Forced Oscillations of Half-Linear Elliptic Equations via ...downloads.hindawi.com/journals/ijde/2010/520486.pdf · Forced oscillations of superlinear elliptic equations of the form

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2010, Article ID 520486, 9 pagesdoi:10.1155/2010/520486

Research ArticleForced Oscillations of Half-Linear EllipticEquations via Picone-Type Inequality

Norio Yoshida

Department of Mathematics, University of Toyama, Toyama 930-8555, Japan

Correspondence should be addressed to Norio Yoshida, [email protected]

Received 3 September 2009; Revised 10 December 2009; Accepted 11 December 2009

Academic Editor: Yuri V. Rogovchenko

Copyright q 2010 Norio Yoshida. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Picone-type inequality is established for a class of half-linear elliptic equations with forcing term,and oscillation results are derived on the basis of the Picone-type inequality. Our approach is toreduce the multi-dimensional oscillation problems to one-dimensional oscillation problems forordinary half-linear differential equations.

1. Introduction

The p-Laplacian Δpv = ∇ · (|∇v|p−2∇v) arises from a variety of physical phenomena such asnon-Newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinearelasticity, glaciology, and petroleum extraction (cf. Dıaz [1]). It is important to study thequalitative behavior (e.g., oscillatory behavior) of solutions of p-Laplace equations withsuperlinear terms and forcing terms.

Forced oscillations of superlinear elliptic equations of the form

∇ ·(A(x)|∇v|α−1∇v

)+ C(x)|v|β−1v = f(x)

(β > α > 0

)(1.1)

were studied by Jaros et al. [2], and the more general quasilinear elliptic equation with first-order term

∇ ·(A(x)|∇v|α−1∇v

)+ (α + 1)B(x) ·

(|∇v|α−1∇v

)+ C(x)|v|β−1v = f(x) (1.2)

2 International Journal of Differential Equations

was investigated by Yoshida [3], where the dot (·) denotes the scalar product. There appearsto be no known oscillation results for the case where α = β. The techniques used in [2, 3] arenot applicable to the case where α = β.

The purpose of this paper is to establish a Picone-type inequality for the half-linearelliptic equation with the forcing term:

P[v] := ∇ ·(A(x)|∇v|α−1∇v

)+ (α + 1)B(x) ·

(|∇v|α−1∇v

)+ C(x)|v|α−1v = f(x), (1.3)

and to derive oscillation results on the basis of the Picone-type inequality. The approach usedhere is motivated by the paper [4] in which oscillation criteria for second-order nonlinearordinary differential equations are studied. Our method is an adaptation of that used in [5].Since the proofs of Theorems 2.2–3.3 are quite similar to those of [5, Theorems 1–4], we willomit them.

2. Picone-Type Inequality

Let G be a bounded domain in Rn with piecewise smooth boundary ∂G. It is assumed that

α > 0 is a constant, A(x) ∈ C(G; (0,∞)), B(x) ∈ C(G;Rn), C(x) ∈ C(G;R), and f(x) ∈C(G;R).

The domain DP (G) of P is defined to be the set of all functions v ∈ C1(G;R) with theproperty that A(x)|∇v|α−1∇v ∈ C1(G;Rn) ∩ C(G;Rn).

Lemma 2.1. If v ∈ DP (G) and |v| ≥ k0 for some k0 > 0, then the following Picone-type inequalityholds for any u ∈ C1(G;R):

− ∇ ·(uϕ(u)

A(x)|∇v|α−1∇v

ϕ(v)

)

≥ −A(x)∣∣∣∣∇u − u

A(x)B(x)

∣∣∣∣α+1

+(C(x) − k−α

0

∣∣f(x)∣∣)|u|α+1

+A(x)

[∣∣∣∣∇u − u

A(x)B(x)

∣∣∣∣α+1

+ α∣∣∣uv∇v

∣∣∣α+1

− (α + 1)(∇u − u

A(x)B(x)

)·Φ

(uv∇v

)]

− uϕ(u)ϕ(v)

(P[v] − f(x)

),

(2.1)

where ϕ(s) = |s|α−1s (s ∈ R) and Φ(ξ) = |ξ|α−1ξ (ξ ∈ Rn).

