International Journal of Differential Equations
Volume 2010 (2010), Article ID 520486, 9 pages
doi:10.1155/2010/520486
Research Article

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

Norio Yoshida

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 𝑝 -Laplacian Δ 𝑝 𝑣 = ∇ ⋅ ( | ∇ 𝑣 | 𝑝 − 2 ∇ 𝑣 ) 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 𝑝 -Laplace equations with superlinear terms and forcing terms.

Forced oscillations of superlinear elliptic equations of the form | | | | ∇ ⋅ 𝐴 ( 𝑥 ) ∇ 𝑣 𝛼 − 1 ∇ 𝑣 + 𝐶 ( 𝑥 ) | 𝑣 | 𝛽 − 1 𝑣 = 𝑓 ( 𝑥 ) ( 𝛽 > 𝛼 > 0 ) ( 1 . 1 )

were studied by Jaroš 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 − 1 𝑣 = 𝑓 ( 𝑥 ) ( 1 . 2 )

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

The purpose of this paper is to establish a Picone-type inequality for the half-linear elliptic 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 − 1 𝑣 = 𝑓 ( 𝑥 ) , ( 1 . 3 ) 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 𝐺 be a bounded domain in ℝ 𝑛 with piecewise smooth boundary 𝜕 𝐺 . It is assumed that 𝛼 > 0 is a constant, 𝐴 ( 𝑥 ) ∈ 𝐶 ( 𝐺 ; ( 0 , ∞ ) ) , 𝐵 ( 𝑥 ) ∈ 𝐶 ( 𝐺 ; ℝ 𝑛 ) , 𝐶 ( 𝑥 ) ∈ 𝐶 ( 𝐺 ; ℝ ) , and 𝑓 ( 𝑥 ) ∈ 𝐶 ( 𝐺 ; ℝ ) .

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

Lemma 2.1. If 𝑣 ∈ 𝒟 𝑃 ( 𝐺 ) and | 𝑣 | ≥ 𝑘 0 for some 𝑘 0 > 0 , then the following Picone-type inequality holds for any 𝑢 ∈ 𝐶 1 ( 𝐺 ; ℝ ) : | | | | − ∇ ⋅ 𝑢 𝜑 ( 𝑢 ) 𝐴 ( 𝑥 ) ∇ 𝑣 𝛼 − 1 ∇ 𝑣 | | | | 𝑢 𝜑 ( 𝑣 ) ≥ − 𝐴 ( 𝑥 ) ∇ 𝑢 − | | | | 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) 𝛼 + 1 + 𝐶 ( 𝑥 ) − 𝑘 0 − 𝛼 | | | | | 𝑓 ( 𝑥 ) 𝑢 | 𝛼 + 1 | | | | 𝑢 + 𝐴 ( 𝑥 ) ∇ 𝑢 − | | | | 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) 𝛼 + 1 | | | 𝑢 + 𝛼 𝑣 | | | ∇ 𝑣 𝛼 + 1 𝑢 − ( 𝛼 + 1 ) ∇ 𝑢 − 𝑢 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) ⋅ Φ 𝑣 − ∇ 𝑣 𝑢 𝜑 ( 𝑢 ) [ 𝑣 ] 𝜑 ( 𝑣 ) ( 𝑃 − 𝑓 ( 𝑥 ) ) , ( 2 . 1 ) where 𝜑 ( 𝑠 ) = | 𝑠 | 𝛼 − 1 𝑠 ( 𝑠 ∈ ℝ ) and Φ ( 𝜉 ) = | 𝜉 | 𝛼 − 1 𝜉 ( 𝜉 ∈ ℝ 𝑛 ) .

