Uniqueness of large solutions for non-monotone nonlinearities

Nonlinear Analysis: Real World Applications 47 (2019) 291–305

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

Uniqueness of large solutions for non-monotone nonlinearities✩ Julián López-Gómez a ,∗, Luis Maire b a

Instituto de Matemática Interdisciplinar (IMI), Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain b Matemática Aplicada, Ciencia e Ingeniería de Materiales y Tecnología Electrónica, Universidad Rey Juan Carlos, Calle Tulipán s/n, 28933-Móstoles, Madrid, Spain

article

info

Article history: Received 1 March 2018 Received in revised form 12 November 2018 Accepted 13 November 2018 Available online xxxx

abstract In this paper we construct a general class of non-monotone positive nonlinearities, f (u), for which the differential equation u′′ = f (u) has a unique large solution in (−T, T ). In most of the available literature it is required f (u) to be strictly increasing in order to establish the uniqueness of the large solution. © 2018 Elsevier Ltd. All rights reserved.

Keywords: Large positive solutions Uniqueness Multiplicity Dynamical approach

1. Introduction The main goal of this paper is analyzing the existence and uniqueness of positive solutions for the singular boundary value problem ⎧ ⎨ u′′ = f (u), t ∈ [0, T ), (1.1) ⎩ u′ (0) = 0, u(T ) = +∞, where T > 0 and f ∈ C 1 [0, +∞) satisfies f (0) = 0. So, 0 is a constant solution of u′′ = f (u). By reflection around t = 0, these solutions provide us with the positive large solutions of the singular problem ⎧ ⎨ u′′ = f (u), t ∈ (−T, T ), (1.2) ⎩ u(−T ) = u(T ) = +∞. ✩ Supported by the Ministry of Economy and Competitiveness under Research Grant MTM2015-65899-P and by the IMI of Complutense University. ∗ Corresponding author. E-mail addresses: [email protected] (J. López-Gómez), [email protected] (L. Maire).

https://doi.org/10.1016/j.nonrwa.2018.11.003 1468-1218/© 2018 Elsevier Ltd. All rights reserved.

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By a positive large solution of (1.1) it is meant any positive solution in [0, T ) such that u′ (0) = 0 and lim u(t) = +∞. t↑T

Although the singular problem (1.2) has been widely studied in the literature, almost all available results imposed f ≥ 0 in [0, ∞), or f ≥ 0 for sufficiently large u (see the results in our list of references, [1–25]). In this paper we are not imposing any special restriction on the sign of f , except occasionally. As for arbitrary f (u) the existence and multiplicity of positive solutions of (1.1) might depend on the length of the interval, T , it is very natural to analyze the existence of positive explosive solutions of the associated Cauchy problem ⎧ ⎨ u′′ = f (u), (1.3) ⎩ u(0) = x > 0, u′ (0) = 0, where x > 0 is regarded as parameter. Since f ∈ C 1 [0, +∞), by the main existence theorem for C 1 nonlinearities, it becomes apparent that, for every x > 0, there exists a maximal existence time Tmax (x) ∈ (0, +∞] such that the initial value problem (1.3) possesses a maximal positive solution, u(t), t ∈ [0, Tmax (x)). Moreover, since f (0) = 0, by continuous dependence it is apparent that lim Tmax (x) = +∞. x↓0

(1.4)

Consequently, much within the spirit of [18, Sect. 2], the unique solution of (1.3) provides us with a solution of the singular problem (1.1) if, and only if, ∫ θ { } x ∈ D := x > 0 : f (x) > 0 and f > 0 for all θ > x (1.5) x

and Tmax (x) < +∞. In this case, according to [18, Lem. 2.2], the maximal existence time admits the following representation ∫ +∞ dθ 1 √∫ Tmax (x) = √ , x ∈ D. (1.6) θ 2 x f x However, in spite of the huge available literature on large solutions, there are still many open questions concerning some very hidden properties of Tmax as a function of f . Probably, because the authors of most of the existing studies in this field, strongly influenced by the pioneering works of J. B. Keller [11] and R. Osserman [21], have adopted an extremely narrow point of view requiring the function f to be increasing, plus some special growth rate at infinity, in order to get the uniqueness of the solution of the singular problem ⎧ ⎨ ∆u = f (u) in Ω , ⎩ u = +∞ on ∂Ω , where Ω ⊂ RN , N ≥ 1, is a bounded smooth domain. Actually, it has not been until very recently, in [18], that some examples of non-increasing (pulse-type) positive nonlinearities, f (u), for which multiplicity of large solutions of (1.2) can occur have been given. Rather astonishingly, for every integer n ≥ 1, there exists a positive f (u) for which (1.1) possesses, at least, 2n + 1 solutions. A. Aftalion, M. Del Pino and R. Letelier [1] had analyzed the role played by the change of sign of f (u) on the number of positive large solutions of u′′ = f (u), but this is a problem of a very different nature, as discussed by J. L´opez-G´omez, A. Tellini and F. Zanolin [19], where it was established that the number of large positive solutions of

