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Oscillation of second-order elliptic equations with damping terms
J. Math. Anal. Appl. 317 (2006) 349–363 www.elsevier.com/locate/jmaa
Oscillation of second-order elliptic equations with damping terms Zhi-Ting Xu Department of Mathematics, South China Normal University, Guangzhou 510631, PR China Received 5 September 2004 Available online 3 November 2005 Submitted by Steven G. Krantz
Abstract Some oscillation criteria are given for the second-order elliptic differential equation with damping terms N
N Di aij (x)Dj y + bi (x)Di y + C(x, y) = 0,
i,j =1
x ∈ Ω(r0 ).
i=1
The results are extensions of modified Riccati technique and include earlier results of Noussair and Swanson, Xu, and Zhang et al. 2005 Elsevier Inc. All rights reserved. Keywords: Oscillation; Elliptic differential equations; Damping terms; Modified Riccati technique
1. Introduction The question of oscillation of the second-order linear ordinary differential equation x (t) + p(t)x(t) = 0,
p ∈ C [t0 , ∞), R ,
(E1 )
has been studied by numerous authors by various techniques. In particular, many oscillation criteria have been found which involve the behavior of the integral of the coefficient function p.
E-mail address: [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.10.002
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In 1950, Leighton [7] proved the first well-known oscillation criterion. Specifically, he showed that (E1 ) is oscillatory if t p(s) ds = ∞.
lim
t→∞
(C1 )
t0
Obviously, condition (C1 ) is not necessary for the oscillation of (E1 ). Actually, if t p(s) ds < ∞,
lim
t→∞ t0
(E1 ) may still be oscillatory (see, for example, [5,10]). By constructing function sequence, in 1977, Kamenev [8] gave some beautiful oscillation criteria for (E1 ) under the assumption that p(t) is an “integrally small” coefficient, that is ∞ (C2 ) P (t) = p(s) ds converges. t
Note that in the original work of Leighton [7] and Kamenev [8], they studied the more general equation than (E1 ). But condition (C1 ) or (C2 ) is the base one. The results of Leighton and Kamenev have been later developed also for various type of equation namely discrete, half-linear, functional differential equation, etc. (see, for example, [1,2,6]). In 1980, employing an N -dimensional vector Riccati transformation, Noussair and Swanson [9] first extended the Leighton theorem to the semilinear elliptic equation N
Di aij (x)Dj y + C(x, y) = 0,
x ∈ Ω(r0 ).
(E2 )
i,j =1
The survey paper by Swanson [11] contains a complete bibliography up to 1979. Recently, this technique was explored further by Xu [12] and Zhang et al. [17], who established Kamenev-type oscillation criteria [7] for (E2 ), respectively. For (E2 ), as more late work in this direction we refer the reader to the papers [13–16] and references cited therein. In this paper we will study oscillation of the second-order elliptic differential equation with damping terms N
N Di aij (x)Dj y + bi (x)Di y + C(x, y) = 0,
i,j =1
x ∈ Ω(r0 ),
(1.1)
i=1
in an exterior domain Ω(r0 ) ⊆ RN , where x = (xi )N i=1 ∈ Ω(r0 ), N 2, Di y = ∂y/∂xi for all i, Ω(r0 ) = {x ∈ RN : x r0 } for some r0 > 0, · is the usual Euclidean norm in RN . Throughout this paper, we assume that the following conditions hold: 1+ν (A1 ) A = (aij ) is a real symmetric positive definite matrix function with aij ∈ Cloc (Ω(r0 ), R) for all i, j , ν ∈ (0, 1). Denote by λmax (x) ∈ C(Ω(r0 ), R+ ) the largest eigenvalue of the matrix A. Suppose that there exists a function λ ∈ C([r0 , ∞), R+ ) such that
λ(r) max λmax (x) |x|=r
for r r0 ;
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ν (Ω(r ) × R, R) with C(x, −y) = −C(x, y) for all x ∈ RN and y ∈ R. Suppose (A2 ) C ∈ Cloc 0 ν (Ω(r ), R) and f ∈ C(R, R) ∪ C 1 (R − {0}, R) with that there exist functions p ∈ Cloc 0 yf (y) > 0 and f (y) k > 0 whenever y = 0 such that
C(x, y) p(x)f (y) (A3 )
for all x ∈ Ω(r0 ), y > 0;
1+ν bi ∈ Cloc (Ω(r0 ), R)
for all i. 2+ν (Ω(r0 ), R) is called a solution of (1.1) if y(x) satisfies (1.1) As usual, a function y ∈ Cloc for all x ∈ Ω(r0 ). We restrict our attention only to the nontrivial solution of (1.1), i.e., to the solution y(x) such that sup{|y(x)|: x ∈ Ω(r)} > 0 for every r r0 . Regarding the question of existence of solution of (1.1) we refer the reader to the monograph [3]. A nontrivial solution y(x) of (1.1) is said to be oscillatory in Ω(r0 ) if the set {x ∈ Ω(r0 ): y(x) = 0} is unbounded, otherwise it is said to be nonoscillatory. (1.1) is called oscillatory if all its nontrivial solutions are oscillatory.
The aim of this paper is to study oscillation properties of (1.1) via modified Riccati technique and obtain extensions of Leighton [7] and Kamenev [8] for this equation, thereby improving results of Noussair and Swanson [9], Xu [12] and Zhang et al. [17]. It is to emphasized that the obtained oscillation criteria here are new even for (E2 ). Examples are inserted in the text to illustrate the relevance of theorems. To the best of our knowledge very little is known about the oscillation of (1.1) when the functions bi and p have variable sign on Ω(r0 ). 2. Notations and lemmas It will be convenient to make use of the following notations in the remainder of this paper. For given constant l > 1, let l ∗ = l/(l − 1) > 1 denote the conjugate number to l, Ω(a, b) = x ∈ RN : a x b , Sr = x ∈ RN : x = r , 1 T −1 1 B A B− Di bi , 4k 2k N
θ (x) = p(x) −
i=1
B T = {bi (x), . . . , bN (x)}, dσ denotes the spherical integral element in RN , ωN = 2π N/2 /Γ (N/2) denotes the surface measure of unit sphere. Lemma 2.1. (See [4].) It holds 1 1 α · β α2 + β2 2 2
S1
dσ =
(2.1)
N for every α, β ∈ RN , α = (αi )N i=1 , β = (βi )i=1 .
The following lemma is a modified version of Lemma 1 in [9]. Lemma 2.2. Let φ ∈ C 1 ([r0 , ∞), R+ ). Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all x ∈ Ω(r1 ) (r1 r0 ), then the N -dimensional vector function W(x) is well defined on Ω(r1 ) by
N N 1 1 W(x) = Wi (x) i=1 , Wi (x) = φ x aij (x)Dj y + bi (x) (2.2) f (y) 2k j =1
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and satisfies the following inequality: kW(x)2 φ (x) div W(x) −φ x θ (x) − + W (x) · µ(x), φ(x)λ(x) φ(x) where µ(x) = x/r is the outward unit normal to Sr , r = x.
