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Dependence of blowup rate oflarge solutions of semilinear ellipticequations, on the curvature of theboundaryCatherine Bandle a & Moshe Marcus ba Mathematisches Institut, Universität Basel, Rheinsprung 21,CH-4051 Basel, Switzerlandb Department of Mathematics, Israel Institute of Technology,Technion City, 32000 Haifa, Israel E-mail:c Communicated by B.Gustafsson

Version of record first published: 05 Mar 2007.

To cite this article: Catherine Bandle & Moshe Marcus (2004): Dependence of blowup rate of largesolutions of semilinear elliptic equations, on the curvature of the boundary, Complex Variables,Theory and Application: An International Journal, 49:7-9, 555-570

To link to this article: http://dx.doi.org/10.1080/02781070410001731729

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Complex Variables,Vol. 49, No. 7–9, 10 June–15 July 2004, pp. 555–570

Dependence of Blowup Rate of Large Solutionsof Semilinear Elliptic Equations, on the Curvatureof the Boundary*

CATHERINE BANDLEyand MOSHEMARCUS

z

yMathematisches Institut,Universita« t Basel, Rheinsprung 21,CH-4051Basel, SwitzerlandzDepartment of Mathematics, Israel Institute of Technology,Technion City, 32000 Haifa, Israel

Communicatedby B.Gustafsson

(Received 5 March 2004)

Let D be a smooth bounded domain in RN . Let f be a positive monotone increasing function on R which

satisfies the Keller–Osserman condition. It is well-known that the solutions of 4u ¼ f ðuÞ, which blow up atthe boundary behave, to a first order approximation, like a function of dist ðx, @DÞ. In this paper we showthat the second order approximation depends on the mean curvature of @D. This paper is an extensionof results in [4] which dealt with radially symmetric solutions. It extends also the results in [5] for f¼ t p.

Keywords: A priori estimates; Upper and lower solutions; Mean curvature

AMS Subject Classifications: 35B40; 35J25; 35J65

1. INTRODUCTION

In this paper we study the asymptotic behavior of solutions of the problem

4u ¼ f ðuÞ in D, uðxÞ ! 1 as x ! @D, ð1:1Þ

where D � RN is a bounded domain and f(t) is a positive non-decreasing function

satisfying the Keller–Osserman condition:

ðtÞ :¼

Z 1

t

dsffiffiffiffiffiffiffiffiffiffiffi2FðsÞ

p <1, F 0 ¼ f 8t > 0 : ðA-1Þ

Solutions of (1.1) are called large solutions.

*To the memory of our friend and colleague Matts Essen.yCorresponding author. E-mail: catherine.bandle@unibas.chzE-mail: marcusm@math.technion.ac.il

ISSN 0278-1077 print: ISSN 1563-5066 online � 2004 Taylor & Francis Ltd

DOI: 10.1080/02781070410001731729

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It is well-known that if D satisfies an inner and outer sphere condition, this is anecessary and sufficient condition for the existence of a solution blowing up at @D[9], [10]. To a first order approximation, the asymptotic behavior of large solutionsat the boundary is the same as that of the solutions of the one-dimensional problem

�00ðxÞ ¼ f ð�ðxÞÞ for x > 0,�ðxÞ ! 1 as x ! 0: ð1:2Þ

Note that the inverse of is a solution of (1.2).It turns out that [2,3], independent of the geometry of @D,

lim�ðxÞ!0

½uðxÞ�=�ðxÞ ¼ 1, �ðxÞ :¼ dist ðx, @DÞ: ð1:3Þ

Under the additional assumption

lim inft!1

ð�tÞ= ðtÞ > 1, 8� 2 ð0,1Þ, ðA-2Þ

it follows that, if � is a solution of (1.2),

uðxÞ

�ð�ðxÞÞ! 1 as �ðxÞ ! 0: ð1:4Þ

Secondary effects, such as the behavior of uðxÞ ��ð�ðxÞÞ as x ! @D, were studied in [4],[7] and [8] under various general assumptions on the nonlinearity f. The role of thegeometry of the domain or, more specifically, the role of the boundary curvature,was first studied in [4] for large solutions in annular domains, under weak assumptionson the nonlinearity. The following result was obtained, for radial solutions u in anannulus fR0 < jxj < R1g:

uðrÞ ¼ � R1 � r�N � 1

2R1ð1þ oð1ÞÞ

Z R1

r

�ð�ðsÞÞ ds

� �ð1:5Þ

as jxj ¼ r ! R1, if blowup occurs at the outer boundary and

uðrÞ ¼ � r� R0 þN � 1

2R0ð1þ oð1ÞÞ

Z r

R0

�ð�ðsÞÞ ds

� �ð1:6Þ

as r ! R0, if blowup occurs at the inner boundary, where

�ðtÞ :¼

R t0

ffiffiffiffiffiffiffiffiffiffiffi2FðsÞ

pds

FðtÞ: ð1:7Þ

Note that assumption (A-1) implies that

limt!1

�ðtÞ ¼ limt!1

ffiffiffiffiffiffiffiffiffiffiffi2FðtÞ

pf ðtÞ

¼ 0: ð1:8Þ

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Therefore the termsR R1

r �ð�ðsÞÞ ds andR rR0

�ð�ðsÞÞ ds in (1.5) and (1.6) represent sec-ondary effects in the blowup behavior of the solution. Observe also that these effectsdepend on the curvature of the boundary where the blowup occurs, namely, 1=R1 in(1.5) and �1=R0 in (1.6).

The results of [4] were extended to arbitrary bounded smooth domains, for equationswith power nonlinearities (in [5],[1]) and with the exponential nonlinearity (in [1]).These results can be stated as follows.

Let �ðxÞ denote the projection of x to @D and let H0ð�Þ be the mean curvature of@D at �. Then, for f ðtÞ ¼ tp, p>1,

uðxÞ ¼ �ð�ðxÞÞ 1þN � 1

pþ 3H0ð�ðxÞÞ�ðxÞ þ oð�ðxÞÞ

� �as x ! @D,

�ð�Þ ¼ cp�� 2

p�1, cp :¼ ðp� 1Þ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðpþ 1Þ

p� �� 2ðp�1Þ

and for f ðtÞ ¼ et

uðxÞ ¼ log2

�2ðxÞþ ðN � 1ÞH0ð�ðxÞÞ�ðxÞ þ oð�ðxÞÞ as x ! @D:

It is interesting to note that the asymptotic behavior in strips is of the formuðxÞ ¼ logð2=�2Þ � ðða�Þ2=6Þ þ oða�2Þ where a depends on the width of the strip.The size of the domain appears therefore in the third term.

In the present paper we present a result on secondary effects in blowup, for arbitrarybounded smooth domains and for a general class of nonlinearities including in parti-cular the power and exponential nonlinearities as well as nonlinearities of the formet

k

, k>0. For the statement of this result we need some additional notation. Put

BðtÞ ¼f ðtÞffiffiffiffiffiffiffiffiffiffiffi2FðtÞ

p ¼d

dt

ffiffiffiffiffiffiffiffiffiffiffi2FðtÞ

p, ð1:9Þ

JðtÞ ¼N � 1

2

Z t

0

�ð�ðsÞÞ ds ð1:10Þ

We observe that

��0ðtÞJðtÞ � ðN � 1Þt�ðtÞ: ð1:11Þ

Indeed, by (1.2) ��0 ¼ffiffiffiffiffiffiffiffiffiffiffiffi2Fð�Þ

pand by the monotonicity of F

�ðtÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffi2=FðtÞ

pt:

Hence

JðtÞ � �ðN � 1Þ

Z t

0

�ðsÞ

�0ðsÞds � �

N � 1

�0ðtÞ

Z t

0

�ðsÞ ds,

which implies (1.11).

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Our main result can be stated as follows:

Let D be a bounded domain of class C4. Suppose that f satisfies conditions (A-1),(A-2) and

ðiÞ lim�!0

Bð�ð�ð1þ oð1ÞÞÞÞ

Bð�ð�ÞÞ¼ 1,

ðiiÞ lim supt!1

BðtÞ�ðtÞ <1:

ðBÞ

Then the large solution u satisfies

juðxÞ ��ð��H0Jð�ÞÞj � �ð�Þoð�Þ, as � ¼ �ðxÞ ! 0: ð1:12Þ

We know that Jð�Þ ¼ oð�Þ however, under condition (B) we obtain a more precise esti-mate, namely,

Jð�Þ ¼ Oð�2Þ: ð1:13Þ

In the case of annular domains inequality (1.12) follows from (1.5) and (1.6), if oneassumes, as in [4], that f satisfies the additional condition

lim sup�!1, �!0

�0ð��Þ=�0ð�Þ <1, ðA-3Þ

In the case of general domains, still assuming (A-3), [4, Thm. 4] yields,

juðxÞ ��ð�ðxÞÞj � c��ð�Þ, ð1:14Þ

which is weaker than (1.12).The proof is based on a construction of upper and lower solutions in a thin strip

parallel to the boundary; the construction is inspired by the formulas developed in[4] and from an idea of [5].