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u′′ = λu + f (u) grows to infinity as λ ↑ +∞ for a class of f (u)’s changing of sign. Naturally, these results complement substantially the standard uniqueness results available in the literature. This paper is a natural continuation of [18] where we are going to benefit of some new hidden properties of the maximal existence time, Tmax , in order to get some new, fairly astonishing, uniqueness results for a large class of non-monotone positive nonlinearities f . As in the available literature the monotonicity of f (u) is always required in order to get the existence and the uniqueness of the large solution of ∆u = f (u), our new findings in this paper should deserve a huge attention, as they show how the monotonicity of f (u) is far from necessary for the uniqueness of the solution of (1.1). Up to the best of our knowledge, these are the first available uniqueness results in the literature for non-monotone nonlinearities. This paper is distributed as follows. Section 2 begins by studying the map Tmax (x) and establishing that if f (u) > 0 for all u > 0, then either Tmax (x) = +∞ for all x ∈ D, or Tmax (x) < +∞ for all x ∈ D. As illustrated by an example constructed in Section 2, this nice global property might fail to be true if f (u) changes of sign. Thus, for these nonlinearities, the classical (global) existence condition of Keller–Osserman cannot be imposed, though large solutions might still exist. Then, it is proven that, under the appropriate conditions whenever f (u) > 0 for all u > 0, 1 ′ Tmax (x) = − √ 2 2

∫

+∞

x

f (θ) − f (x) (∫ )3/2 dθ, θ f x

x ∈ D.

(1.7)

′ through (1.7), we can construct a class of nonIn Section 3, by studying some hidden properties of Tmax monotone nonlinearities, f (u), for which the uniqueness of a solution for (1.1) still holds. Essentially, for ′ (x) < 0 for all x > 0, increasing nonlinearities satisfying the Keller–Osserman conditions one has that Tmax and our examples in Section 3 show that f can be perturbed into a non-monotone function in such a way that Tmax still remains strictly decreasing. No previous result of this type is available in the literature.

2. Some important properties of Tmax The next result sharpens [18, Lem. 2.2] establishing the validity of (1.6) also for the case when Tmax (x) = +∞. Lemma 2.1.

Suppose x ∈ D. Then, u′ (t) > 0 for all t ∈ (0, Tmax (x)) and 1 Tmax (x) = √ 2

∫

+∞

x

dθ √∫ , θ f x

(2.1)

regardless of the value of Tmax (x) ∈ (0, +∞], where u(t) stands for the unique solution of (1.3). Proof . The first assertion is a direct consequence of [18, Lem. 2.2], as well as the validity of (2.1) when Tmax (x) < +∞. Suppose Tmax (x) = +∞. Then, the next limit is well defined u(+∞) := lim u(t) ∈ (0, +∞]. t↑+∞

Since x ∈ D, the next integral is also well defined ∫ +∞ 1 dθ √∫ I(x) := √ ∈ (0, +∞]. θ 2 x f x

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On the other hand, multiplying u′′ = f (u) by u′ and integrating yields ∫ u(t) (u′ (t))2 = f for all 0 ≤ t < Tmax (x) = +∞. 2 x Hence, for every t > 0, we have that ∫ t ∫ u(t) ∫ t u′ (s) 1 dθ 1 √∫ √∫ ds = √ . t= ds = √ θ u(s) 2 2 0 x 0 f f x x Thus, letting t ↑ ∞, we find that 1 +∞ = √ 2

∫

u(+∞)

x

dθ √∫ θ x

1 ≤√ 2 f

∫

+∞

x

dθ √∫ θ x

= I(x). f

Therefore, I(x) = +∞ = Tmax (x), which ends the proof.

■

The next result establishes an important property of the maximal existence time, Tmax . Proposition 2.2. Suppose Tmax (x) = +∞ for some x ∈ D. Then, Tmax (y) = +∞ for every y > x such that y ∈ D. Thus, if, in addition, f > 0, then, either Tmax (y) = +∞ for all y ∈ D, or Tmax (y) < +∞ for all y ∈ D. Proof . According to Lemma 2.1, 1 Tmax (x) = √ 2

∫

+∞

x

dθ √∫ θ x

Moreover, owing to

= +∞.

(2.2)

f

√ θ−x 1 1 lim √∫ = lim √ ∫ =√ θ↓x θ↓x θ θ f (x) ( x f )/(θ − x) f x

and f (x) > 0, there exist ε > 0 and a constant C > 0 such that 1 √∫ θ x

≤√ f

C θ−x

for every θ ∈ [x, x + ε).

(2.3)

Thus, it becomes apparent that the integral of (2.2) is convergent at x. Hence, it follows from (2.2) that ∫ +∞ dθ √∫ = +∞ for all y > x. θ y f x Consequently, thanks to Lemma 2.1, for every y ∈ D such that y > x, we have that ∫ +∞ ∫ +∞ ∫ +∞ 1 dθ 1 dθ 1 dθ √∫ √∫ √ √∫ Tmax (y) = √ ≥√ = = +∞. ∫θ θ y θ 2 y 2 2 y y f f + f f y y x x ∫y Note that x f > 0 because x ∈ D. This ends the proof of the first assertion. To conclude the proof of the proposition, suppose f > 0 and there exists z ∈ D with Tmax (z) < +∞. Then, thanks to a result of S. Dumont et al. [7], collected as Lemma 3.2 in [18], lim inf Tmax (y) = 0.

