Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations

In this paper we discuss the ordering properties of positive radial solutions of the equation Δpu(x)+k|x|δuq-1(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta _p u(x)+ k |x|^{\delta } u^{q-1}(x)=0 \end{aligned}$$\end{document}where x∈Rn,n>p>1,k>0,δ>-p,q>p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x \in {\mathbb {R}}^n, n>p>1, k>0, \delta>-p, q>p$$\end{document}. We are interested both in regular ground states u (GS), defined and positive in the whole of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{document}, and in singular ground states v (SGS), defined and positive in Rn\{0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n \setminus \{0\}$$\end{document} and such that lim|x|→0v(x)=+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim _{|x| \rightarrow 0} v(x)=+\infty $$\end{document}. A key role in this analysis is played by two bifurcation parameters pJL(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{JL}(\delta )$$\end{document} and pjl(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{jl}(\delta )$$\end{document}, such that pJL(δ)>p∗(δ)>pjl(δ)>p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{JL}(\delta )>p^*(\delta )>p_{jl}(\delta )>p$$\end{document}: pJL(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^{JL}(\delta )$$\end{document} generalizes the classical Joseph–Lundgren exponent, and pjl(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{jl}(\delta )$$\end{document} its dual. We show that GS are well ordered, i.e. they cannot cross each other if and only if q≥pJL(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \ge p^{JL}(\delta )$$\end{document}; this way we extend to the p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} case the result proved in Miyamoto (Nonlinear Differ Equ Appl 23(2):24, 2016), Miyamoto and Takahashi (Arch Math Basel 108(1):71–83, 2017) for the p≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \ge 2$$\end{document} case. Analogously we show that SGS are well ordered, if and only if q≤pjl(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \le p_{jl}(\delta )$$\end{document}; this latter result seems to be known just in the classical p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} and δ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =0$$\end{document} case, and also the expression of pjl(δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{jl}(\delta )$$\end{document} has not appeared in literature previously.


Introduction
In this paper we continue the discussion started by Miyamoto [25] and Miyamoto et al [26] concerning the ordering properties of radial solutions for a class of quasilinear elliptic equations, including p-Laplacian and k-Hessian.
Let us start our discussion from the p-Laplace setting, i.e. we consider radial solutions u(|x|) = u(r) for the following equation where Δ p (u) = div(∇u|∇u| p−2 ). In the whole paper we always assume the following relations for the parameters H k > 0, n > p, δ > −p and q > p.
In fact for any d > 0 and any L > 0 there is a unique regular solution u(r; d) and a unique fast decay solution v(r; L) to (1.2), see Proposition 1.5 and Theorem 1.6.
A Ground State (GS) and a Singular Ground State (SGS) are respectively a regular and a singular solution of (1.2) which are positive for any r > 0; it is easy to check that in both cases lim r→+∞ u(r) = 0.
It is well known that the structure of positive solutions to (1.1) undergoes several bifurcations as q passes through some critical exponents, in particular If q > p * (δ) there exists an explicitly known SGS with slow decay u(r; ∞) := P x r −α where P x is defined in (2.3). Further, if q ≥ p * (δ) all the regular solutions are GS. In particular we have the following classical results see e.g. [12, §2].
In order to complete the picture we recall that if q = p * (δ) then regular solutions are GS with fast decay and in fact they are explicitly known, see e.g. [25,Eq. (1.17)] and there is a two parameter family of SGS with slow decay, see e.g. [12].
We think it is worthwhile to point out that p * := p * (0) is related to the continuity of the trace operator in L q , while p * := p * (0) is the Sobolev critical exponent so it is the upper bound for the compactness of the embedding of L q in W 1,p .
In the classical p = 2 and δ = 0 case the intersection properties of positive solutions depend on two further critical exponents, i.e.