The paper is organized as follows. In Section 2 we present certain formulas involvingthe Laplacian in normal (or flow) coordinates. These formulas are known, but as wecould not locate a reference which includes the necessary details, we provide a briefderivation. Section 3 contains the construction of upper and lower solutions andin Section 4 we derive the main result stated above and some related results such asgradient estimates.

2. PRELIMINARIES

Let us first introduce some notation. Let D � RN be a domain with �:¼@D2C4. Put

�ðxÞ :¼ dist ðx, @DÞ

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and

D� :¼ fx 2 D : �ðxÞ < �g, �� :¼ fx 2 D : �ðxÞ ¼ �g:

Let � be a generic point on �, denote by ð�1, . . . , �N�1Þ its coordinates in some localchart and by �xxð�Þ its representation in Cartesian coordinates. It is well known thatif D is of class Ck, k� 1, then there exists a positive number �0 such that:

(a) For every x 2 D�0 there exists a unique point � 2 @D such that jx� �j ¼ �ðxÞ. Thispoint will be denoted by �ðxÞ.

(b) The mapping x ! ð�ðxÞ, �ðxÞÞ is a Ck diffeomorphism of D�0 onto ð0, �0Þ ��. Thismapping will be denoted by �.

(c) For every x 2 D�0

�ðxÞ ¼ ð�, �Þ¼)x ¼ xð�Þ þ n� ð2:1Þ

where n is the inward normal at �.

From (2.1)

dx ¼ ðx�i þ �n�i Þ d�i þ nd�:

Hence, using Einstein’s summation convention,

jdxj2 ¼ gij � 2�Lij þ �2ðn�i , n�j Þ

� �d�id�j þ d�2, ð2:2Þ

where

gij ¼ ðx�i , x�j Þ and Lij ¼ �1

2½ðn, x�i Þ þ ðn, x�j Þ�,

i, j ¼ 1; . . . ,N�1, are the components of the metric tensor and the second fundamentalform of � respectively. Further, Weingarten’s formula yields

n�j ¼ �gskLsjx�k where gij ¼ ðgijÞ�1:

Substituting this expression into (2.2) we obtain

jdxj2 ¼ gijd�id�j þ d�2, ð2:3Þ

where

gij ¼ gij � 2�Lij þ �2LmjLkig

mk: ð2:4Þ

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Let

ðgijÞ ¼ ðgijÞ�1, g :¼ det ðgijÞ:

Then, in terms of the curvilinear coordinates ð�, �Þ, the Laplace operator in D�0 obtainsthe form

4x ¼1ffiffiffig

p@

@�igij

ffiffiffig

p @

@�j

� �þ

1ffiffiffig

p@

@�

ffiffiffig

p @

@�

� �: ð2:5Þ

This expression becomes particularly simple if the set of coordinates � is chosen sothat the curves �i ¼ constant, i ¼ 1, . . . ,N�1 are the curves defined by the vectorfields of the principal curvatures on �. For this choice of � we obtain

gij ¼ Lij ¼ 0 if i 6¼ j, gii ¼1

���iLii,

where ���i ¼ ���ið�Þ, i ¼ 1, . . . ,N�1, are the principal curvatures of � at �. In this case

jdxj2 ¼XN�1

i¼1

giið1� � ���iÞ2d�2i þ d�2 ¼ gijd�id�j þ d�2 ð2:6Þ

and

g ¼YN�1

i¼1

gii ¼YN�1

i¼1

giið1� � ���iÞ2: ð2:7Þ

Therefore, by (2.5),

4x ¼ 4� þ4� ð2:8Þ

where

4� :¼1ffiffiffig

pXN�1

i¼1

@

@�i

ffiffiffig

p

giið1� 2 ���i�þ ���2i �2Þ

@

@�i

� �ð2:9Þ

4� :¼1ffiffiffig

p@

@�

ffiffiffig

p @

@�

� �¼@2

@�2� ðN � 1ÞH

@

@�, ð2:10Þ

and

H :¼1

N�1

XN�1

i¼1

���i1� ���i�

, ð2:11Þ

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in D�0 . For � 2 ð0, �0Þ, Hð�, �Þ is the mean curvature on ��. The mean curvature on �will be denoted by H0. Accordingly

