J Ér Ôme

V Étois

EXISTENCE OF SHARP ASYMPTOTIC PROFILES OF SINGULAR SOLUTIONS TO AN ELLIPTIC EQUATION WITH A SIGN-CHANGING NON-LINEARITY

classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy-Sobolev growth

where B denotes the open unit ball centred at 0 in R n for n ≥ 3, s ∈ (0, 2), 2 (s) := 2(ns)/(n -2), µ > 0 and q > 1. For q ∈ (1, 2 -1) with 2 = 2n/(n -2), it was shown in the op. cit. that the positive solutions with a non-removable singularity at 0 could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for µ > 0, we prove that for every q ∈ (2 (s) -1, 2 -1) there exist infinitely many positive solutions satisfying |x| s/(q-2 (s)+1) u(x) → µ -1/(q-2 (s)+1) as |x| → 0, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with lim inf |x|→0 |x| (n-2)/2 u(x) = 0 and lim sup |x|→0 |x| (n-2)/2 u(x) ∈ (0, ∞) if (and only if) q ∈ (2 -2, 2 -1).

INTRODUCTION AND MAIN RESULTS

The Hardy-Sobolev inequality is obtained by interpolating between the Sobolev inequality (s = 0) and the Hardy inequality (s = 2): For every s ∈ (0, 2) and n ≥ 3, there exists a positive constant K s,n such that

R n |∇u| 2 dx ≥ K s,n R n |x| -s |u| 2 (s) dx 2 2 (s) for all u ∈ C ∞ c (R n ),
where 2 (s) := 2(ns)/(n -2) denotes the critical Hardy-Sobolev exponent. The critical Sobolev exponent 2 corresponds to 2 (s) with s = 0. Recent results and challenges on the Hardy-Sobolev inequalities are surveyed by Ghoussoub-Robert in [START_REF] Ghoussoub | Sobolev inequalities for the Hardy-Schrödinger operator: extremals and critical dimensions[END_REF], see also [START_REF] Ghoussoub | The Hardy-Schrödinger operator with interior singularity: the remaining cases[END_REF]. For s ∈ (0, 2), the best Hardy-Sobolev constant K s,n is attained by a one-parameter family (U η ) η>0 of functions

(1)

U η (x) := c n,s η n-2 2 η 2-s + |x| 2-s -n-2 2-s for x ∈ R n ,
where c n,s := ((ns)(n -2)) 1/(2 (s)-2) is a positive normalising constant. The functions U η are the only positive non-singular solutions of the equation (see Chen-Lin [START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF] and Chou-Chu [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF])

(2)

-∆U = |x| -s U 2 (s)-1 in R n \ {0}.
Moreover, any positive C 2 (R n \ {0}) singular solution U of ( 2) is radially symmetric around 0 and v(t) = e -(n-2)t/2 U(e -t ) is a positive periodic function of t in R (see Hsia-Lin-Wang [START_REF] Hsia | Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations[END_REF]). The isolated singularity problem has been studied extensively, see Véron's monograph [START_REF] Véron | Singularities of solutions of second order quasilinear equations[END_REF]. Recent works of the first author and her collaborators such as [START_REF] Chang | Singular solutions for divergence-form elliptic equations involving regular variation theory: existence and classification[END_REF][START_REF] Cîrstea | A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials[END_REF][START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF] give a full classification of the isolated singularities for various classes of elliptic equations.

In this paper, we settle an open question arising from [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF] with regard to the existence of all the singular profiles at zero for the positive solutions of the perturbed non-linear elliptic equation [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] -∆u = |x| -s u 2 (s)-1µu q for x ∈ B(0, R) \ {0}, where µ is a positive parameter, q > 1 and s ∈ (0, 2). By B(0, R) we denote the open ball in R n (n ≥ 3) centred at 0 with radius R > 0. The first two authors have proved in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF] that the positive singular solutions of (3) can exhibit up to three types of singular profiles at zero in a suitable range for q:

• A (ND) type profile (for "Non Differential") if

lim |x|→0 |x| s q-(2 (s)-1) u(x) = µ - 1 q-(2 (s)-1)
.

(ND)

• A profile of (MB) type (for "Multi-Bump") in the sense that there exists a sequence (r k ) k≥0 of positive numbers decreasing to 0 such that r k+1 = o(r k ) as k → +∞ and

u(x) = (1 + o(1)) ∞ ∑ k=0
U r k (x) as |x| → 0, where U η is as in [START_REF] Bidaut-Veron | A new dynamical approach of Emden-Fowler equations and systems[END_REF].

(MB)

• A profile of (CGS) type (for "Caffarelli-Gidas-Spruck") if there exists a positive periodic function

v ∈ C ∞ (R) such that lim |x|→0 |x| n-2 2 u(x) -v(-log |x|) = 0.
(CGS)

The case q = 2 -1 in (3) was fully dealt with in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF]. Hence, in the sequel we assume that q = 2 -1. We recall the relevant classification result from [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF]:

Theorem 1.1 ( [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF]). Let u ∈ C ∞ (B(0, R) \ {0}) be an arbitrary positive solution to (3).

• If q > 2 -1, then 0 is a removable singularity;

• If 2 (s) -1 < q < 2 -1, then either 0 is a removable singularity, or u develops a profile of type (CGS), (MB) or (ND); • If 1 < q ≤ 2 (s) -1, then either 0 is a removable singularity, or u has a profile of type (CGS) or (MB).

Moreover, if u develops a profile of (MB) type, then 2 -2 < q < 2 -1.

However, no examples of the three singular profiles of Theorem 1.1 were given in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF], leaving open the question of their existence. In the present paper, we fill this gap by proving the following: Theorem 1.2. The three singular profiles of Theorem 1.1 actually do exist.

The existence assertion of Theorem 1.2 is a corollary of the following precise result: Theorem 1.3. Equation (3) admits positive radially symmetric solutions developing (CGS), (MB) and (ND) profiles in the exact range of parameters given by Theorem 1.1. More precisely, when q ∈ (1, 2 -1), there exists R 0 > 0 such that for every R ∈ (0, R 0 ), the following hold:

(i) For every γ > 0, there exists a unique positive radial solution u γ of (3) with a removable singularity at 0 and lim |x|→0 u γ (x) = γ. (ii) If q > 2 -2, then (3) has at least a positive (MB) solution.

(iii) For every positive singular solution U of (2), there exists a unique positive radial (CGS) solution u of (3) with asymptotic profile U near zero. (iv) If q > 2 (s) -1, then (3) admits infinitely many positive (ND) solutions.

Remark 1.4. If q ∈ (1, 2 (s) -1), then all positive radial solutions of (3) extend as positive radial solutions in R n \ {0}. For q ∈ [2 (s) -1, 2 -1), any positive radial non-(ND) solution u of (3) extends as a positive radial solution at least in B(0, R * ) \ {0} with R * independent of u (see Lemma 3.2).

From the three singular profiles of (3), only the (CGS) type is reminiscent of the asymptotics of the local singular solutions for the Yamabe problem in the case of a flat background metric (µ = s = 0) studied in Caffarelli-Gidas-Spruck [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] (see also Korevaar-Mazzeo-Pacard-Schoen [START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF] for a refined asymptotics and Marques [START_REF] Marques | Isolated singularities of solutions to the Yamabe equation[END_REF] for the case of a general background metric). But for µ > 0, the introduction of the perturbation term in (3) yields two new singular profiles: the (ND) and (MB) types.

