International Journal of Differential Equations

Volume 2010, Article ID 520486, 9 pages

http://dx.doi.org/10.1155/2010/520486

## Forced Oscillations of Half-Linear Elliptic Equations via Picone-Type Inequality

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

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

Academic Editor: Yuri V. Rogovchenko

Copyright © 2010 Norio Yoshida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 to reduce the multi-dimensional oscillation problems to one-dimensional oscillation problems for ordinary half-linear differential equations.

#### 1. Introduction

The -Laplacian arises from a variety of physical phenomena such as non-Newtonian fluids, reaction-diffusion problems, flow through porous media, nonlinear elasticity, glaciology, and petroleum extraction (cf. Díaz [1]). It is important to study the qualitative behavior (e.g., oscillatory behavior) of solutions of -Laplace equations with superlinear terms and forcing terms.

Forced oscillations of superlinear elliptic equations of the form

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

was investigated by Yoshida [3], where the dot () denotes the scalar product. There appears to be no known oscillation results for the case where . The techniques used in [2, 3] are not applicable to the case where .

The purpose of this paper is to establish a Picone-type inequality for the half-linear elliptic equation with the forcing term: and to derive oscillation results on the basis of the Picone-type inequality. The approach used here is motivated by the paper [4] in which oscillation criteria for second-order nonlinear ordinary 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 will omit them.

#### 2. Picone-Type Inequality

Let be a bounded domain in with piecewise smooth boundary . It is assumed that is a constant, , , , and .

The domain of is defined to be the set of all functions with the property that .

Lemma 2.1. *If and for some , then the following Picone-type inequality holds for any :
**
where and .*

*Proof. *The following Picone identity holds for any :
(see, e.g., Yoshida [6, Theorem 1.1]). Since , we obtain
and therefore
Combining (2.2) with (2.4) yields the desired inequality (2.1).

Theorem 2.2. *Let be a constant. Assume that there exists a nontrivial function such that on and
**
Then for every solution of (1.3), either has a zero on or
*

#### 3. Oscillation Results

In this section we investigate forced oscillations of (1.3) in an exterior domain in , that is, for some , where

It is assumed that is a constant, , , , and .

The domain of is defined to be the set of all functions with the property that .

A solution of (1.3) is said to be *oscillatory* in if it has a zero in for any , where

Theorem 3.1. *Assume that for any and any there exists a bounded domain such that (2.5) holds for some nontrivial satisfying on . Then for every solution of (1.3), either is oscillatory in or
*

Theorem 3.2. *Assume that for any and any there exists a bounded domain such that
**holds for some nontrivial satisfying on . Then for every solution of (1.3), either is oscillatory in or satisfies (3.3).*

Let denote the spherical mean of over the sphere . We define and by

Theorem 3.3. *If the half-linear ordinary differential equation
**
is oscillatory at for any , then for every solution of (1.3), either is oscillatory in or satisfies (3.3).*

Oscillation criteria for the half-linear differential equation (3.6) were obtained by numerous authors (see, e.g., Došlý and Řehák [7], Kusano and Naito [8], and Kusano et al. [9]).

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

Corollary 3.4. *If
**for any , then for every solution of (1.3), either is oscillatory in or satisfies (3.3).*

*Proof. *The conclusion follows from the Leighton-Wintner oscillation criterion (see Došlý and Řehák [7, Theorem 1.2.9]).

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

Corollary 3.5. *Assume that eventually and suppose that satisfies
**and satisfies
**for any , where
**Then for every solution of (1.3), either is oscillatory in or satisfies (3.3).*

Corollary 3.6. *Assume that eventually and suppose that satisfies
**and satisfies either
**or
**for any , where
**Then for every solution of (1.3), either is oscillatory in or satisfies (3.3).*

*Remark 3.7. *If the following hypotheses are satisfied:
then we observe that eventually.

*Example 3.8. *We consider the half-linear elliptic equation
where , , , , , and . It is easy to verify that
and therefore
for any . Hence, from Corollary 3.4, we see that for every solution of (3.16), either is oscillatory in or satisfies (3.3).

*Example 3.9. *We consider the half-linear elliptic equation
where , , , , , and . It is easily checked that
and therefore eventually by Remark 3.7. Furthermore, we observe that
for any . Hence we obtain
It follows from Corollary 3.5 that for every solution of (3.19), either is oscillatory in or satisfies (3.3).

*Example 3.10. *We consider the half-linear elliptic equation
where , , , , , and is a bounded function. It is easy to see that
and hence eventually by Remark 3.7. Since is bounded, there exists a constant such that . Moreover, we see that
If , then
and if , then
for any . Corollary 3.6 implies that for every solution of (3.23), either is oscillatory in or satisfies (3.3).

#### References

- J. I. Díaz,
*Nonlinear Partial Differential Equations and Free Boundaries. Vol. I. Elliptic Equation*, vol. 106 of*Research Notes in Mathematics*, Pitman, Boston, Mass, USA, 1985. View at Zentralblatt MATH · View at MathSciNet - J. Jaroš, T. Kusano, and N. Yoshida, “Picone-type inequalities for half-linear elliptic equations and their applications,”
*Advances in Mathematical Sciences and Applications*, vol. 12, no. 2, pp. 709–724, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Yoshida, “Picone-type inequalities for a class of quasilinear elliptic equations and their applications,” in
*Differential & Difference Equations and Applications*, pp. 1177–1185, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Google Scholar · View at MathSciNet - W.-T. Li and X. Li, “Oscillation criteria for second-order nonlinear differential equations with integrable coefficient,”
*Applied Mathematics Letters*, vol. 13, no. 8, pp. 1–6, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Yoshida, “Forced oscillation criteria for superlinear-sublinear elliptic equations via Picone-type inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 363, no. 2, pp. 711–717, 2010. View at Publisher · View at Google Scholar - N. Yoshida, “Oscillation of half-linear partial differential equations with first order terms,”
*Studies of the University of Žilina*, vol. 17, no. 1, pp. 177–184, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Došlý and P. Řehák,
*Half-Linear Differential Equations*, vol. 202 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2005. View at Zentralblatt MATH · View at MathSciNet - T. Kusano and Y. Naito, “Oscillation and nonoscillation criteria for second order quasilinear differential equations,”
*Acta Mathematica Hungarica*, vol. 76, no. 1-2, pp. 81–99, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kusano, Y. Naito, and A. Ogata, “Strong oscillation and nonoscillation of quasilinear differential equations of second order,”
*Differential Equations and Dynamical Systems*, vol. 2, no. 1, pp. 1–10, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kusano, J. Jaroš, and N. Yoshida, “A Picone-type identity and Sturmian comparison and oscillation theorems 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Yoshida,
*Oscillation Theory of Partial Differential Equations*, World Scientific, Hackensack, NJ, USA, 2008. View at MathSciNet