Radially Symmetric Solutions of Δ𝑤+|𝑤|^𝑝−1𝑤=0

Abstract

We investigate solutions of 𝑤^′′+((𝑁−1)/𝑟)𝑤^′+|𝑤|^𝑝−1𝑤=0,𝑟>0 and focus on the regime 𝑁>2 and 𝑝>𝑁/(𝑁−2). Our advance is to develop a technique to efficiently classify the behavior of solutions on (𝑟_min,𝑟_max), their maximal positive interval of existence. Our approach is to transform the nonautonomous 𝑤 equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the 𝑤 equation and describe their asymptotic behavior. In the subcritical case 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2) there is a well-known closed-form singular solution, 𝑤₁, such that 𝑤₁(𝑟)→∞ as 𝑟→0⁺ and 𝑤₁(𝑟)→0 as 𝑟→∞. Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies𝑤(𝑟_𝑖)=𝑤₁(𝑟_𝑖) for infinitely many values 𝑟_𝑖>0. At the critical value 𝑝=(𝑁+2)/(𝑁−2) there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime 𝑝>(𝑁+2)/(𝑁−2) we prove the existence of a family of “super singular” sign changing singular solutions.

1. Introduction

In this paper we investigate the behavior of solutions of Δ𝑤+|𝑤|^𝑝−1𝑤=0, (1.1) where 𝑤=𝑤(𝑥₁,…,𝑥_𝑁), 𝑁>1 and 𝑝>1. Solutions of (1.1) are time-independent solutions of the nonlinear heat equation 𝜕𝑤|𝑤|𝜕𝑡=Δ𝑤+^𝑝−1𝑤. (1.2)

Equation (1.1) has been widely studied as a canonical model for Δ𝑢+𝑓(𝑢)=0, (1.3) where 𝑓(𝑢)>0 is superlinear [1–6].

Our focus is on radially symmetric solutions of (1.1) which have the form 𝑤=𝑤(𝑟), where 𝑟=(𝑥²₁+⋯+𝑥²_𝑁)^1/2, and satisfy 𝑤^′′+𝑁−1𝑟𝑤^′+|𝑤|^𝑝−1𝑤=0,𝑟>0. (1.4)

We distinguish two classes of solutions of (1.4). The first is nonsingular solutions which are bounded at 𝑟=0 and satisfy (𝑤(0),𝑤^′(0))=(𝑤₀,0), where 𝑤₀ is finite. The second class consists of singular solutions that are unbounded at 𝑟=0. Equation (1.4) has the known positive singular solution 𝑤₁(𝑟)=2(𝑁−2)𝑝−1−4𝑝−1^{21/(𝑝−1)}𝑟^{−2/(𝑝−1)}𝑁,𝑁>2,𝑝>.𝑁−2 (1.5)

Previous Results
(i) The positive singular solution 𝑤₁(𝑟) has played a central role in analyzing (1.2). For example, when appropriately chosen, similarity solution methods show how 𝑤(𝑥₁,…,𝑥_𝑁,𝑡)→𝑐𝑤₁(𝑟) as 𝑡→∞, where 𝑐>0 is a constant [2, 5, 7]. (ii) Chen and Derrick [8] developed comparison methods to describe the time evolution of solutions of 𝜕𝑢(𝑢),𝜕𝑡=Δ𝑢+𝑓 (1.6) where 𝑓(𝑢) is superlinear [1–6]. Their approach is to let positive, time independent solutions act as upper and/or lower bounds for initial values of solutions of (1.6). Their comparison technique allows them to prove either global existence or finite time blowup of solutions. (iii) For the case 𝑝=(𝑁+2)/(𝑁−1) Caffarelli et al. [9] describe the asymptotic behavior of nonnegative solutions of (1.1) that have an isolated singularity at the origin. (iv) Galaktionov [10] studied sign changing singular solutions of (1.4) on the restricted interval 0<𝑟≤1. He set 𝑤(𝑟)=𝑟^{−2/(𝑝−1)}𝜙(−ln(𝑟)) and derived an ODE for 𝜙(𝑠), 𝑠=−ln(𝑟). He let 𝜙(0)=0, varied 𝜙^′(0), and gave a numerical study of sign changing solutions on 0≤𝑠<∞. (v) Other studies of nonsingular solutions of (1.4) have used Pohozaev identities, together with integral estimates which involve the independent variable [1, 2, 4].

Specific Aims
Our goal is to develop techniques to efficiently classify the behavior of solutions of (1.4) on (𝑟_min,𝑟_max), their maximal positive interval of existence. We study the behavior of solutions which are positive on (𝑟_min,𝑟_max) and also sign changing solutions. In particular, our specific aims are the following.

Specific Aim I
Do positive singular solutions exist, other than 𝑤₁(𝑟), for which (𝑟_min,𝑟_max)=(0,∞)? What is their asymptotic behavior as 𝑟→0⁺, and as 𝑟→∞? In Section 2 we prove the existence of a second singular solution, 𝑤₂(𝑟), (see bottom right panel of Figure 1), which exists on (0,∞). Also, we prove the asymptotic behavior of this solutions as 𝑟→0⁺ and as 𝑟→∞. This result is new and different from previous analyses. In addition, in the conclusion we suggest a possible application for the role of this new solution in analyzing the time-dependent behavior of the full PDE (1.2).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Subcritical example: 𝑁=3, 𝑝=4. ((a) and (b)) The unstable manifold (𝐴1 and 𝐴2) and stable manifold (𝐵1 and 𝐵2) lead from (0,0) into the (ℎ,ℎ′) phase plane. ((c)–(f)) (c) and (e) show the ℎ components of solutions on 𝐴1, 𝐴2, 𝐵1, 𝐵2; (d) and (f) show corresponding solutions of (1.4): 𝑤0 is generated by 𝐴1, 𝑤2(𝑟) is the new positive singular solution generated by 𝐵1, and 𝑤1(𝑟)=(2/9)1/3𝑟−2/3.

Specific Aim II
Do sign changing solutions exist for which (𝑟_min,𝑟_max)=(0,∞)? What is their asymptotic behavior as 𝑟→0⁺, and as 𝑟→∞? In Section 4 we prove the existence of a large amplitude sign changing solution in the (ℎ,ℎ^′) phaseplane (see top left panel of Figure 4). This solution forms a large amplitude outward spiral as the independent variable 𝜏 decreases. Such global analysis has not previously been achieved.

Our Approach
Standard methods to analyze solutions of (1.4) include Pohozaev integral estimates, or topological shooting. Obtaining global results with such methods is difficult since (1.4) is nonautonomous. Thus, to successfully address the issues in Specific Aims I-II, our advance is to develop a two-step approach which significantly simplifies the analysis. The first step is to transform the nonautonomous ODE (1.4) into a simpler, autonomous ODE. Let 𝑤(𝑟) solve (1.4), and define [10, 11] ℎ(𝜏)=𝑤exp(𝜏)𝑤₁exp(𝜏),−∞<𝜏<∞. (1.7)
Then ℎ(𝜏) solves ℎ^′′+𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′+2(𝑁−2)𝑝−1²𝑁𝑝−|ℎ|𝑁−2^𝑝−1ℎ−ℎ=0. (1.8)
Because (1.8) is autonomous, we can apply phase plane techniques to analyze the behavior of solutions. The second step of our approach is to use the “inverse” formula 𝑤(𝑟)=𝑤₁(𝑟)ℎ(ln(𝑟)),0<𝑟<∞ (1.9) to analyze corresponding solutions of the 𝑤 equation (1.4). For example, in Section 2 we analyze (1.8) in the subcritical range 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2) and prove that there is a nonmonotonic heteroclinic orbit (labeled 𝐵₁ in Figure 1) leading from (1,0) to (0,0) in the (ℎ,ℎ^′) phase plane. We then use (1.9) to show that, corresponding to this heteroclinic orbit, there is an entire continuum of new positive singular solutions of (1.4). Let 𝑤₂(𝑟) denote a member of this continuum (Figure 1, 3rd row). Then 𝑤₂(𝑟) intertwines with 𝑤₁(𝑟) infinitely often as 𝑟→0⁺. That is, there are infinitely many positive values {𝑟_𝑖}, with 𝑟_𝑖→0 as 𝑖→∞, such that 𝑤₂𝑟_𝑖=𝑤₁𝑟_𝑖,𝑖≥1. (1.10)
Furthermore, there is a value 𝐷>0 such that 𝑤₂(𝑟)𝑤₁(𝑟)→1as𝑟→0⁺,𝑤₂(𝑟)𝑤₁(𝑟)∼𝐷𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2))}→0as𝑟→∞. (1.11)
Thus, 𝑤₂(𝑟)∼𝑤₁(𝑟) as 𝑟→0⁺, but 𝑤₂(𝑟)→0 faster than 𝑤₁(𝑟) as 𝑟→∞. To our knowledge, this family of solutions has not previously been reported.
In the conclusion, Section 5, we state an open problem which gives a conjecture for the role that 𝑤₂(𝑟) might play in the analysis of the full time-dependent PDE (1.2).
In Sections 3 and 4 we use similar techniques to prove the existence of new families of solutions in the critical case 𝑝=(𝑁+2)/(𝑁−2), and the supercritical regime 𝑝>(𝑁+2)/(𝑁−2), respectively. In particular, in Section 4 we prove the existence of a continuum of “super singular” sign changing solutions, each of which exists on an interval of the form (𝑟_min,∞). For these solutions it remains a challenging, and important, open problem (see Open Problems I and II in Section 4) to prove whether 𝑟_min=0 or 𝑟_min>0.

2. The Subcritical Case: 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2)

In this section we consider the parameter regime 𝑁>2 and 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2). In this range we first analyze solutions of the ℎ equation (1.8) and then show how these solutions translate into corresponding solutions of the 𝑤 equation (1.4). The remainder of this section consists of the following.(I) Lemmas 2.1 and 2.2 state fundamental properties of solutions of (1.8) that satisfy |ℎ|<1 on an interval [𝜏₀,∞) for some 𝜏₀∈ℝ. These properties will be applied in the proof of Lemma 2.3 which asserts that there exists a solution ℎ₂ of (1.8) such that ℎ₂→0 monotonically as 𝜏→∞. In Lemma 2.4 we show that ℎ^′₂(𝜏)ℎ₂(𝜏)→−𝑁−2𝑁𝑝−1𝑝−𝑁−2as𝜏→∞. (2.1) This will be used to prove Lemma 2.5 which shows that the asymptotic behavior of ℎ₂ is ℎ₂(𝜏)∼𝐷exp−𝑁−2𝑁𝑝−1𝑝−𝜏𝑁−2as𝜏→∞, (2.2) for some constant 𝐷>0.(II) Solutions along the unstable manifold, 𝐴₁, described in part (i) of Theorem 2.9, translate into nonsingular solutions of (1.4). Of particular importance is the heteroclinic orbit solution, 𝐵₁, described in part (ii) of Theorem 2.9. Theorem 2.10 asserts that the trajectory 𝐵₁ generates an entire continuum of strictly positive singular solutions of (1.4), each of which intertwines infinitely often with 𝑤₁(𝑟) as 𝑟→0⁺. To our knowledge, this family of solutions is new and has not previously been reported.(III)Numerical Experiments. Figure 1 shows solutions of (1.4) and (1.8) when (𝑁,𝑝)=(3,4). However, it must be emphasized that it is illegitimate to claim that numerical results are rigorous proofs. Complete analytical proofs are needed to determine properties of solutions of (1.4) and (1.8).

