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Publicly Available Published by De Gruyter March 20, 2016

Uniqueness of solutions to singular p-Laplacian equations with subcritical nonlinearity

Bevin Maultsby

Abstract

We present a geometric approach to the study of quasilinear elliptic p-Laplacian problems on a ball in n using techniques from dynamical systems. These techniques include a study of the invariant manifolds that arise from the union of the solutions to the elliptic PDE in phase space, as well as variational computations on two vector fields tangent to the invariant manifolds. We show that for a certain class of nonlinearities f with subcritical growth relative to the Sobolev critical exponent p*, there can be at most one such solution satisfying Δpu+f(u)=0 on a ball with Dirichlet boundary conditions.

1 Introduction

In this paper, we consider solutions u:Ωn to the quasilinear elliptic equation

(1.1){-Δpu=f(u)in Ω,u>0in Ω,u=0on Ω,

for different classes of nonlinear functions f(u). The domain Ω is a ball about the origin of radius R in n, n2, and 1<p2. We are interested in regular solutions, meaning u(0)=d>0. The p-Laplacian is defined by

(1.2)Δpu=div(|u|p-2u).

When p=2, (1.2) is the regular uniformly-elliptic Laplacian operator. In general, solutions to (1.1) are considered in the weak sense because they belong to C1,α(Ω) for some α>0. Many of the results on the uniqueness or symmetry properties of (1.1) in the uniformly elliptic case p=2 rely on classical elliptic principles such as the maximum principle. These principles do not apply directly in (1.1) when the operator is singular (p(1,2)) or degenerate (p>2).

We prove the uniqueness of positive radial solutions to (1.1) on a ball with Dirichlet boundary conditions for a class of nonlinearities f that includes nonnegative functions like f(u)=uq and the model sign-changing function f(u)=uq-u, where q<p*-1. The proof is in the spirit of the Clemons–Jones geometric proof of the case p=2 ([5]) but must address the singularity of the operator Δp, and includes a larger class of nonlinearities than those in [5].

The nonlinearity fC0[0,)C1(0,) satisfies the following conditions:

  1. f(0)=0, and there exists θ0 so that f(θ)=0 and f(u)>0 on (θ,). If θ>0, then f(u)<0 on (0,θ).

  2. The quantity K(u)=uf(u)/f(u) is nonincreasing on (0,θ) and (θ,).

  3. There is a value q1>p satisfying

    (1.3)p(n-1)n-p<q1<p*=pnn-p,

    where p* is the Sobolev critical exponent, and

    (1.4)limuf(u)uq1-1=>0.

    If θ>0, then there is a value q2 with q1>q2p, so that

    limu0f(u)uq2-1=ν<0.

The requirement (R) says that if f changes signs, then f behaves asymptotically like f(u)=uq1-1-νuq2-1, q1>q2>p, ν>0. With these conditions on the nonlinearity established, this paper presents the phase space of solutions to (1.1) on a ball in n and proves the following theorem.

Theorem 1

If 1<p2, (1.3) is satisfied, and f satisfies (F1), (F2), and (R), then for any positive radius R of Ω, there is at most one radial solution to (1.1).

2 Background

In this section we discuss previous results for (1.1) for p(1,2) and p=2, where the domain is typically a ball with Dirichlet boundary conditions.

With the technique of moving parallel planes, Serrin showed in [23] that if Ω is a smooth bounded domain and u is a positive solution to Δu+1=0 in Ω, with u=0 on Ω and with the outward normal vector uν constant on Ω, then Ω is necessarily a ball and u is a radial function. The method of moving parallel planes was also used by Gidas, Ni and Nirenberg [13] to determine that in the ball Ω={𝐱n:|𝐱|<R}, if uC2(Ω¯) is a positive solution of

(2.1)Δu+f(u)=0with u=0 on |𝐱|=R,

with f of class C1, then u is radially symmetric and decreasing. Part of the power of this result stems from the fact that they make no assumptions on the nonlinear term f(u) except that fC1.

In general, existence and uniqueness results for (1.1) require more restrictive conditions on f(u). The prototypical example is the Lane-Emden equation f(u)=uq, for q>1. If q=(n+2)/(n-2)=2*-1, then Δu+f(u)=0 is a version of the Yamabe problem from differential geometry. This particular exponent is a critical threshold for f, as demonstrated by Pohozaev [22], who determined that for any open, star-shaped domain about the origin, Δu+uq=0 with u|Ω=0 has a positive solution only if q<2*-1.

The topology of the domain is important, and there may be a positive solution to Δu+uq=0 on a different domain, such as an annulus. Pohozaev proved that positive solutions to Δu+uq=0 must satisfy the Pohozaev identity

(2.2)Ω(2nq+1-(n-2))uq+1𝑑x=Ω|u|2(xν)𝑑S.

If the domain is star-shaped, the right-hand side of (2.2) is always positive. The left-hand side, however, is negative if q>2*-1. By the Sobolev embedding theorem,

W1,2(Ω)Lq(Ω)

is a continuous embedding if q(n+2)/(n-2), with strict inequality resulting in a compact embedding. The lack of a compact embedding leads to nonexistence of solutions in [22].

Variational methods often show the existence of minimizers to certain functionals. For example, the critical exponent nonlinearity f(u)=λu+|u|p*-2u arises in the general Yamabe problem. For the semilinear case p=2, Brezis and Nirenberg [4] used the energy functional

E(u)=12Ω|u|2𝑑x-λ2Ω|u|2𝑑x-12*Ω|u|2*𝑑x

to show a solution must exist if λ is smaller than the first eigenvalue of Δ.

When f(u) satisfies |f(u)|Cuq-1, C>0, the question of whether f is subcritical (q<p*), critical (q=p*), or supercritical (q>p*) may alter not only when a solution exists but whether or not is unique. In the case p=2, Ni and Nussbaum [21] determined that solutions to Δu+f(u)=0 with f(u)=uq-1+u are not necessarily unique in the supercritical case q>p*.

Uniqueness of positive solutions to Δu+f(u)=0 has been addressed by many authors; the first was Coffman [6] for the subcritical case n=3 and f(u)=u3-u. McLeod and Serrin [20] showed uniqueness results for f(u)=uq-u for certain q, which were generalized by Kwong [17] to 1<q<(n+2)/(n-2); the method of Kwong was generalized and simplified by [19]. Other authors who investigated uniqueness of Δu+f(u)=0 with f subcritical, critical, or supercritical include Kwong and Zhang [18], who proved uniqueness in a ball by using a Sturm comparison principle.

Clemons and Jones illustrated this last uniqueness result with a geometric approach in [5] by recasting Δu+f(u)=0 as a dynamical system. The union of all solution forms a two-dimensional invariant manifold; showing uniqueness of a solution to the Dirichlet equation is interpreted as showing that the rotation of the manifold can be controlled. In this paper, we will use similar geometric methods to prove a uniqueness results for the singular p-Laplacian for a large class of f. These results also extend the geometric proof of [5] to a larger class of nonlinearities.

2.1 Results for the singular p-Laplacian

We are interested in radially symmetric solutions to (1.1), Damascelli and Pacella ([7, 8]) considered positive solutions to (1.1) on a ball with Dirichlet boundary conditions. They determined that if p(1,2), f is locally Lipschitz continuous in (0,) and either

  1. f(u)0 for u0, or

  2. there exists a constant β0>0 and a continuous, positive (except at the origin), nondecreasing function β:[0,β0] with f(u)+β(u)0 for all u[0,β0] and

    β(0)=0and0β01(sβ(s))1/p𝑑s=,

then u must be radially symmetric and ur<0 for r>0. The proof uses a modified moving plane method reminiscent of [13].