H0ð�Þ ¼ Hð0, �Þ ¼1

N�1

XN�1

i¼1

���ið�Þ:

3. UPPER AND LOWER SOLUTIONS

In this section we construct upper and lower solutions in a strip parallel to theboundary.

THEOREM 1 Assume (A-1) and (B). For � 2 � and � 2 ð0, �0Þ put

v �ð�, �Þ :¼ � ��N � 1

2ðH0ð�Þ � �Þ

Z �

0

�ð�ðsÞÞ ds

� �ð3:1Þ

where � is a solution of the one-dimensional problem (1.2). Then, there exists �0 > 0 anda decreasing function � : ð0, �0Þ� ð0,1Þ such that �ð�Þ ! 0 as �! 0 and, for every� 2 ð0, �0Þ, the function vþ (resp. v�) with � ¼ �ð�Þ, is an upper (resp. lower) solution inD�, of the equation

�4uþ f ðuÞ ¼ 0: ð3:2Þ

These functions will be denoted by v�� .

Remark Since � is monotone increasing, it is clear that, for every � 2 ð0, �0Þ and everysufficiently small positive �, such that

� >N � 1

2ðH0ð�Þ � �Þ

Z �

0

�ð�ðsÞÞ ds 8� 2 ð0, �Þ,

we have

vþð�, �Þ > v�ð�, �Þ 8�2�, �2ð0, �Þ: ð3:3Þ

Proof Given a real number � (not necessarily positive) put,

ð�, �Þ :¼ �� ðH0ð�Þ þ �ÞJð�Þ

Then v ¼ � and

4v ¼ �0ðÞ4 þ�00ðÞjrj2 ð3:4Þ

jrj2 ¼XN�1

i¼1

ð�i Þ2

giið1� 2 ���i�þ ���2i �2Þþ ð�Þ

2: ð3:5Þ

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Observe that

�i ¼ �Jð�ÞðH0Þ�i ð�Þ,

� ¼ 1�N � 1

2ðH0ð�Þ þ �Þ�ð�ð�ÞÞ

4� ¼ �Jð�Þ4�H0ð�Þ

4� ¼ �N � 1

2ðH0ð�Þ þ �Þ�

0ð�ð�ÞÞ�0ð�Þ

� ðN � 1ÞHðxÞ 1�N � 1

2ðH0ð�Þ þ �Þ�ð�ð�ÞÞ

� �:

Since �0 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffi2Fð�Þ

pit follows that

�0ð�Þ�0 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffi2Fð�Þ

p ffiffiffiffiffiffiffiffiffiffiffi2FðtÞ

pFðtÞ

�f ðtÞ

F2ðtÞ

Z t

0

ffiffiffiffiffiffiffiffiffiffiffi2FðsÞ

pds

!

¼ �2þ 2f ð�Þffiffiffiffiffiffiffiffiffiffiffiffiffi2Fð�Þ

p �ð�Þ ¼ 2ð�1þ Bð�Þ�ð�ÞÞ:

Then

4� ¼ ðN � 1Þ½H0ð�Þ �HðxÞ þ �� ð3:6Þ

� ðN � 1ÞðH0ð�Þ þ �ÞBð�Þ�ð�Þ

þðN � 1Þ2

2HðxÞðH0ð�Þ þ �Þ�ð�Þ

¼ ðN � 1Þð�� ðH0ð�Þ þ �ÞBð�Þ�ð�ÞÞ þOð�Þ:

If @D is sufficiently smooth (e.g. C 4) it follows that

4� ¼ Jð�Þk1ð�, �Þ,XN�1

i¼1

ð�i Þ2

giið1� 2 ���i�þ ���2i �2Þ

¼ J2ð�Þk2ð�, �Þ

where k1 and k2 are bounded functions in D�0 . Inserting all these expressions up to thoseof order Oð�Þ into (3.4) and substituting �00ðÞ by f ðvÞ we find