The following two technical lemmas are used to help prove that ℎ₂→0 monotonically as 𝜏→∞.

Lemma 2.1. Suppose that ℎ is a nonconstant solution of (1.8) such that 0≤ℎ(𝜏₀)<1 and ℎ^′(𝜏₀)≥0 for some 𝜏₀∈ℝ. Then there exists a constant Δ>0 such that ℎ(𝜏₀+Δ)=1. Likewise, if −1<ℎ(𝜏₀)≤0 and ℎ^′(𝜏₀)≤0, then there exists a constant Δ>0 such that ℎ(𝜏₀+Δ)=−1.

Proof. The proof of this lemma relies on the property 0<ℎ(𝜏)<1,ℎ^′(𝜏)>0impliesthatℎ^′′(𝜏)>0 (2.3) which is an immediate consequence of the case assumption 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2) and (1.8). Since ℎ is a nonconstant solution, then uniqueness of solutions [12, Chapter 1] implies that either ℎ(𝜏₀)>0 or ℎ^′(𝜏₀)>0. This and (1.8) imply that ℎ^′′(𝜏₀)>0. Hence, 0<ℎ(𝜏)<1 and ℎ^′(𝜏)>0 on an interval (𝜏₀,𝜏₀+𝜎) provided that 𝜎>0 is sufficiently small. By (2.3), ℎ^′′(𝜏)>0 on (𝜏₀,𝜏₀+𝜎). The increasing values of ℎ^′(𝜏)>0 on (𝜏₀,𝜏₀+𝜎) imply the existence of Δ>0 such that ℎ(𝜏₀+Δ)=1.
In a similar manner, −1<ℎ(𝜏₀)≤0 and ℎ^′(𝜏₀)≤0 implies the existence of a value Δ>0 such that ℎ(𝜏₀+Δ)=−1. This completes the proof.

The following lemma is also used to show that ℎ₂→0 monotonically as 𝜏→∞.

Lemma 2.2. Suppose that ℎ is a nonconstant solution of (1.8) such that |ℎ|<1 on [𝜏₀,∞). Then ℎ(𝜏)≠0 on [𝜏₀,∞).

Proof. Suppose that ℎ(𝜏₁)=0 for some 𝜏₁≥𝜏₀. Since ℎ is a nonconstant solution of (1.8), then ℎ^′(𝜏₁)≠0. By Lemma 2.1, there exists a constant Δ>0 such that ℎ(𝜏₁+Δ)=1 if ℎ^′(𝜏₁)>0 and ℎ(𝜏₁+Δ)=−1 if ℎ^′(𝜏₁)<0. This contradicts the assumption that |ℎ|<1 on [𝜏₀,∞) and concludes the proof of the lemma.

Lemma 2.3. There exists a solution ℎ₂ of (1.8) such that(a)lim_𝜏→∞(ℎ₂(𝜏),ℎ^′₂(𝜏))=(0,0), and(b)ℎ₂(𝜏)>0 and ℎ^′₂(𝜏)<0 on [0,∞).

Proof. A linearization of (1.8) around the constant solution ℎ≡0 gives ℎ^′′+𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′−2(𝑁−2)𝑝−1²𝑁𝑝−𝑁−2ℎ=0. (2.4)
The eigenvalues associated with (2.4) are 𝜆₁=2𝑝−1>0,𝜆₂=−𝑁−2𝑁𝑝−1𝑝−𝑁−2<0. (2.5)
Since 𝜆₂<0, the Stable Manifold Theorem [12, Chapter 13] ensures that there exists a one-dimensional stable manifold containing (0,0) in the (ℎ,ℎ^′) phase plane. Let ℎ₂ be a solution of (1.8) such that (ℎ₂(0),ℎ^′₂(0)) is a point on the stable manifold. That is, lim_𝜏→∞ℎ₂(𝜏),ℎ^′₂(𝜏)=(0,0). (2.6)
Thus, ℎ₂ satisfies part (a).
We now show that ℎ₂ can be chosen to satisfy (b). Since solutions of (1.8) are translation-invariant, there is no loss in generality in assuming that |ℎ₂(𝜏)|<1 on [0,∞). Combining this with Lemma 2.2 yields ℎ₂(𝜏)≠0 on [0,∞). According to the fact that ℎ₂ and −ℎ₂ are both solutions of (1.8) that satisfy (2.6), we may also assume that ℎ₂(𝜏)>0 on [0,∞). Lemma 2.1 implies that ℎ^′₂(𝜏)<0 on [0,∞). This proves (b) and concludes the proof of this lemma.

Description of the Stable Manifold
Throughout the remainder of this section we let ℎ₂ denote the solution of (1.8) that satisfies 0<ℎ₂(𝜏)<1,ℎ^′₂(𝜏)<0on[0,∞). (2.7)
Furthermore, we define 𝐵₁=ℎ₂(𝜏),ℎ^′₂(𝜏)∣𝜏∈𝜏_min,∞, (2.8) where (𝜏_min,∞) denotes the maximal interval of existence of the solution ℎ₂. In Lemma 2.6 we will show that 𝜏_min=−∞. In addition, we define the negative counterpart of 𝐵₁ by 𝐵₂=−ℎ₂(𝜏),−ℎ^′₂(𝜏)∣𝜏∈𝜏_min,∞. (2.9)
The top row of Figure 1 depicts 𝐵₁ and 𝐵₂ for 𝑁=3 and 𝑝=4.

Asymptotic Behavior of ℎ₂
To state Lemma 2.4 correctly we need to derive basic properties of the functional 𝐻₂(𝜏)=exp(−𝜆₂𝜏)ℎ₂(𝜏). It follows that 𝐻₂ satisfies 𝐻₂^′′−(𝑁−2)𝐻^′₂2()=−𝑁−2𝑝−1²𝑁𝑝−ℎ𝑁−2^2𝑝−1𝐻₂. (2.10)
Also, ℎ₂(𝜏)>0 for all 𝜏≥0 implies that 𝐻₂(𝜏)>0∀𝜏≥0. (2.11)
It follows from (2.10) and (2.11) that 𝐻^′₂(𝜏)>0∀𝜏≥0. (2.12)
By (2.12), all that remains is to show that 𝐻₂(𝜏) is bounded above. That is the purpose of the following two lemmas.

Lemma 2.4. The solution 𝐻₂ of (2.10) satisfies lim_𝜏→∞𝐻^′₂(𝜏)𝐻₂(𝜏)=0. (2.13)
Moreover, lim_𝜏→∞ℎ^′₂(𝜏)ℎ₂(𝜏)=𝜆₂. (2.14)

Proof. Define 𝜌=𝐻^′₂/𝐻₂. Our computations show that 𝜌^′+𝜌²+𝜆₂−𝜆₁𝜌=𝜆₁𝜆₂ℎ₂^𝑝−1. (2.15)
Alternatively, (2.15) can be written as 𝜌^′=−𝜌−𝜇₋𝜌−𝜇₊, (2.16) where 𝜇₋ and 𝜇₊ are defined by 𝜇_±=𝜆₁−𝜆₂±𝜆₁−𝜆₂²+4𝜆₁𝜆₂ℎ₂^𝑝−12. (2.17)
Since ℎ₂(𝜏)→0 as 𝜏→∞, it follows from (2.17) that lim_𝜏→∞𝜇₋=0,lim_𝜏→∞𝜇₊=𝜆₁−𝜆₂. (2.18)
Thus, it is sufficient to show that 0<𝜌(𝜏)≤𝜇₋ for all 𝜏≥0. We accomplish this by process of elimination.
First, 𝜌≤0 is impossible due to (2.11) and (2.12). A consequence of (2.16) is that 𝜌 increases to 𝜇₊ whenever 𝜇₋<𝜌≤𝜇₊ and 𝜌 decreases to 𝜇₊ whenever 𝜌>𝜇₊. In either case, if 𝜌(0)>𝜇₋, then lim_𝜏→∞𝜌=lim_𝜏→∞𝜇₊=𝜆₁−𝜆₂. (2.19)
Since 𝜌=𝐻^′₂/𝐻₂=(ℎ^′₂/ℎ₂)−𝜆₂, then (2.19) implies that lim_𝜏→∞ℎ^′₂ℎ₂=𝜆₁. (2.20)
The fact that 𝜆₁>0 contradicts (2.7). Therefore, 𝜌>𝜇₋ is impossible. This leaves 0<𝜌≤𝜇₋ as the only possibility. Consequently, 0≤lim_𝜏→∞𝜌=lim_𝜏→∞𝐻^′₂𝐻₂=lim_𝜏→∞𝜇₋=0. (2.21)
Combining this result with 𝐻^′₂/𝐻₂=(ℎ^′₂/ℎ₂)−𝜆₂ yields lim_𝜏→∞(ℎ^′₂(𝜏)/ℎ₂(𝜏))=𝜆₂. This completes the proof of the lemma.

We now prove that lim_𝜏→∞ℎ₂(𝜏)exp(−𝜆₂𝜏)=𝐷 for some constant 𝐷>0.

Lemma 2.5. The solution 𝐻₂ of (2.10) satisfies lim_𝜏→∞𝐻₂(𝜏)=𝐷 for some finite constant 𝐷>0.

