# Radially Symmetric Solutions of a Nonlinear Elliptic Equation

Authors: Edward P. Krisner; William C. Troy
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We investigate the existence and asymptotic behavior of positive, radially symmetric singular solutions ofw′′+((N−1)/r)w′−|w|p−1w=0, r>0. We focus on the parameter regime N>2 and 1<p<N/(N-2) where the equation has the closed form, positive singular solution w1=(4-2(N-2)(p-1)/(p-1)2)1/(p-1)r-2/(p-1), r>0. Our advance is to develop a technique to efficiently classify the behavior of solutions which are positive on a maximal positive interval (rmin,rmax). Our approach is to transform the nonautonomous w equation into an autonomous ODE. This reduces the problem to analyzing the behavior of solutions in the phase plane of the autonomous equation. We then show how specific solutions of the autonomous equation give rise to the existence of several new families of singular solutions of the w equation. Specifically, we prove the existence of a family of singular solutions which exist on the entire interval (0,∞), and which satisfy 0<w(r)<w1(r) for all r>0. An important open problem for the nonautonomous equation is presented. Its solution would lead to the existence of a new family of “super singular” solutions which lie entirely above w1(r).


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## Body

## 1. Introduction

We investigate the behavior of solutions of(1.1)Δw-|w|p-1w=0,
where w=w(x1,…,xN), N>1 and p>1. Solutions of (1.1) are time-independent solutions of the nonlinear heat equation(1.2)∂w∂t=Δw-|w|p-1w.
In the mid 1980’s, Brezis et al. [1], and Kamin and Peletier [2], investigated the existence and asymptotic behavior of positive, time-dependent singular solutions of (1.2). This led to the classical 1989 study by Kamin et al. [3], whose goal was to completely classify all positive, time-dependent solutions of (1.2). A natural extension of their study is to classify positive, time-independent solutions. Such solutions play an important role in analyzing the large time behavior of solutions of the time-dependent equation (1.2) (e.g., see the discussion following (1.5) below). Thus, in this paper, our goal is to extend the results in [1–3], and develop a method to efficiently classify the behavior of positive, time-independent solutions of (1.1). Our focus is on radially symmetric solutions, which have the form w=w(r), where r=(x12+⋯+xN2)1/2, and satisfy(1.3)w′′+N-1rw′-|w|p-1w=0,r>0.
Equation (1.3) has the closed form, positive singular solution (see Figure 2)(1.4)w1(r)=(4-2(N-2)(p-1)(p-1)2)1/(p-1)r2/(1-p),N>2,1<p<NN-2.A Related Equation
A second, widely studied nonlinear heat equation is(1.5)∂v∂t=Δv+|v|p-1v.
Equation (1.5) has the closed form, stationary, positive singular solution
(1.6)v1(r)=(2(N-2)(p-1)-4(p-1)2)1/(p-1)r2/(1-p),N>2,NN-2<p<N+2N-2.
This well-known singular solution plays an important role in the analysis of blowup of solutions of (1.5). For example, when v(x1,…,xN,0) is appropriately chosen, similarity solution methods developed by Haraux and Weissler [4], and Souplet and Weissler [5], show how v(x1,…,xN,t)→cv1(r) as t→∞, where c>0 is a constant [4, 5]. In 1999, Chen and Derrick [6] developed comparison methods to determine the large time behavior of solutions of the general equation
(1.7)∂w∂t=Δw+f(w),
where f(w) is super linear, as in (1.7) and (1.5). Their approach is to let positive, time independent solutions act as upper and/or lower bounds for initial values of solutions of (1.7). Their comparison technique allows them to prove either global existence or finite time blowup of solutions. It is hoped that the methods described above, combined with the new singular solutions found in this paper, will lead to future analytical insights into the behavior of solutions of the time-dependent equation (1.2).Specific Aims
We have three specific aims. The first two are listed below. The third is given later in this section. We assume throughout thatN>2 and 1<p<N/(N-2), the parameter regime where w1(r) exists. In order to study properties of positive solutions of (1.3), our approach is to let r0>0 be arbitrarily chosen and analyze solutions with initial values
(1.8)w(r0)=α>0,w′(r0)=β∈R.
Let (rmin,rmax) denote the largest interval containing r0 over which the solution of (1.3)–(1.8) is positive.Specific Aim 1.
For each solution of (1.3)–(1.8), prove whether rmin=0 or rmin>0, and determine limr→rmin+(w(r),w′(r)).Specific Aim 2.
For each solution of (1.3)–(1.8), prove whether rmax<∞ or rmax=∞, and determine limr→rmax-(w(r),w′(r)).Analytical Methods
To address the issues raised in Specific Aims1 and 2, we need to determine the behavior of each solution of (1.3)–(1.8) over the entire interval (rmin,rmax), where
(1.9)rmin⁡=inf⁡{r̂∈(0,r0)∣w(r)>0∀r∈(r̂,r0]},rmax⁡=sup⁡{r̂>r0∣w(r)>0∀r∈[r0,r̂)}.Numerical Experiments
In Figure1, we set (N,p,r0,α)=(3,2,2,2) and illustrate solutions of (1.3)–(1.8) for various β values. For example, when β≤0, panels (a)–(d) show that both rmin=0 and rmin>0 are possible, and that
(1.10)w′(r)<0∀r∈(rmin⁡,r0),lim⁡r→rmin⁡+⁡(w(r),w′(r))=(∞,-∞).
Panels (a)–(f) and also Figure 2 show that solutions can satisfy either rmax<∞ or rmax=∞.Solutions of (1.3)–(1.8) for various β values when (N,p,r0,α)=(3,2,2,2).