(2.3)
Proof. In view of (A1 ) and (A2 ), without loss of generality, let us consider that y = y(x) > 0 on Ω(r1 ). It holds
N φ (x) f (y) Wi (x)µi (x) + φ x − 2 Di y Di Wi (x) = aij (x)Dj y φ(x) f (y) j =1
N 1 1 + Di aij (x)Dj y + Di bi (x) . f (y) 2k j =1
From this equation and (1.1), we have
φ (x) f (y) div W(x) = W (x) · µ(x) + φ x − 2 (A∇y)T A−1 (A∇y) φ(x) f (y)
N N 1 1 − bi (x)Di y + C(x, y) + Di bi (x) , f (y) 2k i=1
(2.4)
i=1
−1 −1 where ∇y = (Di y)N i=1 . Since λmax (x) is the smallest eigenvalue of A (x), then 2 −1 (A∇y)T A−1 (A∇y) λ−1 x A∇y2 , max (x)A∇y λ
and from (2.4), (A2 ), it follows that
T 1 φ (x) 1 div W(x) W (x) · µ(x) + φ x −k A∇y A−1 A∇y φ(x) f (y) f (y)
N 1 1 B T ∇y + Di bi (x) − p(x) − f (y) 2k i=1
φ (x) W (x) · µ(x) = φ(x) T W(x) W(x) 1 1 −1 + φ x −k − B A − B φ(x) 2k φ(x) 2k
N W(x) 1 1 − p(x) − B T A−1 − B + Di bi (x) φ(x) 2k 2k i=1
φ (x) k W(x)2 = W (x) · µ(x) − − φ x θ (x). φ(x) φ(x)λ(x) Thus, (2.3) holds, this competes the proof. 2 Remark 2.1. The key point to note is that the term 1/(2k)bi (x) appearing in Wi (x) is very important. Without this term, our method cannot be applied to damped elliptic equation (1.1) (cf. [9,11–17]).
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3. Main results In this section, we will establish some oscillation criteria for (1.1). The first one is a Leightontype oscillation criteria [7] for (1.1). Theorem 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1. If l λ(x) 2 φ x θ (x) − lim φ x dx = ∞ r→∞ 4k φ(x)
(3.1)
Ω(r0 ,r)
and r lim
r→∞
s 1−N ds = ∞. φ(s)λ(s)
(3.2)
r0
Then (1.1) is oscillatory. Proof. Suppose, by contradiction, that y = y(x) is a solution of (1.1) which is positive on Ω(r1 ) for some r1 r0 . Then W(x) is defined by (2.2) on Ω(r1 ). From (2.3), using integration over the domain Ω(r1 , r) and Gauss divergence theorem, follows W(x) · µ(x) dσ − W(x) · µ(x) dσ Sr
Sr1
φ (x) k W(x)2 W(x) · µ(x) − − φ x θ (x) dx. φ(x) φ(x)λ(x)
(3.3)
Ω(r1 ,r)
Then, by Lemma 2.1, φ (x) k W(x)2 W(x) · µ(x) − φ(x) φ(x)λ(x) 2 k W(x)2 l 1 2k − ∗ λ x φ x W(x) · µ(x) − W(x) = lφ(x)λ(x) 2k 2 l φ(x)λ(x) k W(x)2 l λ(x) 2 φ x − ∗ . 4k φ(x) l φ(x)λ(x) This inequality together with (3.3) yields W(x) · µ(x) dσ − W(x) · µ(x) dσ Sr
−
Sr1
k l∗
W(x)2 dx φ(x)λ(x)
Ω(r1 ,r)
− Ω(r1 ,r)
l λ(x) 2 φ x θ (x) − φ x dx. 4k φ(x)
(3.4)
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
In view of (3.1), there exists r2 r1 such that, for r r2 , l λ(x) 2 φ x θ (x) − φ x dx − W(x) · µ(x) dσ 0, 4k φ(x) Sr1
Ω(r1 ,r)
and (3.4) implies k W(x) · µ(x) dσ − ∗ l Sr
W(x)2 dx, φ(x)λ(x)
(3.5)
Ω(r1 ,r)
for r r2 . Application of the Hölder inequality yields 2 µ(x)2 dσ W(x)2 dσ W(x) · µ(x) dσ Sr
Sr
= ωN r N −1
Sr
W(x)2 dσ.
(3.6)
Sr
Denote
H (r) :=
W(x)2 dx. φ(x)λ(x)
Ω(r1 ,r)
Then, by (3.6), H (r) =
1 φ(r)λ(r)
W(x)2 dσ
Sr
r 1−N ωN φ(r)λ(r)
2 W(x) · µ(x) dσ
.
(3.7)
Sr
Combining (3.5) and (3.7), we obtain the following inequality: 1 k 2 r 1−N H 2 (r), H (r) ωN l ∗ φ(r)λ(r) i.e., 1 ωN
k l∗
2
H (r) r 1−N 2 . φ(r)λ(r) H (r)
Integration of this inequality over [r1 , ∞) gives the divergent integral on the left-hand side, according to (3.2), and the convergent integration on the right-hand side. This contradiction completes the proof. 2 Remark 3.1. Theorem 3.1 generalizes and improves Theorem 4 for (E2 ) in [9]. It may happen that condition (3.1) fails to hold. Consequently, Theorem 3.1 do not apply. In the remainder of this paper we treat this case and give Kamenev-type oscillation criteria [8] for (1.1). For this reason, we begin with a useful lemma which has been discussed in [12] for (E2 ). It is similar to Hartman’s Lemma (see [5, p. 365]) for second-order linear ordinary differential equation.
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Lemma 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) holds, and l λ(x) 2 φ x θ (x) − φ x dx lim r→∞ 4k φ(x)
(3.8)
Ω(r0 ,r)
converges. Suppose (1.1) has a nonoscillatory solution y = y(x) for all x ∈ Ω(r1 ) (r r1 ), and let W(x) be defined by (2.2) on Ω(r1 ). Then the following is true for r r1 , l λ(x) 2 W(x) · µ(x) dσ φ x θ (x) − φ x dx 4k φ(x) Sr
Ω(r)
+
∞
k ωN l ∗
τ 1−N φ(τ )λ(τ )
2 W(x) · µ(x) dσ
r
dτ.
(3.9)
Sτ
Proof. Proceeding as in the proof of Theorem 3.1, we know (3.4) holds. It follows, for b r r1 , l λ(x) 2 φ x θ (x) − W(x) · µ(x) dσ + φ x dx 4k φ(x) Sb
Ω(r,b)
+
k ωN l ∗
b
τ 1−N φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ
Sτ
W(x) · µ(x) dσ.
(3.10)
Sr
Now, we claim ∞
r 1−N φ(r)λ(r)
r1
2 W(x) · µ(x) dσ
dr < ∞.
(3.11)
dr = ∞,
(3.12)
Sr
Otherwise ∞ r1
r 1−N φ(r)λ(r)
2 W(x) · µ(x) dσ
Sr
then there exist a number r2 r1 , noting that (3.8) and (3.12), from (3.10), we have, for r r2 ,
k W(x) · µ(x) dσ − 2 ωN l ∗
Sb
b r
τ 1−N φ(τ )λ(τ )
2 W(x) · µ(x) dσ
dτ.
Sτ
As same as the proof of Theorem 3.1, it is easy to show ∞
s 1−N ds < ∞, φ(s)λ(s)
r2
which contradicts (3.2), then, (3.11) holds. Therefore, from (3.10), for r r1 ,
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
W(x) · µ(x) dσ Sr
lim sup b→∞
+
W(x) · µ(x) dσ + Sb
Ω(r)
∞
k ωN l ∗
l λ(x) 2 φ x θ (x) − φ x dx 4k φ(x)
τ 1−N φ(τ )λ(τ )
r
2
W(x) · µ(x) dσ
dτ.
