Weighted p-Laplacian problems on a half-line

∗

Paul A. Binding † Department of Mathematics and Statistics University of Calgary Calgary, Alberta, Canada T2N 1N4 Patrick J. Browne Department of Mathematics and Statistics University of Saskatchewan Saskatoon, Saskatchewan, Canada S7N 5E6 Bruce A. Watson ‡ School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa September 9, 2015

Abstract We study the weighted half-line eigenvalue problem −(|y 0 (x)|p−1 sgn y 0 (x))0 = (p − 1)(λr(x) − q(x))|y(x)|p−1 sgn y(x),

0 ≤ x < ∞,

for 1 < p < ∞, with initial condition y 0 (0) sin α = y(0) cos α, α ∈ [0, π), using a modified Pr¨ ufer angle φ(λ, x). The eigenvalues λk , k ≥ 0, with λk → ∞ as k → ∞, are characterized by φ(λk , x) → (k + 1)πp −, and φ(λ, x) → (k + 1)πp + if λk < λ < λk+1 , as x → ∞. We allow the weight r to be locally integrable and definite, semidefinite or indefinite. In the first two cases, the sequence of eigenvalues accumulates at one of ±∞, and in the third, the sequence accumulates at both ±∞. In all cases, solutions y are nonoscillatory on (0, ∞) for all λ. ∗

Keywords: p-Laplacian, singular eigenvalue problems, Pr¨ ufer angle, indefinite weight. Mathematics subject classification (2010): 34L30, 34B16, 34B24. † Research supported in part by the NSERC of Canada ‡ Research conducted while visiting University of Calgary. Supported in part by the Centre for Applicable Analysis and Number Theory and by NRF grant IFR2011032400120.

1

1

Introduction

Our aim is to study the weighted half-line eigenvalue problem given by −(|y 0 (x)|p−1 sgn y 0 (x))0 = (p − 1)(λr(x) − q(x))|y(x)|p−1 sgn y(x),

0 ≤ x < ∞, (1.1)

for 1 < p < ∞, with an initial condition y 0 (0) sin α = y(0) cos α,

α ∈ [0, π).

(1.2)

Initially we assume r(x) > 0, and we write s(x) = r(x)1/p ,

(1.3)

0 < s ∈ ACloc and q ∈ L1,loc on (0, ∞)

(1.4)

for all x ≥ 0. We also assume

so r is also locally absolutely continuous. [In Section 5 we shall be able to weaken the assumpions on r – see the end of this section]. We say that λ ∈ R is an Reigenvalue of (1.1)-(1.2) with eigenfunction y if y, y 0 ∈ ACloc , ∞ y satisfies (1.1)-(1.2) and 0 r|y|p is finite. Of particular interest will be results such as characterization of the eigenvalues λk (k ≥ 0, with λk → ∞ as k → ∞) via φ(λk , x) → (k + 1)πp −, and φ(λ, x) → (k + 1)πp + if λk < λ < λk+1 , as x → ∞, (1.5) by which we mean that φ(λk , x) (resp. φ(λ, x)) approaches (k + 1)πp from below (resp. above), as x tends to ∞. Here πp is a constant, dependent on which modifications φ(λ, x) 2π of the Pr¨ ufer angle is used. In our case πp = p sin(π/p) , and in all cases π2 = π. Previous works that we know concerning (perhaps weaker versions of) (1.5) have been for the case of the weight function r ≡ 1, and we shall now review this area. Results along the lines of (1.5) were given by Crandall and Reno [10] for p = 2, with a more complete (and apparently independent) study by Brown and Reichel [9] for p > 1. In both works, q was assumed continuous with q(x) → ∞ as x → ∞ together with various additional conditions (later removed in [7]). Computational usefulness of (1.5) was one motivation, and we remark that the differential inequality analyses in several of the works under review can be used to provide rates and regions of attraction of φ to multiples of πp . We remark that various authors, for example of [7]-[9], treat “radial” p-Laplacian problems which can be transformed to the form (1.1), and then fall within our framework, as will be seen in Section 5. Note that in the above references q is bounded below. Assuming this and p = 2, the authors of [2] showed (1.5) to be equivalent to Molˇcanov’s condition on q, and therefore

