U-+0 as IxJ-+oo. (1.3) or of standing wave solutions for non-linear Schriidinger
equations. (Of .... RADIAL SOLUTIONS OF dld+f(Z.l)=O. 373 there is a sequence
...
JOURNAL
OF DIFFERENTIAL
EQUATIONS
83, 368-378 (1990)
Radial Solutions of Au + f( u) = 0 with Prescribed Numbers of Zeros* W. C. TROY,+ AND F. B. WEISSLER+
KEVIN MCLEOD,
Department of Mathematics, University of Wisconsin, Milwaukee, Wisconsin 53201; tDepartment of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260; and Departement de MathPmariques, UniuersilP de Nantes, 44072 Names Cedex 03, France Received January 10, 1989
1. INTRODUCTION
We consider the initial-boundary value problem U-1 u” + u’ +f( 24)= 0
for r>O
(1.1)
as r-+03.
(1.2)
r
u’(0) = 0,
u(r) --f 0
Here n > 1 is a real parameter, and the function f satisfies the following hypotheses: (f 1) f: R + R is locally Lipschitz continuous, (f2) uf(u) < 0 for (~1 small, 24# 0, (f3) There are values /I > 0 and j?’ < 0 such that F(u) < 0
on
(0, PI,
f(u)
’ 0
on
F(u) 0. Then u(rO) is a local maximum for
371
RADIAL SOLUTIONS OF du+f(u)=O
u and so U(Y)< u(rO) for r slightly larger than rO. Suppose that at some subsequent value of r, say rl , we have u(rl ) = u(rO). Then we would have Q(rl) = h’(r, 1’ + f’(u(r,)) 2 J’lu(rd) = Q(rd, contradicting the fact that Q strictly decreases.It follows that u(r) < u(rO) for all r > rO. Similarly, iff(u,) < 0 we would have deduced that u(r) < u(rO) for all r > rO. The same type of argument also shows that any solution of (1.1) is bounded and must therefore be defined for all r > 0. We observe finally that if u(r) = 0 for some finite value r > 0 then Q(r) = $u’(r)’ > 0, while if u(r) + 0 as r + 0 then Q(r) +O. Thus, if the energy of a solution ever goes negative, that solution cannot subsequently change sign or decay to zero at co. (It is easy to see that it must approach a zero of f as r + co.) In particular, since Q(0) = F(u(0, a)) ~0 for O 0, define u(y, 2) = l-21(p-
‘)u (;,
;121Cp-19.
(2.2)
Then as A -+ co, u(r, 1) + w(r) uniformly on compact subsets of [0, co), where w is the solution of (lS)-(1.6). For 1
