Classification of radial solutions to equations related to Caffarelli–Kohn–Nirenberg inequalities

Abstract

This article studies the qualitative and quantitative properties of radial solutions to an elliptic equation related to the Euler–Lagrange equations for certain sharp Caffarelli–Kohn–Nirenberg inequalities. Namely, we examine the equation

$$\begin{aligned} -\hbox {div} (|x|^{a} D u ) = |x|^{b}u^{p}, ~ u > 0,\, \text{ in } \mathbb {R}^N, \end{aligned}$$

where $p > 1$, $N \ge 2$, $N-2+a \ge 0$ and $b > - N$. The main results establish the properties of radially symmetric solutions including existence, uniqueness, and classification results as well as results on the asymptotic and intersecting behaviour of such solutions.

1 Introduction

We consider the following nonlinear elliptic equation and its positive solutions,

$$\begin{aligned} \hbox {div}( |x|^{a} D u) + |x|^{b}u^p = 0, ~ u > 0, \, \text { in } \mathbb {R}^N, \end{aligned}$$

(1.1)

where throughout the paper $N \ge 2$, $p > 1$, $b > -N$ and $N- 2 + a \ge 0$. Thus, $a, b > -N$, which ensure the coefficients are locally integrable. In certain cases, Eq. (1.1) is the Euler–Lagrange equation for the following Caffarelli–Kohn–Nirenberg inequalities established in [5] (see [6, 8, 36] for further background discussion).

Theorem A

Let $p \ge 1$$N \ge 3$ and $b, a - 2 > -N$. There exists a positive constant $\mathcal {N} = \mathcal {N}(N,a, b)$, depending only on N, a and b, such that for each $u \in C_{c}^{1}(\mathbb {R}^N)$,

$$\begin{aligned} \mathcal {N} \Big ( \int _{\mathbb {R}^N} {|x|^{b}|u|^{p+1} } \,{\text {d}}x \Big )^{2/(p+1)} \le \int _{\mathbb {R}^N} |x|^{a} |Du|^2 \,{\text {d}}x, \end{aligned}$$

(1.2)

where

$$\begin{aligned} a - 2 \le 2b/(p+1) \le a ~\quad \text{ and } \quad ~ \frac{N + b}{p+1} + 1 = \frac{N+a}{2}. \end{aligned}$$

This family of estimates is also called the Hardy–Sobolev inequalities, since the endpoint cases recover the classical Sobolev and Hardy inequalities. Likewise, Eq. (1.1) becomes the so-called Lane–Emden equation if $a,b=0$ and the Hardy–Hénon equation if $a = 0$. In addition to being connected with sharp Sobolev and Caffarelli–Kohn–Nirenberg inequalities, the Lane–Emden and Hardy–Hénon equations arise in numerous fundamental problems, e.g. in the study of astrophysical models [23], the Yamabe problem from conformal geometry [1, 26, 35, 37], a priori bounds and blow-up analysis for general elliptic problems [19, 30, 31], and the properties of solutions to time-dependent problems [2, 20, 32, 33, 40]. As the literature devoted to the two equations alone is quite large, listing all relevant references is not feasible though some notable references, in addition to the papers mentioned shortly below, are [10, 13, 15, 22, 27, 28, 34, 38, 39].

The aim of this paper is to complement previous existence and non-existence results for problem (1.1) by establishing a detailed asymptotic analysis of its radially symmetric solutions. In addition to the aforementioned applications, our motivation for obtaining such results for the elliptic problem is motivated by their importance in understanding the finite-time blow-up of solutions and the stability of positive steady states to closely related parabolic equations [20, 34, 40].

We now introduce some basic terminology used in this paper. We let $\varOmega$ denote any open subset of $\mathbb {R}^N$ containing the origin. For $q \ge 1$, the weighted Lebesgue space $L^{q}_{b}(\varOmega )$ denotes the collection of functions f such that $|x|^{b/q}f \in L^{q}(\varOmega )$ with norm $\Vert f\Vert _{L^{q}_{b} (\varOmega ) } = (\int _{\varOmega } |x|^b |f|^q \,{\text {d}}x)^{1/q}$. The space $L^{q}_{b, loc}(\varOmega )$ denotes the collection of functions f that are locally in $L^{q}_{b}(\varOmega )$, i.e. $f \in L^{q}_{b}(K)$ for every compact subset K of $\varOmega$. We denote by $H^{1,a}(\varOmega )$ the function space given by the completion of $C_{c}^{\infty }(\varOmega )$ with respect to the norm

$$\begin{aligned} \Vert f\Vert ^2 = \Vert f \Vert _{L^{2}_{b}(\varOmega )} + \Vert Df \Vert _{L^{2}_{b}(\varOmega )}. \end{aligned}$$

We let $H^{1,a}_{loc}(\varOmega )$ denote the space of functions locally in $H^{1,a}(\varOmega )$ in the same sense as with the weighted Lebesgue spaces. By a standard variational formulation [6, 10], the optimizers associated with the sharp constant in (1.2) naturally belong to such weighted function spaces and satisfy (1.1) in some weak sense—a term we define more precisely shortly as follows. Generally speaking, we focus mainly on locally bounded (radial) solutions satisfying Eq. (1.1) in the following distributional sense.

Definition

We say a positive function u is a weak solution of problem (1.1) if $u \in H^{1,a}_{loc}(\mathbb {R}^N) \cap L^{\infty }_{loc}(\mathbb {R}^N)$ and

$$\begin{aligned} \int _{\mathbb {R}^N} Du \cdot D \varphi |x|^a \, {\text {d}}x = \int _{\mathbb {R}^N} \varphi |x|^{b} u^{p} \,{\text {d}}x ~ \forall \quad \text{ non-negative } \varphi \in C^{\infty }_{c}(\mathbb {R}^N). \end{aligned}$$

(1.3)

Owing to elementary elliptic regularity theory, weak solutions are at least of the class $C^{2}(\mathbb {R}^N \backslash \{0\} )$ [9, 11]. Due to this, and unless we specify otherwise, we assume solutions are suitably regular in the following sense.

Definition

We say a positive function u is a regular solution of problem (1.1) if u belongs to $C^{2}(\mathbb {R}^N\backslash \{0\} ) \cap C(\mathbb {R}^N)$ and satisfies the equation pointwise everywhere in $\mathbb {R}^N \backslash \{0\}$. Furthermore, we say a regular solution u of (1.1) is stable if

$$\begin{aligned} \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x \ge p\int _{\mathbb {R}^N} |x|^b |u|^{p-1} \varphi ^2 \,{\text {d}}x \text{ for } \text{ all } \varphi \in C^{1}_{c}(\mathbb {R}^N). \end{aligned}$$

(1.4)

As with the model equations, several distinct critical exponents arise when studying the existence of the different notions of solution to problem (1.1). Not surprisingly, we shall see that these critical exponents also appear when we characterize the quantitative and qualitative properties of the radial solutions. The first set of exponents are the familiar Serrin-type exponent $p_\mathrm{se}(a,b)$ and Sobolev-type critical exponent $p_{S}(a,b)$, which are defined by

$$\begin{aligned} p_\mathrm{se}(a,b) = \left\{ \begin{array}{l@{\quad }l} +\infty , &{} \text{ if } \,\, N - 2 + a = 0, \\ \dfrac{N + b}{N - 2 + a}, &{} \text{ if } \,\, N - 2 + a > 0, \end{array} \right. \end{aligned}$$