2

(1.1), (1.2) is equivalent to an eigenvalue problem for a self-adjoint operator on L2 (0, ∞) with a discrete spectrum (which is bounded below). For general p > 1, spectral theory is less developed, but we note that (1.5), with λ−1 := −∞, guarantees that solutions y of (1.1), (1.2) have finitely many zeros, and hence are nonoscillatory on (0, ∞), for all λ ∈ R. Such results, for an extension of Molˇcanov’s condition allowing q to be unbounded below, were given in [1]. Oscillation theory on (0, ∞) for related problems is discussed at length in [11]. In [7] and [9], modified Pr¨ ufer methods were used to investigate the case q(x) → −∞ as x → ∞, leading to a discrete set of eigenvalues accumulating at −∞ as well as at +∞. In this case, an extra condition on y is needed as x → ∞, and all solutions are oscillatory on (0, ∞). The case q(x) → ∞ as x → ∞ is also treated in [7] and more general conditions are analysed in [3, 8] where a finite λe exists so that solutions y are oscillatory for λ > λe and nonoscillatory for λ < λe . In this work we shall consider versions of (1.5) for cases where the weight function r in (1.1) is definite, semidefinite and indefinite. In the first two cases, there are sequences of eigenvalues accumulating at one of ±∞, and in the third, there are sequences accumulating at both ±∞. In all cases, solutions y of (1.1), (1.2) are nonoscillatory on (0, ∞) for all λ, and the above eigenvalue sequences are indexed by oscillation count. As far as we know, our results on (1.5) are new even for the Sturm-Liouville case p = 2. Our conditions on q(x), r(x) allow considerable variation as x → ∞, but are related to those in [12, 13, 14] for the case p = 2. Our plan is as follows. In Section 2 we adopt a generalised form, given via assumptions (2.1)-(2.3) on q and s, of Hinton’s discrete spectrum conditions [12] for positively weighted Sturm-Liouville problems. These are somewhat complicated, and we use two transformations in Section 2 and Section 3, the second being of generalised Pr¨ ufer type, to deduce a relatively simple differential inequality. In Section 4 we modify previous analyses to handle weaker decay of eigenfunctions y(x) as x → ∞ and we deduce (1.5) for the set of all eigenvalues λk , k = 0, 1, 2, . . . . A summary of the results we need later is given in Corollary 4.6. In Section 5 we relax our conditions on the weight function to r ∈ L1,loc , provided that there is an s obeying the conditions from the previous sections and for which |r| ≤ sp . In particular, we allow r to be locally integrable and definite, semidefinite or indefinite, and to vanish on sets of positive measure. Even when p = 2, this enables us to give new results. For example, when r is definite (resp. indefinite) it is standard to study the problem in a Hilbert (resp. Krein) space given by an r-weighted L2 space, but this construction requires r not to vanish on a set of positive measure. A key tool is the two-parameter embedding −(|y 0 |p−1 sgn y 0 )0 = (p − 1)(λr − q − µsp )|y|p−1 sgn y

(1.6)

where s is as above. Then eigenvalues µk (λ) exist for each k and each λ ∈ R. By

3

combining techniques from [1], [4] and the previous sections with (1.6) we establish the existence of discrete sets of eigenvalues of (1.1), (1.2) satisfying weakened versions of (1.5) in Theorem 5.3. Various improvements are given under stronger conditions on the coefficients, and in particular if (1.1), (1.2) is left definite in the sense of Definition 5.7, then (1.5) holds whenever λ, λk > 0.

2

A Transformation

From now on we assume: I There exists M > 0 so that |s0 (x)| ≤ M s(x)2 ,

for a.e.

II There are 0 > 0 and C > 0 so that Z x+0 /s(x) − q s(x) < C, sp x

x ≥ 0.

for all

x ≥ 0.

(2.1)

(2.2)

III For all  satisfying 0 <  < 0 , Z

x+/s(x)

lim s(x)

x→∞

x

q+ = ∞. sp

(2.3)

Here we use the convention that q = q + − q − , with q ± (x) = max{±q(x), 0}. In the case p = 2, Hinton used similar (but stronger) conditions in [12] (see also M¨ uller-Pfeiffer [14]). Next we give an inequality similar to that of [12]: this is fundamental to the change of variables applied later. Lemma 2.1 For each η and b satisfying η ∈ (0, 1/4M ) and b ≥ (1 − 4M η)−1 ,

(2.4)

we have 1 s(u) <
whenever

x − η/s(x) ≤ u ≤ x + η/s(x).

(2.5)

Proof: Assume (2.4) and write δ = 4M η. Suppose that there exist x and y with x − η/s(x) ≤ y ≤ x + η/s(x) and s(y) s(x) − 1 ≥ δ.

4

Let g(u) = s(u) s(x) , then, since g(x) = 1, there is u with x − η/s(x) ≤ u ≤ x + η/s(x) for which |g(u) − 1| = δ and |g(y) − 1| < δ for all y between x and u. Without loss of generality we assume u > x. We now observe that δ = |g(u) − 1| Z u g 0 = Z xu ≤ |g 0 | Zxu 0 |s (y)| = dy s(x) x Z u s(y)2 dy ≤ M x s(x)  Z u s(y) 2 = M s(x) dy s(x) x < M s(x)|u − x|(1 + δ)2 ≤ M η(1 + δ)2 (1 + δ)2 = δ 4 < δ, a contradiction. From this it follows that 1−δ <

s(u) <1+δ s(x)

for x − η/s(x) ≤ u ≤ x + η/s(x). Thus (2.5) holds for all b ≥ max{1 + δ, (1 − δ)−1 } = (1 − δ)−1 .

Lemma 2.2 There exists G > 0 so that s(x)−1 ≤ M x + G,

Proof: From (2.1)

for all

x ≥ 0.

(2.6)

x s(x)−1 0 ≤ M x

and the lemma follows with G = s(0)−1 . Define the variable t in terms of x by Z t = t(x) =

s. 0

5

x

(2.7)

Lemma 2.3 Let x and t be related by (2.7). Then t → ∞ if and only if x → ∞. Proof: Since s is continuous, t(x) is defined and continuous for all x ≥ 0. Thus t(x) is finite for x finite and so t(x) → ∞ implies x → ∞. Moreover from (2.6) we have Z x Mx + G dξ 1 →∞ t(x) ≥ = ln M ξ + G M G 0 as x → ∞. Thus, if x → ∞ then t(x) → ∞.

3

Generalized Pr¨ ufer Angle

As in [5], we write [f ]β = |f |β sgn f and define sinp (x) to be the solution of −([y 0 (x)]p−1 )0 = (p − 1)[y(x)]p−1 ,

0 ≤ x < ∞,

obeying the initial conditions sinp (0) = 0,

sin0p (0) = 1.