if n ≤ 10. (1.4) Notice that 2 < 2 * (0) < 2 jl < 2 * (0) < 2 JL . The latter, 2 JL , is the so called Joseph-Lundgren exponent and it was introduced in [23], while 2 jl was introduced in [4]: 2 JL and 2 jl are used to determine the intersection properties respectively of regular and singular solutions of (1.2). Namely if 2 * < q < 2 JL all the GS of Theorem A intersect each other indefinitely, while if q ≥ 2 JL they are well ordered, i.e. if d 2 > d 1 then u(r; d 2 ) > u(r; d 1 ), for any r ≥ 0. These results are crucial for determining the long time behaviour of positive solutions of the following parabolic problem ∂u ∂t (t, x) = Δu(t, x) + |x| δ u q−1 (t, x) u(0, x) = U 0 (x), (1.5) where U 0 (x) need not be radial. We emphasise that (1.5) is a simple model for an explosion: u describes the temperature and the non linearity |x| δ u q−1 represents a spatial-dependent esothermic reaction. The two main expected behaviors are the blowing up in finite time of the solution (the explosion takes place) or the convergence to zero of the solution (the temperature is too low to initiate the explosion). If 2 * < q < 2 JL , using the intersection property in Theorem A, it is possible to construct sub and super solutions and to show that the radial GS are on the threshold between blowing up and fading solutions, see [2,19,31] for details. Further, if q ≥ 2 JL , the ordering property described in Theorem B is essential to prove that GS enjoy some stability in appropriated L ∞ weighted space, see again [2,19,31]. In fact Theorem B was a key stone for a flourishing of interesting papers, concerning the possible rate of convergence either to the null solution or to the GS, or to determine the speed of the blow up of the solutions of (1.5), see e.g. [9,21,27] and references therein, see also Corollary 3.6 below. The generalisation of 2 JL to a p-Laplace context firstly appeared in [3] (see also [15] and [25]). Its complicated expression can be found explicitly in [25, (1.13)] and it is obtained as the largest solution of a quadratic equation, see Sect. 4.1. The exponent 2 jl , is the bifurcation parameter for the ordering properties of SGS. This characteristic is again crucial for (1.5): in fact Sato and Yanagida in the interesting paper [28] managed to prove local uniqueness of singular solutions of (1.5) (assuming a priori that the type of singularity is preserved for any t ≥ 0) and some stability properties of SGS with slow and fast decay when p = 2, δ = 0 and 2 * (0) < q < 2 jl , see also [22] and Corollary 3.7 below.
One achievement of this article is to obtain the expression of 2 jl in a p-Laplace context: it is denoted by p jl (δ) and it is the smallest solution of the quadratic equation mentioned above, see Sect. 4.1 and in particular equation (4.4). For any value of the parameters we have: One of our main results is the following We emphasise that this result has already been proved in [31] in the case p = 2, and in [25,26] when p > 2. Here we extend it to the 1 < p < 2 case too. After the paper was completed we were informed that Theorem 1.1 was recently proved by Guo and Zhou in the 1 < p < 2 case too, see [20], which, however, is in Chinese. Theorem 1.1 is in some sense optimal. In fact we have the following known result, see e.g. [15,25]. Theorem C. Assume H, p * (δ) < q < p JL (δ) and consider (1.2). Then for any R > 0 and 0 < d 1 < d 2 the function u(r; d 1 ) − u(r; d 2 ) changes sign infinitely many times when r ≥ R.
In the classical p = 2 case Theorem C has been proved in [19,31] and it was an important ingredient to prove the weak asymptotic stability of the GS, u(r; d), for (1.5), and several nice results concerning the rate of convergence of the solutions of (1.5), see e.g. [27] and references therein.
We give a second main contribution concerning the dual situation, i.e. the ordering properties of SGS in the subcritical case. Theorem 1.2. Assume H, p * (δ) < q ≤ p jl (δ), q ≥ 2 and consider the SGS with fast decay v(r; L). If 0 < L 1 < L 2 , then 0 < v(r; L 1 ) < v(r; L 2 ) for any r > 0.
We emphasize that the rather weak condition q ≥ 2 in Theorems 1.1 and 1.2 seems to be technical. However it is always satisfied if p ≥ 2.
As far as we are aware the ordering properties of SGS are known just when p = 2 and δ = 0 (see [4,Proposition 2.5]). One of the main purposes of this paper is to generalise this result to the p > 1 and δ > −p case. In fact in the p = 2 case Theorem 1.2 can be obtained from Theorem 1.1 by combining Kelvin inversion and Fowler transformation, however as far as we are aware the result has not yet appeared in literature (we provide a short proof in this easier case at the end of Sect. 3). Anyway in the general p > 1 case the Kelvin inversion is not available, so we have to argue differently, in fact even the exponent p jl (δ) was known just in the case p = 2 and δ = 0.
Also in this context the result is optimal, i.e. we have the following result which seems to be new, as far as we are aware.
We wish to spend few lines to point out that, if δ ≤ 0 and p = 2, positive solutions to (1.1) have to be radial, see e.g. [17], while if δ > 0 even in the p = 2 case we may have non-radial positive solutions, see e.g. [7] where the authors constructed a positive non-radial singular solution of the Hénon equation in R n \ {0}; see also [30] for an example concerning a non-radial regular solution in a ball in the critical case.
In fact there are some symmetry results also in the p > 1 case, when δ = 0: e.g. radial symmetry is ensured in the critical case for any p > 1, see [29], and by requiring 1 < p ≤ 2 and some a priori estimates on the asymptotic behavior of positive solutions, see [6].
The results concerning existence and ordering properties of GS and SGS are summed up in Table 1.
Let us denote by B the unit ball in R n . Following [25], as a Corollary of Theorems 1.1 and C we also find the bifurcation diagram for the positive radial solutions of the following problem with Dirichlet boundary conditions. Abusing the notation we denote once again by U (x) = U (r) when r = |x| since U is radially symmetric; in fact (1. show that U (r; D) < 0 for any 0 < r < ρ(D; λ); then, following [25], we see that for any D we have a value λ = λ(D) for which (1.6) admits a solution.