(a)(b)(c)(d)(e)(f)Figure 2
Solutions of (1.3)–(1.8) such that rmin=0 when (N,p)=(3,2); w1(r) is defined in (1.4); w2(r) and w3(r) satisfy parts (ii) and (iii) of Specific Aim 3. Also, w2(r) satisfies part (i) of Theorem 2.2, with rmin=0 and rmax=∞. w3(r) is a solution satisfying part (iv) of Theorem 2.2, with rmin=0 and rmax=2.2.Remark 1.1.
It must be emphasized that it is illegitimate to claim that numerical results are rigorous proofs. Complete analytical proofs are needed to determine properties such as (1.10).We now give a brief discussion of (1.10) which demonstrates the difficulties that arise in studying only the w equation to resolve Specific Aims 1 and 2. The proof of the first property in (1.10) follows from (1.3), which implies that w′′(r)>0 at any r∈(rmin,r0], where w′(r)=0 and w(r)>0. However, the fact that w′(r)<0 for all (rmin,r0) is not sufficient by itself to prove whether rmin=0 or rmin>0. Nor does it prove the second part of (1.10), that limr→rmin+(w(r),w′(r))=(∞,-∞). In fact, our study suggests that there is a unique βcrit<0 where rmin=0 (see Figure 1(d)), and that rmin>0 at all other negative β values. The proof of these claims requires the development of further estimates. Such estimates might be obtained using Pohozaev-type identities [7] or topological shooting techniques [8]. Once the location of rmin and limr→rmin+(w(r),w′(r)) have been determined, we need to turn our attention to the interval r>r0. As Figure 1 shows, there are several different types of behavior when r>r0. For example, consider the solutions in panels (a), (b), and (c) in Figure 1. In each case,(1.11)rmin⁡>0,lim⁡r→rmin⁡+⁡w(r)=∞.
When r>r0, panels (a), (b), and (c) show three different behaviors of solutions, namely,(1.12)rmax⁡<∞,lim⁡r→rmax⁡-⁡(w(r),w′(r))=(0,-.25),rmax⁡=∞,lim⁡r→rmax⁡-⁡(w(r),w′(r))=(0,0),rmax⁡<∞,lim⁡r→rmax⁡-⁡(w(r),w′(r))=(∞,∞).
These results lead to the following analytical challenge: given only the fact that a solution satisfies property (1.11) when r<r0, how can we prove which of the possibilities (1.12) occurs when r>r0? It is not at all clear how to answer this question using standard methods such as Pohozaev identities or topological shooting.Solutions withrmin=0
It is particularly important to understand the global behavior of solutions for whichrmin=0 since such solutions may play an important role in analyzing the asymptotic behavior of blowup of solutions of the time-dependent equation (1.2). Figure 1(d) shows one such solution for which rmin=0. This solution lies entirely abovew1(r), that is, w(r)>w1(r) for all r∈(0,rmax). Figure 2 shows two other solutions, labeled w2(r) and w3(r), for which rmin=0. These solutions lie entirely beloww1(r) on (0,rmax). Our computations indicate that w2(r) satisfies rmax=∞, and that rmax<∞ for w3(r). These numerical experiments lead toSpecific Aim 3.
LetN>2 and 1<p<N/(N-2). Prove that there are at least three families of solutions, other than w1(r), with rmin=0. The solutions in these families have the following properties:(i)
(see Figure1(d)). For each α0>w1(r0) there exists β0<0 such that if w0(r) is the solution of (1.3) with (w0(r0),w0′(r0))=(α0,β0), then rmin=0, rmax<∞,(1.13)lim⁡r→0+⁡(w(r),w′(r))=(∞,-∞),lim⁡r→rmax⁡+-⁡(w(r),w′(r))=(∞,∞).(ii)
(See Figure2). For each α2∈(0,w1(r0)) there exists β2<0 such that if w2(r) is the solution of (1.3) with (w2(r0),w2′(r0))=(α2,β2), then rmin=0, rmax=∞,(1.14)0<w2(r)<w1(r)∀r>0,lim⁡r→0+⁡(w2(r),w2′(r))=(∞,-∞),(w2(r),w2′(r))~(w1(r),w1′(r))asr⟶∞.(iii)
(See Figure2). For each α3∈(0,w1(r0)) there exists β3<0 such that if w3(r) is the solution of (1.3) with (w3(r0),w3′(r0))=(α3,β3), then rmin=0,rmax<∞,(1.15)lim⁡r→0+⁡(w3(r),w3′(r))~(w1(r),w1′(r))asr⟶0+,w(rmax⁡)=0,w(rmax⁡′)<0.Our Analytical Approach
Our goal is to develop techniques to efficiently prove the existence of solutions of thew equation (1.3) satisfying the properties described in Specific Aims 1, 2, and 3. Our experience shows that the analysis of (1.3) is especially complicated since useful estimates must include the independent variable r. Our advance is to significantly simplify the analysis by transforming (1.3) into an equation which is autonomous, that is, independent or r. For this, let w(r) denote any solution of (1.3), and define
(1.16)h(τ)=w(exp⁡⁡(τ))w1(exp⁡⁡(τ)),-∞<τ<∞.
Then h(τ) solves
(1.17)h′′+N-2p-1(p-N+2N-2)h′+2(N-2)(p-1)2(p-NN-2)(|h|p-1-1)h=0.Remark 1.2.