Definition 3.1 By ‘the initial value problem (1.1), (3.1)’ we mean (1.1) with the initial conditions y 0 (0) = cos α,

y(0) = sin α,

(3.1)

where α is as given in (1.2). From [5] it is known that (1.1), (3.1) has a unique solution on each compact interval containing 0 and hence has a unique solution in [0, ∞). Since (1.1), (1.2) are homogeneous, non-trivial solutions of (1.1), (1.2) can be scaled to be a solution of (1.1), (3.1). We define the modified Pr¨ ufer angle φ(λ, x) to be continuous in x and such that s(x)y(x) sin0p φ(λ, x) = y 0 (x) sinp φ(λ, x),

(3.2)

with initial value φ(λ, 0) = α ˜ ∈ [0, πp ) where s(0) sin α sin0p α ˜ = cos α sinp α ˜.

(3.3)

Here y is the solution of the initial value problem (1.1), (3.1). From [5, Equation (2.4)] with f = 1/s (and where S 0 (θ)p−1 should be [S 0 (θ)]p−1 ) it follows that   λr − q s0 0 p 0 p φ = s | sin φ| + | sin φ| + [sin0p φ]p−1 sinp φ (3.4) p p sp s    λr − q p ≤ s | sinp φ| + 1 + M , (3.5) sp

6

by | sinp φ|p +| sin0p φ|p = 1 and (2.1). With x and t related by (2.7), we write Φ(λ, t), Q(t), S(t) for φ(λ, x(t)), q(x(t)), s(x(t)) respectively, and also ˙ = ∂∂t . Then, since | sin0p Φ|, | sinp Φ| ≤ 1, we have Φ˙ ≤ (1 + M ) − (QS −p − λ)| sinp Φ|p .

(3.6)

This inequality is of the type considered in [3, Section 2], with (in the notation of that reference) D = 1 + M, b(t) = 0, g(t) = QS −p − λ h(Φ) = | sinp Φ|p , Ω = πp . In related circumstances [1, 3], conditions labelled (M+) and (B–), similar to those of Molˇcanov [13] and Brinck [6], have been central. We next establish the analogues of these conditions for g. First we discuss condition (M+). Lemma 3.2 For each sufficiently small  > 0, Z t+ lim g + = ∞.

(3.7)

t→∞ t

Proof: Choose η and b = (1 − 4M η)−1 to satisfy (2.4) and write  = bη. Note that  may be made arbitrarily small by choosing η small enough. Now Z x(t+) = s(u) du x(t)

and by Lemma 2.1 we have Z x(t)+/bs(x(t)) s(u) du ≤ bs(x(t)) x(t)

 == bs(x(t))

Z

x(t+)

s(u) du x(t)

giving x(t) +

 ≤ x(t + ). bs(x(t))

(3.8)

Applying (3.8) and then Lemma 2.1 we obtain Z t+ Z x(t)+/(bs(x(t))) + g (τ ) dτ ≥ (qs−p − λ)+ (ξ)s(ξ) dξ t

x(t)

≥

s(x(t)) b

= −

Z

x(t)+/(bs(x(t)))

(q + s−p − |λ|) dξ

x(t)

|λ| s(x(t)) + b2 b

7

Z

x(t)+/(bs(x(t)))

x(t)

q + s−p dξ.

Now, by Lemma 2.3, Z lim

t→∞ t

t+

s(x) |λ| g (τ ) dτ ≥ − 2 + lim x→∞ b b +

x+/(bs(x)

Z x

q+ dξ, sp

and then (3.7) follows from (2.3) provided  < 0 . We now consider condition (B–) for g. Lemma 3.3 For some sufficiently small  > 0, Z t+ g− t

is bounded for t ∈ [0, ∞). Proof: Again we choose b = (1 − 4M η)−1 but this time we write  = η/b. Again we can make  arbitrarily small (and, in particular, to satisfy 0 < b < min{0 , 1/4M },

(3.9)

where 0 is as in (2.2)), by choosing η small enough. Note that (3.9) and b = (1−4M η)−1 also guarantee (2.4). Using inequalities from Lemma 2.1 complementary to those of the previous proof, we see that Z x(t)+b/s(x(t)) Z x(t+) s(x(t)) b s(u) du ≥ == s(u) du b s(x(t)) x(t) x(t) which gives x(t) +

b ≥ x(t + ). s(x(t))

Thus, for  > 0 satisfying (3.9), we can use Lemma 2.1 to give Z t+ Z x(t)+b/s(x(t)) − g (τ ) dτ ≤ (qs−p − λ)− s(ξ) dξ t

x(t)

Z

x(t)+b/s(x(t))

≤ bs(x(t))

(q − s−p + |λ|) dξ

x(t)

≤ b(C + b0 |λ|), proving the required uniform boundedness in t. Corollary 3.4 (a) Φ(λ, t) increases with t through multiples of πp . (b) As t increases from 0 to ∞, Φ(λ, t) is bounded above and is ultimately trapped in a band strictly between two consecutive multiples of πp . Proof: Result (a) follows from [5, Lemma 2.3], while (b) follows from part (a), (3.6), Lemmas 3.2, 3.3 and [3, Lemma 2.3].

8

4

− Λn, Λ+ n and Λn

Define Λn = {λ ∈ R : nπp < Φ(λ, t) < (n + 1)πp for all t large}, Λ+ n = {λ ∈ Λn : Φ(λ, t) → (n + 1)πp as t → ∞}, Λ− n = {λ ∈ Λn : Φ(λ, t) → nπp as t → ∞}. Observe that Λn = {λ ∈ R : nπp < φ(λ, x) < (n + 1)πp for all x large}, and, by Corollary 3.4,

∞ [

Λn = R. Because of [1, Lemma 2.4] we have

n=0 + Λn = Λ− n ∪ Λn

and as before, because Φ is monotonic in λ, the sets Λn , Λ± n are convex and so are + − intervals or empty. In particular if λ ∈ Λn , ξ ∈ Λn and ζ ∈ Λn+1 then λ < ξ < ζ. From [5, Lemma 2.5] we have Φ(λ, t) → ∞ as λ → ∞ for each t > 0 so each set Λn is bounded above.