4v ¼ f ðvÞ þ�0ðÞ�ðN � 1Þ�� BðvÞJ2k2

þ ðN � 1ÞðH0ð�Þ þ �Þ�ð�ÞðBðvÞ � Bð�ÞÞ þOð�Þ�: ð3:7Þ

Since vð�, �Þ ¼ �ð�ð1þ oð1ÞÞÞ, assumption (B) implies that

�ð�ð�ÞÞ�Bðvð�, �ÞÞ � Bð�ð�ÞÞ

�¼ oð1Þ�ð�ð�ÞÞBð�ð�ÞÞ ¼ oð1Þ, ð3:8Þ

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as �! 0. Further, assumption (B)(ii) is equivalent to

supð0, �Þ

jd�ð�ðÞÞ=dj <1, 8� > 0: ð3:9Þ

Indeed, as �0 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2F �

p,

d�ð�ðÞÞ

d¼

ffiffiffi2

pðF �Þ

3=2� f

R�ðÞ0 FðsÞds

ðF �Þ2

�0ðÞ

¼ �2þ 2ðB�Þ �ðÞ:

Therefore, for every �>0, there exist M� > 0 such that

� �ðÞ � M�, ð3:10Þ

and consequently

JðÞ �N � 1

2M�

2, ð3:11Þ

for every 2 ð0, �Þ. Therefore

Bðvð�, �ÞÞJ2ð�Þ � Oð1ÞBð�ð�ÞÞ�2�2ð�ð�ÞÞ ¼ oð�2Þ: ð3:12Þ

The conclusion of the theorem follows from (3.7), (3.8) and (3.12). g

For easy reference we state separately the following result which was established inthe proof of the previous theorem.

LEMMA 1 Condition (B) (ii) is equivalent to the boundedness of the derivative of thefunction � � in every interval ð0, �Þ, �> 0. Consequently this condition implies (3.10)and (3.11).

4. MAIN RESULTS

Throughout this section u(x) will denote a solution of (1.1). By the classical methodof upper and lower solutions Theorem 1 implies that, for every � 2 ð0, �0Þ there existsa solution u� of (3.2) in D� such that

v�� � u�ðxÞ � vþ� ð4:1Þ

where v�� and vþ� are the lower and upper solutions constructed in that theorem.

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We recall that by [4, Thm. 4], if f satisfies conditions (A-1), (A-2) and (A-3),then (1.14) holds. This result follows from (1.11) and the estimate

juðxÞ ��ð�ðxÞÞj � �c�0ð�ðxÞÞJð�ðxÞÞ

� c�ðxÞ�ð�ðxÞÞ as x ! @D ð4:2Þ

where c* and c depend only on f. The next theorem provides a sharper estimatethan (4.2).

THEOREM 2 Assume (A-1)–(A-3) and (B). If u(x) is a solution of (1.1), then

juðxÞ ��ð��H0Jð�ÞÞj � oð�Þ�ð�Þ, as � ¼ �ðxÞ ! 0: ð4:3Þ

Proof By (4.2)

juðxÞ ��ð�Þj � �c�0ð�ÞJð�Þ in D�, ð4:4Þ

where the constant c* is independent of u. Let u� be the solution in D� defined above.By (3.1) and the mean value theorem we have

vþ� ðxÞ ¼ �ð��H0JÞ ��0ð��þÞ�Jð�Þ ð4:5Þ

¼ �ð�Þ ��0ð ~��þÞðH0 þ �ÞJð�Þ,

and

v�� ðxÞ ¼ �ð��H0JÞ þ�0ð���Þ�Jð�Þ ð4:6Þ

¼ �ð�Þ ��0ð ~��þÞðH0 � �ÞJð�Þ,

where ��� (resp. ~���) is a point between ��H0J and �� ðH0 � �Þ J (resp. between � and�� ðH0 � �ÞJ). Therefore, ~��� and ��� are of the order �ð1þ oð1ÞÞ as �! 0.