International Journal of Differential Equations 3

Proof. The following Picone identity holds for any u ∈ C1(G;R):

− ∇ ·(uϕ(u)

A(x)|∇v|α−1∇v

ϕ(v)

)

= −A(x)∣∣∣∣∇u − u

A(x)B(x)

∣∣∣∣α+1

+ C(x)|u|α+1

+A(x)

[∣∣∣∣∇u − u

A(x)B(x)

∣∣∣∣α+1

+ α∣∣∣uv∇v

∣∣∣α+1

− (α + 1)(∇u − u

A(x)B(x)

)·Φ

(uv∇v

)]

− uϕ(u)ϕ(v)

(P[v] − f(x)

) − uϕ(u)ϕ(v)

f(x)

(2.2)

(see, e.g., Yoshida [6, Theorem 1.1]). Since |v| ≥ k0, we obtain

∣∣ϕ(v)∣∣ = |v|α ≥ kα0 , (2.3)

and therefore

∣∣∣∣uϕ(u)ϕ(v)

f(x)∣∣∣∣ ≤ |u|α+1k−α

0

∣∣f(x)∣∣. (2.4)

Combining (2.2) with (2.4) yields the desired inequality (2.1).

Theorem 2.2. Let k0 > 0 be a constant. Assume that there exists a nontrivial function u ∈ C1(G;R)such that u = 0 on ∂G and

MG[u] :=∫

G

[A(x)

∣∣∣∣∇u − u

A(x)B(x)

∣∣∣∣α+1

− (C(x) − k−α

0

∣∣f(x)∣∣)|u|α+1]dx ≤ 0. (2.5)

Then for every solution v ∈ DP (G) of (1.3), either v has a zero on G or

|v(x0)| < k0 for some x0 ∈ G. (2.6)

3. Oscillation Results

In this section we investigate forced oscillations of (1.3) in an exterior domain Ω in Rn, that

is, Ω ⊃ Er0 for some r0 > 0, where

Er = {x ∈ Rn; |x| ≥ r} (r > 0). (3.1)

It is assumed that α > 0 is a constant, A(x) ∈ C(Ω; (0,∞)), B(x) ∈ C(Ω;Rn), C(x) ∈ C(Ω;R),and f(x) ∈ C(Ω;R).


The domain DP (Ω) of P is defined to be the set of all functions v ∈ C1(Ω;R) with theproperty that A(x)|∇v|α−1∇v ∈ C1(Ω;Rn).

A solution v ∈ DP (Ω) of (1.3) is said to be oscillatory in Ω if it has a zero in Ωr for anyr > 0, where

Ωr = Ω ∩ Er. (3.2)

Theorem 3.1. Assume that for any k0 > 0 and any r > r0 there exists a bounded domainG ⊂ Er suchthat (2.5) holds for some nontrivial u ∈ C1(G;R) satisfying u = 0 on ∂G. Then for every solutionv ∈ DP (Ω) of (1.3), either v is oscillatory in Ω or

lim inf|x|→∞

|v(x)| = 0. (3.3)

Theorem 3.2. Assume that for any k0 > 0 and any r > r0 there exists a bounded domain G ⊂ Er

such that

MG[u] :=∫

G

[2αA(x)|∇u|α+1 −

(C(x) − 2αA(x)−α|B(x)|α+1 − k−α

0

∣∣f(x)∣∣)|u|α+1

]dx ≤ 0 (3.4)

holds for some nontrivial u ∈ C1(G;R) satisfying u = 0 on ∂G. Then for every solution v ∈ DP (Ω)of (1.3), either v is oscillatory in Ω or satisfies (3.3).

Let {Q(x)}(r) denote the spherical mean of Q(x) over the sphere Sr = {x ∈ Rn; |x| =

r}. We define p(r) and qk0(r) by

p(r) = {2αA(x)}(r),

qk0(r) ={C(x) − 2αA(x)−α|B(x)|α+1 − k−α

0

∣∣f(x)∣∣}(r).

(3.5)

Theorem 3.3. If the half-linear ordinary differential equation

(rn−1p(r)

∣∣y′∣∣α−1y′)′

+ rn−1qk0(r)∣∣y∣∣α−1y = 0 (3.6)

is oscillatory at r = ∞ for any k0 > 0, then for every solution v ∈ DP (Ω) of (1.3), either v isoscillatory in Ω or satisfies (3.3).

Oscillation criteria for the half-linear differential equation (3.6) were obtainedby numerous authors (see, e.g., Dosly and Rehak [7], Kusano and Naito [8], andKusanoet al. [9]).


Now we derive the following Leighton-Wintner-type oscillation result.

Corollary 3.4. If

∫∞

r0

(1

rn−1p(r)

)1/α

dr = ∞,

∫∞rn−1qk0(r)dr = ∞ (3.7)

for any k0 > 0, then for every solution v ∈ DP (Ω) of (1.3), either v is oscillatory in Ω or satisfies(3.3).

Proof. The conclusion follows from the Leighton-Wintner oscillation criterion (see Dosly andRehak [7, Theorem 1.2.9]).

