© 2000 by London Mathematical Society
© The London Mathematical Society
On Positive Multipeak Solutions of a Nonlinear Elliptic Problem
School of Mathematics, University of New South Wales Sydney, NSW 2052, Australia
School of Mathematics and Statistics, University of Sydney Sydney, NSW 2006, Australia
Received 19 August 1997. Revision received 24 June 1999.
In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem
–
2
u+u=Q(x)|u|q–2u, x
RN,
uH1(RN) (1.1)
where = 
Ni=1
2/x2i is the Laplace operator in RN, 2 < q < 
for N = 1, 2, 2 < q < 2N/(N–2) for N
3, and Q(x) is a bounded positive continuous function on RN satisfying the following conditions.
(Q1) Q has a strict local minimum at some point x0RN, that is, for some > 0
Q(x)>Q(x0)
for all 0 < |x–x0| < .
(Q2) There are constants C, 
> 0 such that
|Q(x)–Q(y)|
C|x–y|
for all |x–x0| , |y–y0| .
Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in RN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as 
0 everywhere else in RN.