Hartman – Wintner type theorem for PDE with p−Laplacian

Robert Mařík,

Author supported by the grant No. 384/1999 of Development of Czech Universities.

September 21, 2000

Abstract The well known Hartman–Wintner oscillation criterion is extended to

the PDE

div(||∇u||^{p−2}∇u) + c(x)|u|^{p−2}u = 0, p > 1.  (E)

The condition on the function c(x) under which (E) has no solution positive for large ||x||, i.e. ∞ belongs to the closure of the set of zeros of every solution defined on the domain Ω = {x ∈ ℝ^{n} : ||x|| > 1}, is derived.

Keywords. p−Laplacian, positive solution, Riccati equation.

1 Introduction

Let us consider the following partial differential equation with p−Laplacian

div(||∇u||^{p−2}∇u) + c(x)|u|^{p−2}u = 0  (1.1)

where p > 1, x = (x_{1}, x_{2}, …, x_{n}), || · || is the usual Euclidean norm in ℝ^{n} and ∇ is the usual nabla operator. Define the sets Ω(a) = {x ∈ ℝ^{n} : a ≤ ||x||}, Ω(a, b) = {x ∈ ℝ^{n} : a ≤ ||x|| ≤ b}. The function c(x) is assumed to be integrable on every compact subset of Ω(1). Under solution of the equation (1.1) we understand every absolutely continuous function u : Ω(1) → ℝ such that ||∇u||^{p−2} \frac{∂u}{∂x_{i}} is absolutely continuous with respect to x_{i} and u satisfies the equation (1.1) almost everywhere on Ω(1).

Equation (1.1) appears for example in the study of non-Newtonian fluids, nonlinear elasticity and in glaciology. Special cases of the equation (1.1) are the linear Schrödinger equation

Δu + c(x)u = 0  (1.2)

if p = 2, the half–linear ordinary differential equation

(|u′|^{p−2}u′)′ + c(x)|u|^{p−2}u = 0  ′ = \frac{d}{dx}  (1.3)

This paper is in final form and no version of it will be submitted for publication elsewhere.

if n = 1, and the ordinary differential equation

u′′ + c(x)u = 0  (1.4)

if both n = 1 and p = 2 holds.

Remark that if c(x) is radial function, i.e., c(x) = c̃(||x||), then the equation for radial solution u(x) = ũ(||x||) of the equation (1.1) becomes

(r^{n−1}|ũ′|^{p−2}ũ′)′ + r^{n−1}c̃(r)|ũ|^{p−2}ũ = 0  ′ = \frac{d}{dr} ,  (1.5)

which can be transformed into the equation (1.3).

This paper is motivated by the papers [1, 4] and [5, 6], where the Riccati technique is used to establish oscillation criteria for the equation (1.3) and (1.2), respectively.

The well-known result from the theory of second order ODE is the following theorem.

Theorem (Hartman–Wintner). If either

−∞ < lim inf_{t→∞} \frac{1}{t} ∫_{1}^{t} ∫_{1}^{s} c(ξ) dξ ds < lim sup_{t→∞} \frac{1}{t} ∫_{1}^{t} ∫_{1}^{s} c(ξ) dξ ds ≤ ∞, (1.6)

or

lim_{t→∞} \frac{1}{t} ∫_{1}^{t} ∫_{1}^{s} c(ξ) dξ ds = ∞,  (1.7)

then the equation (1.4) is oscillatory.

This theorem is proved using Riccati technique in [3, Chap. XI]. The aim of this paper is to extend this statement to the equation (1.1). Another statement of this type was proved in [2, Theorem 3.4] under additional condition p ≥ n+1. Here we prove a similar criterion, without the restriction on p.

We use the following function C(t):

C(t) = \frac{p − 1}{t^{p−1}} ∫_{1}^{t} s^{p−2} ∫_{Ω(1,s)} ||x||^{1−n}c(x) dx ds .  (1.8)

2 Main results

First we introduce main ideas from the Riccati technique.