(3.13)
Sτ
If lim supr→∞ Sr W(x) · µ(x) dσ < 0, then there exist two numbers δ < 0 and r3 r2 such that Sb W(x) · µ(x) dσ < δ for b r3 , it follows from (3.2), for r b, ∞
τ 1−N φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
∞ dτ δ
Sτ
which contradicts (3.11). Thus lim supr→∞ (3.9) holds. This proves the lemma. 2
Sr
2
τ 1−N dτ = ∞, φ(τ )λ(τ )
r
W(x) · µ(x) dσ 0, it follows from (3.13) that
Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold, we define a function sequence as follows, for r r0 , l λ(x) 2 φ x θ (x) − φ x dx; Q0 (r) = 4k φ(x) Ω(r) ∞
s 1−N Q2 (s)+ ds; φ(s)λ(s) 0
Q1 (r) = r
∞ Qn+1 (r) = r
2 s 1−N k Q0 (s) + Q (s) ds, n φ(s)λ(s) ωN l ∗ +
n = 1, 2, . . . ,
(3.14)
where g(r)+ = [g(r)]+ = max{g(r), 0}. By induction method, it is easy to prove that (3.14) is a nondecreasing sequence, that is Qn+1 (r) Qn (r)
for r r0 , n = 1, 2 . . . .
(3.15)
Theorem 3.2. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all r ∈ Ω(r1 ) (r1 r0 ), then (3.14) exists and converges, that is lim Qn (r) = Q(r)
n→∞
for r r1 .
(3.16)
Proof. Without loss of generality, let us consider y = y(x) > 0 for x ∈ Ω(r1 ), then the result of Lemma 3.1 holds, and (3.9) implies that W(x) · µ(x) dσ Q0 (r). (3.17) Sr
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357
Note that (3.9) and (3.18) imply that ∞ Q1 (r)
τ 1−n φ(τ )λ(τ )
2 W(x) · µ(x) dσ
r
dτ < ∞.
(3.18)
Sτ
Then, from (3.14) and (3.18), we get k W(x) · µ(x) dσ Q0 (r) + Q1 (r). ωn l ∗ Sr
Thus, by Lemma 3.1, ∞ Q2 (r)
τ 1−n φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ < ∞.
Sτ
By induction method, we can easily obtain ∞ Qm (r)
τ 1−n φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ < ∞.
(3.19)
Sτ
Noting that (3.15) and (3.19), we see that (3.14) is nondecreasing and bounded. So (3.14) exists and converges, that is, (3.16) holds. Therefore, theorem is proved. 2 As an immediate consequence of Theorem 3.2, we have Theorem 3.3. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose for sequence (3.14), one of the following conditions is satisfied: (1) There is a positive integer m 1 such that Q1 (r), . . . , Qm−1 (r) exist, but Qm (r) does not exist; ∗ (2) {Qi (r)}∞ i=1 exists, but there is a r b for an arbitrarily large b r0 such that ∗ limn→∞ Qn (r ) = ∞. Then (1.1) is oscillatory. Remark 3.2. For (E2 ), Theorem 3.2 improves Theorem 1 in [12], Theorem 3.3 extends Theorem 2 in [12] and Theorem 2.2 in [17]. Corollary 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If −1
r s 1−N ωN l ∗ α 1 Q0 (r) ds for some α > . k φ(s)λ(s) 4 r0
Then (1.1) is oscillatory.
(3.20)
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Proof. From (3.14), we have −1 −1
r
r ∗ 2 1−N ∗ 2 1−N ω l s l s ω N N ds ds = c0 α , Q1 (r) α 2 k φ(s)λ(s) k φ(s)λ(s) r0
c0 = α. Then
ωN l ∗ Q2 (r) c1 α k
r0
2 r
s 1−N ds φ(s)λ(s)
−1 ,
c1 = (1 + c0 )2 α.
r0
By induction, it is easy to show that −1 r s 1−N ωN l ∗ 2 ds , Qm+1 (r) cm α k φ(s)λ(s)
cm = (1 + cm−1 )2 α.
r0
Since {cm }∞ n=0 (1 + cm−1 )2 α, oscillatory.
is nondecreasing, then limm→∞ cm = c or ∞. Noting that α > 1/4 and cm = we get limm→∞ cm = ∞. So, limm→∞ Qm (r) = ∞. By Theorem 3.3(2), (1.1) is 2
Remark 3.3. Corollary 3.1 includes Theorem 2.3 for (E2 ) in [17]. By an argument similar to the proof of Corollary 3.1, we easily prove the following corollary. Corollary 3.2. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If ∞
s 1−N ωN l ∗ α Q20 (s)+ ds Q0 (r) φ(s)λ(s) k
1 for some α > . 4
(3.21)
r
Then (1.1) is oscillatory. Theorem 3.4. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all r ∈ Ω(r1 ) (r1 r0 ), then
r 4k s 1−N Q0 (s)+ ds lim sup Q(r) exp < ∞, (3.22) ωN l ∗ φ(s)λ(s) r→∞ r1
where Q(r) is defined in (3.4). Proof. By Lemma 3.1, we have W(x) · µ(x) dσ Q0 (r) + U (r), Sr
where U (r) =
k ωN l ∗
∞ r
τ 1−N φ(τ )λ(τ )
2
W(x) · µ(x) dσ Sτ
dτ.
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
Observing that U (r) = −
r 1−N ∗ ωN l φ(r)λ(r) k
2 W(x) · µ(x) dσ
−
359
r 1−N 4k Q0 (r)+ U (r), ∗ ωN l φ(r)λ(r)
Sr
then
4k U (r) U (r1 ) exp − ωN l ∗
r
s 1−N Q0 (s)+ ds . φ(s)λ(s)
(3.23)
r1
On the other hand, U (r)
k ωN l ∗
∞ r
s 1−N k Q20 (r)+ ds = Q1 (r). φ(s)λ(s) ωN l ∗
By induction method, we have k U (r) Qm (r), m = 1, 2, . . . . ωN l ∗ Noting that (3.23), we have
r 4k s 1−N ωN l ∗ Q U (r1 ), (s) ds Qm (r) exp 0 + ωN l ∗ φ(s)λ(s) k
m = 1, 2, . . . .
(3.24)
r1
By Theorem 3.2, {Qm (r)}∞ m=1 converges, then, (3.24) follows
r s 1−N 4k Q0 (s)+ ds lim Qm (r) exp m→∞ ωN l ∗ φ(s)λ(s)
4k = Q(r) exp ωN l ∗
r1
r
s 1−N ωN l ∗ Q0 (s)+ ds U (r1 ). φ(s)a(s) k
r1
Let r → ∞ in above inequality, we get (3.22) holds.
2
By Theorem 3.4, we have Theorem 3.5. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose for sequence (3.14), one of the following conditions is satisfied: (1) If Qi (r), i = 1, 2, . . . , m, exist, and
r 4k s 1−N Q0 (s)+ ds lim sup Qm (r) exp = ∞; ωN l ∗ φ(s)λ(s) r→∞
(3.25)
r0
(2) If (3.16) holds, and
4k lim sup Q(r) exp ωN l ∗ r→∞
Then (1.1) is oscillatory.
r r0
s 1−N Q0 (s)+ ds φ(s)λ(s)
= ∞.