Proof. Because of (2.12) it is sufficient to show that 𝐻₂(𝜏) is bounded above on [0,∞). For a contradiction, assume that lim_𝜏→∞𝐻₂(𝜏)=∞. (2.22)
By (2.13) there is a value 𝜏₀>0 such that 𝐻^′₂/𝐻₂<−𝜆₂(𝑝−1)/2𝑝 for all 𝜏≥𝜏₀. Hence, 𝐻^′₂+(𝜆₂(𝑝−1)/2𝑝)𝐻₂<0 for all 𝜏≥𝜏₀ which yields 𝐻₂(𝜏)≤𝐾₂−𝜆exp²𝑝−1𝜏2𝑝∀𝜏>𝜏₀, (2.23) where 𝐾₂=𝐻₂(𝜏₀)exp(𝜆₂(𝑝−1)𝜏₀/2𝑝).
Integrating (2.10) over [𝜏₀,𝜏] gives 𝐻^′₂−(𝑁−2)𝐻₂=𝐾₁−2(𝑁−2)𝑝−1²𝑁𝑝−𝑁−2^𝜏_𝜏0ℎ₂^𝑝−1𝐻₂𝑑𝜂, (2.24) where 𝐾₁=𝐻^′₂(𝜏₀)−(𝑁−2)𝐻₂(𝜏₀). Next, we use (2.23) to show that ∫^∞_𝜏0|ℎ₂|^𝑝−1𝐻₂𝑑𝜂 exists and is finite. Subsequently, we divide both sides of (2.24) by 𝐻₂ and let 𝜏→∞.
Combining (2.23) with the fact that ℎ₂(𝜏)=exp(𝜆₂𝜏)𝐻₂(𝜏)>0 for all 𝜏≥0 yields ℎ₂(𝜏)^𝑝−1𝐻₂(𝜏)=exp𝜆₂𝑝−1𝜏𝐻^𝑝₂(𝜏)≤exp𝜆₂𝑝−1𝜏𝐾^𝑝₂−𝜆exp²𝑝−12𝜏=𝐾^𝑝₂𝜆exp²𝑝−12𝜏. (2.25)
This implies that ^∞_𝜏0ℎ₂^𝑝−1𝐻₂𝑑𝜂<∞. (2.26)
Dividing both sides of (2.24) by 𝐻₂ and letting 𝜏→∞ we obtain −(𝑁−2)=0 as a consequence of (2.13), (2.22), and (2.26). This is an obvious contradiction. Thus, there exists a finite constant 𝐷>0 such that lim_𝜏→∞ℎ₂(𝜏)exp−𝜆₂𝜏=lim_𝜏→∞𝐻₂(𝜏)=𝐷. (2.27)
This concludes the proof of the lemma.

Conclusion. It follows from (2.27) that the asymptotic behavior of ℎ₂ described in (2.2) is now proved.

The next lemma shows that ℎ₂ exists on ℝ and that ℎ₂(𝜏)→1 as 𝜏→−∞.

Lemma 2.6. The solution ℎ₂ of (1.8) is defined on ℝ and satisfies lim_{𝜏→−∞}(ℎ₂(𝜏),ℎ^′₂(𝜏))=(1,0). Furthermore, there is a decreasing sequence {𝜏_𝑘}, with lim_𝑘→∞𝜏_𝑘=−∞, such that ℎ𝜏_𝑘=1∀𝑘≥1,ℎ^′𝜏_𝑘<0if𝑘isodd,ℎ^′𝜏_𝑘>0if𝑘iseven. (2.28)

Proof. Let (𝜏_min,∞) denote the maximal interval of existence of ℎ₂. We claim that 𝜏_min=−∞. To prove this, we make use of the functional 1𝑆=2ℎ^′2+2(𝑁−2)𝑝−1²𝑁𝑝−|ℎ|𝑁−2^𝑝+1−ℎ𝑝+12², (2.29) and the region 𝑈 defined by 𝑈=ℎ,ℎ^′∣ℎ≥0,𝑆≤0. (2.30)
Let 𝑈⁰ denote the interior of 𝑈 (Figure 1, upper right). Note that the constant solution (ℎ(𝜏),ℎ^′(𝜏))≡(1,0)∈𝑈⁰. Since 𝑝<(𝑁+2)/(𝑁−2), a differentiation of (2.29) gives 𝑆^′=−𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′2≥0∀𝜏∈𝜏_min,∞. (2.31)
We conclude from (2.2), (2.6), (2.14), (2.29), and (2.31) that, when ℎ=ℎ₂, 𝑆(∞)=0,𝑆^′(𝜏)≥0,𝑆(𝜏)<0∀𝜏∈𝜏_min,∞. (2.32)
Therefore, (ℎ₂(𝜏),ℎ^′₂(𝜏)) is uniformly bounded on (𝜏_min,∞). From this and standard theory it follows that 𝜏_min=−∞. Thus, for the solution ℎ₂, we conclude that 𝑆(𝜏)<0forall𝜏∈ℝ and that ℎ₂(𝜏),ℎ^′₂(𝜏)∈𝑈⁰∀𝜏∈(−∞,∞). (2.33)
From (2.31) we also conclude that (1.8) has no periodic solutions. In addition, the constant solution (ℎ,ℎ^′)≡(1,0) is the only constant solution in 𝑈⁰. Thus, it follows from (2.33) and standard phase plane arguments that ℎ₂(𝜏)>0∀𝜏∈ℝ,lim_{𝜏→−∞}ℎ₂(𝜏),ℎ^′₂(𝜏)=(1,0). (2.34)
Finally, we need to determine precisely how solutions approach (ℎ,ℎ^′)=(1,0) as 𝜏→−∞. For this a linearization of (1.8) around the constant solution ℎ≡1 gives ℎ^′′+𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′+2(𝑁−2)𝑁𝑝−1𝑝−𝑁−2ℎ=0. (2.35)
The eigenvalues associated with (2.35) are complex, with positive real parts. Thus, solutions with initial values on the curve 𝐵₁ must spiral into the constant solution (ℎ,ℎ^′)=(1,0) as 𝜏→−∞. Property (2.28) follows as a consequence. This completes the proof of the lemma.

In Lemma 2.3, the eigenvalue 𝜆₂<0 defined in (2.5) led to the existence of the solution ℎ₂ of (1.8). The decay rate of ℎ₂ as 𝜏→∞ is described in Lemmas 2.3–2.5. The methods used to prove Lemmas 2.3–2.5 can be applied to the eigenvalue 𝜆₁>0 defined in (2.5) to result in the following lemma.

Lemma 2.7. There exists a solution ℎ₁ of (1.8) such that(a)lim_{𝜏→−∞}(ℎ₁(𝜏),ℎ^′₁(𝜏))=(0,0),(b)ℎ₁(𝜏)>0 and ℎ^′₁(𝜏)>0 on (−∞,0],(c)lim_{𝜏→−∞}(ℎ^′₁(𝜏)/ℎ₁(𝜏))=𝜆₁, and(d)lim_{𝜏→−∞}ℎ₁(𝜏)exp(−𝜆₁𝜏)=𝑐 for some constant 𝑐>0.

Description of the Unstable Manifold
As shown in the proof of Lemma 2.3, the solution ℎ₁ is chosen so that (ℎ₁(0),ℎ^′₁(0)) is a point on the unstable manifold of the constant solution (ℎ,ℎ^′)≡(0,0). As depicted in Figure 1 we let 𝐴₁ denote the component of the unstable manifold in the ℎ>0 and ℎ^′>0 quadrant. Also, we denote the component of the unstable manifold in the ℎ<0 and ℎ^′<0 component by 𝐴₂. Precisely, 𝐴₁=ℎ₁(𝜏),ℎ^′₁(𝜏)∣𝜏∈−∞,𝜏_max, (2.36) where (−∞,𝜏_max) denotes the maximal interval of existence of ℎ₁. Noting that −ℎ₁ is also a solution of (1.8) such that lim_{𝜏→−∞}(−ℎ₁(𝜏),−ℎ^′₁(𝜏))=(0,0) we can define 𝐴₂=−ℎ₁(𝜏),−ℎ^′₁(𝜏)∣𝜏∈−∞,𝜏_max. (2.37)
In the next lemma, we continue our analysis of the solution ℎ₁ as 𝜏→𝜏⁻_max.

Lemma 2.8. There is an increasing, positive sequence {𝜏_𝑘} such that ℎ^′₁𝜏_𝑘=0∀𝑘≥1,ℎ₁𝜏_𝑘>1if𝑘isodd,ℎ₁𝜏_𝑘<−1if𝑘iseven. (2.38)

Proof. The corresponding solution 𝑤₀(𝑟) of (1.4) satisfies 𝑤₀(𝑟)=ℎ₁((𝑟2()ln))𝑁−2𝑝−1−4𝑝−1^{21/(𝑝−1)}𝑟^{−2/(𝑝−1)},𝑟>0, (2.39) and it follows from parts (b) and (d) of Lemma 2.7 and (2.39) that 𝑤₀(𝑟)>0forall𝑟>0, and 𝑤₀(𝑟)→𝑤₀2()=𝑐𝑁−2𝑝−1−4𝑝−1^{21/(𝑝−1)}>0as𝑟→0. (2.40)
It follows from standard theory that solutions that are bounded at 𝑟=0 must satisfy 𝑤^′(0)=0. Thus, for solutions of (1.8) such that (ℎ(0),ℎ^′(0))∈𝐴₁, and satisfying parts (b) and (d) of Lemma 2.7, the corresponding solution 𝑤₀(𝑟) of (1.4) is nonsingular and satisfies 𝑤₀(0)=𝑤₀>0, 𝑤^′₀(0)=0. Haraux and Weissler [2] showed that 𝑤₀(𝑟) has at least one positive zero. Chen et al. [1] proved that 𝑤₀(𝑟) has infinitely many positive zeros. These results, and the fact that ℎ₁=𝑤/𝑤₁, imply that solutions of (1.8) satisfying (ℎ₁(0),ℎ^′₁(0))∈𝐴₁ have infinitely many positive zeros. Thus, there is an increasing, positive sequence {𝜏_𝑘}, where ℎ₁ attains a positive relative maximum when 𝑘≥1 is odd, and a negative relative minimum when 𝑘 is even. It follows from (1.8) that ℎ₁𝜏_𝑘>1when𝑘isodd,ℎ₁𝜏_𝑘<−1when𝑘iseven. (2.41)
This concludes the proof of the lemma.

The following theorem summarizes our results obtained thus far. In particular, part (i) of the following theorem summarizes the results of Lemmas 2.7 and 2.8 regarding the solution ℎ₁. Part (ii) summarizes the results of Lemmas 2.3–2.6 regarding the solution ℎ₂.

