ERRATA TO “SOLUTIONS OF SUPER-LINEARELLIPTIC EQUATIONS AND THEIR MORSEINDICES, I,” DUKE MATH. J. 94 (1998), 141–157

SALEM REBHI

We first point out that the second estimate in Lemma 2.6 (p. 150) is false. Indeed, letGBC

1

.x; y/ be the Green function of BC1 D ¹x 2RN ; jxj< 1;xN > 0º. We then have

��yGBC1

.x; y/D ıy.x/; in BC1 and G.x;y/D 0 if y 2 @BC1 :

HenceZ@BC

1

@GBC

1

.x; y/

@nyd� D

ZBC

1

�GBC

1

.x; y/dy D�1; 8x 2BC1 ;

which implies that

Z@BC

1

@2GBC

1

.x; y/

@xN @nyd� D 0; 8x 2BC1 :

Then we cannot have

@2GBC

1

.x; y/

@xN @ny> 0 on @BC1 D S

C1 [ .� \B1/

wherever the point x 2BC1 is.However, the first estimate of [1, Lemma 2.6] on SC1 D ¹x 2 @B

C1 ; xN > 0º holds

true. That is, when x D .x0; xN /, there exists � > 0 and c > 0 such that if jx0j �1=2; 0 < xN < � , we have

0 <@2G

BC

1

.x; y/

@xN @ny� c; 8y 2 SC1 : (0.1)

Since in the proof of [1, Lemma 2.7] we used the second estimate of Lemma 2.6(see line 5, p.153), we indicate how to overcome this problem.

The function w D 1C v � u satisfies ��w D 0 in RNCw � 0. For R > 0 and for

all x 2BCR , with 0 < xN <R� and 0 < jx0j< R2

, we have

DUKE MATHEMATICAL JOURNALVol. 162, No. 1, © 2013 DOI 10.1215/00127094-1959328Received 23 July 2012.2010 Mathematics Subject Classification. Primary 35J60.

199

200 SALEM REBHI

w.x/D

Z�R

@GBC

R

.x; y/

@nyd�y C

ZSC

R

@GBC

R

.x; y/

@nyw.y/d�y ;

where BCR D ¹.x0; xN / 2R

N ; xN > 0º, �R D ¹xN D 0º\BR, and GBC

R

is the Green

function of BCR . Thus,

@w

@xN.x/D

Z�R

@2GBC

R

.x; y/

@xN @nyd�y C

ZSC

R

@2GBC

R

.x; y/

@xN @nyw.y/d�y D .I /C .II/:

Using the fact that GBC

R

.x; y/ D R2�NGBC

1

. xR; yR/ and w � 0, we derive that

for all R > 0, 0 < xN <R� , and 0 < jx0j< R2

, .II/ is positive.

However, we prove that .I / tends to 0 as R!1. Indeed, let � 1 in BCR . Thenit satisfies � D 0, and so we have

1D

Z�R

@GBC

R

.x; y/

@nyd�y C

ZSC

R

@GBC

R

.x; y/

@nyd�y :

This implies that

Z�R

@2GBC

R

.x; y/

@xN @nyd�y D�

ZSC

R

@2GBC

R

.x; y/

@xN @nyd�y D�

1

R

ZSC

1

@2GBC

1

. xR; yR/

@xN @nyd�1:

Thanks to (0.1), we see that .I / tends to 0 as R!1. We then get that

@w

@xN.x/� 0; 8x 2RNC :

From this point the proof of Lemma 2.7 proceeds in the same way as in [1].

Acknowledgment. The author would like to thank Professor A. Bahri for fruitful dis-cussions.

Reference

[1] A. HARRABI, S. REBHI, and A. SELMI, Solutions of superlinear elliptic equations andtheir Morse indices, II, Duke Math. J. 94 (1998), 159–179. MR 1635912.DOI 10.1215/S0012-7094-98-09407-8. (199, 200)

Département de Mathématiques, Faculté des Sciences de Tunis, Université Elmanar, Campus

universitaire 2092, Tunis, Tunisia; salem_rebhi@yahoo.ca