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.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Recommended by Yuri V. Rogovchenko

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

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

1. Introduction

The p -Laplacian _Δp 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 [figure omitted; refer to PDF]

were studied by Jaros et al. [2], and the more general quasilinear elliptic equation with first-order term [figure omitted; refer to PDF]

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: [figure omitted; refer to PDF] 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 ^...n with piecewise smooth boundary ∂G . It is assumed that α>0 is a constant, A(x)∈C(G¯;(0,∞)) , B(x)∈C(G¯;^...n ) , C(x)∈C(G¯;...) , and f(x)∈C(G¯;...) .

The domain _...9F;P (G) of P is defined to be the set of all functions v∈^C1 (G¯;...) with the property that A(x)|∇v^|α-1 ∇v∈^C1 (G;^...n )∩C(G¯;^...n ) .

Lemma 2.1.

If v∈_...9F;P (G) and |v|≥_k0 for some _k0 >0 , then the following Picone-type inequality holds for any u∈^C1 (G;...) : [figure omitted; refer to PDF] where [straight phi](s)=^|s|α-1 s (s∈...) and Φ(ξ)=|ξ^|α-1 ξ (ξ∈^...n ) .

Proof.

The following Picone identity holds for any u∈^C1 (G;...) : [figure omitted; refer to PDF] (see, e.g., Yoshida [6, Theorem 1.1]). Since |v|≥_k0 , we obtain [figure omitted; refer to PDF] and therefore [figure omitted; refer to PDF] 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¯;...) such that u=0 on ∂G and [figure omitted; refer to PDF] Then for every solution v∈_...9F;P (G) of (1.3), either v has a zero on G¯ or [figure omitted; refer to PDF]

3. Oscillation Results

In this section we investigate forced oscillations of (1.3) in an exterior domain Ω in ^...n , that is, Ω⊃_Er0 for some _r0 >0 , where [figure omitted; refer to PDF]

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

The domain _...9F;P (Ω) of P is defined to be the set of all functions v∈^C1 (Ω;...) with the property that A(x)|∇v^|α-1 ∇v∈^C1 (Ω;^...n ) .

A solution v∈_...9F;P (Ω) of (1.3) is said to be oscillatory in Ω if it has a zero in _Ωr for any r>0 , where [figure omitted; refer to PDF]

Theorem 3.1.

Assume that for any _k0 >0 and any r>_r0 there exists a bounded domain G⊂_Er such that (2.5) holds for some nontrivial u∈^C1 (G¯;...) satisfying u=0 on ∂G . Then for every solution v∈_...9F;P (Ω) of (1.3), either v is oscillatory in Ω or [figure omitted; refer to PDF]

Theorem 3.2.

Assume that for any _k0 >0 and any r>_r0 there exists a bounded domain G⊂_Er such that [figure omitted; refer to PDF] holds for some nontrivial u∈^C1 (G¯;...) satisfying u=0 on ∂G . Then for every solution v∈_...9F;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 _Sr ={x∈^...n ; |x|=r} . We define p(r) and _qk0 (r) by [figure omitted; refer to PDF]

Theorem 3.3.

If the half-linear ordinary differential equation [figure omitted; refer to PDF] is oscillatory at r=∞ for any _k0 >0 , then for every solution v∈_...9F;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., Doslý and Rehák [7], Kusano and Naito [8], and Kusano et al. [9]).

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

Corollary 3.4.

If [figure omitted; refer to PDF] for any _k0 >0 , then for every solution v∈_...9F;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 Doslý and Rehák [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. Doslý and Rehák [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 [figure omitted; refer to PDF] and _qk0 (r) satisfies [figure omitted; refer to PDF] for any _k0 >0 , where [figure omitted; refer to PDF] Then for every solution v∈_...9F;P (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 [figure omitted; refer to PDF] and _qk0 (r) satisfies either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] for any _k0 >0 , where [figure omitted; refer to PDF] Then for every solution v∈_...9F;P (Ω) of (1.3), either v is oscillatory in Ω or satisfies (3.3).

Remark 3.7.

If the following hypotheses are satisfied: [figure omitted; refer to PDF] then we observe that _qk0 (r)>0 eventually.

Example 3.8.

We consider the half-linear elliptic equation [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and therefore [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] and therefore _qk0 (r)>0 eventually by Remark 3.7. Furthermore, we observe that [figure omitted; refer to PDF] for any _k0 >0 . Hence we obtain [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] 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| , and f(x) is a bounded function. It is easy to see that [figure omitted; refer to PDF] and hence _qk0 (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 [figure omitted; refer to PDF] If α>1 , then [figure omitted; refer to PDF] and if 0<α<1 , then [figure omitted; refer to PDF] 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).

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