Lemma 4.1 (i) If λ ∈ Λ− n for some n and y is the solution of the initial value problem (1.1), (3.1), then sy ∈ / Lp (0, ∞). (ii) If λ ∈ Λ+ n for some n and y is the solution of the initial value problem (1.1), (3.1), then for each D > 0 there exists Xn+ so that s(x)|y(x)| ≤

s(Xn+ )|y(Xn+ )|



Mx + G M Xn+ + G

−D/M for all

x ≥ Xn+ . (4.1)

(iii) If λ ∈ Λ+ n for some n and y is the solution of the initial value problem (1.1), (3.1), then sy ∈ Lp (0, ∞) and y(x) → 0 as x → ∞. Proof: For brevity we denote sin0p Φ/ sinp Φ by cotp Φ. (i) By hypothesis, cotp Φ(λ, t) → +∞ as t → ∞ and so for any given K > 0 there is Xn− so that y 0 (x(t)) > K, for t ≥ t(Xn− ). s(x(t))y(x(t)) Equivalently, for x ≥ Xn− ,

y 0 (x) K > Ks(x) ≥ y(x) Mx + G

9

by Lemma 2.2, and without loss of generality we may take y, y 0 > 0, so y(Xn− )

y(x) ≥



Mx + G M Xn− + G

K/M .

Thus K

p

p

≥

s(x) y(x) and the result follows by taking

K M

y(Xn− )p

(M x + G)p( M −1) , (M Xn− + G)Kp/M

(4.2)

≥ 1 − p1 .

(ii) Now, analogously to (i), cotp Φ(λ, t) → −∞ as t → ∞ and for D > 0 there is Xn+ so that for x ≥ Xn+ , y 0 (x) < −D − M, s(x)y(x)

(4.3)

and without loss of generality we may take y > 0 and y 0 < 0. Observe that (sy)0 s0 y 0 = + ≤ (M − D − M )s = −Ds, sy s y by (2.1). Thus  Z s(x)y(x) ≤ s(Xn+ )y(Xn+ ) exp −D

x

 s

Xn+

≤

s(Xn+ )y(Xn+ )



Mx + G M Xn+ + G

−D/M .

(iii) Taking D > M/p in (4.1) we obtain sy ∈ Lp (0, ∞). From (2.6) and (4.1), |y(x)| ≤ B(M x + G)1−D/M for a suitable constant B > 0 and x ≥ Xn+ . Now choose D > M to give y(x) → 0 as x → ∞. Lemma 4.2 If λ ∈ Λ+ n and y is the solution of the initial value problem (1.1), (3.1), then y 0 is bounded.

Proof: As in the proof of Lemma 4.1, for each D > 0 there exists Xn+ so that for x ≥ Xn+ , we may (after scaling, if necessary) take y 0 (x) < 0.

y(x) > 0,

(4.4)

From (1.1), |y 0 | is continuous and thus bounded on [0, Xn+ ]. Also y 0 (x) < 0 for x ≥ Xn+ giving y 0 (x) bounded above on [0, ∞). It remains only to show that y 0 (x) is bounded

10

below for large x. Suppose instead that there is a sequence xj → ∞ as j → ∞ with y 0 (xj ) → −∞ as j → ∞ and xj > Xn+ for j ∈ N. Choose η to satisfy 0 < bη < 0 and (2.5), where 0 is as in (2.2). In what follows we let 0 < z < η/s(x) and we write u = x − η/s(x). Integrating (1.1) from x − z > Xn+ to x and using (4.4) we obtain Z x 0 p−1 0 p−1 (λr − q)y p−1 |y (x)| − |y (x − z)| = (p − 1) Zx−z x (|λ|sp + q − )y p−1 ≤ (p − 1) x−z  Z x q− |λ| + p sp y p−1 . ≤ (p − 1) (4.5) s u From (4.1), for ξ ≥ Xn+ , s(ξ)y(ξ) ≤ s(Xn+ )y(Xn+ ),

(4.6)

and thus by Lemma 2.1 and (4.6),   Z x Z x q− q− p p−1 = |λ| + p (sy)(p−1) s |λ| + p s y s s u u  Z x q− |λ| + p (sy)p−1 ≤ bs(x) s u Z u+ s(u) η − !
s(x) s(u) q s(x) + + p−1 ≤ b η|λ| + s(u) s(X )y(X ) n n s(u) sp u ! Z u+ bη −
p−1 s(u) q ≤ b η|λ| + bs(u) s(Xn+ )y(Xn+ ) p s u ≤ b(η|λ| + bC)(s(Xn+ )y(Xn+ ))p−1 , where C is as in (2.2). Hence, from (4.5), there is a constant C(λ) > 0 so that |y 0 (x)|p−1 − |y 0 (x − z)|p−1 ≤ C(λ). In the notation of (2.5), choose D so that ηD > bs(Xn+ )y(Xn+ ) and fix j so large that |y 0 (xj )|p−1 ≥ C(λ) + Dp−1 . For 0 < z < η/s(xj ) with xj − z ≥ Xn+ , |y 0 (xj )|p−1 ≤ |y 0 (xj − z)|p−1 + C(λ). Now, recalling (4.4), we have y 0 (xj − z) ≤ −D,

whenever

0 < z < η/s(xj ).