(3.26)
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Theorem 3.6. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If
r s τ 1−N 4k Q0 (τ )+ dτ ds < ∞ exp − lim r→∞ ωN l ∗ φ(τ )λ(τ ) r0
r0
and there exists m 1 such that r lim sup Qm (s) ds = ∞. r→∞
(3.27)
(3.28)
r0
Then (1.1) is oscillatory. Proof. Proceeding as in the proof of Theorem 3.4, we have (3.24) holds, that is
r s 1−N ωN l ∗ 4k Qm (r) U (r1 ) exp − Q0 (s)+ ds . k ωN l ∗ φ(s)λ(s) r1
Noting that (3.27) and (3.28), let r → ∞ in above inequality, we get a contradiction. This contradiction proves our theorem. 2 Remark 3.4. The results in this paper are still true if we replace condition f (y) k > 0 for y = 0 with the following one f (y) k > 0 for y = 0. y But p(x) should be nonnegative on Ω(r0 ) in this case. In the following we illustrate our theorems with some examples which do not be covered by earlier results in [9–17]. Example 3.1. Consider the elliptic equation y +
∂y ∂y 1 1 3 + ε sinx y + y 3 = 0, + + 1/2 1/2 2x x ∂x1 x ∂x2
where x ∈ Ω(1), N = 2, A = diag(1, 1), ε ∈ R, k = 1, and 3 + ε sinx 1 1 , p(x) = , B= , 1/2 1/2 2x x x 1 + ε sinx x1 + x2 θ (x) = + . 2x 4x5/2 Let φ(r) = 1/r and l = 4, then l λ(x) 2 φ x θ (x) − φ x dx lim r→∞ 4k φ(x) Ω(1,r)
r = 2π lim
r→∞ 1
1 + ε sin s 1 − 2 ds = ∞, 2s s
(3.29)
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
361
and r lim
r→∞
s 1−N ds = lim r→∞ φ(s)λ(s)
1
r ds = ∞. 1
Thus, by Theorem 3.1, (3.29) is oscillatory. Example 3.2. Consider the elliptic equation 1 ∂y ∂ 1 ∂y 1 ∂y ∂ 1 ∂y + + + ∂x1 x2 ∂x1 ∂x2 x2 ∂x2 x3 ∂x1 x3 ∂x2 10 − 4x7/4 + 5x3/4 sinx f (y) = 0, + 4x4
(3.30)
where x ∈ Ω(1), N = 2, f (y) 1 = k > 0, and 1 1 1 1 , B= , A = diag , , x2 x2 x3 x3 10 − 4x7/4 + 5x3/4 sinx p(x) = , 4x4 θ (x) =
8 − 4x7/4 cosx + 5x3/4 sinx 3(x1 + x2 ) + . 4x4 2x5
Let φ(r) = 1 and l > 1, then 1 sin r Q0 (r) = θ (x) dx = 2π 2 + 5/4 , r r Ω(1,r)
consequently,
Q0 (r)+
2π sin r , r 5/4
0,
r ∈ I, otherwise,
where I = [2nπ, (2n + 1)π], n = 1, 2, . . . . So, for any r, there exists integer J 1 such that for r ∈ [2J π, (2J + 1)π], ∞
∞ (2n+1)π
s 1−N Q20 (s)+ ds 4π 2 φ(s)λ(s) n=J
r
∞
sin2 s ds π 2 3/2 s
(2n+1/6)π
n=J (2n+5/6)π
2nπ
∞
1 1 − = 2π 1/2 (2n + 1/6) (2n + 5/6)1/2 n=J ∞ 1 o 3/2 . = 2π 3/2 n 3/2
n=J
For above r, we have 1 = r
∞ r
∞ 2(n+1)π
1 ds 2 s n=J
2nπ
∞ 1 1 1 1 1 . = o ds = 2 2π n(n + 1) 2π s n2 n=J
1 s 3/2
ds
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
Thus, for sufficiently large r, l ∗ > 1 and α > 1/4, we have ∞
s 1−N 1 ωN l ∗ α Q20 (s)+ ds Q0 (r). φ(s)λ(s) r k
r
Hence (3.21) holds. So (3.30) is oscillatory by Corollary 3.2. Example 3.3. Consider the elliptic equation ∂y ∂ 1 ∂y ∂ 1 ∂y 1 ∂y + + + √ ∂x1 x ∂x1 ∂x2 x ∂x2 x x + 1 ∂x1 ∂x1 1 + ε cosx 1 + 4ε sinx 5 y + y + + = 0, 4x(1 + x) x(1 + x)2 √ where x ∈ Ω(1), N = 2, 0 ε < 1 − 2π/4, and 1 1 1 1 , , , B= , A = diag √ √ x x x x + 1 x x + 1) 2 + 4ε sinx 1 + ε cosx p(x) = , + 4x(1 + x) x(1 + x)2 and θ (x) =
1 + ε cosx 1 (3x + 2)(x1 + x2 ) ε sinx + + . √ x(1 + x) x(1 + x) 2 (x x + 1)3
Let φ(r) = 1 and l ∗ = π/2, then 2π(1 + ε cos r) , 1+r ∞ ∞ 1 + ε cos s 2 4π 2 (1 − ε)2 2 2 . Q1 (r) = Q0 (s)+ ds = 4π ds 1+s 1+r Q0 (r) =
r
r
Thus r Q1 (s) ds = ∞,
lim
r→∞ 1
and r lim
r→∞
4k exp − ωN l ∗
1
r = lim
r→∞ 1
r lim
r→∞ 1
s 1
8 exp − π
τ 1−N Q0 (τ )+ dτ ds φ(τ )λ(τ )
s 1
1 + ε cos τ 1+τ
8 exp − (1 − ε)2 π
s 1
dτ ds +
1 dτ ds 1+τ
(3.31)
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
r lim
r→∞
363
(1 + s)− π (1−ε) ds < ∞. 8
2
1
Hence, all conditions of Theorem 3.6 hold, then (3.31) is oscillatory. References [1] R.P. Agarwal, S.R. Grace, D. O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht, 2000. [2] R.P. Agarwal, S.R. Grace, D. O’Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic, Dordrecht, 2002. [3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. [4] G. Hardy, J.E. Littlewood, G. Pólya, Inequalities, second ed., Cambridge Univ. Press, Cambridge, 1999. [5] P. Hartman, Ordinary Differential Equations, Wiley–Interscience, New York, 1964. [6] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Argument, Dekker, New York, 1987. [7] W. Leighton, The detection of oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950) 57–67. [8] I.V. Kamenev, Oscillation of solutions of a second order differential equation with an “integrally small” coefficient, Differ. Uravn. 13 (1977) 2141–2148 (in Russian). [9] E.S. Noussair, C.A. Swanson, Oscillation of semilinear elliptic inequalities by Riccati transformation, Canad. J. Math. 32 (4) (1980) 908–923. [10] C.A. Swanson, Comparison and Oscillatory Theory of Linear Differential Equations, Academic Press, New York, 1968. [11] C.A. Swanson, Semilinear second order elliptic oscillation, Canad. Math. Bull. 22 (1979) 139–157. [12] Z.T. Xu, Oscillation of second order elliptic partial differential equations with a “weakly integrally small” coefficient, J. Sys. Math. Sci. 18 (4) (1998) 478–484 (in Chinese). [13] Z.T. Xu, Oscillation of second order nonlinear differential elliptic equations, Appl. Math. J. Chinese Univ. Ser. A 17 (1) (2002) 29–36 (in Chinese). [14] Z.T. Xu, Riccati technique and oscillation of semilinear elliptic equations, Chinese J. Contemp. Math. 24 (4) (2003) 329–340. [15] Z.T. Xu, D.K. Ma, B.G. Jia, Oscillation theorems for elliptic equations of second order, Acta Math. Sci. Ser. A 24 (2) (2004) 144–151 (in Chinese). [16] Z.T. Xu, H.Y. Xing, Kamenev-type oscillation criteria for semilinear elliptic differential equations, Northeast. Math. J. 22 (2) (2004) 153–160. [17] B.G. Zhang, T. Zhao, B.S. Lalli, Oscillation criteria for nonlinear second order elliptic differential equations, Chinese Ann. Math. Ser. B 1 (17) (1996) 89–102.