The effect of transformation (1.16) is to change (1.3) into (1.17). Transformation (1.16) is similar to the classical Emden-Fowler transformation y=w/t, x=1/t, which changes the Emden-Fowler equation
(1.18)y′′=Axnym
to the new equation
(1.19)w′′=At-n-m-3wm.Because (1.17) is autonomous, we can apply phase plane techniques to prove the behavior of its solutions. We then use the “inverse” formula(1.20)w(r)=h(ln⁡(r))w1(r),0<r<∞
to determine the global behavior of corresponding solutions of the w equation (1.3). In Section 2, we demonstrate the utility of this two step procedure. First, in Theorem 2.1, we analyze the h equation (1.17), and prove the existence and global behavior of four new classes of solutions. Secondly, in Theorem 2.2, we demonstrate how these families generate four new families of singular solutions of the w equation (1.3). In parts (i), (iii), and (iv) of Theorem 2.2 we show how the formula w(r)=h(ln(r))w1(r) can be efficiently used to prove the precise asymptotic behavior of each solution as r→rmin+, and as r→rmax-. These three solutions satisfy parts (i), (ii), and (iii) of Specific Aim 3. The final family of solutions in Theorem 2.2 (see part (ii)), is a family of “super singular solutions,” which satisfy rmax=∞,(1.21)w(r)>w1(r)∀r∈(rmin⁡,∞),lim⁡r→rmin⁡+⁡w(r)w1(r)=∞.
However, it remains a challenging open problem (see Open Problems 1 and 2 in Section 2) to prove whether rmin=0 or rmin>0. If the first possibility holds, then we have a fourth family of singular solutions, other than w1(r), which satisfy rmin=0.


## 2. The Main Result

In this section, we show how to make use of the autonomoush equation (1.17) to address the issues raised in Specific Aims 1, 2, and 3 for solutions of the nonautonomous w equation (1.3). In particular, our technique shows how the analysis of a solution of (1.17) can be used to completely determine the behavior of the corresponding solution of the w equation (1.3) on the maximal interval (rmin,rmax), where w is positive. To demonstrate the utility of our method, we restrict our focus to four specific branches of solutions of the h equation (1.17). Our approach consists of two steps.First, in Theorem2.1, we classify the behavior of solutions of (1.17) whose trajectories lie on the stable and unstable manifolds leading to and from the constant solution (h,h′)=(1,0) in the (h,h′) plane. The stable manifold has two components, B1 nd C1, and the unstable manifold has two components, D1 and E1. Solutions on B1, C1, D1, andE1 are illustrated in Figure 4(a).Secondly, in Theorem2.2, we make use of the link(2.1)w(r)=h(ln⁡(r))w1(r),
to show how solutions with initial values on B1, C1, D1, and E1 translate into four new continuous families of singular solutions of the w equation (1.3). For three of the four cases, we completely prove the behavior of solutions of the w equation on the maximal interval (rmin,rmax), where they are positive. For the fourth case, it remains a challenging open problem (see Open Problems 1 and 2 below) to prove the asymptotic behavior of the solution at the left end point r=rmin. The important consequences of resolving these open problems is described in Section 3.Theorem 2.1.
LetN>2 and 1<p<N/(N-2). Then(i)
There is a one-dimensional stable manifoldΓ of solutions of (1.17) leading to (1,0) in the (h,h′) phase plane. One component, B1, of Γ points into the region h<1, h′>0. If (h(0),h′(0))∈B1, then(2.2)0<h(τ)<1,0<h′(τ)<N-2p-1(NN-2-p)h(τ)∀τ∈R,(2.3)lim⁡τ→-∞⁡(h(τ),h′(τ))=(0,0),lim⁡τ→-∞⁡h′(τ)h(τ)=N-2p-1(NN-2-p),(2.4)lim⁡τ→∞⁡(h(τ),h′(τ))=(1,0).(ii)
The second component,C1, of Γ points into the region h>1, h′<0 of the (h,h′) plane. If a solution satisfies (h(0),h′(0))∈C1, and (τmin,∞) is its interval of existence, then(2.5)h(τ)>1,h′(τ)<0,h′′(τ)>0∀τ∈(τmin⁡,∞),(2.6)lim⁡τ→∞⁡(h(τ),h′(τ))=(1,0)lim⁡τ→τmin+⁡h(τ)=∞.(iii)
There is a one-dimensional unstable manifoldΩ of solutions of (1.17) leading from (1,0) into the (h,h′) plane. One component, D1, of Ω points into the region h>1, h′>0. If a solution satisfies (h(0),h′(0))∈D1, and (-∞,τmax) is its interval of existence, then τmax<∞,(2.7)h(τ)>1,h′(τ)>0,h′′(τ)>0∀τ∈(-∞,τmax⁡),(2.8)lim⁡τ→-∞⁡(h(τ),h′(τ))=(1,0),lim⁡τ→τmax⁡-⁡(h(τ),h′(τ))=(∞,∞).(iv)
The second component,E1, of Ω points into the region h<1,h′<0 of the (h,h′) plane. If (h(0),h′(0))∈E1, with 0<h(0)<1 and h′(0)<0, then there exists a value τ*>0 such that(2.9)0<h(τ)<1,h′(τ)<0∀τ∈(-∞,τ*),lim⁡τ→-∞⁡(h(τ),h′(τ))=(1,0),h(τ*)=0,h′(τ*)<0.Proof of (i).
We need to prove properties (2.2)–(2.4). The first step is to linearize (1.17) around the constant solution (h,h′)=(0,0). This gives
(2.10)h′′+N-2p-1(p-N+2N-2)h′-2(N-2)(p-1)2(p-NN-2)h=0.
The eigenvalues associated with (2.10) satisfy
(2.11)μ1=N-2p-1(NN-2-p)>0,μ2=2p-1>0.
We will make use of the observation that (1.17) can be written as
(2.12)h′′-(μ1+μ2)h′+μ1μ2h=μ1μ2|h|p-1h.