Several authors have examined existence and uniqueness questions for the p-Laplacian equation (1.1) with different choices of nonlinearity f, different domains (typically all of n, a ball of radius R, or an annulus), and different boundary conditions (usually Dirichlet); we refer here to [15, 9, 14]. Guedda and Veron [15] determined criteria for existence of positive solutions to

Δpu+up*-1+a(x)up-1=0,

in a bounded open subset of n with Dirichlet boundary conditions. Their result can be seen as an extension of the Brezis and Nirenberg [4] result. Erbe and Tang [9] proved uniqueness to (1.1) with f(u)=uq for q is subcritical. They also proved uniqueness holds for f(u)=uq1+λuq2, with q1>q2 and λ>0, if the quantity

ξx2+λσx+λ2v

is positive for all x>0, where

ξ=-(n-p-npq1+1),
σ=v(q2-q1+1)+ξ(q1-q2+1),
v=-(n-p-npq2+1).

Gonçalves and Alves [14] used minimax arguments on an energy functional to study existence of solutions to (1.1) with f(u)=up*-1+h(x)uq in n.

Recently, existence of solutions has been studied in a geometric framework, notably by Franca ([10, 11, 12]). He used an Emden–Fowler transformation to show the existence or nonexistence of ground states and singular ground states (1.1) for positive solutions to (1.1) in n. The function he studied is a Pohozaev function which is related to the Hamiltonian structure that arises in phase space; we will discuss this in Section 4.4.

Adimurthi and Yadava [1] investigated uniqueness of

-Δpu=uq+λ|u|p-2u

in both a ball and an annulus in n by using a Pohozaev-type identity. In particular, we note that for the ball with Dirichlet boundary conditions, they established uniqueness with λ0, 1<q+1(np)(n-p), p<n.

Aftalion and Pacella [2, 3] investigated uniqueness of positive radial solutions to problem (1.1) with fC0[0,)C1(0,) satisfying the following conditions:

  1. f(0)=0, with some θ>0 so that f<0 in (0,θ) and f>0 in (θ,),

  2. the expression

    (2.3)K(u):=uf(u)f(u)

    is nonincreasing in (θ,),

  3. the quantity

    (2.4)uf(u)-(p-1)f(u)

    is positive for u>0.

The prototypical nonlinearity satisfying (AP1)–(AP3) is f(u)=uq-up-1, where q>p-1. With an additional requirement on the growth of f near zero, [3] show that (1.1) has at most one weak radial solution if p2 by using a variant of the maximum principle and a suitable implicit function theorem.

In this paper, we drop condition (AP3): the quantity

(2.5)uf(u)-(p-1)f(u)

may change signs at some value of u0, and f may be nonnegative. Moreover, our proof is geometric, and both quantities (2.3) and (2.4) will emerge in a physical interpretation of nonuniqueness as quantities that determine how a particular invariant manifold bends.

3 Properties of the nonlinear function f(u)

Let us remark on a few properties of the function K(u) defined in (2.3). Under hypothesis (R),

(3.1)limuuf(u)f(u)=limuuq1-1f(u)f(u)(q1-1)uq1-2(q1-1)=q1-1,

and similarly K(u)q2-1 as u0. By (F1), it follows that K(u) as uθ+, and if θ>0, K(u)- as uθ-. Together with (F2) and (R), the following remarks on K(u) are true:

  1. K(u)q1-1 for u>θ,

  2. if θ>0, then K(u)q2-1 for u(0,θ),

  3. if 0<a<θ<b, then K(b)>K(a).

Conditions (F1) and (F2) are similar to hypotheses in [3]. However, the case θ=0 allows f to be a nonnegative function, and we do not require (AP3), as the quantity (2.4) may be positive, negative, or zero for different values of u.

Any polynomial of the form

(3.2)f(u)=uq1-1-νuq2-1,ν0,q1>q2p,

satisfies (F1), (F2) and (R), including the representative example f(u)=uq-up-1, q>p-1, from [3]. However, many basic nonlinearities satisfy our requirements without satisfying (AP1)–(AP3), for example

  1. Lane–Emden positive functions f(u)=uq1-1,

  2. sign-changing functions where the slower growth term is superlinear; for example, f(u)=u3-u2, where p and n must be chosen to satisfy (1.3),

  3. nonnegative functions like f(u)=us1ν+us2, where s1,s2,ν>0, with condition (R) satisfied for q1-1=s1-s2.

As we are interested in positive solutions solving the Dirichlet problem, we do not specify f(u) for u<0. However, to show existence of solutions to (1.1) using an Emden–Fowler approach, it is often necessary for f to be odd. One way to ensure this condition holds for functions of the form (3.2) is to write f as

(3.3)f(u)=|u|q1-2u-ν|u|q2-2u,ν0,q1>q2p.

Lastly, requiring q1<p* makes the growth rate of f subcritical; the inequalities in (1.3) will be elaborated on in Section 4.4; see Figure 3.

As the regular Laplacian operator corresponding to p=2 is well studied, we will focus on 1<p<2, when the p-Laplacian is singular. However, the proof in this paper covers the case p=2 for f nonnegative; a class of nonlinearities that was not addressed by [5].

4 The phase space

In the case p=2, Gidas, Ni and Nirenberg [13] proved that if fC1, solutions to (1.1) must be radially symmetric and monotone decreasing. This result does not extend immediately to nonuniformly elliptic case p2. The following theorem, a corollary to the work of Damascelli and Pacella ([7, 8]), establishes that any solution to (1.1) with f(u) in our class of nonlinearities must be radially symmetric and monotonically decreasing.

Theorem 1

Any positive solution u to (1.1) with f(u) satisfying (F1), (F2) and (R) is radially symmetric and monotonically decreasing.

Proof.

If f is nonnegative, then the result is automatic by (DP1). If f changes signs at θ>0, then let

β(u)=uq1-1-f(u),

where q1>q2p satisfy condition (R). Then β(u) is continuous with β(0)=0 and β(u)>0 for 0<u<θ, and f(u)+β(u)=uq1-1>0 for all u>0. As f(0)=0 and f(u)<0 for u<θ, there is some u^(0,θ) so that f(u) is nonincreasing on [0,u^]. As a result, β(u) is a nondecreasing function on [0,u^].

To use (DP2), it remains to check that there is some β0>0 so that

0β01(sβ(s))1/p𝑑s=0β01(sq1-sf(s))1/p𝑑s=.

All terms in the denominator are positive for any s<θ. Moreover, if s1, then q1>q2 implies

1(sq1-sf(s))1/p1(sq2(1-f(s)sq2-1))1/p.

By condition (R),

lims0-f(s)sq2-1=|ν|>0.

Hence for any ϵ>0 there is a δ>0 so that if s<δ, -f(s)/sq2-1<|ν|+ϵ. For s<δ,

1(sq2(1-f(s)sq2-1))1/p>1(sq2(1+|ν|+ϵ))1/p.

Let β0=min{u^,1,δ} so that the above inequalities are valid for all s(0,β0). Then

0β01(sβ(s))1/p𝑑s1(1+|ν|+ϵ)1/p0β01sq2/p𝑑s=,

as q2p. Hence all solutions must be radial and monotone decreasing. ∎

4.1 Dynamical system in (u,ω,r)-coordinates

Radial solutions u to (1.1) can be rewritten in terms of r=|𝐱| to obtain the following ODE:

(4.1)(u|u|p-2)+n-1ru|u|p-2+f(u)=0,

with =ddr. Radial symmetry implies u(0)=0, and the boundary condition u|Ω=0 can be written simply as u(R)=0, where R is the radius of Ω. Setting ω=u|u|p-2rp-1 yields a first-order ODE for ω,

(r1-pω)+r-p(n-1)ω+f(u)=0,

and therefore the (u,ω,r)-system can be written as

u=ωr|ω|2-pp-1,
ω=(p-n)ωr-rp-1f(u),
r=1.

