(Krakow)On the oscillatory properties oî solutions

Séria I: PRACE MATEM AT Y CZNE X IX (1976)

Eu g e n i t j s z W a c h n i c k i

(Krakow)

O n the oscillatory properties oî solutions oî certain elliptic equations

In the paper we shall prove th at ни (1er certain assumptions con­

cerning the coefficients a{, i = 1, 2, ..., p, the solutions of the equation (1) Apu{X) + al Ap~l u { X) +. . . + apu{X) = 0

are oscillatory. We suppose th a t the algebraic equation

(

) zp -{-a1zp~1A~-.-A-(ip = 0

has only simple negative roots. We shall give an example of equation (1) for which there exists non oscillatory solutions.

The three-dimensional case was studied in paper [1].

1. We give now definitions and theorem which will be used in the sequel.

De f i n i t i o n

1 ([2]). A solution u(X) of equation (1), defined and class G2p in F n ( n > 1 ) will be called oscillatory if the exterior of every n-dimensional ball contains a zero of u(X) and the set of the zeros of u(X) has no interior points.

Let — c\, —c l , . . . , —cp be simple negative roots of equation (2), where c1} c2, ..., cp are positive numbers. Then equation (1) is of the form (3) ( A+c t ) ( A + 4 ) . . . ( A + c 2 p)u(X) = 0 ,

where сг- > 0? c^ Cj for i ^ j, i ? j — 1? , p, X (x^, x2, • • • ? ^n)‘

Th e o r e m 1 ( [ 7 ] , p . 2 0 1 ) .

For every solution u(X) of equation

there exists a unique system o f the functions щ{Х), i = 1, 2, . . p, such that

p

(I) u(X) ^ u {(X) and Аи{(Х) + (% u{(X)

Let u( X) denote an arbitrary solution of equation (3) and let {щ{Х), i = 1, 2, . . . , p] , be a convenient system of the functions satisfying con­

ditions (4).

Let K ( X 0, R ) denote a ball of radius R and with the centre A 0.

Let S ( X

, R) denote the sphere K ( X 0, R).

Applying to the functions %(X) the mean value theorem ([3], p. 280) we get

(5) f u{(X)dS = ÛnR n~1ui {X0)p{Rci),

S (X 0,R)

where Qn denotes the surface area of the ^-dimensional unit sphere;

p(Rct) = (4В

()'2- ”»

Г(

/2)«7(

_2)/2(Д

(),

J(n_

)/2(-Rct-) being the convenient Bessel functions. In particular case n — 3 we have piRcf) = —— sinRfy.

From (4) and (5) we get

p

(6)

u ( X ) d S = QnR n~1^ ? ui {X0)p{Rci)

S { Xq, R ) i= 1

for every point X 0e E n and every R > 0.

De f in it io n

2 ([4], p. 199). The function f(s) defined and continuous in the interval (

is called uniformly almost periodic function (in the sense of Bohr) if for every e >

there exists a number L >

such th a t in every interim! [a, a -f L] there exists a number r for which

l / ( * + * ) - / ( * ) ! < «

for every se( -

From Definition 2 follows th a t every periodic and continuous function in ( —

is an uniformly almost periodic function.

Theorem 2 ([4], p. 200).

The finite sum of uniformly almost periodic functions is also uniformly almost periodic function.

Theorem 3 ([4], p. 202).

I f f(t) is uniformly almost periodic function f(t) ф 0 and non-negative (non-positive) in the interval { —

then

T T

l i m f № ) dt > 0 ( l i m 4 г f f ( t ) d t < ° ) - T-+ oo J- J

0

0

2. Now we shall prove three lemmas. Let

p

9 (t) = ^ 4 -sin (c o ^ + a).

i=i

L^emma

1. For every system of numbers bi , coi , i = 1, 2, ..., p, a, for

p

which £ b\ > 0, œi Ф 0, со{ ф coj, i ф j, i, j = 1, 2, ..., p, there exists

г—1

the positive and negative value of the function g{t) for

166 E. W a c h n i c k i

P ro o f. By Theorem 2 g(t) is uniformly almost periodic function.

We suppose th at g(t) > 0 for every te ( — oo,

From Theorem 3 follows th at

1 T

lim ~rr f °*

У —> o o L J

I t is easy to observe th at lim

T-+CQ

1 T

J

^g(t)dt

0

_ lir a T f h

T o.b

i= l г

[cos (о»г-Т + a) — cos a] = 0

T—>oo

and we get the contradition. Similarly the assumption th a t g(t) < 0 for every #e( —

conducts to contradiction.

Le m m a 2.

Let f(t) be uniformly almost periodic function and there exists a positive and negative value of f(t) for te ( — oo, oo), Let h(t) be con­

tinuous for te (

oo,

and limü(tf) = 0 as t->oo. Then exists a root of the equation f ( t ) Jr h(t) = 0 in every interval (A, oo), A > 0.