Oscillation of second-order elliptic equations with damping terms Zhi-Ting Xu Department of Mathematics, South China Normal University, Guangzhou 510631, PR China Received 5 September 2004 Available online 3 November 2005 Submitted by Steven G. Krantz
Abstract Some oscillation criteria are given for the second-order elliptic differential equation with damping terms N
N Di aij (x)Dj y + bi (x)Di y + C(x, y) = 0,
i,j =1
x ∈ Ω(r0 ).
i=1
The results are extensions of modified Riccati technique and include earlier results of Noussair and Swanson, Xu, and Zhang et al. 2005 Elsevier Inc. All rights reserved. Keywords: Oscillation; Elliptic differential equations; Damping terms; Modified Riccati technique
1. Introduction The question of oscillation of the second-order linear ordinary differential equation x (t) + p(t)x(t) = 0,
p ∈ C [t0 , ∞), R ,
(E1 )
has been studied by numerous authors by various techniques. In particular, many oscillation criteria have been found which involve the behavior of the integral of the coefficient function p.
E-mail address: [email protected]. 0022-247X/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2005.10.002
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
In 1950, Leighton [7] proved the first well-known oscillation criterion. Specifically, he showed that (E1 ) is oscillatory if t p(s) ds = ∞.
lim
t→∞
(C1 )
t0
Obviously, condition (C1 ) is not necessary for the oscillation of (E1 ). Actually, if t p(s) ds < ∞,
lim
t→∞ t0
(E1 ) may still be oscillatory (see, for example, [5,10]). By constructing function sequence, in 1977, Kamenev [8] gave some beautiful oscillation criteria for (E1 ) under the assumption that p(t) is an “integrally small” coefficient, that is ∞ (C2 ) P (t) = p(s) ds converges. t
Note that in the original work of Leighton [7] and Kamenev [8], they studied the more general equation than (E1 ). But condition (C1 ) or (C2 ) is the base one. The results of Leighton and Kamenev have been later developed also for various type of equation namely discrete, half-linear, functional differential equation, etc. (see, for example, [1,2,6]). In 1980, employing an N -dimensional vector Riccati transformation, Noussair and Swanson [9] first extended the Leighton theorem to the semilinear elliptic equation N
Di aij (x)Dj y + C(x, y) = 0,
x ∈ Ω(r0 ).
(E2 )
i,j =1
The survey paper by Swanson [11] contains a complete bibliography up to 1979. Recently, this technique was explored further by Xu [12] and Zhang et al. [17], who established Kamenev-type oscillation criteria [7] for (E2 ), respectively. For (E2 ), as more late work in this direction we refer the reader to the papers [13–16] and references cited therein. In this paper we will study oscillation of the second-order elliptic differential equation with damping terms N
N Di aij (x)Dj y + bi (x)Di y + C(x, y) = 0,
i,j =1
x ∈ Ω(r0 ),
(1.1)
i=1
in an exterior domain Ω(r0 ) ⊆ RN , where x = (xi )N i=1 ∈ Ω(r0 ), N 2, Di y = ∂y/∂xi for all i, Ω(r0 ) = {x ∈ RN : x r0 } for some r0 > 0, · is the usual Euclidean norm in RN . Throughout this paper, we assume that the following conditions hold: 1+ν (A1 ) A = (aij ) is a real symmetric positive definite matrix function with aij ∈ Cloc (Ω(r0 ), R) for all i, j , ν ∈ (0, 1). Denote by λmax (x) ∈ C(Ω(r0 ), R+ ) the largest eigenvalue of the matrix A. Suppose that there exists a function λ ∈ C([r0 , ∞), R+ ) such that
λ(r) max λmax (x) |x|=r
for r r0 ;
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
351
ν (Ω(r ) × R, R) with C(x, −y) = −C(x, y) for all x ∈ RN and y ∈ R. Suppose (A2 ) C ∈ Cloc 0 ν (Ω(r ), R) and f ∈ C(R, R) ∪ C 1 (R − {0}, R) with that there exist functions p ∈ Cloc 0 yf (y) > 0 and f (y) k > 0 whenever y = 0 such that
C(x, y) p(x)f (y) (A3 )
for all x ∈ Ω(r0 ), y > 0;
1+ν bi ∈ Cloc (Ω(r0 ), R)
for all i. 2+ν (Ω(r0 ), R) is called a solution of (1.1) if y(x) satisfies (1.1) As usual, a function y ∈ Cloc for all x ∈ Ω(r0 ). We restrict our attention only to the nontrivial solution of (1.1), i.e., to the solution y(x) such that sup{|y(x)|: x ∈ Ω(r)} > 0 for every r r0 . Regarding the question of existence of solution of (1.1) we refer the reader to the monograph [3]. A nontrivial solution y(x) of (1.1) is said to be oscillatory in Ω(r0 ) if the set {x ∈ Ω(r0 ): y(x) = 0} is unbounded, otherwise it is said to be nonoscillatory. (1.1) is called oscillatory if all its nontrivial solutions are oscillatory.
The aim of this paper is to study oscillation properties of (1.1) via modified Riccati technique and obtain extensions of Leighton [7] and Kamenev [8] for this equation, thereby improving results of Noussair and Swanson [9], Xu [12] and Zhang et al. [17]. It is to emphasized that the obtained oscillation criteria here are new even for (E2 ). Examples are inserted in the text to illustrate the relevance of theorems. To the best of our knowledge very little is known about the oscillation of (1.1) when the functions bi and p have variable sign on Ω(r0 ). 2. Notations and lemmas It will be convenient to make use of the following notations in the remainder of this paper. For given constant l > 1, let l ∗ = l/(l − 1) > 1 denote the conjugate number to l, Ω(a, b) = x ∈ RN : a x b , Sr = x ∈ RN : x = r , 1 T −1 1 B A B− Di bi , 4k 2k N
θ (x) = p(x) −
i=1
B T = {bi (x), . . . , bN (x)}, dσ denotes the spherical integral element in RN , ωN = 2π N/2 /Γ (N/2) denotes the surface measure of unit sphere. Lemma 2.1. (See [4].) It holds 1 1 α · β α2 + β2 2 2
S1
dσ =
(2.1)
N for every α, β ∈ RN , α = (αi )N i=1 , β = (βi )i=1 .
The following lemma is a modified version of Lemma 1 in [9]. Lemma 2.2. Let φ ∈ C 1 ([r0 , ∞), R+ ). Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all x ∈ Ω(r1 ) (r1 r0 ), then the N -dimensional vector function W(x) is well defined on Ω(r1 ) by
N N 1 1 W(x) = Wi (x) i=1 , Wi (x) = φ x aij (x)Dj y + bi (x) (2.2) f (y) 2k j =1
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
and satisfies the following inequality: kW(x)2 φ (x) div W(x) −φ x θ (x) − + W (x) · µ(x), φ(x)λ(x) φ(x) where µ(x) = x/r is the outward unit normal to Sr , r = x.
