusion; Large Solutions; Keller–Osserman Condition

MSC 2010: 45P05; 35B40; 35J61

1 Introduction

In this work, we analyze the existence of large solutions to the following semilinear nonlocal problem:

(1.1) { - ∫ ℝ N J ⁢ ( x - y ) ⁢ ( u ⁢ ( y ) - u ⁢ ( x ) ) ⁢ 𝑑 y + h ⁢ ( u ⁢ ( x ) ) = f ⁢ ( x ) , x ∈ Ω , u = g , x ∈ ℝ N ∖ Ω .

Here the kernel J is a smooth probability density, f is a continuous function, g∈L1⁢(ℝN∖Ω), and h is an increasing continuous function.

By large solutions we understand solutions satisfying

(1.2) lim x → ∂ ⁡ Ω , x ∈ Ω ⁡ u ⁢ ( x ) = + ∞ .

This interpretation of large solutions was originated by the works of Keller [6] and Osserman [9] who separately proved that the Cauchy problem -Δ⁢u+h⁢(u)=0 has no entire solutions (solutions well defined in the whole ℝN) if h is non-decreasing and verifies

(1.3) ∫ 0 ∞ ( ∫ 0 x h ⁢ ( z ) ⁢ 𝑑 z ) - 1 2 ⁢ 𝑑 x < ∞ .

Moreover, Keller shows that in this case, for any bounded domain Ω, there exists a solution to -Δ⁢u+h⁢(u)=0 in Ω such that (1.2) holds. Condition (1.3) is known as the Keller–Osserman condition and in fact it is a necessary and sufficient condition if in addition h∈C1⁢([0,+∞)) and h⁢(0)=0.

The existence of this kind of solutions has been analyzed by a large number of authors for a wide variety of diffusions and nonlinearities; see for instance the compiling paper by Bandle and Marcus [1], the books by López-Gómez [8] and Véron [11], and the references therein.

In [5], García-Melián and Sabina de Lis use the monotonicity methods developed in [7] to study large solutions to the problem

{ Δ ⁢ u = λ ⁢ f ⁢ ( u ) 1 + 1 | Ω | ⁢ ∫ Ω g ⁢ ( u ) ⁢ 𝑑 x , x ∈ Ω , u = ∞ , x ∈ ∂ ⁡ Ω .

Notice the nonlocal character of the above absorption term. However, the problem under consideration here corresponds to a nonlocal diffusion operator, competing with a local absorption given by the function h⁢(u).

On the other hand, nonlocal diffusion problems have attracted a great interest over the last few years, though within this context of large solutions, the literature is not so rich to our knowledge. In [2], Chen, Felmer and Quaas change the diffusion by the fractional Laplacian and consider the absorption h⁢(u)=|u|p-1⁢u. Precisely, they study the existence of large solutions to

{ - ∫ ℝ N u ⁢ ( x + y ) - u ⁢ ( x ) | y | 2 ⁢ α + N ⁢ 𝑑 y + | u | p - 1 ⁢ u = 0 , x ∈ Ω , u = 0 , x ∈ ℝ N ∖ Ω ,

with α∈(0,1). They prove that if p∈(1+2⁢α,p*⁢(α)), there exists a unique large solution, whose precise asymptotic behavior close to the boundary is given by dist⁢(x,∂⁡Ω)-γ with γ=2⁢α/(p-1).

In [10], Rossi and Topp show the existence and uniqueness of large solutions for the problem

∫ | y | ≤ ρ ⁢ ( x ) u ⁢ ( x + y ) - u ⁢ ( x ) | y | 2 ⁢ α + N ⁢ 𝑑 y + | u | p - 1 ⁢ u = 0 , x ∈ Ω ,

where ρ⁢(x)=Λ⁢dist⁢(x,∂⁡Ω)σ with 0<Λ<1. Since Λ<1, the integration is performed within Ω, thus no condition is needed in the complementary set. They prove that for certain relations of the parameters α, σ, Λ and p there exists a unique large solution, which approaching the boundary behaves as

dist ⁢ ( x , ∂ ⁡ Ω ) - σ ⁢ ( α - 2 ) + 2 p - 1 .

We wish to emphasize here that in both of the mentioned problems, the respective authors consider a singular symmetric kernel, while our kernel is smooth and not necessarily symmetric. Furthermore, the operator associated to our kernel (which is integrable) is not even of differential nature, contrary to what happens for the fractional Laplacian. On the contrary, if J is integrable, we have a zero-order operator competing with a nonlinear term h⁢(u). In our opinion, that is precisely the reason behind the breakage of the Keller–Osserman condition. An interesting question arising here is if being integrable is a necessary and sufficient requirement on J for the rupture of the Keller–Osserman condition. In Section 5, we answer part of the question for symmetric kernels belonging to L1⁢(ℝn).