(2.4)

y→+∞

Therefore, by the first assertion, Tmax (x) < +∞ for all x ∈ D. This ends the proof.

■

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The alternative of the statement of Proposition 2.2 holds true because we assumed f > 0. The next example shows that it might fail to be true if f (u) changes sign. Example 2.3. Consider the function F : [0, +∞) → [0, +∞) defined by ⎧ ⋃ 3 ⎪ (n − n22 , n + n22 ), if u ∈ [0, +∞) \ ⎨ u n≥2 F (u) := ⎪ 2 2 ⎩ g , n + 2 ) for some n ≥ 2, if u ∈ (n − n n2 n where, for every n ≥ 2, the function gn is defined through ⎧ 2 1 ⎪ ⎪ gn,1 (u) if u ∈ (n − 2 , n − 2 ), ⎪ ⎪ n n ⎪ ⎨ 1 1 1 gn (u) := 1+ 2 if u ∈ [n − 2 , n + 2 ], ⎪ n n n ⎪ ⎪ ⎪ 2 1 ⎪ ⎩ gn,2 (u) if u ∈ (n + 2 , n + 2 ), n n with gn,1 and gn,2 smooth functions defined in such a way that F ∈ C 2 and 2 3 1 ) ≥ gn,1 (u) ≥ 1 + 2 2 n n 1 2 3 1 + 2 ≤ gn,2 (u) ≤ (n + 2 ) n n ′ Setting f := F in (1.3), the next result holds. (n −

Lemma 2.4.

for all u ∈ (n −

2 ,n n2

−

1 ), n2

for all u ∈ (n +

1 ,n n2

+

2 ). n2

Suppose f = F ′ . Then, (0, 1] ⊂ D, Tmax (1) = +∞ and Tmax (x) < +∞ for all x ∈ (0, 1).

Proof . By the definition of f , the following holds true: • f (1) = F ′ (1) = 3 > 0. ∫θ • 1 f = F (θ) − F (1) = F (θ) − 1 > 0 for all θ > 1. Hence, by the definition of D, it is apparent that 1 ∈ D. Thus, according to Lemma 2.1, ∫ +∞ ∫ +∞ 1 1 dθ dθ √∫ √ Tmax (1) = √ =√ θ 2 1 2 1 F (θ) − 1 f 1 1 ∫ ∫ 1 dθ 1 ∑ n+ n2 dθ √ √ ≥√ ⋃ =√ 1 1 1 2 2 F (θ) − 1 (n− 2 ,n+ 2 ) 1+ 1 −1 n≥2 n− 2 n≥2 n

n

n2

n

1 ∑ 2n √ ∑ 1 =√ = +∞. = 2 n 2 n≥2 n2 n≥2 Now, pick x ∈ (0, 1). Then, by construction, • f (x) = F ′ (x) = 3x2 > 0. ∫θ • x f = F (θ) − x3 > 0 for all θ > x. Thus, x ∈ D and, thanks to Lemma 2.1, ∫ +∞ 1 dθ √ √ Tmax (x) = 2 x F (θ) − x3 ∫ 1 =√ ⋃ 2 [x,+∞)\ n≥2 (n− 22 ,n+ n

√ 2 ) n2

2 ∫ dθ 1 ∑ n+ n2 dθ √ +√ . 3 3 2 2 θ −x F (θ) − x3 n≥2 n− 2 n

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Note that

∫

√

[x,+∞)\

⋃

(n− 22 ,n+ 22 ) n≥2 n n

dθ < θ 3 − x3

∫

+∞

√

x

dθ < +∞. θ 3 − x3

Moreover, taking into account that F (θ) − x3 > 1 − x3 > 0

( 2 2) for every n ≥ 2 and θ ∈ n − 2 , n + 2 , n n

we find that ∑∫ n≥2

n+ 22 n

n− 22 n

∑∫

dθ

√ ≤ F (θ) − x3 n≥2

n+ 22 n

n− 22 n

√

∑ 1 dθ 4 < +∞ =√ 1 − x3 1 − x3 n≥2 n2

and therefore, Tmax (x) < +∞ for all x ∈ (0, 1). This ends the proof.

■

As pointed us by the reviewer, since, in general, Tmax is a lower-semicontinuous function of x, by Lemma 2.4, we can infer that lim Tmax (x) = +∞. x↑1

The next result establishes that, under very general assumptions on f (u), Tmax (x) is a differentiable function ′ (x). of x and, in addition, it provides us with Tmax Proposition 2.5.

Consider the function G(x, τ ) defined through (∫ τ )−1/2 G(x, τ ) := f (x + t)dt , x ∈ D, τ ∈ (0, +∞).