(2.3)
Proof. In view of (A1 ) and (A2 ), without loss of generality, let us consider that y = y(x) > 0 on Ω(r1 ). It holds
N φ (x) f (y) Wi (x)µi (x) + φ x − 2 Di y Di Wi (x) = aij (x)Dj y φ(x) f (y) j =1
N 1 1 + Di aij (x)Dj y + Di bi (x) . f (y) 2k j =1
From this equation and (1.1), we have
φ (x) f (y) div W(x) = W (x) · µ(x) + φ x − 2 (A∇y)T A−1 (A∇y) φ(x) f (y)
N N 1 1 − bi (x)Di y + C(x, y) + Di bi (x) , f (y) 2k i=1
(2.4)
i=1
−1 −1 where ∇y = (Di y)N i=1 . Since λmax (x) is the smallest eigenvalue of A (x), then 2 −1 (A∇y)T A−1 (A∇y) λ−1 x A∇y2 , max (x)A∇y λ
and from (2.4), (A2 ), it follows that
T 1 φ (x) 1 div W(x) W (x) · µ(x) + φ x −k A∇y A−1 A∇y φ(x) f (y) f (y)
N 1 1 B T ∇y + Di bi (x) − p(x) − f (y) 2k i=1
φ (x) W (x) · µ(x) = φ(x) T W(x) W(x) 1 1 −1 + φ x −k − B A − B φ(x) 2k φ(x) 2k
N W(x) 1 1 − p(x) − B T A−1 − B + Di bi (x) φ(x) 2k 2k i=1
φ (x) k W(x)2 = W (x) · µ(x) − − φ x θ (x). φ(x) φ(x)λ(x) Thus, (2.3) holds, this competes the proof. 2 Remark 2.1. The key point to note is that the term 1/(2k)bi (x) appearing in Wi (x) is very important. Without this term, our method cannot be applied to damped elliptic equation (1.1) (cf. [9,11–17]).
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353
3. Main results In this section, we will establish some oscillation criteria for (1.1). The first one is a Leightontype oscillation criteria [7] for (1.1). Theorem 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1. If l λ(x) 2 φ x θ (x) − lim φ x dx = ∞ r→∞ 4k φ(x)
(3.1)
Ω(r0 ,r)
and r lim
r→∞
s 1−N ds = ∞. φ(s)λ(s)
(3.2)
r0
Then (1.1) is oscillatory. Proof. Suppose, by contradiction, that y = y(x) is a solution of (1.1) which is positive on Ω(r1 ) for some r1 r0 . Then W(x) is defined by (2.2) on Ω(r1 ). From (2.3), using integration over the domain Ω(r1 , r) and Gauss divergence theorem, follows W(x) · µ(x) dσ − W(x) · µ(x) dσ Sr
Sr1
φ (x) k W(x)2 W(x) · µ(x) − − φ x θ (x) dx. φ(x) φ(x)λ(x)
(3.3)
Ω(r1 ,r)
Then, by Lemma 2.1, φ (x) k W(x)2 W(x) · µ(x) − φ(x) φ(x)λ(x) 2 k W(x)2 l 1 2k − ∗ λ x φ x W(x) · µ(x) − W(x) = lφ(x)λ(x) 2k 2 l φ(x)λ(x) k W(x)2 l λ(x) 2 φ x − ∗ . 4k φ(x) l φ(x)λ(x) This inequality together with (3.3) yields W(x) · µ(x) dσ − W(x) · µ(x) dσ Sr
−
Sr1
k l∗
W(x)2 dx φ(x)λ(x)
Ω(r1 ,r)
− Ω(r1 ,r)
l λ(x) 2 φ x θ (x) − φ x dx. 4k φ(x)
(3.4)
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
In view of (3.1), there exists r2 r1 such that, for r r2 , l λ(x) 2 φ x θ (x) − φ x dx − W(x) · µ(x) dσ 0, 4k φ(x) Sr1
Ω(r1 ,r)
and (3.4) implies k W(x) · µ(x) dσ − ∗ l Sr
W(x)2 dx, φ(x)λ(x)
(3.5)
Ω(r1 ,r)
for r r2 . Application of the Hölder inequality yields 2 µ(x)2 dσ W(x)2 dσ W(x) · µ(x) dσ Sr
Sr
= ωN r N −1
Sr
W(x)2 dσ.
(3.6)
Sr
Denote
H (r) :=
W(x)2 dx. φ(x)λ(x)
Ω(r1 ,r)
Then, by (3.6), H (r) =
1 φ(r)λ(r)
W(x)2 dσ
Sr
r 1−N ωN φ(r)λ(r)
2 W(x) · µ(x) dσ
.
(3.7)
Sr
Combining (3.5) and (3.7), we obtain the following inequality: 1 k 2 r 1−N H 2 (r), H (r) ωN l ∗ φ(r)λ(r) i.e., 1 ωN
k l∗
2
H (r) r 1−N 2 . φ(r)λ(r) H (r)
Integration of this inequality over [r1 , ∞) gives the divergent integral on the left-hand side, according to (3.2), and the convergent integration on the right-hand side. This contradiction completes the proof. 2 Remark 3.1. Theorem 3.1 generalizes and improves Theorem 4 for (E2 ) in [9]. It may happen that condition (3.1) fails to hold. Consequently, Theorem 3.1 do not apply. In the remainder of this paper we treat this case and give Kamenev-type oscillation criteria [8] for (1.1). For this reason, we begin with a useful lemma which has been discussed in [12] for (E2 ). It is similar to Hartman’s Lemma (see [5, p. 365]) for second-order linear ordinary differential equation.
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Lemma 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) holds, and l λ(x) 2 φ x θ (x) − φ x dx lim r→∞ 4k φ(x)
(3.8)
Ω(r0 ,r)
converges. Suppose (1.1) has a nonoscillatory solution y = y(x) for all x ∈ Ω(r1 ) (r r1 ), and let W(x) be defined by (2.2) on Ω(r1 ). Then the following is true for r r1 , l λ(x) 2 W(x) · µ(x) dσ φ x θ (x) − φ x dx 4k φ(x) Sr
Ω(r)
+
∞
k ωN l ∗
τ 1−N φ(τ )λ(τ )
2 W(x) · µ(x) dσ
r
dτ.
(3.9)
Sτ
Proof. Proceeding as in the proof of Theorem 3.1, we know (3.4) holds. It follows, for b r r1 , l λ(x) 2 φ x θ (x) − W(x) · µ(x) dσ + φ x dx 4k φ(x) Sb
Ω(r,b)
+
k ωN l ∗
b
τ 1−N φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ
Sτ
W(x) · µ(x) dσ.
(3.10)
Sr
Now, we claim ∞
r 1−N φ(r)λ(r)
r1
2 W(x) · µ(x) dσ
dr < ∞.
(3.11)
dr = ∞,
(3.12)
Sr
Otherwise ∞ r1
r 1−N φ(r)λ(r)
2 W(x) · µ(x) dσ
Sr
then there exist a number r2 r1 , noting that (3.8) and (3.12), from (3.10), we have, for r r2 ,
k W(x) · µ(x) dσ − 2 ωN l ∗
Sb
b r
τ 1−N φ(τ )λ(τ )
2 W(x) · µ(x) dσ
dτ.
Sτ
As same as the proof of Theorem 3.1, it is easy to show ∞
s 1−N ds < ∞, φ(s)λ(s)
r2
which contradicts (3.2), then, (3.11) holds. Therefore, from (3.10), for r r1 ,
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
W(x) · µ(x) dσ Sr
lim sup b→∞
+
W(x) · µ(x) dσ + Sb
Ω(r)
∞
k ωN l ∗
l λ(x) 2 φ x θ (x) − φ x dx 4k φ(x)
τ 1−N φ(τ )λ(τ )
r
2
W(x) · µ(x) dσ
dτ.
(3.13)
Sτ
If lim supr→∞ Sr W(x) · µ(x) dσ < 0, then there exist two numbers δ < 0 and r3 r2 such that Sb W(x) · µ(x) dσ < δ for b r3 , it follows from (3.2), for r b, ∞
τ 1−N φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
∞ dτ δ
Sτ
which contradicts (3.11). Thus lim supr→∞ (3.9) holds. This proves the lemma. 2
Sr
2
τ 1−N dτ = ∞, φ(τ )λ(τ )
r
W(x) · µ(x) dσ 0, it follows from (3.13) that
Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold, we define a function sequence as follows, for r r0 , l λ(x) 2 φ x θ (x) − φ x dx; Q0 (r) = 4k φ(x) Ω(r) ∞
s 1−N Q2 (s)+ ds; φ(s)λ(s) 0
Q1 (r) = r
∞ Qn+1 (r) = r
2 s 1−N k Q0 (s) + Q (s) ds, n φ(s)λ(s) ωN l ∗ +
n = 1, 2, . . . ,
(3.14)
where g(r)+ = [g(r)]+ = max{g(r), 0}. By induction method, it is easy to prove that (3.14) is a nondecreasing sequence, that is Qn+1 (r) Qn (r)
for r r0 , n = 1, 2 . . . .