By (4.5), (4.6) and assumption (A-3),

j�ð�Þ � v�� ðxÞj � �c1�0ð�ÞJð�Þ in D�:

where c1 is a positive constant independent of �. Hence, by (4.4),

juðxÞ � v�� ðxÞj � �c�0ð�ÞJð�Þ in D� ð4:7Þ

for some constant c independent of �. Consider the domains

Dþ� ¼ fx 2 R

N : dist ðx,DÞ < �g, D�� ¼ fx 2 D : dist ðx, @DÞ > �g

with 0 < � < �. Clearly

D�� � D � Dþ

� , @D�� ¼ �� � D�:

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Let U�� be a large solution in D�

� . By the maximum principle we have

U�� � u � Uþ

� in D�� :

If Uþ :¼ lim�!0 Uþ� and U� :¼ lim�!0 U

�� , then

U� � u � Uþ:

It is obvious that Uþ is the maximal solution of (3.2) in D. It is easy to see that U� is alarge solution and, consequently, it is the minimal large solution of (3.2) in D. (To verifythe last assertion: let U�,M be the solution of (3.2) in Dþ

� such that U�,M ¼ M on @Dþ� .

Here M is a fixed positive number. In view of the smoothness of the boundary,the Hopf maximum principle implies that UM ¼ lim�!0 U�,M is the solution of (3.2)in D which satisfies U¼M on the boundary.)

For � 2 ð0, �Þ put

Vþ�, �ðxÞ :¼ � �� �� ðH0ð�Þ þ �Þ Jð�� �Þð Þ

where � ¼ � ð�, �Þ is chosen so that Vþ�, � is an upper solutions in D� nD�. It is easy to

see that

lim�!0

� ð�, �Þ ¼ � ð�Þ, lim�!0

Vþ�, � ¼ vþ� :

Put

L�, � ¼ maxfðUþ � Vþ�, �ÞþðxÞ : x 2 �� g,

where aþ :¼ maxða, 0Þ, and consider the function

h�, � :¼ L�, � 2! � 1ð Þ

�1ð�=�þ 1Þ! � 1ð Þ,

where ! is a number in (0,1). It satisfies h�, � ¼ 0 on @D, h�, � ¼ L�, � on �� and, for �sufficiently small,

4h�, � � 0 in D�:

Consequently the function

u�, � :¼ Vþ�, � þ h�, �

is an upper solution of (3.2) in the ‘‘strip’’ D�nD�,

u�, � � Uþ on ��, u�, � ¼ 1 on ��:

Hence

Uþ � u�, � � Vþ�, � þ h�, � in D�nD�:

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We observe that, by (4.7),

lim�!0

L�, � ¼ L� ¼ maxfðUþ � vþ� ÞþðxÞ : x 2 �� g � �c�0ð�ÞJð�Þ,

while

lim�!0

h�, � ¼ L� 2! � 1ð Þ�1

ð�=�þ 1Þ! � 1ð Þ � �c�0ð�ÞJð�Þ

��:

Hence, letting �! 0 we obtain,

Uþ � vþ� �c�0ð�ÞJð�Þ

�� in D�: ð4:8Þ

This inequality and (4.5) imply

Uþ � � ð��H0JÞ ��0ð��þÞ�ð�ÞJð�Þ �c�0ð�ÞJð�Þ

�� in D�: ð4:9Þ

By (1.11) we have �c�0ð�ÞJð�Þ � cðN � 1Þ��ð�Þ. Now, given a>0, choose � suffi-ciently small so that �ð�Þ � a=2, and after that choose �a sufficiently small so that(with the value of � already chosen) cðN � 1Þ�ð�Þ � ða=2Þ�ð�Þ for � � �a. Then,using (A-3),

Uþ � �ð��H0JÞ þ ða=2Þ��0ð�ÞJð�Þð1þ oð1ÞÞ þ�ð�Þ�

�in D�:

Finally this implies

Uþ � �ð��H0JÞ þ oð1Þ�ð�Þ�, ð4:10Þ

where oð1Þ ! 0 as �! 0. Hence

u � �ð��H0JÞ þ oð1Þ�ð�Þ�: ð4:11Þ

In order to derive a similar lower estimate, put

~LL�, � ¼ maxfðv�� �U�� ÞþðxÞ : x 2 �� g,

and consider the function

~hh�, � :¼ ~LL�, � 2! � 1ð Þ

�1ð�=�þ 1Þ! � 1ð Þ,

where ! is a number in (0, 1). It satisfies ~hh�, � ¼ 0 on @D, ~hh�, � ¼ L�, � on �� and, for �sufficiently small,