This system is undefined when r=0; by introducing a new independent variable t and parametrizing r as r(t)=et, we blow up the singularity at r=0 into an invariant plane {r=0}. The resulting first-order system is

(4.2)u˙=ω|ω|2-pp-1,
(4.3)ω˙=(p-n)ω-rpf(u),
(4.4)r˙=r,

with =ddt. Solutions to (4.1) can now be viewed as trajectories in the phase space of (4.2)–(4.4).

In phase space, an initial condition at r=0 corresponds to the limit of a solution to (4.2)–(4.4) as t-. Suppose a solution satisfies the boundary condition u(0)=0 and has an initial value u(0)=a, where a>0; for this hypothesis to be satisfied in phase space, a trajectory must have as its limit the point (a,0,0) on the u-axis. Each such point (a,0,0) is a fixed point of (4.2)–(4.4); linearization about (a,0,0) yields

(4.5)(01p-1|ω|2-pp-10-rpf(u)p-n-prp-1f(u)001)|(a,0,0)=(0000p-n0001).

For 1<p<2 and n2, there is one zero eigenvalue, one negative eigenvalue, and one positive eigenvalue. (We will not concern ourselves with (4.5) in the case p=2 as this is the well-understood case.) Hence each (a,0,0) has a one-dimensional stable manifold Wps((a,0,0)), a one-dimensional unstable manifold Wpu((a,0,0)), and a one-dimensional center manifold Wpc((a,0,0)).

Figure 1 The center, stable, and unstable manifolds for sixteen radial solutions to (4.2)–(4.4) with evenly spaced initial values u⁢(0)∈[0,5]${u(0)\in[0,5]}$ on the domain R=0.25${R=0.25}$. The dotted line in the plane {r=0}${\{r=0\}}$ is the u-axis (the leftmost point along this axis is the origin); this line forms the center manifold of each point (a,0,0)${(a,0,0)}$. The S-shaped curves in {r=0}${\{r=0\}}$ are the stable manifolds of each point (a,0,0)${(a,0,0)}$, and each curve moving into r>0${r>0}$ space is an unstable manifold for one of the initial conditions (a,0,0)${(a,0,0)}$. Parameter values are p=1.8${p=1.8}$ and n=3${n=3}$, and the nonlinear function is f⁢(u)=u3-u${f(u)=u^{3}-u}$.

Figure 1

The center, stable, and unstable manifolds for sixteen radial solutions to (4.2)–(4.4) with evenly spaced initial values u(0)[0,5] on the domain R=0.25. The dotted line in the plane {r=0} is the u-axis (the leftmost point along this axis is the origin); this line forms the center manifold of each point (a,0,0). The S-shaped curves in {r=0} are the stable manifolds of each point (a,0,0), and each curve moving into r>0 space is an unstable manifold for one of the initial conditions (a,0,0). Parameter values are p=1.8 and n=3, and the nonlinear function is f(u)=u3-u.

An eigenvector for the eigenvalue 0 is parallel to the u-axis; by invariant manifold theory, the u-axis is the (global) center manifold Wpc((a,0,0)) for each a. The global stable manifold to (a,0,0) is the vertical line a={(a,ω,0):ω} in the case p=2; for 1<p<2, the stable manifold is tangent to the vertical vector (0,1,0) at (a,0,0). The S-shape of the stable manifold in the plane {r=0} when p2 is due to the presence of |ω|2-pp-1 in (4.2)–(4.4). See Figure 1 for a picture illustrating these manifolds at several values of a>0.

Hence the initial condition a plays the role of a parameter, and we examine how Wpu((a,0,0)) behaves for different values of a. The winding behavior of the two-dimensional “center-unstable” manifold (together with the boundary formed by the r-axis)

Wpu,c=a0Wpu((a,0,0))

is tracked in Section 6 until it intersects the plane {r=R}; see Figure 2. This intersection is nonempty: as t grows large in system (4.2)–(4.4), r=et will grow large as well. In fact, Wpu,c must contain the point (0,0,R)Wpu((0,0,0)).

Figure 2 Two viewpoints of thirty unstable manifolds for solutions to (4.2)–(4.4) on the domain R=1${R=1}$ with evenly spacedinitial values u⁢(0)=a∈[0,5]${u(0)=a\in[0,5]}$. Each unstable manifold is part of the global center-unstable manifold Wpu,c${W_{p}^{u,c}}$. Parameter values are p=1.8${p=1.8}$ and n=3${n=3}$, and the nonlinear function is f⁢(u)=u3-u${f(u)=u^{3}-u}$.
Figure 2 Two viewpoints of thirty unstable manifolds for solutions to (4.2)–(4.4) on the domain R=1${R=1}$ with evenly spacedinitial values u⁢(0)=a∈[0,5]${u(0)=a\in[0,5]}$. Each unstable manifold is part of the global center-unstable manifold Wpu,c${W_{p}^{u,c}}$. Parameter values are p=1.8${p=1.8}$ and n=3${n=3}$, and the nonlinear function is f⁢(u)=u3-u${f(u)=u^{3}-u}$.

Figure 2

Two viewpoints of thirty unstable manifolds for solutions to (4.2)–(4.4) on the domain R=1 with evenly spacedinitial values u(0)=a[0,5]. Each unstable manifold is part of the global center-unstable manifold Wpu,c. Parameter values are p=1.8 and n=3, and the nonlinear function is f(u)=u3-u.

Any point lying on Wpu,c is part of a solution that tends to (a,0,0), where a0, as t-. These solution trajectories, determined by the choice of a, foliate Wpu,c. Any such solution can be denoted by (u(t,a),ω(t,a),r(t)). To be more concise, we will occasionally write the above solution as (u(t),ω(t),r(t))a to mean (u(t),ω(t),r(t))a(a,0,0) as t-.

4.2 Dynamical system in Emden–Fowler coordinates

The Emden–Fowler transformation for (4.1) is obtained using the relation

(4.6)y=rλu,λ.

Let z=r(p-1)λω; in terms of the original solution u, z can be written

(4.7)z=u|u|p-2r(p-1)(λ+1).

In the uniformly elliptic case p=2, (4.7) does not reduce to the same z in the Emden–Fowler coordinate system in the proof by Clemons and Jones in [5] but leads to an alternate and equally useful set of equations.

The distance r from the origin is again parametrized as r(t)=et to obtain

(4.8)y˙=λy+z|z|2-pp-1,
(4.9)z˙=((p-1)λ-n+p)z-rp+λ(p-1)f(r-λy),
(4.10)r˙=r,

where =t has the same meaning as in (4.2)–(4.4). As before, the limit r0 is equivalent to t-.

A lower bound for the Emden–Fowler parameter λ is 0; otherwise, y will diverge in (4.6) as r0. We will not require λ to be nonnegative, however, and will note the effect that setting λ<0 has on the dynamics of (4.8)–(4.10) in Section 4.3.

For (4.8)–(4.10) to exist when r=0, the quantity rp+λ(p-1)f(r-λy) must exist as t-. This requirement leads to an upper bound on λ, defined in the following theorem in terms of p and q1.

Theorem 2

Theorem 2 (Maximal Emden–Fowler parameter)

The Emden–Fowler system (4.8)–(4.10) exists as t- if 0λλ^, where

λ^=pq1-p.

Proof.

As t-, (4.9) converges if and only if rp+λ(p-1)f(r-λy) is defined. We recast (4.6) as u=r-λy and treat y as independent of λ0. Suppose that as r0, |y|. Then u=r-λy. From (R),

limr0rp+λ(p-1)f(r-λy)rp-λ(q1-p)yq1-1=limr0rp+λ(p-1)f(r-λy)rp+λ(p-1)(r-λy)q1-1=.