Proof. The following Picone identity holds for any 𝑢 ∈ 𝐶 1 ( 𝐺 ; ℝ ) : | | | | − ∇ ⋅ 𝑢 𝜑 ( 𝑢 ) 𝐴 ( 𝑥 ) ∇ 𝑣 𝛼 − 1 ∇ 𝑣 | | | | 𝑢 𝜑 ( 𝑣 ) = − 𝐴 ( 𝑥 ) ∇ 𝑢 − | | | | 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) 𝛼 + 1 + 𝐶 ( 𝑥 ) | 𝑢 | 𝛼 + 1 | | | | 𝑢 + 𝐴 ( 𝑥 ) ∇ 𝑢 − | | | | 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) 𝛼 + 1 | | | 𝑢 + 𝛼 𝑣 | | | ∇ 𝑣 𝛼 + 1 𝑢 − ( 𝛼 + 1 ) ∇ 𝑢 − 𝑢 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) ⋅ Φ 𝑣 − ∇ 𝑣 𝑢 𝜑 ( 𝑢 ) [ 𝑣 ] 𝜑 ( 𝑣 ) ( 𝑃 − 𝑓 ( 𝑥 ) ) − 𝑢 𝜑 ( 𝑢 ) 𝜑 ( 𝑣 ) 𝑓 ( 𝑥 ) ( 2 . 2 ) (see, e.g., Yoshida [6, Theorem  1.1]). Since | 𝑣 | ≥ 𝑘 0 , we obtain | | | | 𝜑 ( 𝑣 ) = | 𝑣 | 𝛼 ≥ 𝑘 𝛼 0 , ( 2 . 3 ) and therefore | | | | 𝑢 𝜑 ( 𝑢 ) 𝑓 | | | | ≤ 𝜑 ( 𝑣 ) ( 𝑥 ) | 𝑢 | 𝛼 + 1 𝑘 0 − 𝛼 | | 𝑓 | | ( 𝑥 ) . ( 2 . 4 ) Combining (2.2) with (2.4) yields the desired inequality (2.1).

Theorem 2.2. Let 𝑘 0 > 0 be a constant. Assume that there exists a nontrivial function 𝑢 ∈ 𝐶 1 ( 𝐺 ; ℝ ) such that 𝑢 = 0 on 𝜕 𝐺 and 𝑀 𝐺 [ 𝑢 ] ∶ = 𝐺 | | | | 𝑢 𝐴 ( 𝑥 ) ∇ 𝑢 − | | | | 𝐴 ( 𝑥 ) 𝐵 ( 𝑥 ) 𝛼 + 1 − 𝐶 ( 𝑥 ) − 𝑘 0 − 𝛼 | | | | 𝑓 ( 𝑥 ) | 𝑢 | 𝛼 + 1 𝑑 𝑥 ≤ 0 . ( 2 . 5 ) Then for every solution 𝑣 ∈ 𝒟 𝑃 ( 𝐺 ) of (1.3), either 𝑣 has a zero on 𝐺 or | | 𝑣 𝑥 0 | | < 𝑘 0 f o r s o m e 𝑥 0 ∈ 𝐺 . ( 2 . 6 )

3. Oscillation Results

In this section we investigate forced oscillations of (1.3) in an exterior domain Ω in ℝ 𝑛 , that is, Ω ⊃ 𝐸 𝑟 0 for some 𝑟 0 > 0 , where 𝐸 𝑟 = { 𝑥 ∈ ℝ 𝑛 ; | 𝑥 | ≥ 𝑟 } ( 𝑟 > 0 ) . ( 3 . 1 )

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

The domain 𝒟 𝑃 ( Ω ) of 𝑃 is defined to be the set of all functions 𝑣 ∈ 𝐶 1 ( Ω ; ℝ ) with the property that 𝐴 ( 𝑥 ) | ∇ 𝑣 | 𝛼 − 1 ∇ 𝑣 ∈ 𝐶 1 ( Ω ; ℝ 𝑛 ) .