(2.5)

0

Suppose that there exist x ∈ D, ε > 0, C > 0 and φ(τ ) ∈ L1 (C, +∞) such that ⏐} ⏐ { ⏐ ⏐ ∂G (ξ, τ )⏐⏐ ≤ φ(τ ) if |ξ − x| ≤ ε, τ ≥ C. max G(ξ, τ ), G3 (ξ, τ ), ⏐⏐ ∂τ Then, Tmax is differentiable at x ∈ D, and ∫ +∞ ∫ +∞ −1 f (τ + x) − f (x) −1 f (θ) − f (x) ′ Tmax (x) = √ (∫ )3/2 dθ. (∫ τ )3/2 dτ = √ θ 2 2 0 2 2 x f (t + x)dt f 0 x

(2.6)

(2.7)

Proof . Since D is open, by choosing a sufficiently small ε > 0 we can get [x − ε, x + ε] ⊂ D. Thus, by Lemma 2.1, as soon as |ξ − x| ≤ ε, Tmax (ξ) is well defined and is given by ∫ +∞ 1 dθ √∫ . (2.8) Tmax (ξ) = √ θ 2 ξ f ξ

So, by performing the change of variable θ = τ + ξ in (2.8), we can infer that ∫ +∞ ∫ +∞ ∫ +∞ 1 dτ dτ 1 1 √∫ √∫ Tmax (ξ) = √ =√ G(ξ, τ ) dτ. =√ τ ξ+τ 2 0 2 0 2 0 f (ξ + t)dt f (s)ds ξ

(2.9)

0

Thanks to (2.3), the integral in (2.9) is convergent at τ = 0. Thus, since G(ξ, ·) ∈ L1 (C, +∞), it also converges at τ = +∞ and hence, Tmax (ξ) < +∞ if |ξ − x| ≤ ε. Since the derivative of G(x, τ ) with respect to x is given by (∫ τ )−3/2 ∂G 1 (x, τ ) = − f (t + x)dt (f (τ + x) − f (x)), (2.10) ∂x 2 0

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it suffices to show that differentiating under the integrand sign in (2.9) up to get (2.7) is worth-legal. Indeed, by the mean value theorem, we have that, for every h ∈ (−ε, ε) there exists ξ := ξ(x, h) > 0 such that |ξ − x| < h and G(x + h, τ ) − G(x, τ ) ∂G = (ξ, τ ). h ∂x In order to apply the theorem of dominated convergence of Lebesgue, one should make sure that these quotients are uniformly dominated, in |ξ − x| ≤ ε, by some function in L1 (0, +∞) with respect to τ . To analyze the behavior of ∂G ∂x (ξ, τ ) for τ ∼ 0, τ > 0, we first observe that, rearranging terms, (2.10) can be expressed as ∂G 1 f (τ + ξ) − f (ξ) 1 √∫ . (ξ, τ ) = − ∫ τ τ ∂x 2 0 f (t + ξ)dt f (t + ξ)dt 0

Thus, using (2.3) and taking into account that f (τ + ξ) − f (ξ) lim ∫ τ = lim τ ↓0 τ ↓0 f (t + ξ)dt 0

f (τ +ξ)−f (ξ) τ

1 τ

∫τ 0

f (t + ξ)dt

=

f ′ (ξ) , f (ξ)

it becomes apparent that there exist η > 0 and a constant C1 > 0 such that ⏐ ∂G ⏐ C1 ⏐ ⏐ (ξ, τ )⏐ ≤ √ ⏐ ∂x τ

for all τ ∈ [0, η) and ξ ∈ [x − ε, x + ε].

Consequently, ∫ 0

η

∫ η ⏐ ⏐ ∂G dτ √ ⏐ ⏐ √ = 2C1 η. (ξ, τ )⏐ dτ ≤ C1 ⏐ ∂x τ 0

As far as concerns the uniform integrability at τ = +∞ of ∂G ∂x (ξ, τ ) for |ξ − x| ≤ ε one can argue as follows. Since (∫ τ (∫ τ )−3/2 )−3/2 ∂G 1 f (ξ) (ξ, τ ) = − f (t + ξ)dt f (τ + ξ) + f (t + ξ)dt ∂x 2 2 0 0 f (ξ) ∂G 3 (ξ, τ ) + (G(ξ, τ )) , = ∂τ 2 due to (2.6), we find that, for every ξ ∈ [x − ε, x + ε] and τ ≥ C, ( ) ⏐ ∂G ⏐ ⏐ ∂G ⏐ f (ξ) 1 ⏐ ⏐ ⏐ ⏐ 3 (ξ, τ )⏐ ≤ ⏐ (ξ, τ )⏐ + (G(ξ, τ )) ≤ φ(τ ) + max f φ(τ ) ∈ L1 (C, +∞). ⏐ ∂x ∂τ 2 2 [x−ε,x+ε] Therefore, by the theorem of dominated convergence, (2.7) holds. The proof is complete.

■

3. Uniqueness results Throughout this section we will assume that f (u) > 0 for all u > 0. Then, ∫

θ

f >0

for all θ > x > 0

x

and hence, D = (0, +∞). Thus, by Lemma 2.1, (2.1) holds for all x > 0. The following result sharpens, very ′ substantially, the first assertion of [18, Th. 3.1] by establishing, under weaker conditions, that Tmax (x) < 0 for all x > 0 provided f (u) is nondecreasing and satisfies (3.1).

