#### 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).