4 ~hh�, � � 0 in D�:

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Consequently the function

u�, � :¼ v�� � ~hh�, �

is a lower solution of (3.2) in the ‘‘strip’’ D�,

u�, � � U�� on ��, u�, � ¼ 1 on @D:

Hence

U�� � u�, � in D�:

We observe that, by (4.2),

lim�!0

~LL�, � ¼ ~LL� ¼ maxfðv�� �U�ÞþðxÞ : x 2 �� g � c��ð�Þ,

and hence

lim�!0

~hh�, � � c��ð�Þ:

Therefore, letting �! 0 we obtain,

U� � v�� � c��ð�Þ, in D�: ð4:12Þ

This inequality and (4.6) imply

U� � �ð��H0jÞ ��0ð���Þ�ð�ÞJð�Þ � c��ð�Þ in D�: ð4:13Þ

Therefore, as before,

u � �ð��H0JÞ þ oð1Þ�ð�Þ�: ð4:14Þ

This completes the proof of the theorem. g

In [4] it was proved that under the assumptions (A-1), (A-2), (A-3) and

limt!1

t4=FðtÞ ¼ 0, ðA-4Þ

u behaves asymptotically like

uðxÞ ��ð�ðxÞÞ ! 0 as x ! @D: ð4:15Þ

In view of Theorem 2 it is possible to improve (4.15). For this purpose let us introducethe condition

lim supt!1

t4=FðtÞ <1: ðA-4Þ0

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COROLLARY 1 Let the assumptions of the previous theorem and (A-4)0 hold. Then

uðxÞ ��ð�ðxÞÞ ��0ð�ðxÞÞH0ð�Þ

Z �

0

�ð�ðsÞÞds ! 0 as x ! @D:

Proof Because of Theorem 2 we only have to show that ��ð�Þ < const. for � < �0.By the rule of Bernoulli-l’Hospital and (A-4)0 we have

lim�!0

��ð�Þ ¼ limt!1

t

Z 1

t

dsffiffiffiffiffiffiffiffiffiffiffi2FðsÞ

p ¼ limt!1

t2ffiffiffiffiffiffiffiffiffiffiffi2FðtÞ

p <1:

The conclusion now follows from (4.3). g

It is well-known [3] that for domains with positive mean curvature H0 the followinginequality holds

jruj2 � 2ðFðuÞ � FðuminÞÞ:

Under the conditions of Corollary 1 we have

u� ¼ ��0ð��H0jÞ 1�H0N � 1

2�ð�ð�ÞÞ

� �ð1þ oð1ÞÞ:

From these observations we deduce that jruj � u�. This indicates that the level surfacesof u tend to parallel surfaces near the boundary.

Observe that this corollary is an improvement of (4.15) only in the case wherelim supt!1t4=FðtÞ > 0.

Example 1 Let f ðtÞ ¼ t p. Then �ð�Þ ¼ cp��2=p�1 and �ðtÞ ¼ ð2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðpþ 1Þ

p=pþ 3Þ t1�p=2.

Theorem 2 implies

uðxÞ ¼ � ��ðN � 1Þðp� 1Þ

2ðpþ 3ÞH0ð�Þ�

2

� �ð1þ oð�ÞÞ

¼ �ð�Þ 1þN � 1

pþ 3H0ð�Þ�þ oð�Þ

� �as �! 0:

This result was already obtained in [1] and [5]. Notice that for p>3

lim�!0

ðuðxÞ � cp��2=p�1Þ ¼ 0 ð4:16Þ

and for p¼ 3

lim �!0 uðxÞ �ffiffiffi2

p��1

� �¼

ffiffiffi2

p

6ðN � 1ÞH0ð�Þ: ð4:17Þ

(4.16) was already obtained in [8], [4] whereas (4.17) seems to be new.