Next, a linearization of (1.17) about the constant solution (h,h′)=(1,0) gives
(2.13)h′′+N-2p-1(p-N+2N-2)h′+2(N-2)p-1(p-NN-2)(h-1)=0.
Define k=-2/(p-1). Then (2.13) becomes
(2.14)h′′+γh′+2(γ-k)(h-1)=0,
where
(2.15)γ=N-2p-1(p-N+2N-2)<0,γ-k=N-2p-1(p-NN-2)<0.
Thus, the eigenvalues associated with (2.13) and (2.14) satisfy
(2.16)λ1=-γ-γ2-8(γ-k)2<0,λ2=-γ+γ2-8(γ-k)2>0.
It follows from (2.16) and the Stable Manifold Theorem that there is a one-dimensional stable manifold Γ of solutions leading to (1,0) in the (h,h′) phase plane. Additionally,
(2.17)lim⁡τ→∞⁡h′(τ)h(τ)-1=λ1
for (h(τ),h′(τ))∈Γ. Thus, for sufficiently large τ, solutions on Γ satisfy h(τ)>1 if h′(τ)<0 and h(τ)<1 if h′(τ)>0. Let B1 denote the component of Γ pointing into the region h<1, h′>0 of the (h,h′) plane. Assume that (h(0),h′(0))∈B1. Then (2.4) holds. It remains to prove (2.2)-(2.3). Because of (2.11) and (2.17), and the translation invariance of (2.12), we can choose 1-h(0)>0 and h′(0)>0 small enough so that
(2.18)0<h(τ)<1,0<h′(τ)<μ1h(τ)∀τ∈[0,∞).
The definition of B1, together with (2.18), imply that the maximal interval of existence is of the form (τmin,∞), where τmin<0.
Next, we show thatB1⊂Uo, where Uo is the bounded open triangular region
(2.19)Uo={(h1,h2)∣0<h1<1,0<h2<μ1h1}.
Figure 4(b) shows Uo when (N,p)=(3,2). Because of (2.18), it suffices to show that (h(τ),h′(τ))∈Uo for all τ∈(τmin,0]. For contradiction, assume that (h(τ),h′(τ)) leaves Uo at some point in (τmin,0). Define
(2.20)H=dhdτ-μ1h.
It follows from (2.12) that H satisfies
(2.21)H′-μ2H=μ1μ2|h|p-1h.
Suppose that (h(τ),h′(τ)) leaves Uo across the line H=0. That is, (see Figure 3(a)) suppose that there exists τ0∈(τmin,0) such that
(2.22)H(τ)<0,0<h(τ)<1on⁡(τ0,0),H(τ0)=0.
If h(τ0)=0, then (2.20) implies that h′(τ0)=0, contradicting uniqueness of the constant solution (h,h′)=(0,0). Thus, h(τ0)>0. Also, (2.22) implies that
(2.23)H′(τ0)≤0.
The fact that h(τ0)>0, combined with (2.21), results in
(2.24)H′(τ0)=μ1μ2(h(τ0))p>0,
contradicting (2.23). Thus, (h(τ),h′(τ)) can only leave Uo across the line segment 0<h<1,h′=0. If so, there is a τ1∈(τmin,0) such that
(2.25)H(τ)<0,0<h(τ)<1,h′(τ)>0∀τ∈(τ1,0),(2.26)0<h(τ1)<1,h′(τ1)=0,
as depicted in the right panel of Figure 3. Hence,
(2.27)h′′(τ1)≥0.
It follows from (2.12) and (2.26) that
(2.28)h′′(τ1)=μ1μ2((h(τ1))p-1-1)h(τ1)<0,
contradicting (2.27). We conclude that (h(τ),h′(τ)) cannot leave Uo on (τmin,∞), hence B1⊂Uo as claimed. Moreover, since (h(τ),h′(τ)) is bounded, then τmin=-∞ follows from standard ODE theory. Thus, (h(τ),h′(τ))∈Uo for all τ∈ℝ, and, therefore, h′(τ)>0 for all τ∈ℝ.Proof of the first part (2.3).
First, we prove thath→0+ as τ→-∞. Since h′(τ)>0 and 0<h(τ)<1 on ℝ, then 0≤h¯<1 where, h¯=limτ→-∞h. To obtain a contradiction suppose that h¯>0. Then 0<h¯<1 and (2.12) yield
(2.29)d2hdτ2-(μ1+μ2)dhdτ⟶μ1μ2(h¯p-1-1)h¯<0asτ⟶-∞.
It follows from (2.29) that h′(τ)-(μ1+μ2)h(τ)→∞ as τ→-∞ which contradicts the fact that Uo is bounded and (h(τ),h′(τ))∈Uo for all τ∈ℝ. Thus, h(τ)→0+ as τ→-∞. Next, we show that h′(τ)→0+ as τ→-∞. Note that 0<h′(τ)<μ1h(τ) on (-∞,0] is an immediate consequence of H(τ)<0 and h′(τ)>0 on (-∞,0]. Therefore, h′(τ)→0+ as τ→-∞ follows from the fact that h(τ)→0+ as τ→-∞.Proof of second part of (2.3).
Finally, we need to prove thatρ=(h′/h)→μ1 as τ→-∞. The definition of ρ together with (2.12) gives
(2.30)ρ′+ρ2-(μ1+μ2)ρ=μ1μ2(hp-1-1).
We now show that ρ→μ1 monotonically as τ→-∞. Differentiating (2.30) yields
(2.31)ρ′′+(2ρ-μ1-μ2)ρ′=μ1μ2(p-1)hp-2h′.
Hence, if ρ′(τ*)=0 for some τ*∈ℝ, then
(2.32)ρ′′(τ*)=μ1μ2(p-1)hp-2(τ*)h′(τ*)>0.