(1.5)

and

$$\begin{aligned} p_{S}(a,b) = \left\{ \begin{array}{l@{\quad }l} +\infty , &{} \text{ if } \,\, N - 2 + a = 0, \\ \dfrac{N+2+2b - a}{N - 2 + a}, &{} \text{ if } \,\, N - 2 + a > 0. \end{array} \right. \end{aligned}$$

(1.6)

We define a third exponent, a Joseph–Lundgren-type critical exponent $p_{JL}(a,b)$, by

$$\begin{aligned} p_{JL}(a,b) = \left\{ \begin{array}{l@{\quad }l} +\infty , &{} \text{ if } \,\, N \le 10 + 4 b - 5a, \\ P_{+}, &{} \text{ if } \,\, N > 10 + 4b - 5a, \end{array} \right. \end{aligned}$$

(1.7)

where

$$\begin{aligned} P_{+} = \frac{ (N - 2 + a)^2 - 2(2 + b - a)(N + b) + 2\sqrt{(2+b - a)^3 (2(N+b) - (2 + b - a) )} }{ (N - 2 + a)(N - 10 - 4b + 5a)}. \end{aligned}$$

Thus, for $N \ge 2$ and $b> a - 2 > -N$, there holds

$$\begin{aligned} 1< p_\mathrm{se}(a,b)< p_{S}(a,b) < p_{JL}(a,b) \le +\infty . \end{aligned}$$

Under suitable assumptions, $p_{JL}(a,b)$ is the largest positive zero of the map

$$\begin{aligned} f(p) = p\Big ( \frac{2+b - a}{p-1} \Big ) \Big (N- 2 + a - \frac{2+b - a}{p-1} \Big ) - \Big (\frac{N-2+a}{2}\Big )^2. \end{aligned}$$

(1.8)

Remark 1

The approach (and further details) in deriving $p_{JL}(a,b)$ from (1.8) may be found in [9, 11, 12]. For $a,b = 0$, this critical exponent first appeared in [24], and its connection to finite Morse index and stable solutions along with other related results can be found in [12] (also see [11, 20]). Alternatively, $p_{JL}(a,b)$ may be derived from a stability analysis of an ODE system related to (1.1) [for details, see (2.19) in the proof of Lemma 3 below].

The terminology adopted for the three critical exponents is inspired by the similar terms coined for the Lane–Emden equation ($a,b = 0$) and the Hardy–Hénon equation ($a = 0$). Not surprisingly, just as with those two model cases, the Serrin-type, the Sobolev-type and the Joseph–Lundgren-type exponents are the dividing numbers for the sharp existence of singular, regular, and stable solutions for problem (1.1), respectively. This collection of sharp existence results is already well-established, but we summarize them in the following theorem; meanwhile, for more details on the special cases and other closely related results, the interested reader is referred to [16, 29, 31] for results pertaining to the Serrin exponent, [3, 4, 7, 14, 17, 18] for the Sobolev exponent, and [9, 12, 20, 24] for the Joseph–Lundgren exponent.

Theorem B

Let $p > 1$, $N \ge 2$ and suppose that $b> a - 2 > -N$. Then,

• (a) Problem (1.1) admits a $C^{2}$ classical solution in $\mathbb {R}^N \backslash \{0\}$ if and only if $p > p_\mathrm{se}(a,b)$.

• (b) Problem (1.1) admits a regular solution if and only if $p \ge p_{S}(a,b)$.

• (c) Problem (1.1) admits a non-trivial (non-negative or not) stable solution if and only if $p \ge p_{JL}(a,b)$.

Part (a) of Theorem B is a special case of Theorem 4.1 in [21], and we point out the result extends to unbounded solutions. The proof of part (b) may be found in [22], and part (c) was established in [11]. We may extend the assertion in part (b) to the borderline case $N - 2 + a = 0$. That is, since $p_{S}(a,b) = +\infty$ if $N - 2 + a = 0$, we show no regular solutions exist for each $1< p < \infty$ when $N - 2 + a = 0$ (see [11] and Proposition 1 below, although a slightly more general result is provided). It should be understood that part (c) in Theorem B is asserting that no non-trivial stable solutions exist for all $1< p < \infty$ for spatial dimension within the range $0 < N-2+a \le 4(2 + b - a)$. As a consequence of Theorem B and the above remarks, we assume hereafter that $N - 2 + a > 0$. The following result partly justifies the assumptions we place on the weighted coefficients of (1.1). The result can be found in [11], but we provide a proof for completeness sake.

Theorem 1

Let $N \ge 2$, $p > 1$, $b > - N$ and $a - 2> -N$. Then, (1.1) does not admit any weak solution provided that $b \le a - 2$.

This result indicates that a necessary condition for the existence of solutions to (1.1) is $b> a - 2 > -N$, which we assume hereafter unless specified otherwise.

Remark 2

Let us assume $p > p_\mathrm{se}(a,b)$, and thus, $N - 2 + a> \beta > 0$, where

$$\begin{aligned} \beta := \frac{2 + b - a}{p-1}. \end{aligned}$$

(1.9)

The function $u(x) = U_{s}(r)$, where $r = |x|$ and

$$\begin{aligned} U_{s}(r) = \Big [ \beta ( N - 2 + a - \beta ) \Big ]^{\frac{1}{p-1}} r^{-\beta }, \end{aligned}$$

(1.10)

defines an unbounded classical solution of (1.1) in punctured space $\mathbb {R}^N \backslash \{0\}$. We shall see that this singular solution plays an important role in the asymptotic properties of the radially symmetric solutions.

Our main goal in this paper is to supplement Theorem B with a comprehensive analysis of positive radial solutions of (1.1), but let us clarify our notion of a radial solution here. The following observation motivates our definition below. If we set

$$\begin{aligned} r = |x| > 0 ~ \text{ and } ~ \omega = x/|x| \in \mathbb {S}^{N-1} := \{ \omega \in \mathbb {R}^N \, : \, |\omega | = 1 \}, \end{aligned}$$

(1.11)

and write $u(x) = v(r,\omega )$, elementary calculations will reveal that $v = v(r,\omega )$ is a smooth solution of

$$\begin{aligned} r^{a}\Big ( \frac{\partial ^2 v}{\partial r^2} + \Big ( \frac{N - 1 + a}{r} \Big ) \frac{\partial v}{\partial r} + \frac{1}{r^2}\Delta _{\omega }v\Big ) + r^{b}v^p = 0, \, v > 0, \, \text{ in } (0,\infty ) \times \mathbb {S}^{N-1}, \end{aligned}$$

(1.12)

where $\Delta _{\omega }$ is the Laplace–Beltrami operator on the unit sphere $\mathbb {S}^{N-1}$.