(3.15)
Theorem 3.2. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all r ∈ Ω(r1 ) (r1 r0 ), then (3.14) exists and converges, that is lim Qn (r) = Q(r)
n→∞
for r r1 .
(3.16)
Proof. Without loss of generality, let us consider y = y(x) > 0 for x ∈ Ω(r1 ), then the result of Lemma 3.1 holds, and (3.9) implies that W(x) · µ(x) dσ Q0 (r). (3.17) Sr
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
357
Note that (3.9) and (3.18) imply that ∞ Q1 (r)
τ 1−n φ(τ )λ(τ )
2 W(x) · µ(x) dσ
r
dτ < ∞.
(3.18)
Sτ
Then, from (3.14) and (3.18), we get k W(x) · µ(x) dσ Q0 (r) + Q1 (r). ωn l ∗ Sr
Thus, by Lemma 3.1, ∞ Q2 (r)
τ 1−n φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ < ∞.
Sτ
By induction method, we can easily obtain ∞ Qm (r)
τ 1−n φ(τ )λ(τ )
r
2 W(x) · µ(x) dσ
dτ < ∞.
(3.19)
Sτ
Noting that (3.15) and (3.19), we see that (3.14) is nondecreasing and bounded. So (3.14) exists and converges, that is, (3.16) holds. Therefore, theorem is proved. 2 As an immediate consequence of Theorem 3.2, we have Theorem 3.3. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose for sequence (3.14), one of the following conditions is satisfied: (1) There is a positive integer m 1 such that Q1 (r), . . . , Qm−1 (r) exist, but Qm (r) does not exist; ∗ (2) {Qi (r)}∞ i=1 exists, but there is a r b for an arbitrarily large b r0 such that ∗ limn→∞ Qn (r ) = ∞. Then (1.1) is oscillatory. Remark 3.2. For (E2 ), Theorem 3.2 improves Theorem 1 in [12], Theorem 3.3 extends Theorem 2 in [12] and Theorem 2.2 in [17]. Corollary 3.1. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If −1
r s 1−N ωN l ∗ α 1 Q0 (r) ds for some α > . k φ(s)λ(s) 4 r0
Then (1.1) is oscillatory.
(3.20)
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Proof. From (3.14), we have −1 −1
r
r ∗ 2 1−N ∗ 2 1−N ω l s l s ω N N ds ds = c0 α , Q1 (r) α 2 k φ(s)λ(s) k φ(s)λ(s) r0
c0 = α. Then
ωN l ∗ Q2 (r) c1 α k
r0
2 r
s 1−N ds φ(s)λ(s)
−1 ,
c1 = (1 + c0 )2 α.
r0
By induction, it is easy to show that −1 r s 1−N ωN l ∗ 2 ds , Qm+1 (r) cm α k φ(s)λ(s)
cm = (1 + cm−1 )2 α.
r0
Since {cm }∞ n=0 (1 + cm−1 )2 α, oscillatory.
is nondecreasing, then limm→∞ cm = c or ∞. Noting that α > 1/4 and cm = we get limm→∞ cm = ∞. So, limm→∞ Qm (r) = ∞. By Theorem 3.3(2), (1.1) is 2
Remark 3.3. Corollary 3.1 includes Theorem 2.3 for (E2 ) in [17]. By an argument similar to the proof of Corollary 3.1, we easily prove the following corollary. Corollary 3.2. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If ∞
s 1−N ωN l ∗ α Q20 (s)+ ds Q0 (r) φ(s)λ(s) k
1 for some α > . 4
(3.21)
r
Then (1.1) is oscillatory. Theorem 3.4. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose (1.1) has a nonoscillatory solution y = y(x) = 0 for all r ∈ Ω(r1 ) (r1 r0 ), then
r 4k s 1−N Q0 (s)+ ds lim sup Q(r) exp < ∞, (3.22) ωN l ∗ φ(s)λ(s) r→∞ r1
where Q(r) is defined in (3.4). Proof. By Lemma 3.1, we have W(x) · µ(x) dσ Q0 (r) + U (r), Sr
where U (r) =
k ωN l ∗
∞ r
τ 1−N φ(τ )λ(τ )
2
W(x) · µ(x) dσ Sτ
dτ.
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
Observing that U (r) = −
r 1−N ∗ ωN l φ(r)λ(r) k
2 W(x) · µ(x) dσ
−
359
r 1−N 4k Q0 (r)+ U (r), ∗ ωN l φ(r)λ(r)
Sr
then
4k U (r) U (r1 ) exp − ωN l ∗
r
s 1−N Q0 (s)+ ds . φ(s)λ(s)
(3.23)
r1
On the other hand, U (r)
k ωN l ∗
∞ r
s 1−N k Q20 (r)+ ds = Q1 (r). φ(s)λ(s) ωN l ∗
By induction method, we have k U (r) Qm (r), m = 1, 2, . . . . ωN l ∗ Noting that (3.23), we have
r 4k s 1−N ωN l ∗ Q U (r1 ), (s) ds Qm (r) exp 0 + ωN l ∗ φ(s)λ(s) k
m = 1, 2, . . . .
(3.24)
r1
By Theorem 3.2, {Qm (r)}∞ m=1 converges, then, (3.24) follows
r s 1−N 4k Q0 (s)+ ds lim Qm (r) exp m→∞ ωN l ∗ φ(s)λ(s)
4k = Q(r) exp ωN l ∗
r1
r
s 1−N ωN l ∗ Q0 (s)+ ds U (r1 ). φ(s)a(s) k
r1
Let r → ∞ in above inequality, we get (3.22) holds.
2
By Theorem 3.4, we have Theorem 3.5. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. Suppose for sequence (3.14), one of the following conditions is satisfied: (1) If Qi (r), i = 1, 2, . . . , m, exist, and
r 4k s 1−N Q0 (s)+ ds lim sup Qm (r) exp = ∞; ωN l ∗ φ(s)λ(s) r→∞
(3.25)
r0
(2) If (3.16) holds, and
4k lim sup Q(r) exp ωN l ∗ r→∞
Then (1.1) is oscillatory.
r r0
s 1−N Q0 (s)+ ds φ(s)λ(s)
= ∞.