This implies that ρ′ has at most one zero on ℝ. Furthermore,
(2.33)0<ρ(τ)=h′(τ)h(τ)<μ1∀τ∈R
since
(2.34)(h(τ),h′(τ))∈Uo∀τ∈R.
Thus, ρ¯=limτ→-∞ρ exists and 0≤ρ¯≤μ1. Moreover, the fact that ρ¯ is finite ensures the existence of an unbounded decreasing sequence {τn} such that limτn→-∞ρ′(τn)=0. Substituting
(2.35)lim⁡τn→-∞⁡ρ′(τn)=0=lim⁡τn→-∞⁡h(τn),ρ¯=lim⁡τn→-∞⁡ρ
into (2.30) results in
(2.36)ρ¯2-(μ1+μ2)ρ¯+μ1μ2=0.
The bound 0≤ρ¯≤μ1 and (2.36) imply that ρ¯=μ1. Thus, ρ→μ1 as τ→-∞ as claimed.(a) solid curve illustrates properties (2.22). (b) solid curve illustrates properties (2.25) and (2.26).
(a)(b)N=3, p=2. Row 1: Solutions on the stable and unstable manifolds associated with (h,h′)≡(±1,0) and (h,h′)≡(0,0). Rows 2 and 3: h components on A1, A2, B1, B2 and w components along A1 and B1; w0(r) is bounded at r=0, w1(r)=2r-2 is the known singular solution; w2(r) is the new, positive singular solution corresponding to heteroclinic orbit B1.
(a)(b)(c)(d)(e)(f)Proof of (ii).
It follows from the Stable Manifold Theorem and (2.17) that there is a second component, C1, of Γ which points into the region h>1,h′<0 of the (h,h′) plane (Figure 4(a)). Thus, if (h(0),h′(0))∈C1, and h(0)-1>0 is sufficiently small, then
(2.37)h(τ)>1h′(τ)<0∀τ∈[0,∞),(2.38)lim⁡τ→∞⁡(h(τ),h′(τ))=(1,0).
Let (τmin,∞) denote the interval of existence of this solution. It remains to prove (2.5) and the second part of (2.6), that is, that
(2.39)h(τ)>1,h′(τ)<0,h′′(τ)>0∀τ∈(τmin⁡,∞),(2.40)lim⁡τ→τmin⁡+⁡h(τ)=∞.
Let (τ*,∞) denote the maximal subinterval of (τmin,∞) such that h′(τ)<0 for all τ∈(τ*,∞). From the definition of τ* and (2.37), it follows that h(τ)>1 for all τ>τ*. Next, we prove that τ*=τmin. Suppose, for contradiction, that τ*>τmin. Then
(2.41)h(τ*)>1h′(τ*)=0,h′′(τ*)≤0.
From (1.17), and the fact that h(τ*)>1 and h′(τ*)=0, it follows that
(2.42)h′′(τ*)=2(N-2)(p-1)2(p-NN-2)(|h(τ*)|p-1-1)h(τ*)>0,
which contradicts (2.41). We conclude that τ*=τmin, hence h(τ)>1 and h′(τ)<0 for all τ∈(τmin,∞). Finally, suppose that h′′(τ̂)=0 at some τ̂∈(τmin,∞). A differentiation (1.17) gives
(2.43)h′′′(τ̂)=-2(N-2)(p-1)2(p-NN-2)(p|h(τ̂)|p-1-1)h′(τ̂)<0.
Thus, since h′′′<0 whenever h′′=0, we conclude that h′′(τ)<0 for all τ>τ̂. This implies that h′(∞)<0, contradicting (2.38). Therefore, it must be the case that h′′(τ)>0 for all τ∈(τmin,∞). This completes the proof of (2.39). It then follows from (2.39) and standard theory that limτ→τmin+h(τ)=∞, and (2.40) is proved.Open Problem 1.
The issue of whetherτmin=-∞ or τmin>-∞ remains unresolved. Its resolution may lead to new classes of solutions of the w equation (1.3). Precise details of the implications for solutions of (1.3) are given below, both in the proof of Theorem 2.2, and in the discussion which follows its proof.Proof of (iii).
It follows from (2.16) and the Stable Manifold Theorem that there is a one-dimensional unstable manifold Ω of solutions leading from (1,0) into the (h,h′) plane. Additionally, solutions on Ω satisfy
(2.44)lim⁡τ→-∞⁡h′(τ)h(τ)-1=λ2>0.
Thus, for sufficiently large τ, solutions on Ω satisfy h(τ)>1 if h′(τ)>0, and h(τ)<1 if h′(τ)<0. Let D1 denote the component of Ω pointing into the region h>1, h′>0 of the (h,h′) plane (Figure 4(a)). Let (h(0),h′(0))∈D1. Then limτ→-∞(h(τ),h′(τ))=(1,0), hence, the first part of (2.8) is proved. Next, because of (2.44) and the translation invariance of (2.12), we can choose h(0)-1>0 and h′(0)>0 small enough so that
(2.45)h(τ)>1,h′(τ)>0,∀τ∈(-∞,0].
The interval of existence of this solution is of the form (-∞,τmax), where τmax>0. It remains to prove that finite time blowup occurs, that is, that τmax<∞, and
(2.46)h(τ)>1,h′(τ)>0,h′′(τ)>0∀τ∈(-∞,τmax⁡),(2.47)lim⁡τ→τmax⁡-⁡(h(τ),h′(τ))=(∞,∞).