By combining Theorem 3.3 with the results of [8, 9], we obtain Hille-Nehari-typecriteria for (1.3) (cf. Dosly and Rehak [7, Section 3.1], Kusano et al. [10], and Yoshida [11,Section 8.1]).

Corollary 3.5. Assume that qk0(r) ≥ 0 eventually and suppose that p(r) satisfies

∫∞

r0

(1

rn−1p(r)

)1/α

dr = ∞, (3.8)

and qk0(r) satisfies

lim infr→∞

(P(r))α∫∞

r

sn−1qk0(s)ds >αα

(α + 1)α+1, (3.9)

for any k0 > 0, where

P(r) =∫ r

r0

(1

sn−1p(s)

)1/α

ds. (3.10)

Then for every solution v ∈ DP (Ω) of (1.3), either v is oscillatory in Ω or satisfies (3.3).

Corollary 3.6. Assume that qk0(r) ≥ 0 eventually and suppose that p(r) satisfies

∫∞

r0

(1

rn−1p(r)

)1/α

dr < ∞, (3.11)

and qk0(r) satisfies either

∫∞(π(r))α+1rn−1qk0(r)dr = ∞ (3.12)


or

lim infr→∞

1π(r)

∫∞

r

(π(s))α+1sn−1qk0(s)ds >( α

α + 1

)α+1(3.13)

for any k0 > 0, where

π(r) =∫∞

r

(1

sn−1p(s)

)1/α

ds. (3.14)

Then for every solution v ∈ DP (Ω) of (1.3), either v is oscillatory in Ω or satisfies (3.3).

Remark 3.7. If the following hypotheses are satisfied:

C(x) − 2αA(x)−α|B(x)|α+1 > 0(eventually

),

lim|x|→∞

∣∣f(x)∣∣C(x) − 2αA(x)−α|B(x)|α+1

= 0,(3.15)

then we observe that qk0(r) > 0 eventually.

Example 3.8. We consider the half-linear elliptic equation

∇ ·(A(x)|∇v|α−1∇v

)+ (α + 1)B(x) ·

(|∇v|α−1∇v

)+ C(x)|v|α−1v = f(x), x ∈ Ω, (3.16)

where n = 2, Ω = E1, A(x) = 2|x|−1, B(x) = 2|x|−1−α/(α+1)(cos |x|, sin |x|), C(x) = |x|−1(5/2 +sin |x|), and f(x) = |x|−1e−|x|. It is easy to verify that

∫∞

1

(1

rp(r)

)1/α

dr = ∞,

qk0(r) =1r

(12+ sin r − k−α

0 e−r),

(3.17)

and therefore

∫∞rqk0(r)dr =

∫∞(12+ sin r − k−α

0 e−r)dr = ∞ (3.18)

for any k0 > 0. Hence, from Corollary 3.4, we see that for every solution v of (3.16), either vis oscillatory in Ω or satisfies (3.3).


∇ ·(A(x)|∇v|α−1∇v

)+ (α + 1)B(x) ·

(|∇v|α−1∇v

)+ C(x)|v|α−1v = f(x), x ∈ Ω, (3.19)


where n = 2, Ω = E1, A(x) = 2|x|−1, B(x) = |x|−α/(α+1)(sin |x|, cos |x|), C(x) = 3 + cos |x|, andf(x) = e−|x| sin |x|. It is easily checked that

lim|x|→∞

∣∣f(x)∣∣C(x) − 2αA(x)−α|B(x)|α+1

= lim|x|→∞

e−|x||sin|x||2 + cos|x| = 0, (3.20)

and therefore qk0(r) > 0 eventually by Remark 3.7 . Furthermore, we observe that

∫∞

1

(1

rp(r)

)1/α

dr = ∞,

qk0(r) = 2 + cos r − k−α0 e−r |sin r|,

(P(r))α = 2−(α+1)(r − 1)α,

∫∞

r

sqk0(s)ds =∫∞

r

s(2 + cos s − k−α

0 e−s|sin s|)ds

≥∫∞

r

s(1 − k−α

0 e−s)ds = ∞

(3.21)

for any k0 > 0. Hence we obtain

lim infr→∞

(P(r))α∫∞

r

sn−1qk0(s)ds = ∞. (3.22)

It follows from Corollary 3.5 that for every solution v of (3.19), either v is oscillatory in Ω orsatisfies (3.3).


