The existence and blow-up rate of large solutions of one-dimensional -Laplacian equations

Nonlinear Analysis: Real World Applications 13 (2012) 665–676

Contents lists available at SciVerse ScienceDirect

Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

The existence and blow-up rate of large solutions of one-dimensionalp-Laplacian equations

Lei Wei ∗, Jiang ZhuSchool of Mathematical Sciences, Xuzhou Normal University, Xuzhou 221116, PR China

a r t i c l e i n f o

Article history:Received 16 January 2011Accepted 8 August 2011

Keywords:Large solutionsBlow-up ratep-LaplacianSupersolutionSubsolution

a b s t r a c t

In this paper, we consider the existence, uniqueness and blow-up rate of positive solutionsof the singular boundary value problem

(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > 0,u(0) = +∞, u(+∞) = 0,

where f is a nondecreasing and continuous function such that f (0) ≥ 0, and h is anon-negative function satisfying the Keller–Osserman condition. Our results extend someprevious findings of [S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimensional problems on the half-line,J. Differential Equations 244 (2008) 3180–3203] in a certain sense.

Crown Copyright© 2011 Published by Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we mainly study the following problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > 0,u(0) = +∞, u(+∞) = 0, (1.1)

where f is a continuous function such that f (0) ≥ 0 and 1 < p ≤ 2. Throughout this paper, we will use the followingconditions on f and h:

(f1) f ∈ C[0, +∞) is nondecreasing and satisfies f (t) > 0 for each t > 0;(h1) h ∈ C1

[0, +∞) satisfies h(0) = 0, h′(u) > 0 for any u > 0, and the Keller–Osserman condition∫+∞

t

∫ s

th(τ )dτ

−1p

ds < +∞, ∀t > 0; (1.2)

(f2) limt↓0f (t)tα = A, where α ≥ 0 and A > 0;

(h2) limt→+∞h(t)tq = B, where B > 0 and q > p − 1.

Boundary blow-up problems have been studied by many authors and in many contexts for a long time. Generally,solutions of boundary blow-up problems are said to be large solutions. The pioneering research work on boundary blow-upproblems can go back to [1,2]. In [1,2], the following boundary blow-up problem is considered

∆u = h(u), x ∈ Ω,u = +∞, x ∈ ∂Ω,

Authors were supported by the foundation of Jiangsu University Natural Science 11KJB110015.∗ Corresponding author.

E-mail addresses:[email protected] (L. Wei), [email protected] (J. Zhu).

1468-1218/$ – see front matter Crown Copyright© 2011 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2011.08.007

http://dx.doi.org/10.1016/j.nonrwa.2011.08.007

http://www.elsevier.com/locate/nonrwa

http://www.elsevier.com/locate/nonrwa

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.nonrwa.2011.08.007

666 L. Wei, J. Zhu / Nonlinear Analysis: Real World Applications 13 (2012) 665–676

whereΩ ⊂ RN is a bounded and smooth domain, and a necessary and sufficient condition of the existence is supplied. Moreprecisely, hmust satisfy the Keller–Osserman condition∫

+∞

1

dt√H(t)

< +∞, H(t) =

∫ t

0h(s)ds.

Concerning the recent studies on boundary blow-up problems, we can refer to [3–13,1,14–19]. In particular, in [16],López-Gómez studied the existence, uniqueness and the blow-up rate of large solutions of

u′′= f (t)uq, t > 0,

u(0) = +∞, u(+∞) = 0,

where f ∈ C[0, +∞) satisfies f (t) ≥ f (s) > 0 for t ≥ s > 0, and q > 1. In [4], Cano-Casanova and López-Gómez studiedthe existence, uniqueness and the blow-up rate of large solutions of

u′′= f (t)h(u), t > 0,

u(0) = +∞, u(+∞) = 0, (1.3)

where f and h satisfy the following conditions:(i) f ∈ C[0, +∞) satisfies f (t) ≥ f (s) > 0 for t ≥ s > 0;(ii) h ∈ C1

[0, +∞) satisfies h(0) = h′(0) = 0, h′(u) > 0 for u > 0, and the Keller–Osserman condition∫+∞

z

[∫ x

zh(s)ds

]−12

dx < +∞.

Under the conditions (i) and (ii), Cano-Casanova and López-Gómez obtained the existence and uniqueness of large solutionsof (1.3). Further, suppose that the following conditions are satisfied

(iii) g(u) :=h(u)u satisfies g ′(u) ≥ 0 for all u > 0 and, for some q > 1,

limt→+∞

h(u)uq

> 0;

(iv) the limit

I0 = limt↓0

F(t)F ′′(t)(F ′(t))2

∈ (0, +∞)

is well defined for some R > 0, where F(t) is defined by

F(t) =

∫ R

t

∫ s

0[f (τ )]

1q+1 dτ

−q+1q−1

ds, t ∈ (0, R],

then Cano-Casanova and López-Gómez proved that the unique large solution l of (1.3) satisfies

limt↓0

l(t)F(t)

= I−

qq−1

0

q + 1q − 1

q+1q−1

H−1

q−1 .

