On some oscillation problems lor elliptic equations of fourth order

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)

ANN ALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)

E. Śliwiński (Kraków)

On some oscillation problems lor elliptic equations of fourth order

The purpose of this note is to present some results concerning the oscillation of the solutions of certain elliptic equations of fourth order.

First we treat the case of two variables and then the general w-dimen­

sional case (n > 3).

Let us consider the differential equation

д I д и ( х , у ) \ д I d u { x , y ) \

(1) + 5^ ) +

+p( r ) u( x , у) == 0 where

d2 d2

A = — — + — - and r = (x2+ y 2) ox2 dy2

A non-vanishing solution of class C74 of equation (1) will be called oscillatory if is defined for r > B 0 and changes the sign in each set {X: r ^ N} where N ^_R0. We say (1) is an oscillatory equation if it has at least one oscillatory solution.

We shall look for a solution of (1) in the form и = u(r). By a simple computation we may check that every solution u(r) of (1) satisfies the ordinary self-adjoint equation

(2) \ru" (r))"+{u(r) [rq(r) — r~l^\Y+rp{r)u(r) = 0.

We shall make use of the following lemma of Hille [1]:

Le m m a 1 . I f h{x) is a continuous function of constant sign in an in-

CO

terval ж > a > 0 and if j xh(x)dx < ^{o o ,} then there exists an integral y(x) a

of the equation y" (x)-\-h(x)y(x) = 0 satisfying the limit condition Ищу (ж) = 1.

AO—>-oo

200 E. Ś1 i w i ń s k i

Lemma 2. I f 1° p(r)eC, q{r)€<3x for r > a > 0, 20 L(r) = g(r) — §r *

oo

does not change its sign for г > a, 3° the integral j L(r)dr is convergent, a

then there exists a transformation

(3) ~ r = g ( t ) > 0

such that

(4) g' ( t ) > 0, ' g{t)e[a, oo) for t ^ A > 0 and such that equation (2) will be transformed into

(6) {B(t)w"(t))" + P(t)w(t) = 0.

Proof. Let ns consider the equation

(6) (ro' (rj)'+(rq(r) — r~l)o{r) = 0, r > 0. (6) is equivalent to

(7) о" (г)+г~г o'(r)+(q(r) — r~2)o(r) = 0.

The substitution

(8) o" (r) = r~1/2v(r)

converts (7) into

(9) v" (r)-j-v(r)L(r) — 0.

In virtue of Lemma 1 and (8) it exists a solution v(r) of (9) such that limv(r) = 1 and o(r) > 0 for r > A. If g(t) is the function inverse to

Г-+ 0 0

r

t =

j

then the transformation (3) satisfies conditions (4). Indeed, we have Г

F{r) = J u~ll2v{u)du, F'(r) > 0 A

for r ^ A , and moreover for r ^ r0 ^ A we have

r r r

J \u ~ l,2du < J u~ll2v(u)du < j l u ~ ll2du,

r0 rQ r0

r r

i f u~12du < F ( r ) ~ F(r0) J u ~ ll2du.

ro ro

and hence

Oscillation problems for elliptic equations 201

We see that lim F (r) = ©o and that g(t) satisfies (4). When we apply Г—+ОЭ

the transformation (3) to the equation (2), we have (10) \g{t)o3{g(t))w,’(t)]''+{ag{tj)-1p(g{t))w(t) = 0, where w(t) = и (g(t)). If we substitute

R(t) = g(t)a’ {g.(t)), P(t) = (c(g(t)))-'p(g(t)),

then equation (10) will take the form (5).

We now shall use the following lemmas of hTehari and Leighton [2].

Lemma 3. I f the coefficients of equation (5) satisfy the conditions:■

1° R(t)eC*, 2°E(t) > 0, 3 °P(t)eC°, ± ° P ( t ) > 0 for t ^ a > 0, 5° there exists a constant a > 0 such that

lim supr'2-a.S(ć) < 1, lim inff~“P(tf) > a%

<—>oo <->oo

then equation (4) is oscillatory.

Lemma 4. Lei Pit) and B(t) satisfy the following conditions 6° P(t) еС%, 7° P(t) > 0, 8° P (t) e(7°, 9°P(t) < 0 for t ^ a > 9 , 10° for same a > 0 we have limsnpt-2_ajR(i) < 1, then equation (5) is oscillatory.

t—>oo

Theorem 1. Let us assume that:

(a) theie exists a real number a such that

limsup/ 2 aP(£) < 1, liminf[ — f aP{t)] > -^(1 — a2)2,

<—>oo <—>00

where B{t) = g{t)az(g(t)), P(t) = (a(g{t) )~ap (#(*)), (b) p{r) < 0, p(r)eC .for r > 0,

(c) qftyeC1 and L(r) does not change the sign for r ^ a > 0 and

CO

J rL(i)dr < oo,

a

then the equation (1) is oscillatory.