Finally, integration with respect to z from 0 to η/s(xj ), (2.5) and (4.6) give s(xj )y(xj ) ≤ s(xj )y(xj − η/s(xj )) − ηD ≤ bs(Xn+ )y(Xn+ ) − ηD < 0,

11

contradicting y(xj ) > 0. In order to show that Λ+ n consists of at most one point, we also need the following. Lemma 4.3 For λ ∈ Λ+ n and y the solution of the initial value problem (1.1), (3.1), we have q − |y|p ∈ L1 [0, ∞), Proof: Let 0 > 0 and C > 0 be as in (2.2) and x0 = Xn+ > 0 of Lemma 4.1 with Dp/M ≥ 2. If 0 <  < min{0 , 1/(4M )}, then defining xn+1 = xn + /s(xn ),

(4.7)

we have xn → ∞ as n → ∞. [Otherwise xn % l ∈ R as n → ∞. Since 1/s is continuous taking n → ∞ in (4.7) we obtain the contradiction l = l + /s(l).] With x = xn in (2.2), as 0 <  < 0 , Z xn+1 − q s(xn ) ≤ C, sp xn and thus Z

∞

q − |y|p ≤

x1

∞  X

Z

xn+1

s(xn )

n=1 ∞ X

≤ C

n=1

xn

q− sp



1 max (s|y|)p s(xn ) [xn ,xn+1 ]

1 max (s|y|)p . s(xn ) [xn ,xn+1 ]

Moreover, by (4.1), there is a constant K so that max (s|y|)p ≤ Kx−2 n ,

for all

[xn ,xn+1 ]

n ∈ N.

Hence by (2.5) Z

∞

x1

q − |y|p ≤

∞ X

CK s(xn+1 )x2n+1 n=0

Z xn+1 ∞ X s(xn )−1 dx ≤ bCK x − xn xn x2n+1 n=0 n+1 ∞ Z bCK X xn+1 dx ≤ from (4.7)  x2 n=0 xn Z bCK ∞ dx = < ∞, 2  x0 x and

R x1 0

q − |y|p < ∞ then gives the result.

12

Theorem 4.4 Each set Λ+ n consists of at most one point. Proof: If there are λ, µ ∈ Λ+ k with λ < µ then φ(λ, x) < φ(µ, x) < (k + 1)πp ,

x ∈ [0, ∞),

for all

(4.8)

where φ(λ, x) is as in (3.2). Let y be the solution of the initial value problem (1.1), (3.1), and z be the solution of this initial value problem but with λ replaced by µ. Set x0 = 0 if k = 0 and x0 to be a solution of φ(λ, x0 ) = kπp otherwise. Let v be the solution of (1.1) on [x0 , ∞) with λ replaced by µ satisfying the initial condition v(x0 ) = y(x0 ),

v 0 (x0 ) = y 0 (x0 ).

(4.9)

For x ∈ [x0 , ∞) denote by ψ(x) the modified Pr¨ ufer angle corresponding to v with ψ(x0 ) = φ(λ, x0 ) − kπp .

(4.10)

Then φ(λ, x) − kπp < ψ(x) ≤ πp ,

for all

x ∈ (x0 , ∞).

(4.11)

Now v and y have the same (constant) sign in (x0 , ∞), which we assume to be positive, and by (4.11) and the definition of Λ+ k ψ(x) → πp −,

as x → ∞.

(4.12)

Thus Lemma 4.2 can be applied to v on [x0 , ∞) showing that v 0 is bounded, so by (4.12), v(x) → 0 as x → ∞. Similarly y(x) → 0 as x → ∞, so for small  > 0, the function w=

yp (v + )p−1

satisfies w(x) → 0 as x → ∞. Again, as Lemma 4.2 applies to both v and y, it follows that w0 (x) → 0 as x → ∞. Combining Lemma 2.2 and (4.1) we obtain y δ ∈ L1 [x0 , ∞) for all δ > 0 and since y 0 and v 0 are bounded, we now have w, w0 ∈ L1 [x0 , ∞). From Lemma 4.3, q − y p ∈ L1 (0, ∞) and Lemma 4.1 (ii), 0 ≤ ry p ≤ (sy)p ∈ L1 (0, ∞), so the proof can now be completed as for [1, Theorem 3.3], with λ and µ replaced by λr and µr respectively.

Theorem 4.5 The set Λn 6= φ so λn := sup Λ− n is finite, n = 0, 1, 2, . . . . Also Λ− n = (λn−1 , λn ), λ−1 = −∞.

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Λ+ n = {λn },

Λn = (λn−1 , λn ] where

Proof: Since Φ obeys the differential inequality (3.6) and it has been established in Theorem 4.4 that Λ+ n has at most one element, the theorem follows as in the proof of [3, Theorem 3.12]. In consequence of the above results we can give the following summary.

Corollary 4.6 Assume that q, r and s satisfy (1.3), (1.4) and (2.1)-(2.3). Then the sequence (λn ) increases strictly to ∞ as n → ∞. Solutions, y, decay faster than any power, while sy, y 0 ∈ Lp (0, ∞) and thus each λn is an eigenvalue with eigenfunction yn having precisely n zeros in (0, ∞). The corresponding modified Pr¨ ufer angle ϕ(λn , x) → (n + 1)πp − as x → ∞, while for λ 6= λn , ϕ(λ, x) → k(λ)πp +, say, where k(λ) ≤ n

if

λ < λn ,

and

k(λ) ≥ n + 1

if

λ > λn .