In fact, themain results of [4] is a substantial improvement of themain results of [16]. In this paper, our results extend themain results of [4] in a certain sense. Our idea is inspired by [4,16]. The problem studied in [4,16] is a semilinear equationwith boundary blow-up, so the regularity of solutions and maximum principle can be directly used to get some results.But, in this paper, the equation is a quasilinear equation, so some auxiliary results cannot been directly established by theregularity of solutions and maximum principle. In addition, in order to obtain some useful conclusions, we need to useintegral methods and fixed points theory. In this paper, the proof of the existence is based on the method of supersolutionand subsolution for the p-Laplacian operator. For the comparison principle of p-Laplacian operator, we can refer to[13,20,21]. The next theorem collects the main findings of the present paper.

Theorem 1. Suppose that f and h satisfy (f1) and (h1). Then, (1.1) has a unique positive solution, denoted by l. Further, supposethat f and h satisfy (f2) and (h2), respectively, then the unique positive solution l satisfies

limt↓0

l(t)

t−p+α

q−p+1=

1AB

p + α

q − p + 1

p−1 α + q + 1q − p + 1

(p − 1)

1q−p+1

. (1.4)

The paper is organized as follows. In Section 2, we will prove the existence and the uniqueness of large solutions of (1.1).In Section 3, we will give the exact blow-up rate of the large solution at t = 0.

L. Wei, J. Zhu / Nonlinear Analysis: Real World Applications 13 (2012) 665–676 667

2. Existence and uniqueness

Theorem 2. Suppose that (f1) and (h1) hold. Then, the problem (1.1) possesses a unique positive solution l(t). Moreover, l′(t) < 0for all t > 0.

Proof. For any M > 0 and L > 0, we firstly consider the auxiliary problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > 0,u(0) = M, u(L) = 0. (2.1)

It is clear that (0,M) is an ordered sub–supersolution pair of (2.1). It follows from the method of subsolution andsupersolution that (2.1) has a solution u ∈ C1

[0, L] such that 0 ≤ u(t) ≤ M for all t ∈ [0, L]. We claim that

u(t) > 0, ∀t ∈ [0, L). (2.2)

In fact, if (2.2) is not true, then

0 < t0 = inft ∈ [0, L] : u(t) = 0 < L, u(t0) = 0, u′(t0) = 0.

So, u satisfies(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), 0 < t < t0,u(t0) = 0, u′(t0) = 0. (2.3)

For t ∈ [0, t0), by integrating from t to t0, we obtain

|u′(t)|p−2u′(t) = −

∫ t0

tf (τ )h(u(τ ))dτ . (2.4)

As f (τ ) > 0 and h(u(τ )) > 0 for all τ ∈ (0, t0), then (2.4) implies u′(t) < 0 for all t ∈ [0, t0). So, (2.4) implies

|u′(t)|p−1=

∫ t0

tf (τ )h(u(τ ))dτ , ∀t ∈ [0, t0]. (2.5)

LetM0 = sup0≤t≤t0 |u′(t)|. By 1 < p ≤ 2, we can see

|u′(t)|Mp−20 ≤

∫ t0

tf (τ )h(u(τ ))dτ .

Thus, we have

|u′(t)| ≤ M2−p0

∫ t0

tf (τ )h(u(τ ))dτ

≤ f (t0)M2−p0

∫ t0

t[h(u(τ )) − h(u(t0))]dτ

≤ CM2−p0

∫ t0

tu(τ )dτ = CM2−p

0

∫ t0

t

∫ τ

t0u′(s)ds

dτ

≤ CM2−p0

∫ t0

t

∫ t0

τ

|u′(s)|dsdτ , (2.6)

where C = f (t0) · maxs∈[0,M] h′(s). It follows from u′(t) < 0 for any t ∈ [0, t0) and (2.5) that |u′(t)| is decreasing in [0, t0].Therefore, (2.6) implies

|u′(t)| ≤ CM2−p0 L

∫ t0

t|u′(τ )|dτ . (2.7)

So, by virtue of Gronwall’s Lemma, we obtain u′(t) = 0 for all t ∈ [0, t0], which is impossible.In addition, it is easy to see u′(L) ≤ 0. If u′(L) = 0, similarly to the above proof we can obtain a contradiction, so

u′(L) < 0. We claim that u′(t) < 0 for all t ∈ [0, L]. If it is not so, then there exists t0 ∈ [0, L) such that u′(t0) = 0. Denotet = supt ∈ [0, L) : u′(t) = 0. Now, let v(t) = u(t) − u(t), then v′(t) = u′(t) = 0. Hence, v satisfies

(|v′(t)|p−2v′(t))′ = f (t)h(v(t) + u(t)), t < t < L,v(t) = 0, v′(t) = 0.