If the limit of rp-λ(q1-p)yq1-1 is finite as r0, then the limit of rp+λ(p-1)f(r-λy) is finite as well. Therefore, in order for (4.8)–(4.10) to exist at r=0, λ must satisfy

λpq1-p=:λ^.

4.3 Wpu,c under the Emden–Fowler transformation

Let T[] be the Emden–Fowler transformation (4.6) from (u,ω,r) to (y,z,r), and let T[Wpu,c]=W~p. If λ satisfies 0<λλ^, then any point T[(u(t),ω(t),r(t))a] on W~p now satisfies

(y(t),z(t),r(t))(0,0,0)as t-.

Thus choosing λ(0,λ^] has the effect of “blowing down” the u-axis. As a consequence, it no longer makes sense to parametrize solutions via their limit as t-. To employ a similar notion, the notation y(t,a) and (y(t),z(t),r(t))a will mean that the solution (u(t),ω(t),r(t)) obtained from y=rλu satisfies (u(t),ω(t),r(t))(a,0,0) as t-.

Linearization of (4.8)–(4.10) at the origin yields

(4.11)(λ000(p-1)λ-n+p0001)if p2,(λ100λ-n+20001)if p=2,

with eigenvalues {λ,(p-1)λ-n+p,1}. As λλ^, the lower bound on q1 in (1.3) implies that (p-1)λ-n+p<0. If p(1,2), the eigenvectors are parallel to the axes; see Figure 1.

Assuming the critical exponent inequalities (1.3) are satisfied and p(1,2), we describe below the behavior of the invariant manifold W~p.

4.3.1 Invariant manifold structure if λ>λ^

We will never need this case in this paper as we require λλ^. As it may be interesting in future problems, should one choose λ>λ^, then the limit of (y˙,z˙,r˙) as t- is undefined. However, the manifold Wpu,c derives from the (u,ω,r)-system and therefore exists independently of λ. For any ϵ>0, T[Wpu,c{ϵrR}] is a two-dimensional manifold with boundary. Selecting λ>λ^ and defining Wp~ as

(4.12)Wp~=T[Wpu,c{ϵrR}]

yields a well-defined two-dimensional manifold with boundary in (y,z,r)-space.

4.3.2 Invariant manifold structure if 0<λλ^

If λ(0,λ^], then (4.11) has one negative eigenvalue and two positive eigenvalues. Under the Emden–Fowler transformation, W~p is a two-dimensional unstable manifold of the origin.

4.3.3 Invariant manifold structure if λ=0

If λ=0, then the Emden–Fowler transformation is simply u=y. Hence W~p is identical to Wpu,c, a two-dimensional center-unstable manifold.

4.3.4 Invariant manifold structure if λ<0

If λ<0, then there is one positive eigenvalue of (4.11) and two negative eigenvalues, {λ,(p-1)λ-n+p}. Thus all trajectories on the plane {r=0} tend to the origin as t. In this case, W~p transforms to a two-dimensional stable-unstable manifold composed of the unstable manifold of the origin and the one-dimensional stable of the origin in {r=0} associated with the eigenvalue λ and tangent to the y-axis at the origin.

As in Section 4.3.1, if we select λ<0, then we define Wp~ by (4.12) for an appropriately small ϵ>0.

4.4 Existence of solutions

At this point, we have not specified any particular λ; the choice we make to demonstrate the existence of solutions is the upper bound λ=λ^. This selection characterizes Wp~ as described above in Section 4.3.2, and moreover, this choice is ideal as (4.8)–(4.10) simplifies to

y˙=λ^y+z|z|2-pp-1,
z˙=((p-1)λ^-n+p)z-rp+λ^(p-1)f(r-λ^y),
r˙=r,

which in the invariant plane {r=0} reduces to

(4.13)y˙=λ^y+z|z|2-pp-1,
(4.14)z˙=(pλ^-n+p)z-λ^z-rp+λ^(p-1)f(r-λ^y)|r=0.

If it were the case that pλ^-n+p=0, then this system would be Hamiltonian in the plane {r=0} with

H(y,z)=λ^yz+p-1p|z|pp-1+rp+λ^(p-1)f(r-λ^y)|r=0dy.

However, whenever λ=λ^,

pλ^-n+p>-p2p-q1+pq1p-q1+p=0.

The resulting behavior of Wu((0,0)) in system (4.13)–(4.14) produces a “bowtie” as seen in Figure 3. Existence of solutions to problem (1.1) follows whenever the structure of the stable and unstable manifolds is in the configuration of Figure 3 (c): the stable manifold is trapped inside of the curve H(y,z)=0 while the unstable manifold appears to spiral outwards. It is a result of Franca [10] that this occurs for a large class of nonlinearities.

Figure 3 illustrates precisely why (1.3) must be satisfied for existence. The different dynamics corresponding to different values of q1, with (n,p)=(3,1.8), in the {r=0}-plane are pictured. The switching of roles between the stable and unstable manifolds of (0,0) as q1 is varied to be below, in and above the inequalities in (1.3).

As Theorem 1 is concerned with uniqueness, rather than existence, we will not explore existence further here.

5 Variational equations

For any time τ, we define the intersection curve by

(5.1)Cτ=Wpu,c{r=r(τ)}.

For any chosen u with initial condition u0>0, let Cτ(u0) be the truncated intersection curve defined by

Cτ(u0)={(u(τ),ω(τ),r(τ))aCτ:a[0,u0]}.

These curves lie in the {r=r(τ)}-plane. The curve defined by

γ(τ,u0)={(u(t),ω(t),r(t))u0:t(-,τ)},

which we call a solution trajectory, tends to (u0,0,0) as t- and intersects Cτ(u0) at u(τ,u0). Examples of both of these curves are sketched in Figure 4.

5.1 Variational equations

For any t, the curve Ct from (5.1) can be parametrized by initial conditions a via

Ct(a)=(u(t,a),ω(t,a),r(t)),a0.

Taking the derivative along Ct(a) with respect to a yields a family of tangent vectors:

(5.2)(δu(t,u0),δω(t,u0),0):=dCtda(u0)=(u(t,a)a,ω(t,a)a,r(t)a)|a=u0.

In the {r=0}-plane, the center manifold is parametrized by

C{r=0}(a)=(a,0,0),

and thus

limt-(δu(t,u0),δω(t,u0))=(1,0).
Figure 3

Varying q1 with (n,p)=(3,1.8) and f(u)=u3-u to show the stable and unstable manifolds of (y,z,r)=(0,0,0) in the {r=0}-plane. (a) is less than the lower bound, (b) is the lower bound, (c) is within the bounds, (d) is at the Sobolev criticalexponent p*, and (e) is above p*. In (a) and (b), the origin is a source; for (c)–(e), the origin is a hyperbolic saddle point.Figures (c) and (e) show the behavior switching between the stable and unstable manifolds. As each case demonstrates, if 1<p<2, then (4.11) implies that the unstable manifold is tangent to the y-axis at (0,0).

(a) q1=2.5${q_{1}=2.5}$

(a)

q1=2.5

(b) q1=3${q_{1}=3}$

(b)

q1=3

(c) q1=4${q_{1}=4}$

(c)

q1=4

(d) q1=4.5${q_{1}=4.5}$

(d)

q1=4.5

(e) q1=5${q_{1}=5}$

(e)

q1=5

The variational equations that describe how such a family of tangent vectors is carried under the flow of (4.2)–(4.4) are

(5.3)δu˙=1p-1|ω|2-pp-1δω,
(5.4)δω˙=(p-n)δω-u(rpf(u))δu-r(rpf(u))δr,
(5.5)δr˙=δr.

In particular, the tangent vectors in (5.2) satisfy

δu˙=1p-1|ω|2-pp-1δω,
δω˙=(p-n)δω-rpf(u)δu,
δr˙=0.

