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 Δpv=∇·(|∇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)

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)

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),
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 ℝn with piecewise smooth boundary ∂G. It is assumed that α>0 is a constant, A(x)∈C(G¯;(0,∞)), B(x)∈C(G¯;ℝn), C(x)∈C(G¯;ℝ), and f(x)∈C(G¯;ℝ).

The domain 𝒟P(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;ℝn)∩C(G¯;ℝn).

Lemma 2.1.

If v∈𝒟P(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∇vφ(v))≥-A(x)|∇u-uA(x)B(x)|α+1+(C(x)-k0-α|f(x)|)|u|α+1+A(x)[|∇u-uA(x)B(x)|α+1+α|uv∇v|α+1-(α+1)(∇u-uA(x)B(x))·Φ(uv∇v)]-uφ(u)φ(v)(P[v]-f(x)),
where φ(s)=|s|α-1s(s∈ℝ) and Φ(ξ)=|ξ|α-1ξ(ξ∈ℝn).

Proof.

The following Picone identity holds for any u∈C1(G;ℝ):
-∇·(uφ(u)A(x)|∇v|α-1∇vφ(v))=-A(x)|∇u-uA(x)B(x)|α+1+C(x)|u|α+1+A(x)[|∇u-uA(x)B(x)|α+1+α|uv∇v|α+1-(α+1)(∇u-uA(x)B(x))·Φ(uv∇v)]-uφ(u)φ(v)(P[v]-f(x))-uφ(u)φ(v)f(x)
(see, e.g., Yoshida [6, Theorem 1.1]). Since |v|≥k0, we obtain
|φ(v)|=|v|α≥k0α,and therefore
|uφ(u)φ(v)f(x)|≤|u|α+1k0-α|f(x)|.
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
MG[u]:=∫G[A(x)|∇u-uA(x)B(x)|α+1-(C(x)-k0-α|f(x)|)|u|α+1]dx≤0.
Then for every solution v∈𝒟P(G) of (1.3), either v has a zero on G¯ or
|v(x0)|<k0forsomex0∈G.

3. Oscillation Results

In this section we investigate forced oscillations of (1.3) in an exterior domain Ω in ℝn, that is, Ω⊃Er0 for some r0>0, where
Er={x∈ℝn;|x|≥r}(r>0).

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

The domain 𝒟P(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;ℝn).

A solution v∈𝒟P(Ω) of (1.3) is said to be oscillatory in Ω if it has a zero in Ωr for any r>0, where
Ωr=Ω∩Er.

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¯;ℝ) satisfying u=0 on ∂G. Then for every solution v∈𝒟P(Ω) of (1.3), either v is oscillatory in Ω or
lim inf|x|→∞|v(x)|=0.

Theorem 3.2.

Assume that for any k0>0 and any r>r0 there exists a bounded domain G⊂Er such that
M̃G[u]:=∫G[2αA(x)|∇u|α+1-(C(x)-2αA(x)-α|B(x)|α+1-k0-α|f(x)|)|u|α+1]dx≤0holds for some nontrivial u∈C1(G¯;ℝ) satisfying u=0 on ∂G. Then for every solution v∈𝒟P(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∈ℝn;|x|=r}. We define p(r) and qk0(r) by
p(r)={2αA(x)}¯(r),qk0(r)={C(x)-2αA(x)-α|B(x)|α+1-k0-α|f(x)|}¯(r).

Theorem 3.3.

If the half-linear ordinary differential equation
(rn-1p(r)|y′|α-1y′)′+rn-1qk0(r)|y|α-1y=0
is oscillatory at r=∞ for any k0>0, then for every solution v∈𝒟P(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∞(1rn-1p(r))1/αdr=∞,∫∞rn-1qk0(r)dr=∞for any k0>0, then for every solution v∈𝒟P(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 qk0(r)≥0 eventually and suppose that p(r) satisfies
∫r0∞(1rn-1p(r))1/αdr=∞,and qk0(r) satisfies
lim infr→∞(P(r))α∫r∞sn-1qk0(s)ds>αα(α+1)α+1,for any k0>0, where
P(r)=∫r0r(1sn-1p(s))1/αds.Then for every solution v∈𝒟P(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
∫r0∞(1rn-1p(r))1/αdr<∞,and qk0(r) satisfies either
∫∞(π(r))α+1rn-1qk0(r)dr=∞or
lim infr→∞1π(r)∫r∞(π(s))α+1sn-1qk0(s)ds>(αα+1)α+1for any k0>0, where
π(r)=∫r∞(1sn-1p(s))1/αds.Then for every solution v∈𝒟P(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)-2αA(x)-α|B(x)|α+1>0(eventually),lim|x|→∞|f(x)|C(x)-2αA(x)-α|B(x)|α+1=0,then we observe that qk0(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,
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∞(1rp(r))1/αdr=∞,qk0(r)=1r(12+sinr-k0-αe-r), and therefore
∫∞rqk0(r)dr=∫∞(12+sinr-k0-αe-r)dr=∞ 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,
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)-2αA(x)-α|B(x)|α+1=lim|x|→∞e-|x||sin|x||2+cos|x|=0,and therefore qk0(r)>0 eventually by Remark 3.7. Furthermore, we observe that
∫1∞(1rp(r))1/αdr=∞,qk0(r)=2+cosr-k0-αe-r|sinr|,(P(r))α=2-(α+1)(r-1)α,∫r∞sqk0(s)ds=∫r∞s(2+coss-k0-αe-s|sins|)ds≥∫r∞s(1-k0-αe-s)ds=∞for any k0>0. Hence we obtain
lim infr→∞(P(r))α∫r∞sn-1qk0(s)ds=∞.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,
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
C(x)-2αA(x)-α|B(x)|α+1=e2|x|-2αe|x|,lim|x|→∞|f(x)|C(x)-2αA(x)-α|B(x)|α+1=lim|x|→∞|f(x)|e2|x|-2αe|x|=0,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
∫1∞(1rp(r))1/αdr=∫1∞(14er)1/2dr<∞,π(r)=α2-2/αe-r/α,qk0(r)=e2r-2αer-k0-α{|f(x)|}¯(r)≥e2r-2αer-k0-αM.If α>1, then
∫∞(π(r))α+1rqk0(r)dr≥αα+122(α+1)/α∫∞r(e((α-1)/α)r-2αe-r/α-k0-αMe-((α+1)/α)r)dr=∞,and if 0<α<1, then
lim infr→∞1π(r)∫r∞(π(s))α+1sqk0(s)ds≥lim infr→∞αα4er/α∫r∞s(e((α-1)/α)s-2αe-s/α-k0-αMe-((α+1)/α)s)ds≥lim infr→∞αα4er/α∫r∞(e((α-1)/α)s-2αe-s/α-k0-αMe-((α+1)/α)s)ds=lim infr→∞αα4(α1-αer-2αα-k0-αMαα+1e-r)=∞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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