Then we have this result concerning the shape of λ(D), which is a generalisation of [25,Theorem B], [26,Theorem B].
where D k is the set of the critical points of λ(D). In particular D k is a local minimum point for k even and a local maximum point for k odd.
Proof. The proof is a consequence of Theorem 1.1 and can be obtained by a straightforward repetition of the argument developed in [25,Theorem B] for the case p ≥ 2.
To complete the picture we recall that if p < q ≤ p * (δ) then there is a Λ such that the Dirichlet problem associated to (1.6) admits two solutions if 0 < λ < Λ, one solution for λ = Λ and no solution if λ > Λ, see [25,Theorem B], [8,Theorem 4]. Now we briefly observe how our discussion can be used to obtain information on the analogous problems where the p-Laplace operator is replaced by the K-Hessian. Following [25] we denote by {λ j } N j=1 the set of the eigenvalues of the Hessian matrix D 2 u in R N . Let K ∈ {1, 2, . . . , N}; the K-Hessian operator is defined by so that the classical Laplace operator is S 1 (D 2 u) = Δu, while the Monge-Ampere operator is S N (D 2 u) = det(D 2 (u)). Let us set Then, following the introduction of [25] (see page 16) the radial solutions of the K-Hessian equation A proof of this result can be found in [16,Appendix] and in [25]. Analogously we have the following. Theorem 1.6. Assume H, q > p * (δ). Then, for any L > 0 there exists a fast decay solution v(r; L) of (1.2) and it is unique and C 2 for r large enough.
The existence part is in fact already known, see, e.g. [12], but the uniqueness is new as far as we are aware, and it will be proved in Sect. 4.2.1 Let us recall some well known facts. Proof. From a straightforward application of De l'Hospital rule we find Then the statement easily follows.
and this concludes the proof.
Most of the proofs of this article rely on a change of variable known as Fowler transformation, which enables us to use phase plane analysis, and to profit of invariant manifold theory and dynamical systems techniques. The plan of the paper is as follows. In Sect. 2 we introduce the Fowler transformation for p-Laplace equations and system (2.2). In §2.1 we define the critical exponents p JL (δ) and p jl (δ) even if the lengthy computation needed for their evaluation is postponed to §4.1; further we explore the dynamics of (2.2) in a neighborhood of the critical point P , corresponding to singular and slow decay solutions of (1.2). In §2.2 we introduce the unstable manifold W u + and the stable manifold W s + of the origin of (2.2) corresponding to regular and fast decay solutions of

Fowler transformation
In this section we introduce a change of variables known as Fowler transformation which allows to pass from the non-autonomous singular ODE (1.2) to a two dimensional autonomous dynamical systems. Hence we set and we obtain Here and later " · " stands for d dt , while " " stands for d dr . In the whole paper we denote by φ(t; Q) = (x(t; Q), y(t; Q)) the trajectory of (2.2) passing through Q at t = 0, omitting the dependence on Q when it is not needed.
This change of variables was developed by Fowler in the 30s and extended to the p-Laplace case by Bidaut-Veron in [1] and independently by Franca in [10].
One of the main advantage in studying (2.2) lies in the fact that the system is autonomous and we can profit of phase plane techniques and of invariant manifold theory.
Remark 2.1. The well known scaling invariance property of (1.2) here is translated in the fact that (2.2) is autonomous. The property that if u(r) is a solution of (1.2) then u(cr)c α solves (1.2) for any c > 0, here becomes the fact that if However in the p = 2 case we have the following problem.
is just Holder continuous on the x axis if p > 2 and on the y axis if p < q < 2.

Definition of the critical exponents and their dynamical interpretation
As we said in the introduction equation (1.2) and system (2.2) change their characteristics as q crosses some critical values, see (1.3).

NoDEA
Ordering properties for p-Laplace equations Page 9 of 36 54 Let df dφ (P ) be the linearisation of (2.2) on P , i.e.
Let T and D be the trace and the determinant of df dφ (P ) respectively, we easily see that for a detailed computation. Hence, T < 0 if and only if α < n−p p or equivalently q > p * (δ). Thus the critical point P is asymptotically unstable if p * (δ) < q < p * (δ), a center if q = p * (δ), asymptotically stable if q > p * (δ): in this case (1.2) is respectively subcritical, critical or supercritical (see Theorems A, B).
From a straightforward computation we see that when λ 1 and λ 2 are real and distinct their eigenvectors are Notice that m 1 ≥ m 2 > 0, and we have the same expression if either p * (δ) < q ≤ p jl (δ), or q ≥ p JL (δ). If λ 1 = λ 2 , i.e. when q ∈ {p jl (δ); p JL (δ)}, their geometric multiplicity is 1, so the unique eigenvector of df and it is again given by (2.7), and df dφ (P ) has a nilpotent part (hence the corresponding linear differential equation is resonant).

