( 2 ) 0, 0 ,

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE VIII (1964)

E. Ś

(Kraków)

O n some oscillation problems for the equation A(n)u —f{r )u = 0 in a three-dimensional space

The purpose of this paper is to give some theorems concerning the existence of oscillatory solutions of the equation

(1 ) A(n)u —f(V x 2Jr y 2JrZ2)u — 0

where A(n) denotes the n-times iterated Laplace operator. Our consi­

deration will be based on results contained in [1] and [2]. We give now definitions and theorem which will be used in the sequel.

D

1. A solution u(x, у , z) — 0 of equation (1), defined and of class C2n in an unbounded domain B, whose complement WB is compact, will be called oscilatory if the exterior of every sphere con­

tains a zero of и and the set of the zeros of и has no interior points.

D

2. A bounded domain G will be called a proper knot domain of the function и if u(x, у , z) — 0 in the interior of G and u(x, у , z)

= 0 on the boundary F(G) of G.

D

3. A solution y(t) Ф 0 of the equation

( 2 ) -~~+<p(t)y = dny 0, t ^ a > 0 ,

dt

defined and*of class Cn in (a,

will be called oscillatory if for every b > a there is at least one zero of y(t) in (b,

and the set of the zeros of y(t) has no accumulation points in (a,

T

A

B

[1]. I f cp(t) in equa­

tion ( 2) is positive, if n is even and if O O

j tn~2cp(t)dt —

a

then every solution y{t) Ф 0 of equation (2 ) is oscillatory.

T

K

[2]. I f n is even and m is a positive inte­

ger then there exist a function <p(t) and two поп-trivial solutions y x(t) and

120 E. Śliw ińsk i

у 2(t) of the corresponding equation ( 2) such that between every two suecesive zeros of y t (t) there are at least m zeros of y 2(t).

We shall prove the following

T

1. I f the coefficient f(r) in (1) is positive and if

OG

J r2n~2f(r)d r = oo,

a

then there exists an oscillatory solution of equation (1 ).

P r o o f. Let

u ( x , y , z ) = u(Vx2-\-y2-\-z2)

be a non-constant solution of equation (1 ) which depends only on r

= vof + y2Ą-z2. We put v{r) = ru(r). A simple computation shows that the function u(r) satisfies the equation

(3) u{2n){r)-\--- u^2n~^(r)-\-f(r)u(r) = 0, r

whence it follows that v(r) satisfies the equation

(4) v^2n)(r)-}-f(r)v(r) = 0.

According to the theorem of Ananeva and Balaganskii every non trivial solution v(r) of (4) is oscillatory. Thus the function

u(r) = r~lv{r) is an oscillatory solution of equation (1 ).

C

. I f rx < r2 < < rv < ... is the sequence of zeros of v{r), then the knot domains of the function u(r) are the spherical rings

rv < r < r v+1, v — 1 , 2 , ...

T

2. For every positive integer m there exist a coefficient f(r) and two oscillatory solutions ux and uz of the corresponding equation (1 ) such that every knot domain of иг contains at least m knot domains of u2.

P r o o f. We make the substitution applied in the proof of Theorem 1 , and we obtain our result by the theorem of Kondratev.

References

[1] Г. А н а н е в а и В. Б а л а г а н с к и й , решений некоторых дифференциальных уравнений вышего порядка, Успехи Математ. Наук, т. X I V , выд. 1 (85) (1959), рр.

135-140.

[2] В. К о н д р а т ь е в , О нулевовых решениях уравнения у ^ + р ( х ) у — 0, Докл.

Акад. Наук 120 (1958), рр. 1085-1090. •