If we rewrite problem (1.1) in the form

(1.4) - ∫ Ω J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y + u ⁢ ( x ) + h ⁢ ( u ⁢ ( x ) ) = f ⁢ ( x ) + ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y ,

it is clear that the datum g does not play any role for the existence of large solutions. Recall that g∈L1⁢(ℝN∖Ω), hence the last term is bounded, thus not relevant in the analysis of the blow-up. On the other hand, this hypothesis on g is essential to give sense to the equation.

Now, we seek large solutions for the corresponding Keller–Osserman problem

- ∫ Ω J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y + u ⁢ ( x ) + h ⁢ ( u ⁢ ( x ) ) = 0 .

We will see that any continuous large solution must be bounded from below and that u∉L1⁢(Ω). As a result, the convolution term does not make sense and none large solution exists. This is the core of the following theorem.

Theorem 1.1.

Let f be a continuous bounded function, g∈L1⁢(RN∖Ω) and h continuous and increasing. Then there does not exist any continuous large solution for problem (1.1).

According to this theorem, there does not exist any absorption term capable of compensating this kind of nonlocal diffusion, if we wish that (1.2) holds. This fact not only exhibits another important difference concerning the behavior of solutions to this kind of nonlocal problems with respect to models involving nonlocalities of fractional type. It indeed constitutes a notorious novel result within the theory of large solutions.

In consequence, if we look for large solutions to our problem, the function f must be required from now on to satisfy

(1.5) lim x → ∂ ⁡ Ω , x ∈ Ω ⁡ f ⁢ ( x ) = + ∞ .

Let us now determine a necessary and sufficient condition for our problem that replaces (1.3). For this purpose, it will be useful to introduce the function

(1.6) H ( s ) = { h - 1 ⁢ ( s ) if h ⁢ ( s ) s → ∞ as s → ∞ , s if h ⁢ ( s ) s is bounded,

to unify the subsequent notation.

Turning back our attention to equation (1.4), it is clear that the terms u and h⁢(u) compete for the role of providing large solutions. More concisely, we show the following result.

Theorem 1.2.

Let g∈L1⁢(RN∖Ω), let f be a continuous function verifying (1.5), let h be a continuous and increasing function, and let H be defined as in (1.6). Then problem (1.1) admits large solutions if and only if

(1.7) ∫ Ω H ⁢ ( f ⁢ ( x ) ) ⁢ 𝑑 x ≤ C .

Remark 1.3.

We note that, contrarily to what occurs for local diffusions, the presence of the term u in the equation allows us to obtain large solutions, even when the absorption is null or bounded. We would like to emphasize the surprising difference with respect to its local linear counterpart -Δ⁢u⁢(x)=f⁢(x) if x∈Ω and u=∞ on ∂⁡Ω, for which no solution exists under assumption (1.5).

Our second result deals with the blow-up rate for large solutions obtained as an approximation procedure. This approach returns the minimal large solution u in the sense that any large solution v satisfies v≥u. Regarding the maximal solution, for local problems it is usually constructed as the limit of large solutions to the problem settled in a certain subdomain Ωε⊂⊂Ω. However, in our case f is bounded in Ωε and by Theorem 1.1 no large solution exists in Ωε. For this reason, we need to consider a family of functions fε that blow up on the boundary ∂⁡Ωε to guarantee the existence of large solutions for the approximating problem; see Remark 3.7 below.

Theorem 1.4.

Let u be a large solution. Then there exist two positive constants such that

(1.8) H ⁢ ( δ ⁢ f ⁢ ( x ) ) - C 1 ≤ u ⁢ ( x ) ≤ H ⁢ ( f ⁢ ( x ) ) + C 2 ,

where the parameter δ is given by

δ = { 1 if h ⁢ ( s ) s → ∞ as s → ∞ , 1 1 + C if h ⁢ ( s ) s < C .

Now we focus on the uniqueness of large solutions. Accordingly to the hypotheses of h and J, we perform two different arguments to accomplish the uniqueness result.

Theorem 1.5.

In the hypothesis of Theorem 1.2, if either f≥0 and the function s→h⁢(s)/s is unbounded and nondecreasing, or J is a symmetric function, then the large solution is unique.

Remark 1.6.

A typical example of explosive source is f=dist⁢(x,∂⁡Ω)-γ. In this case, if we take h⁢(s)=|s|p-1⁢s, by condition (1.7) there exist large solutions if and only if γ<r:=max⁡(1,p). Furthermore, close to the boundary large solutions behave as

u ∼ dist ⁢ ( x , ∂ ⁡ Ω ) - γ r .

However, if we choose h⁢(s)=es, there are no large solutions, since condition (1.7) reads as

∫ Ω H ⁢ ( f ⁢ ( x ) ) ⁢ 𝑑 x = - γ ⁢ ∫ Ω log ⁡ ( dist ⁢ ( x , ∂ ⁡ Ω ) ) ⁢ 𝑑 x = ∞ .