A solution 𝑣 ∈ 𝒟 𝑃 ( Ω ) of (1.3) is said to be oscillatory in Ω if it has a zero in Ω 𝑟 for any 𝑟 > 0 , where Ω 𝑟 = Ω ∩ 𝐸 𝑟 . ( 3 . 2 )

Theorem 3.1. Assume that for any 𝑘 0 > 0 and any 𝑟 > 𝑟 0 there exists a bounded domain 𝐺 ⊂ 𝐸 𝑟 such that (2.5) holds for some nontrivial 𝑢 ∈ 𝐶 1 ( 𝐺 ; ℝ ) satisfying 𝑢 = 0 on 𝜕 𝐺 . Then for every solution 𝑣 ∈ 𝒟 𝑃 ( Ω ) of (1.3), either 𝑣 is oscillatory in Ω or l i m i n f | 𝑥 | → ∞ | | | | 𝑣 ( 𝑥 ) = 0 . ( 3 . 3 )

Theorem 3.2. Assume that for any 𝑘 0 > 0 and any 𝑟 > 𝑟 0 there exists a bounded domain 𝐺 ⊂ 𝐸 𝑟 such that 𝑀 𝐺 [ 𝑢 ] ∶ = 𝐺 2 𝛼 | | | | 𝐴 ( 𝑥 ) ∇ 𝑢 𝛼 + 1 − 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 − 𝑘 0 − 𝛼 | | | | | 𝑓 ( 𝑥 ) 𝑢 | 𝛼 + 1 𝑑 𝑥 ≤ 0 ( 3 . 4 ) holds for some nontrivial 𝑢 ∈ 𝐶 1 ( 𝐺 ; ℝ ) satisfying 𝑢 = 0 on 𝜕 𝐺 . Then for every solution 𝑣 ∈ 𝒟 𝑃 ( 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 𝑆 𝑟 = { 𝑥 ∈ ℝ 𝑛 ; | 𝑥 | = 𝑟 } . We define 𝑝 ( 𝑟 ) and 𝑞 𝑘 0 ( 𝑟 ) by 𝑝 ( 𝑟 ) = { 2 𝛼 𝐴 𝑞 ( 𝑥 ) } ( 𝑟 ) , 𝑘 0 ( 𝑟 ) = 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 − 𝑘 0 − 𝛼 | | | | 𝑓 ( 𝑥 ) ( 𝑟 ) . ( 3 . 5 )

Theorem 3.3. If the half-linear ordinary differential equation 𝑟 𝑛 − 1 | | 𝑦 𝑝 ( 𝑟 ) | | 𝛼 − 1 𝑦 + 𝑟 𝑛 − 1 𝑞 𝑘 0 | | 𝑦 | | ( 𝑟 ) 𝛼 − 1 𝑦 = 0 ( 3 . 6 ) is oscillatory at 𝑟 = ∞ for any 𝑘 0 > 0 , then for every solution 𝑣 ∈ 𝒟 𝑃 ( Ω ) of (1.3), either 𝑣 is oscillatory in Ω 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 ∞ 𝑟 0 1 𝑟 𝑛 − 1 𝑝 ( 𝑟 ) 1 / 𝛼 𝑑 𝑟 = ∞ , ∞ 𝑟 𝑛 − 1 𝑞 𝑘 0 ( 𝑟 ) 𝑑 𝑟 = ∞ ( 3 . 7 ) for any 𝑘 0 > 0 , then for every solution 𝑣 ∈ 𝒟 𝑃 ( 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 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 𝑞 𝑘 0 ( 𝑟 ) ≥ 0 eventually and suppose that 𝑝 ( 𝑟 ) satisfies ∞ 𝑟 0 1 𝑟 𝑛 − 1 𝑝 ( 𝑟 ) 1 / 𝛼 𝑑 𝑟 = ∞ , ( 3 . 8 ) and 𝑞 𝑘 0 ( 𝑟 ) satisfies l i m i n f 𝑟 → ∞ ( 𝑃 ( 𝑟 ) ) 𝛼 ∞ 𝑟 𝑠 𝑛 − 1 𝑞 𝑘 0 𝛼 ( 𝑠 ) 𝑑 𝑠 > 𝛼 ( 𝛼 + 1 ) 𝛼 + 1 , ( 3 . 9 ) for any 𝑘 0 > 0 , where 𝑃 ( 𝑟 ) = 𝑟 𝑟 0 1 𝑠 𝑛 − 1 𝑝 ( 𝑠 ) 1 / 𝛼 𝑑 𝑠 . ( 3 . 1 0 ) Then for every solution 𝑣 ∈ 𝒟 𝑃 ( Ω ) of (1.3), either 𝑣 is oscillatory in Ω or satisfies (3.3).