An important novelty in this paper is the existence of infinitely many positive radial (ND) solutions for (3) when q ∈ (2 (s) -1, 2 -1). To our best knowledge, there are no previous existence results known for this type of singularities, which arise as a consequence of studying [START_REF] Caffarelli | Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth[END_REF] with a critical Hardy-Sobolev growth (i.e., s ∈ (0, 2)) rather than with a critical Sobolev growth (s = 0). Since (4) fails for the (ND) solutions, neither Pohozaev-type arguments nor Fowler-type transformations relevant for (CGS) or (MB) profiles can be used. Specific to the (ND) solutions, the first term in their asymptotics arises from the competition generated in the right-hand side of (3) and not directly from the differential structure. To overcome this obstacle, we rewrite the radial form of (3) as a dynamical system using an original transformation involving three variables, see [START_REF] Cîrstea | A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials[END_REF]. The variable X 1 in ( 10) is suggestive of a second order term in the asymptotics of the (ND) solutions, which will make apparent the differential structure of our equation in a dynamical systems setting. Nevertheless, by linearising the flow around the critical point, we find a positive eigenvalue, a null one and a negative eigenvalue so that we cannot apply the classical Hartman-Grobman theorem. Instead, we shall use Theorem 7.1 in the Appendix, which invokes the notion of center-stable manifold and ideas of Kelley [START_REF] Kelley | The stable, center-stable, center, center-unstable, unstable manifolds[END_REF].

For 1 < q < 2 -1, Theorem 1.1 yields that every positive non-(ND) solution of (3) satisfies

(4) lim sup |x|→0 |x| n-2 2 u(x) < ∞.
Moreover, (4) holds for every positive solution of (3) when q ∈ (1, 2 (s) -1]. Note that (4) is crucial for Pohozaev type arguments [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF], on the basis of which we prove in Sect. 3 the non-existence of smooth positive solutions for (3), subject to u = 0 on ∂ B(0, R).

Theorem 1.5. Let µ > 0 and s ∈ (0, 2) be arbitrary. Let Ω be a smooth bounded domain in R n (n ≥ 3) such that 0 ∈ Ω. Assume that Ω is star-shaped with respect to 0. Then, for every q ∈ (1, 2 (s) -1], there are no positive smooth solutions for the problem Motivated by the problem of finding a metric conformal to the flat metric of R n such that K(x) is the scalar curvature of the new metric, Chen-Lin [START_REF] Chen | Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent[END_REF][START_REF] Chen | Estimates of the conformal scalar curvature equation via the method of moving planes[END_REF][START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF] and Lin [START_REF] Lin | Estimates of the scalar curvature equation via the method of moving planes[END_REF] analysed the local behaviour of the positive singular solutions u ∈ C 2 (B(0, 1) \ {0}) to ( 6)

(5) -∆u = |x| -s u 2 (s)-1 -µu q in Ω \ {0}, u = 0 on ∂ Ω. If q ∈ (2 (s) -1, 2 -1), then (
-∆u = K(x) u 2 -1 in B(0, 1) \ {0},
where K is a positive continuous function on B(0, 1) in R n (n ≥ 3) with K(0) = 1. Moreover, K was always assumed to be a C 1 function on B(0, 1) \ {0} such that

(7) 0 < L := lim inf |x|→0 |x| 1-|∇K(x)| ≤ L := lim sup |x|→0 |x| 1-|∇K(x)| < ∞ for some > 0.
In the above-mentioned works (see also Lin-Prajapat [START_REF] Lin | Asymptotic symmetry of singular solutions of semilinear elliptic equations[END_REF] and Taliaferro-Zhang [START_REF] Taliaferro | Asymptotic symmetries for conformal scalar curvature equations with singularity[END_REF]), the following question was investigated: Under what conditions on K, the positive singular solutions of (2) with s = 0 are asymptotic models at zero for the positive singular solutions of (6)? This question was settled positively in any of the following situations: Extra conditions in situation (b) are needed to guarantee a positive answer to the above question. Otherwise, for every 0 < < (n -2)/2, Chen-Lin [START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF]Theorem 1.6] provided general positive radial functions K(r) non-increasing in r = |x| ∈ [0, 1] with K(0) = 1 such that [START_REF] Chen | Estimate of the conformal scalar curvature equation via the method of moving planes[END_REF] holds and ( 6) has a positive singular solution with lim inf |x|→0 |x| (n-2)/2 u(x) = 0.

The importance of condition [START_REF] Chen | Estimate of the conformal scalar curvature equation via the method of moving planes[END_REF] in settling the above question can be inferred from our next result as a by-product of Theorem 1.3(ii): For every 0 < < min{(n -2)/2, 2} and s ∈ (0, 2) \ { }, we construct a positive continuous function K on B(0, R) for some R > 0 with K(0) = 1 such that exactly one inequality in [START_REF] Chen | Estimate of the conformal scalar curvature equation via the method of moving planes[END_REF] fails, yet generating for (8) a positive singular solution, the asymptotics of which at zero cannot be modelled by any positive singular solution of (2).

Corollary 1.6. For every 0 < < min{(n -2)/2, 2} and s ∈ (0, 2) \ { }, there exist R > 0 and a positive C 1

- function K on B(0, R)\{0} in R n (n ≥ 3) with K < lim |x|→0 K(x) = 1 on B(0, R)\{0} such that 0 = L < L < ∞ if < s and 0 < L < L = ∞ if > s, yet (8)
-

∆u = K(x)|x| -s u 2 (s)-1 in B(0, R) \ {0}
admits a positive singular solution with lim inf |x|→0 |x| (n-2)/2 u(x) = 0.

Structure of the paper. In Sect. 2, we prove Theorem 1.3(iv) on the existence of infinitely many positive (ND) solutions for (3). In Sect. 3, we establish Theorem 1.5, together with uniform a priori estimates for the positive radial solutions of (3) satisfying (4) (see Proposition 3.1). In Sect. 4, by setting u(r) = y(ξ ) with ξ = r (2-s)/2 , we reduce the assertion of Theorem 1.3(i) on removable singularities to the existence and uniqueness of the solution for (44) on an interval [0, T ]. The latter follows from Biles-Robinson-Spraker [2, Theorems 1 and 2]. In Sect. 5, after giving the proof of Corollary 1.6, we use an argument influenced by Chen-Lin [START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF] to prove the existence of (MB) solutions for (3) in the whole possible range q ∈ (2 -2, 2 -1).

In Sect. 6, with a dynamical system approach, we prove Theorem 1.3(iii): the positive singular solutions of (2) serve as asymptotic models for the positive radial (CGS) solutions of (3). For a dynamical approach to Emden-Fowler equations and systems, see Bidaut-Véron-Giacomini [START_REF] Bidaut-Veron | A new dynamical approach of Emden-Fowler equations and systems[END_REF]. The results in this paper give the existence and profile at infinity for the positive solutions to

-∆ ũ = |x| -s ũ2 (s)-1 -µ|x| (n-2)q-(n+2) ũq for |x| > 1/R by using the Kelvin transform ũ(x) = |x| 2-n u(x/|x| 2 )
, where u is a positive solution of (3).

(ND) SOLUTIONS

In this section, we let q ∈ (2 (s) -1, 2 -1) and prove Theorem 1.3(iv), restated below.

Proposition 2.1. Assume that q ∈ (2 (s)-1, 2 -1). Then, there exists R 0 > 0 such that for every R ∈ (0, R 0 ), equation (3) admits infinitely many positive (ND) solutions.