We define two curves in the tangent bundle to Wpu,c as follows: for any point (u(τ),ω(τ),r(τ))aCτ(u0), the tangent vectors from (5.2) form the following curve:

(5.6)SCτ(u0)={δu(τ,a),δω(τ,a),0):a[0,u0]}.

Similarly, for each point (u(t),ω(t),r(t))u0 along a single solution trajectory γ(τ,u0), the tangent vectors (δu(t,u0),δω(t,u0),0) defined by (5.2) form the curve

(5.7)Sγ(τ,u0)={(δu(t,u0),δω(t,u0),0):t(-,τ]}.

Figure 4 illustrates the tangent vectors that define these curves.

Figure 4 The planes {r=0}${\{r=0\}}$ and {r=r⁢(τ)}${\{r=r(\tau)\}}$ with the solution curve γ⁢(τ,u0)${\gamma(\tau,u_{0})}$ and the truncated intersection curve Cτ⁢(u0)${C_{\tau}(u_{0})}$. Thetangent vector fields that form the curves SCτ⁢(u0)${S_{C_{\tau}(u_{0})}}$ and Sγ⁢(τ,u0)${S_{\gamma(\tau,u_{0})}}$ defined by (5.6) and (5.7), respectively, are sketched as well.

Figure 4

The planes {r=0} and {r=r(τ)} with the solution curve γ(τ,u0) and the truncated intersection curve Cτ(u0). Thetangent vector fields that form the curves SCτ(u0) and Sγ(τ,u0) defined by (5.6) and (5.7), respectively, are sketched as well.

5.2 Variational equations under the Emden–Fowler transformation

Recalling that T is the Emden–Fowler transformation, the intersection curve in Emden–Fowler coordinates is T(Ct). By construction, T(Ct) is a curve lying in W~p in the {r=r(t)}-plane. As before, this is parametrized by a. Differentiating with respect to initial condition yields

aT(Ct(a))=(rλ0λrλ-1u0r(p-1)λ(p-1)λr(p-1)λ-1ω001)(δuδω0),

which defines the tangent vectors

(δy(t,u0),δz(t,u0),0):=(rλδu,r(p-1)λδω,0).

Under the Emden–Fowler transformation, the variational equations for the (y,z,r)-system are given by

(5.8)δy˙=λδy+1p-1|z|2-pp-1δz,
(5.9)δz˙=((p-1)λ-n+p)δz-y(rp+λ(p-1)f(r-λy))δy-r(rp+λ(p-1)f(r-λy))δr,
(5.10)δr˙=δr.

In particular, the vector field (δy(t),δz(t),0)u0 satisfies (5.8)–(5.10) with δr0. There are three cases for limt-δy:

  1. if λ>0, then δy0,

  2. if λ=0, then δy1,

  3. if λ<0, then the limit of δy is undefined.

In cases (1) and (2), the relation z=r(p-1)λω implies that δz0 as t-. In case (3), although (δy,δz) is undefined in the limit (more precisely, |δy|), the tangent vector field δu exists independently of λ, and for any ϵ>0, T[(δu,δω,0)|rϵ]=(δy,δz,0)|rϵ is a well-defined vector field.

5.3 Winding of admissible curves

We define a continuous angle measure ϑ:Cτ(a) so that ϑ(Cτ(a)) is on the appropriate branch of arctan(δω(τ,a)/δu(τ,a)), see Figure 5. For 1<p<2, along the invariant line {(0,0,r):r0}, the first component δu has δu˙0 by (5.3). Hence δu1 along {(0,0,r):r0}, and therefore (5.11) below is defined for every τ. Thus

(5.11)ϑ(Cτ(0))arctan(δω(τ,0)δu(τ,0))(-π2,π2).

(The case p=2 is done in [16].)

The “winding number” of SCτ(α) along the intersection curve Cτ(α) is defined by

(5.12)I(Cτ(α))=12(-2ϑ(Cτ(α))π+1);

the symbol denotes the greatest integer function. This quantity counts the number of net crossings (with clockwise about the origin crossings positive and counterclockwise about the origin crossings negative) of the δω-axis in the (δu,δω)-plane. See Figure 5 for a demonstration.

Figure 5 On the left, an imagined SCτ⁢(α^)${S_{C_{\tau}(\hat{\alpha})}}$ with ten selected points. On the right, for each point’s estimated angle ϑ, the winding number I⁢(cτ⁢(a))${I(c_{\tau}(a))}$ is computed from (5.12) in the third column.

Figure 5

On the left, an imagined SCτ(α^) with ten selected points. On the right, for each point’s estimated angle ϑ, the winding number I(cτ(a)) is computed from (5.12) in the third column.

We use the word homotopic for curves to refer to the notion of being path-homotopic (i.e. homotopic preserving endpoints) in the punctured plane 2\{0}. The winding number I is then invariant for homotopic curves. Let us consider the piecewise-defined curves

{(0,0,r):0rr(τ)}Cτ(α)and{(a,0,0):0aα}γ(τ,α).

As they form the boundary of the region

{0rr(τ)}{0<a<αWpu((a,0,0))},

there is a piecewise smooth path homotopy between these two curves. Thus the winding number along them must be the same. However, δu1 along both pieces {(a,0,0):0aα} and {(0,0,r):0rr(τ)}. Thus any winding behavior happens along Cτ(α) and γ(τ,α) and I(SCτ(α))=I(Sγ(τ,α)).

With this construction, we can now state a result connecting the algebraic winding number of δu and the number of zeros of δu along γ(τ,α). The following lemma is similar to [16, Proposition 3.5].

Lemma 1

For any trajectory (u(t),ω(t),r(t))α at time t=τ, I(Sγ(τ,α)) is the exact number of zeros of δu(t,α) for -<tτ.

This is not immediate as the winding number is a lower bound on the number of times δu=0. To prove this lemma, therefore, we must show that along γ(τ,α), the winding curve can only cross the axis {δu=0} in one direction, namely in a manner clockwise about the origin.

Remark

When examining Figures 47 and 10, it is important to remember that δu˙ is differentiation of δu with respect to time, and not with respect to the initial condition, a. Therefore, it is generally not possible to determine δu˙ when examining an r= constant plane.

Figure 6 An imagined Sγ⁢(τ,α)${S_{\gamma(\tau,\alpha)}}$ from Lemma 1 satisfying δ⁢u˙=1p-1⁢|ω|2-pp-1⁢δ⁢ω${\dot{\delta u}=\frac{1}{p-1}|\omega|^{\frac{2-p}{p-1}}\delta\omega}$. The open circle on the δ⁢u${\delta u}$-axis indicates that the limit as t→-∞${t\to-\infty}$ of Sγ⁢(τ,α)${S_{\gamma(\tau,\alpha)}}$ is (1,0)${(1,0)}$. The closed circle on the δ⁢ω${\delta\omega}$-axis indicates that at the moment this winding number iscomputed for this particular trajectory, δ⁢u⁢(τ,α)=0${\delta u(\tau,\alpha)=0}$ with δ⁢ω⁢(τ,α)>0${\delta\omega(\tau,\alpha)>0}$. The winding number of the curve Sγ⁢(τ,α)${S_{\gamma(\tau,\alpha)}}$ in this case is 4, the algebraic number of zeros for δ⁢u${\delta u}$.

Figure 6

An imagined Sγ(τ,α) from Lemma 1 satisfying δu˙=1p-1|ω|2-pp-1δω. The open circle on the δu-axis indicates that the limit as t- of Sγ(τ,α) is (1,0). The closed circle on the δω-axis indicates that at the moment this winding number iscomputed for this particular trajectory, δu(τ,α)=0 with δω(τ,α)>0. The winding number of the curve Sγ(τ,α) in this case is 4, the algebraic number of zeros for δu.

Proof.