The stable and unstable manifolds W s + and W u −
Now we turn to consider the stability properties of the origin for system (2.2). In this subsection we assume for simplicity that (2.2) is C 1 , cf. Remark 2.2, so that the origin is a saddle (a discussion of the non-smooth case is postponed to Sect. 4.2). So we can define the following sets This Lemma easily follows from the Hartman-Grobman Theorem, see e.g. [18, §1.3].
It is easy to check that W u (respectively W s ) is split by the origin in two connected components: since we are interested in definitively positive solutions, we consider the one leaving the origin and entering x ≥ 0, denoted by W u + (respectively W s + ). In fact we have the following Lemma From the Hartman-Grobman theory we also get the following useful result, see again [18, §1.3]. Lemma 2.6. Assume q > p * (δ) and that (2.2) is C 1 . Then W u + is tangent in the origin to the x axis, while W s + is tangent to the y axis if 1 < p < 2 and to the line Then we immediately obtain the following.
Using standard tools of dynamical system theory and some integral estimates, we find the following correspondences between trajectories of (2.2), and solutions of the original equation (1.2).  (1.2) correspond to the trajectories φ(t) converging to P respectively as t → −∞ or as t → +∞; thus lim r→0 u(r)r α = P x and lim r→+∞ u(r)r α = P x respectively. No other solutions of (1.2) definitively positive either for r small or for r large exist.
is positive and decreasing for 0 < r 1. Using standard tools of invariant manifold theory, see e.g. [18, §1.3], we see that the trajectory in W u + has the same asymptotic behaviour as a trajectory of the unstable space of the linearisation of system (2.2) and the claim concerning regular solutions is proved. Now we turn to consider a fast decay solution v(r; L) of (1.2) and the corresponding trajectory φ(t) of (2.2).
If v(r; L) is a fast decay solution then from Lemma 1.8 it follows that lim r→+∞ v (r; L)r n−1 Viceversa assume that lim t→+∞ φ(t) = (0, 0). Since φ(t) ∈ W s + we easily see that y(t) < 0 < x(t) when t 0, hence u(r) is positive and decreasing for r 1. From Lemma 1.8 we see that lim r→+∞ v(r)r n−p p−1 exists and it is either a positive constant or +∞.
Again from standard tools of invariant manifold theory, [18, §1.3], we see that the trajectories in W s + satisfy lim t→+∞ φ(t) e −γt = n−p p−1 L, for some L > 0. Further W s + is tangent to the y axis in the origin hence we find where we used the fact that β−γ = n−1. Hence we obtain lim r→+∞ |v (r)|r When q = p * the results in Lemma 2.8 need to be modified slightly. In fact in this case (2.2) is Hamiltonian and admits periodic trajectories which correspond to singular solutions u(r) of (1.2) which have a slightly different behaviour. We remand the interested reader, e.g., to [   Proof. Let x(t; Q u ), Q u ∈ W u + , be the trajectory of (2.2) corresponding to u(r; 1). From (2.1), Lemma 2.8 and using the invariance for t-translations of (2.2), for any d > 0, we can write where r = e t and Q := x(−τ ; Q u ) ∈ W u + (since W u + is invariant, see Lemma 2.5). Then passing to the limit as r tends to 0 we find Similarly let v(r; 1) be a fast decay solution of (1.2) and let x(t; Q s ) be the corresponding trajectory of (2.2), so that Q s ∈ W s + . Reasoning as above we see that φ τ (t) := φ(t − τ ; Q s ) is a trajectory of (2.2) and φ τ (0) = R ∈ W s + , see again Lemma 2.5. So φ τ (t; R) corresponds to a fast decay solution v(r; L) of (1. where we have used the same idea as in (2.9) Hence passing to the limit as r → +∞ on the left hand side and as re −τ → +∞ on the right hand side we find for any r > 0. The proof of this result is rather technical, and in fact the part concerning W u + and regular solutions can be found in literature, with some effort, even in a non-autonomous context, see [11] and references in Sect. 4.2. We give a new and shorter proof suitable for this simpler autonomous context in Sect. 4.2 for completeness.

Pohozaev function and heteroclinic connections
Let us introduce the following energy like function which is closely related to the Pohozaev identity, see e.g. [13, §2], or [10, §2]: From a straightforward computation it is easy to check that when q = p * (δ) then H(x, y) is a first integral, see e.g. [13, §2]. In general we have the following result.
We provide here the proof of this known result for the convenience of the reader, however see, e.g. [13, §2], for a proof in a non autonomous context.
