1. In the paper we shall give any theorem concerning the oscillatory properties of the solutions of th e metaharmonic equation

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X IX (1976) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X IX (1976)

F. B

(Krakôw)

The mean value theorem and oscillatory properties oî certain elliptic equations in three dimensional space

1. In the paper we shall give any theorem concerning the oscillatory properties of the solutions of th e metaharmonic equation

(1) {Д + с\) ... (A + c*)u(X ) = О, X = (a?1? x 2, xz), c{ being positive different constants.

2. We shall use the definition of the oscillatory solution of equation (!) according paper [1]. The following theorems will be applied:

T

1 ([4], p. 208). For every solution u (X ) of the equation there exists a unique system of the functions u{{X), i = 1 , n, being the solutions o f the equations

(2) Ащ {Х) + с]и{{Х) = 0

such that

П

<3) u (X ) = 2 4 ( X ) .

i=l

Let S ( X Q, B) denote the sphere with radius В centered at X 0.

T

2 ([2], p. 230). For every continuous solution u (X ) of equation (2) holds the mean value formula

(4) {4:71В)-1 j f U i(Y)dS = ut {X 0) {BcJ-hinBCi.

S ( X 0,B )

From Theorems 1, 2 follows th a t for every continuous solution u (X ) of equation ( 1 ) we obtain

П

<5) ( 47 Æ 2) - 1 / / «(Y )dS = У ( е гГ Ч ( Х 0) sin Be,.

S { X 0,R ) г = 1

T

3 ([3], p. 215). I f the function f(t) is uniformly almost periodic,

fit) ^ 0 ) for t e ( —

-f-

then for every a e ( —

+

14 F. B a r a n s k i

the inequality holds :

а + Т

lim (T )-1 J f(t)d t > О (< 0), as Т-^оо.

а

3. Now we shall prove the following

T

4. Every non-trivial solution u (X ) of equation (1) continuous in the space E 3 is oscillatory in E 3,

P ro o f. Let X 0 be a point such th a t u { X 0) ф 0. The function

П

F (B ) — ^ щ (Х 0) (ci)“ 1 sinJ2ci

i = 1

is uniformly almost periodic with respect to B e ( — oo, + oo) [3],

a + T n

lim (T )-1 J J h (

{)_ 1 %( 1 0 ) 8

Е

{( 1 £ = 0 when T->oo.

a i — l

Prom Theorem 3 follows th a t the function F (B ) changes the sign in the interval <a, + oo) for [every a. Hence [there exists a number B 0€ (a, +oo) such th a t

F {B 0) = (éTiPo ) - 1 j f u (Y )d S = 0.

fif(X0,Bo)

Prom th e last formula follows th a t the function u(X ) changes the sign on the sphere S ( X 0, B 0).

Since the function u (X ) is analytic th at the set of its zeros does not contain any domain.

References

[1] F. B a ra n sk i, O wlasnoéciaeh oscylacyjnych i liniach wçzlôw rozwiqzan pewnych rôwnan rozniczJcowych czqstlcowych typu eliptycznego, Prace Mat. 7 (1962), p. 71-96.

[2] R. C ourant and D. H ilb ert, Methods of mathematical physics, vol II, New York 1961.

[3] R. S. G-uter, L. D. K u d riacew i В. M. L ew itan , Elementy tieorii funkcij, Moskwa 1963.

[4] I. N. V ekua, New methods for solving elliptic equations, Amsterdam 1960.