As δu˙=1p-1|ω|2-pp-1δω and (δu,δω)=(0,0) is invariant, then whenever r>0, the relation

δω=0δu˙=0

implies that any time Sγ(τ,α) crosses the δu-axis, then δu˙=0. Hence the curve Sγ(τ,α) must be perpendicular to the δu-axis at any such crossing. Furthermore, as the sign of δω and δu˙ must be the same, then δu must be increasing in the first and second quadrants, and decreasing in the third and fourth quadrants. Thus if it crosses the δω-axis with δω<0, it must be crossing from the fourth quadrant to the third quadrant, and if it crosses the δω-axis with δω>0, it must be crossing from the second quadrant to the first quadrant. Therefore, the winding is clockwise about the origin and the winding number I(Sγ(τ,α)) is equal to the exact number of zeros of δu. An example of an Sγ(τ,α) that follows these guidelines is pictured in Figure 6. ∎

5.4 Two vector fields

Following the methodology in [5], a normal vector in the dual space to T(y,z,r)aWp~ is

(5.13)(δy*(t,a),δz*(t,a),δr*(t,a))=(y˙,z˙,r˙(t))×(δy,δz,δr)|(t,a).

We are solely interested in the third component, δr*, whose sign indicates whether the normal vector points forward toward increasing r, or backwards toward the origin. The third component of the normal vector (δy*,δz*,δr*) has derivative

(5.14)(δr*)=(pλ-n+p)δr*+rp+λ(p-1)((p+λ(p-1))f(u)-λuf(u))δy.

Let us now define a second vector field that does not appear in [5]:

(y(t,a),(p-1)z(t,a),r(t,a))×(δy(t,a),δz(t,a),δr(t,a)).

Analogous to (5.13), we are particularly interested in the third component

Wp(t,a):=yδz-(p-1)zδy|(t,a).

The choice of λ for δr* will depend on cases; for Wp we set the Emden–Fowler parameter λ to be λ^. Omitting in the notation below the dependence on time and the initial condition u(0)=a, we obtain

Wp˙=(pλ^-n+p)Wp+rp+λ^(p-1)((p-1)f(u)-uf(u))δy,

which is also a linear differential equation.

To derive integral expressions for δr* and Wp, we first show that as t-, the quantity e(n-p-pλ)tδr* tends to 0 for all λ.

Lemma 2

The vector component δr*(t,a) has an integral expression for all values of the Emden–Fowler parameter λR.

As the choice of the parameter λ can result in Emden–Fowler coordinates which are not necessarily defined in the limit, the proof of Lemma 2 requires a careful examination of u, y, and λ, done in full detail below.

Proof.

Employing the expression y=rλu to write y˙=λrλu+rλ+1u, and recalling that r=et and δy=rλδu, we obtain

e(n-p-pλ)tδr*=y˙δzr-(n-p-pλ)-z˙δyr-(n-p-pλ)
=(λrλu+rλ+1u)δzr-(n-p-pλ)-[((p-1)λ-n+p)z-rp+λ(p-1)f(u)]δyr-(n-p-pλ)
=rλ+n-p-pλ(λu+ru)δz-[((p-1)λ-n+p)u|u|p-2r(p-1)(λ+1)-rp+λ(p-1)f(u)]rλδur-(n-p-pλ)
=rn-p(λu+ru)δω-[((p-1)λ-n+p)u|u|p-2rn-1-rnf(u)]δu.

As ua, u0, rn-p0, and δω0 (and δω is independent of λ), the first term vanishes in the limit for all λ. For the second term, u|u|p-2,rn-1,rn0, f(u)f(a), and δu1. Hence

[((p-1)λ-n+p)u|u|p-2rn-1-rnf(u)]δu0

independently of the chosen parameter λ. ∎

In the linear differential equation for δr* in (5.14), let

Q(u,λ)=(p+λ(p-1))f(u)-λuf(u).

By Lemma 2, along a given trajectory (y(t,a),z(t,a),r(t)) the normal vector component δr*(t) for that trajectory is given by the integral

δr*(t,a)=e(pλ-n+p)t-te-(pλ-n+p)sr(s)p+λ(p-1)Q(u(s,a),λ)δy(s,a)𝑑s.

As r(t)=et and y=rλu, this can be written as

(5.15)δr*(t,a)=r(t)pλ-n+p-tr(s)nQ(u(s,a),λ)δu(s,a)𝑑s.

After the conversion from y back to u, the Emden–Fowler transformation only manifests itself in (5.15) in the λ-dependence of Q(u,λ).

For Wp(t,a), observe

limt-Wp(t,a)e(n-p-pλ^)t=limt-rn-p-pλ^(yδz-(p-1)zδy)
=limt-rn-p-pλ^(rλ^uδz-(p-1)zrλ^δu)
=limt-rλ^+n-p-pλ^[uδz-(p-1)zδu].

The quantity λ^+n-p-pλ^ is positive by (1.3); thus rλ^+n-p-pλ^0. Moreover, z0, ua, δz0, and δu1, and consequently

limt-Wpe(n-p-pλ^)t=0,

and therefore,

Wp(t,a)=r(t)p-n+pλ^-tr(s)n[(p-1)f(u(s,a))-u(s,a)f(u(s,s))]δu(s,a)𝑑s.

In terms of K(u) from hypothesis (F2), the above is equivalent to

Wp(t,a)=r(t)p-n+pλ^-tr(s)nf(u(s,a))[(p-1)-K(u(s,a))]δu(s,a)𝑑s.

For both vector components, once a particular trajectory has been identified, we will frequently suppress the dependence on the initial condition u(0)=a in (5.15) and write uδu to mean u(s,a)δu(s,a).

6 The geometry of uniqueness

For a given radius R>0, let αT(1) be the initial condition of the first positive solution on the intersection curve CT, T=lnR, to vanish on the boundary of Ω=BR(0):

(6.1)αT(1)=min{a>0:u(T,a)=0,u(t,a)>0 for all t<T}.

There fails to be a unique solution to (1.1) if CT contains a second point with initial condition

(6.2)αT(2)=min{a>αT(1):u(T,a)=0,u(t,a)>0 for all t<T}

so that

(u(T),ω(T),r(T))αT(1)=(0,β1,r(T))and(u(T),ω(T),r(T))αT(2)=(0,β2,r(T)),

with β1β2.

6.1 Transversality of the first intersection

By construction we have δu(T,αT(1))0, as δu(T,αT(1))>0 implies that αT(1) is not the minimal initial condition for t=T. Thus there are three possible configurations: either δu(T,αT(1))=0, or δu(T,αT(1))<0 with either δω(T,αT(1))<0 or δω(T,αT(1))>0. Therefore the truncated intersection curve CT(αT(1)) must be homotopic to one of the three possible configurations pictured in Figure 7.

By Lemma 1, the corresponding number of zeros for δu(T,αT(1)) is either 1 (Figure 7 (a) and (b)) or 2 (Figure 7 (c)). Figure 7 (a) depicts a transversal intersection; in Figure 7 (b) and (c) the intersection is tangent with either a winding number of 1, which we refer to as “underrotation” as the manifold appears to fold back on itself, or 2, which we call “overrotation” as the manifold is curling inward. With Lemmas 1 and 2, we prove that CT(αT(1)) must be homotopy equivalent to Figure 7 (a).

Figure 7

For any T>0, the truncated intersection curve CT(αT(1)) in the plane {r=eT} must be homotopic to one of the three configurations above. For each, the tangent vector (δu(T,αT(1)),δω(T,αT(1))) is included. (a) A transversal intersection with δu(T,αT(1))<0. The winding number I(CT(αT(1))) is 1. (b) A tangential intersection with δu(T,αT(1))=0 and δω(T,αT(1))<0. The winding number for this case, referred to as “underrotation”, is 1. (c) A tangential intersection with δu(T,αT(1))=0 and δω(T,αT(1))>0. The winding for this case, “overrotation”, is 2.