The proof of Proposition 2.1 takes place in several steps. First, we reformulate the radial form of (3) as a first order autonomous differential system using a new transformation, see [START_REF] Cîrstea | A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials[END_REF].

2.1. Formulation of our problem as a dynamical system. We first assume that u is a positive radial (ND) solution of (3). We define [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF] ϑ := s q -2 (s) + 1

,

β := (q -1) ϑ 2 -1, ζ := 2 (s) -2 q -2 (s) + 1 .
We introduce a new transformation involving three functions X 1 , X 2 and X 3 as follows

(10) X 1 (t) = t 1 -µr s u q-2 (s)+1 , X 2 (t) = 1 t , X 3 (t) = ru (r) u(r) + ϑ ,
where t := r -β and β , ϑ are given by [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF]. Since u is a positive radial (ND) solution of (3), that is,

lim r→0 + r ϑ u(r) = µ -1/(q-2 (s)+1) , it follows that (11) 1 -X 1 (t)X 2 (t) = µr s u(r) q-2 (s)+1 > 0 for all t ∈ [2R -β , ∞), X 1 (t)X 2 (t) → 0 as t → ∞.
If we set X = (X 1 , X 2 , X 3 ), then, as one easily checks, we have that ( 12)

X (t) = (H 1 ( X(t)), H 2 ( X(t)), H 3 ( X(t)))
for all t ∈ [2R -β , ∞), where H 1 , H 2 and H 3 are real-valued functions defined on R 3 by ( 13)

       H 1 (ξ 1 , ξ 2 , ξ 3 ) := ξ 1 ξ 2 + β -1 (q -2 (s) + 1)(1 -ξ 1 ξ 2 )ξ 3 , H 2 (ξ 1 , ξ 2 , ξ 3 ) := -ξ 2 2 , H 3 (ξ 1 , ξ 2 , ξ 3 ) := β -1 µ -ζ ξ 1 (1 -ξ 1 ξ 2 ) ζ + + β -1 ξ 2 (ξ 3 -ϑ )(ξ 3 -ϑ + n -2).
By ξ + we mean the positive part of ξ . We define

Y := (Y 1 ,Y 2 ,Y 3 ), where Y (t) = X(t + 2R -β ) for all t ≥ 0.
Then, [START_REF] Ghoussoub | Sobolev inequalities for the Hardy-Schrödinger operator: extremals and critical dimensions[END_REF] gives that

Y (t) = (H 1 ( Y (t)), H 2 ( Y (t)), H 3 ( Y (t))) for all t ∈ [0, ∞).
To get more regularity, for any ε ∈ (0, 1), we choose Ψ ε ∈ C 1 (R) such that Ψ ε (t) = t ζ for all t ≥ ε. By choosing ε 0 ∈ (0, 1) small enough and using [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF], we find that ( 14)

Y (t) = (H 1 ( Y (t)), H 2 ( Y (t)), H 3,Ψ ε ( Y (t))) for all t ∈ [0, ∞)
for every ε ∈ (0, ε 0 ), where the function H 3,Ψ ε : R 3 → R is defined by

H 3,Ψ ε (ξ 1 , ξ 2 , ξ 3 ) := β -1 µ -ζ ξ 1 Ψ ε (1 -ξ 1 ξ 2 ) + β -1 ξ 2 (ξ 3 -ϑ )(ξ 3 -ϑ + n -2).
2.2. Existence of solutions for [START_REF] Hsia | Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations[END_REF]. Using ϑ , β and ζ in (9), we define ϒ and Γ by ( 15)

ϒ := µ ζ /2 q -2 (s) + 1 and Γ := ϑ (n -2 -ϑ ) µ ζ . Lemma 2.2. Let q ∈ (2 (s) -1, 2 -1) and ε ∈ (0, 1). Fix Ψ ε ∈ C 1 (R) such that Ψ ε (t) = t ζ for all t ≥ ε.
For every δ > 0 small, there exist r 0 ∈ (0, δ /2) and a Lipschitz function w

: [0, r 0 ] × [-r 0 , r 0 ] → [-r 0 , r 0 ]
such that for any (Y 2,0 , Z 3,0 ) ∈ (0, r 0 ] × [-r 0 , r 0 ], the system (14) subject to the initial condition

(16) Y (0) = (ϒ(Z 3,0 -w(Y 2,0 , Z 3,0 )) + ΓY 2,0 ,Y 2,0 , w(Y 2,0 , Z 3,0 ) + Z 3,0 ) has a solution Y (t) = (Y 1 (t),Y 2 (t),Y 3 (t)) for all t ≥ 0 satisfying (17) lim t→+∞ Y (t) = (0, 0, 0).
Moreover, we have Y 2 (t) = 1/(t +Y -1 2,0 ) for all t ≥ 0.

Proof. Since Ψ ε (1) = 1, we find one critical point (0, 0, 0) for [START_REF] Hsia | Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations[END_REF]. Linearising the flow around (0, 0, 0), we get one unstable eigenvalue λ 1 = µ -ζ /2 β -1 q -2 (s) + 1 with associated eigenvector (ϒ, 0, 1), one null eigenvalue with associated eigenvector (Γ, 1, 0) and one stable eigenvalue -λ 1 with associated eigenvector (ϒ, 0, 1). For Z = (Z 1 , Z 2 , Z 3 ), using a change of coordinates

(18) Y = (ϒ(Z 1 -Z 3 ) + ΓZ 2 , Z 2 , Z 1 + Z 3 ), i.e., Z = Y 1 -ΓY 2 + ϒY 3 2ϒ ,Y 2 , ΓY 2 + ϒY 3 -Y 1 2ϒ
, we bring the system (14) to a diagonal form, namely

Z (t) = (λ 1 Z 1 (t) + h 1 ( Z(t)), -Z 2 2 (t), -λ 1 Z 3 (t) + h 3 ( Z(t) (19)
)) for all t ≥ 0. For any δ > 0 small, the functions h 1 and h 3 are C 1 on the ball B δ (0) in R 3 centred at 0 with radius δ . Moreover, for some constant C 1 > 0, the functions h 1 and h 3 satisfy [START_REF] Taliaferro | Asymptotic symmetries for conformal scalar curvature equations with singularity[END_REF] |h

1 ( ξ )| + |h 3 ( ξ )| ≤ C 1 3 ∑ j=1 ξ 2 j and |∇h 1 ( ξ )| + |∇h 3 ( ξ )| ≤ C 1 3 ∑ j=1 |ξ j | for all ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ B δ (0)
. By [START_REF] Lin | Asymptotic symmetry of singular solutions of semilinear elliptic equations[END_REF], proving Lemma 2.2 is equivalent to showing that for every small δ > 0, there exist r 0 ∈ (0, δ /2) and a Lipschitz map w :

[0, r 0 ] × [-r 0 , r 0 ] → [-r 0 , r 0 ] such that for all (Y 2,0 , Z 3,0 ) ∈ (0, r 0 ] × [-r 0 , r 0 ], the system (19) subject to (21) Z(0) = (w(Y 2,0 , Z 3,0 ),Y 2,0 , Z 3,0 )
has a solution Z(t) for all t ≥ 0 with lim t→+∞ Z(t) = (0, 0, 0). Linearising the flow for ( 19) around (0, 0, 0) yields one null eigenvalue, and the classical Hartman-Grobman theorem does not apply to [START_REF] Marques | Isolated singularities of solutions to the Yamabe equation[END_REF]. In Appendix, using the notion of center-stable manifold and inspired by Kelley [START_REF] Kelley | The stable, center-stable, center, center-unstable, unstable manifolds[END_REF], we prove Theorem 7.1 that can be applied to [START_REF] Marques | Isolated singularities of solutions to the Yamabe equation[END_REF] due to [START_REF] Taliaferro | Asymptotic symmetries for conformal scalar curvature equations with singularity[END_REF]. This ends the proof.