Theorem 2.9. Let 𝑁>2 and 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2).(i)There is a one-dimensional unstable manifold of solutions of (1.8) leading from (0,0) in the (ℎ,ℎ^′) phase plane. One component, 𝐴₁, points into the positive quadrant, and its negative counterpart, 𝐴₂, points into the negative quadrant. If (ℎ(0),ℎ^′(0))∈𝐴₁, and ℎ(0)>0 is sufficiently small, then there is a value 𝑐>0 such that ℎ(𝜏)>0∀𝜏∈(−∞,0],ℎ(𝜏)∼𝑐exp2𝜏𝑝−1as𝜏→−∞. (2.42)
Furthermore, there is an increasing, positive sequence {𝜏_𝑘} such that ℎ^′𝜏_𝑘=0∀𝑘≥1,ℎ𝜏_𝑘>1if𝑘isodd,ℎ𝜏_𝑘<−1if𝑘iseven. (2.43)(ii) There is a one-dimensional stable manifold of solutions leading to (0,0) in the (ℎ,ℎ^′) phase plane. One component, 𝐵₁, points into the ℎ>0, ℎ^′<0 quadrant of the phase plane, and its negative counterpart, 𝐵₂, points into the ℎ<0, ℎ^′>0 quadrant. If (ℎ(0),ℎ^′(0))∈𝐵₁, then ℎ(𝜏)>0∀𝜏∈ℝ,lim_{𝜏→−∞}ℎ(𝜏),ℎ^′(𝜏)=(1,0),lim_𝜏→∞ℎ(𝜏),ℎ^′(𝜏)=(0,0). (2.44)
 Also, there is a decreasing sequence {𝜏_𝑘}, with lim_𝑘→∞𝜏_𝑘=−∞, such that ℎ𝜏_𝑘=1∀𝑘≥1,ℎ^′𝜏_𝑘<0if𝑘isodd,ℎ^′𝜏_𝑘>0if𝑘iseven. (2.45)
 Finally, there is a constant 𝐷>0 such that ℎ(𝜏)∼𝐷exp−𝑁−2𝑁𝑝−1𝑝−𝜏𝑁−2as𝜏→∞. (2.46)

Solutions of the 𝑤 Equation
Below, in Theorem 2.10, we show how to combine part (ii) of Theorem 2.9 with the formula 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟), (2.47) to prove the existence and asymptotic behavior of a new family of singular solutions of the 𝑤 equation (1.4). Our approach is to let (ℎ(0),ℎ^′(0)) be an arbitrarily chosen element of the continuous curve 𝐵₁. Since 𝑟=𝑒^𝜏, the initial conditions for the corresponding solution of (1.4) are given at 𝑟=𝑒⁰=1 and satisfy 𝑤(1)=ℎ(0)𝑤₁(1),𝑤^′(1)=ℎ^′(0)𝑤(1)+ℎ(0)𝑤^′(1). (2.48)
Since (ℎ(0),ℎ^′(0))∈𝐵₁, and 𝐵₁ is a continuous curve, then (2.48) generates an entire continuum of solutions of the 𝑤 equation. We show how these solutions intertwine with 𝑤₁(𝑟) infinitely often as 𝑟→0⁺. In addition, we show how to prove the limiting behavior of each solution at both ends of (𝑟_min,𝑟_max), its maximal positive interval of existence.

Theorem 2.10 (a continuum of new singular solutions of (1.4)). Let 𝑁>2 and 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2). Let 𝑤₁(𝑟) denote the positive singular solution of (1.4) defined in (1.5), and let ℎ₂(𝜏) be a solution of (1.8) which satisfies (ℎ₂(0),ℎ^′₂(0))∈𝐵₁ in part (ii) of Theorem 2.9. The corresponding solution 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟) of (1.4) has initial values 𝑤₂(1)=ℎ₂(0)𝑤₁(1),𝑤^′₂(1)=ℎ^′₂(0)𝑤₁(1)+ℎ₂(0)𝑤^′₁(1). (2.49)
Furthermore, (0,∞) is the maximal interval of existence of 𝑤₂(𝑟), and there is a decreasing positive sequence, {𝑟_𝑘}, with lim_𝑘→∞𝑟_𝑘=0, such that 0<𝑤₂(𝑟)<𝑤₁(𝑟)∀𝑟>𝑟₁,𝑤₂𝑟_𝑘=𝑤₁𝑟_𝑘∀𝑘≥1,lim_𝑟→0+𝑤₂(𝑟)𝑤₁(𝑟)𝑤=1,²(𝑟)𝑤₁(𝑟)∼𝐷𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2))}as𝑟→∞. (2.50)(2.51)

Numerical Example
In Figure 1 we let (𝑁,𝑝)=(3,4) so that 𝜆₂=−1/3. The stable manifold 𝐵₁ (third row, left panel) is generated by solution ℎ₂(𝜏) of (1.8) with (ℎ₂(0),ℎ^′₂(0))=(.93,.07). The right panel shows the corresponding solution 𝑤₂(𝑟) of (1.4). For this example the asymptotic properties (2.51) become 𝑤₂(𝑟)∼𝑤₁(𝑟)=29^1/3𝑟^−2/3as𝑟→0⁺,𝑤₂(𝑟)∼.05𝑟as𝑟→∞. (2.52)

Proof of Theorem 2.10. Let ℎ₂ denote a solution of (1.8) which satisfies part (ii) of Theorem 2.9. The solution of (1.4) corresponding to ℎ₂ is 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟). (2.53)
It follows from (2.45) in Theorem 2.9 that the sequence {𝑟_𝑘} defined by 𝑟_𝑘=exp𝜏_𝑘∀𝑘≥1 (2.54) is positive and decreasing in 𝑘, with lim_𝑘→∞𝑟_𝑘=0, and 𝑤₂𝑟_𝑘=𝑤₁𝑟_𝑘∀𝑘≥1. (2.55)
Next, it follows from (2.53), and the definition 𝜏=ln(𝑟), that 𝑤₂(𝑟)𝑤₁(𝑟)=ℎ₂(ln(𝑟))=ℎ₂(𝜏). (2.56)
Since 𝑟=𝑒^𝜏→0 as 𝜏→−∞, and since ℎ₂(𝜏)→1 as 𝜏→−∞, it follows from (2.56) that lim_𝑟→0+𝑤₂(𝑟)𝑤₁(𝑟)=1. (2.57)
This proves the first part of (2.51). It remains to prove the asymptotic behavior of the solutions as 𝑟→∞. For this we combine property (2.46) in Theorem 2.9 with (2.56) and substitute 𝜏=ln(𝑟) to obtain 𝑤₂(𝑟)𝑤₁(𝑟)∼𝐷𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2))}as𝑟→∞. (2.58)
This completes the proof of Theorem 2.10.

3. The Critical Case: 𝑝=(𝑁+2)/(𝑁−2)

In this section we investigate the behavior of solutions of (1.4) and (1.8) when 𝑁>2 and 𝑝=(𝑁+2)/(𝑁−2). In this case (1.4) and (1.8) become 𝑤^′′+𝑁−1𝑟𝑤^′+|𝑤|^4/(𝑁−2)𝑤=0,𝑟>0,𝑑²ℎ𝑑𝜏²+𝑁−22²|ℎ|^4/(𝑁−2)ℎ−ℎ=0, (3.1)(3.2) and (1.5) reduces to 𝑤₁(𝑟)=𝑁−22^(𝑁−2)/2𝑟^{−(𝑁−2)/2},𝑟>0. (3.3)

The remainder of this section consists of the following.(I)Theorem 3.1 gives a complete classification of solutions of (3.2).(II)In Theorem 3.2 we show how to combine the results of Theorem 3.1 with the formula 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟) to obtain a continuum of new positive singular solutions of (3.1), and also a continuum of new sign changing singular solutions.(III)Figures 2 and 3 illustrate our results when (𝑁,𝑝)=(3,5).

Figure 2 

Critical case: 𝑁=3, 𝑝=5. Periodic and homoclinic orbits of (3.2) proved in Theorem 3.1.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3 

Critical case: 𝑁=3, 𝑝=5. ((a), (c), and (e)) Homoclinic and periodic orbit solutions of (3.2) proved in Theorem 3.1. ((b), (d), and (f)) The corresponding solutions of (3.1) proved in Theorem 3.2: 𝑤1(𝑟)=1/√2𝑟 is the known singular solution, 𝑤2(𝑟) is a new positive singular solution, and 𝑤3(𝑟) is a new sign changing singular solution.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 

Supercritical example: 𝑁=3, 𝑝=7. ((a) and (b)) Trajectories of the unstable manifold (𝐴1 and 𝐴2) and stable manifold (𝐵1 and 𝐵2) leading from (0,0) into the (ℎ,ℎ′) plane. ((c)–(f)) The ℎ components of solutions along 𝐴1, 𝐴2, 𝐵1, 𝐵2 (left) and corresponding 𝑤 components along 𝐴1 and 𝐵1 (right): 𝑤0(𝑟) is bounded at 𝑟=0, 𝑤1(𝑟)=(2/9)1/6𝑟−1/3 is the known singular solution, and 𝑤2(𝑟) is the new sign changing singular solution corresponding to 𝐵1.

Theorem 3.1. Let 𝑁>2 and 𝑝=(𝑁+2)/(𝑁−2). Each solution of (3.2) satisfies 12ℎ^′2+𝑁−22²𝑁−2|ℎ|2𝑁^{2𝑁/(𝑁−2)}−ℎ²2=𝐸, (3.4) where 𝐸 is a constant. Define 𝐸_∗=−(𝑁−2)²/4𝑁.(i)If 𝐸<𝐸_∗, then there are no real solutions of (3.2) which satisfy (3.4).(ii)If 𝐸=𝐸_∗, then solutions of (3.2) are constant, and either (ℎ(𝜏),ℎ^′(𝜏))=(1,0) or (ℎ(𝜏),ℎ^′(𝜏))=(−1,0) for all 𝜏∈ℝ.(iii)If 𝐸_∗<𝐸<0, then solutions of (3.2) are nonconstant, periodic, they have one sign, and the interior of their trajectories in the (ℎ,ℎ^′) phase plane contains one of the constant solutions (±1,0).(iv)If 𝐸=0, then there is a one parameter family of solutions ℎ_𝜅(𝜏)(𝜏)=𝜅2𝑁exp𝑁(𝑁−2)+|𝜅|^4/(𝑁−2)exp(2𝜏)^(𝑁−2)/2,𝜅≠0,𝜏∈ℝ. (3.5)
Depending on the sign of 𝜅, these solutions are either strictly positive or negative. Their trajectories form homoclinic orbits in the (ℎ,ℎ^′) phase plane, with one of the constant solutions (±1,0) in their interior, and lim_𝜏→±∞ℎ(𝜏),ℎ^′(𝜏)=(0,0). (3.6)(v) If 𝐸>0 solutions of (3.2) are nonconstant, periodic, they change sign, and the interior of their trajectories in the (ℎ,ℎ^′) phase plane contains all three constant solutions (±1,0) and (0,0).

Proof. Let 𝑄=(1/2)(ℎ^′)²+((𝑁−2)/2)²(((𝑁−2)/2𝑁)|ℎ|^{2𝑁/(𝑁−2)}−ℎ²/2). Then 𝑄^′=0forall𝜏, hence 12ℎ^′2+𝑁−22²𝑁−2|ℎ|2𝑁^{2𝑁/(𝑁−2)}−ℎ²2=𝐸∀𝜏, (3.7) where 𝐸 is a constant, and (3.4) is proved. Properties (i)–(v) follow from (3.7).

Numerical Experiments
Figure 2 illustrates homoclinic orbits solutions, and also periodic solutions, of (3.2) in the (ℎ,ℎ^′) plane when (𝑁,𝑝)=(3,5). Graphs of the ℎ components of these solutions are shown in the left column of Figure 3. The corresponding solutions of the 𝑤 equation (3.1) are shown in the right column of Figure 3. Proofs of their existence are given below in Theorem 3.2.