Proof. For any solution u(r) of (1.2) we can define the following Pohozaev function One of the main tool in the analysis of this equation is the well known Pohozaev identity, see, e.g., [24], that in this context reads as follows:

and it is monotone decreasing if q > p * (δ). Further observe that if u(r) is a regular solution and v(r) is a fast decay solution we have
(2.14) Then we go back to system (2.2) and we see that if φ(t) = (x(t), y(t)) is the trajectory of (2.2) corresponding to u(r) we have H(x(t), y(t)) = P (u(e t ), u (e t ), e t )e (α+γ)t , (2.15) where H is the function defined in (2.12). Let φ u (t), φ v (t) be the trajectories of (2.2) corresponding to the regular and the fast decay solution u(r) and v(r) of (1.2). Assume to fix the ideas that q > p * (δ); then from (2.13) and (2.14) we find that P (u(r), u (r), r) < 0 < P (v(r), v (r), r) for any r > 0. Hence from (2.15) we see that H(φ u (t)) < 0 < H(φ v (t)) for any t ∈ R, so the Lemma follows. The case p < q < p * (δ) is analogous.
With the same argument we easily see that if q = p * (δ) then H is a first integral; hence W u + and W s + are obtained as the subset of the 0-level set of H which is contained in x ≥ 0.
Then from an elementary analysis of the phase portrait we obtain the following, cf. Fig. 1.
To complete the picture we observe that if p * (δ) < q < p * (δ) then W u + is made up by an unbounded trajectory which converges to the origin as t → −∞ and rotates clockwise indefinitely as t increases. Similarly if q > p * (δ) then W s + is made up by an unbounded trajectory which converges to the origin as t → +∞ and rotates clockwise indefinitely as t decreases. If q = p * (δ) then W u + = W s + and it is the graph of an homoclinic trajectory and coincide with the set {Q = (Q x , Q y ) | H(Q) = 0, Q x > 0}, see again Fig. 1. We do not give a full fledged proof of this known facts (which will not be used in this article) remanding the interested reader, e.g., to [10]. Proof of Theorems C and 1. 3 Theorems C and 1.3 simply follow putting together Lemma 2.8, Proposition 2.12 and Lemma 2.3.

In this section we show that if
+ is a graph on π. Whence Theorems 1.1 and 1.2 easily follow.
The idea is inspired by the work by Miyamoto [25] and it is obtained by constructing suitable positively and negatively invariant sets.
Let q > p. If q ≥ p JL (δ), q ≥ 2 and p > 1 then both the lines r 1 and r 2 intersect the x positive semi-axis respectively in the points S i x = (X i , 0), for i = 1, 2, with X 1 ≥ X 2 > 0. If p * (δ) < q ≤ p jl (δ) and p ∈ (1, 2] then the semi-line r 1 and r 2 intersect the y positive semi-axis in the points The proof is based on a geometric argument. We first observe that on the positive x-semi-axis we haveẋ > 0 andẏ < 0. We restrict to consider the 4 th quadrant so the x-nullcline is and it is represented in Fig. 2 for p ∈ (1, 2) and for p ≥ 2.
The y-nullcline is (see Fig. 3) soẏ > 0 if y < − k |γ| x q−1 .  Figure 2. The x-nullcline in the case p ∈ (1, 2) and p ≥ 2 respectively Figure 3. The y-nullcline when q > 2 on the left and when q ∈ (1, 2) on the right Since q > p, the x-nullcline is below the y-nullcline when while we have the opposite situation for x > P x . The two nullclines, the vertical line x = P x and the horizontal line y = P y define 8 regions in the fourth quadrant (see Fig. 4). In order to conclude the proof we consider two cases. Case 1: q > p > 1, q ≥ 2 and q ≥ p JL (δ). In this case the fixed point P is a stable node; so there are solutions of the linearised system which converge to P along the semi-lines r 1 and r 2 ; correspondingly there are solutions of the nonlinear system which converge to P tangentially to r 1 and r 2 . Then the semi-lines r i , spanned by the eigenvectors v i , must lie inside the regions of the space which allow convergence to P . By the analysis of the vector field we conclude that there are only two regions in which we can find solutions converging to P , that is region 3 and 7 (see Fig. 4). Whence both r 1 and r 2 are in region 3 and, since the y-nullcline is concave, the lines r 1 and r 2 both intersect the x-axis at a certain X i > 0. We note that this is not ensured when q < 2 since the y-nullcline is convex (a priori r i may intersect the x-axis for a negative value of x). Case 2: p ∈ (1, 2], p < p * (δ) < q ≤ p jl (δ). In this case the fixed point P is an unstable node; reasoning as above we see that the semi-lines r i must lie inside the regions of the space which allow convergence to P in the past, i.e. regions 1 and 5. Then, r 1 and r 2 lie in region 1, below the convex x-nullcline; hence r 1 and r 2 both intersect the y-negative semi-axis. We note that this argument does not work for p > 2 since the x-nullcline is concave.
Let us denote by S x the intersection between the semi-line r 2 and the x axis, and by S y the intersection between the semi-line r 1 and the y axis, see Proof. We divide the proof into two parts.
(i). On the x-nullcline and on the segment OS x the vector field points inside A + . It remains to check the vector field on the segment P S x which lies on the line r 2 .