2.3. Proof of Proposition 2.1. For fixed ε ∈ (0, 1), we choose

Ψ ε ∈ C 1 (R) such that Ψ ε (t) = t ζ for all t ≥ ε. Let δ ∈ (0, (1 -ε) 1/2
). Let r 0 ∈ (0, δ /2) and w : [0, r 0 ] × [-r 0 , r 0 ] → [-r 0 , r 0 ] be given by Lemma 2.2. We fix Y 2,0 := r 0 /2. Then for any fixed Z 3,0 ∈ [-r 0 , r 0 ], the system [START_REF] Hsia | Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations[END_REF], subject to the initial condition [START_REF] Korevaar | Refined asymptotics for constant scalar curvature metrics with isolated singularities[END_REF] has a solution Y (t) for all t ≥ 0 such that (17) holds. Moreover, we find that Y 2 (t) = 1/(t +Y -1 2,0 ) for all t ≥ 0. Let t 0 > 0 be large such that Y (t) ∈ B δ (0) for all t ≥ t 0 . Using that 0 < ε < 1δ 2 , for all t ≥ t 0 , we get that

1-Y 1 (t)Y 2 (t) > ε so that Ψ ε (1-Y 1 (t)Y 2 (t)) = (1-Y 1 (t)Y 2 (t)) ζ . Hence, we have H 3,Ψ ε ( Y (t)) = H 3 ( Y (t)
) for all t ≥ t 0 . For every t ≥ T := t 0 +Y -1 2,0 , we define X(t) by X(t) := Y (t -Y -1 2,0 ), which yields that X 2 (t) = 1/t. Then, X(t) is a solution of the system (12) for all t ≥ T such that lim t→∞ X(t) = (0, 0, 0). With ϑ and β be given by [START_REF] Chou | On the best constant for a weighted Sobolev-Hardy inequality[END_REF] and t := r -β , we define u(r) as in [START_REF] Cîrstea | A complete classification of the isolated singularities for nonlinear elliptic equations with inverse square potentials[END_REF]. Then u is a positive radial (ND) solution of (3) with R := T -1/β . The above construction leads to an infinite number of positive radial (ND) solutions for (3) by varying Z 3,0 in [-r 0 , r 0 ]. This completes the proof.

CONSEQUENCES OF POHOZAEV'S IDENTITY

In this section, using Pohozaev's identity, we prove Theorem 1.5, followed by uniform a priori estimates for the positive radial solutions of (3) satisfying (4) (see Proposition 3.1).

Let u be any positive solution of (3) with q ∈ (1, 2 -1) such that (4) holds. As in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF], for every r ∈ (0, R), we denote by P (q) r (u) the Pohazev-type integral associated to u, namely ( 22)

P (q) r (u) := ∂ B(0,r) (x, ν) |∇u| 2 2 - u 2 (s) 2 (s)|x| s + µ u q+1 q + 1 -T (x, u) ∂ ν u dσ ,
where T (x, u) = (x, ∇u(x)) + (n -2)u(x)/2. Here, ν denotes the unit outward normal at ∂ B(0, r). Assuming u satisfies (4), it was shown in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF] that there exists lim r→0 + P (q) r (u) := P (q) (u) and ( 23)

P (q) (u) ≥ 0
with strict inequality if and only if u is a (CGS) solution of (3). We refer to P (q) (u) as the asymptotic Pohozaev integral. We introduce the notation (24) λ := (n -2)(2 -1q)/2 and c µ,q,n := λ µ/(q + 1).

Both λ and c µ,q,n are positive by the assumption q ∈ (1, 2 -1).

3.1. Proof of Theorem 1.5. Let q ∈ (1, 2 -1). Suppose that (5) admits a positive smooth solution u satisfying [START_REF] Chang | Singular solutions for divergence-form elliptic equations involving regular variation theory: existence and classification[END_REF]. From u = 0 on ∂ Ω, we have ∇u = (∂ ν u) ν for x ∈ ∂ Ω, where ν denotes the unit outward normal at ∂ Ω. For every r > 0 small, by applying the Pohozaev identity as in [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF]Proposition 6.1] for

ω = ω r = Ω \ B(0, r), we get that (25) - 1 2 ∂ Ω (x, ν)|∇u| 2 dσ = P (q) r (u) + c µ,q,n ω r u q+1 dx.
By letting r → 0 + in (25) and using (23), we arrive at

(26) - 1 2 ∂ Ω (x, ν)|∇u| 2 dσ = P (q) (u) + c µ,q,n Ω u q+1 dx ≥ 0.
Since Ω is star-shaped with respect to the origin, we have (x, ν) > 0 on ∂ Ω. Then, (26) can only hold when ∇u ≡ 0 on ∂ Ω and u ≡ 0 in Ω. Hence, (5) has no positive smooth solutions satisfying (4). Using the comments before statement of Theorem 1.5, we finish the proof.

3.2.

Uniform a priori estimates. Let q ∈ (1, 2 -1). For the positive radial solutions u of ( 3) satisfying (4), we derive uniform a priori estimates. These are crucial for proving the existence of (MB) solutions in Proposition 5.1 and (CGS) solutions in Proposition 6.1. We define

(27)      R(u) := sup{R > 0 : u is a positive radial solution of (3)}, z(r) := r n-2 2 u(r) for r ∈ (0, R), F 0 (ξ ) := (n -2) 2 4 ξ 2 - 2 2 (s) ξ 2 (s) for ξ ≥ 0.
If u has a removable singularity at 0 or u is a solution of (MB) type, then lim inf r→0 + z (r) = 0. If u is a (CGS) solution, then from [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF], we can derive that

(28) 0 < lim inf r→0 + z (r) ≤ [(n -2)/2] 2/(2 (s)-2) := M 0 .
For R > 0, we also define

(29) F R (ξ ) := (n -2) 2 4 - 2 2 (s) ξ 2 (s)-2 + 2µR λ ξ q-1 q + 1 for ξ ≥ 0.
For F 0 given by ( 27), let Λ 0 denote the unique positive solution of F 0 (ξ ) = 0, that is

(30) Λ 0 := [(n -2)(n -s)/4] 1 2 (s)-2 .
For any Λ > Λ 0 , we have

F 0 (Λ) < 0. Let R Λ denote the unique R > 0 for which F R (Λ) = 0: (31) R Λ := - (q + 1)F 0 (Λ) 2µΛ q+1 1 λ > 0.
Moreover, it holds

(32) R Λ = sup ξ > 0 : F R Λ (t) > 0 for all t ∈ (0, ξ ) .
Proposition 3.1 (Uniform a priori estimates). Let q ∈ (1, 2 -1). Then for every Λ > Λ 0 , there exists R Λ > 0 as in (31) such that any positive radial solution of (3) with R ∈ (0, R Λ ) satisfying (4) can be extended as a positive radial solution of (3) in B(0, R Λ ] \ {0} and

(33) r n-2 2 u (r) < Λ for all r ∈ (0, R Λ ].
Let ω n-1 denote the volume of the Euclidean (n -1)-sphere S n-1 in R n . Let λ and c µ,q,n be given by (24). For q > 2 (s) -1, we define q as follows (34)

q := (2 -s)(q + 1) (n -s)(q -1) (n -2)(n -s)(q -1) 4(q -2 (s) + 1) - q-2 (s)+1 2 (s)-2
.