Definition

In view of the above, when referring to a radial or a radially symmetric solution u of (1.1), we mean $u(|x|) = v_{\alpha }(r)$ where $\alpha$ is some positive real and

$$\begin{aligned} v_{\alpha } \in C^{2}((0,\infty )) \cap C( [0,\infty ) ) \end{aligned}$$

satisfies the initial-value problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{{\text {d}}^2 v_{\alpha }}{{\text {d}} r^2} + \Big ( \frac{N-1 + a}{r} \Big ) \frac{{\text {d}} v_{\alpha }}{{\text {d}} r} + r^{b-a}v_{\alpha }^{p} = 0, ~ v_{\alpha }(r)> 0, ~ r> 0, \\ v_{\alpha }(0) = \alpha > 0. \end{array}\right. \end{aligned}$$

In our remaining two theorems, Theorem B dictates we take $p \ge p_{S}(a,b)$. Now, our second theorem indicates that all radial solutions can be classified in the critical case $p = p_{S}(a,b)$. At the same time, it also shows these solutions decay near infinity with the fast rate $N-2+a$, while radial solutions in the supercritical setting decay with the slow rate $\beta$ (as $N - 2 + a > \beta$).

Theorem 2

Let $N \ge 2$, $b> a - 2 > -N$, and suppose that $p \ge p_{S}(a,b)$. Then for each $\alpha > 0$, there exists a unique radially symmetric solution of (1.1) with $u(0) = \alpha$, and this solution satisfies

$$\begin{aligned} u(x) = \alpha v_{1}(\alpha ^{1/\beta } |x|). \end{aligned}$$

Given a radially symmetric solution u of (1.1), if $p > p_{S}(a,b)$, then

$$\begin{aligned} \lim _{|x| \longrightarrow +\infty } |x|^{\beta } u(x) = \Big [ \beta (N - 2 + a - \beta ) \Big ]^{\frac{1}{p-1}}, \end{aligned}$$

and u is stable provided that $p \ge p_{JL}(a,b)$.

If $p = p_{S}(a,b)$, then

$$\begin{aligned} v_{1}(r) = \Big ( \frac{(N+b)(N - 2 + a)}{ (N+b)(N-2 + a) + r^{2 + b - a } } \Big )^{\frac{N-2 + a}{2 + b - a} } \end{aligned}$$

(1.13)

and thus

$$\begin{aligned} \lim _{|x| \longrightarrow +\infty } |x|^{N-2 + a}u(x) = \frac{1}{\alpha }\Big ( (N+b)(N - 2 + a) \Big )^{\frac{N-2 + a}{2 + b - a} }. \end{aligned}$$

More can be said regarding the behaviour of radial solutions, namely on their intersecting properties. The subsequent theorem may be regarded as a refined asymptotic analysis of radial solutions for (1.1). It confirms that any radial solution will intersect another or the singular solution $U_s$ finitely many times if $p = p_{S}(a,b)$, while it will intersect either solution infinitely many times if $p_{S}(a,b)< p < p_{JL}(a,b)$. If $p \ge p_{JL}(a,b)$, no intersection occurs; therefore, the radially symmetric solutions of (1.1) are ordered according to their initial values. Here, we let $u_{\alpha }$ denote the unique radially symmetric solution of (1.1) with initial condition $u(0) = \alpha$. We define the intersection number or the zero number between two distinct solutions $u_{\alpha _1}$ and $u_{\alpha _2}$ in $(0,\infty )$ by

$$\begin{aligned} \mathcal {Z}_{(0,\infty )}(u_{\alpha _1} - u_{\alpha _2}) := \#\{ r \in (0,\infty ) \, | \, u_{\alpha _1}(r) = u_{\alpha _2}(r) \}. \end{aligned}$$

Theorem 3

Let $N \ge 2$, $b> a - 2 > -N$ and $p \ge p_{S}(a,b)$, and suppose $\alpha _1, \alpha _2 > 0$ is a pair of distinct reals.

• (a) If $p_{S}(a,b)< p < p_{JL}(a,b)$, then

  $$\begin{aligned} \mathcal {Z}_{(0,\infty )}(u_{\alpha _1} - u_{\alpha _2}) = \infty \text{ and } \mathcal {Z}_{(0,\infty )}( u_{\alpha _1} - U_{s} ) = \infty . \end{aligned}$$

• (b) If $p \ge p_{JL}(a,b)$, then $\mathcal {Z}_{(0,\infty )}(u_{\alpha _1} - u_{\alpha _2}) = 0$, i.e. $\alpha _1 < \alpha _2$ implies

  $$\begin{aligned} u_{\alpha _1}(r) < u_{\alpha _2}(r) ~ \text{ for } \text{ all } ~ r \ge 0. \end{aligned}$$

• (c) If $p = p_{S}(a,b)$, then

  $$\begin{aligned} \mathcal {Z}_{(0,\infty )}(u_{\alpha _1} - u_{\alpha _2}) = 1 \text{ and } \mathcal {Z}_{(0,\infty )}( u_{\alpha _1} - U_{s} ) = 2. \end{aligned}$$

We emphasize that the general existence and classification results in the case $p = p_{S}(a,b)$ were obtained earlier in [6]. The classification of finite-energy entire solutions of Eq. (1.1) in the ‘symmetry region’ (which includes the optimizers) was later established in [10]. This resolved a longstanding conjecture regarding the optimal symmetry range for optimizers and, in particular, complements the symmetry breaking observed in [6, 13]. We also point out that the existence and slow decay property of radial solutions in the supercritical range $p > p_{S}(a,b)$ given in Theorem 2 were obtained in [11]; however, we provide the complete proofs here for completeness and because our approach gives a unified approach for obtaining both sets of theorems.

The remainder of this paper is organized into two main sections. The first, Sect. 2, contains the proof of Theorem 1 then establishes several preliminary results concerning the local existence and local asymptotics of radially symmetric solutions. The section further provides a key phase plane stability analysis of a closely related first-order ODE system which we invoke to establish the refined asymptotic study of radial solutions to (1.1). The proofs of Theorems 2 and 3 are then given in the remaining section, Sect. 3.

2 Preparations and the Proof of Theorem 1

2.1 Proof of Theorem 1 and Proposition 1

Proof of Theorem 1

If u is a positive weak solution of (1.1), then elliptic regularity theory ensures $u \in C^{\infty }(\mathbb {R}^N \backslash \{0\})$ and u satisfies (1.1) pointwise everywhere in any punctured ball domain $B_{R}(0)\backslash \{0\}$. Writing $u = u(r,\theta )$ in polar coordinates so that $u(r,\theta )$ satisfies (1.12), we then set

By Jensen’s inequality,

(2.1)

Then, (2.1) implies that $U(r) > 0$ satisfies the differential inequality

$$\begin{aligned} -\Big ( U''(r) + \frac{N-1 + a}{r}U'(r) \Big ) \ge r^{b - a}U(r)^p ~ \text{ for } 0< r < R. \end{aligned}$$

(2.2)

This implies that, for $0< r < R$,

$$\begin{aligned} -(r^{N-1+a}U'(r))' \ge r^{N - 1 + b}U(r)^p. \end{aligned}$$

(2.3)

Since $U > 0$, we obtain $-(r^{N-1+a}U'(r))' > 0$, and thus,

$$\begin{aligned} r^{N-1 + a}U'(r) \longrightarrow \ell ~ \text{ as } r \longrightarrow 0^{+}, \end{aligned}$$

where $-\infty < \ell \le \infty$. We infer that $\ell \le 0$. Otherwise, if $0 < \ell \le +\infty$, then we can find $\delta > 0$ and $r_{\delta } > 0$ such that

$$\begin{aligned} U'(r) \ge \delta r^{-(N+a-1)} ~ \text{ for } 0< r < r_{\delta }. \end{aligned}$$

Integrating this in $(r_0, r)$ where $0< r_0< r < r_{\delta }$ yields

$$\begin{aligned} U(r) \ge \delta \int _{r_0}^{r} t^{-(N+a-2) } \, \frac{\hbox {d}t}{t}. \end{aligned}$$

As $N - 2 + a > 0$, sending $r_0 \longrightarrow 0^{+}$ leads to an impossibility.