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Example 2 Let f ðtÞ ¼ et. Theorem 2 does not depend on the particular choice of asolution of (1.2). If we take �ð�Þ ¼ logð2=�2), then �ðtÞ ¼ 23=2ðe�t=2 � e�tÞ and

�ð��H0JÞ ¼ log2

�� ðN � 1ÞHðð�2=2Þ � ðffiffiffi2

p�3=6Þ

� �2¼ log

2

�2þ ðN � 1ÞH0ð�Þ�þOð�2Þ as �! 0:

Theorem 2 implies

uðxÞ ¼ log2

�2ðxÞþ ðN � 1ÞH0ð�ðxÞÞ�ðxÞ þ oð�ðxÞÞ log

2

�2: ð4:18Þ

The error term can be improved considerably by taking a more general solutionof (1.2), namely

�að�Þ ¼

loga2

sinh2ða�=ffiffiffi2

pÞ

solving �02 ¼ 2ðe� þ a2Þ

loga2

sin2ða�=ffiffiffi2

pÞ

solving �02 ¼ 2ðe� � a2Þ.

8>>><>>>:

For small values of a� we get by a straightforward computation

��a ð�Þ ¼ log

2

�2�ða�Þ2

6þOðða�Þ3Þ

and

Jð�Þ ¼N � 1

2�2 þOðða�Þ3Þ

Hence the upper and lower solutions defined in (3.1) assume the form

v�ð�, �Þ ¼ log2

�2� ðN � 1ÞH0ð�Þ�þ ��

2 �ða�Þ2

6þ oðða�Þ2Þ:

In view of (4.4) it is possible to choose a ¼ c=ffiffiffi�

psuch that for large c the following

inequality holds v� � u � vþ on ��. Consequently by the maximum principle thisinequality is valid in the whole parallel strip D�. The same argument as for Theorem 2yields

uðxÞ ¼ log2

�2ðxÞþ ðN � 1ÞH0ð�ðxÞÞ�ðxÞ þ oð�ðxÞÞ as x ! @D:

This result was already stated in [1]. The proof indicated there however has a gap.

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Acknowledgement

The research for this paper was done when the first author C.B. visited the Technion.She would like to express her gratitude for the hospitality and the stimulatingatmosphere.

References

[1] C. Bandle (2003). Asymptotic behaviour of large solutions of quasilinear elliptic problems. ZAMP, 54,1–8.

[2] C. Bandle and M. Essen, (1994). On the solutions of quasilinear elliptic problems with boundary blow-up.Symposia Mathematica, 35, 93–111.

[3] C. Bandle and M. Marcus (1992). Large solutions os semilinear elliptic equations: existence, uniquenessand asymptotic behaviour. J. d’Anal. Math., 58, 9–24.

[4] C. Bandle and M. Marcus (1998). On second- order effects in the boundary behaviour of large solutionsof semilinear elliptic problems. Diff. Int. Equ., 11, 23–34.

[5] M. del Pino and R. Letelier (2002). The influence of domain geometry in boundary blow-up elliptic prob-lems. Nonlinear Anal. TMA, 48, 897–904.

[6] D. Gilbarg and N.S. Trudinger (1998). Elliptic partial differential equations of second order. Springer.[7] A. Greco and G. Porru (1997). Asymptotic estimates and convexity of large solutions to semilinear ellip-

tic equations. Diff. Int. Equ., 10, 219–229.[8] A.C. Lazer and P.J. McKenna (1994). Asymptotic behaviour of solutions of boundary blow up problems.

Diff. Int. Equ., 7, 1001–1019.[9] J.B. Keller (1957). On solutions of 4u ¼ f ðuÞ. Comm. Pure Appl. Math., 10, 503–510.

[10] R. Osserman (1957). On the inequality 4u � f ðuÞ. Pac. J. Math., 7, 1641–1647.

APPENDIX (B)(i) is satisfied if

limt!1

Ff 0

f 2¼ � 6¼

1

2: ð4:19Þ

Proof We know that Kð�Þ :¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Fð�ð�ÞÞ

p=f ð�ð�ÞÞ ! 0 as �! 0. Hence

K 0ð0Þ ¼ lim�!0

Kð�Þ

�¼ 1� lim

�!0

2Fð�ð�ÞÞf 0ð�ð�ÞÞ

f 2ð�ð�ÞÞ¼ 1� 2�:

If (4.19) holds, then 1=Bð�ð�ÞÞ ¼ Kð�Þ and therefore Bð�ð�ÞÞ � ð1� 2�Þ�1=�. Therefore(B)(i) holds. g

The condition (4.19) is very general. It holds for instance for f ðtÞ ¼ t log pðtÞ, p > 1,f ðtÞ ¼ et

p

. If � ¼ 1=2 then FðtÞ ! t2 as t ! 1. These functions however do notsatisfy (A-1).

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