(3.26)
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
Theorem 3.6. Let φ ∈ C 1 ([r0 , ∞), R+ ) and l > 1 such that (3.2) and (3.8) hold. If
r s τ 1−N 4k Q0 (τ )+ dτ ds < ∞ exp − lim r→∞ ωN l ∗ φ(τ )λ(τ ) r0
r0
and there exists m 1 such that r lim sup Qm (s) ds = ∞. r→∞
(3.27)
(3.28)
r0
Then (1.1) is oscillatory. Proof. Proceeding as in the proof of Theorem 3.4, we have (3.24) holds, that is
r s 1−N ωN l ∗ 4k Qm (r) U (r1 ) exp − Q0 (s)+ ds . k ωN l ∗ φ(s)λ(s) r1
Noting that (3.27) and (3.28), let r → ∞ in above inequality, we get a contradiction. This contradiction proves our theorem. 2 Remark 3.4. The results in this paper are still true if we replace condition f (y) k > 0 for y = 0 with the following one f (y) k > 0 for y = 0. y But p(x) should be nonnegative on Ω(r0 ) in this case. In the following we illustrate our theorems with some examples which do not be covered by earlier results in [9–17]. Example 3.1. Consider the elliptic equation y +
∂y ∂y 1 1 3 + ε sinx y + y 3 = 0, + + 1/2 1/2 2x x ∂x1 x ∂x2
where x ∈ Ω(1), N = 2, A = diag(1, 1), ε ∈ R, k = 1, and 3 + ε sinx 1 1 , p(x) = , B= , 1/2 1/2 2x x x 1 + ε sinx x1 + x2 θ (x) = + . 2x 4x5/2 Let φ(r) = 1/r and l = 4, then l λ(x) 2 φ x θ (x) − φ x dx lim r→∞ 4k φ(x) Ω(1,r)
r = 2π lim
r→∞ 1
1 + ε sin s 1 − 2 ds = ∞, 2s s
(3.29)
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
361
and r lim
r→∞
s 1−N ds = lim r→∞ φ(s)λ(s)
1
r ds = ∞. 1
Thus, by Theorem 3.1, (3.29) is oscillatory. Example 3.2. Consider the elliptic equation 1 ∂y ∂ 1 ∂y 1 ∂y ∂ 1 ∂y + + + ∂x1 x2 ∂x1 ∂x2 x2 ∂x2 x3 ∂x1 x3 ∂x2 10 − 4x7/4 + 5x3/4 sinx f (y) = 0, + 4x4
(3.30)
where x ∈ Ω(1), N = 2, f (y) 1 = k > 0, and 1 1 1 1 , B= , A = diag , , x2 x2 x3 x3 10 − 4x7/4 + 5x3/4 sinx p(x) = , 4x4 θ (x) =
8 − 4x7/4 cosx + 5x3/4 sinx 3(x1 + x2 ) + . 4x4 2x5
Let φ(r) = 1 and l > 1, then 1 sin r Q0 (r) = θ (x) dx = 2π 2 + 5/4 , r r Ω(1,r)
consequently,
Q0 (r)+
2π sin r , r 5/4
0,
r ∈ I, otherwise,
where I = [2nπ, (2n + 1)π], n = 1, 2, . . . . So, for any r, there exists integer J 1 such that for r ∈ [2J π, (2J + 1)π], ∞
∞ (2n+1)π
s 1−N Q20 (s)+ ds 4π 2 φ(s)λ(s) n=J
r
∞
sin2 s ds π 2 3/2 s
(2n+1/6)π
n=J (2n+5/6)π
2nπ
∞
1 1 − = 2π 1/2 (2n + 1/6) (2n + 5/6)1/2 n=J ∞ 1 o 3/2 . = 2π 3/2 n 3/2
n=J
For above r, we have 1 = r
∞ r
∞ 2(n+1)π
1 ds 2 s n=J
2nπ
∞ 1 1 1 1 1 . = o ds = 2 2π n(n + 1) 2π s n2 n=J
1 s 3/2
ds
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Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
Thus, for sufficiently large r, l ∗ > 1 and α > 1/4, we have ∞
s 1−N 1 ωN l ∗ α Q20 (s)+ ds Q0 (r). φ(s)λ(s) r k
r
Hence (3.21) holds. So (3.30) is oscillatory by Corollary 3.2. Example 3.3. Consider the elliptic equation ∂y ∂ 1 ∂y ∂ 1 ∂y 1 ∂y + + + √ ∂x1 x ∂x1 ∂x2 x ∂x2 x x + 1 ∂x1 ∂x1 1 + ε cosx 1 + 4ε sinx 5 y + y + + = 0, 4x(1 + x) x(1 + x)2 √ where x ∈ Ω(1), N = 2, 0 ε < 1 − 2π/4, and 1 1 1 1 , , , B= , A = diag √ √ x x x x + 1 x x + 1) 2 + 4ε sinx 1 + ε cosx p(x) = , + 4x(1 + x) x(1 + x)2 and θ (x) =
1 + ε cosx 1 (3x + 2)(x1 + x2 ) ε sinx + + . √ x(1 + x) x(1 + x) 2 (x x + 1)3
Let φ(r) = 1 and l ∗ = π/2, then 2π(1 + ε cos r) , 1+r ∞ ∞ 1 + ε cos s 2 4π 2 (1 − ε)2 2 2 . Q1 (r) = Q0 (s)+ ds = 4π ds 1+s 1+r Q0 (r) =
r
r
Thus r Q1 (s) ds = ∞,
lim
r→∞ 1
and r lim
r→∞
4k exp − ωN l ∗
1
r = lim
r→∞ 1
r lim
r→∞ 1
s 1
8 exp − π
τ 1−N Q0 (τ )+ dτ ds φ(τ )λ(τ )
s 1
1 + ε cos τ 1+τ
8 exp − (1 − ε)2 π
s 1
dτ ds +
1 dτ ds 1+τ
(3.31)
Z.T. Xu / J. Math. Anal. Appl. 317 (2006) 349–363
r lim
r→∞
363
(1 + s)− π (1−ε) ds < ∞. 8
2
1
Hence, all conditions of Theorem 3.6 hold, then (3.31) is oscillatory. References [1] R.P. Agarwal, S.R. Grace, D. O’Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht, 2000. [2] R.P. Agarwal, S.R. Grace, D. O’Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic, Dordrecht, 2002. [3] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. [4] G. Hardy, J.E. Littlewood, G. Pólya, Inequalities, second ed., Cambridge Univ. Press, Cambridge, 1999. [5] P. Hartman, Ordinary Differential Equations, Wiley–Interscience, New York, 1964. [6] G.S. Ladde, V. Lakshmikantham, B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Argument, Dekker, New York, 1987. [7] W. Leighton, The detection of oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950) 57–67. [8] I.V. Kamenev, Oscillation of solutions of a second order differential equation with an “integrally small” coefficient, Differ. Uravn. 13 (1977) 2141–2148 (in Russian). [9] E.S. Noussair, C.A. Swanson, Oscillation of semilinear elliptic inequalities by Riccati transformation, Canad. J. Math. 32 (4) (1980) 908–923. [10] C.A. Swanson, Comparison and Oscillatory Theory of Linear Differential Equations, Academic Press, New York, 1968. [11] C.A. Swanson, Semilinear second order elliptic oscillation, Canad. Math. Bull. 22 (1979) 139–157. [12] Z.T. Xu, Oscillation of second order elliptic partial differential equations with a “weakly integrally small” coefficient, J. Sys. Math. Sci. 18 (4) (1998) 478–484 (in Chinese). [13] Z.T. Xu, Oscillation of second order nonlinear differential elliptic equations, Appl. Math. J. Chinese Univ. Ser. A 17 (1) (2002) 29–36 (in Chinese). [14] Z.T. Xu, Riccati technique and oscillation of semilinear elliptic equations, Chinese J. Contemp. Math. 24 (4) (2003) 329–340. [15] Z.T. Xu, D.K. Ma, B.G. Jia, Oscillation theorems for elliptic equations of second order, Acta Math. Sci. Ser. A 24 (2) (2004) 144–151 (in Chinese). [16] Z.T. Xu, H.Y. Xing, Kamenev-type oscillation criteria for semilinear elliptic differential equations, Northeast. Math. J. 22 (2) (2004) 153–160. [17] B.G. Zhang, T. Zhao, B.S. Lalli, Oscillation criteria for nonlinear second order elliptic differential equations, Chinese Ann. Math. Ser. B 1 (17) (1996) 89–102.