This last example illustrates another important difference with respect to the (local) Laplacian, which admits large solutions with exponential absorption.

Remark 1.7.

Following step by step the proofs of the above theorems, a similar result can be established if we assume that f blows up only at a certain Γ⊂∂⁡Ω.

In this case there exist large solutions if and only if (1.7) holds. Moreover, u blows up only at Γ, with blow-up rate given also by (1.8).

Let us conclude this introduction by specifying which notion of solution we are using along this paper.

Definition 1.8.

We say that u is a classical solution of (1.1) if u∈C⁢(Ω) and if it satisfies (1.1) pointwise.

This paper is organized as follows: In Section 2 we show that there does not exist any large solution to (1.1) whenever f is bounded, namely we prove Theorem 1.1. Section 3 includes some preliminary results on existence and comparison of solutions when f is bounded. Subsequently, we prove Theorem 1.2 with the use of approximation arguments. Section 4 contains our uniqueness results, and finally Section 5 is devoted to extend the breakage of the Keller–Osserman condition to integrable and symmetric kernels.

2 Breakage of the Keller–Osserman Condition

We devote this section to prove Theorem 1.1. First, we make the following observations:

Any classical large solution is bounded from below. Notice that it is a continuous function diverging to infinity as x approaches the boundary.
Moreover, if f is a bounded function, then none classical large solution of (1.1) belongs to L1⁢(Ω). In fact, assuming that u∈L1⁢(Ω), from (1.4) we infer that

u ⁢ ( x ) + h ⁢ ( u ⁢ ( x ) ) = f ⁢ ( x ) + ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y + ∫ Ω J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y

≤ ∥ f ∥ L ∞ ⁢ ( Ω ) + ∥ J ∥ L ∞ ⁢ ( ℝ N ) ⁢ ∥ g ∥ L 1 ⁢ ( ℝ N ∖ Ω ) + ∥ J ∥ L ∞ ⁢ ( ℝ N ) ⁢ ∥ u ∥ L 1 ⁢ ( Ω ) .

Since the function s→s+h⁢(s) is increasing and it goes to infinity, the above estimate makes the occurrence of (1.2) impossible.

Ad contrarium, let us admit that a classical large solution u exists. From (ii) we know that necessarily u∉L1⁢(Ω). This fact means that there must exist a point x0∈∂⁡Ω such that

∫ Ω ∩ B δ ⁢ ( x 0 ) u ⁢ ( y ) ⁢ 𝑑 y = ∞

for some δ>0. Let us fix x∈Ω∩Bδ⁢(x0) and δ small enough ensuring that

(2.1) J ⁢ ( x - y ) ≥ α > 0 , y ∈ Ω ∩ B δ ⁢ ( x 0 ) .

Taking into account that u is bounded from below, we obtain that

∫ Ω J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y = ∫ Ω ∖ ( Ω ∩ B δ ⁢ ( x 0 ) ) J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y + ∫ Ω ∩ B δ ⁢ ( x 0 ) J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 y

≥ - C ⁢ ∥ J ∥ L 1 ⁢ ( ℝ n ) + α ⁢ ∫ Ω ∩ B δ ⁢ ( x 0 ) u ⁢ ( y ) ⁢ 𝑑 y = ∞ .

By (1.4), this implies that u⁢(x)=∞, hence no large solution exists. ∎

Remark 2.1.

Notice that the argument above does not require the continuity of J. Then if the kernel J∈L1⁢(ℝn) satisfies (2.1), any large solution must belong to L1⁢(Ω).

3 Existence of Large Solutions

From now on, we will turn our attention to precise necessary and sufficient conditions for the existence of large solutions. This aim will be accomplished by approximation arguments. Therefore, we start by showing the existence of (bounded) solutions when f is bounded via the sub-supersolution method. Thus, we first prove a comparison result.

Lemma 3.1.

Let u¯ and u¯ be a classical bounded supersolution and subsolution, respectively. Then u¯≤u¯.

Proof.

Defining w⁢(x)=u¯⁢(x)-u¯⁢(x), we get that

- ∫ Ω J ⁢ ( x - y ) ⁢ ( w ⁢ ( y ) - w ⁢ ( x ) ) ⁢ 𝑑 y + w ⁢ ( x ) ⁢ ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ 𝑑 y + h ⁢ ( u ¯ ⁢ ( x ) ) - h ⁢ ( u ¯ ⁢ ( x ) ) ≤ 0 .

In order to get a contradiction, we define K=supΩ⁡w and assume that K>0.

(i) If there exists x0∈Ω such that w⁢(x0)=K, then, evaluating the previous expression at x0, we observe that the first two terms are non-negative. Hence h⁢(u¯⁢(x0))-h⁢(u¯⁢(x0))≤0. The application of the monotonicity of h leads to the desired contradiction.