Corollary 3.6. Assume that 𝑞 𝑘 0 ( 𝑟 ) ≥ 0 eventually and suppose that 𝑝 ( 𝑟 ) satisfies ∞ 𝑟 0 1 𝑟 𝑛 − 1 𝑝 ( 𝑟 ) 1 / 𝛼 𝑑 𝑟 < ∞ , ( 3 . 1 1 ) and 𝑞 𝑘 0 ( 𝑟 ) satisfies either ∞ ( 𝜋 ( 𝑟 ) ) 𝛼 + 1 𝑟 𝑛 − 1 𝑞 𝑘 0 ( 𝑟 ) 𝑑 𝑟 = ∞ ( 3 . 1 2 ) or l i m i n f 𝑟 → ∞ 1 𝜋 ( 𝑟 ) ∞ 𝑟 ( 𝜋 ( 𝑠 ) ) 𝛼 + 1 𝑠 𝑛 − 1 𝑞 𝑘 0 𝛼 ( 𝑠 ) 𝑑 𝑠 > 𝛼 + 1 𝛼 + 1 ( 3 . 1 3 ) for any 𝑘 0 > 0 , where 𝜋 ( 𝑟 ) = ∞ 𝑟 1 𝑠 𝑛 − 1 𝑝 ( 𝑠 ) 1 / 𝛼 𝑑 𝑠 . ( 3 . 1 4 ) Then for every solution 𝑣 ∈ 𝒟 𝑃 ( 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 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 > 0 ( e v e n t u a l l y ) , l i m | 𝑥 | → ∞ | | | | 𝑓 ( 𝑥 ) 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 = 0 , ( 3 . 1 5 ) then we observe that 𝑞 𝑘 0 ( 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 − 1 𝑣 = 𝑓 ( 𝑥 ) , 𝑥 ∈ Ω , ( 3 . 1 6 ) where 𝑛 = 2 , Ω = 𝐸 1 , 𝐴 ( 𝑥 ) = 2 | 𝑥 | − 1 , 𝐵 ( 𝑥 ) = 2 | 𝑥 | − 1 − 𝛼 / ( 𝛼 + 1 ) ( c o s | 𝑥 | , s i n | 𝑥 | ) , 𝐶 ( 𝑥 ) = | 𝑥 | − 1 ( 5 / 2 + s i n | 𝑥 | ) , and 𝑓 ( 𝑥 ) = | 𝑥 | − 1 𝑒 − | 𝑥 | . It is easy to verify that ∞ 1 1 𝑟 𝑝 ( 𝑟 ) 1 / 𝛼 𝑞 𝑑 𝑟 = ∞ , 𝑘 0 1 ( 𝑟 ) = 𝑟 1 2 + s i n 𝑟 − 𝑘 0 − 𝛼 𝑒 − 𝑟 , ( 3 . 1 7 ) and therefore ∞ 𝑟 𝑞 𝑘 0 ( 𝑟 ) 𝑑 𝑟 = ∞ 1 2 + s i n 𝑟 − 𝑘 0 − 𝛼 𝑒 − 𝑟 𝑑 𝑟 = ∞ ( 3 . 1 8 ) for any 𝑘 0 > 0 . Hence, from Corollary 3.4, we see that for every solution 𝑣 of (3.16), either 𝑣 is oscillatory in Ω or satisfies (3.3).