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Theorem 3.1. Suppose that f (u) > 0 for all u > 0, Tmax (x0 ) < +∞ at some x0 > 0, and, for every x > 0, there exist constants ε = ε(x) > 0, c1 = c1 (x) > 0 and c2 = c2 (x) > 0 such that f (ξ + τ ) ≤ c1 (x) ∫ ξ+τ f (t) dt ξ

if |ξ − x| ≤ ε and τ > c2 (x).

(3.1)

′ Then, for every x > 0, Tmax (x) < +∞ and Tmax (x) is given by (2.7). Consequently, if, in addition, f is nondecreasing, then ′ Tmax (x) < 0 for every x > 0. (3.2)

Proof . Note that condition (3.1) holds if, for instance, there exists p > 1 such that f (u) = up , or f (u) = eu , for large u. By Proposition 2.2, Tmax (x) < +∞ for all x > 0. Equivalently, ∫

+∞

G(x, τ )dτ < +∞ for all x > 0,

(3.3)

0

where τ

(∫ G(x, τ ) :=

)−1/2 f (x + t)dt .

0

On the other hand, for any given x > 0, ε ∈ (0, x) and ξ ∈ [x − ε, x + ε], since f ≥ 0, we have that ∫

τ

∫

ξ+τ

f (ξ + t) dt =

∫ f=

∫

ξ x−ε+τ

≥

∫

x−ε+τ

f+

ξ

0

x+ε

∫

x+ε

ξ+τ

f+

x+ε x+ε+τ −2ε

f=

∫ ∫

f= x+ε

f

x−ε+τ τ −2ε

f (x + ε + t) dt 0

for all τ > 2ε. Hence, G(ξ, τ ) ≤ G(x + ε, τ − 2ε),

for all ξ ∈ [x − ε, x + ε], τ > 2ε.

Therefore, setting ψ(τ ) := G(x + ε, τ − 2ε), we obtain, by (3.3), that G(ξ, τ ) ≤ ψ(τ ) ∈ L1 (2ε, +∞)

for all ξ ∈ [x − ε, x + ε] and τ > 2ε.

(3.4)

Moreover, the function ψ is decreasing because, since f > 0, we have that ∂G (x + ε, τ − 2ε) ∂τ (∫ τ −2ε )−3/2 1 =− f (x + ε + t)dt f (x + ε + τ ) < 0 2 0

ψ ′ (τ ) =

for all τ > 2ε.

(3.5)

Thus, from (3.3) and (3.5), it is apparent that lim ψ(τ ) = 0.

τ →+∞

Consequently, there exists C1 ≥ 2ε such that ψ(τ ) ≤ 1 for all τ > C1 , and combining this estimate together with (3.4) yields G3 (ξ, τ ) ≤ ψ 3 (τ ) ≤ ψ(τ ) for all ξ ∈ [x − ε, x + ε] and τ > C1 . (3.6)

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Lastly, according to (3.4) and (3.1) ⏐ ⏐ ⏐ 1 ⏐ ∂G f (ξ + τ ) c1 ⏐ ⏐ ≤ ψ(τ ) ⏐ ∂τ (ξ, τ )⏐ = 2 G(ξ, τ ) ∫ ξ+τ 2 f (s) ds

for all τ > C := max{C1 , c2 },

299

(3.7)

ξ

uniformly in ξ ∈ [x − ε, x + ε]. Therefore, by (3.4), (3.6) and (3.7), it becomes apparent that the auxiliary function φ(τ ) := λψ(τ ), τ > C, λ > 0, satisfies (2.6) for sufficiently large λ > 0. Finally, Proposition 2.5 provides us with the desired result. The proof is complete. ■ The next result sharpens all previous available one-dimensional uniqueness results as it does not require the strict monotonicity of f (u). Corollary 3.2. Suppose f (0) = 0, f (u) > 0, f ′ (u) ≥ 0 for all u > 0, Tmax (x0 ) < +∞ for some x0 > 0 and ′ (x) < 0 for all x > 0. Therefore, for every T > 0, the problem (3.1) holds. Then, Tmax (x) < +∞ and Tmax (1.1) possesses a unique positive solution. ′ Proof . The fact that Tmax (x) < +∞ and Tmax (x) < 0 for all x > 0 is a direct consequence of Theorem 3.1. Hence, due to (1.4) and (2.4), for every T > 0 there exists a unique x = x(T ) such that

Tmax (x(T )) = T. This ends the proof.