Solutions of the 𝑤 Equation
We now show how to combine the results of Theorem 3.1 with the formula 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟), (3.8) to prove the existence and asymptotic behavior of solutions of (3.1). First, part (ii) of Theorem 3.1 shows that when 𝐸=𝐸_∗=−(𝑁−2)²/4𝑁, then ℎ≡±1. This and (3.8) imply that the corresponding solutions of (3.1) are 𝑤(𝑟)=±𝑤₁(𝑟). Below, in Theorem 3.2, we show how parts (iii)–(v) of Theorem 3.1 generate continuous families of strictly positive solutions of (3.1), and also a family of sign changing singular solutions.

Theorem 3.2. Let 𝑁>2 and 𝑝=(𝑁+2)/(𝑁−2).
(a) A Continuum of Positive Nonsingular Solutions. For each 𝜅>0 (3.1) has the nonsingular solution 𝑤₀(𝑟)=𝜅𝑁(𝑁−2)𝑁(𝑁−2)+|𝜅|^4/(𝑁−2)𝑟^{2(𝑁−2)/2},𝑟>0. (3.9)
(b) A Continuum of Positive “Interlacing” Singular Solutions. Let ℎ₂(𝜏) be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (iii) of Theorem 3.1. The corresponding solution 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟) of (3.1) has initial values 𝑤₂(1)=ℎ₂(0)𝑤₁(1),𝑤^′₂(1)=ℎ^′₂(0)𝑤₁(1)+ℎ₂(0)𝑤^′₁(1), (3.10) and its interval of existence is (0,∞). Furthermore, the solution 𝑤₂(𝑟) interlaces with 𝑤₁(𝑟); that is, there is a positive increasing sequence, {𝑟_𝑘}, with lim_{𝑘→−∞}𝑟_𝑘=0 and lim_𝑘→∞𝑟_𝑘=∞ such that 𝑤₂𝑟_𝑘=𝑤₁𝑟_𝑘,−∞<𝑘<∞,lim_𝑟→0+𝑤₂(𝑟)=∞,lim_𝑟→∞𝑤₂(𝑟)=0. (3.11)(3.12)
(c) A Continuum of Sign Changing Singular Solutions. Let ℎ₃(𝜏) be a member of the family of positive, sign changing periodic solutions of (3.2) which satisfy part (v) of Theorem 3.1. The corresponding solution 𝑤₃(𝑟)=ℎ₃(ln(𝑟))𝑤₁(𝑟) of (3.1) has initial values 𝑤₃(1)=ℎ₃(0)𝑤₁(1),𝑤^′₃(1)=ℎ^′₃(0)𝑤₁(1)+ℎ₃(0)𝑤^′₁(1), (3.13) and its interval of existence is (0,∞). Furthermore, the solution 𝑤₃(𝑟) changes sign infinitely often as follows: there is an positive increasing sequence, {𝑟_𝑖}, with lim_{𝑖→−∞}𝑟_𝑖=0 and lim_𝑖→∞𝑟_𝑖=∞ such that w₃𝑟_𝑖=𝑤₁𝑟_𝑖if𝑖iseven,𝑤₃𝑟_𝑖=−𝑤₁𝑟_𝑖if𝑖isodd,limsup_𝑟→0+𝑤₃(𝑟)=∞,liminf_𝑟→0+𝑤₃(𝑟)=−∞,lim_𝑟→∞𝑤₃(𝑟)=0. (3.14)(3.15)

Remarks. (i) The solutions given in (3.9) were first derived by Joseph and Lundgren [3]. (ii) To our knowledge, the singular solutions described in parts (b) and (c) have not previously been reported.

Proof of Theorem 3.2. Part (a). For each 𝜅>0, let ℎ_𝜅(𝜏) denote the solution given in (3.5) in Theorem 3.1. Setting 𝜏=ln(𝑟) in (3.5) gives ℎ_𝜅(ln(𝑟))=𝜅2𝑁𝑟𝑁(𝑁−2)+|𝜅|^4/(𝑁−2)𝑟^{2(𝑁−2)/2},𝑟>0. (3.16)
Next, substitute (3.3) and (3.16) into (3.8) and obtain 𝑤₀(𝑟)𝑁()=𝜅𝑁−2𝑁(𝑁−2)+|𝜅|^4/(𝑁−2)𝑟^{2(𝑁−2)/2},𝑟>0. (3.17)
 Part (b). Let ℎ₂(𝜏) be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (iii) of Theorem 3.1. The trajectory of (ℎ₂,ℎ^′₂) lies in the positive quadrant of the (ℎ,ℎ^′) plane and surrounds the constant solution (ℎ,ℎ^′)=(1,0). Thus, there are values 𝐿₂>𝐿₁>0 and a positive increasing sequence {𝜏_𝑘}, such that lim_{𝑘→−∞}𝜏_𝑘=−∞,lim_𝑘→∞𝜏_𝑘=∞,0<𝐿₁<ℎ₂(𝜏)<𝐿₂∀𝜏∈ℝ,ℎ₂𝜏_𝑘=1∀𝑘. (3.18)(3.19)
The solution of the 𝑤 equation (3.1) corresponding to ℎ₂ is 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟)𝑟>0. (3.20)
Define 𝑟_𝑘=𝑒^𝜏𝑘forall𝑘. It follows from (3.18)-(3.19)-(3.20) that lim_{𝑘→−∞}𝑟_𝑘=0 and lim_𝑘→∞𝑟_𝑘=∞, and 𝑤₂𝑟_𝑘=ℎ₂𝜏_𝑘𝑤₁𝑟_𝑘=𝑤₁𝑟_𝑘∀𝑘. (3.21)
This proves property (3.11). It remains to prove property (3.12). For this we combine (3.19) with (3.20), and the fact that 𝜏=ln(𝑟), to conclude that 0<𝐿₁𝑤₁(𝑟)<𝑤₂(𝑟)<𝐿₂𝑤₁(𝑟)∀𝑟>0. (3.22)
It follows from (3.3) and (3.22) that lim_𝑟→0+𝑤₂(𝑟)=∞,lim_𝑟→∞𝑤₂(𝑟)=0. (3.23)
This completes the proof of property (3.12).
 Part (c). Let ℎ₃(𝜏) be a member of the continuum of positive periodic solutions of (3.2) which satisfy part (v) of Theorem 3.1. The trajectory of (ℎ₃,ℎ^′₃) surrounds the constant solutions (ℎ,ℎ^′)=(0,0) and (ℎ,ℎ^′)=(±1,0) in the (ℎ,ℎ^′) plane. Thus, there exists a value 𝐿>0, and a positive increasing sequence {𝜏_𝑖}, such that −𝐿<ℎ₃(𝜏)<𝐿∀𝜏∈ℝ,lim_{𝑖→−∞}𝜏_𝑖=−∞,lim_𝑖→∞𝜏_𝑖=∞,ℎ₃𝜏_𝑖=−1if𝑖isodd,ℎ₃𝜏_𝑖=1if𝑖iseven. (3.24)(3.25)(3.26)
The solution of the 𝑤 equation (3.1) corresponding to ℎ₃ is 𝑤₃(𝑟)=ℎ₃(ln(𝑟))𝑤₁(𝑟)𝑟>0. (3.27)
Define 𝑟_𝑖=𝑒^𝜏𝑖forall𝑖. It follows from (3.24)-(3.25)-(3.27) that lim_{𝑖→−∞}𝑟_𝑖=0, lim_𝑖→∞𝑟_𝑖=∞, and 𝑤₃𝑟_𝑖=−𝑤₁𝑟_𝑖if𝑖isodd,𝑤₃𝑟_𝑖=𝑤₁𝑟_𝑖if𝑖iseven. (3.28)
This proves property (3.14). It remains to prove property (3.15). For this we combine (3.24) with (3.27), and the fact that 𝜏=ln(𝑟), to conclude that −𝐿𝑤₁(𝑟)<𝑤₃(𝑟)<𝐿𝑤₁(𝑟)∀𝑟>0. (3.29)
It follows from (3.3), (3.28), and (3.29) that limsup_𝑟→0+𝑤₃(𝑟)=∞,liminf_𝑟→0+𝑤₃(𝑟)=−∞,lim_𝑟→∞𝑤₃(𝑟)=0. (3.30)
This completes the proof of property (3.15) and of Theorem 3.2.

4. The Supercritical Case: 𝑝>(𝑁+2)/(𝑁−2)

In this section we investigate the behavior of solutions of (1.4) and (1.8) when 𝑁>2 and 𝑝>(𝑁+2)/(𝑁−2). The remainder of the section consists of the following.(I)Theorem 4.1 classifies the behavior of solutions of (1.8). Again, we focus on solutions whose trajectories in the (ℎ,ℎ^′) phase plane form the stable and unstable manifolds of solutions associated with the constant solution (0,0). Part (ii) of Theorem 4.1 gives a detailed proof that solutions on the stable manifold form an outgoing spiral in the (ℎ,ℎ^′) phase plane as 𝜏 decreases from 𝜏=∞. The proof is sufficiently general as to include Galaktionov’s numerical observation of large amplitude oscillations [10]. For these spiraling solutions it remains a challenging open problem to prove their asymptotic behavior at the left endpoint 𝜏_min of their interval of existence (see Open Problem I after the statement of Theorem 4.1).(II)Theorem 4.5 shows how to combine the results of Theorem 4.1 with the formula 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟) to obtain a continuum of positive nonsingular solutions of (3.1). In addition, we prove the existence of a continuum of new sign changing, “super singular” solutions which, to our knowledge, have not previously been reported. For these sign changing solutions it remains a challenging open problem to prove their asymptotic behavior at the left endpoint 𝑟_min of their interval of existence (see Open Problem II after the statement of Theorem 4.5).(III)Figure 4 illustrates the behavior of solutions when (𝑁,𝑝)=(3,7).

Theorem 4.1. Let 𝑁>2 and 𝑝>(𝑁+2)/(𝑁−2).(i)There is a one-dimensional unstable manifold of solutions of (1.8) leading from (0,0) into the (ℎ,ℎ^′) phase plane. One component, 𝐴₁, points into the positive quadrant, and its negative counterpart, 𝐴₂, points into the negative quadrant (Figure 4, upper left). If (ℎ(0),ℎ^′(0))∈𝐴₁, then ℎ(𝜏)>0forall𝜏∈ℝ, and there is a constant 𝑐>0 such that ℎ(𝜏)∼𝑐exp2𝜏𝑝−1as𝜏→−∞,lim_𝜏→∞ℎ(𝜏),ℎ^′(𝜏)=(1,0). (4.1)(ii)There is a one-dimensional stable manifold of solutions leading to (0,0) in the (ℎ,ℎ^′) phase plane. One component, 𝐵₁, points into the ℎ>0, ℎ^′<0 quadrant of the phase plane, and its negative counterpart, 𝐵₂, points into the ℎ<0, ℎ^′>0 quadrant (Figure 4, upper left). Additionally, if ℎ(0)>0 is sufficiently small and (ℎ(0),ℎ^′(0))∈𝐵₁, then there is a 𝐷>0 such that ℎ(𝜏)>0∀𝜏≥0,ℎ(𝜏)∼𝐷exp−𝑁−2𝑁𝑝−1𝑝−𝜏𝑁−2as𝜏→∞. (4.2)
Let ℎ=((𝑝+1)/2)^1/(𝑝−1). There is a negative decreasing sequence {𝜏_𝑁} such that ℎ^′𝜏_𝑁=0∀𝑁≥1,ℎ𝜏_𝑁>ℎif𝑁isodd,ℎ𝜏_𝑁<−ℎif𝑁iseven. (4.3)
Moreover, |ℎ(𝜏_𝑁)| increases as 𝑁 increases.