Suppose that there exists a number a ∈ ℝ^{+} and a solution u of (1.1) which is positive on Ω(a). The vector function w = \frac{||∇u||^{p−2}∇u}{|u|^{p−2}u} is defined on Ω(a) and solves the Riccati type equation

div w + c(x) + (p − 1)||w||^{q} = 0,  (2.1)

where q is the conjugate number to p (i.e. \frac{1}{p} + \frac{1}{q} = 1 holds). Really, direct computation shows

div w = div(||∇u||^{p−2}∇u)u^{1−p} − (p − 1)||∇u||^{p−2}u^{−p}⟨∇u, ∇u⟩

= −c(x) − (p − 1)\frac{||∇u||^{p}}{u^{p}}

= −c(x) − (p − 1)||w||^{q}.

The following Lemma plays a crucial role in our consideration. It is a straighforward generalization of [3, Lemma 7.1, Chap XI.]

Lemma 2.1. Let w be the solution of (2.1) defined on Ω(a) for some a > 1. The following statements are equivalent:

(i)

∫_{Ω(a)} ||x||^{1−n}||w||^{q} dx < ∞;  (2.2)

(ii) there exists a finite limit

lim_{t→∞} C(t) = C_{0};  (2.3)

(iii)

lim inf_{t→∞} C(t) > −∞,  (2.4)

where the function C(t) is defined by (1.8).

Our main theorem now follows from Lemma 2.1.

Theorem 2.2 (Hartman–Wintner type oscillation criterion). If either

−∞ < lim inf_{t→∞} C(t) < lim sup_{t→∞} C(t) ≤ ∞

or

lim_{t→∞} C(t) = ∞,

then the equation (1.1) has no positive solution positive on Ω(a) for any a > 1.

Proof. It follows from the assumptions of the theorem that lim inf_{t→∞} C(t) > −∞. If there would exist a number a > 1 such that (1.1) has a solution positive on Ω(a), then Theorem 2.1 would imply that there exists a finite limit lim_{t→∞} C(t). This contradiction ends the proof.

Corollary 2.3 (Leighton–Wintner type criterion). If

lim_{t→∞} ∫_{Ω(1,t)} ||x||^{1−n}c(x) dx = ∞,  (2.5)

then equation (1.1) has no positive solution on Ω(a) for any a > 1.

Proof. If (2.5) holds, then lim_{t→∞} C(t) = ∞ and the statement follows from Theorem 2.2.

Remark. For the equation (1.2) were the results from this paper proved in [5]. Criteria analogous to the second part of Theorem 2.2 and Corollary 2.3 were proved in [2] without the term ||x||^{1−n} but under additional conditions p ≥ n+1 and p ≥ n, respectively.

Proof of Lemma 2.1. First we multiply the Riccati equation (2.1) by ||x||^{1−n} and integrate on Ω(a, t). Application of the identity

||x||^{1−n} div w = div(||x||^{1−n}w) − (1 − n)||x||^{−n}⟨w, j⟩,

and Gauss divergence theorem yields

∫_{||x||=t} ||x||^{1−n}⟨w, j⟩ dσ − ∫_{||x||=a} ||x||^{1−n}⟨w, j⟩ dσ

− (1 − n) ∫_{Ω(a,t)} ||x||^{−n}⟨w, j⟩ dx + (p − 1) ∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx

+ ∫_{Ω(a,t)} ||x||^{1−n}c(x) dx = 0, (2.6)

where ∫ · dσ denotes the surface integral, j is the unit outside normal vector to the sphere in ℝ^{n} and ⟨·, ·⟩ is the usual scalar product.