Let $\ell \le 0$. Therefore, we have that $U'(r) < 0$. Indeed, there exists a positive constant c and $r_1 > 0$ such that

$$\begin{aligned} U(r) \ge c \text{ for } 0< r < r_1. \end{aligned}$$

(2.4)

Choose a small $r_0 \in (0,r_1)$. By integrating (2.3) in $(r_0, r) \subset (r_0, r_1)$ and since U is monotone decreasing in this interval of integration, we obtain

$$\begin{aligned} -r^{N-1+a}U'(r) \ge {}&r_{0}^{N-1+a}U'(r_0) - r^{N-1+a}U'(r) \nonumber \\ \ge {}&U(r)^p \int _{r_0}^{r} t^{N+b} \, \frac{\hbox {d}t}{t} \text{ for } 0< r_0< r < r_1. \end{aligned}$$

(2.5)

Note that if $N + b \le 0$, then sending $r_0 \longrightarrow 0^{+}$ leads to a contradiction, since the integral on the right diverges. Now, as $N + b > 0$, we may integrate then send $r_0 \longrightarrow 0^+$ in (2.5) to get

$$\begin{aligned} -U'(r)U(r)^{-p} \ge (N+b)^{-1} r^{1 + b - a} ~ \text{ for } 0< r < r_1. \end{aligned}$$

That is, for each $\epsilon _0 > 0$,

$$\begin{aligned} (p-1)^{-1} \Big ( U(r)^{-(p-1)} \Big )' \ge (N+b)^{-1}r^{1+b - a} ~ \text{ for } 0< \epsilon _0< r < r_1. \end{aligned}$$

Integrating once again in the interval $(\epsilon _0, r)$ yields

$$\begin{aligned} U(r)^{-(p-1)} \ge \frac{p-1}{N+b} \int _{\epsilon _0}^{r} t^{2 + b - a} \,\frac{\hbox {d}t}{t}. \end{aligned}$$

Now, after sending $\epsilon _0 \longrightarrow 0^{+}$ in the last estimate, the resulting integral diverges if $b \le a - 2$. This is impossible due to (2.4).

Hence, we arrive at a contradiction, and this completes the proof of the theorem. $\square$

Proposition 1

Let $p > 1$, $N\ge 2$ and $b > a - 2$. If $N - 2 + a > 0$ and $p \le p_\mathrm{se}(a,b)$, then (1.1) admits no regular solution. If $N - 2 + a = 0$, then the same conclusion holds for each $1< p < \infty$.

Proof

We assume u is a regular solution of (1.1).

• Step 1 Integral Estimates.

Fix $R>0$ and choose a test function $\xi \in C^{\infty }_{c}(B_{2}(0))$ such that $0 \le \xi \le 1$ and $\xi \equiv 1$ in $B_{1}(0)$. Take $\varphi (x) = \xi (x/R)^{2p^{*}}$, where $p^{*} = p/(p-1)$. It is simple to check there exists a constant $C>0$ such that

$$\begin{aligned} |\hbox {div} (|x|^a D {} \varphi )| =&\, |x|^a \Big | \Delta \varphi + a |x|^{-2}(x \cdot D\varphi ) \Big | \\ \le {}\,&\frac{C|x|^a}{R^2} \xi \Big (\frac{x}{R}\Big )^{2(p^{*}-1)} \Big (\xi \Big (\frac{x}{R}\Big )\Big |\Delta \xi \Big (\frac{x}{R}\Big )\Big | + \Big |D\xi \Big (\frac{x}{R}\Big )\Big |^2 + \Big |D\xi \Big (\frac{x}{R}\Big )\Big | \Big ) \\ \le {}\,&CR^{-2} |x|^a \varphi (x)^{1/p} ~ \text{ for } ~ R \le |x| < 2R. \end{aligned}$$

Multiplying the equation in (1.1) by $\varphi$ then integrating over $\mathbb {R}^N$, integration by parts and Hölder’s inequality imply

$$\begin{aligned} \int _{\mathbb {R}^N} \varphi |x|^b u^{p} \,{\text {d}}x = {}&\int _{\mathbb {R}^N} u (-\hbox {div}(|x|^a D\varphi ) \,{\text {d}}x \le \frac{C}{R^2} \int _{\varOmega _R} \varphi ^{1/p} |x|^a u \,{\text {d}}x \\ \le {}&\frac{C}{R^2}\Big (\int _{\varOmega _R} |x|^{(a - b/p)p^{*}} \,{\text {d}}x\Big )^{1/p^{*}}\Big ( \int _{\varOmega _R} \varphi |x|^b u^{p} \,{\text {d}}x \Big )^{1/p} \\ \le {}&CR^{-2 + \frac{N}{p^{*}} + a - \frac{b}{p} } \Big ( \int _{\varOmega _R} \varphi |x|^b u^{p} \,{\text {d}}x \Big )^{1/p}, \end{aligned}$$

where $1/p + 1/p^{*} = 1$ and $\varOmega _R := B_{2R}(0) \backslash B_{R}(0)$. It follows that there exists a constant $C>0$ depending on N, p, a and b such that

$$\begin{aligned} \int _{B_{R}(0)} |x|^b u^p \,{\text {d}}x \le CR^{-2 + \frac{N}{p^{*}} + a - \frac{b}{p} }\Big ( \int _{B_{2R}(0) \backslash B_{R}(0)} |x|^b u^p \,{\text {d}}x \Big )^{1/p}, \end{aligned}$$

(2.6)

and

$$\begin{aligned} \int _{B_{R}(0)} |x|^b u^{p} \,{\text {d}}x \le CR^{N + p^{*}\left( a-\frac{b}{p} - 2\right) }. \end{aligned}$$

(2.7)

• Step 2 If $N - 2 + a \ge 0$ and $p < p_\mathrm{se}(a,b) \le + \infty$, then $N + p^{*}(a-\frac{b}{p} - 2) < 0.$ Sending $R \longrightarrow \infty$ in (2.7) will show $u \equiv 0$. If $N - 2 + a > 0$ and $p = p_\mathrm{se}(a,b)$, then the same reasoning yields $u \in L^{p}_{b}(\mathbb {R}^N)$. Then, sending $R \longrightarrow \infty$ in (2.6) allows us to conclude that $\Vert u\Vert _{L^{p}_{b}(\mathbb {R}^N)} = 0$. In either case, we arrived at a contradiction, and this completes the proof.