Remarks. The proof of part (i) of Theorem 4.1 uses straightforward phase plane type arguments. The proof of (ii) is admittedly more technical. Our numerical experiments (Figure 4, lower left) indicate that the amplitudes of the oscillations of the solutions described in part (ii) grow without bound as 𝜏 decreases. It remains a challenging open problem to determine whether these solutions exist on the entire interval (−∞,∞), or if they exist only on a semi-infinite interval of the form (𝜏_min,∞). These fundamental theoretical questions are summarized in the following.
Open Problem I (Super Singular Solutions). Let {𝜏_𝑁} denote the decreasing sequence described in part (ii) of Theorem 4.1. Prove whether 𝜏_min=−∞, or 𝜏_min>−∞. Second, prove whether limsup_𝑁→∞|ℎ(𝜏_𝑁)| is finite or infinite. Our numerical study suggests that limsup_𝑁→∞|ℎ(𝜏_𝑁)|=∞.

Proof of Theorem 4.1. Part (i). First, note that properties of solutions on the component 𝐴₁ of the unstable manifold leading from (0,0) into the positive quadrant of the (ℎ,ℎ^′) phase plane are the same as those seen in Lemma 2.7. From these properties it again follows that if (ℎ(0),ℎ^′(0))∈𝐴₁, and ℎ(0)>0 is sufficiently small, then ℎ(𝜏)>0forall𝜏≤0, and ℎ(𝜏)∼𝑐exp𝜆₁𝜏,ℎ^′(𝜏)ℎ(𝜏)→𝜆₁as𝜏→−∞ (4.4) for some 𝑐>0, where 𝜆₁=2/(𝑝−1). This proves the first part of (4.1). To complete the proof of (4.1) recall that the functional 𝑆 and the region 𝑈 defined in the proof of Lemma 2.6 are 1𝑆=2ℎ^′2+2(𝑁−2)𝑝−1²𝑁𝑝−|ℎ|𝑁−2^𝑝+1−ℎ𝑝+12²,𝑈=ℎ,ℎ^′∣ℎ≥0,𝑆≤0. (4.5)(4.6)
Again, let 𝑈⁰ denote the interior of 𝑈, and note that 𝑈⁰ contains one constant solution, (ℎ,ℎ^′)=(1,0). A differentiation of (4.5) gives 𝑆^′=−𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′2≤0. (4.7)
Thus, if (ℎ(0),ℎ^′(0))∈𝐴₁, and ℎ(0)>0 is sufficiently small, we conclude from (4.4)–(4.7) (Figure 4, upper right) that ℎ(𝜏) exists for all 𝜏∈ℝ, and 𝑆(−∞)=0,𝑆^′(𝜏)<0∀𝜏≤0,𝑆(𝜏)<0∀𝜏∈ℝ. (4.8)
Therefore, ℎ(𝜏),ℎ^′(𝜏)∈𝑈⁰∀𝜏∈ℝ. (4.9)
From (4.7) we conclude that (1.8) has no periodic solutions. Also, a linearization of (1.8) around the constant solution (ℎ,ℎ^′)=(1,0) shows that (1,0) is an asymptotically stable equilibrium point in the (ℎ,ℎ^′) phase plane. Thus, if ℎ(𝜏) is a solution of (1.8) with initial condition (ℎ(0),ℎ^′(0))∈𝐴₁, it follows from (4.7)–(4.9), and standard phase plane arguments, that ℎ(𝜏)>0∀𝜏∈ℝ,lim_𝜏→∞ℎ(𝜏),ℎ^′(𝜏)=(1,0). (4.10)
Part (ii). It follows from (2.5) that there is a one-dimensional stable manifold of solutions leading to (0,0) in the (ℎ,ℎ^′) phase plane. As in Theorem 2.9, one component, 𝐵₁, points into the quadrant ℎ>0, ℎ^′<0, and along 𝐵₁ solutions satisfy lim_𝜏→∞ℎ(𝜏),ℎ^′(𝜏)=(0,0),lim_𝜏→∞ℎ^′(𝜏)ℎ(𝜏)=𝜆₂<0, (4.11) where 𝜆₂=−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2)). Thus, if ℎ₂(𝜏) is a solution of (1.8) with ℎ₂(0)>0 sufficiently small and (ℎ₂(0),ℎ^′₂(0))∈𝐵₁, there is a 𝐷>0 such that ℎ₂(𝜏)>0,ℎ^′₂(𝜏)<0∀𝜏≥0,ℎ₂(𝜏)∼𝐷exp𝜆₂𝜏as𝜏→∞. (4.12)(4.13)
Along this solution 𝑆 (Figure 4, upper right) satisfies 𝑆(∞)=0,𝑆^′(𝜏)<0,𝑆(𝜏)>0∀𝜏≥0. (4.14)
To complete the proof of (ii) let ℎ=((𝑝+1)/2)^1/(𝑝−1) be the unique positive ℎ value where 𝑆=ℎ^′=0. We need to prove that (ℎ₂(𝜏),ℎ^′₂(𝜏)) rotates counterclockwise around (1,0) and that (ℎ₂(𝜏),ℎ^′₂(𝜏)) generates an outwardly growing spiral as 𝜏 decreases. For this we show that there is a decreasing sequence of negative values {𝜏_𝑁} such that ℎ^′₂𝜏_𝑁=0,ℎ₂𝜏_𝑁+1>ℎ₂𝜏_𝑁∀𝑁≥1, (4.15) and that ℎ₂𝜏_𝑁>ℎif𝑁isodd,ℎ₂𝜏_𝑁<−ℎif𝑁iseven. (4.16)
The proof of (4.15)-(4.16) is in two steps. First, we prove three technical results, Lemmas 4.2–4.4. Secondly, we use these lemmas to follow (ℎ₂(𝜏),ℎ^′₂(𝜏)) as 𝜏 decreases. The proofs of Lemmas 4.2 and 4.4 are straightforward. The proof of Lemma 4.3 is admittedly technical.

Lemma 4.2. Let 𝜏∈(−∞,0). (a) If a solution of (1.8) satisfies 0≤ℎ(𝜏)<ℎ=𝑝+12^1/(𝑝−1),ℎ^′(𝜏)<0,𝑆(𝜏)>0, (4.17) then there is a 𝜏<𝜏 such that ℎ^′(𝜏)<0∀𝜏∈[𝜏,𝜏],ℎ(𝜏)=ℎ. (4.18)
(b) If a solution of (1.8) satisfies −ℎ<ℎ(𝜏)≤0,ℎ^′(𝜏)>0,𝑆(𝜏)>0, (4.19) then there is a 𝜏<𝜏 such that ℎ^′(𝜏)>0∀𝜏∈[𝜏,𝜏],ℎ(𝜏)=−ℎ. (4.20)

Proof. (a) Let 𝛿=2𝑆(𝜏)>0. Since (4.7) implies that 𝑆^′(𝜏)≤0forall𝜏, then 𝑆 increases monotonically as 𝜏 decreases. It follows from the definition of 𝑆 in (4.5) that ℎ^′≤−√𝛿 for 𝜏≤𝜏 as long as 0≤ℎ(𝜏)<ℎ. An integration of ℎ^′≤−√𝛿 shows that ℎ(𝜏)=ℎ at some first 𝜏∈[𝜏+(ℎ(𝜏)−ℎ)/√𝛿,𝜏). This proves (4.18). The proof of part (b) is the same and is omitted.

Lemma 4.3. Let 𝜏∈(−∞,0). (a) If a solution of (1.8) satisfies ℎ(𝜏)=ℎ=𝑝+12^1/(𝑝−1),ℎ^′(𝜏)<0, (4.21) then there is a 𝜏^∗<𝜏 such that ℎ^′(𝜏)<0∀𝜏∈𝜏^∗,𝜏,ℎ^′𝜏^∗=0,ℎ𝜏^∗>ℎ. (4.22)
(b) If a solution of (1.8) satisfies ℎ(𝜏)=−ℎ,ℎ^′(𝜏)>0, (4.23) then there is a 𝜏^∗<𝜏 such that ℎ^′(𝜏)>0∀𝜏∈𝜏^∗,𝜏,ℎ^′𝜏^∗=0,ℎ𝜏^∗<−ℎ. (4.24)

