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

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_𝑝𝑣=∇⋅(|∇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−1𝑣=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−1𝑣=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−1𝑣=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∈𝐶¹(𝐺;ℝ) with the property that 𝐴(𝑥)|∇𝑣|^𝛼−1∇𝑣∈𝐶¹(𝐺;ℝ^𝑛)∩𝐶(𝐺;ℝ^𝑛).

Lemma 2.1. If 𝑣∈𝒟_𝑃(𝐺) and |𝑣|≥𝑘₀ for some 𝑘₀>0, then the following Picone-type inequality holds for any 𝑢∈𝐶¹(𝐺;ℝ): −∇⋅(𝑢𝜑(𝑢)𝐴(𝑥)|∇𝑣|^𝛼−1∇𝑣)𝑢𝜑(𝑣)≥−𝐴(𝑥)|∇𝑢−𝐴(𝑥)𝐵(𝑥)|^𝛼+1+(𝐶(𝑥)−𝑘₀^−𝛼|𝑓(𝑥)|)|𝑢|^𝛼+1𝑢+𝐴(𝑥)[|∇𝑢−𝐴(𝑥)𝐵(𝑥)|^𝛼+1𝑢+𝛼|𝑣∇𝑣|^𝛼+1𝑢−(𝛼+1)(∇𝑢−𝑢𝐴(𝑥)𝐵(𝑥))⋅Φ(𝑣−∇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∈ℝ^𝑛).

Proof. The following Picone identity holds for any 𝑢∈𝐶¹(𝐺;ℝ): −∇⋅(𝑢𝜑(𝑢)𝐴(𝑥)|∇𝑣|^𝛼−1∇𝑣)𝑢𝜑(𝑣)=−𝐴(𝑥)|∇𝑢−𝐴(𝑥)𝐵(𝑥)|^𝛼+1+𝐶(𝑥)|𝑢|^𝛼+1𝑢+𝐴(𝑥)[|∇𝑢−𝐴(𝑥)𝐵(𝑥)|^𝛼+1𝑢+𝛼|𝑣∇𝑣|^𝛼+1𝑢−(𝛼+1)(∇𝑢−𝑢𝐴(𝑥)𝐵(𝑥))⋅Φ(𝑣−∇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≥𝑘^𝛼₀, (2.3)and therefore |𝑢𝜑(𝑢)𝜑(𝑣)𝑓(𝑥)|≤|𝑢|^𝛼+1𝑘₀^−𝛼|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 𝑀_𝐺[𝑢]∶=_𝐺𝑢[𝐴(𝑥)|∇𝑢−𝐴(𝑥)𝐵(𝑥)|^𝛼+1−(𝐶(𝑥)−𝑘₀^−𝛼|𝑓(𝑥)|)|𝑢|^𝛼+1]𝑑𝑥≤0. (2.5) Then for every solution 𝑣∈𝒟_𝑃(𝐺) of (1.3), either 𝑣 has a zero on 𝐺 or |𝑣(𝑥₀)|<𝑘₀forsome𝑥₀∈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_𝑟0 for some 𝑟₀>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∈𝐶¹(Ω;ℝ) with the property that 𝐴(𝑥)|∇𝑣|^𝛼−1∇𝑣∈𝐶¹(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 𝑘₀>0 and any 𝑟>𝑟₀ there exists a bounded domain 𝐺⊂𝐸_𝑟 such that (2.5) holds for some nontrivial 𝑢∈𝐶¹(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 liminf_|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⊂𝐸_𝑟 such that 𝑀_𝐺[𝑢]∶=_𝐺[2^𝛼𝐴(𝑥)|∇𝑢|^𝛼+1−(𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1−𝑘₀^−𝛼|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∈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_𝑘0(𝑟) by 𝑝(𝑟)={2^𝛼𝑞𝐴(𝑥)}(𝑟),_𝑘0(𝑟)={𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1−𝑘₀^−𝛼|𝑓(𝑥)|}(𝑟). (3.5)