We rewrite (2.2) in the following forṁ where , and φ = (x, y). We observe thatψ(t) = P + v 1 e λ1t is a solution of the linear systemψ
(ii) We observe that on the negative y-axis the vector field points outside A − . The rest of the proof is identical to case (i).
From Lemma 2.7 we know that there is a neighborhood Ω of the origin such that W u loc ⊂ A + , and if Q ∈ W u loc then φ(t; Q) ∈ W u loc ⊂ A + for any t ≤ 0. From Proposition 3.2 we see that φ(t; Q) ∈ A + for any t ≥ 0. Hence from Lemma 2.5 we see that φ(t; Q) ∈ A + for any t ∈ R and W u + ⊂ A + . Thuṡ x(t; Q) > 0 for any t ∈ R and the claim easily follows from Lemma 2.5. Now let 0 < d 1 < d 2 < +∞; let φ(t; Q(d i )) be the trajectory of (2.2) corresponding to the regular solution u(r, d i ) of (1.2), for i = 1, 2, so that Q( ) < x(t; Q(d 2 )) < P x for any t ∈ R; then Theorem 1.1 immediately follows.
Analogously let p * (δ) < q ≤ p jl (δ): from Lemma 2.7 we see that W s loc ⊂ A − , for a suitable neighborhood Ω of the origin. Using again Lemma 2.7 and Proposition 3.2 and reasoning as above we see that if Q ∈ W s + then φ(t; Q) = (x(t; Q), y(t; Q)) is such thatẋ(t; Q) < 0 for any t ∈ R, and W s + is a graph on π. Now let 0 < L 1 < L 2 < +∞, and let φ(t; Q(L i )) be the trajectory of (2.2) corresponding to the fast decay solution v(r, L i ) of (1.2), for i = 1, 2, so that From Lemma 2.9 we see that 0 < Q x (L 1 ) < Q x (L 2 ) and that 0 < x(t; Q(L 1 ) < x(t; Q(L 2 ) < P x for any t ∈ R; then Theorem 1.2 immediately follows. Now we turn to consider the p > 2 case: in this setting we use the argument developed by Miyamoto in [25]. So, with a slight adaption of [25] we rewrite (2.2) as follows: We denote byφ(t; Q) = (x(t; Q),z(t; Q)) a solution of (3.6) leaving from Q at t = 0. Moreover, we rewrite (3.6) asφ(t) =f (φ), i.e.f (φ) is the right hand side of (3.6). The pointP = (P x , 0) (where P x is given in (2.3)) is the critical point of system (3.6) corresponding to the critical point P of (2.2). Notice that from a lengthy but straightforward computation we find df dφ We emphasize that df dφ has the same trace T and determinant D as df dφ (P ), cf. (2.5). Hence, as it has to be expected,P has the same stability properties as P , i.e. we have the following.
Further its eigenvalues are the λ i given in (2.6), and whenP is a node the eigenvectors are given byṽ i = (1, λ i ).
Passing from (2.2) to (3.6) the 1 dimensional manifolds W u + and W s + are driven into the 1 dimensional manifoldsW u + andW s + by a global diffeomorphism (which brings the nullclineẋ = 0 into the x axis, and it is linear when p = 2). Obviously the trajectories inW u + andW s + correspond respectively to regular and fast decay solutions of (1.2) and all the results in §2 hold forW u + andW s + too, apart from the ones regarding the tangent in the origin. We omit the computation which is quite similar to the one carried on for system (2.2), see also the analogous computation performed in [25, Lemma 2.5]. Let us setr Notice that the semi-line˜ of (3.6) corresponds to the x-positive semi-axis of (2.2), so to solutions u(r) of (1.2) such that u (r) = 0. We stress that if p * (δ) < q ≤ p jl (δ) then λ 2 > λ 1 > 0; hencer 1 lies in the semiplane z < 0 and it intersects the z negative semi-axis in a pointS z = (0, Z − ), with Z − < 0, while if q ≥ p JL (δ) then λ 1 < λ 2 < 0; hencer 2 lies in the z > 0 semiplane and it intersects˜ in the pointS . If p * (δ) < q ≤ p jl (δ) we denote byB − the compact set enclosed by the segmentsS z O,S zP and OP . Similarly if q ≥ p JL (δ) we denote byB + the compact set enclosed by the segments OP , OS andS P (see Fig. 6).
Proof. The flow of (3.6) on the segment of the x positive semi-axis between the origin andP points upwards, this can be checked directly; similarly we see that on the z-negative semi-axis the flow points towards x < 0. We recall that the line˜ of (3.6) corresponds to the x positive semi-axis of (2.2): hence the vector field points towards the interior ofB + . It remains to check the vector field onr i . Observe first thatψ i (t) =P + v i e λi = (P x + e λit , λ i e λit ), for i = 1, 2 is a solution of the autonomous linear equatioṅ and Now we prove the claim. Let s ∈ (0, P x ]; we define the following function: By constructionc(s) evaluatesφ(t; Q) −ψ 1 (Q) for Q = (s; λ 1 (s − P x )) ∈r.