Proof. (a) The first step is to assume, for contradiction, that 𝜏^∗ does not exist and that ℎ^′(𝜏)<0∀𝜏∈(𝑇,𝜏), (4.25) where (𝑇,𝜏) denotes the maximal left interval of existence of the solution. We claim that 𝑇=−∞. Suppose that 𝑇>−∞. Then standard theory implies that ℎ^′ must be unbounded on (𝑇,𝜏). To show that this cannot happen we use the function 𝐻=(ℎ^′)². Then 𝐻 satisfies 𝐻^′−2𝜆₁+𝜆₂𝐻=𝜆₁𝜆₂|ℎ|^𝑝−1−1ℎℎ^′>0∀𝜏∈(𝑇,𝜏), (4.26) since 𝜆₁𝜆₂<0, ℎ(𝜏)>1 and ℎ^′<0forall𝜏∈(𝑇,𝜏). Integrating (4.26) gives 0<𝐻(𝜏)≤𝐻(𝜏)exp−2𝜆₁+𝜆₂(𝜏−𝜏)∀𝜏∈(𝑇,𝜏). (4.27)
From (4.27) it follows that ℎ^′ is bounded on (𝑇,𝜏), contradicting the previous conclusion that ℎ^′ is unbounded on (𝑇,𝜏). We conclude that 𝑇=−∞. Therefore, (4.25) becomes ℎ^′(𝜏)<0∀𝜏∈(−∞,𝜏). (4.28)
Next, to obtain a contradiction of (4.28) we need to prove two technical properties. The first is that ℎ^′′(𝜏)≥0,ℎ^′(𝜏)≤ℎ^′(𝜏)<0∀𝜏≤𝜏,lim_{𝜏→−∞}ℎ(𝜏)=∞. (4.29)
To prove the first part of (4.29), differentiate (1.8) and get ℎ^′′′+𝑁−2𝑝−1𝑝−𝑁+2ℎ𝑁−2^′′=2𝑁−2𝑝(𝑁−2)𝑝−1²𝑝|ℎ|^𝑝−1−1ℎ^′>0 (4.30) for 𝜏≤𝜏 as long as ℎ(𝜏)>ℎ>1. If there is an 𝑎≤𝜏, where ℎ^′′(𝑎)<0, then an integration of (4.30) gives ℎ^′′(𝜏)≤ℎ^′′(𝑎)exp𝑁−2𝑝−1𝑝−𝑁+2()𝑁−2𝑎−𝜏<0∀𝜏≤𝑎. (4.31)
An integration of (4.31) gives ℎ^′(𝜏)>0 for 𝜏≪𝑎, contradicting (4.28). We conclude that ℎ^′′(𝜏)≥0forall𝜏≤𝜏, and the first part of (4.29) is proved. In turn, this implies that ℎ^′(𝜏)≤ℎ^′(𝜏)<0∀𝜏≤𝜏,lim_{𝜏→−∞}ℎ(𝜏)=∞, (4.32) and the proof of (4.29) is complete. The second property we need is 𝜌(𝜏)=ℎ^′(𝜏)ℎ(𝜏)≥−𝑚=min𝜆₂,𝜌(𝜏)∀𝜏≤𝜏, (4.33) where 𝜌(𝜏) satisfies the ODE 𝜌^′+𝜌−𝜆₁𝜌−𝜆₂=𝜆₁𝜆₂|ℎ|^𝑝−1. (4.34)
Property (4.33) follows immediately from (4.34), since (4.34) implies that 𝜌^′<0 whenever 𝜌≤𝜆₂.
We now show how to use properties (4.29) and (4.33) to obtain a contradiction of (4.28). First, an integration of (4.34) gives 𝜌(𝜏)−𝜆₂𝑒^−∫_𝜏𝜏_{(𝜌−𝜆1)𝑑𝑠}^′=𝜆₁𝜆₂|ℎ|^𝑝−1𝑒^−∫_𝜏𝜏_{(𝜌−𝜆1)𝑑𝑠},∀𝜏≤𝜏. (4.35)
Integrating both sides of (4.35) from 𝜏 to 𝜏, we obtain 𝜌(𝜏)=𝜆₂+I₁(𝜏)+I₂(𝜏), (4.36) where I₁(𝜏)=𝜌(𝜏)−𝜆₂𝑒^∫_𝜏𝜏_{(𝜌−𝜆1)𝑑𝑠},I₂(𝜏)=−𝑒^∫_𝜏𝜏_{(𝜌−𝜆1)𝑑𝑠}𝜆₁𝜆_2𝜏𝜏|ℎ|^𝑝−1𝑒^−∫_𝜂𝜏_{(𝜌−𝜆1)𝑑𝑠}𝑑𝜂. (4.37)(4.38)
Our goal is to prove that lim_{𝜏→−∞}I₁=0,lim_{𝜏→−∞}I₂=∞. (4.39)
Once we prove these properties, we combine (4.39) with (4.36) and conclude that 𝜌=ℎ^′/ℎ>0 when 𝜏≪𝜏. Since ℎ(𝜏)>0, this implies that ℎ^′>0 when 𝜏≪𝜏, which contradicts (4.28).
To prove the first part of (4.39), evaluate the right side of (4.37) and get I₁(𝜏)=𝜌(𝜏)−𝜆₂ℎ(𝜏)ℎ(𝜏)𝑒^−𝜆1(𝜏_−𝜏)→0as𝜏→−∞, (4.40) since 𝜆₁>0 and 0<ℎ(𝜏)/ℎ(𝜏)≤1forall𝜏≤𝜏. To prove the second part of (4.39), recall from (4.32) that lim_{𝜏→−∞}ℎ(𝜏)=∞. Thus, since 𝜌<0 and 𝜆₁>0, we conclude that _𝜏𝜏|ℎ|^𝑝−1𝑒^−∫_𝜂𝜏_{(𝜌−𝜆1)𝑑𝑠}𝜏𝑑𝜂≥_𝜏|ℎ|^𝑝−1𝑑𝜂→∞as𝜏→−∞. (4.41)
Next, it follows from (4.33) that −𝜆₁>𝜌−𝜆₁≥−𝑚−𝜆₁. This, (4.41), and the facts that 𝜆₁𝜆₂<0 and 𝑒^∫_𝜏𝜏_{(𝜌−𝜆1)𝑑𝑠}→0 as 𝜏→−∞ allow us to apply L’Hospital’s Rule to I₂(𝜏). This gives lim_{𝜏→−∞}I₂(𝜏)=lim_{𝜏→−∞}𝜆₁𝜆₂|ℎ(𝜏)|^𝑝−1𝜌(𝜏)−𝜆₁=∞. (4.42)
This proves part (a). The proof of part (b) is the same and is omitted.

Lemma 4.4. Let 𝜏^∗∈(−∞,0). (a) If a solution of (1.8) satisfies ℎ𝜏^∗>ℎ=𝑝+12^1/(𝑝−1),ℎ^′𝜏^∗=0, (4.43) then there is a 𝜏^∗∗<𝜏^∗ such that ℎ^′(𝜏)>0∀𝜏∈𝜏^∗∗,𝜏^∗,ℎ𝜏^∗∗=0. (4.44)
(b) If a solution of (1.8) satisfies ℎ𝜏^∗<−ℎ,ℎ^′𝜏^∗=0, (4.45) then there is a 𝜏^∗∗<𝜏^∗ such that ℎ^′(𝜏)<0∀𝜏∈𝜏^∗∗,𝜏^∗,ℎ𝜏^∗∗=0. (4.46)

Proof. (a) Since ℎ>1, it follows from (1.8) that ℎ^′′(𝜏^∗)<0. Thus, there is an 𝑎<𝜏^∗ such that ℎ^′(𝜏)>0∀𝜏∈𝑎,𝜏^∗,ℎ<ℎ(𝑎)<ℎ𝜏^∗. (4.47)
Since ℎ<ℎ(𝑎), a calculation shows that |ℎ|^𝑝+1−ℎ𝑝+12²≤|ℎ|^𝑝+1(𝑎)−ℎ𝑝+1²(𝑎)2whenℎ∈[0,ℎ(𝑎)]. (4.48)
Recall from (4.7) that 𝑆^′(𝜏)≤0forall𝜏. Thus, 𝑆(𝜏)≥𝑆(𝑎)forall𝜏≤𝑎, and (4.48) implies that ℎ^′(𝜏)²2−ℎ^′(𝑎)²2≥−𝜆₁𝜆₂|ℎ|^𝑝+1(𝑎)−ℎ𝑝+1²(𝑎)2−|ℎ|^𝑝+1(𝜏)+ℎ𝑝+1²(𝜏)2≥0 (4.49) for 𝜏≤𝑎 as long as 0≤ℎ(𝜏)≤ℎ(𝑎). Integrating ℎ^′(𝜏)≥ℎ^′(𝑎)>0 shows that there is a 𝜏^∗∗∈[𝑎−ℎ^′(𝑎)/ℎ(𝑎),𝑎) such that ℎ^′(𝜏)≥ℎ^′(𝑎)>0∀𝜏∈𝜏^∗∗,𝑎,ℎ𝜏^∗∗=0. (4.50)
This proves part (a). The proof of part (b) is the same and is omitted.
We now return to the proof of Theorem 4.1. It remains to prove that (ℎ₂(𝜏),ℎ^′₂(𝜏)) rotates counterclockwise around (0,0) in the (ℎ,ℎ^′) plane as 𝜏 decreases from 𝜏=0. To accomplish this we use Lemmas 4.2–4.4 to show that (ℎ₂(𝜏),ℎ^′₂(𝜏)) passes infinitely often through the sets 𝑅₁=ℎ,ℎ^′∣0≤ℎ<ℎ,ℎ^′𝑅<0,𝑆>0₂=ℎ,ℎ^′∣ℎ=ℎ,ℎ^′𝑅<0₃=ℎ,ℎ^′∣ℎ>ℎ,ℎ^′𝑅=0₄=ℎ,ℎ^′∣ℎ=0,ℎ^′𝑅>0₅=ℎ,ℎ^′∣ℎ=−ℎ,ℎ^′𝑅>0₆=ℎ,ℎ^′∣ℎ<−ℎ,ℎ^′𝑅=0₇=ℎ,ℎ^′∣ℎ=0,ℎ^′<0. (4.51)
Recall that (ℎ₂(0),ℎ^′₂(0))∈𝐵₁ and that (4.12) is satisfied, consequently (ℎ(0),ℎ^′(0))∈𝑅₁. Lemma 4.2 implies that there is a first 𝜏<0 such that (ℎ(𝜏),ℎ^′(𝜏))∈𝑅₂, that is, ℎ(𝜏)=ℎ and ℎ^′(𝜏)<0. This and Lemma 4.3 imply that there is a 𝜏₁<𝜏 such that ℎ^′₂(𝜏)<0forall𝜏∈(𝜏₁,𝜏], ℎ^′₂(𝜏₁)=0, and ℎ₂(𝜏₁)>ℎ. Thus, (ℎ(𝜏₁),ℎ^′(𝜏₁))∈𝑅₃. It follows from Lemma 4.4 that there is a 𝜏^∗∗<𝜏₁ such that ℎ^′₂(𝜏)>0forall𝜏∈[𝜏^∗∗,𝜏₁) and ℎ₂(𝜏^∗∗)=0, hence (ℎ₂(𝜏^∗∗),ℎ^′₂(𝜏^∗∗))∈𝑅₄. This and part (b) of Lemma 4.2 imply that there is a 𝑏<𝜏^∗∗ such that ℎ^′₂(𝜏)>0forall𝜏∈[𝑏,𝜏^∗∗] and ℎ₂(𝑏)=−ℎ, hence (ℎ(𝑏),ℎ^′(𝑏))∈𝑅₅. This and part (b) of Lemma 4.3 imply that there is a 𝜏₂<𝑏 such that ℎ^′₂(𝜏)>0forall𝜏∈(𝜏₂,𝑏], ℎ^′₂(𝜏₂)=0, and ℎ₂(𝜏₂)<−ℎ, hence (ℎ₂(𝜏₂),ℎ^′₂(𝜏₂))∈𝑅₆. This and part (b) of Lemma 4.4 imply that there is a 𝑐<𝜏₂ such that ℎ^′₂(𝜏)<0forall𝜏∈[𝑐,𝜏₂) and ℎ₂(𝑐)=0, hence (ℎ₂(𝑐),ℎ^′₂(𝑐))∈𝑅₇. We have shown how (ℎ₂(𝜏),ℎ^′₂(𝜏)) passes sequentially through 𝑅₁,…,𝑅₇ as 𝜏 decreases. Since 𝑅₇ is contained in 𝑅₁ it follows from a repetition of the steps given above, and mathematical induction, that (ℎ₂(𝜏),ℎ^′₂(𝜏)) passes sequentially through 𝑅₁,…,𝑅₇ infinitely often as 𝜏 decreases from 𝜏=0. This produces a decreasing sequence {𝜏_𝑁} where ℎ^′₂(𝜏_𝑁)=0forall𝑁≥1, ℎ₂𝜏_𝑁>ℎwhen𝑁isodd,ℎ₂𝜏_𝑁<−ℎwhen𝑁iseven. (4.52)
Finally, since 𝑆 increases as 𝜏 decreases, it follows that |ℎ(𝜏_𝑁)| increases as 𝑁 increases. This completes the proof of Theorem 4.1.