A key tool in proving Proposition 3.1 is given by Lemma 3.2, which is of interest in its own. Lemma 3.2. Let q ∈ (1, 2 -1). Let u be a positive radial solution of (3) satisfying (4).

(a) For all r ∈ (0, R), the functions z and F r (z) in ( 27) and (29), respectively satisfy

(35) z 2 (r) F r (z(r)) = 2P (q) (u) ω n-1 + [rz (r)] 2 + 2c µ,q,n r 0 ξ n-1 u q+1 (ξ ) dξ . (b) If R < +∞, then lim inf r R u(r) > 0 and lim sup r R u(r) = +∞. (c) If 1 < q < 2 (s) -1, then R = +∞. (d) If q = 2 (s) -1, then R ≥ (1/µ) 1/s . (e) If q ∈ (2 (s) -1, 2 -1), then R > ( q /µ) 1/λ
, where q is given by (34).

Remark 3.3. We have q → 1 as q 2 (s) -1 and using F 0 in (27), we get

(36) q = q + 1 2 sup Λ∈(Λ 0 ,∞) -F 0 (Λ) Λ q+1 .
Proof. From our assumptions, it follows that lim r→0 + r n u q+1 (r) = 0.

Proof of (a). Since u is a radial solution of (3), the Pohozaev-type integral P

(q) r (u) satisfies (37) 2P (q) r (u) ω n-1 = -[rz (r)] 2 + z 2 (r) F r (z(r)) for all r ∈ (0, R).
By the Pohozaev identity, see [START_REF] Cîrstea | Sharp asymptotic profiles for singular solutions to an elliptic equation with a sign-changing non-linearity[END_REF]Proposition 6.1], for every 0 < r 1 < r < R, we find that ( 38)

P (q) r (u) -P (q) r 1 (u) = ω n-1 c µ,q,n r r 1 ξ n-1 u q+1 (ξ ) dξ .
Letting r 1 → 0 + in (38), for any r ∈ (0, R), we find that (39)

P (q) r (u) = P (q) (u) + ω n-1 c µ,q,n r 0 ξ n-1 u q+1 (ξ ) dξ .
Then we conclude (35) by using (37) and (39). The proof of (a) is now complete.

Proof of (b). Assume that R < +∞. To prove that lim inf r R u(r) > 0, we proceed by contradiction. Assume that for a sequence (r k ) k≥1 of positive numbers with r k R as k → ∞, we have lim k→∞ u(r k ) = 0, that is lim k→∞ z(r k ) = 0. We let r = r k in (35), then pass to the limit k → ∞ to obtain a contradiction. For the other claim in (b), assume that lim sup r R u(r) < +∞. Then lim sup r R z(r) < +∞ since R < +∞. By the classical ODE theory, it follows that lim sup r R |u (r)| = ∞. On the other hand, by (35), we get that lim sup r R |rz (r)| < +∞, which shows that lim sup r R |u (r)| < +∞. This contradiction completes the proof of (b).

Proof of (c). Let q < 2 (s) -1. If R < ∞, then there exists a sequence (r k ) k≥1 in (0, R) with lim k→∞ r k = R and lim k→∞ z(r k ) = +∞. By letting r = r k in (35) and k → ∞, the left-hand side of (35) diverges to -∞ as k → ∞, which is a contradiction. This proves that R = +∞.

Proof of (d). Let q = 2 (s) -1. We argue by contradiction. Assume that R < (1/µ) 1/s . Then, there exists (r k ) k≥1 in (0, R) with lim k→∞ r k = R and lim k→∞ z(r k ) = +∞. Since r λ = r s < Rs for all r ∈ (0, R), from (35) and the definition of F r in (29) (with R = r), we have

(40) (n -2) 2 z 2 (r k ) 4 - 2 (1 -µ Rs ) z 2 (s) (r k ) 2 (s) > 0 for all k ≥ 1.
By letting k → ∞ in (40) and using that 1µ Rs > 0, we get that the left-hand side of (40) tends to -∞ as k → ∞. This contradiction proves that R ≥ (1/µ) 1/s .

Proof of (e). Let q ∈ (2 (s) -1, 2 -1). To prove R > ( q /µ) 1/λ with q as in (34), it suffices to assume R < +∞. Let F R be the function F R in (29) with R = R. We distinguish two cases: CASE 1: If u has a removable singularity at 0, or u is a (MB) solution, then lim inf r→0 + z(r) = 0 using that z(r) = r n-2

2 u(r). Since lim sup r R z(r) = +∞, to ensure (35) for a positive radial solution u of (3) which is not (CGS) nor (ND), it is necessary to have

(41) F R(ξ ) > 0 for all ξ ∈ [0, ∞).
We next study the monotonicity of F R. We see that F R has only one positive critical point ξ c defined by 42) and (28), we get

(42) ξ c := (2 -s)(q + 1) µ(n -s)(q -1) Rλ 1 q-2 (s)+1 . Moreover, ξ c is a global minimum point for F on [0, ∞). Thus, (41) holds if and only if F R(ξ c ) > 0, which corresponds to R > ( q /µ) 1/λ . CASE 2: If u is a radial (CGS) solution of (3) then we need F R(ξ ) > 0 for every ξ ≥ lim inf r→0 + z(r). If M 0 in (28) satisfies M 0 ≤ ξ c then R > ( q /µ) 1/λ is necessary to have F R(ξ ) > 0 for every ξ ∈ [lim inf r→0 + z(r), +∞). If M 0 > ξ c , then from (
(43) Rλ > (2 -s)(q + 1) µ(n -s)(q -1) n -2 2 - 2(q-2 (s)+1) 2 (s)-2
, which again implies R > ( q /µ) 1/λ .

We have established the assertion of (e) in both Cases 1 and 2.

This completes the proof of Lemma 3.2.

Proof of Proposition 3.1. For any q ∈ [2 (s) -1, 2 -1), we denote R * = R * (q) as follows

R * := (1/µ) 1/s if q = 2 (s) -1, ( q /µ) 1/λ if 2 (s) -1 < q < 2 -1.
Let Λ > Λ 0 be fixed. Let u be any positive radial solution of (3) with R ∈ (0, R Λ ) such that (4) holds. From Lemma 3.2, the maximum radius of existence R = R(u) for 36) and (31), we have R Λ ≤ R * for all 2 (s) -1 < q < 2 -1. When q = 2 (s) -1, then using the definition of F 0 and R * , we see easily that R Λ < R * . Hence, we can extend u as a positive radial solution of (3) in B(0, R Λ ] \ {0} for all 1 < q < 2 -1.

u satisfies R = +∞ if 1 < q < 2 (s) -1, R ≥ R * for q = 2 (s) -1 and R > R * for 2 (s) -1 < q < 2 -1. From (
We now prove (33). Assume by contradiction that (33) fails, that is, z(r 0 ) ≥ Λ for some r 0 ∈ (0, R Λ ], where z(r) := r n-2

2 u(r) is defined as in (27). Since z(r 0 ) ≥ Λ > Λ 0 > M 0 , the Mean Value Theorem, together with (32) and (28), gives that there exists r 1 ∈ (0, r 0 ) such that z (r 1 ) = Λ. Hence, using Lemma 3.2(a), we find that 0 = Λ 2 F R Λ (Λ) > 0. This contradiction ends the proof of Proposition 3.1.