Let (-∞,τ*) denote the maximal subinterval of (-∞,τmax) such that h′(τ)>0 for all τ∈(-∞,τ*). It follows from (2.45) and the definition of τ* that h(τ)>1 for all τ∈(-∞,τ*). We claim that τ*=τmax. Suppose, for contradiction, that τ*<τmax. Then
(2.48)h(τ*)>1,h(τ*)=0,h′′(τ*)≤0.
However, (1.17) and the fact that h(τ*)>1 and h′(τ*)=0, imply that
(2.49)h′′(τ*)=2(N-2)(p-1)2(p-NN-2)(|h(τ*)|p-1-1)h(τ*)>0,
contradicting (2.48). Thus, τ*=τmax, hence h(τ)>1 and h′(τ)>0 for all τ∈(-∞,τmax). Also, it follows exactly as in the Proof of (ii) that h′′(τ) does not change sign on (-∞,τmax), and that h′′(τ)>0 for all τ∈(-∞,τmax). This completes the proof of (2.46). Next, we prove that τmax<∞. Suppose, however, that τmax=∞. Then h′′(τ)>0 for all τ∈(-∞,∞). This implies that
(2.50)h′(τ)≥h′(0)>0∀τ≥0,lim⁡τ→∞⁡h(τ)=∞.
To use (2.50) to contradict the assumption that τmax<∞, we analyze
(2.51)S=(h′)22+2(N-2)(p-1)2(p-NN-2)(hp+1p+1-h22),
which satisfies
(2.52)S′=N-2p-1(N+2N-2-p)(h′)2.
Since h′(τ)≥h′(0)>0 for all τ≥0, it follows from an integration of (2.52) that S(τ)→∞ as τ→∞. These, (2.51) and the fact that h(τ)→∞ as τ→∞, imply that there is a τ1≥0 such that S(τ)≥0 for all τ≥τ1, that is, that
(2.53)(h′)2≥2(N-2)(p-1)2(NN-2-p)hp+1p+1∀τ≥τ1.
An integration of (2.53) gives
(2.54)(h(τ))(1-p)/2≤(h(τ1))(1-p)/2+a(1-p)2(τ-τ1),τ≥τ1,
where a=(2(N-2)/(p-1)(p-1)2)(N/(N-2)-p))1/2>0 since 1<p<N/(N-2). The right side of (2.54) is negative when τ>τ2=τ1+(2(p-1)/a)(h(τ1))(1-p)/2. Thus, (2.54) reduces to (h(τ))(1-p)/2<0 when τ>τ2, a contradiction. We conclude that τmax<∞, as claimed. Since τmax<∞, it follows from (1.17), (2.7), and standard theory that (h(τ),h′(τ))→(∞,∞) as τ→τmax-. This proves property (2.47).Proof of (iv).
It follows from the Stable Manifold Theorem and (2.44) that there is a second component, E1, of Ω which points into the region 0<h<1, h′<0 of the (h,h′) plane. Thus, if (h(0),h′(0))∈E1, and 1-h(0)>0 is sufficiently small, then
(2.55)0<h(τ)<1,h′(τ)<0∀τ∈(-∞,0],lim⁡τ→-∞⁡(h(τ),h′(τ))=(1,0).
Define
(2.56)τ*=sup⁡{τ̂>0∣0<h(τ)<1,h′(τ)<0∀τ∈[0,τ̂)}.
We need to prove that τ*<∞, that h(τ*) and h′(τ*) are finite,
(2.57)h(τ*)=0,h′(τ*)<0.
For this, integrate (1.17) and get
(2.58)h′(τ)eAτ=h′(0)+B∫0τeAη(|h(η)|p-1-1)h(η)dη,0≤τ<τ*,
where A=((N-2)/(p-1))(p-(N+2)/(N-2))<0 and B=(2(N-2)/(p-1)2)(N/(N-2)-p)>0. Because (|h|p-1-1)h>-1 for all h∈[0,1], it follows that
(2.59)∫0τeAη(|h(η)|p-1-1)h(η)dη≥-∫0τeAηdη=-1A(eAτ-1)∀τ∈[0,τ*).
Combining (2.58) and (2.59) gives
(2.60)0>h′(τ)eAτ≥h′(0)-BA(eAτ-1)∀τ∈[0,τ*).
We conclude from (2.60) that if τ* is finite, then h(τ) and h′(τ) are bounded on the closed interval [0,τ*]. This, (2.58), the definition of τ*, and (2.60) imply that h′(τ*)<0 and h(τ*)=0 if τ* is finite. Thus, (2.57) is proved if τ* is shown to be finite. We assume, for contradiction, that τ*=∞. Then
(2.61)0<h(τ)<1,h′(τ)<0∀τ≥0.
Since the integral term in (2.58) is negative for all τ≥0, then (2.58) reduces to h′(τ)eAτ≤h′(0) for all τ≥0. An integration gives
(2.62)h(τ)≤h(0)-h′(0)A(e-Aτ-1)∀τ∈[0,∞).
The right side of (2.62) is negative when τ>-(1/A)ln(h(0)A/h′(0)+1), contradicting (2.61). We conclude that τ*<∞, as claimed. This completes the proof of Theorem 2.1.Solutions of thew equation
Below, in Theorem2.2, we show how to combine parts (i)–(iv) of Theorem 2.1 together with the formula
(2.63)w(r)=h(ln⁡(r))w1(r),
to generate new families of solutions of the w equation (1.3). In each of the four cases (i)–(iv), we show how to use (2.63) to prove the existence of an entire continuum of new singular solutions of (1.3). In each case, our approach is to let (h(0),h′(0)) be an arbitrarily chosen element of one of the four continuous curves B1,C1,D1 or E1. Since r=eτ, the initial conditions for the corresponding solution of (1.3) are given at r=e0=1, and satisfy
(2.64)w(1)=h(0)w1(1),w′(1)=h′(0)w(1)+h(0)w′(1).