$\square$

2.2 Local existence of radially symmetric solutions

Recall to find radially symmetric solutions of (1.1), we consider the second-order equation

$$\begin{aligned} v''(r) + \Big ( \frac{N - 1 + a}{r} \Big ) v'(r) + r^{b-a}v(r)^p = 0, \, v(r) > 0, \, \text{ in } (0,r_{*}) \subset \mathbb {R}. \end{aligned}$$

(2.8)

Therefore, given $\alpha > 0$, our goal is to eventually find a unique global solution v of (2.8) with $v(0) = \alpha$ and $r_{*} = +\infty$. Moreover, the following Emden–Fowler-type transformation will be useful. If we set

$$\begin{aligned} t = \log r \, \text{ and } \, w(t) = r^{\beta }v(r), \end{aligned}$$

(2.9)

then w satisfies

$$\begin{aligned} w'' + \varLambda _1 w' - \varLambda _2 w + w^p = 0, \, w > 0, \, \text{ for } t \in (-\infty , \log r_*), \end{aligned}$$

(2.10)

where

$$\begin{aligned} \varLambda _1 = N - 2 + a - 2\beta ~ \text{ and } ~ \varLambda _2 = \beta (N - 2 + a - \beta ). \end{aligned}$$

Here, the notation $'$ will either denote $\hbox {d}/\hbox {d}r$ or $\hbox {d}/\hbox {d}t$ when the context is clear, and we note that $p \ge p_{S}(a,b)$ implies that $\varLambda _1 \ge 0$ and $\varLambda _2 > 0$. Another key component of our proofs of the main results will rely on the following energy functional

$$\begin{aligned} E(w) := E(w,w') = \frac{1}{2}(w')^2 - \frac{\varLambda _2}{2}w^2 + \frac{1}{p+1}w^{p+1}, \end{aligned}$$

(2.11)

which satisfies $E(0,0) = 0$ and

$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}t}E(w(t)) = -\varLambda _{1} (w'(t) )^2 \le 0 \text{ for } \text{ all } t. \end{aligned}$$

(2.12)

Lemma 1

Let $N \ge 2$, $b> a - 2 > -N$ and suppose that $p \ge p_{S}(a,b)$. For each $\alpha > 0$, the initial-value problem (2.8) with initial condition $v(0) = \alpha$ admits a unique local solution.

Proof

The existence of a local solution will follow from a standard fixed point argument, but we shall first require the following.

• Step 1 For each local solution v of (2.8) with $v(0) = \alpha$, we necessarily have that

  $$\begin{aligned} \lim _{r \rightarrow 0^{+}} r^{N-1 + a}v'(r) = 0. \end{aligned}$$

  (2.13)

The proof of this claim has many points in common with the proof of Theorem 1. First, observe that from (2.8), v must satisfy

$$\begin{aligned} -(r^{N-1+a}v'(r) )' = r^{N - 1 + b}v(r)^p> 0 \, \text{ for } r > 0. \end{aligned}$$

(2.14)

Thus, $r^{N-1+a}v'(r)$ is monotone decreasing, and therefore,

$$\begin{aligned} \lim _{r \rightarrow 0^{+}} r^{N-1+a}v'(r) = \ell \in (-\infty , \infty ]. \end{aligned}$$

(2.15)

We assert that $\ell = 0$. Otherwise, if $\ell \in (0,\infty ]$, then we can find $0< c < \ell$ and $r_0 > 0$ such that

$$\begin{aligned} v'(r) \ge c r^{-(N-1 + a)} \, \text{ for } 0< r < r_0. \end{aligned}$$

By choosing $0<\epsilon _0< r < r_0$, integrating the above inequality over $(\epsilon _0, r)$ leads to

$$\begin{aligned} v(r) \ge v(r) - v(\epsilon _0) \ge \frac{c}{N - 2 + a}\Big ( \epsilon _{0}^{-(N-2+a)} - r_{0}^{-(N-2+a)} \Big ). \end{aligned}$$

Sending $\epsilon _0 \longrightarrow 0^+$ above will lead to an obvious contradiction. Now, let us assume $\ell \in (-\infty , 0)$. In this case, we can find $r_1 > 0$ such that

$$\begin{aligned} r^{N-1+a}v'(r)< \ell /2< 0\, \text{ for } 0< r < r_1. \end{aligned}$$

Integrating this over $(r, r_1)$ and rearranging terms gives us

$$\begin{aligned} v(r) \ge v(r_1) - \frac{\ell }{2(N-2+a)}\Big ( r^{-(N-2+a)} - r_{1}^{-(N-2+a)} \Big ) \, \text{ for } 0< r < r_1. \end{aligned}$$

Hence, for all suitably small $0< r < r_{1}$,

$$\begin{aligned} v(r) \ge C_1 r^{-(N-2+a)}, \end{aligned}$$

(2.16)

where $C_1$ is some positive constant. On the other hand, if we choose $0< \epsilon _1< r < r_1$, noting that $\epsilon _{1}^{N-1+a}v'(\epsilon _1) \le 0$, and integrating (2.14) over the interval $(\epsilon _1, r)$, we get

$$\begin{aligned} - r^{N-1+a}v'(r) \ge \epsilon _{1}^{N-1+a}v'(\epsilon _1) - r^{N-1+a}v'(r) = \int _{\epsilon _1}^{r} s^{N-1+b} v(s)^p \,\hbox {d}s. \end{aligned}$$

From this, after sending $\epsilon _1 \longrightarrow 0$, we obtain

$$\begin{aligned} -r^{N-1+a}v'(r) \ge \frac{r^{N+b}}{N+b}v(r)^p \, \text{ for } 0< r < r_1; \end{aligned}$$

which further implies there is a constant $C_2 > 0$ such that

$$\begin{aligned} v(r)^{-(p-1)} \ge C_2 r^{2+b - a} \, \text{ for } 0< r < r_1. \end{aligned}$$

Thus,

$$\begin{aligned} v(r) \le C_3 r^{-\beta } \, \text{ for } 0< r < r_1. \end{aligned}$$

Combining this with (2.16) proves there is a constant $C>0$ such that

$$\begin{aligned} r^{N-2 + a - \beta } \ge C \, \text{ for } \text{ all } \text{ suitably } \text{ small } r > 0. \end{aligned}$$

But this cannot happen since $N - 2 + a > \beta$ due to $p \ge p_{S}(a,b) > p_\mathrm{se}(a,b)$. Hence, $\ell = 0$ and (2.13) follows.