(2.8)


For all t ∈ [t, L], we have

|v′(t)|p−2v′(t) =

∫ t

tf (τ )h(u(τ ))dτ ,

which implies that v′(t) > 0 for each t ∈ (t, L] and v′(t) is increasing in [t, L]. Therefore, by u′(t) = 0, we obtain u′(L) > 0,which is a contradiction to u′(L) < 0. This shows that u′(t) < 0 for all t ∈ [0, L].

We now prove the uniqueness of positive solutions of (2.1). Suppose that u1 and u2 are positive solutions of (2.1) andT = t ∈ [0, L] : u1(t) > u2(t) = ∅. Let (a, b) be a connected component of T , then u1 and u2 satisfy

(|u′

1(t)|p−2u′

1(t))′≥ (|u′

2(t)|p−2u′

2(t))′, t ∈ (a, b),

u1(a) = u2(a), u1(b) = u2(b),

but the comparison principle implies u2 ≥ u1 in [a, b], so we obtain a contradiction. This shows the uniqueness, and nowwe denote the unique positive solution of (2.1) by u[M,L].

LetM1 ≥ M2 > 0 and L1 ≥ L2 > 0, then we claim

0 < u[M2,L2](t) ≤ u[M1,L1](t) ≤ M1, ∀t ∈ [0, L2]. (2.9)

Indeed, u[M1,L1] and 0 is an ordered super–subsolution pair of the problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (0, L2),u(0) = M2, u(L2) = 0, (2.10)

so the uniqueness of the positive solution of (2.10) implies u[M2,L2](t) ≤ u[M1,L1](t) in [0, L2]. Therefore, for anyM > 0,

uM(t) = limL↑+∞

u[M,L](t),

is well defined for all t ∈ [0, +∞), and 0 < uM(t) ≤ M in [0, +∞). For each compact interval [a, b] ⊂ [0, +∞), we canchoose L > b. Since u′

[M,L](t) is an increasing function in [0, L], we have

|u′

[M,L](b)|p−1

≤

M − u[M,L](b)

b

p−1

<

Mb

p−1

.

Therefore, for t ∈ [a, b],

|u′

[M,L](t)|p−2u′

[M,L](t) = |u′

[M,L](b)|p−2u′

[M,L](b) +

∫ t

bf (τ )h(u[M,L](τ ))dτ ,

implies

|u′

[M,L](t)|p−1

= |u′

[M,L](b)|p−1

+

∫ b

tf (τ )h(u[M,L](τ ))dτ

≤

Mb

p−1

+ f (b)h(M)(b − a).

So, u[M,L]|[a,b] is bounded in C1[a, b], moreover, u[M,L]|[a,b] is equicontinuous. It is easy to see|u′

[M,L](t1)|p−1

− |u′

[M,L](t2)|p−1 =

|u′

[M,L](t1)|p−2u′

[M,L](t1) − |u′

[M,L](t2)|p−2

|u′

[M,L](t2)|

=

∫ t1

t2f (τ )h(u[M,L](τ ))dτ

≤ f (b)h(M)|t1 − t2|, ∀t1, t2 ∈ [a, b]. (2.11)

Since the function r(t) = t1

p−1 is uniformly continuous in a given compact interval, (2.11) implies that u′

[M,L]|[a,b] isequicontinuous. By Ascoli–Arzela theorem, uM provides us with a positive solution of

(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > 0; u(0) = M.

Denote ω := limt↑+∞ uM(t). We claim ω = 0. If ω > 0, then there exists t0 > 0 such that uM(t) ≥ω2 for all t ≥ t0. Since

|u′

M(t)|p−2u′

M(t) = |u′

M(t0)|p−2u′

M(t0) +

∫ t

t0f (τ )h(uM(τ ))dτ

≥ |u′

M(t0)|p−2u′

M(t0) + f (t0)hω

2

(t − t0),


we can get that u′

M(t) > 0 for sufficiently large t . This is a contradiction to u′

M(t) ≤ 0. Therefore, uM is a positive solution ofthe problem

(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > 0,u(0) = M, u(+∞) = 0. (2.12)

Similarly to the proof of u′

[M,L](t) < 0 in [0, L], we can also obtain u′

M(t) < 0 in [0, +∞). Now, we show the uniquenessof positive solutions of (2.12). Suppose that u1 and u2 are positive solutions of (2.12) and satisfy t ∈ [0, +∞) : u1(t) >u2(t) = ∅. We can still obtain u′

1(t) < 0 and u′

2(t) < 0 in [0, +∞) by virtue of the above methods. Let t0 ∈ (0, +∞) besuch that

u1(t0) − u2(t0) = maxu1(t) − u2(t) : t ∈ [0, +∞).

Obviously,

u′

1(t0) = u′

2(t0), (u1 − u2)′′(t0) ≤ 0.

It follows from the sign of u′

1(t) and u′

2(t) that

−((−u′

1(t))p−1)′ = f (t)h(u1(t)), t > 0,

−((−u′

2(t))p−1)′ = f (t)h(u2(t)), t > 0.