Because the curves B1, C1, D1, and E1 are continuous, this technique generates four new continua of solutions of the w equation. In addition, for cases (i), (iii), and (iv), our analytical technique allows us to completely resolve the issues raised in Specific Aims 1, 2, and 3 in Section 1. That is, for each of the solutions described in (i), (iii), and (iv) we show how to efficiently prove the limiting behavior of the solution at both ends of the maximal interval (rmin,rmax), where it is positive. For part (ii), our analysis of the behavior of solutions at rmin is incomplete, and this leads to Open Problem 2 which is stated at the end of the proof of (ii). This problem is directly related to Open Problem 1 described above at the end of the proof of part (ii) of Theorem 2.1.
Open Problem3 Prove the existence of other families of solutions of (1.3). For example, the existence and limiting behavior of the solutions labeled (c) and (f) in Figure 1 have not yet been proved. It is our hope that our analytical techniques can be extended to prove the existence and limiting behavior of these and many other new families of solutions.Theorem 2.2.
LetN>2 and 1<p<N/(N-2), and let w1(r) denote the positive singular solution of (1.3) defined in (1.4).(1) A Continuum of Singular Solutions Generated byB1
Leth2(τ) denote a solution of (1.17) which satisfies (h2(0),h2′(0))∈B1 in part (i) of Theorem 2.1. The corresponding solution w2(r)=h2(ln(r))w1(r) of (1.3) has initial values
(2.65)w2(1)=h2(0)w1(1),w2′(1)=h2′(0)w(1)+h2(0)w′(1),
and satisfies
(2.66)0<w2(r)<w1(r)∀r>0,w2(r)w1(r)⟶1asr⟶∞,w2(r)~(4-2(N-2)(p-1)(p-1)2)1/(p-1)r-(N-2)asr⟶0+.
Figures 2 and 4(d) show solutions of (1.3) with these properties.(2) A Continuum of Singular Solutions Generated byC1
Leth3(τ) denote a solution of (1.17) which satisfies (h3(0),h3′(0))∈C1 in part (ii) of Theorem 2.1. The corresponding solution w3(r)=h3(ln(r))w1(r) of (1.3) has initial values
(2.67)w3(1)=h3(0)w1(1),w3′(1)=h3′(0)w(1)+h3(0)w′(1).
Let (rmin,rmax) be the maximal interval where w3(r)>0. Then rmax=∞,
(2.68)w3(r)>w1(r)∀r>rmin⁡,(2.69)lim⁡r→rmin⁡+⁡w3(r)w1(r)=∞,lim⁡r→∞⁡w3(r)w1(r)=1.
Figure 1(b) shows a solution of (1.3) with these properties.(3) A Continuum of Singular Solutions Generated byD1
Leth4(τ) denote a solution of (1.17) which satisfies (h4(0),h4′(0))∈D1 in part (iii) of Theorem 2.1. The corresponding solution w4(r)=h4(ln(r))w1(r) of (1.3) has initial values
(2.70)w4(1)=h4(0)w1(1),w4′(1)=h4′(0)w(1)+h4(0)w′(1).
Let (rmin,rmax) be the maximal interval where w4(r)>0. Then rmin=0 and rmax<∞,
(2.71)w4(r)>w1(r)∀r∈(0,rmax⁡),(2.72)lim⁡r→0+⁡w4(r)w1(r)=1,lim⁡r→rmax⁡⁡w4(r)=∞.
Figure 1(d) shows a solution of (1.3) with these properties.(4) A Continuum of Singular Solutions Generated byE1
Leth5(τ) denote a solution of (1.17) which satisfies (h5(0),h5′(0))∈E1 in part (iv) of Theorem 2.1. The corresponding solution w5(r)=h5(ln(r))w1(r) of (1.3) has initial values
(2.73)w5(1)=h5(0)w1(1),w5′(1)=h5′(0)w(1)+h5(0)w′(1).
Let (rmin,rmax) be the maximal interval where w5(r)>0. Then rmin=0 and rmax<∞,
(2.74)0<w5(r)<w1(r)∀r∈(0,rmax⁡),(2.75)lim⁡r→0+⁡w5(r)w1(r)=1,lim⁡r→rmax⁡w4(r)=0.
Figure 2 shows a solution of (1.3) with these properties.Proof of (1).
Leth2 denote a solution of (1.3) which satisfies part (i) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to h2 is
(2.76)w2(r)=h2(ln⁡(r))w1(r).
It follows from (2.2) in Theorem 2.1 that 0<h2(ln(r))<1 for all r>0. This, as well as (2.76), implies that
(2.77)0<w2(r)<w1(r)∀r>0
(see Figure 4(d)). We claim that w2 is singular at r=0. The first step in proving this claim is to observe that (2.3) and (2.11) imply that (h2′(τ)/h2(τ))→μ1 as τ→-∞. Thus, ln(h2(τ))~μ1τ as τ→-∞. This and the fact that τ=ln(r) lead to
(2.78)h2(τ)=h2(ln⁡(r))~rμ1asr→0+.
Substituting (1.4) and (2.78) into (2.76) gives
(2.79)w2(r)~(4-2(N-2)(p-1)(p-1)2)1/(p-1)rμ1-μ2asr⟶0+.
Our claim that w2 is singular at r=0 follows from (2.79) and the fact that μ1-μ2=2-N<0. It remains to determine the asymptotic behavior of w2(r) as r→∞. Since h2(ln(r))→1- as r→∞, then (w2(r)/w1(r))→1 as r→∞. This completes the proof of properties (2.66).Proof of (2).