• Step 2 Consider the integral operator $T: C([0,r_{*} ]) \longrightarrow C([0,r_{*}] )$, where

  $$\begin{aligned} Tv(r) = \alpha - \int _{0}^r \int _{0}^{t} \Big ( \frac{s^{N - 1 + b}}{t^{N- 1 + a}} \Big ) v(s)^{p} \,\hbox {d}s \, \hbox {d}t. \end{aligned}$$

From (2.8), we see that $-(r^{N-1+a}v'(r))' = r^{N-1+b}v(r)^p$. By choosing $0< \epsilon < r$ and integrating the previous equation in the interval $(\epsilon , r)$, we obtain

$$\begin{aligned} \epsilon ^{N-1+a}v'(\epsilon ) - r^{N-1+a}v'(r) = \int _{\epsilon }^{r} s^{N-1+b}v(s)^p \,\hbox {d}s. \end{aligned}$$

(2.17)

In view of (2.13), sending $\epsilon \longrightarrow 0^+$ leads to

$$\begin{aligned} - r^{N-1+a}v'(r) = \int _{0}^{r} s^{N-1+b}v(s)^p \,\hbox {d}s ~ \text{ for } r > 0. \end{aligned}$$

From this, we see that v(r) is a local solution of (2.8) with $v(0) = \alpha$ if and only if it is a fixed point of T. Indeed, the existence of a unique positive fixed point of T follows from the contraction mapping principle provided $r_{*} > 0$ is chosen sufficiently small. $\square$

Lemma 2

For each $\alpha > 0$, let $v= v(r)$ be the unique local solution of (2.8) with $v(0) = \alpha$. If w(t) is defined by (2.9), then $(w, w')$ converges to (0, 0) as $t \longrightarrow -\infty$.

Proof

Clearly, w tends to 0 as $t \longrightarrow -\infty$. By rewriting the equation in (2.8) into

$$\begin{aligned} -(r^{N-1+a}v'(r))' = r^{N-1+b}v(r)^p \end{aligned}$$

and keeping (2.13) in mind, we obtain

$$\begin{aligned} -r^{N-1+a}v'(r) = \int _{0}^{r} s^{N-1+b}v(s)^{p} \, \hbox {d}s > 0. \end{aligned}$$

Thus,

$$\begin{aligned} 0 < -rv'(r) = \frac{1}{r^{N-2+a}} \int _{0}^{r} s^{N-1+b}v(s)^p \,\hbox {d}s \le Cr^{2 + b - a} ~ \text{ for } \text{ small } r > 0. \end{aligned}$$

In view of $w'(t) = r^{\beta }( \beta v(r) + rv'(r))$, we conclude that

$$\begin{aligned} |w'(t)| \le C(w(t) + r^{\beta + 2 + b - a} ), \end{aligned}$$

where $\beta + 2 + b - a = p\beta > 0$, and thus, $w'(t) \longrightarrow 0$ as $t\longrightarrow -\infty$. $\square$

2.3 Stability analysis of radially symmetric solutions

By taking $x_{1} = w$ and $x_{2} = w'$, we may convert Eq. (2.10) into the two-dimensional nonlinear system

$$\begin{aligned} \begin{pmatrix} x_{1}' \\ x_{2}' \end{pmatrix} = F(x_1,x_2) := \begin{pmatrix} 0 &{} 1 \\ \varLambda _2 &{} -\varLambda _1 \end{pmatrix} \begin{pmatrix} x_{1} \\ x_{2} \end{pmatrix} - \begin{pmatrix} 0 \\ x_{1}^{p} \end{pmatrix}. \end{aligned}$$

(2.18)

The equilibria of (2.18) are

$$\begin{aligned} (x_1,x_2) = \mathcal {O} := (0,0) \text{ and } (x_1, x_2) = \mathcal {S}:= (\varLambda _{2}^{\frac{1}{p-1}}, 0). \end{aligned}$$

To study their stability, we note that

$$\begin{aligned} A:= DF(0,0) = \begin{pmatrix} 0 &{} 1 \\ \varLambda _2 &{} -\varLambda _1 \end{pmatrix} ~ \text{ and } ~ B:= DF(\varLambda _{2}^{\frac{1}{p-1}},0) = \begin{pmatrix} 0 &{} 1 \\ -\varLambda _{2} (p-1) &{} -\varLambda _1 \end{pmatrix}. \end{aligned}$$

The eigenvalues and corresponding eigenvectors of A are

$$\begin{aligned} \lambda _{\pm } = -\frac{\varLambda _1}{2} \pm \frac{\sqrt{\varLambda _{1}^2 + 4\varLambda _2 }}{2}, ~ v_{\pm } = \begin{pmatrix} \lambda _{\mp } \\ -\varLambda _2 \end{pmatrix}, \end{aligned}$$

and for B, they are

$$\begin{aligned} \lambda _{\pm } = -\frac{\varLambda _1}{2} \pm \frac{\sqrt{\varLambda _{1}^2 - 4\varLambda _2 (p-1) }}{2}, ~ v_{\pm } = \begin{pmatrix} \lambda _{\mp } \\ \varLambda _{2}(p-1) \end{pmatrix}. \end{aligned}$$

Lemma 2 indicates we should examine system (2.18) subject to the conditions

$$\begin{aligned} x_1 > 0 \, \text{ and } \, (x_1,x_2) \longrightarrow (0,0) \, \text{ as } \, t \longrightarrow -\infty . \end{aligned}$$

Now, given a solution $(x_1,x_2)$ of (2.18) we consider its orbit, which we denote by $\Gamma$, that emanates from the origin (0, 0) in the right half-space $x_1 > 0$ of the $x_1 x_2$-plane. This trajectory $\Gamma$ represents the graph of a solution of (2.10) on the $w w'$-plane. Due to $E(0,0) = 0$, $E(0,y) > 0$ for all $y\ne 0$, and from the monotonicity of the energy functional indicated by (2.12), we first conclude that $\Gamma$ cannot hit the $x_2$-axis and so this orbit must remain in the right half-space; and secondly, the corresponding solution w(t) must exist globally in $\mathbb {R}$ and both w(t) and $w'(t)$ remain bounded for all $t \in \mathbb {R}$. It follows that $w'$ and $w''$ vanish as $t \longrightarrow \infty$ and therefore $\Gamma$ must converge to either (0, 0) or $(\varLambda _{2}^{1/(p-1)}, 0)$ (for more details, the reader is referred to, e.g. Lemmas 3.1–3.2 in [25]).

We now establish a phase plane analysis for system (2.18), and this will comprise an important ingredient in our proofs of Theorems 2 and 3.

Lemma 3

Let $N \ge 2$, $b> a - 2 > -N$ and suppose that $p \ge p_{S}(a,b)$.

• (a) If $p > p_{S}(a,b)$, then $\mathcal {O} = (0,0)$ is a saddle and $\mathcal {S}$ is stable. Furthermore,

  • (i) if $p_{S}(a,b)< p < p_{JL}(a,b)$, then $\mathcal {S} = (\varLambda _{2}^{1/(p-1)}, 0)$ is a stable focus and $\Gamma$ spirals towards $\mathcal {S}$.

  • (ii) If $p \ge p_{JL}(a,b)$, then $\Gamma$ is a heteroclinic orbit in the first quadrant $\{x_1, x_2 > 0\}$ from $\mathcal {O}$ to $\mathcal {S}$, and $x_1$ is monotone increasing along the orbit $\Gamma$ as t increases from $-\infty$ to $+\infty$.

• (b) If $p = p_{S}(a,b)$, then $\Gamma$ is a homoclinic orbit of the stable equilibrium $\mathcal {O}$.

Proof

Case (a)$p > p_{S}(a,b)$.

The stability of the equilibria just follows from the fact that the eigenvalues of A are real with $\lambda _{-}< 0 < \lambda _{+}$ and the eigenvalues of B have negative real parts. To get a better picture of the flow generated by system (2.18), observe the $x_1$-axis is the vertical nullcline and since $\varLambda _1 > 0$ in this case, the horizontal nullcline is given by the curve $x_2 = -\varLambda _{1}^{-1} x_{1} (x_{1}^{p-1} - \varLambda _2)$ that lies on right half-plane and passes through both equilibria.