■

Throughout the rest of this section we are going to construct a general class of non-monotone positive functions, f (u), for which the uniqueness of the large solutions of u′′ = f (u) in any interval is preserved. This is the first result of this nature available in the literature. Subsequently, it is appropriate to emphasize the dependence of the maximal existence time, Tmax (x), on f . So, Tf,max (x) will stand for the maximal existence time of the (unique) solution of (1.3). Similarly, we will make explicit the dependence of the set D defined in (1.5) on f by setting Df , instead of D. Let f ∈ C 1 [0, +∞) be satisfying (3.1) such that f (0) = 0, f (u) > 0 and f ′ (u) ≥ 0 for all u > 0, and Tf,max (x0 ) < +∞ for some x0 > 0. Then, by Theorem 3.1, (3.2) holds true. Next, fix 0 < a < b and consider any sequence of test functions, (φn )n≥1 ⊂ C0∞ (R) with supp φn = [a, b], n ∈ N, such that φn ≤ φn+1 for all n ≥ 1 and

lim φn = 0 in [a, b],

n→+∞

(3.8)

for which there exists a (uniform Lipschitz) constant L > 0 such that |φn (x) − φn (y)| ≤ L|x − y| for all x, y ∈ R, n ≥ 1.

(3.9)

Later, we will give a particular example. Finally, define fn := f + φn ,

n ≥ 1.

(3.10)

By Dini’s criterion, the convergence in (3.8) is uniform in x ∈ R. Hence, without loss of generality, we can assume that f (x) fn (x) = f (x) + φn (x) > for all x > 0 and n ≥ 1. (3.11) 2

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Fig. 3.1. The graphs of f and fn , and the construction of x1 (n).

In particular fn (x) > 0 for all x > 0 and n ≥ 1 and hence, Dfn = (0, +∞) for all n ≥ 1. Moreover, from (3.11) we deduce that ∫ +∞ ∫ +∞ √ 1 dθ dθ √∫ √∫ Tfn ,max (x0 ) = √ < = 2 Tf,max (x0 ) < +∞ θ θ 2 x0 x0 f f x0

n

x0

for all n ≥ 1. Thus, by Theorem 3.1, Tfn ,max (x) < +∞ for all x > 0 and n ≥ 1. Furthermore, by the first assertion of Theorem 3.1, ∫ +∞ −1 fn (θ) − fn (x) ′ Tfn ,max (x) = √ x > 0, n ≥ 1. (3.12) (∫ )3/2 dθ, θ 2 2 x f x n Since f = fn in [b, +∞) for all n ≥ 1, from (3.12) the next result is evident. Lemma 3.3.

′ For every x ≥ b and n ≥ 1, Tf′n ,max (x) = Tf,max (x) < 0.

Similarly, the next result of technical nature holds. Lemma 3.4.

Setting x1 (n) := min{x > 0 : fn (x) = min fn },

n ≥ 1,

[a,b]

we have that Tf′n ,max (x) < 0

for all n ≥ 1 and x ∈ (0, x1 (n)).

Moreover, the mapping n ↦→ x1 (n) is increasing and lim x1 (n) = a.

n→+∞

Proof . Pick n ∈ N. By the construction of x1 (n), it is apparent that fn (θ) ≥ fn (x) for all x ∈ (0, x1 (n)) and θ > x. Fig. 3.1 illustrates the construction of x1 (n) and might help the reader to realize what is going on. Thus, from (3.12) we can infer that Tf′n ,max (x) < 0 if x ∈ (0, x1 (n)). The last assertion of the statement is obvious by the construction of fn and the definition of x1 (n). ■

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301

Furthermore, the next convergence result holds. For every x > 0 we have that

Lemma 3.5.

′ lim Tf′n ,max (x) = Tf,max (x).

n→+∞

Moreover, this convergence is uniform on compact subsets of (0, +∞). Proof . For every x > 0 and n ≥ 1, it follows from (3.12) that ∫ ∫ +∞ ⏐ 1 ⏐⏐ +∞ f (θ)−f (x)+φn (θ)−φn (x) ⏐ f (θ)−f (x) ′ dθ − |Tf′n ,max (x) − Tf,max (x)| = √ ⏐ (∫ )3/2 (∫ )3/2 dθ⏐ θ θ 2 2 x x fn f x x ⏐ ⏐ ⏐∫ ⏐ ∫ ∫ +∞ ⏐ +∞ ⏐ +∞ ⏐ ⏐ f (θ)−f (x) |φn (θ)−φn (x)| f (θ)−f (x) ≤⏐ (∫ (∫ )3/2 dθ⏐ + (∫ )3/2 dθ − )3/2 dθ. θ θ θ ⏐ x ⏐ x x fn f fn ⏐ ⏐ x x x

(3.13)

By (3.11), we have that ∫ 1 θ fn ≥ f for all θ > x > 0 and n ≥ 1. 2 x x Thus, for every x > 0, it becomes apparent that ∫

θ

f (θ)−f (x) )3/2 ≤ 23/2 (∫ )3/2

f (θ)−f (x)

(∫

θ x

fn

θ

x

(3.14)

for all θ > x and n ≥ 1.