“(i)=>(ii)” Suppose that (2.2) holds. The Hölder inequality implies that

∫_{Ω(a,t)} ||x||^{1−n}|⟨w, j⟩| dx ≤ (∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx )^{1/q}(∫_{Ω(a,t)} ||x||^{1−n−p} dx )^{1/p}

≤ (∫_{Ω(a)} ||x||^{1−n}||w||^{q} dx )^{1/q}(ω_{n} ∫_{a}^{t} s^{−p} ds )^{1/p},

where ω_{n} is the measure of surface of the n−dimensional unit sphere in ℝ^{n}. Hence

∫_{Ω(a)} ||x||^{−n}⟨w, j⟩ dx ≤ ∞.  (2.7)

Denote

Ĉ = −(p − 1) ∫_{Ω(a)} ||x||^{1−n}||w||^{q} dx + ∫_{||x||=a} ||x||^{1−n}⟨w, j⟩ dσ

+ (1 − n) ∫_{Ω(a)} ||x||^{−n}⟨w, j⟩ dσ + ∫_{Ω(1,a)} ||x||^{1−n}c(x) dx .

Below we will show that Ĉ = C_{0}. The equation (2.6) can be written in the form Ĉ − ∫_{Ω(1,t)} ||x||^{1−n}c(x) dx = ∫_{||x||=t} ||x||^{1−n}⟨w, j⟩ dσ

− (p − 1) ∫_{Ω(t)} ||x||^{1−n}||w||^{q} dx + (1 − n) ∫_{Ω(t)} ||x||^{−n}⟨w, j⟩ dx . (2.8)

Multiplying (2.8) by t^{p−2}, integrating over [a, t] and multiplying by \frac{p−1}{t^{p−1}} we obtain

Ĉ − (\frac{a}{t})^{p−1}[Ĉ − C(a)] − C(t) = \frac{p − 1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{||x||=s} ||x||^{1−n}⟨w, j⟩ dσ ds

− \frac{(p − 1)^{2}}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{Ω(s)} ||x||^{1−n}||w||^{q} dx ds

+ \frac{(1 − n)(p − 1)}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{Ω(s)} ||x||^{−n}⟨w, j⟩ dx ds . (2.9)

The second and the third integral on the right hand side tend to zero as t tends to infinity in view of (2.2) and (2.7). The Hölder inequality implies

|\frac{1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{||x||=s} ||x||^{1−n}⟨w, j⟩ dσ ds |

≤ \frac{1}{t^{p−1}} ∫_{a}^{t} s^{p−2}(∫_{||x||=s} ||x||^{1−n}||w||^{q} dσ )^{1/q}ω_{n}^{1/p} ds

≤ \frac{ω_{n}^{1/p}}{t^{p−1}} (∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx )^{1/q}(∫_{0}^{t} s^{p^{2}−2p} ds )^{1/p}

≤ \frac{ω_{n}^{1/p}}{(p − 1)^{2/p}} (∫_{Ω(a)} ||x||^{1−n}||w||^{q} dx )^{1/q}t^{\frac{(p−1)^{2}}{p} −(p−1)}  (2.10)

and the first integral in (2.9) tends to zero too. Hence

lim_{t→∞} C(t) = Ĉ = C_{0}.  (2.11)

The implication “(ii)=>(iii)” is trivial.

“(iii)=>(i)” Suppose, by contradiction, that (2.4) holds and

∫_{Ω(a)} ||x||^{1−n}||w||^{q} dx = +∞.  (2.12)

Multiplication of (2.6) by t^{p−2}, integration over the interval [a, b] and multiplication by t^{1−p} gives

\frac{1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{||x||=s} ||x||^{1−n}⟨w, j⟩ dσ ds

+ \frac{p − 1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{1−n}||w||^{q} dx ds

− \frac{1 − n}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{−n}⟨w, j⟩ dx ds

= \frac{1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ds ∫_{||x||=a} ||x||^{1−n}⟨w, j⟩ dσ

− \frac{1}{t^{p−1}} ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{1−n}c(x) dx ds . (2.13)

Define the function

v(t) := (p − 1) ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{1−n}||w||^{q} dx ds .