(4.13)

Proof: The Λn are non-empty (by Theorem 4.5) and disjoint (by definition), so the first sentence follows. From (4.1) we have sy ∈ Lp (0, ∞), and (except for y 0 ∈ Lp (0, ∞)) the second sentence follows from Theorem 4.5 and definitions. Now multiply (1.1) by y and integrate from 0 to b to give Z 0

b

|y 0 |p ≤ (p − 1)

b

Z 0

b (λ|sy|p + q − |y|p ) + [y 0 ]p−1 y 0 .

As b → ∞, the right hand side has a finite limit by sy ∈ Lp (0, ∞), Lemma 4.3, Lemma 4.1 (iii) and Lemma 4.2. Thus y 0 ∈ Lp (0, ∞). The final sentence follows from the definitions of Λ± n. Note: λn is an increasing function of q and decreasing in α and r; also if m > n then between any two zeros of yn , ym has a zero. Moreover, if q is replaced by q˜ > q, then between any two zeros of yn , y˜n has a zero, cf. [1, Theorem 4.3].

5

Removal of sign restrictions on r

We now consider (1.1)-(1.2) without the requirement that r be positive. We assume that r is measurable and Z ∞ |r| > 0, (5.1) 0

which is needed to ensure that (1.1)-(1.2) is a non-trivial eigenvalue problem in λ. We also assume that |r| ≤ sp

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(5.2)

where q, s continue to satisfy conditions (1.4) and (2.1)-(2.3). To aid our analysis we consider the two parameter equation −(|y 0 |p−1 sgn y 0 )0 = (p − 1)(λr − µsp − q)|y|p−1 sgn y,

on

[0, ∞),

(5.3)

with initial condition (1.2). It should be noted that assumptions (2.2) and (2.3) also hold with q replaced by q − λr, in view of (5.2). Thus the theory of the previous sections is applicable with q replaced by q−λr and λr by −µsp where −µ is the eigenparameter. For each λ ∈ R, we denote the resulting nth eigenvalue by µn (λ). In this context, Corollary 4.6 gives µ0 (λ) > µ1 (λ) > · · · > µn (λ) → −∞.

(5.4)

Remark: As indicated R ∞ above, for each λ ∈ R, µn (λ) obeys (5.2) and (5.3) for some function y satisfying 0 |sy|p < ∞, and for brevity we shall call µn (λ) an R“s-eigenvalue”. ∞ Those λ for which µn (λ) = 0 also provide s-eigenvalues for (1.1)-(1.2). If 0 |r| |y|p < ∞ (as required in Section 2) then such a λ will now be called an “r-eigenvalue”. In Sections 2–4, this was equivalent to being an s-eigenvalue, but such equivalence will need extra arguments below in view of our weakened assumptions on r. We shall adapt material from [1, Section 5] on truncation of one parameter p-Laplacian eigenvalue problems from (0, ∞) to (0, X) with finite X, the boundary condition at X being taken as the Dirichlet condition y(X) = 0. We write µnX (λ) for the nth eigenvalue corresponding to (5.4) of the X-truncated problem. Our adaptation consists of replacing weight function 1 by sp . Indeed the arguments used for [1, Theorem 5.2] can be carried over directly since s is positive and continuous on [0, ∞). Accordingly we shall apply the above results from [1] to our sp -weighted situation.

Lemma 5.1 µnX (λ) % µn (λ) as X → ∞. Lemma 5.2 For all integers n ≥ 0, λ ∈ R and X ≥ 0, µnX (λ) is C 1 in λ, µ0nX (λ)

RX =

R0X 0

r|y|p |sy|p

∈ [−1, 1],

(5.5)

where y is an eigenfunction for (λ, µnX (λ)).

Proof: Most of this follows from [4, Theorem 3.1]. The final contention of (5.5) comes from (5.2).

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Let m = min{n ∈ Z|n ≥ 0, ∃λ ∈ R such that µn (λ) < 0} and let λ∗ ∈ R be taken so that µm (λ∗ ) < 0. We shall give some possible choices of λ∗ below. In the light of (5.4), it follows that µn (λ∗ ) < 0 for all n ≥ m. We remark that m ≥ 0 is sometimes called the minimal oscillation number, and we refer to Theorem 5.8 and the ensuing remarks for cases when m = 0. Our first result on existence of eigenvalues λ is for the range λ > λ∗ .