So, we have

0 < f (t0)[h(u1(t0)) − h(u2(t0))] = −((−u′

1(t))p−1)′|t=t0 + ((−u′

2(t))p−1)′|t=t0

= (p − 1)(−u′

1(t0))p−2(u1 − u2)

′′(t0) ≤ 0,

which is impossible. Therefore, the uniqueness of positive solutions of (2.12) is proved.For any M1 ≥ M2 > 0 and L > 0, u[M2,L] ≤ u[M1,L] implies

uM2(t) ≤ uM1(t), ∀t ∈ [0, +∞).

For any L > ϵ > 0, it is easy to show that(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (ϵ, L),u(ϵ) = +∞, u(L) = +∞

(2.13)

has a nonnegative solution (refer to [22]), denoted by wL. For any M > 0 and L > 0, choose sufficiently small δ > 0such that

uM ≤ M ≤ wL, ∀t ∈ (ϵ, ϵ + δ] ∪ [L − δ, L).

So, wL provides a supersolution of the problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (ϵ + δ, L − δ),u(ϵ + δ) = uM(ϵ + δ), u(L − δ) = uM(L − δ).

(2.14)

Similarly to the above proof of the uniqueness, we can also obtain (2.14) has a unique positive solution. The method ofsupersolution and subsolution shows that uM ≤ wL holds in [ϵ + δ, L − δ]. Further, we have that uM ≤ wL holds in (ϵ, L).Since uM is nondecreasing onM , by virtue of the arbitrariness of ϵ > 0, the following limit function

l(t) = limM↑+∞

uM(t), ∀t ∈ (0, +∞) (2.15)

is well defined. It follows from the Ascoli–Arzela theorem that l solves (1.1). Since uM(t) > 0 and uM is nondecreasing onM , l is a positive solution of (1.1). In addition, we can also obtain l′(t) < 0 for all t ∈ (0, +∞).

Assume that w is an arbitrary positive solution of (1.1), then for sufficiently small ϵ > 0, w is a supersolution of thefollowing problem

(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > ϵ,u(ϵ) = w(ϵ), u(L) = 0,

so the uniqueness of positive solutions of the above problem implies w(t) ≥ u[M,L](t) for all t ∈ [ϵ, L]. So, we have

w(t) ≥ u[M,L](t), ∀t ∈ (0, L].

Therefore, as L is arbitrary, it follows thatw ≥ uM and, hence,w ≥ l. This shows that l is a minimal positive solution of (1.1).Let denote it by lmin = l.


Now, we prove the existence of the maximal positive solution of (1.1). We consider the following problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > ϵ,u(ϵ) = +∞, u(+∞) = 0. (2.16)

Then, (2.16) has a minimal positive solution. We denote it by lϵmin. The unique positive solution of the following problem isdenoted by uϵ

M(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > ϵ,u(ϵ) = M, u(+∞) = 0. (2.17)

For any 0 < ϵ < ϵ, we claim that lϵmin ≤ lϵmin holds in [ϵ, +∞). The following problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t > ϵ,u(ϵ) = lϵmin(ϵ), u(+∞) = 0 (2.18)

has a unique positive solution, i.e., lϵmin|[ϵ,+∞) is the unique positive solution. It is clear that (0, lϵmin) is an orderedsub–supersolution pair of (2.18), which implies lϵmin ≤ lϵmin in (ϵ, +∞). Therefore,

lmax(t) = limϵ↓0

lϵmin(t), ∀t ∈ (0, +∞)

is well defined. Then, lmax provides a positive solution of (1.1). In order to show the uniqueness of the positive solutions of(1.1), we only need to show that for any positive solution l of (1.1), l ≥ lmax holds in (0, +∞). Suppose that l is an arbitrarypositive solution of (1.1), then we can prove l′(t) < 0 for any t > 0. Let l(t) = l(t − ϵ), t > ϵ, so l(t) > l(t) for all t > ϵ.Then, l is a supersolution of (2.17), which implies that uϵ

M ≤ l for all t > ϵ and M > 0. Therefore, we have lϵmin ≤ l for allt > ϵ. Consequently, passing to the limit as ϵ ↓ 0, we see that for each t > 0

l(t) ≥ limϵ↓0

lϵmin(t) = lmax(t).

Theorem 3. Suppose that (f1) and (h1) hold, and there exist L > 0 and M > 0 such that(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (0, L),u(0) = +∞, u(L) = M (2.19)

has a positive solution Z(t) with Z ′(L) < 0. Then, problem (2.19) has a unique positive solution.

Proof. Suppose u is a positive solution of(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (0, L),u(0) = γ , u(L) = M.