(ii) Now we admit that there exists a sequence xn→x∈∂⁡Ω such that w⁢(xk)→K. Using the dominate convergence theorem and the fact that w is bounded, we can pass to the limit in the previous expression to infer

- ∫ Ω J ⁢ ( x 0 - y ) ⁢ ( w ⁢ ( y ) - K ) ⁢ 𝑑 y + K ⁢ ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ 𝑑 y + h ⁢ ( u ¯ ⁢ ( x 0 ) ) - h ⁢ ( u ¯ ⁢ ( x 0 ) ) ≤ 0 .

The contradiction now follows by arguing as in the previous step. ∎

Remark 3.2.

We point out that the comparison lemma also holds even assuming that u¯→∞ as x→∂⁡Ω.

The solution to the following problem will be useful to construct suitable sub- and supersolutions to (1.1) when f is bounded.

Lemma 3.3.

There exists a classical non-negative bounded solution of

(3.1) - ∫ Ω J ⁢ ( x - y ) ⁢ w ⁢ ( y ) ⁢ 𝑑 y + w ⁢ ( x ) = 1 .

Proof.

The existence is obtained by the sub-supersolution method. First, we note that w¯=0 is a subsolution. To find a positive supersolution we consider two cases.

(i) J has compact support. Define w¯=k⁢ϕ1, where ϕ1 is the first eigenfunction of the operator

ℒ = - ∫ Ω J ⁢ ( x - y ) ⁢ u ⁢ ( y ) ⁢ 𝑑 x + u ⁢ ( x )

which is continuous and strictly positive in Ω¯; see [4]. Since ϕ⁢(x)>0 in Ω¯, we can take k large enough to get

- ∫ Ω J ⁢ ( x - y ) ⁢ w ¯ ⁢ ( y ) ⁢ 𝑑 y + w ¯ ⁢ ( x ) = λ 1 ⁢ k ⁢ w ¯ ⁢ ( x ) > 1 .

(ii) If the support of J is unbounded, we take w¯=k. Notice that since Ω is bounded,

sup x ∈ Ω ⁡ ∫ Ω J ⁢ ( x - y ) ⁢ 𝑑 y = η < 1 .

Then if k is large enough, we have

- ∫ Ω J ⁢ ( x - y ) ⁢ w ¯ ⁢ ( y ) ⁢ 𝑑 y + w ¯ ⁢ ( x ) = - η ⁢ k + k ≥ 1 . ∎

Theorem 3.4.

Let f be a continuous bounded function. Then there exists a bounded classical solution of (1.1).

Proof.

Let w be the function given in the previous lemma. We claim that the functions

u ¯ ⁢ ( x ) = λ ⁢ w ⁢ ( x ) , u ¯ ⁢ ( x ) = - μ ⁢ w ⁢ ( x ) ,

with λ and μ being positive parameters, are a supersolution and a subsolution to problem (1.1), respectively.

Moreover, clearly u¯≤u¯, and hence the sub-supersolution method (see [3]) guarantees the existence of a solution such that

u ¯ ⁢ ( x ) ≤ u ⁢ ( x ) ≤ u ¯ ⁢ ( x ) .

Furthermore, the continuity of f and (1.4) ensure that u+h⁢(u) is continuous. Thus, u is continuous.

We now proceed with the claim. To see that u¯ is a supersolution, take into account that

- ∫ Ω J ⁢ ( x - y ) ⁢ u ¯ n ⁢ ( y ) ⁢ 𝑑 y + u ¯ n ⁢ ( x ) + h ⁢ ( u ¯ n ⁢ ( x ) ) = λ + h ⁢ ( λ ⁢ w ⁢ ( x ) ) ≥ λ + h ⁢ ( 0 ) ,

while on the other hand

f ⁢ ( x ) + ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y ≤ ∥ f ∥ L ∞ ⁢ ( Ω ) + ∥ J ∥ L ∞ ⁢ ( ℝ N ) ⁢ ∥ g ∥ L 1 ⁢ ( ℝ N ∖ Ω ) .

Taking λ large enough, we obtain that u¯ is a supersolution. The fact that u¯ is a subsolution follows in the same way. ∎

We are almost ready to characterize the existence of large solutions. Let us first state a regularity result, which follows by similar arguments as in the proof of Theorem 1.1.

Lemma 3.5.

If u is a large solution of (1.1), then u∈L1⁢(Ω).

Proof of Theorem 1.2.

In order to prove the existence of a large solution, let us perform the truncation

f n ( x ) = { f ⁢ ( x ) if ⁢ f ⁢ ( x ) ≤ n , n if ⁢ f ⁢ ( x ) > n ,

on the function f and define un as a solution to

(3.2) - ∫ Ω J ⁢ ( x - y ) ⁢ u n ⁢ ( y ) ⁢ 𝑑 y + u n ⁢ ( x ) + h ⁢ ( u n ⁢ ( x ) ) = f n ⁢ ( x ) + ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y .