As $\mathcal {O}$ is a saddle point, while noting the direction of the corresponding eigenvectors of A, and because the flow along nullclines changes precisely across the line $x_1 = \varLambda _{2}^{1/(p-1)}$, we see that the flow generated by the system ensures $\Gamma$ emanates and increases away from the origin initially in the first quadrant. Although we know $\Gamma$ converges to $\mathcal {S}$, we shall confirm only two scenarios arise. Namely, either $\Gamma$ spirals towards $\mathcal {S}$ if $p < p_{JL}(a,b)$, or else $x_1$ increases along $\Gamma$, which follows the horizontal nullcline as it converges to $\mathcal {S}$.

Subcase (i)$p_{S}(a,b)< p < p_{JL}(a,b)$. Indeed, the eigenvalues of B are complex conjugates with negative real parts. Hence, $\mathcal {S}$ is a stable spiral.

Subcase (ii)$p \ge p_{JL}(a,b)$. We defined the Joseph–Lundgren-type exponent $p_{JL}(a,b)$ to ensure that $p \ge p_{JL}(a,b)$ implies that the eigenvalues of B are real, since

$$\begin{aligned} \varLambda _{1}^{2} \ge 4\varLambda _2 (p-1). \end{aligned}$$

(2.19)

Since $\mathcal {O}$ is a saddle point and noting the direction of its unstable eigenvector, $\Gamma$ increases away from the origin in the first quadrant initially. Now, along the line

$$\begin{aligned} x_2 = -\frac{\varLambda _1}{2}\left( x_1 - \varLambda _{2}^{\frac{1}{p-1}}\right) \text{ for } x_1 < \varLambda _{2}^{\frac{1}{p-1}} \end{aligned}$$

(2.20)

that passes through the equilibrium point $\mathcal {S}$, we show $\Gamma$ must remain in the first quadrant but then decreases towards $\mathcal {S}$. Indeed, $\Gamma$ always lies above the $x_1$-axis due to the increasing direction along this vertical nullcline for $x_1 < \varLambda _{2}^{1/(p-1)}$, and on the line (2.20) the slope of the trajectory satisfies

$$\begin{aligned} \frac{x_{2}'}{x_{1}'} = {}&-\varLambda _1 - \frac{x_1}{x_2}\Big (x_{1}^{p-1} - \varLambda _2 \Big ) = -\varLambda _{1} + \frac{2x_1 (x_{1}^{p-1} - \varLambda _2)}{\varLambda _1 \left( x_1 - \varLambda _{2}^{\frac{1}{p-1}} \right) } \\ = {}&-\varLambda _{1} + \frac{2}{\varLambda _1} x_{1}(p-1)\xi ^{p-2} ~ \left( \text{ for } \text{ some } ~ x_1< \xi< \varLambda _{2}^{\frac{1}{p-1}} \right) \\ < {}&-\varLambda _{1} + \frac{2}{\varLambda _1} (p-1) \varLambda _2 \le -\frac{\varLambda _1}{2}, \end{aligned}$$

where we used the mean value theorem and the fact that $\frac{2}{\varLambda _1} (p-1) \varLambda _2 \le \frac{\varLambda _1}{2}$ thanks to (2.19). This guarantees the incoming trajectory remains below the line (2.20). Thus, we deduce that $\Gamma$ remains in the first quadrant and, as $t \longrightarrow +\infty$, $x_{1}(t)$ increases along $\Gamma$ as it tends to $\varLambda _{2}^{1/(p-1)}$.

Case (b)$p = p_{S}(a,b)$. In this case, the eigenvalues of B are purely complex conjugates. Since $\varLambda _1 = 0$ in this case, there holds $E'(w(t)) = 0$ so that E is constant along any solution; that is, the orbit is given by the level curve

$$\begin{aligned} E(x_1,x_2) \equiv 0 = E(0,0). \end{aligned}$$

(2.21)

Since $E(0,x_2) = x_{2}^2 /2 > 0$ for all $x_2 \ne 0$ and

$$\begin{aligned} E(\varLambda _{2}^{\frac{1}{p-1}}, 0) = \left( \frac{1}{1+p} - \frac{1}{2}\right) \varLambda _{2}^{\frac{p+1}{p-1}} < 0, \end{aligned}$$

the orbit $\Gamma$ lies on the curve (2.21), tending away $\mathcal {O}$ and traversing about $\mathcal {S}$ clockwise as it must converge back towards $\mathcal {O}$ as $t\longrightarrow \infty$. Hence, $\Gamma$ is a homoclinic orbit. $\square$

3 Proof of Theorems 2 and 3

We are now in the position to prove the remaining theorems.

3.1 Proof of Theorem 2

Suppose that $N \ge 2$, $b> a - 2 > - N$ and $p \ge p_{S}(a,b)$.

• Step 1 (Existence) Given $\alpha > 0$, the initial value problem,

  $$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{{\text {d}}^2 v}{{\text {d}} r^2} + \Big ( \frac{N-1 + a}{r} \Big ) \frac{{\text {d}} v}{{\text {d}} r} + r^{b-a}v^{p} = 0, ~ v(r)> 0, ~ r> 0, \\ v(0) = \alpha > 0, \end{array}\right. \end{aligned}$$

  (3.1)

admits a unique local solution of class $C^{2}((0,r_{*} ))\cap C([0,r_{*}))$ by Lemma 1. We denote this unique solution by $v_{\alpha }$. By taking the Emden–Fowler-type transformation $w_{\alpha }(t) = r^{\beta }v_{\alpha }(r)$ and recalling the discussion in the previous section, we can extend $w_{\alpha }$ globally, and therefore, we may extend $v_{\alpha }$ to a unique solution with $r_{*} = +\infty$.

• Step 2 (Uniqueness) Let $u = u(|x|)$ be a radially symmetric solution of problem (1.1). Indeed, u satisfies (3.1) with $\alpha = u(0)$, and therefore, $v_{\alpha }(r) = u(|x|)$. Observe that (1.1) is invariant under the scaling

  $$\begin{aligned} u(x) \longrightarrow u_{\lambda }(x) := \lambda u(\lambda ^{1/\beta } x), \end{aligned}$$

i.e. for each $\lambda > 0$, $u_{\lambda }$ remains a solution provided that u is a solution of (1.1). Consequently, $u(0) v_{1}(u(0)^{1/\beta }r )$ satisfies the same initial-value problem as $v_{\alpha }$, and hence, ODE uniqueness theory ensures that

$$\begin{aligned} v_{\alpha }(r) = \alpha v_{1}(\alpha ^{1/\beta } r). \end{aligned}$$

(3.2)

In particular, if $p = p_{S}(a,b)$ and $\alpha = 1$, then a routine calculation reveals $v_{1}(r)$ defined by (1.13) is the unique solution of (3.1) with initial value $\alpha = 1$.