(2.20)

The existence of aminimal andof amaximal positive solution follows the schemeof the existence proof of Theorem2. Indeed,it is easy to see, for every γ > 0, the problem (2.20) has a unique positive solution (similarly to the proof of Theorem 2), andwe denote it by zγ . Obviously, we can see

zγ (t) ≤ wL(t), ∀t ∈ (ϵ, L),

where wL is a positive solution of (2.13) (The proof of Theorem 2 can show that the nonnegative solution wL of (2.13) ispositive.). Then, by the arbitrariness of ϵ > 0 and the regularity arguments,

zmin(t) = limγ→+∞

zγ (t), ∀t ∈ (0, L]

provides us with a positive solution of (2.19) and the solution is a minimal positive solution. For sufficiently small ϵ > 0,we denote by zϵ

min the minimal positive solution of(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (ϵ, L),u(ϵ) = +∞, u(L) = M.

(2.21)

We define

zmax(t) = limϵ↓0

zϵmin(t), ∀t ∈ (0, L],

then zmax provides us with a maximal positive solution of (2.19).Since Z is a positive solution of (2.19) with Z ′(L) < 0, we have

Z(L − ϵ) > Z(L)


for sufficiently small ϵ > 0. Define an auxiliary function

Z(t) = Z(t − ϵ), ∀t ∈ [ϵ, L],

then Z satisfies

Z(ϵ) = +∞, Z(L) = Z(L − ϵ) > Z(L) = M.

Thus, Z provides us with a supersolution of (2.21), which implies that

Z(t) ≥ zϵmin(t), ∀t ∈ (ϵ, L].

Therefore, we have

Z(t) = limϵ↓0

Z(t) ≥ limϵ↓0

zϵmin(t) = zmax(t).

So, Z = zmax.In order to show the uniqueness of positive solutions of (2.19), we only need to prove

zmin = zmax.

Suppose that zmin = zmax, then z ′

min(L) ≥ 0. By z ′max(L) = Z ′(L) < 0, we can choose β < 0 such that β ∈ (z ′

max(L), z′

min(L)),and we consider following problem

(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (0, L),u(L) = M, u′(L) = β.

(2.22)

Obviously, zmin and zmax provide us with an ordered sub–supersolution pair of the equation in (2.22). We claim that (2.22)has a positive solution V (t) satisfying

zmin(t) ≤ V (t) ≤ zmax(t), ∀t ∈ (0, L). (2.23)

In fact, if w(t) is a positive solution of (2.22), then we have

|w′(t)|p−2w′(t) = |β|p−2β +

∫ t

Lf (s)h(w(s))ds, t ∈ (0, L).

So, we see

(−w′(t))p−1= |β|

p−1+

∫ L

tf (s)h(w(s))ds,

i.e.,

−w′(t) =

|β|

p−1+

∫ L

tf (s)h(w(s))ds

1p−1

.

Further, for all τ ∈ (0, L), it follows that

w(τ) = M +

∫ L

τ

|β|

p−1+

∫ L

tf (s)h(w(s))ds

1p−1

dt.

Now, for each ϵ > 0, we define an operator Nϵ : C[ϵ, L] → C[ϵ, L] by

(Nϵv)(t) = M +

∫ L

τ

|β|

p−1+

∫ L

tf (s)h(v(s))ds

1p−1

dt, ∀τ ∈ (ϵ, L). (2.24)

In ordered interval [zmin, zmax], it is clear that Nϵ is an increasing operator. We also see z ′max(τ ) < 0 for all τ ∈ (0, L]. Since

zmax(L) = M and z ′max(L) < β < 0, we have for all τ ∈ (ϵ, L)

zmax(τ ) = M +

∫ L

τ

|z ′

max(L)|p−1

+

∫ L

tf (s)h(zmax(s))ds

1p−1

dt

≥ M +

∫ L

τ

|β|

p−1+

∫ L

tf (s)h(zmax(s))ds

1p−1

dt

= Nϵzmax(τ ). (2.25)


In addition, for zmin, it is clear that there exists a unique t0 ∈ (0, L] such that z ′

min(t0) = 0. If t0 = L, then

|z ′

min(t)|p−2z ′

min(t) = −

∫ L

tf (s)h(zmin(s))ds,

so

|z ′

min(t)|p−1

≤ |β|p−1

+

∫ L

tf (s)h(zmin(s))ds and z ′

min(t) < 0, t ∈ (0, L).

Further, we can see

zmin(τ ) ≤ M +

∫ L

τ

|β|

p−1+

∫ L

tf (s)h(zmin(s))ds

1p−1

dt, ∀τ ∈ (0.L), (2.26)

which implies

Nϵzmin ≥ zmin.

If t0 < L, then for any τ ∈ (0, t0] we can obtain

zmin(τ ) ≤ M +

∫ t0

τ

∫ t0

tf (s)h(zmin(s))ds

1p−1

dt,

further, (2.26) holds. It is easy to see z ′

min(τ ) > 0 as τ ∈ (t0, L]. So for each t ∈ (t0, L], we have

0 < |z ′

min(t)|p−2z ′

min(t) = |z ′

min(L)|p−2z ′

min(L) −

∫ L

tf (s)h(zmin(s))ds,

which implies that

z ′

min(t) =

(z ′

min(L))p−1

−

∫ L

tf (s)h(zmin(s))ds

1p−1

.