The function v satisfies

\frac{v(t)}{t^{p−1}} → ∞ for t → ∞.  (2.14)

Because of the right hand side of the equality (2.13) is bounded from above, there exists t_{a} such that the right hand side of (2.13) is less than \frac{v(t)}{3t^{p−1}} for t ≥ t_{a}. Now we have from (2.13)

\frac{2}{3}v(t) < |∫_{a}^{t} s^{p−2} ∫_{||x||=s} ||x||^{1−n}⟨w, j⟩ dσ ds |

+ |(1 − n) ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{−n}⟨w, j⟩ dx ds | (2.15)

for t ≥ t_{a}. The same way as in the inequality (2.10) gives

|∫_{a}^{t} s^{p−2} ∫_{||x||=s} ||x||^{1−n}⟨w, j⟩ dσ ds |

≤ (∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx )^{1/q} \frac{ω_{n}^{1/p}t^{\frac{(p−1)^{2}}{p}}}{(p − 1)^{2/p}} = K(tv′(t))^{1/q}, (2.16)

where K = ω_{n}^{1/p}(p − 1)^{− \frac{2}{p} − \frac{1}{q}} . The Hölder inequality gives

|(1 − n) ∫_{a}^{t} s^{p−2} ∫_{Ω(a,s)} ||x||^{−n}⟨w, j⟩ dx ds |

≤ (n − 1) ∫_{a}^{t} s^{p−2}(∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx )^{1/q}(∫_{1}^{∞} ω_{n}ξ^{−p} dξ )^{1/p} ds

≤ (n − 1)(∫_{a}^{t} s^{p−2} ∫_{Ω(a,t)} ||x||^{1−n}||w||^{q} dx ds )^{1/q}
× (∫_{1}^{∞} ω_{n}s^{−p} ds )^{1/p}(∫_{0}^{t} s^{p−2} ds )^{1/p}

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 18, p. 6

= (n − 1)(\frac{v(t)}{p − 1})^{1/q} \frac{t^{\frac{p−1}{p}} ω_{n}^{1/p}}{(p − 1)^{2/p}}

= \frac{(n − 1)ω_{n}^{1/p}}{(p − 1)^{\frac{1}{q} + \frac{2}{p}}} v^{1/q}(t)t^{\frac{p−1}{p}} .  (2.17)

In view of the fact (2.14) there exists a number t_{b} ≥ t_{a} such that

\frac{(n − 1)ω_{n}^{1/p}}{(p − 1)^{\frac{1}{q} + \frac{2}{p}}} t^{\frac{p−1}{p}} ≤ \frac{1}{3}v^{1/p}(t)  (2.18)

for t ≥ t_{b}. Combining (2.15), (2.16), (2.17) and (2.18) we get

\frac{1}{3}v(t) ≤ K(tv′(t))^{1/q}

for t ≥ t_{b}. From here

\frac{v′(t)}{v^{q}(t)} ≥ \frac{1}{t}(\frac{1}{3 K} )^{q}

for t ≥ t_{b}. Integration of this inequality from t_{b} to ∞ gives a convergent integral on the left hand side and divergent integral on the right hand side. This contradiction ends the proof.

References

[1] T. Chantladze, N. Kandelaki, A. Lomtatidze, Oscillation and nonoscillation criteria of a second order linear equation, Georgian Math. J., 6, No. 5, (1999), 401–414.

[2] O. Došlý, R. Mařík, Nonexistence of the positive solutions of partial differential equations with p−Laplacian, to appear in Math. Hung.

[3] P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York, 1964.

[4] N. Kandelaki, A. Lomtatidze, D. Ugulava, On the oscillation and nonsocillation of a second order half–linear equation, to appear in Georgian Math. J. (2000).

[5] R. Mařík, Oscillation criteria for Schrödinger PDE, to appear in Adv. Math. Sci. Comp.

[6] U.–W. Schminke, The lower spectrum of Schrödinger operators, Arch. Rational Mech. Anal. 75, (1989), 147–155.

Robert Mařík

Masaryk University, Department of Mathematics Janáčkovo nám. 2A, 662 95 Brno, Czech Republic

   E-mail: marik@math.muni.cz