Theorem 3.3. If the half-linear ordinary differential equation (𝑟^𝑛−1𝑝(𝑟)|𝑦^′|^𝛼−1𝑦^′)^′+𝑟^𝑛−1𝑞_𝑘0(𝑟)|𝑦|^𝛼−1𝑦=0 (3.6) is oscillatory at 𝑟=∞ for any 𝑘₀>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 ^∞_𝑟0(1𝑟^𝑛−1)𝑝(𝑟)^1/𝛼𝑑𝑟=∞,^∞𝑟^𝑛−1𝑞_𝑘0(𝑟)𝑑𝑟=∞ (3.7)for any 𝑘₀>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_𝑘0(𝑟)≥0 eventually and suppose that 𝑝(𝑟) satisfies ^∞_𝑟0(1𝑟^𝑛−1)𝑝(𝑟)^1/𝛼𝑑𝑟=∞, (3.8)and 𝑞_𝑘0(𝑟) satisfies liminf_𝑟→∞(𝑃(𝑟))^𝛼∞_𝑟𝑠^𝑛−1𝑞_𝑘0𝛼(𝑠)𝑑𝑠>^𝛼(𝛼+1)^𝛼+1, (3.9)for any 𝑘₀>0, where 𝑃(𝑟)=^𝑟_𝑟0(1𝑠^𝑛−1)𝑝(𝑠)^1/𝛼𝑑𝑠. (3.10)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_𝑘0(𝑟)≥0 eventually and suppose that 𝑝(𝑟) satisfies ^∞_𝑟0(1𝑟^𝑛−1)𝑝(𝑟)^1/𝛼𝑑𝑟<∞, (3.11)and 𝑞_𝑘0(𝑟) satisfies either ^∞(𝜋(𝑟))^𝛼+1𝑟^𝑛−1𝑞_𝑘0(𝑟)𝑑𝑟=∞ (3.12)or liminf_𝑟→∞1𝜋(𝑟)^∞_𝑟(𝜋(𝑠))^𝛼+1𝑠^𝑛−1𝑞_𝑘0𝛼(𝑠)𝑑𝑠>()𝛼+1^𝛼+1 (3.13)for any 𝑘₀>0, where 𝜋(𝑟)=^∞_𝑟(1𝑠^𝑛−1)𝑝(𝑠)^1/𝛼𝑑𝑠. (3.14)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^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1>0(eventually),lim_|𝑥|→∞|𝑓(𝑥)|𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1=0, (3.15)then we observe that 𝑞_𝑘0(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/𝛼𝑞𝑑𝑟=∞,_𝑘01(𝑟)=𝑟(12+sin𝑟−𝑘₀^−𝛼𝑒^−𝑟), (3.17) and therefore ^∞𝑟𝑞_𝑘0(𝑟)𝑑𝑟=^∞(12+sin𝑟−𝑘₀^−𝛼𝑒^−𝑟)𝑑𝑟=∞ (3.18) for any 𝑘₀>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−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|, and 𝑓(𝑥)=𝑒^−|𝑥|sin|𝑥|. It is easily checked that lim_|𝑥|→∞|𝑓(𝑥)|𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1=lim_|𝑥|→∞𝑒^−|𝑥||sin|𝑥||2+cos|𝑥|=0, (3.20)and therefore 𝑞_𝑘0(𝑟)>0 eventually by Remark 3.7. Furthermore, we observe that ^∞₁(1)𝑟𝑝(𝑟)^1/𝛼𝑞𝑑𝑟=∞,_𝑘0(𝑟)=2+cos𝑟−𝑘₀^−𝛼𝑒^−𝑟|sin𝑟|,(𝑃(𝑟))^𝛼=2^−(𝛼+1)(𝑟−1)^𝛼,_∞𝑟𝑠𝑞_𝑘0(𝑠)𝑑𝑠=^∞_𝑟𝑠(2+cos𝑠−𝑘₀^−𝛼𝑒^−𝑠≥|sin𝑠|)𝑑𝑠_∞𝑟𝑠(1−𝑘₀^−𝛼𝑒^−𝑠)𝑑𝑠=∞ (3.21)for any 𝑘₀>0. Hence we obtain liminf_𝑟→∞(𝑃(𝑟))^𝛼∞_𝑟𝑠^𝑛−1𝑞_𝑘0(𝑠)𝑑𝑠=∞. (3.22)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−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)=𝑒^2|𝑥|, and 𝑓(𝑥) is a bounded function. It is easy to see that 𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1=𝑒^2|𝑥|−2^𝛼𝑒^|𝑥|,lim_|𝑥|→∞|𝑓(𝑥)|𝐶(𝑥)−2^𝛼𝐴(𝑥)^−𝛼|𝐵(𝑥)|^𝛼+1=lim_|𝑥|→∞|𝑓(𝑥)|𝑒^2|𝑥|−2^𝛼𝑒^|𝑥|=0, (3.24)and hence 𝑞_𝑘0(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)𝑟𝑝(𝑟)^1/𝛼𝑑𝑟=^∞₁(14𝑒^𝑟)^1/2𝑑𝑟<∞,𝜋(𝑟)=𝛼2^−2/𝛼𝑒^−𝑟/𝛼,𝑞_𝑘0(𝑟)=𝑒^2𝑟−2^𝛼𝑒^𝑟−𝑘₀^−𝛼{|𝑓(𝑥)|}(𝑟)≥𝑒^2𝑟−2^𝛼𝑒^𝑟−𝑘₀^−𝛼𝑀. (3.25)If 𝛼>1, then ^∞(𝜋(𝑟))^𝛼+1𝑟𝑞_𝑘0𝛼(𝑟)𝑑𝑟≥^𝛼+12^{2(𝛼+1)/𝛼∞}𝑟(𝑒^{((𝛼−1)/𝛼)𝑟}−2^𝛼𝑒^−𝑟/𝛼−𝑘₀^−𝛼𝑀𝑒^{−((𝛼+1)/𝛼)𝑟})𝑑𝑟=∞, (3.26)and if 0<𝛼<1, then liminf_𝑟→∞1𝜋(𝑟)^∞_𝑟(𝜋(𝑠))^𝛼+1𝑠𝑞_𝑘0(𝑠)𝑑𝑠≥liminf_𝑟→∞𝛼^𝛼4𝑒^𝑟/𝛼∞_𝑟𝑠(𝑒^{((𝛼−1)/𝛼)𝑠}−2^𝛼𝑒^−𝑠/𝛼−𝑘₀^−𝛼𝑀𝑒^{−((𝛼+1)/𝛼)𝑠})𝑑𝑠≥liminf_𝑟→∞𝛼^𝛼4𝑒^𝑟/𝛼∞_𝑟(𝑒^{((𝛼−1)/𝛼)𝑠}−2^𝛼𝑒^−𝑠/𝛼−𝑘₀^−𝛼𝑀𝑒^{−((𝛼+1)/𝛼)𝑠})𝑑𝑠=liminf_𝑟→∞𝛼^𝛼4(𝛼𝑒1−𝛼^𝑟−2^𝛼𝛼−𝑘₀^−𝛼𝑀𝛼𝑒𝛼+1^−𝑟)=∞ (3.27)for any 𝑘₀>0. Corollary 3.6 implies that for every solution 𝑣 of (3.23), either 𝑣 is oscillatory in Ω or satisfies (3.3).