Hence, for any τ ∈ (t0, L], we have that

zmin(τ ) = zmin(L) −

∫ L

τ

(z ′

min(L))p−1

−

∫ L

tf (s)h(zmin(s))ds

1p−1

dt

≤ M +

∫ L

τ

|β|

p−1+

∫ L

tf (s)h(zmin(s))ds

1p−1

dt.

Therefore, above arguments imply

Nϵzmin ≥ zmin.

Together with (2.25), it follows that

Nϵ : [zmin, zmax] → [zmin, zmax].

By virtue of Ascoli–Arzela theorem, we can get that

Nϵ : [zmin, zmax][ϵ,L] → [zmin, zmax]

[ϵ,L]

is compact. Therefore, Nϵ has a fixed point. We denote it by Vϵ . Similarly to the proof of Theorem 2, i.e., by a compactnessargument involving the Ascoli–Arzela theorem, Vϵ has a convergent subsequence in C1(0, L], without loss of generality,Vϵ → V in C1(0, L], then V is a positive solution of (2.22) and satisfies (2.23). Therefore, V (t) solves (2.19) and satisfiesV ′(L) < 0, which implies V = zmax. But V ′(L) > z ′

max(L), so we obtain a contradiction.

Remark 1. Suppose that (f1) and (h1) hold. Then according to Theorem 3, for each L > 0, the following problem(|u′(t)|p−2u′(t))′ = f (t)h(u(t)), t ∈ (0, L),u(0) = +∞, u(L) = l(L) (2.27)

has a unique positive solution, where l is the unique positive solution of (1.1).


3. The exact blow-up rate of l at t = 0

Proposition 1. For any α ≥ 0(|u′(t)|p−2u′(t))′ = γ tαuq, t > 0,u(0) = +∞, u(+∞) = 0 (3.1)

has a unique positive solution, moreover the solution is

Wγ (t) =

1γ

p + α

q − p + 1

p−1 α + q + 1q − p + 1

(p − 1)

1q−p+1

t−p+α

q−p+1 .

Proof. By calculating,Wγ (t) is a positive solution of (3.1). Theorem2 implies the uniqueness of positive solution of (3.1).

Now, we give an estimate of blow-up rate of l(t) near t = 0.

Proposition 2. Suppose (fi) (hi) (i = 1, 2) hold. Then, there exist δ > 0, ϵ > 0, K > 0 such that

ϵWAB(t) ≤ l(t) ≤ KWAB(t), t ∈ (0, δ]. (3.2)

Proof. It follows from (f2) and (h2) that there exist δ > 0 andM0 > 0 such that

(AB − ϵ)tαuq≤ f (t)h(u) ≤ (AB + ϵ)tαuq, t ∈ (0, δ], u ∈ [M0, +∞).

We denote the unique positive solutions of(|u′(t)|p−2u′(t))′ = (AB − ϵ)tαuq, t > 0,u(0) = +∞, u(+∞) = 0 (3.3)

and (|u′(t)|p−2u′(t))′ = (AB + ϵ)tαuq, t > 0,u(0) = +∞, u(+∞) = 0 (3.4)

by W1 and W2, respectively. By Remark 1, for any δ > 0, the problem(|u′(t)|p−2u′(t))′ = f (t)h(u), t ∈ (0, δ),u(0) = +∞, u(δ) = l(δ) (3.5)

has a unique positive solution l. Consider the auxiliary problem(|u′(t)|p−2u′(t))′ = f (t)h(u), t ∈

1n, δ

,

u(n−1) = W2(n−1), u(δ) = l(δ).(3.6)

We can choose sufficiently small 0 < δ < 1 and suitable 1 ≥ ϵ0 > 0 such that

minl(δ),W2(δ) > M0, M0 ≤ ϵ0W2(δ) ≤ l(δ).

Further, we obtain ϵ0W2(t) ≥ M0 for all t ∈ (0, δ]. So we have

(|(ϵ0W ′

2)|p−2(ϵ0W ′

2))′= ϵ

p−10 (|W ′

2|p−2W ′

2)′

= ϵp−q−10 (AB + ϵ)tα(ϵ0W2)

q

≥ f (t)h(ϵ0W2), (3.7)

which implies that ϵ0W2 is a subsolution of (3.6). Choose K0 > 1 such that K0W1(δ) ≥ maxl(δ),M0, then K0W1(t) ≥ M0for all t ∈ (0, δ]. Hence, we have

(|(K0W ′

1)|p−2(K0W ′

1))′= K p−1

0 (|W ′

1|p−2W ′

1)′

= K p−q−10 (AB − ϵ)tα(K0W1)

q

≤ f (t)h(K0W1), (3.8)


which implies that K0W1 is a supersolution of the equation in (3.6). So, (ϵ0W2, K0W1) provides us with an orderedsub–supersolution pair of (3.6). Thus, (3.6) possesses a positive solution un such that