Theorem 5.3 Assume n ≥ m (as defined above), (2.1)-(2.3), (5.2) for s ∈ ACloc , and (5.1). R∞ (i) If 0 r+ > 0 then there exist two sequences λn+ ≥ λn− > λ∗ , both increasing to ∞ with n, such that µn (λn± ) = 0 (so λn± are s-eigenvalues), and if µn (λ) = 0 with λ > λ∗ , then λ ∈ En := [λn− , λn+ ]. (ii) Corollary 4.6 holds with λn replaced by λn− , and by λn+ , except that (4.13) should be replaced by k(λ) ≤ n if λ∗ < λ < λn− and k(λ) ≥ n + 1 if λ > λn+ . ˜ > λ∗ Proof: (i) First we note by [4, Theorem 4.1] that for any X0 > 0 there exists λ ˜ > 0. Now letting b increase from X0 to ∞, we see from Lemma 5.1 that with µnX0 (λ) ˜ µn (λ) > 0. From Lemma 5.1 and (5.5), the function µn has Lipschitz constant 1, and, in particular, is continuous. Thus, since µn (λ∗ ) ≤ µm (λ∗ ) < 0, the set Sn = {λ > λ∗ : µn (λ) = 0} is non-empty and closed. This allows us to define λn− (resp. λn+ ) as min Sn (resp. max Sn ). (ii) Evidently λ∗ < λ < λn− implies µn (λ) < 0, so µn+1 (λ) < 0, and also µn+1 (λn− ) < 0 by (5.4). Thus λ(n+1)− > λn− and similarly λ(n+1)+ > λn+ . Moreover for each λ ∈ R the µn (λ) have no finite accumulation as n → ∞, and therefore nor do the λn− , by virtue of (5.4). This establishes the first sentence of the analogue of Corollary 4.6. The second follows directly as in the previous sections, and similarly ϕ(λn± , x) → (n + 1)πp − as x → ∞ since µn (λn± ) = 0. Next λ∗ < λ < λn− as x → ∞ implies 0 < −µn (λ) so ϕ(λ, x) → k(λ)πp + where k(λ) ≤ n as in Corollary 4.6. Similarly λ > λn+ implies 0 > −µn (λ) so k(λ) ≥ n + 1. Remarks: (a) An important case of the above result is right semi-definiteness (RSD), where r ≥ 0 a.e.

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Corollary 5.4 Assume RSD. Then we can take λ∗ = −∞ in Theorem 5.3, and the set of s-eigenvalues with index n equals En . This follows because, by (5.5), the µnX are nondecreasing, and hence so is µn by Lemma 5.1.

(b) An example where equivalence of r- and s-eigenvalues can be obtained is as follows. For some X > 0, we assume the conditions on r, s in Section 2 for x ≥ X, and on (0, X) we assume the same smoothness conditions on r, we amend r(x) > 0 to 0 ≤ r(x) ≤ r(X), and we take s as the constant r(X)1/p . Then the conditions on r, s of this section are satisfied, continuity (and hence boundedness on [0, X]) of eigenfunctions y shows that r-eigenvalues are s-eigenvalues, and the converse follows from (5.2).

(c) As indicated in Section 1, several results have been obtained for equations involving the radial p-Laplacian, in the form ∂ −ξ 1−n ∂(ξ n−1 [∂y]p−1 ) = (λ − κ)[y]p−1 , ∂ = , (5.6) ∂ξ ∂y(0) = 0. (5.7) If n < p then a modified Sturm transformation (cf. [5, 11]) x = Aξ a leads to (1.1) with r(x) = B p xb where a, A, b, B > 0. This problem satisfies the conditions of Section 2, except that r(0) = 0. Thus it obeys RSD, and is moreover a special case of (b) above, for any finite X > 0. In fact s-integrability of eigenfunctions coincides not only with r-integrability as in (b), but also with ξ n−1 -integrability, as required in [7, 8, 9]. If n = p then the modified Sturm transformation is of logarithmic form, but the ideas are similar. If n > p then the problem is of limit circle type [9, Remark 7] and thus cannot be transformed directly into our framework.

(d) If r takes negative (or both) signs then there are analogous eigenvalue sequences accumulating at −∞ (or ±∞). A formal statement is as follows. R∞ Corollary 5.5 (i) If 0 r− > 0 then there exist sequences of s-eigenvalues λ− n− ≤ − λn+ < λ∗ , both decreasing to −∞ as n → +∞, such that if µn (λ) = 0 with λ < λ∗ , then − λ ∈ [λ− n− , λn+ ]. (ii) If r takes both signs on sets of positive measure, then λ∗ can be taken as a minimiser of µm , and eigenvalue sequences λn± → ∞ as in Theorem 5.3, and λ− n± → −∞ as above, both exist. The proof follows from that of Theorem 5.3(i) and the identity λr = (−λ)(−r). Analogues of Theorem 5.3(ii) will be left to the reader.

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The next result explores equivalence of r- and s-eigenvalues in greater generality.

Theorem 5.6 In addition to the assumptions of Theorem 5.3, suppose that there is a p sequence of non-overlapping intervals P Ij = [cdj , cj + `j ] on which |r| ≥ aj s , j = 1, 2, . . . , where aj , cj > 0, lim cj = ∞ and j aj `j cj = ∞ for some real d. Then the r- and j→∞

s-eigenvalues of (1.1)-(1.2) coincide.

Proof: If λ is not an s-eigenvalue of (1.1)-(1.2), then µ = 0 is not an s-eigenvalue of (5.3)-(1.2). From Lemma 4.1 (applied to the above µ problem for fixed λ), 0 ∈ Λ− n for some n. Reasoning as for (4.2), we obtain s(x)p |y(x)|p ≥ Cxd

(5.8)

K for x sufficiently large, where C > 0 and d = p( M − 1) can be made arbitrarily large. Without loss of generality, we can assume that (5.8) holds on all Ij .

It follows that XZ j

Ij

|r||y|p ≥

XZ j

aj sp |y|p ≥ C

X

Ij

aj `j cdj = ∞.

j

Thus |r||y|p ∈ / L1 , so µ = 0 is not an r-eigenvalue of (5.3)-(1.2), and therefore λ is not an r-eigenvalue of (1.1)-(1.2). Conversely, if λ is an s-eigenvalue of (1.1)-(1.2), then it is an r-eigenvalue by (5.2). We give two examples of the above equivalence, assuming the standing conditions of this section on r, s. (i) Let r = sp on [X, ∞) for some X > 0. This is similar to (b) above except that r is now integrable, with no restrictions on sign or vanishing, on a set of finite measure. Theorem 5.6 applies with aj = `j = 1, cj = j and d = 1. (ii) For some a > 0, let |r| ≥ asp hold on Ij = [j, j + j −2 ], j = 1, 2, . . . , In this case, r may change sign (or vanish) arbitrarily often except on the the Ij , which have a union of finite measure. On the Ij , r is bounded away from zero, so if it changes sign there then it must be discontinuous. Theorem 5.6 applies with aj = 1, `j = j −2 , cj = j and d = 2. We conclude by specialising to a left definite situation.