Since the non-linear part ofc(s) just depend on the presence ofg in (3.6) we havec 2 (0) =c 2 (0) = 0; furtherc 1 (s) ≡ s − s = 0. To prove the claim it is sufficient to show thatc 2 (s) < 0 for any s ∈ (0, P x ]. Once again, since all the linear terms cancel out, it is enough to differentiatẽ where we have set for simplicity A(s) := αs − λ 1 (s − P x ). Differentiating the previous expression we find Hence we find ds 2 (s) = d 2G ds 2 (s) < 0 for any 0 < s < P x and the claim is proved. Now, using the claim, we easily conclude in both the cases: q ≥ p JL (δ) and p * (δ) < q ≤ p jl (δ) Remark 3.5. The second part of the Lemma can be obtained by [25,Lemma 2.6] and in fact our proof is a slight simplification and a geometrical interpretation of the one by Miyamoto. The first part is obtained using Miyamoto's idea in the subcritical context.
Proof of Theorems 1.1 and 1.2 in the p > 2 case We develop the proof just for Theorem 1.2, Theorem 1.1 is analogous. Let p * (δ) < q ≤ p jl (δ): we claim thatW s + is a graph on π : ∈ Ω for any t ≥ 0}, for a suitable neighbourhood Ω of the origin. In fact applying Lemma 2.7 and Proposition 2.10 to W s + and passing to system (3.6), we can choose Ω so that if Q ∈W s loc , theñ φ(t; Q) = (x(t; Q),ỹ(t; Q)) ∈B − for any t ≥ 0. Further from Proposition 3.4 we see thatφ(t; Q) ∈B − for any t ≤ 0. Henceẋ(t; Q) < 0 for any t ∈ R, and W s + is a graph on π.
Then we easily conclude the proof repeating the argument of the proof of the 1 < p ≤ 2 case.
From Theorems 1.1 and 1.2 , we easily deduce the following results which are useful in a parabolic context.
Let O(r a ) as r → +∞ (respectively as r → 0) denote a function such that O(r a )r −a has a finite limit, possibly null as r → +∞ (respectively as r → 0). Corollary 3.6. Assume H and q > p JL (δ), q ≥ 2, then for any > 0 the GS have the following expansion as r → +∞: In the p = 2, δ = 0 case Corollary 3.6 was an essential ingredient to construct sub-super solutions for (1.5); then these sub-super solutions allowed to prove interesting results concerning the rate of convergence of the solutions of (1.5) to the stationary GS, see e.g. [9,19,21] and references therein.
Proof. Since u(r; d) is monotone increasing in d for any r ≥ 0, we immediately find that w(r; d 0 ) ≥ 0.
We claim that, as r → +∞, we find where c(1) < 0 and > 0 is an arbitrarily small positive constant. Then from (2.10) we see that (3.14) holds and c(d) = c(1)d −|λ2|/α , hence c(d) < 0, c (d) > 0, and we prove the Corollary. Now we prove the claim, using a new geometrical idea. Let φ(t) = (x(t), y(t)) be the trajectory corresponding to u(r; 1). From standard facts of invariant manifold theory, see e.g. [5, §13.4], we know that any trajectory converging to P as t → +∞ satisfies (3.16) where R(t) = O(e (2λ2+ )t ), and (a(1), b(1)) ∈ R 2 . So we immediately see that u(r; 1) can be expanded as in (3.15), but we have to show that c(1) < 0. Generically, i.e. if b(1) = 0, a trajectory converging to P is tangent to the line y = −m 2 (x − P x ) + P y , cf. (2.7), but if b(1) = 0 then it is tangent to y = −m 1 (x−P x )+P y . However from an inspection of the proof of Theorem 1.1 and of the positively invariant regions for 1 < p ≤ 2 and for p > 2, we see that just the former possibility takes place, whence b(1) = 0. Further, since x(t) − P x < 0 for any t ∈ R we see that c(1) < 0 and the claim is proved. To construct W u + we look for a positively invariant triangular like set E u , then we conclude with a topological argument based on Wazewski's principle.
Let We now consider the curve y = −εx q−1 where ε > 0 is a small constant to be fixed below. Let us set B u y = −ε(A x ) q−1 and B u = (A x , B u y ); we define the following sets: (4.22) Then E u ∪ {(0, 0)} is the compact set enclosed by A, B u and U, see Fig. 7; we have the following. Proof. If Q ∈ U the proof is obtained simply by observing that, by construction, we are in the set whereẋ > 0. If Q ∈ A it follows from a straightforward computation, or from the fact that we are on the branch of the nullcineẋ = 0 between the origin and A (so on the left of P In fact, using (2.2) and the fact that y = −εx q−1 , we find Hence D < 0 if ε < ε u , so (4.23) follows and the Lemma is proved.
