RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO

Sefie II, Tomo XLV (1996), pp. 116-118

REMARKS ON A THEOREM OF GIDAS, NI, AND NIRENBERG

ALICE SIMON - PETER VOLKMANN

The theorem ment ioned in the title is the fol lowing one (cf.

[1]):

THEOREM. Let u (x ) be a positive, C 2 solution o f

- A u + m2u = g(u) in R n, n > 2, m > O

with u(x ) --* 0 as Ix[--+ oo and g continuous, g(u) = O(u~) , ot > 1

near u = O. On the interval 0 < s < uo = m a x u ( x ) , assume

g(s) = gl (s) + g2(s)

nondecreasing and gl ~ C 1 satisfying, f o r some C > O, with g2

p > l ,

(1) [gl(u) - g l (v) l ~ f l u - vl/I l ogmin (u , v)l p (0 < u, v < uo).

Then u(x ) is spherically symmetr ic about some point in R n

and Ur < 0 for r > 0, where r is the radial coordinate about that

point. Furthermore,

l im r(n-1)/2eru(r) = Iz > O. r---~ Oo

REMARKS ON A THEOREM OF GIDAS, NI, AND NIRENBERG 117

Remarks . A continuous function g l " [0, u0] ~ R satisfying (1)

is necessarily constant: In fact, (1) implies

[g l (u ) -- gl(V)[ ~ C ( u - o ) / l l o g v l p (0 < v < u < uo),

and v ~ 0 yields g l ( u ) = g l ( 0 ) ( 0 < u < u o ) .

So it is very surprising that even in recent papers the above

theorem has been repeated without changing condition (1) (cf. [3],

[4]). On the other hand, the proof given in [1] works with (1)

replaced by

(2) [gl(u) - gl(v)l < Clu - ol/I logmax(u , v)l p (0 < u, o ~ u0)

(with the right-hand side being + ~ , if the denominator vanishes).

Therefore we believe, it might have been the original intention of

Gidas, Ni, and Nirenberg, to state their theorem with (2) instead of

(1). A condition similar to (2) has already been used by Isselkou

[2].

Let us finally observe that there are many functions which

satisfy (2), e.g. all functions gl : [0, u0] -+ R such that

Ig l (u ) - gl(V)l ~ Llu q - vq[ (0 < u, v < uo)

for some L > 0, q > 1.

REFERENCES

[1] Gidas B., Ni Wei-Ming, Nirenberg L., Symmetry of positive solutions of nonlinear elliptic equations in R n. Math. Anal. and Appl., part A, Advances in Math. Suppl. Studies 7 A, Academic Press 1981, 369-402.

[2] Isselkou Ould Ahmed Izid Bih, Probldmes semi-lin~aires elliptiqes dans des domaines born~s et non born~s de R 2. Th~se, Universit6 d'OrlEans 1991.

[3] Li Cong-Ming, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains. Commun. Partial Differential Equations, 16 (1991), 585-615.

118 ALICE S I M O N - PETER V O L K M A N N

[4] Li u Ni Wei-Ming, Radial symmetry of positive solutions of nonlinear elliptic equations in R n. Ibid. 18 (1993), 1043-1054

Pervenuto il 5 d icembre 1994.

Alice Simon D~partement de Math~matiques Universitd d'Orlgans, BP 6759

45067 Orleans Cedex 2, France

Peter Volkmann Mathematisches lnstitut I

Universitiit Karlsruhe Postfach 6980

76128 Karlsruhe, Germany