P roof. From Lemmas 2 and 4 it follows that there exists an oscilla­

tory solution w(t) of (5). Let tlt t2, ... denote a sequence of zeros of w(t) arranged in such a way that U -> oo. By Lemma 2 we have rn = g (tn) -> oo and U(rn) = u(g(tn)) = w(tn) = 0.

Let G(r) denote the function inverse to g(r). We see that JJ(r)

= v[G(r)) — u(x, y) is an oscillating solution of (1). Moreover, u(x, у) = 0 on each circle x2 -f y2 = r2n.

2 0 2 E. Ś l i w i ń s k i

Theorem 2. Assume that p(r) is a positive continuous function for r ^ A > 0, L(r) is also continuous and does not change the sign for r > a

> 0. I f we have

limsupt~*~aB(t) < 1, liminftf2_aP(i) > a2,

t—> 00 t—УОО

then equation (1) is oscillatory.

The proof is an analogous to that of Theorem 1 (we use Lemma 3).

Let us now turn over to the case when the number of independent variables is larger than 2. We shall deal with the equation

П д I d u \

AAn+l- d Ai{r)^}+v(r)u=()

where

= и ( х г, . x n), A

v J L

dxi2 ’

i —

1

1/2

Every solution u(x) of the form u(x) = U(r) satisfies the ordinary equation

(11) Ui y ( r ) + 2 ( n ~ l ) r ~ 1U " ' ( r ) + { n - l ) { n - 3 ) r - 2U"{r)+

+ ( n - l ) ( n - 3 ) r ~ 3U'(r)+q(r) V " ( r ) +( n - l ) q ( r ) r ~ l U’(r)+

+ q' (r)U’(r)+p(r)U(r) = 0 .

The transformation V (?) — v(r)r^1 n)l2 does not change the zeros of U{r) and changes (1 1) to the self-adjoint equation

(12) vIY{r)+{[q(r) — (n— l )( n — 3)2~1r~2]v'{r)y+d(r)v(r) = 0 where

d(r) = l%~lr~4,( n—l )(n — 3)(n2+ l n — 2 9) +l ~lr~2(n—±)2q(r) —

— 2~1r~1(n—l)q'(r).

Lemma 5. We assume that q(r) is a continuously differentiable func­

tion in some interval [r, oo), where a > 0. I f the function I(r) = q(r) —

oo

— 2~1r~2(n — l ) ( n — 3) does not change its sign and the integral j I(r)rdr a

is finite, then there exists a transformation r = g(t) satisfying the conditions g'(t) > 0, t ^ A, A = const > 0: g(t) oe when t -> cx> and such that the substitution r — g(t) transforms (12) into the equation

(13) (R(t)w"(t))" + P(t)w(t) = 0, where w(t) = w[g(t)).

Oscillation 'problems for elliptic equations 2 0 3

Proof. Wo take into consideration a solution a(r) of the equation (14) a" (r) + 1 (r) a (r) = 0.

In virtue of Lemma 1 there exists a non-zero solution a(r) of (14) which is defined in some interval r > a, such that a(r) -> 1 when r oo. Then desired transformation is inverse to the following

Г

t — J <j(u)du = F(r), A = a (a).

A

If we substitute r = g(t) in the equation (12) then we obtain the equation (13), where R{t) = g(t)a*(g(t)), P(t) = (a(g(t)))~1 d(g(t)).

Theorem 3. Let us suppose that the assumptions of Lemma 3 are satisfied and that there exists a real number a such that

lim supr'2~ajB(i) < 1, liminff2-a( — P{t)) > 16_ i(l —a2)2.

t—>oo <—>00

Then equation (1) is oscillatory.

The proof is analogous to that of Theorem 1.

Theorem 4. I f the assumption of Lemma 3 remain valid and if there exists a number a such that

limsupt_2_a-R(i) < 1, liminfJ2-a.R(t) > a2,

t—>0O <—>00

then equation (1) is oscillatory.

A proof may be performed in a way analogous to that of Theorem 2.

R eferences

[1] E. I lille , Nonoscillation theorems, Trans. Arner. Math. Soc. 64 (1948), pp. 234-252.

[2] W. L e ig h t o n and Z. N e h a r i, On the oscillation of self adjoint linear differ­

ential equations of the fourth order, Trans. Amer. Math. Soc. 89 (1958), pp. 325-377.