REMOVABLE SINGULARITIES

The assertion of Theorem 1.3(i) follows from Lemma 3.2 and Lemma 4.1 below. Lemma 4.1. For q > 1 and every γ ∈ (0, ∞), there exists R > 0 such that (3) has a unique positive radial solution u γ with a removable singularity at 0 and lim r→0 + u γ (r) = γ.

Proof. Fix γ ∈ (0, ∞) arbitrarily. We consider the following initial value problem:

(44) y (ξ ) + a y (ξ )/ξ + 4(y 2 (s)-1 -µ ξ 2s 2-s y q )/(2 -s) 2 = 0 for ξ > 0, y(0) = γ, y (0) = 0,
where we denote a := (2ns -2)/(2s). By Biles-Robinson-Spraker [2, Theorems 1 and 2], for every γ > 0, there exists a unique positive solution y γ of (44) on some interval [0, T ] with T > 0. A solution y of (44) is defined in [START_REF] Biles | A generalization of the Lane-Emden equation[END_REF] as follows: (c) y satisfies the initial conditions in (44). Since a > 1, the function ξ -→ ξ a y γ (ξ ) is absolutely continuous on [0, T ]. From (44), we have

ξ a y γ (ξ ) = - 4 (2 -s) 2 ξ a y 2 (s)-1 γ -µ ξ 2s 2-s y q γ a.e. in [0, T ].
Thus, for all ξ ∈ [0, T ], we find that

ξ a y γ (ξ ) = - 4 (2 -s) 2 ξ 0 t a y 2 (s)-1 γ (t) -µ t 2s 2-s y q γ (t) dt.
By the property (a) for y γ , we find that -s) and lim r→0 + u γ (r) = γ. This proves the existence claim.

y γ ∈ C 2 (0, T ] satisfies the ODE in (44) on (0, T ]. The change of variable u γ (r) = y γ (ξ ) with ξ = r (2-s)/2 yields that u γ is a positive radial C 2 (0, R]-solution of (3) with R = T 2/(2
We now show the uniqueness claim: any positive radial C 2 (0, R]-solution u of (3) for some R > 0 such that lim r→0 + u(r) = γ must coincide with u γ on their common domain of existence. Indeed, using the change of variable u(r) = y(ξ ) with ξ = r (2-s)/2 , we get that y ∈ C 2 (0, R (2-s)/2 ] satisfies the differential equation in (44) for all ξ ∈ (0, R (2-s)/2 ) and lim ξ →0 + y(ξ ) = lim r→0 + u(r) = γ. Hence, y can be extended by continuity at 0 by defining y(0) = γ. To conclude that y is a solution of (44) on [0, R (2-s)/2 ] in the sense of [START_REF] Biles | A generalization of the Lane-Emden equation[END_REF], that is, y satisfies properties (a)-(c) stated above with T = R (2-s)/2 , it suffices to show that (45) y (ξ ) → 0 and y

(ξ ) → -2 γ 2 (s)-1 /[(n -s)(2 -s)] as ξ → 0 + .
This would give that y ∈ C 2 [0, R (2-s)/2 ], and then, by applying Theorem 2 in [START_REF] Biles | A generalization of the Lane-Emden equation[END_REF], we conclude that y = y γ on [0, min{T, R (2-s)/2 }], proving our uniqueness assertion.

We prove (45). Since u is a positive radial solution of (3) with lim r→0 + u(r) = γ, we have

(46) r -(n-1-s) r n-1 u (r) = -u 2 (s)-1 + µr s u q → -γ 2 (s)-1 as r → 0 + .
Hence, the function r -→ r n-1 u (r) is decreasing on some interval (0, r 0 ) for small r 0 > 0. Thus, there exists

lim r→0 + r n-1 u (r) = θ ∈ (-∞, ∞].
We next show that θ = 0. Assume by contradiction that θ = 0. Then choosing min{θ , 0} < c < max{θ , 0}, we find that h(r) = u(r) + c(n -2) -1 r 2-n is decreasing (respectively, increasing) on (0, r 1 ) for r 1 > 0 small when θ < 0 (respectively, when θ > 0). Since lim r→0 + h(r) = -∞ if θ < 0 and lim r→0 + h(r) = +∞ if θ > 0, we arrive at a contradiction. This proves that lim r→0 + r n-1 u (r) = 0. Hence, by (46), we get that lim r→0 + r s-1 u (r) = -γ 2 (s)-1 /(ns). Coming back to the ξ variable, we obtain (45). This ends the proof of Lemma 4.1.

(MB) SOLUTIONS

In Sect. 5.1 we prove Corollary 1.6. In Sect. 5.2 we prove Theorem 1.3(ii) given as Proposition 5.1.

5.1. Proof of Corollary 1.6. For every 0 < < min{(n -2)/2, 2}, we set q := 2 -1 -2 /(n -2) so that q ∈ (2 -2, 2 -1) with q > 1. Then, for every s ∈ (0, 2), Theorem 1.3(ii) yields a positive radial (MB) solution u MB of (3) for some R > 0. We define z(r) = r (n-2)/2 u MB (r) for r ∈ (0, R). Since z * := lim sup r→0 + z(r) ∈ (0, ∞) and z * = lim inf r→0 + z(r) = 0, the asymptotics of u MB at zero is different from that of any positive singular solution of (2). By defining K(r) = 1µr s u MB (r) q-2 (s)+1 and C s, := 2(s -)/(n -2) for r = |x| ∈ (0, R),

we see that u = u MB is a positive singular solution of [START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF]. Moreover, we find that (47)

|r 1-K (r)| = µ[z(r)] C s, +C s, rz (
r)/z(r) for all r ∈ (0, R).

We have C s, > 0 when < s and C s, < 0 when > s. With L and L as in [START_REF] Chen | Estimate of the conformal scalar curvature equation via the method of moving planes[END_REF], we prove that

(48) L = 0 < L < ∞ if ∈ (0, s), whereas 0 < L < L = ∞ if > s.
Indeed, since P (q) (u MB ) = 0 and z * < ∞, Lemma 3.2(a) yields that (49) lim sup

r→0 + r|z (r)|/z(r) < ∞ and F 0 (z(r)) -[rz (r)] 2 → 0 as r → 0 + ,
where F 0 is given by (27). Hence, L = 0 and L < ∞ if ∈ (0, s). Since z * = 0, for every ρ ∈ (0, z * ), there exists a sequence {r k } of positive numbers decreasing to 0 as k → ∞ such that lim k→∞ z(r k ) = ρ. Then, by (49), we have lim k→∞ (r k z (r k )) 2 = F 0 (ρ). For suitable ρ, using r k in (47), we get that L > 0 for ∈ (0, s), and correspondingly 0

< L < ∞ for > s. It remains to show that L = ∞ if > s. Assuming the contrary, R k z (R k )/z(R k ) → -/C s, for every sequence {R k } of positive numbers decreasing to 0 such that lim k→∞ z(R k ) = 0. Lemma 3.2(a) gives that F R k (z(R k )) ≥ [R k z (R k )/z(R k )] 2 . Letting k → ∞, we would have (n -2) 2 /4 ≥ 2 /C 2 s,
, which is a contradiction with s > 0. Thus, (48) holds and K satisfies the properties in Corollary 1.6.

Existence of (MB) solutions.