• Step 3 If $p > p_{S}(a,b)$, then Lemma 3 shows that any trajectory, including the solution curve $(w,w')$ where $w(t) = r^{\beta }v_{\alpha }(r)$ and $\alpha = u(0)$, must converge to the equilibrium solution $(\varLambda _2^{1/(p-1)}, 0)$. Therefore,

  $$\begin{aligned} \lim _{|x| \longrightarrow +\infty } |x|^{\beta }u(|x|) = \varLambda _2^{\frac{1}{p-1}}. \end{aligned}$$

Now, assume $p \ge p_{JL}(a,b)$ and let u be a radially symmetric solution of (1.1). Thanks to part (a.ii) of Lemma 3, we have that

$$\begin{aligned} u(|x|) \le U_{s}(r) \, \text{ for } \text{ all } \, r > 0 \end{aligned}$$

Hence, for each $\varphi \in C^{1}_{c}(\mathbb {R}^N)$,

$$\begin{aligned}&\int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - {} p\int _{\mathbb {R}^N} |x|^b u^{p-1} \varphi ^2 \,{\text {d}}x \\&\quad \ge {} \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - p\int _{\mathbb {R}^N} |x|^b U_{s}^{p-1} \varphi ^2 \,{\text {d}}x \\&\quad = {} \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - pC_{a,b,N,p}\int _{\mathbb {R}^N} |x|^{-2 + a} \varphi ^2 \,{\text {d}}x, \end{aligned}$$

where

$$\begin{aligned} pC_{a,b,N,p} = p \beta (N - 2 + a - \beta ) \le \Big (\frac{ N - 2 + a}{2}\Big )^2, \end{aligned}$$

and this last inequality holds because $p \ge p_{JL}(a,b)$ implies $f(p) \le 0$, where f(p) was defined in (1.8). Then, from the endpoint case of the Caffarelli–Kohn–Nirenberg inequality (1.2), namely when $N \ge 3$, $b = a - 2$, and $p =1$ there holds

$$\begin{aligned} \mathcal {N}(a,N)\int _{\mathbb {R}^N} |x|^{-2 + a} \varphi ^2 \,{\text {d}}x \le \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x ~ \text{ for } \text{ all } \varphi \in C^{1}_{c}(\mathbb {R}^N), \end{aligned}$$

and the sharp constant is explicitly given by (see [6, 8] for details)

$$\begin{aligned} \mathcal {N}(a,N) = \Big ( \frac{N-2+a}{2} \Big )^2. \end{aligned}$$

Invoking this in the previous chain of estimates yields

$$\begin{aligned}&\int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - {} p\int _{\mathbb {R}^N} |x|^b u^{p-1} \varphi ^2 \,{\text {d}}x \\&\quad \ge {} \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - pC_{a,b,N,p}\int _{\mathbb {R}^N} |x|^{-2 + a} \varphi ^2 \,{\text {d}}x \\&\quad \ge {} \int _{\mathbb {R}^N} |x|^a |D\varphi |^2 \,{\text {d}}x - \mathcal {N}(a,N) \int _{\mathbb {R}^N} |x|^{-2 + a} \varphi ^2 \,{\text {d}}x \ge 0. \end{aligned}$$

This proves u is a stable solution of (1.1). This completes the proof of the theorem. $\square$

3.2 Proof of Theorem 3

We make the same assumptions and adopt the same notation used in Lemma 3. Given $\alpha > 0$, we denote the unique global solution of (3.1) by $v_{\alpha }$ and we set $w_{\alpha }(t) = r^{\beta }v_{\alpha }(r)$, where $t = \log r$. We choose any pair of positive reals $\alpha _1 < \alpha _2$.

Case (a)$p_{S}(a,b)< p < p_{JL}(a,b)$.

From the rescaling (3.2), the graphs of $w_{\alpha }$ and $w_1$ are identical and therefore lie on some orbit $\Gamma$. From part (a.i) of Lemma 3, this trajectory $\Gamma$ emanates away from (0, 0) and spirals towards the stable focus $(\varLambda _2^{1/(p-1)}, 0)$. Specifically, by rescaling, we find a real number $\tau = \tau (\alpha )$ satisfying $\alpha = {\text {e}}^{-\beta \tau (\alpha )}$ so that

$$\begin{aligned} w_{\alpha } (t) = {\text {e}}^{\beta t} \alpha v_{1}(\alpha ^{1/ \beta } {\text {e}}^{t} ) = {\text {e}}^{-\beta \tau + \beta t} v_{1}({\text {e}}^{-\tau } {\text {e}}^{t}) = w_{1}(t - \tau ) ~ \text{ for } \text{ all } t \in \mathbb {R}. \end{aligned}$$

(3.3)

So we have that $v_{\alpha _1}(r) = v_{\alpha _2}(r)$ for some $r > 0$ if and only if

$$\begin{aligned} w_{1}(t - \tau (\alpha _1)) = w_{1}(t - \tau (\alpha _2)) \text{ for } \text{ some } t \in \mathbb {R}. \end{aligned}$$

(3.4)

Since $\Gamma$ spirals towards the non-trivial equilibrium point $\mathcal {S}$, (3.4) of course happens for infinitely many points in t, and thus, the radial solutions intersect one another infinitely many times. Similar reasoning will show that $w_{\alpha _1}(t)$ crosses

$$\begin{aligned} W(t) = r^{\beta }U_{s}(r) \equiv \varLambda _2^{1/(p-1)} \end{aligned}$$

infinitely many times. Hence, $v_{\alpha _1}(r)$ intersects the singular solution $U_{s}(r)$ at infinitely many points.

Case (b)$p \ge p_{JL}(a,b)$.

From (3.3) and part (a.ii) of Lemma 3, we see that $w_{\alpha _1}(t) < w_{\alpha _2}(t)$ for all $-\infty< t < \infty$. That is, $v_{\alpha _1}(r) < v_{\alpha _2}(r)$ for all $r > 0$.

Case (c)$p = p_{S}(a,b)$.

Imitating the earlier argument in part (a) and noting part (b) of Lemma 3, we can show there exists a unique time t such that $w_{1}(t - \tau (\alpha _1)) = w_{1}(t - \tau (\alpha _2)).$ Hence, the corresponding solutions $v_{\alpha _1}(r)$ and $v_{\alpha _2}(r)$ intersect at only one point $\bar{r} > 0$. Alternatively, from the classification result of Theorem 2, we have that $v_{\alpha _1}(r) = v_{\alpha _2}(r)$ if and only if

$$\begin{aligned} \alpha _1 v_{1}( \alpha _{1}^{1/\beta } r) = \alpha _2 v_{1}( \alpha _{2}^{1/\beta } r). \end{aligned}$$

This occurs at exactly one point in $(0,\infty )$; namely, when

$$\begin{aligned} \bar{r} = \Big [ \frac{ (N+b)(N+2- a)((\alpha _2 / \alpha _1 )^{(p-1)/2} - 1 ) }{ \alpha _{2}^{p-1} - (\alpha _1 \alpha _2)^{(p-1)/2} } \Big ]^{\frac{1}{2 + b - a}}. \end{aligned}$$

Similar arguments will show that $w_{\alpha _1}(t) = \varLambda _{2}^{1/(p-1)}$ holds at two distinct times. This shows that each radially symmetric solution $v_{\alpha _1}$ twice interests the singular solution. This completes the proof of the theorem $\square$