Leth3 denote a solution of (1.3) which satisfies part (ii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to h3 is
(2.80)w3(r)=h3(ln⁡(r))w1(r).
Initial conditions (2.67) follow exactly as in the proof of part (i). Let (rmin,rmax) denote the maximal interval over which w3(r)>0. It follows from (2.5) in Theorem 2.1 that rmax=∞ and h4(ln(r))>1 for all r>rmin. This, together with (2.80), implies that
(2.81)w3(r)>w1(r)∀r∈(rmin⁡,∞).
This proves (2.68). Property (2.6) in Theorem 2.1, as well as (2.80), imply that
(2.82)lim⁡r→∞⁡w3(r)w1(r)=lim⁡r→∞⁡h3(ln⁡(r))=1,(2.83)lim⁡r→rmin⁡+⁡w3(r)w1(r)=lim⁡r→rmin⁡+⁡h3(ln⁡(r))=∞.Open Problem 2.
Prove whetherrmin=0 or rmin>0. This problem arises as a direct consequence of Open Problem 1 described above at the end of the proof of part (ii) of Theorem 2.1. If it can be proved that rmin=0, then
(2.84)w3(r)>w1(r)∀r∈(0,∞),lim⁡r→0+⁡w3(r)w1(r)=∞.
Because w3(r)→∞ much faster than w1(r), we refer to any solution satisfying either (2.83) of (2.84) as a Super Singular Solution. This class of solutions has not previously been reported.Proof of (3).
Leth4 denote a solution of (1.3) which satisfies part (iii) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to h4 is
(2.85)w4(r)=h4(ln⁡(r))w1(r).
Initial conditions (2.70) follow exactly as in the proof of part (i). Let (rmin,rmax) denote the maximal interval over which w4(r)>0. It follows from (2.7) in Theorem 2.1 that rmin=0 and rmax<∞, and h4(ln(r))>1 for all r∈(0,rmax). This, together with (2.85), implies that
(2.86)w4(r)>w1(r)∀r∈(0,rmax⁡).
This proves (2.71). Property (2.8), in Theorem 2.1, as well as (2.85), implies that
(2.87)lim⁡r→rmax⁡-⁡w4(r)=lim⁡r→rmax⁡-⁡h4(ln⁡(r))w1(r)=∞,lim⁡r→0+⁡w4(r)w1(r)=lim⁡r→0+⁡h4(ln⁡(r))=1.
This completes the proof of (2.72).Proof of (4).
Leth5 denote a solution of (1.3) which satisfies part (iv) of Theorem 2.1. By (1.20), the solution of (1.3) corresponding to h5 is
(2.88)w5(r)=h5(ln⁡(r))w1(r).
Initial conditions (2.73) follow exactly as in the proof of part (i). Let (rmin,rmax) be the maximal interval over which w5(r)>0. It follows from (2.9) in Theorem 2.1 that rmin=0 and rmax<∞, and h5(ln(r))<1 for all r∈(0,rmax). This, together with (2.88), implies that
(2.89)0<w5(r)<w1(r)∀r∈(0,rmax⁡).
This proves (2.74). Properties (2.9) in Theorem 2.1, and (2.88), imply that
(2.90)lim⁡r→0+⁡w3(r)w1(r)=lim⁡r→0+⁡h3(ln⁡(r))=1,lim⁡r→rmax⁡-⁡w5(r)=lim⁡r→rmax⁡-⁡h5(ln⁡(r))w1(r)=0.
This completes the proof of (2.75). Therefore, Theorem 2.2 is proved.


## 3. Conclusions

In this paper, our analytic advance is the development of methods to efficiently prove the existence and asymptotic behavior of families of positive singular solutions of (1.3). Our approach consists of the following three steps.Step 1.
Transform the nonautonomousw equation (1.3) into the autonomous h equation (1.17) by setting
(3.1)h(τ)=w(exp⁡(τ))w1(exp⁡(τ)),-∞<τ<∞.Step 2.
Analyze the existence and asymptotic behavior of solutions of (1.17) which are positive on a maximal interval (τmin,τmax).Step 3.
For each such solution of theh equation, make use of the inverse transformation
(3.2)w(r)=h(ln⁡(r))w1(r),0<r<∞
to prove the existence and asymptotic behavior of the associated solution (3.2) of the w equation on the maximal interval (rmin,rmax), where w(r)>0.In Section2, we used this three-step procedure (see Theorems 2.1 and 2.2) to prove the existence and asymptotic behavior of three new families of solutions of (1.3). Open Problems 1 and 2 describe a fourth family of solutions whose existence is also proved in these theorems, and which satisfy(3.3)w(r)>w1(r),w′(r)<0∀r∈(rmin⁡,∞),lim⁡r→∞⁡w(r)=0.
The unresolved issue is to prove whether rmin=0 or rmin>0. If rmin=0, then solutions in this family satisfy the limiting property(3.4)lim⁡r→0+⁡w(r)w1(r)=∞.
Thus, as r→0+, these “super singular” solutions approach ∞ much faster than the closed form solution w1(r).Open Problem 3.
Prove the existence and asymptotic behavior of families of positive solutions of (1.3) other than those found in Theorem 2.2. For example, the existence and limiting behavior of the solutions labelled (c) and (f) in Figure 1 have not yet been proved. It is our hope that our techniques can be extended to prove the existence and limiting behavior of these and many other new families of solutions.Open Problem 4.
Determine the role that the singular solutions proved in Theorem2.2 play in the analysis of the full time-dependent PDE (1.2). Can the analytic techniques developed by Souplet and Weissler [5], and those of Chen and Derrick [6], be extended to apply to these new solutions?



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