Let us set W u loc := {Q | φ(t; Q) ∈ E u for any t ≤ 0}. A priori a trajectory may converge to the origin in finite time since local uniqueness on the coordinate axes is not ensured. However, from an inspection of the proof of Proposition 4.2, we find the following. From an elementary analysis of the phase portrait of (2.2) we easily obtain the viceversa.  A and B u ): since U is connected we have found a contradiction. Hence W u loc is a closed non-empty set. Let Q u ∈ (W u loc ∩U), then by construction w u loc := {φ(t; Q u ) | t ≤ 0} is a 1 dimensional manifold and w u loc ⊂ W u loc . Further, from elementary consideration on the phase portrait it is easy to check that W u loc is connected. Lemma 4.7. Assume q > p * (δ), p > 1, then W u + is a 1 dimensional immersed manifold.
Proof. We claim that W u loc ∩ U is a singleton say {Q u }. In fact assume by contradiction that there is P u ∈ (W u loc ∩U), P u = Q u . Let us denote by w u P := {φ(t; P u ) | t ≤ 0}, and by w u Q := {φ(t; Q u ) | t ≤ 0}; then by construction w u P ⊂ W u loc and w u Q ⊂ W u loc . Further w u Q ∩ w u P = ∅, due to local uniqueness of the solutions of (2.2). Finally if R ∈ (w u P ∪ w u Q ) then the trajectory φ(t; R) of (2.2) corresponds to a regular solution u(r; d(R)) of (1.2), where d(R) > 0,  ). Assume to fix the ideas d(Q u ) > d(P u ). Using invariance for t-translations of (2.2) as in (2.9), we find that there areR ∈ w u Q andτ = 1 α ln d(Q u ) d(P u ) > 0 such that u(r; d(R))r α = x(t;R) = x(t −τ ; Q u ) = u(re −τ ; d(Q u ))r α e −ατ .
Hence d(R) = d(Q u )e −ατ = d(P u ). (4.24) But this is a contradiction sinceR ∈ w u Q while P u ∈ w u P and w u Q ∩ w u P = ∅ and d(·) is injective; so the claim is proved and W u loc ∩ U is a singleton. Then we easily see that W u + = w u Q = {φ(t; Q u ) | t ∈ R}, hence W u + is a 1 dimensional immersed manifold.
The construction of the stable set is completely analogous.
We consider the curve x = ε|y| 1/(p−1) where ε > 0 is a small constant to be fixed below. Let us set B s x = ε|A y | 1/(p−1) and B s = (B s x , A y ). Then we define the following sets.  Proof. Observe that if 0 < x < P x the nullclineẋ = 0 lies below the nullclinė y = 0, since q > p, cf Fig. 4; hence if Q ∈ S the proof is obtained simply by observing that we are in the set whereẏ > 0 by construction. If Q ∈ A it follows from the fact that we are on the nullclineẋ = 0 between the origin and A (so on the left of P ).
If Q = (x, y) ∈ B s we have to show that In fact, using (2.2) and the fact that |y| 1/(p−1) = x/ε, we find D s =ε p−1 (γy − kx q−1 ) + (p − 1)x p−2 (αx − |y| 1/(p−1) ) Hence D s < 0 if 0 < ε < ε s , so (4.26) follows and the Lemma is proved. Once again priori a trajectory may converge to the origin in finite time since local uniqueness on the coordinate axes is not ensured. However, from an inspection of the proof of Proposition 4.2, we find the following.  Lemma 4.11. Assume q > p * (δ), then W s + is a 1 dimensional immersed manifold.
Proof. We just need to show that W s loc ∩ S is a singleton, say {Q s }; then we conclude the proof arguing as in Lemma 4.7.
Assume by contradiction that there is P s ∈ (W s loc ∩S), P s = Q s . Denote by w s P := {φ(t; P s ) | t ≥ 0} ⊂ E s , and by w s Q := {φ(t; Q s ) | t ≥ 0} ⊂ E s : by construction w s P ⊂ W s + and w s Q ⊂ W s + , and w s P ∩ w s Q = ∅, due to local uniqueness of the solutions of (2.2) outside the coordinate axes. If R ∈ (w s P ∪ w s Q ) then the trajectory φ(t; R) of (2.2) corresponds to a fast decay solution v(r; L(R)) of (1.2), where L(R) > 0. From Theorem 1.6 we see that L(·) : W s + → [0, +∞) is injective, so in particular L(Q s ) = L(P s ). Assume to fix the ideas L(Q s ) > L(P s ); using invariance for t-translations of (2.2) as in (2.11), we find that there areR ∈ w s Q andτ = But this is a contradiction sinceR ∈ w s Q while P s ∈ w s P , w s Q ∩ w s P = ∅, and L is injective so the Lemma is proved. Remark 4.12. Lemma 2.4 in the general p > 1 case now immediately follows from Lemmas 4.7 and 4.11 , while Lemma 2.7 is easily obtained observing that by construction W u loc ⊂ E u and W s loc ⊂ E s .