P ro o f. There exist the numbers tx, t2 such th a t f ( t x) < 0, g(t2) > 0.

If f(t) is a periodic function which period œ > 0, then tx-\-na>-^.

t2 -j- no) ->

as n

Since \imh(t) = 0 as t

thus l i m[f(tl + no))Ah(tJ Anco)] = f ( t x) < 0,

^У) n-> oo

lim [/(£2-j-?ia>) -\-h(t2Jr noo)'] = f ( t 2) > 0.

7i->oo

Let A denote an arbitrary positive number. From (7) follows th at there exists a positive number n 0 such th a t t{Ano)e (A, oo), i — 1,2, for n > n 0 and

(8) f ( t x + no) -f h(tx -j-nco) < 0, f ( t 2-\-nw) -\-h(t2Ana>) > 0.

By continuity of the function f(t)-\-h(t) and (8) there exists a point tQe (A, oo) for which f ( t 0) + h(t0) = 0.

Let f(t) be not periodic but uniformly almost periodic function.

Let e — ljn, n = 1, 2, ... There exists a sequence {L n} (Ln > 0), such that in every interval [Ln, 2Ln] there exists a number rn for which

Since f(t) is not periodic function, thus L n^ o o as n-^oo ([4], p. 200).

Hence t{ + rn->oo as n ^ o o , i = 1, 2. By (9) and lim^(t) = 0 follows that

t-> oo

lim[f(tx + rn) A h ( t x + rn)] = f ( t x) < 0

И т[/(< а + тп) + Л(*а + тя)] = f ( t 2) > 0.

and

Similarly as in the proof for periodic function we get the thesis of Lemma 2.

From Lemmas 1 and 2 follows

Lemma

3. I f hx(t) is continuous and bounded function for te (

о

then of every A > 0 the interval ( A

contains a zero of the function g{t} + + for an arbitrary biy coi , a satisfying the. assumptions of Lemma 1.

3. We shall prove the following

T

4. Every non-trivial solution of equation (3) of class C2p in E n (n > 1) is oscillatory.

P ro o f. Let u( X) = u(æx, œ2, ..., wn), n ^ l , be an arbitrary non­

trivial solution of equation (3). Let X 0e E n be a point such th at u { X 0) Ф 0.

By Theorem 1 there exists the system of the functions щ(Х), i = 1, 2,

p

such th a t (6) is satisfied and ]? u\ (X0) > 0.

г= 1

By asymptotic formula ([6], p. 219)

^(ra-2)/2 (-®Сг) — we obtain

i / 2 Г I / - ^ ■ sn

r tu

Bci L sin Всл — (П — 3)

Щ J

p

(10) J u ( X ) d S = <3 сс*п~1) и{{Х0)вт1щ~ j +

1 J

*1

+ -g 2

« « д а or*{n+i)rnm ) I,

* ( X 0 , R )

where G is a constant and the functions rn(Bc{) is continuous and bounded for В > 0.

Let

bi = оГ«п- 1)Щ{*о), a>t = c{, a — £тс(3-ю) and

M-R) = 2 V 1("+14 № ) > » № ) - i=1

From (10) and Lemma 3 follows th at, there exists a number B 0e (A, oo), A > 0, such th a t

f u ( X) d S = 0.

S ( X 0 , R 0 )

Hence there exists a point X xe S { X 0, B 0) such th a t u ( X x) = 0 and X x is a exterior point of the ball K ( X 0, A).

Since the function u(X) is analytic ([5], p. 406) in E n, thus the set

of the zeros of the function u( X) has no interior points.

4. Let ns consider the equation

(11) A {A + 2c2)u(æ, y ) — A*u(æ, y) + 2c2 Au(æ, y) = 0 .

The function u( œ, y) = +sine#cos су, к > 1, is the solution of equation (11) and is positive for (a>,y)eEz. The function u(so,y) = smcxco&cy is the oscillatory solution of equation (11). Hence for equation (11) there exist the oscillatory and non-oscillatory solutions.

References

[1] F. B a r a n sk i, The mean value theorem and oscillatory properties of certain elliptic equations in three-dimensional space (unpublished).

[2] — О wlasnosciach oscylacyjnych i liniach wçslow rozwiqzan pewnych rôwnan rôz- niczkowych czqstkowych typu eliptyeznego, Prace Mat. 7 (1962), p. 71-96.

[3] R. C ou ran t, TJrawnienija z czastnymi proizwodnymi, Moskwa 1964.

[4] R. S. G-uter, L. D. K u d r ia c e w , В. M. L e w ita n , Elementy tieorii funkeij, Moskwa 1963.

[5] K. M aurin, Analiza, cz. II, PW N, Warszawa 1971.

[6] G. To Is to w , Szeregi Fouriera, Warszawa 1954.

[7] I. N. V ek u a , New methods for solving elliptic equations, Amsterdam 1967.