Proposition 5.1. Let q ∈ (1, 2 -1). Assuming that q > 2 -2, then for R > 0 small, (3) admits at least a positive radial (MB) solution u, that is,

(50) lim inf r→0 + r n-2
2 u(r) = 0 and lim sup

r→0 + r n-2 2 u(r) ∈ (0, ∞).
Proof. We use an argument inspired by Chen-Lin [START_REF] Chen | On the asymptotic symmetry of singular solutions of the scalar curvature equations[END_REF]. Let (γ i ) i≥1 be an increasing sequence of positive numbers with lim i→∞ γ i = ∞. By Lemmas 4.1 and 3.2, for every i ≥ 1, there exists R i > 0 such that (3), subject to lim |x|→0 + u(x) = γ i , admits a unique positive radial C 2 (0, R i ]-solution u γ i . From now on, we use u i instead of u γ i . Let Λ > Λ 0 be fixed, where Λ 0 is given by (30). By Proposition 3.1, there exists R Λ > 0 such that u i can be extended as a positive radial

C 2 (0, R Λ ] ∩C[0, R Λ ]-solution of (3) in (0, R Λ ] satisfying (51) u i (0) = γ i , r n-2
2 u i (r) ≤ Λ for all r ∈ (0, R Λ ] and every i ≥ 1.

CLAIM: For any u 0 > 0, there exist r 0 ∈ (0, R Λ ) and i 0 ≥ 1 such that u i (r 0 ) ≥ u 0 for all i ≥ i 0 .

We now complete the proof of Proposition 5.1 assuming the Claim. From (51), there exists a subsequence of (u i ), relabelled (u i ), converging uniformly to u ∞ on any compact subset of (0,

R Λ ]. Moreover, u i → u ∞ in C 2 loc (0, R Λ ]
and u ∞ is a radial solution of (3). The above Claim yields lim sup r→0 + u ∞ (r) = ∞, that is, u ∞ has a non-removable singularity at 0. By (51), we get lim sup r→0 + r n-2

2 u ∞ (r) ∈ (0, ∞). Since q < 2 -1, we thus find that u q+1 ∞ ∈ L 1 (B(0, R Λ )).
We have P (q) (u i ) = 0 for all i ≥ 1. By letting u = u i in (39) and (37), then passing to the limit i → +∞, we find that ( 52)

P (q) r (u ∞ ) = c µ,q,n B(0,r) u q+1 ∞ (x) dx for all r ∈ (0, R Λ ].
By letting r → 0 + in (52), we find that P (q) (u ∞ ) = 0. Hence by (23), u ∞ is not a (CGS) solution of (3). As u ∞ does not have a removable singularity at 0, we conclude that u ∞ is a radial (MB) solution of (3), that is u ∞ satisfies (50). This ends the proof of Proposition 5.1.

Proof of the Claim. Suppose the contrary. Then for some u 0 > 0 and any r 0 ∈ (0, R Λ ), there exists a subsequence of u i , relabeled (u i ), such that (53) u i (r 0 ) < u 0 for all i ≥ 1.

We apply the following transformation (54)

w i (t) = r n-2 2 u i (r) with t = log r.
By w i (t) and w i (t), we denote the first and second derivative of w i with respect to t, respectively. Then w i satisfies the equation

(55) w i (t) -f (w i (t)) = µe λt w q i (t) for -∞ < t < log R Λ , where λ := (n -2)(2 -1 -q)/2 and f : [0, ∞) → R is defined by (56) f (ξ ) := (n -2) 2 ξ /4 -ξ 2 (s)-1 for all ξ ≥ 0.
From (51), we have that (57)

w i (t) ∈ (0, Λ] for all t ∈ (-∞, log R Λ ] and i ≥ 1.
The proof of the Claim is now divided into five steps:

Step 5.1. The family (w i (t)) i≥1 is uniformly bounded on (-∞, log R Λ ].

Proof of Step 5.1. Using F R in (29) with R = e t , we define

E i : (-∞, log R Λ ] → R by (58) E i (t) := w i (t) 2 -w 2 i (t) F e t (w i (t)
). We have λ > 0 (since q < 2 -1) and lim t→-∞ w i (t) = 0. By Lemma 3.2(a), we find that (59)

E i (t) = -2c µ,q,n t -∞ e λ ξ w q+1 i (ξ ) dξ and E i (t) = -2c µ,q,n e λt w q+1 i (t) < 0 for all t ∈ (-∞, log R Λ ). It follows that (60) lim t→-∞ w i (t) = lim t→-∞ E i (t) = 0.
From (59), we have

E i < 0 on (-∞, log R Λ ]
. Thus, by (57), we get that (w i (t)) i≥1 is uniformly bounded for t ∈ (-∞, log R Λ ], completing Step 5.1.

Step 5.2. For ε 0 > 0 and r 0 ∈ (0, R Λ ) small such that r

(n-2)/2 0 u 0 < ε 0 /2, we set

F i := {t ∈ (-∞, log r 0 ) : w i (t) ≥ ε 0 } for all i ≥ 1.
Then there exists i 0 ≥ 1 such that w i (log r 0 ) < ε 0 /2 and F i = / 0 for every i ≥ i 0 .

Proof of Step 5.2. For 0

< ε 0 < [(n -2)/2] (n-2)/(2-s) , we define (61) β 0 := 2ε 2 (s)-2 0 n -2 + (n -2) 2 -4ε 2 (s)-2 0 -1 so that 0 < β 0 < n -2 2 is small.
Since β 0 → 0 + as ε 0 → 0 + , we can take ε 0 > 0 small enough such that β 0 is smaller than max{(n -2)/4, 2/q, (2s)/(2 (s) -1)}. Our choice of r 0 and (53) yield that (62)

w i (log r 0 ) = r n-2 2 0 u i (r 0 ) < r n-2 2
0 u 0 < ε 0 /2 for all i ≥ 1. To end Step 5.2, we show by contradiction that there exists i 0 ≥ 1 such that F i = / 0 for all i ≥ i 0 . Indeed, suppose that for a subsequence (w i k ) k≥1 of (w i ) i≥1 , we have (63) w i k (t) < ε 0 for all t ∈ (-∞, log r 0 ] and every k ≥ 1.

Let k ≥ 1 be arbitrary. Using (63) and (61), we infer that (64)

β 0 (n -2 -β 0 ) = ε 2 (s)-2 0 > w 2 (s)-2 i k (t)
for all t ≤ log r 0 . From (55) and (64), we obtain that (65)

w i k (t) > [(n -2)/2 -β 0 ] 2 w i k (t) for all t ≤ log r 0 .
In particular, t -→

w i k (t) is increasing on (-∞, log r 0 ]. Since lim t→-∞ w i k (t) = 0, we find that w i k (t) > 0 for all t ≤ log r 0 . Set G i k (t) := (w i k (t)) 2 -[(n -2)/2 -β 0 ] 2 w 2 i k (t). Using (65), we get that G i k is increasing on (-∞, log r 0 ] and lim t→-∞ G i k (t) = 0. Thus, G i k > 0 on (-∞, log r 0 ], which implies that w i k (t) > [(n -2)/2 -β 0 ] w i k (t) for all t ≤ log r 0 .
Thus, t -→ e -( n-2 2 -β 0) t w i k (t) is increasing on (-∞, log r 0 ]. Using (54) and (62), we find that (66) u i k (r) ≤ c 0 r -β 0 for every r ∈ (0, r 0 ] and all k ≥ 1, where c 0 := (ε 0 /2) r -n-2 2 +β 0 0 . Since β 0 can be made arbitrarily small, it follows from (66) that the righthand side of (3) with u = u i k is uniformly bounded in L p (B(0, r 0 )) for some p > n/2. Then, u i k satisfies (3) in D (B(0, r 0 )) (in the sense of distributions) and (u i k ) k≥1 is uniformly bounded in W 2,p (B(0, r 0 )) for some p > n/2. Hence, (u i k (r)) k≥1 is uniformly bounded in r ∈ [0, r 0 /2], which leads to a contradiction with u i k (0) = γ i k → ∞ as k → ∞. This ends the proof of Step 5.2.