Note that {un} is a family of continuous and bounded functions. Furthermore, by the comparison stated in Lemma 3.1 we have that {un} is an increasing family in n.

In fact, we will see that un is uniformly bounded from above. With this purpose in mind we construct a family of supersolutions to (3.2), which are bounded uniformly in n.

Declare

u ¯ n ⁢ ( x ) := H ⁢ ( f n ⁢ ( x ) ) + λ ⁢ w ,

where λ is a positive parameter, w is the solution to (3.1) and H is defined in (1.6). Let us show that u¯n is the desired supersolution, namely

(3.3) - ∫ Ω J ⁢ ( x - y ) ⁢ u ¯ n ⁢ ( y ) ⁢ 𝑑 y + u ¯ n ⁢ ( x ) + h ⁢ ( u ¯ n ⁢ ( x ) ) ≥ f n ⁢ ( x ) + ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y ,

by choosing λ sufficiently large. Indeed, assumption (1.7) yields

∫ Ω J ⁢ ( x - y ) ⁢ H ⁢ ( f n ⁢ ( y ) ) ⁢ 𝑑 y ≤ ∥ J ∥ L ∞ ⁢ ( Ω ) ⁢ ∫ Ω | H ⁢ ( f n ⁢ ( y ) ) | ⁢ 𝑑 y ≤ K .

Recalling that g∈L1⁢(ℝN∖Ω), we have that (3.3) is fulfilled if

- K + λ + H ⁢ ( f n ⁢ ( x ) ) + h ⁢ ( H ⁢ ( f n ⁢ ( x ) ) + λ ⁢ w ⁢ ( x ) ) ≥ f n ⁢ ( x ) + C g .

Whenever x is taken far away from the boundary, this inequality is trivial for λ large and independent of n (notice that fn⁢(x)=f⁢(x) for n large and x away from ∂⁡Ω). As x approaches the boundary, notice that

H ⁢ ( f n ⁢ ( x ) ) + h ⁢ ( H ⁢ ( f n ⁢ ( x ) ) + λ ⁢ w ⁢ ( x ) ) - f n ⁢ ( x ) ≥ 0 ,

thus the above inequality holds by taking λ>Cg+K.

Furthermore, applying the comparison principle, we infer that

u n ⁢ ( x ) ≤ u ¯ n ⁢ ( x ) ≤ u ¯ := H ⁢ ( f ⁢ ( x ) ) + λ ⁢ w .

As a result, the increasing family {un} is uniformly bounded by u¯, and we can define the pointwise limit

u ⁢ ( x ) = lim n → ∞ ⁡ u n ⁢ ( x ) .

Monotone convergence returns easily that u is a solution of (1.1) verifying

(3.4) u ≤ H ⁢ ( f ⁢ ( x ) ) + λ ⁢ w .

To see that, indeed, u is a large solution, we compare it with an appropriate subsolution. Define

u ¯ n ⁢ ( x ) := H ⁢ ( δ ⁢ f n ⁢ ( x ) ) - λ ⁢ w ⁢ ( x ) ,

where w is once more the nonnegative solution to problem (3.1) and δ>0 will be conveniently chosen.

In order to show that u¯n is a subsolution, we need to verify

(3.5) - ∫ Ω J ⁢ ( x - y ) ⁢ u ¯ n ⁢ ( y ) ⁢ 𝑑 y + u ¯ n ⁢ ( x ) + h ⁢ ( u ¯ n ⁢ ( x ) ) - f n ⁢ ( x ) - ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ g ⁢ ( y ) ⁢ 𝑑 y ≤ 0 .

The first two terms can be estimated thanks to the uniform boundedness of fn from below. In consequence, there exists a constant K independent on n such that

- ∫ Ω J ⁢ ( x - y ) ⁢ u ¯ n ⁢ ( y ) ⁢ 𝑑 y + u ¯ n ⁢ ( x ) ≤ K - λ + H ⁢ ( δ ⁢ f n ⁢ ( x ) ) .

With respect to the third term recall that h is increasing, hence h⁢(u¯n⁢(x))≤h⁢(H⁢(δ⁢fn⁢(x))). Recalling the former considerations and the fact that g∈L1⁢(ℝN∖Ω), we see that inequality (3.5) is equivalent to

C g + K - λ + H ⁢ ( δ ⁢ f n ⁢ ( x ) ) + h ⁢ ( H ⁢ ( δ ⁢ f n ⁢ ( x ) ) ) - f n ⁢ ( x ) ≤ 0 .