ϵ0W2(t) ≤ un(t) ≤ K0W1(t), t ∈

[1n, δ

]. (3.9)

By a compactness argument, we can extract a subsequence of un, say unk , approximating to a solution of (3.6). By theuniqueness, the solution must be l. Therefore, passing to the limit as k → ∞, together with (3.9), we have

ϵ0W2(t) ≤ l(t) ≤ K0W1(t), t ∈ (0, δ]. (3.10)

SinceW1 and W2 solve to (3.3) and (3.4), respectively, by virtue of Proposition 1, there exist ϵ > 0 and K > 0 such that

ϵWAB(t) ≤ l(t) ≤ KWAB(t), t ∈ (0, δ].

Theorem 4. Suppose (fi) (hi) (i = 1, 2) hold. Then

limt↓0

l(t)

t−p+α

q−p+1=

1AB

p + α

q − p + 1

p−1 α + q + 1q − p + 1

(p − 1)

1q−p+1

, (3.11)

where l(t) is the unique positive solution of (1.1).

Proof. We consider an auxiliary function

g(t) =l(t)

WAB(t), t ∈ (0, δ].

By virtue of Proposition 2, we have

ϵ ≤ ξ := lim inft↓0

g(t) ≤ ξ := lim supt↓0

g(t) ≤ K0.

In order to show that limt↓0l(t)

WAB(t)exists, we need to show ξ = ξ . Suppose ξ < ξ . Then, there exist two sequences tn, sn,

n ≥ 1, such that

limn→∞

tn = limn→∞

sn = 0, limn→∞

g(tn) = ξ, limn→∞

g(sn) = ξ,

and, for each n ≥ 1,

g ′(tn) = g ′(sn) = 0, g ′′(tn) ≤ 0, g ′′(sn) ≥ 0. (3.12)

Clearly, we see

l′(t) = g ′(t)WAB(t) + g(t)W ′

AB(t)l′′(t) = g ′′(t)WAB(t) + 2g ′(t)W ′

AB(t) + g(t)W ′′

AB(t).

Since (|l′(t)|p−2l′(t))′ = f (t)h(l(t)) and l′(t) < 0, we have

(p − 1)(−l′(t))p−2l′′(t) = f (t)h(l(t)),

that is

(p − 1)−g ′(t)WAB(t) − g(t)W ′

AB(t)p−2 g ′′(t)WAB(t) + 2g ′(t)W ′

AB(t) + g(t)W ′′

AB(t)

= f (t)h (g(t)WAB(t)) . (3.13)

Thus, by (3.12) and (3.13), we obtain that

g(tn) ≥ g ′′(tn)WAB(tn)W ′′

AB(tn)+ g(tn) =

f (tn)h(g(tn)WAB(tn))(p − 1)[−g(tn)W ′

AB(tn)]p−2W ′′

AB(tn),

g(sn) ≤ g ′′(sn)WAB(sn)W ′′

AB(sn)+ g(sn) =

f (sn)h(g(sn)WAB(sn))(p − 1)[−g(sn)W ′

AB(sn)]p−2W ′′

AB(sn),

further, we have

g(tn)p−1≥

f (tn)h(g(tn)WAB(tn))(p − 1)[−W ′

AB(tn)]p−2W ′′

AB(tn)=

f (tn)h(g(tn)WAB(tn))ABtαn W

qAB(tn)

,

g(sn)p−1≤

f (sn)h(g(sn)WAB(sn))(p − 1)[−W ′

AB(sn)]p−2W ′′

AB(sn)=

f (sn)h(g(sn)WAB(sn))ABsαnW

qAB(sn)

.


Thus, passing to the limit as n → ∞ in above two inequalities, we obtain

ξp−1

≥ ξq, ξ p−1

≤ ξ q.

Consequently,

ξ ≤ 1, ξ ≥ 1,

which contradicts ξ < ξ . Therefore, limt↓0l(t)

WAB(t)exists and ϵ ≤ limt↓0

l(t)WAB(t)

≤ K . Let ρ = limt↓0l(t)

WAB(t), then ϵ ≤ ρ ≤ K .

For a given t0 > 0 (sufficiently small) and an arbitrary τ ∈ (0, t0), we have

l(τ ) = l(t0) +

∫ t0

τ

|l′(t0)|p−1

+

∫ t0

tf (s)h(l(s))ds

1p−1

dt, (3.14)

WAB(τ ) = WAB(t0) +

∫ t0

τ

|W ′

AB(t0)|p−1

+

∫ t0

tABsαW q

AB(s)ds 1

p−1

dt. (3.15)

It is easy to see

limτ↓0

l(τ ) − l(t0)WAB(τ ) − WAB(t0)

= limτ↓0

l(τ )

WAB(τ )= ρ, (3.16)

further we have

limτ↓0

t0τ

|l′(t0)|p−1

+ t0t f (s)h(l(s))ds

1p−1

dt t0τ

|W ′

AB(t0)|p−1 + t0t ABsαW q

AB(s)ds 1

p−1dt

= ρ. (3.17)

By (3.14), (3.15), (3.17) and l’Hôpital’s rule, if

limτ↓0

|l′(t0)|p−1+ t0τ

f (s)h(l(s))ds

|W ′

AB(t0)|p−1 + t0τ

ABsαW qAB(s)ds

exists, then

ρp−1= lim

τ↓0

|l′(t0)|p−1+ t0τ

f (s)h(l(s))ds

|W ′

AB(t0)|p−1 + t0τ

ABsαW qAB(s)ds

.