∇ ·(A(x)|∇v|α−1∇v

)+ (α + 1)B(x) ·

(|∇v|α−1∇v

)+ C(x)|v|α−1v = f(x), x ∈ Ω, (3.23)

where n = 2, Ω = E1, A(x) = |x|−1e|x|, B(x) = |x|−α/(α+1)e|x|(cos |x|, sin |x|), C(x) = e2|x|, andf(x) is a bounded function. It is easy to see that

C(x) − 2αA(x)−α|B(x)|α+1 = e2|x| − 2αe|x|,

lim|x|→∞

∣∣f(x)∣∣C(x) − 2αA(x)−α|B(x)|α+1

= lim|x|→∞

∣∣f(x)∣∣e2|x| − 2αe|x|

= 0,(3.24)


and hence qk0(r) > 0 eventually by Remark 3.7 . Since f(x) is bounded, there exists a constantM > 0 such that |f(x)| ≤ M. Moreover, we see that

∫∞

1

(1

rp(r)

)1/α

dr =∫∞

1

(14er

)1/2

dr < ∞,

π(r) = α2−2/αe−r/α,

qk0(r) = e2r − 2αer − k−α0

{∣∣f(x)∣∣}(r) ≥ e2r − 2αer − k−α0 M.

(3.25)

If α > 1, then

∫∞(π(r))α+1rqk0(r)dr ≥ αα+1

22(α+1)/α

∫∞r(e((α−1)/α)r − 2αe−r/α − k−α

0 Me−((α+1)/α)r)dr = ∞,

(3.26)

and if 0 < α < 1, then

lim infr→∞

1π(r)

∫∞

r

(π(s))α+1sqk0(s)ds

≥ lim infr→∞

αα

4er/α

∫∞

r

s(e((α−1)/α)s − 2αe−s/α − k−α

0 Me−((α+1)/α)s)ds

≥ lim infr→∞

αα

4er/α

∫∞

r

(e((α−1)/α)s − 2αe−s/α − k−α

0 Me−((α+1)/α)s)ds

= lim infr→∞

αα

4

(α

1 − αer − 2αα − k−α

0 Mα

α + 1e−r

)= ∞

(3.27)

for any k0 > 0. Corollary 3.6 implies that for every solution v of (3.23), either v is oscillatoryin Ω or satisfies (3.3).

References

[1] J. I. Dıaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equation, vol. 106 ofResearch Notes in Mathematics, Pitman, Boston, Mass, USA, 1985.

[2] J. Jaros, T. Kusano, and N. Yoshida, “Picone-type inequalities for half-linear elliptic equations andtheir applications,” Advances in Mathematical Sciences and Applications, vol. 12, no. 2, pp. 709–724, 2002.

[3] N. Yoshida, “Picone-type inequalities for a class of quasilinear elliptic equations and their appli-cations,” in Differential & Difference Equations and Applications, pp. 1177–1185, Hindawi PublishingCorporation, New York, NY, USA, 2006.

[4] W.-T. Li and X. Li, “Oscillation criteria for second-order nonlinear differential equations withintegrable coefficient,” Applied Mathematics Letters, vol. 13, no. 8, pp. 1–6, 2000.

[5] N. Yoshida, “Forced oscillation criteria for superlinear-sublinear elliptic equations via Picone-typeinequality,” Journal of Mathematical Analysis and Applications, vol. 363, no. 2, pp. 711–717, 2010.

[6] N. Yoshida, “Oscillation of half-linear partial differential equations with first order terms,” Studies ofthe University of Zilina, vol. 17, no. 1, pp. 177–184, 2003.


[7] O. Dosly and P. Rehak,Half-Linear Differential Equations, vol. 202 of North-Holland Mathematics Studies,Elsevier Science B.V., Amsterdam, The Netherlands, 2005.

[8] T. Kusano and Y. Naito, “Oscillation and nonoscillation criteria for second order quasilineardifferential equations,” Acta Mathematica Hungarica, vol. 76, no. 1-2, pp. 81–99, 1997.

[9] T. Kusano, Y. Naito, and A. Ogata, “Strong oscillation and nonoscillation of quasilinear differentialequations of second order,” Differential Equations and Dynamical Systems, vol. 2, no. 1, pp. 1–10, 1994.

[10] T. Kusano, J. Jaros, and N. Yoshida, “A Picone-type identity and Sturmian comparison and oscillationtheorems for a class of half-linear partial differential equations of second order,” Nonlinear Analysis:Theory, Methods & Applications, vol. 40, no. 1–8, pp. 381–395, 2000.

[11] N. Yoshida, Oscillation Theory of Partial Differential Equations, World Scientific, Hackensack, NJ, USA,2008.

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Forced Oscillations of Half-Linear Elliptic Equations via ...downloads.hindawi.com/journals/ijde/2010/520486.pdf · Forced oscillations of superlinear elliptic equations of the form