Example 3.9. 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 − 1 𝑣 = 𝑓 ( 𝑥 ) , 𝑥 ∈ Ω , ( 3 . 1 9 ) where 𝑛 = 2 , Ω = 𝐸 1 , 𝐴 ( 𝑥 ) = 2 | 𝑥 | − 1 , 𝐵 ( 𝑥 ) = | 𝑥 | − 𝛼 / ( 𝛼 + 1 ) ( s i n | 𝑥 | , c o s | 𝑥 | ) , 𝐶 ( 𝑥 ) = 3 + c o s | 𝑥 | , and 𝑓 ( 𝑥 ) = 𝑒 − | 𝑥 | s i n | 𝑥 | . It is easily checked that l i m | 𝑥 | → ∞ | | | | 𝑓 ( 𝑥 ) 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 = l i m | 𝑥 | → ∞ 𝑒 − | 𝑥 | | | | | s i n | 𝑥 | 2 + c o s | 𝑥 | = 0 , ( 3 . 2 0 ) and therefore 𝑞 𝑘 0 ( 𝑟 ) > 0 eventually by Remark 3.7. Furthermore, we observe that ∞ 1 1 𝑟 𝑝 ( 𝑟 ) 1 / 𝛼 𝑞 𝑑 𝑟 = ∞ , 𝑘 0 ( 𝑟 ) = 2 + c o s 𝑟 − 𝑘 0 − 𝛼 𝑒 − 𝑟 | | | | , s i n 𝑟 ( 𝑃 ( 𝑟 ) ) 𝛼 = 2 − ( 𝛼 + 1 ) ( 𝑟 − 1 ) 𝛼 , ∞ 𝑟 𝑠 𝑞 𝑘 0 ( 𝑠 ) 𝑑 𝑠 = ∞ 𝑟 𝑠 2 + c o s 𝑠 − 𝑘 0 − 𝛼 𝑒 − 𝑠 | | | | ≥ s i n 𝑠 𝑑 𝑠 ∞ 𝑟 𝑠 1 − 𝑘 0 − 𝛼 𝑒 − 𝑠 𝑑 𝑠 = ∞ ( 3 . 2 1 ) for any 𝑘 0 > 0 . Hence we obtain l i m i n f 𝑟 → ∞ ( 𝑃 ( 𝑟 ) ) 𝛼 ∞ 𝑟 𝑠 𝑛 − 1 𝑞 𝑘 0 ( 𝑠 ) 𝑑 𝑠 = ∞ . ( 3 . 2 2 ) It follows from Corollary 3.5 that for every solution 𝑣 of (3.19), either 𝑣 is oscillatory in Ω or satisfies (3.3).