(a)

(a)

(b)

(b)

(c)

(c)

6.1.1 Underrotation

In the underrotation configuration, the single zero for δu(t,αT(1)) on (-,T] occurs when t=T; thus δu(t,αT(1))>0 for all t<T. The following lemma proves that this configuration is impossible.

Lemma 1

There is some t<T so that δu(t,αT(1))=0.

Proof.

We must rule out the underrotation case pictured in Figure 7 (b). Suppressing the initial condition αT(1) and setting λ=λ^ in (5.15) yields

(6.3)δr*(T)=r(T)p-n+pλ^-Tr(s)nλ^f(u)((q1-1)-K(u))δu𝑑s.

By the remarks on K(u) following (3.1), the quantity (q1-1)-K(u) is positive for u<θ and negative for u>θ, hence

f(u)((q1-1)-K(u))0.

(This statement holds for f nonnegative by setting θ=0.) In underrotation, δu>0 for all t<T; thus the integrand of (6.3), and consequently δr*(T), is negative. However, in Emden–Fowler coordinates the underrotation configuration in Figure 7 (b) indicates that

δr*(T)=y˙(T)δz(T)-z˙(T)δy(T)=y˙(T)δz(T)>0,

as y is decreasing and δz(T)=r(p-1)λδω<0 by hypothesis. Thus, δu(t,αT(1)) must vanish before t=T. ∎

6.1.2 Overrotation

As a consequence of Lemma 1, δu(t,αT(1))=0 exactly once before t=T; denote this time by t=τ0. Let u(τ0,αT(1))=u0, which must be positive.

If λ0 can be chosen to force Q(u,λ0) to be zero at t=τ0 when u=u0, then λ0 would be explicitly given by

(6.4)λ0=pf(u0)u0f(u0)-(p-1)f(u0).

The denominator is (AP3). Unlike [3], we will not require it to be nonzero or have a particular sign; if λ0 is defined, however, then u0f(u0)-(p-1)f(u0) must be nonzero. As (6.4) depends on u0, and u0 is not easily determined, let

(6.5)Λ(u)={pK(u)-(p-1),uθ,0,u=θ,

so that Λ(u0)=λ0 and Λ is continuous at u=θ even if f(θ)=0. An example of the graph of this function is given in Figure 8.

Figure 8 The graph of Λ⁢(u)${\Lambda(u)}$ from (6.5), with p=1.8${p=1.8}$, n=3${n=3}$ and f⁢(u)=u3-u2${f(u)=u^{3}-u^{2}}$. This curve graphs the appropriate choice of λ0${\lambda_{0}}$ given u0${u_{0}}$ on the u-axis. The vertical asymptote occurs when u⁢f′⁢(u)-(p-1)⁢f⁢(u)=0${uf^{\prime}(u)-(p-1)f(u)=0}$, while the top horizontal asymptote is the line Λ=p/(q2-p)>λ^${\Lambda=p/(q_{2}-p)>\hat{\lambda}}$. The u-intercept occurs when u0=θ${u_{0}=\theta}$.

Figure 8

The graph of Λ(u) from (6.5), with p=1.8, n=3 and f(u)=u3-u2. This curve graphs the appropriate choice of λ0 given u0 on the u-axis. The vertical asymptote occurs when uf(u)-(p-1)f(u)=0, while the top horizontal asymptote is the line Λ=p/(q2-p)>λ^. The u-intercept occurs when u0=θ.

Lemma 2

The winding number of the truncated intersection curve CT(u(t,αT(1))) must be 1.

Lemma 2 implies that the first solution cannot exhibit overrotation. To prove this lemma, there are three cases to consider:

  1. u0f(u0)-(p-1)f(u0)>0,

  2. u0f(u0)-(p-1)f(u0)<0,

  3. u0f(u0)-(p-1)f(u0)=0.

These three possibilities are highlighted in Figure 8. Case (3) is equivalent to K(u0)=p-1; since p<q1, then remarks (i)–(iii) on K(u) imply that u0<θ. Therefore, for any nonlinearity f(u) satisfying (F1), (F2) and (R), u=θ lies on the u-axis to right of the vertical asymptote. Consequently, the case f nonnegative falls under Case (1) with u0>θ.

The Case (1) region, as indicated by Figure 8, satisfies λ0λ^. This fact is the subject of the next lemma.

Lemma 3

If u0θ, then 0λ0λ^.

Proof.

We note that λ0=0 if and only if u0=θ. By (F2), K(u) is nonincreasing on (θ,), and so Λ(u) is nondecreasing. Thus λ>0 for all u>θ. By remark (i), K(u)q1-1 for all u>θ; thus

λ0=pK(u)-(p-1)p(q1-1)-(p-1)=λ^.
6.1.3 Proof for Case (1)

Whenever λ0 is defined and bounded above by λ^, Q(u,λ0) simplifies to the following expression:

(6.6)Q(u):=Q(u,λ0)=λ0(K(u0)-K(u))f(u).

At first glance, Q(u) appears to change signs twice if u0θ. If u0>θ, however, by (F2) the expression (K(u0)-K(u))f(u) is negative for u<u0 and positive for u>u0; see Figure 9. If u0<θ, then (K(u0)-K(u))f(u) is positive for u<u0 and negative for u>u0. Hence Q(u) changes signs exactly once, at u=u0; see Figure 9 (b).

If f is a nonnegative nonlinearity, then Q(u) is either zero (e.g. for f(u)=uq1-1), or (K(u0)-K(u))f(u) satisfies the same statement as u0>θ above (e.g. for f(u)=us1/(ν+us2)).

Figure 9

This figure assumes u0>θ, with f(u)=u3-u2. (a) A plot of K(u0)-K(u) (dashed, with its vertical asymptoteat u=θ included), and f(u), solid. (b) The product of K(u0)-K(u) and f(u). This product only changes signs at u=u0;for this illustration, u0=2.

(a)

(a)

(b)

(b)

To prove Case (1) is impossible, we consider the behavior of the vector component δr*. Omitting the initial condition αT(1), we find

(6.7)δr*(T)=y˙δz-z˙δy|t=T=y˙(T)δz(T)<0,

as y is decreasing and δz(T)>0 by assumption.

Suppose first that f changes signs at u=θ. If u0θ, then by Lemma 3, λ0 satisfies 0λ0λ^, with λ0=0 if and only if u0=θ. Hence Q(u) in (6.6) changes sign once from positive to negative as u decreases through u0. As a result, the integral expression for δr*,

(6.8)δr*(T)=r(T)pλ-n+p-Tr(s)nQ(u(s))δu(s)𝑑s,

is positive, which contradicts (6.7).

Now suppose u0 in the Case (1) region satisfies u0<θ. Again selecting λ0 to be whatever value of λ forces Q(u,λ) to change signs at u0. Since u0<θ and Λ(u) is a monotone increasing function, λ0<0.

Although the integral definition (6.8) of δr* contains u and δu, which are independent of λ, we take extra care to calculate δr* if u0<θ because the dynamics passed through the Emden–Fowler transformation to arrive at (5.15). This step reconciles the two different ways to calculate δy: the first way is simply δy=rλδu, which exists for all r>0 and all λ. However, δy(t) is also defined by the intersection curve on Wp~, which does not exist for λ<0 in the limit r0, as described in Section 4.3.4.

By Lemma 2, the integral definition in (6.8) solves (5.14) for any λ. Let ϵ>0 be sufficiently small so that the first zero of δu=r-λδy occurs at t=τ0>ln(ϵ). Then the manifold Wp~=T[Wpu,c{ϵrR}] described in Section 4.3.1 captures the sign-switching behavior at τ0. However, for any ϵ0(0,ϵ), we can just as easily construct W~p=T[Wpu,c{ϵ0rR}] so that W~pW~p. Thus

ϵ0ϵr(s)nQ(u(s),λ0)δu(s)𝑑s>0,

as δu>0 for all t<τ0, and λ0<0 and u0<θ implies Q(u(t))=λ0(K(u0)-K(u))f(u) must be positive for all t[ln(ϵ0),ln(ϵ)]. Therefore,

δr*(T)>r(T)pλ-n+pϵTr(s)nQ(u(s),λ)δu(s)𝑑s.