As before, by choosing the parameter λ large enough, the above inequality holds away from the boundary. Close to the boundary we claim that it is possible to find an appropriate value for δ>0, independent of n, such that

H ⁢ ( δ ⁢ f n ⁢ ( x ) ) + h ⁢ ( H ⁢ ( δ ⁢ f n ⁢ ( x ) ) ) - f n ⁢ ( x ) ≤ 0 .

If we take λ>Cg+K, the function u¯n is a subsolution and accordingly, by comparison, un≥u¯n. Since {un} is an increasing sequence, we indeed obtain that

(3.6) u ⁢ ( x ) ≥ u n ⁢ ( x ) ≥ u ¯ n ⁢ ( x ) .

Now we prove the claim arguing by contradiction. Suppose that

lim sup s → ∞ ⁡ ( H ⁢ ( δ ⁢ s ) + h ⁢ ( H ⁢ ( δ ⁢ s ) ) - s ) = 2 ⁢ L > 0 ,

which implies that there exists a sequence sj→∞ such that

(3.7) H ⁢ ( δ ⁢ s j ) + h ⁢ ( H ⁢ ( δ ⁢ s j ) ) - s j > L .

If H⁢(s)=s, the above inequality reads as

h ⁢ ( δ ⁢ s j ) - ( 1 - δ ) ⁢ s j ≥ L ,

or equivalently

h ⁢ ( δ ⁢ s j ) δ ⁢ s j - 1 - δ δ ≥ L δ ⁢ s j .

Since h⁢(s)/s<C, we get the desired contradiction by choosing δ=1/(1+C). Then by comparison

u n ⁢ ( x ) ≥ 1 1 + C ⁢ f n ⁢ ( x ) - λ ⁢ w ⁢ ( x ) .

Passing to the limit in (3.6), we obtain

(3.8) u ⁢ ( x ) ≥ 1 1 + C ⁢ f ⁢ ( x ) - λ ⁢ w ⁢ ( x ) .

If on the contrary H⁢(s)=h-1⁢(s), inequality (3.7) can be expressed as

h - 1 ⁢ ( δ ⁢ s j ) - ( 1 - δ ) ⁢ s j ≥ L .

Recall that h-1⁢(t)/t→0 as t goes to infinity, which contradicts the previous inequality by taking δ∈(0,1). The comparison result implies that

u n ⁢ ( x ) ≥ h - 1 ⁢ ( δ ⁢ f n ⁢ ( x ) ) - λ ⁢ w ⁢ ( x ) for ⁢ δ ∈ ( 0 , 1 ) .

Letting δ→1, we obtain the inequality

u ⁢ ( x ) ≥ u n ⁢ ( x ) ≥ h - 1 ⁢ ( f n ⁢ ( x ) ) - λ ⁢ w ⁢ ( x ) ,

which as n→∞ reads as

(3.9) u ⁢ ( x ) ≥ h - 1 ⁢ ( f ⁢ ( x ) ) - λ ⁢ w ⁢ ( x ) .

We conclude the proof by looking for a nonexistence result. Arguing by contradiction, if we assume that there exists a large solution, the comparison lemma (see Remark 3.2) implies that the lower estimate (3.6) holds. Thus, if (1.7) does not occur, then u∉L1⁢(Ω) and no large solution exists; see Lemma 3.5. ∎

Remark 3.6.

Let us observe that if v is a large solution, then un≤v and by passing to the limit we have u≤v. Thus, u is the minimal large solution.

Proof of Theorem 1.4.

For the minimal large solution, estimate (1.8) is deduced directly from (3.4), (3.8) and (3.9).

For a general large solution v∈L1⁢(ℝn) the lower estimate follows by the definition of the minimal large solution

v ≥ u ≥ H ⁢ ( δ ⁢ f ) - C 1 .

In order to prove the upper estimate we define

Ω ε = { x ∈ Ω : dist ⁢ ( x , ∂ ⁡ Ω ) > ε }

and fε∈C⁢(Ωε) such that fε⁢(x)≥f⁢(x) in Ωε, fε→f as ε→0 and

lim x → ∂ ⁡ Ω ε , x ∈ Ω ε ⁡ f ε ⁢ ( x ) = + ∞ , ∫ Ω ε H ⁢ ( f ε ⁢ ( x ) ) ⁢ 𝑑 x ≤ K < ∞ .

We consider the problem

(3.10) - ∫ Ω ε J ⁢ ( x - y ) ⁢ u ε ⁢ ( y ) ⁢ 𝑑 y + u ε ⁢ ( x ) + h ⁢ ( u ε ⁢ ( x ) ) = f ε ⁢ ( x ) + ∫ ℝ n ∖ Ω ε J ⁢ ( x - y ) ⁢ v ⁢ ( y ) ⁢ 𝑑 y .