Solutions of the 𝑤 Equation
Below, in Theorem 4.5, we show how to combine part (ii) of Theorem 4.1 with the formula 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟), (4.53) to prove the existence and asymptotic behavior of families of nonsingular and singular solutions of the 𝑤 equation (1.4). In particular, in part (b) of Theorem 4.5 we show how a new family of “super singular” solutions is generated. Open Problem II stated after Theorem 4.5 describes important, and as yet unproven, properties of this continuum of singular solutions.

Theorem 4.5. Let 𝑁>2 and 𝑝>(𝑁+2)/(𝑁−2).
(a) A Continuum of Positive Nonsingular Solutions. Let ℎ₀(𝜏) denote a solution of (1.8) which satisfies (ℎ₀(0),ℎ^′₀(0))∈𝐴₁ in part (i) of Theorem 4.1. The corresponding solution 𝑤₀(𝑟)=ℎ₀(ln(𝑟))𝑤₁(𝑟) of (1.4) is strictly positive and satisfies 𝑤₀(1)=ℎ₀(0)𝑤₁(1),𝑤^′₀(1)=ℎ^′₀(0)𝑤₁(1)+ℎ₀(0)𝑤^′₁(1). (4.54)
Furthermore, its interval of existence is (0,∞), 0<𝑤₀(0)<∞,𝑤^′₀(0)=0,lim_𝑟→∞𝑤₀(𝑟)𝑤₁(𝑟)=1. (4.55)
(b) A Continuum of Sign Changing Singular Solutions. Let ℎ₂(𝜏) be a member of the family of “spiraling” solutions of (3.2) which satisfy (ℎ₂(0),ℎ^′₂(0))∈𝐵₁ in part (ii) of Theorem 4.1, and let {𝜏_𝑁} denote the decreasing sequence of 𝜏 values which satisfy property (4.3). The corresponding solution 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟) of (3.1) satisfies 𝑤₂(1)=ℎ₂(0)𝑤₁(1),𝑤^′₂(1)=ℎ^′₂(0)𝑤₁(1)+ℎ₂(0)𝑤^′₁(1). (4.56)
Its interval of existence is of the form (𝑟_min,∞). As 𝑟→∞, 𝑤₂(𝑟)→0 faster than 𝑤₁(𝑟). That is, there exists 𝐷>0 such that. 𝑤₂(𝑟)𝑤₁(𝑟)∼𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−(𝑁/(𝑁−2)))}→0as𝑟→∞. (4.57)
As 𝑟 decreases from ∞, 𝑤₂(𝑟) changes sign infinitely often. That is, along the decreasing sequence {𝑟_𝑁=exp(𝜏_𝑁)}, one has lim_{𝑁→−∞}𝑟_𝑁=𝑟_min, 𝑤₂𝑟_𝑁>𝑤₁𝑟_𝑁if𝑁isodd,𝑤₂𝑟_𝑁<−𝑤₁𝑟_𝑁if𝑁iseven. (4.58)
Open Problem II (Super Singular Solutions). Let {𝑟_𝑁} denote the decreasing sequence in part (b), which satisfies lim_𝑁→∞𝑟_𝑁=𝑟_min. Prove whether 𝑟_min=0 or 𝑟_min>0. Secondly, prove whether limsup_𝑁→∞|𝑤₂(𝑟_𝑁)| is finite or infinite. Our numerical experiments suggest that limsup_𝑁→∞|𝑤₂(𝑟_𝑁)|=∞. As in Open Problem I, our analytical methods have not allowed us to resolve these fundamental theoretical issues.

Proof of Theorem 4.5. Part (a). Let ℎ₀(𝜏) be a solution of (1.8) satisfying (ℎ₀(0),ℎ^′₀(0))∈𝐴₁ in part (i) of Theorem 4.1. The corresponding solution of (1.4) is 𝑤₀(𝑟)=ℎ₀(ln(𝑟))𝑤₁(𝑟), (4.59) where we recall that 𝑤₁(𝑟)=2(𝑁−2)𝑝−1−4𝑝−1^{21/(𝑝−1)}𝑟^{−2/(𝑝−1)},𝑟>0. (4.60)
Because 𝑟=exp(𝜏), the initial point 𝜏=0 for ℎ₀(𝜏) translates to 𝑟=1 for 𝑤₀(𝑟). This, (4.59) and (4.60) give (4.54). Next, it follows from property (4.1) and the fact that 𝜏=ln(𝑟) that there exists 𝑐>0 such that ℎ₀(ln(𝑟))∼𝑐𝑟^2/(𝑝−1)as𝑟→0⁺,lim_𝑟→∞ℎ₀(ln(𝑟))=1. (4.61)
We conclude from (4.59), (4.60), and the first part of (4.61) that 𝑤₀(𝑟)2()→𝑐𝑁−2𝑝−1−4𝑝−1^{21/(𝑝−1)}>0as𝑟→0⁺. (4.62)
It then follows from standard theory that 𝑤^′₀(0)=0. This agrees with a result of Haraux and Weissler (see Theorem  4 in [2]). Finally, we conclude from (4.59), (4.60), and the second part of (4.61) that lim_𝑟→∞𝑤₀(𝑟)𝑤₁(𝑟)=lim_𝑟→∞ℎ₀(ln(𝑟))=1. (4.63)
This completes the proof of part (a).
Part (b). Let ℎ₂(𝜏) be a solution of (1.8) satisfying (ℎ₂(0),ℎ^′₂(0))∈𝐵₁ in part (ii) of Theorem 4.1. The corresponding solution of (1.4) is 𝑤₂(𝑟)=ℎ₂(ln(𝑟))𝑤₁(𝑟). (4.64)
It follows from (4.2) that there exists 𝐷>0 such that ℎ₂(ln(𝑟))∼𝐷𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2))}as𝑟→∞. (4.65)
Combining (4.60), (4.64), and (4.65) gives 𝑤₂(𝑟)𝑤₁(𝑟)∼𝐷𝑟^{−((𝑁−2)/(𝑝−1))(𝑝−𝑁/(𝑁−2))}as𝑟→∞. (4.66)
This proves the first part of property (4.57). Next, let {𝜏_𝑁} denote a sequence of 𝜏 values satisfying property (4.3) in Theorem 4.1. Then {𝜏_𝑁} decreases as 𝑁 increases, with lim_𝑁→∞𝜏_𝑁=𝜏_min, and ℎ₂𝜏_𝑁>ℎif𝑁isodd,ℎ₂𝜏_𝑁<−ℎif𝑁iseven. (4.67)
Define 𝑟_𝑁=exp(𝜏_𝑁)forall𝑁≥1. Setting 𝜏_𝑁=ln(𝑟_𝑁) in (4.67) gives ℎ₂ln𝑟_𝑁>ℎif𝑁isodd,ℎ₂ln𝑟_𝑁<−ℎif𝑁iseven. (4.68)
Finally, we combine with (4.64) and (4.68) and obtain 𝑤₂𝑟_𝑁>ℎ𝑤₁𝑟_𝑁if𝑁isodd,𝑤₂𝑟_𝑁<−ℎ𝑤₁𝑟_𝑁if𝑁iseven. (4.69)
Since ℎ>1, this completes the proof of (4.58) and of Theorem 4.5.

5. Conclusions

In this paper we have developed an analytical method to classify the behavior of radially symmetric, time-independent solutions of the nonlinear heat equation (1.2). These solutions satisfy the ODE (1.4). We have studied solutions which remain strictly positive on their entire intervals of existence, and also solutions which change sign. There have been few analyses in the literature of sign changing solutions. Our analytical method follows a three-step approach:

Step 1. Transform the nonautonomous 𝑤 equation (1.4) into the autonomous ℎ equation (1.8).

Step 2. Analyze (1.8) using phase plane methods.

Step 3. Use the inverse transformation 𝑤(𝑟)=ℎ(ln(𝑟))𝑤₁(𝑟), (5.1) to translate results for (1.8) into new results for the 𝑤 equation (1.4).

Our Advance
This approach has allowed us to prove the existence and asymptotic behavior of several new families of solutions of (1.4). In particular, we mention two important classes of solutions which, to our knowledge, have not previously been reported.(I)When 𝑁>2 and 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2), we proved (see part (ii) of Theorem 2.10) the existence and asymptotic behavior of a continuum of positive, singular solutions which “interlace” with the known singular solutions 𝑤₁(𝑟).(II)When 𝑁>2 and 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2), we proved (see part (ii) of Theorem 4.1) the existence of sign changing solutions of the ℎ equation which form outward spirals in the (ℎ,ℎ^′) phase plane. These solutions transform, by means of (5.1), into “super singular” sign changing solutions of the 𝑤 equation (1.4). Open Problems I and II (see Section 4) summarize important issues for these solutions which have not yet been resolved.
Below, we describe challenging problems for further research.
Open Problem III. When 𝑁/(𝑁−2)<𝑝<(𝑁+2)/(𝑁−2), do the new positive singular solutions, which interlace with 𝑤₁(𝑟), play an important role similar to that of 𝑤₁(𝑟) (e.g., see [5]) in proving the large time behavior of solutions of the time-dependent problem (1.2)?
Open Problem IV. Equation (1.1) is a canonical model for the general equation Δ𝑢+𝑓(𝑢)=0, (5.2) where 𝑓(𝑢)>0 is positive and superlinear [1–5]. A natural extension of our investigation is to use our new singular solutions of (1.4) as a guide in analyzing (5.2) for the existence of new classes of solutions.
Open Problem V. Gazzola and Grunau [13] investigate the behavior of solutions of the biharmonic equation −Δ²𝑢+|𝑢|^𝑝−1𝑢=0. (5.3)
This equation has the singular solution 𝑤₁=𝐴𝑟^{−4/(𝑝−1)}, 𝑟>0. It is hoped that our approach can be used to look for new classes of solutions of (5.3).
Open Problem VI. Develop a comparison technique which allows one to use the new singular solution 𝑤₂(𝑟) to establish blowup of solutions of the full time-dependent PDE.