By (3.16), (f2), (h2) and l’Hôpital’s rule, we have

limτ↓0

|l′(t0)|p−1+ t0τ

f (s)h(l(s))ds

|W ′

AB(t0)|p−1 + t0τ

ABsαW qAB(s)ds

= limτ↓0

f (τ )h(l(τ ))

ABτ αW qAB(τ )

= ρq.

So, we have ρp−1= ρq, which implies ρ = 1. Therefore, we have

limt↓0

l(t)

t−p+α

q−p+1=

1AB

p + α

q − p + 1

p−1 α + q + 1q − p + 1

(p − 1)

1q−p+1

.

Acknowledgments

The first author is very grateful to Professor Mingxin Wang, who leads him into the research of boundary blow-upproblems and gives him some favorable advice. The authors thank the referee for his/her careful reading of the manuscriptand many valuable suggestions.

References

[1] J.B. Keller, On solutions of ∆u = f (u), Comm. Pure Appl. Math. 10 (1957) 503–510.[2] R. Osserman, On the inequality ∆u ≥ f (u), Pacific J. Math. 7 (1957) 1641–1647.[3] C. Bandle, M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior, J. Anal. Math. 58 (1991) 9–21.[4] S. Cano-Casanova, J. López-Gómez, Existence, uniqueness and blow-up rate of large solutions for a canonical class of one-dimension problems on the

half-line, J. Differential Equations 244 (2008) 3180–3203.


[5] F. Cîrstea, Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations, Proc. Lond. Math. Soc. 91 (2005) 459–482.[6] F. Cîrstea, Y. Du, Large solutions of elliptic equations with a weakly superlinear nonlinearity, J. Anal. Math. 103 (2007) 261–277.[7] Y. Du, Z. Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89 (2003) 277–302.[8] Y. Du, Q. Hang, Blow-up solutions for a class of semilinear elliptic and parabolic equations, SIAM J. Math. Anal. 31 (1999) 1–18.[9] H. Li, M. Wang, Existence and uniqueness of positive solutions to the boundary blow-up problem for an elliptic system, J. Differential Equations 234

(2007) 246–266.[10] H. Li, P.Y.H. Pang,M.Wang, Boundary blow-up solutions for logistic-type porousmedia equationswith nonregular source, J. Lond.Math. Soc. 80 (2009)

273–294.[11] J. García-Melián, Nondegeneray and uniqueness for boundary blow-up elliptic problems, J. Differential Equations 223 (2006) 208–227.[12] J. García-Melián, Boundary behavior for large solutions to elliptic equations with singular weights, Nonlinear Anal. 67 (2007) 818–826.[13] J. García-Melián, J.C. Sabina de Lis, Maximum and comparison principles for operators involving the p-Laplacian, J. Math. Anal. Appl. 218 (1998) 49–65.[14] J. López-Gómez, The boundary blow-up rate of large solutions, J. Differential Equations 195 (2003) 25–45.[15] J. López-Gómez,, Uniqueness of large soluions for a class of radially symmetric elliptic equations, in: S. Cano-Casanova, J. López-Gómez, C. Mora-Corral

(Eds.), Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, World Scientific, 2005, pp. 75–110.[16] J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Differential Equations 224 (2006) 385–439.[17] J. López-Gómez, Uniqueness of radially symmetric large solutions, in: Supplement Dedicated to the 6th AIMS Conference, Poitiers, France, Discrete

Contin. Dyn. Syst. (2007) 667–686.[18] L. Wei, The existence of large solutions of semilinear elliptic equations with negative exponents, Nonlinear Anal. 73 (2010) 1739–1746.[19] L. Wei, M. Wang, Existence and estimate of large solutions for an elliptic system, Nonlinear Anal. 70 (2009) 1096–1104.[20] P. Tolksdorf, On the Dirichlet problem for quasilinear equations in domains with conical boundary points, Comm. Partial Differential Equations 8

(1983) 773–817.[21] J.L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202.[22] Z. Guo, J. Shang, Remarks on uniqueness of boundary blow-up solutions, Nonlinear Anal. 66 (2007) 484–497.

The existence and blow-up rate of large solutions of one-dimensional -Laplacian equations

Documents

Transcript of The existence and blow-up rate of large solutions of one-dimensional -Laplacian equations