(3.15)

f

On the other hand, since Tf,max (x) < +∞ for all x > 0, we find from Theorem 3.1 that ∫ +∞ f (θ)−f (x) ′ Tf,max (x) = (∫ )3/2 dθ < +∞ θ

x

x

f

1 and hence, the function θ ↦→ (f∫(θ)−f)(x) 3/2 lies in L (x, +∞). Thus, setting θ x

f

∫

+∞ f (θ)−f (x)

In (x) :=

(∫

x

θ

fn

x

)3/2 dθ,

x > 0,

and taking into account (3.15), it follows from the theorem of dominated convergence that ∫ +∞ f (θ)−f (x) lim In (x) = (∫ )3/2 dθ for all x > 0. n→+∞

θ

x

x

f

Actually, by (3.8), the sequence of continuous functions (In )n≥1 is monotone in n. Thus, thanks to the Dini’s criterion, it becomes apparent that ∫ +∞ f (θ)−f (x) lim In (x) = uniformly on compact subsets of (0, +∞). (3.16) (∫ )3/2 dθ n→+∞

x

θ

x

f

Going back to (3.13), the estimate (3.14) also yields ∫ +∞ ∫ +∞ ∫ |φn (θ)−φn (x)| |φn (θ)−φn (x)| (∫ )3/2 dθ ≤ 23/2 (∫ )3/2 dθ = 23/2 x

θ

x

fn

x

θ

x

f

0

+∞

|φ (τ +x)−φn (x)|

(∫nτ 0

)3/2 dτ

f (t+x)dt

302

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for all n ≥ 1, where we have made the change of variable θ − x = τ in the last integral. Set K ⊂ (0, +∞) any compact subset and consider x0 := min K. Thus, since ∫ τ ∫ τ f (t + x)dt ≥ f (t + x0 )dt 0

0

for all x ∈ K and τ > 0, because f is non-decreasing, we find that ∫ +∞ ∫ +∞ |φ (τ +x)−φn (x)| |φn (τ +x)−φn (x)| (∫ τ )3/2 dτ ≤ ψn (x) := (∫ nτ )3/2 dτ. 0

0

f (t+x)dt

0

0

f (t+x0 )dt

We claim that lim ψn (x) = 0

n→+∞

uniformly on K.

(3.17)

Since the support of φn lies in [a, b], ψn (x) = 0 for all n ≥ 1 and x ∈ K with x ≥ b. So, suppose x ∈ K ∩(0, b). By (3.9), we have that |φn (τ + x) − φn (x)| ≤ Lτ

for all x > 0 and n ≥ 1.

Thus, for every n ≥ 1, x ∈ K ∩ (0, b) and δ > 0, we have that ∫ δ ∫ +∞ |φn (τ +x)−φn (x)| |φ (τ +x)−φn (x)| ψn (x) = (∫ τ (∫ nτ )3/2 dτ + )3/2 dτ 0

∫ ≤ 0

0

f (t+x0 )dt

δ

δ

∫

Lτ

(∫ τ 0

)3/2 dτ +

f (t+x0 )dt

0

+∞

|φ (τ +x)−φn (x)|

(∫ nτ

δ

f (t+x0 )dt

0

)3/2 dτ.

f (t+x0 )dt

Subsequently, for any given ε > 0, we fix δ > 0 satisfying ∫ δ ε Lτ (∫ τ )3/2 dτ < . 2 f (t+x0 )dt 0 0

Note that such an ε exists because f (x0 ) > 0. By (3.8), there exists n0 such that ε |φn (τ + x) − φn (x)| ≤ |φn (τ + x)| + |φn (x)| < ∫ )−3/2 +∞ (∫ τ 2 f (t + x0 )dt dτ δ

0

for all x ∈ (0, b], τ > δ and n ≥ n0 . Therefore, we can conclude that ψn (x) < ε for all n ≥ n0 and x ∈ K ∩ (0, b). ′ Lastly, substituting (3.16) and (3.17) in (3.13), the uniform convergence of Tf′n ,max (x) to Tf,max (x), on compact subsets of x > 0, readily follows. This completes the proof. ■

As a consequence of the previous preparatory results, we will derive two types of uniqueness theorems of large solutions for non monotone nonlinearities f (u). First, we will consider a function f satisfying Eq. (3.1) and ⎧ ′ ⎪ for every x ∈ (0, +∞) \ [a, b], ⎪ ⎨ f (x) > 0 f ′ (x) = 0

⎪ ⎪ ⎩

for every x ∈ [a, b],

Tf,max (a) < +∞,

and a test function φ ∈ C0∞ (R) with supp φ = [a, b] such that, for some xM ∈ (a, b), ⎧ ′ ⎪ ⎪ ⎨ φ (x) < 0 for all x ∈ (a, xM ), φ′ (x) > 0 for all x ∈ (xM , b),

⎪ ⎪ ⎩

min φ = φ(xM ) = −f (a)/2.

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303

Then, we define

1 φ(x), x > 0, n ≥ 1. n Clearly, φn satisfies (3.8) and (3.9). Then, defining fn through (3.10), the next result holds. φn (x) :=

Theorem 3.6.

There exists n0 such that, for every n ≥ n0 , Tf′n ,max (x) < 0

for all x > 0.