However, as Q(u) and δu both change signs from positive to negative at τ0, we conclude that the integrand must always be nonnegative. Hence δr*(T)>0. To verify that this conclusion contradicts (6.7), we ensure that (6.7) is true for λ<0 by converting δr* to an expression containing u, ω, δu, and δω rather than y and z. The cross product at t=τ0 yields

δr*(T)=rpλ(λu+u˙)δω-(λ(p-1)ω+ω˙)δu|t=T=rpλ(u˙δω)|t=T<0.

Thus Case (1) is impossible for sign-changing f.

Now suppose f is nonnegative. Then λ0 in (6.4) must be positive, and by Lemma 3, λ0λ^. It is possible that Q(u)0; in this scenario, we conclude by (5.15) that δr*(T)=0; this contradicts (6.7).

If Q(u) is not identically zero, then the nonincreasing behavior of K(u) implies

(K(u0)-K(u))f(u)δu0.

Hence the sign of

δr*(T)=r(T)pλ-n+p-Tr(s)nQ(u(s),λ)δu(s)𝑑s

is determined by λ0>0; consequently, δr*(T)>0. However, this result contradicts (6.7). Thus Case (1) is impossible.

6.1.4 Proof for Case (2)

This case is the interesting case frequently omitted in the literature, as Λ(u)>λ^. We first verify that this region does lie to the left of u=θ, as seen in Figure 8. Notice

u0f(u0)-(p-1)f(u0)<0f(u0)(K(u0)-(p-1))<0,

and therefore exactly one of f(u0) and K(u0)-(p-1) is negative. If u0>θ, then both are positive. Consequently, u0<θ. As f(u0)<0, it follows that K(u0)-(p-1)>0; moreover, as u is monotone decreasing and K is nonincreasing (or nondecreasing as u decreases), the expression

(6.9)f(u)(K(u)-(p-1)))<0

for all t(τ0,T).

To rule out overrotation in this case, we use Wp(y,z)(t) (with λ=λ^) to write

Wp(T)=r(T)p-n+pλ^(-τ0r(s)nf(u)[(p-1)-K(u)]δu𝑑s+τ0Tr(s)nf(u)[(p-1)-K(u)]δu𝑑s)
(6.10)=(r(T)r(τ0))p-n+pλ^Wp(τ0)+r(T)p-n+pλ^τ0Trnf(u)[(p-1)-K(u)]δu𝑑s.

We can compute Wp(τ0) directly:

Wp(τ0)=y(τ0)δz(τ0)-(p-1)z(τ0)δy(τ0)=y(τ0)δz(τ0),

which must be negative as y>0 and δz<0 at t=τ0.

For t(τ0,T), the component δu<0. This fact together with (6.9) implies that the integrand in (6.10) is negative. Hence Wp(T)<0. Yet at t=T, y(T)=δy(T)=0, and therefore

Wp(T)=y(T)δz(T)-(p-1)z(T)δy(T)=0.

Thus Case (2) is impossible.

6.1.5 Proof for Case (3) (the asymptote case)

In the asymptote case, we consider

u0f(u0)-(p-1)f(u0)=0,

or equivalently,

K(u0)=p-1.

(If f(θ)=0, then Λ=0.)

For this particular region, we write

Wp(T)=(r(τ0)r(T))n-p-pλ^Wp(τ0)+r(T)p-n+pλ^τ0Trnf(u)[(p-1)-K(u)]δu𝑑s.

The last integral is negative, as δu<0 for t(τ0,T) and (6.9) is true. As in Case (2), Wp(τ0)<0. However, this is a contradiction, as Wp(T) must be zero.

6.2 Proof of Theorem 1

Suppose there are two solutions u(t,αT(1)) and u(t,αT(2)) with u(T,αT(i))=0, as in (6.1)–(6.2). By Section 6.1, δu(T,αT(1))<0, and by the construction in (6.2), u(t,αT(2)), δu(T,αT(1))0. We now prove Theorem 1.

Proof.

Let I=[αT(1),αT(2)]; for each aI, u(τ,a)0, see Figure 10. By the Intermediate Value Theorem, for each aI there is some time TaT such that u(Ta,a)=0. Let tI:I be the map that sends each aI to the first time tI(a)T such that u(tI(a),a)=0. This map is well-defined and continuous, and as I is compact, tI(I) is compact. Therefore, tI(I) attains its minimum; let τ=min{tI(a)}aI, and let α^I denote a solution trajectory that satisfies u(τ,α^)=0 with

α^=min{aI:u(τ,a)=0}.

There is a neighborhood Bα^I about α^ so that u(τ,a)|Bα^0; thus

δu(τ,α^)=u(τ,a)a|a=α^=0,

as u has a local minimum at α^. Figure 10 illustrates this for the overrotation case. However, by construction α^ is in fact the first solution on C(τ); in other words, α^=ατ(1) with δu(τ,ατ(1))=0. By Section 6.1, the first intersection solution must intersect transversally, a contradiction. ∎

Figure 10 The setup for Section 6.2 is illustrated. Pictured are the truncated intersection curves Cτ(αT⁢(2))⊂W+0∩{r=r(τ)}${C_{\tau}(\alpha_{T(2)})\subset W_{+}^{0}\cap\{r=r(\tau)\}}$ and CT(αT⁢(2))⊂W+0∩{r=r(T)}${C_{T}(\alpha_{T(2)})\subset W_{+}^{0}\cap\{r=r(T)\}}$. Assuming nonuniqueness, the trajectories (u⁢(t),ω⁢(t),r⁢(t))αT⁢(1)${(u(t),\omega(t),r(t))_{\alpha_{T(1)}}}$ and (u⁢(t),ω⁢(t),r⁢(t))αT⁢(2)${(u(t),\omega(t),r(t))_{\alpha_{T(2)}}}$ each solve the Dirichlet problem when r=R${r=R}$, resulting in the existence of α^∈(αT⁢(1),αT⁢(2))${\hat{\alpha}\in(\alpha_{T(1)},\alpha_{T(2)})}$ and τ<T${\tau<T}$ such that the curve (u⁢(t),ω⁢(t),r⁢(t))α^${(u(t),\omega(t),r(t))_{\hat{\alpha}}}$ intersects {u=0}${\{u=0\}}$ when r=r⁢(τ)${r=r(\tau)}$, as pictured. Moreover, δ⁢u⁢(τ,α^)=0${\delta u(\tau,\hat{\alpha})=0}$ (in this case, as a result of overrotation), violating theresults of Section 6.1.

Figure 10

The setup for Section 6.2 is illustrated. Pictured are the truncated intersection curves Cτ(αT(2))W+0{r=r(τ)} and CT(αT(2))W+0{r=r(T)}. Assuming nonuniqueness, the trajectories (u(t),ω(t),r(t))αT(1) and (u(t),ω(t),r(t))αT(2) each solve the Dirichlet problem when r=R, resulting in the existence of α^(αT(1),αT(2)) and τ<T such that the curve (u(t),ω(t),r(t))α^ intersects {u=0} when r=r(τ), as pictured. Moreover, δu(τ,α^)=0 (in this case, as a result of overrotation), violating theresults of Section 6.1.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-0940363

Funding statement: This work was supported by the National Science Foundation under grant DMS-0940363.

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Received: 2015-11-12
Accepted: 2016-2-1
Published Online: 2016-3-20
Published in Print: 2017-2-1

© 2017 by De Gruyter

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