Since v∈L1⁢(ℝn) and

∫ ℝ n ∖ Ω n J ⁢ ( x - y ) ⁢ v ⁢ ( y ) ⁢ 𝑑 y ≤ ∥ J ∥ L ∞ ⁢ ( ℝ n ) ⁢ ( ∥ v ∥ L 1 ⁢ ( Ω ) + ∥ g ∥ L 1 ⁢ ( ℝ n ∖ Ω ) ) := C v ,

according to Theorem 1.2, if we take λ>K+Cv, there exists a minimal solution of (3.10) satisfying

{ u ε ⁢ ( x ) ≤ H ⁢ ( f ε ⁢ ( x ) ) + λ ⁢ w ⁢ ( x ) , x ∈ Ω ε , u ⁢ ( x ) = v ⁢ ( x ) , x ∈ ℝ n ∖ Ω ε .

Furthermore, since fε⁢(x)≥f⁢(x) in Ωε, it is easy to see that v is indeed a subsolution to (3.10) in Ωε. Moreover, v is bounded in Ωε. Hence, the comparison principle (see Remark 3.2) implies that v⁢(x)≤uε⁢(x), and then

v ⁢ ( x ) ≤ H ⁢ ( f ε ⁢ ( x ) ) + λ ⁢ w ⁢ ( x ) in ⁢ Ω ε .

Since H⁢(fε)→H⁢(f) as ε→0, passing to the limit in the above inequality shows that v verifies (1.8). ∎

Remark 3.7.

Taking

f ε 1 ⁢ ( x ) ≥ f ε 2 ⁢ ( x ) + ∥ J ∥ L ∞ ⁢ ( ℝ n ) ⁢ K ⁢ | Ω ε 2 ∖ Ω ⁢ ε 1 | for ⁢ ε 1 > ε 2 ,

we can easily prove that uε is decreasing in ε. Thus, the maximal large solution can be obtained as the limit of uε.

4 Uniqueness

We devote this part of the work to the analysis of uniqueness of large solutions. We start by treating the case f≥0 and h⁢(s)/s being unbounded and non-decreasing.

Proof of Theorem 1.5 (i).

Suppose that u, v are two solutions to problem (1.1). Denote

𝒜 := { x ∈ Ω : u ⁢ ( x ) > v ⁢ ( x ) } ,

and assume that 𝒜≠∅. Thus, there exists k>1 for which the set

𝒜 k := { x ∈ Ω : w k ⁢ ( x ) := u ⁢ ( x ) - k ⁢ v ⁢ ( x ) > 0 } ≠ ∅ .

The blow-up rates in Theorem 1.4 imply that

lim x → ∂ ⁡ Ω , x ∈ Ω ⁡ w k ⁢ ( x ) H ⁢ ( f ⁢ ( x ) ) = 1 - k < 0 .

Hypothesis (1.5) guarantees that H⁢(f)>0 close to boundary (indeed, H⁢(f)=+∞ on ∂⁡Ω). Accordingly, wk<0 approaching the boundary, thus 𝒜k⊂⊂Ω. Furthermore, there exists x^∈Ω such that wk⁢(x^)=max𝒜k¯⁡wk. The fact that wk≤0 in 𝒜kc entails that x^∈𝒜k and then wk⁢(x^)=maxℝN⁡wk.

On the other hand, recall that h⁢(s) is increasing and we are considering that h⁢(s)/s is nondecreasing. In this case there holds

k ⁢ h ⁢ ( v ⁢ ( x ) ) ≤ h ⁢ ( k ⁢ v ⁢ ( x ) ) ≤ h ⁢ ( u ⁢ ( x ) ) , x ∈ 𝒜 k .

Evaluate now at x^ the equation for wk and use the positivity of f to get

- ∫ ℝ N J ⁢ ( x ^ - y ) ⁢ ( w k ⁢ ( y ) - w k ⁢ ( x ^ ) ) ⁢ 𝑑 y = - h ⁢ ( u ⁢ ( x ^ ) ) + k ⁢ h ⁢ ( v ⁢ ( x ^ ) ) + ( 1 - k ) ⁢ f ⁢ ( x ^ ) ≤ 0 .

However, x^ is a point of maximum, hence

- ∫ ℝ N J ⁢ ( x ^ - y ) ⁢ ( w k ⁢ ( y ) - w k ⁢ ( x ^ ) ) ⁢ 𝑑 y ≥ 0 .

These inequalities imply that wk is a constant function, which is a contradiction to the fact that wk⁢(x^)>0 and wk<0 close to the boundary. ∎

If we assume that the kernel J is symmetric, we can perform an integration by parts in our equation, which allows to prove the uniqueness without extra hypotheses on h and f.

Proof of Theorem 1.5 (ii).

Recall that by Lemma 3.5 any large solution must belong to L1⁢(Ω). The uniqueness is then a direct consequence of the following version of the comparison principle for sub- and supersolutions in L1⁢(Ω). ∎

Lemma 4.1.