Therefore, for every T > 0, the problem ⎧ ⎨ u′′ = f (u), n ⎩ u′ (0) = 0, u(T ) = +∞.

t ∈ [0, T ),

(3.18)

possesses a unique solution for all n ≥ n0 . Proof . Thanks to Lemma 3.3, we already know that Tf′n ,max (x) < 0 for all n ≥ 1 and x > b. Similarly, according to Lemma 3.4, Tf′n ,max (x) < 0 for all n ≥ 1 and x ∈ (0, x1 (1)). Now, consider the compact subinterval of (0, +∞) defined by K := [x1 (1), b] . By Corollary 3.2 and Lemma 3.5, there exists n0 ∈ N such that Tf′n ,max (x) < 0

for all n ≥ n0 and x ∈ K.

Tf′n ,max (x) < 0

for all x > 0 and n ≥ n0 .

Therefore, Finally, since fn (0) = 0, it becomes apparent that lim Tfn ,max (x) = +∞,

n ≥ 1,

lim inf Tfn ,max (x) = 0,

n ≥ 1.

x↓0

while, due to [18, Le. 3.2], x↑+∞

Therefore, for every n ≥ n0 and T > 0 there exists a unique x = x(T ) > 0 such that the unique solution of ⎧ ⎨ u′′ = f (u), n ⎩ u(0) = x > 0, u′ (0) = 0, satisfies Tfn ,max (x(T )) = T, which ends the proof of the theorem.

■

Theorem 3.6 provides us with a family of non-monotone perturbations fn , n ≥ n0 , satisfying 1 ′ φ (x) < 0 if x ∈ (a, xM ), n for which the singular problem (3.18) possesses a unique solution. In this example, is rather remarkable that, by the construction of fn , the interval (a, xM ) where fn′ < 0 can be taken arbitrarily large, however these perturbations have small derivatives in the interval (a, xM ), in the sense that fn′ (x) = f ′ (x) + φ′n (x) =

lim fn′ (x) = 0

n→+∞

uniformly on x ∈ (a, xM ).

Subsequently, we will construct another sequence of perturbations for which this is not true.

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Let us consider p-ordered points in (a, b), a < x1 < · · · < xp < b, and set δk := min{|xi − xj |/k : 1 ≤ i, j ≤ p},

k ∈ N.

Now, consider any sequence (φn )n≥1 ⊂ C0∞ (R) with suppφn = [a, b] satisfying (3.8), (3.9), and such that ⎧ ⎪ for all 1 ≤ i ≤ p, ⎪ ⎨ φn (xi ) = −1/n ⋃p φn (x) = 0 if x ∈ [a, b] \ i=1 (xi − δk /n, xi + δk /n), ⎪ ⎪ ⎩ −1/n ≤ φn (x) ≤ 0 if x ∈ (xi − δk /n, xi + δk /n) for some 1 ≤ i ≤ p, for all n ∈ N. Then, by the mean value theorem, for each integer n > 1, there exist at least p points, ξi ∈ (xi − δk /n, xi ), 1 ≤ i ≤ p, such that fn′ (ξi ) =

fn (xi ) − fn (xi − δk /n) f (xi ) − 1/n − f (xi − δk /n) 1 = =− , xi − xi + δk /n xi − xi + δk /n δk

(3.19)

because f is constant in [a, b]. By the definition of δk , it can be as small as we wish. Thus, the perturbations fn can have arbitrarily large derivatives at some points. Arguing as in the proof of Theorem 3.6, the next result holds. Theorem 3.7. There exists an integer n0 ≥ 1 such that, for every n ≥ n0 and T > 0, the problem (3.18) possesses a unique solution. Therefore, besides satisfying (3.19), the functions fn do preserve the uniqueness for every n ≥ n0 . Final remarks. exchanged by

The monotonicity of the sequence (φn )n≥1 imposed in (3.8) could have been interφn (x) ≥ φn+1 (x),

x > 0, n ≥ 1,

and Theorems 3.6 and 3.7 would still work. Indeed, as the sequence (φn )n≥1 is monotone, the arguments of the proofs of Lemma 3.3, 3.4 and 3.5 can be easily adapted to cover this case because Dini’s criterion still works out. Although Dini’s criterion should not be pivotal in order to get the uniform convergence of Tf′n ,max to ′ Tf,max on compact subsets as n → ∞, we could not avoid it in our argument. Acknowledgments We sincerely thank the reviewers for their careful reading of the paper and for pointing us a technical gap in the proof of the previous Proposition 2.5, which has lead to a reformulation of Proposition 2.5 and Theorem 3.1. So, the final form of this paper is indebted with the excellent work of the reviewers. References [1] A. Aftalion, M. del Pino, R. Letelier, Multiple boundary blow-up solutions for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh A 133 (2003) 225–235. [2] F.C. Cirstea, V. Radulescu, Existence and uniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp. Math. 4 (2002) 559–586. [3] F.C. Cirstea, V. Radulescu, Solutions with boundary blow-up for a class of nonlinear elliptic problems, Houst. J. Math. 29 (2003) 821–829. [4] O. Costin, L. Dupaigne, Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry, J. Differential Equations 249 (2010) 931–964. [5] O. Costin, L. Dupaigne, O. Goubet, Uniqueness of large solutions, J. Math. Anal. Appl. 395 (2012) 806–812.

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