Definition 5.7 The problem (1.1), with s as above and boundary condition (1.2) on [0, ∞), is uniformly left definite (ULD) if there is a constant c > 0 for which Z ∞ Z ∞ p p c s |y| ≤ lq [y] := (|y 0 |p + q|y|p ) + |y(0)|p [cot∗ α0 ]p−1 (5.9) 0

0

18

1 obeying (1.2). Here for all y ∈ Wps 1 0 Wps = {y ∈ Lloc 1 |y ∈ Lp , sy ∈ Lp }

and cot∗ α is cot α for α not an integer multiple of π and is zero at integer multiples of π. ULD implies µ0 (0) ≤ −c in (5.4), by virtue of the variational characterization of µ0 – see [1, Theorem 5.3]. Theorem 5.8 Under the conditions of Theorem 5.3, if (1.1), (1.2) is ULD then λ∗ can be taken as zero, m = 0 and for all n ≥ 0, λn− = λn+ =: λn . Moreover, Corollary 4.6 holds verbatim except that λ is to be positive in (4.13). Proof: We write lqX for lq but with the integral taken from 0 to X. By [4, Theorem 3.1] we have lqX [y] λµ0nX (λ) = R X + µnX (λ) (5.10) p 0 s|y| where y is any eigenfunction of (5.3) on [0, X], (1.2) and y(X) = 0 corresponding to 1. (λ, µnX (λ)). Note that y (extended over (X, ∞) by zero) belongs to Wps Let λn ∈ Sn , i.e. µn (λn ) = 0 and choose X so that µnX (λn ) > −c. Then (5.9) and (5.10) give µ0nX (λ) c µnX (λ) ≥ 2+ λ λ λ2 for all λ ∈ (0, λn ). Thus µnX (λn ) µnX (λ) (λn − λ)c − ≥ λn λ λ2n and as X → ∞ we obtain

µn (λ) (λn − λ)c ≤− <0 λ λ2n

from Lemma 5.1. A similar argument shows that µn (λ) > 0 if λ > λn , so Sn = {λn }, whence λn− = λn+ = λn . The remaining results now follow from Corollary 4.6. Remarks: Similar results to those in Theorem 5.8 are available for λ < 0 – cf. Corollary 5.5. Also, one could allow a more general “translated” form of ULD in Theorem 5.8, with λ replaced by λ − λ∗ , so that µ0 (λ∗ ) < 0, which is equivalent to m = 0. The corresponding modifications of (5.9) and Theorem 5.8 will be left to the reader. Acknowledgment We thank the referee for thoughtful comments that have helped us improve the presentation.

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References [1] P.A. Binding, P.J. Browne, Generalized Pr¨ ufer angle and variational methods for p-Laplacian eigenvalue problems on the half-line, Proc. Edinb. Math. Soc. (2), 51 (2008), 565-579. [2] P.A. Binding, L. Boulton, P.J. Browne, A Pr¨ ufer angle approach to singular Sturm-Liouville problems with Molˇcanov potentials, J. Comput. Appl. Math., 208 (2007), 226-234. [3] P.A. Binding, P.J. Browne, I.M. Karabash, Sturm-Liouville problems for the p-Laplacian on a half-line, Proc. Edinb. Math. Soc. (2), 53 (2010), 271-291. [4] P.A. Binding, P.J. Browne, B.A. Watson, Eigencurves for the p-Laplacian, J. Differential Equations, 255 (2013), 2751-2777. ´ bek, Sturm-Liouville theory for the p-Laplacian, Studia [5] P.A. Binding, P. Dra Scientiarum Math. Hungarica, 40 (2003), 375-396. [6] I. Brinck, Self-adjointness and spectra of Sturm-Liouville operators, Math. Scand., 7 (1959), 219-239. [7] B.M. Brown, M. Eastham, Eigenvalues of the radial p-Laplacian with a potential on (0, ∞), J. Comput. Appl. Math., 208 (2006), 111-119. [8] B.M. Brown, M. Eastham, Bounds for the positive eigenvalues of the p-Laplacian with decaying potentials, J. Math. Anal. Appl., 325 (2007), 734-744. [9] B.M. Brown, W. Reichel, Eigenvalues of the radially symmetric p-Laplacian, J. Lond. Math. Soc., 69 (2004), 657-675. [10] R.E. Crandall, M. Reno, Ground state energy bounds for potentials |x|ν , J. Math. Phys., 23 (1982), 64-70. ˇ a ´ k, Half-linear differential equations, North-Holland Mathe[11] O. Doˇ sly, P. Reh matics Studies, Volume 202, Elsevier, 2005. [12] D.B. Hinton, Molˇcanov’s discrete spectra criterion for a weighted operator, Canad. Math. Bull., 22 (1979), 425-431. ˇanov, Conditions for the discreteness of the spectrum of self-adjoint [13] A.M. Molc second-order differential equations, Trudy Mosk. Mat. Obshc., 2 (1953), 169-199. ¨ ller-Pfeiffer, Spectral theory of ordinary differential operators, Ellis Hor[14] E. Mu wood, Chichester, 1981.

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