Assume that J is symmetric. Let u¯ and u¯ be a classical L1⁢(Ω) supersolution and subsolution of (1.1), respectively. Then u¯≤u¯.

Proof.

The function w⁢(x)=u¯⁢(x)-u¯⁢(x) satisfies

- ∫ Ω J ⁢ ( x - y ) ⁢ ( w ⁢ ( y ) - w ⁢ ( x ) ) ⁢ 𝑑 y + w ⁢ ( x ) ⁢ ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ 𝑑 y + h ⁢ ( u ¯ ⁢ ( x ) ) - h ⁢ ( u ¯ ⁢ ( x ) ) ≤ 0 ,

Multiplying by zM⁢(x)=min⁡{w+⁢(x),M}, where w+=max⁡{0,w}, and integrating in Ω, we obtain

I 1 + I 2 + I 3 ≤ 0

with

I 1 = - ∫ Ω z M ⁢ ( x ) ⁢ ∫ Ω J ⁢ ( x - y ) ⁢ ( w ⁢ ( y ) - w ⁢ ( x ) ) ⁢ 𝑑 y ⁢ 𝑑 x ,

I 2 = ∫ Ω z M ⁢ ( x ) ⁢ w ⁢ ( x ) ⁢ ∫ ℝ N ∖ Ω J ⁢ ( x - y ) ⁢ 𝑑 y ⁢ 𝑑 x ,

I 3 = ∫ Ω ( h ⁢ ( u ¯ ⁢ ( x ) ) - h ⁢ ( u ¯ ⁢ ( x ) ) ) ⁢ z M ⁢ ( x ) ⁢ 𝑑 x .

Using the hypothesis that J is symmetric and since zM∈L∞⁢(Ω), we get that

I 1 = 1 2 ⁢ ∫ Ω ∫ Ω J ⁢ ( x - y ) ⁢ ( w ⁢ ( y ) - w ⁢ ( x ) ) ⁢ ( z M ⁢ ( y ) - z M ⁢ ( x ) ) ⁢ 𝑑 y ⁢ 𝑑 x .

Furthermore, we now recall that the integrands in I2 and I3 are nonnegative, as well as the function T⁢(x,y)=(w⁢(y)-w⁢(x))⁢(zM⁢(y)-zM⁢(x)). As a result, the three integrals above must vanish. From I3=0 we infer that zM≡0, hence w+≡0. This implies that w≤0 as desired. ∎

5 Breakage of the Keller–Osserman Condition for Integrable Symmetric Kernels

We conclude this work by showing that even when we allow the kernel to be singular at the origin it is still not possible to reach a balance between this nonlocal diffusion and any absorption term to obtain large solutions without the action of an explosive source.

More precisely, we consider problem (1.1) with a probability density which is symmetric, singular at the origin and satisfies J⁢(x)>α>0 in a small ball centered at the origin.

As we already observed in Section 2, if a large solution exists, it must be in L1⁢(Ω). However, since J is just integrable, the term J*u in equation (1.1) is not necessarily continuous, hence the solution is not automatically continuous.

On the other hand, since J*u∈L1⁢(Ω), if we admit that f∈L∞⁢(Ω), this yields that h⁢(u)∈L1⁢(Ω). Arguing now as in Lemma 4.1 we obtain a comparison principle, and consequently the uniqueness of L1⁢(Ω) solutions.

At this stage, it is not difficult to see that if h⁢(s)→±∞ as s→±∞, then u¯=-K and u¯=K are a sub- and a supersolution to problem (1.1), respectively. The sub-supersolutions method guarantees the existence of a bounded weak solution. By uniqueness, no large solution exists.

Communicated by Julian Lopez Gomez

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2014-53037-P

Funding source: Agencia Nacional de Promoción Científica y Tecnológica

Award Identifier / Grant number: PICT2014-1771

Funding source: Consejo Nacional de Investigaciones Científicas y Técnicas

Award Identifier / Grant number: PIP 11220150100032CO

Funding statement: The first author is supported by project MTM2014-53037-P (Ministerio de Economía y Competitividad, Spain). The second author is supported by projects ANPCYT PICT2014-1771 and CONICET PIP 11220150100032CO (Argentine).

Acknowledgements

This research was accomplished while the second author was visiting Universidad Complutense de Madrid. She feels really grateful to many people of this institution for their hospitality and nice atmosphere. The authors would like to express their gratitude to the referees for some interesting suggestions that improved and enhanced the former version of this manuscript.

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Received: 2016-09-14

Revised: 2016-11-18

Accepted: 2016-11-28

Published Online: 2017-01-10

Published in Print: 2017-10-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2016-6011

Keywords for this article

Nonlocal Diffusion; Large Solutions; Keller–Osserman Condition

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