Example 3.10. 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 − 1 𝑣 = 𝑓 ( 𝑥 ) , 𝑥 ∈ Ω , ( 3 . 2 3 ) where 𝑛 = 2 , Ω = 𝐸 1 , 𝐴 ( 𝑥 ) = | 𝑥 | − 1 𝑒 | 𝑥 | , 𝐵 ( 𝑥 ) = | 𝑥 | − 𝛼 / ( 𝛼 + 1 ) 𝑒 | 𝑥 | ( c o s | 𝑥 | , s i n | 𝑥 | ) , 𝐶 ( 𝑥 ) = 𝑒 2 | 𝑥 | , and 𝑓 ( 𝑥 ) is a bounded function. It is easy to see that 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 = 𝑒 2 | 𝑥 | − 2 𝛼 𝑒 | 𝑥 | , l i m | 𝑥 | → ∞ | | | | 𝑓 ( 𝑥 ) 𝐶 ( 𝑥 ) − 2 𝛼 𝐴 ( 𝑥 ) − 𝛼 | | | | 𝐵 ( 𝑥 ) 𝛼 + 1 = l i m | 𝑥 | → ∞ | | | | 𝑓 ( 𝑥 ) 𝑒 2 | 𝑥 | − 2 𝛼 𝑒 | 𝑥 | = 0 , ( 3 . 2 4 ) and hence 𝑞 𝑘 0 ( 𝑟 ) > 0 eventually by Remark 3.7. Since 𝑓 ( 𝑥 ) is bounded, there exists a constant 𝑀 > 0 such that | 𝑓 ( 𝑥 ) | ≤ 𝑀 . Moreover, we see that ∞ 1 1 𝑟 𝑝 ( 𝑟 ) 1 / 𝛼 𝑑 𝑟 = ∞ 1 1 4 𝑒 𝑟 1 / 2 𝑑 𝑟 < ∞ , 𝜋 ( 𝑟 ) = 𝛼 2 − 2 / 𝛼 𝑒 − 𝑟 / 𝛼 , 𝑞 𝑘 0 ( 𝑟 ) = 𝑒 2 𝑟 − 2 𝛼 𝑒 𝑟 − 𝑘 0 − 𝛼 | | | | 𝑓 ( 𝑥 ) ( 𝑟 ) ≥ 𝑒 2 𝑟 − 2 𝛼 𝑒 𝑟 − 𝑘 0 − 𝛼 𝑀 . ( 3 . 2 5 ) If 𝛼 > 1 , then ∞ ( 𝜋 ( 𝑟 ) ) 𝛼 + 1 𝑟 𝑞 𝑘 0 𝛼 ( 𝑟 ) 𝑑 𝑟 ≥ 𝛼 + 1 2 2 ( 𝛼 + 1 ) / 𝛼 ∞ 𝑟 𝑒 ( ( 𝛼 − 1 ) / 𝛼 ) 𝑟 − 2 𝛼 𝑒 − 𝑟 / 𝛼 − 𝑘 0 − 𝛼 𝑀 𝑒 − ( ( 𝛼 + 1 ) / 𝛼 ) 𝑟 𝑑 𝑟 = ∞ , ( 3 . 2 6 ) and if 0 < 𝛼 < 1 , then l i m i n f 𝑟 → ∞ 1 𝜋 ( 𝑟 ) ∞ 𝑟 ( 𝜋 ( 𝑠 ) ) 𝛼 + 1 𝑠 𝑞 𝑘 0 ( 𝑠 ) 𝑑 𝑠 ≥ l i m i n f 𝑟 → ∞ 𝛼 𝛼 4 𝑒 𝑟 / 𝛼 ∞ 𝑟 𝑠 𝑒 ( ( 𝛼 − 1 ) / 𝛼 ) 𝑠 − 2 𝛼 𝑒 − 𝑠 / 𝛼 − 𝑘 0 − 𝛼 𝑀 𝑒 − ( ( 𝛼 + 1 ) / 𝛼 ) 𝑠 𝑑 𝑠 ≥ l i m i n f 𝑟 → ∞ 𝛼 𝛼 4 𝑒 𝑟 / 𝛼 ∞ 𝑟 𝑒 ( ( 𝛼 − 1 ) / 𝛼 ) 𝑠 − 2 𝛼 𝑒 − 𝑠 / 𝛼 − 𝑘 0 − 𝛼 𝑀 𝑒 − ( ( 𝛼 + 1 ) / 𝛼 ) 𝑠 𝑑 𝑠 = l i m i n f 𝑟 → ∞ 𝛼 𝛼 4 𝛼 𝑒 1 − 𝛼 𝑟 − 2 𝛼 𝛼 − 𝑘 0 − 𝛼 𝑀 𝛼 𝑒 𝛼 + 1 − 𝑟 = ∞ ( 3 . 2 7 ) for any 𝑘 0 > 0 . Corollary 3.6 implies that for every solution 𝑣 of (3.23), either 𝑣 is oscillatory in Ω or satisfies (3.3).

References

  1. 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
  2. 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 Zentralblatt MATH · View at MathSciNet
  3. 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 MathSciNet
  4. 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
  5. 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
  6. 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 Zentralblatt MATH · View at MathSciNet
  7. 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
  8. 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
  9. 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 Zentralblatt MATH · View at MathSciNet
  10. 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
  11. N. Yoshida, Oscillation Theory of Partial Differential Equations, World Scientific, Hackensack, NJ, USA, 2008. View at MathSciNet