For i ≥ i 0 , we define t i := sup {t ∈ (-∞, log r 0 ) : w i (t) ≥ ε 0 } .

It follows from

Step 5.2 that t i is well-defined and that t i ∈ (-∞, log r 0 ) for all i ≥ i 0 .

Step 5.3. We claim that for every i ≥ i 0 , the function w i is decreasing on [t i , ti ] for some ti ∈ (t i , log r 0 ]. Moreover, by diminishing ε 0 > 0 and r 0 > 0, there exist positive constants c 1 , c 2 independent of ε 0 and i such that

(67) w i (t i ) ≥ c 1 ε q+2 2 0 e λt i 2 and ti -t i ≤ 2 n -2 log ε 0 w i (t i ) + c 2 .
Moreover, if ti < log r 0 , then w i is increasing on [t i , log r 0 ] and

(68) log r 0 -ti ≤ 2 n -2 log w i (log r 0 ) w i (t i ) + c 3 ,
where c 3 > 0 is a constant independent of ε 0 and i. CASE 1: w i (t) = 0 for all t ∈ [t i , log r 0 ). Hence, w i < 0 on [t i , log r 0 ) using that w i < ε 0 on (t i , log r 0 ].

CASE 2: w i (t i ) = 0 for some ti ∈ [t i , log r 0 ). Then, w i < 0 on [t i , ti ) and w i > 0 on (t i , log r 0 ].

In both cases, w i is decreasing on [t i , ti ] such that (1) ti = log r 0 in Case 1;

(2) ti ∈ (t i , log r 0 ) and w i (t) > 0 for all t ∈ (t i , log r 0 ] in Case 2.

Unless explicitly mentioned, the argument below applies for both Case 1 (when ti = log r 0 ) and Case 2 (when ti ∈ (t i , log r 0 )).

From (55), we have that (69)

w i (t) ≥ f (w i (t)) for all t ∈ [t i , log r 0 ].
Thus, using (69), we find that

(70) t -→ w i (t) 2 -F 0 (w i (t)) (a) is non-increasing on [t i , ti ] (in both Case 1 and Case 2); (b) is non-decreasing on [t i , log r 0 ] in Case 2.
Proof of the first inequality in (67). By (62) and w i (t i ) = ε 0 , we infer that there exists t i ∈ (t i , log r 0 ) such that w i ( t i ) = ε 0 /2 and, moreover, ti ∈ ( t i , log r 0 ]. Hence, there exists

ξ i ∈ [t i , t i ] such that -ε 0 /2 = w i ( t i ) -w i (t i ) = w i (ξ i )( t i -t i ).
By Step 5.1, (|w i (t)|) i≥1 is uniformly bounded on (-∞, log R Λ ) so that (71) t it i ≥ cε 0 for some constant c > 0.

From (57), ( 58) and (59), there exists c > 0 such that (72) c w 2 i (t) ≤ E i (t) ≤ E i ( t i ) for every t i < t ≤ log r 0 . Moreover, using (71), together with E i (t i ) < 0 and w i ≥ ε 0 /2 on [t i , t i ], we obtain that (73)

E i ( t i ) = E i (t i ) - 2λ µ q + 1 t i t i
e λt w q+1 i (t) dt ≤ -λ µc ε q+2 0 e λt i 2 q (q + 1) .

Since ti ∈ ( t i , log r 0 ], by combining ( 72) and ( 73), there exists c 1 > 0 such that (74)

w i (t i ) ≥ c 1 ε q+2 2 0 e λt i 2 ,
where c 1 > 0 is independent of ε 0 and i.

Proof of the second inequality in (67). From (70), for all t ∈ [t i , ti ), we have

(75) [w i (t)] 2 -F 0 (w i (t)) ≥ -F 0 (w i (t i )),
which jointly with w i (t) < 0 and F 0 increasing on [0, ε 0 ], yields that (76)

-w i (t) [F 0 (w i (t)) -F 0 (w i (t i ))] -1/2 ≥ 1 for all t ∈ [t i , ti ).
Hence, for all t ∈ [t i , ti ), by integrating (76) over [t, ti ], we get that

(77) ti -t ≤ w i (t) w i (t i ) dη [F 0 (η) -F 0 (w i (t i ))] 1/2 =: D i (t).
We shall prove below that (78)

D i (t) ≤ 2 n -2 log w i (t) w i (t i ) + log 2 + kw 2 (s)-2 i (t)
for all t ∈ [t i , ti ), where k > 0 is a constant independent of ε 0 and i. Then, since w i ≤ ε 0 on [t i , ti ], from (77) an (78), we conclude the proof of the second inequality in (67).

Proof of (78). For every ξ ≥ 0, we define (79)

g i (ξ ) := n -2 2 2 ξ 2 - 2 2 (s) ξ 2 (s) [w i (t i )] 2 (s)-2 .
By a change of variable, we find that (80)

D i (t) = w i (t)/w i (t i ) 1 dξ [g i (ξ ) -g i (1)] 1/2 for all t ∈ [t i , ti ).
By the definition of g i in (79), for each ξ > 1, we have (81) g i (ξ )g i (1)

ξ 2 -1 = n -2 2 2 - 2[w i (t i )] 2 (s)-2 2 (s) ξ 2 (s) -1 ξ 2 -1 .
Since (2 (s) -1) ξ 2 (s) -2 (s) ξ 2 (s)-2 + 1 increases for ξ ≥ 1, we get that ξ 2 (s) -1 is bounded from above by 2 (s) ξ 2 (s)-2 (ξ 2 -1) for all ξ ≥ 1. Hence, for any 1 < ξ ≤ ε 0 /w i (t i ), we find that

(82) [w(t i )] 2 (s)-2 2 (s) ξ 2 (s) -1 ξ 2 -1 ≤ [w i (t i ) ξ ] 2 (s)-2 ≤ ε 2 (s)-2 0 .
Since we fix ε 0 > 0 small, there exists a positive constant k, independent of ε 0 and i, such that

(83) ξ 2 -1 g i (ξ ) -g i (1) 1/2 ≤ 2 n -2 + 2k [w i (t i ) ξ ] 2 (s)-2
for every 1 < ξ ≤ ε 0 /w i (t i ). Since w i (t) ≤ w i (t i ) ≤ ε 0 for each t ∈ [t i , ti ), using (83) in (80), we get

(84) D i (t) ≤ 2 n -2 h i (t) 1 dξ [ξ 2 -1] 1/2 + 2k [w i (t i )] 2 (s)-2 E i (t),
where for every t ∈ [t i , ti ), we define h i (t) and E i (t) by ( 85)

h i (t) := w i (t) w i (t i ) and E i (t) := h i (t) 1 ξ 2 (s)-2 (ξ 2 -1) 1/2 dξ .
A simple calculation gives that there exists C > 0 such that for every t ∈ [t i , ti ), we have

(86) E i (t) ≤ Ch 2 (s)-2 i (t).
Using ( 85) and ( 86) into (84), we reach (78) with k large enough. This completes the proof of the inequalities in (67).

Proof of (68) in Case 2 (when ti ∈ (t i , log r 0 )). Recall that w i is increasing on [t i , log r 0 ] so that using (62), we get that w i (t) ≤ w i (log r 0 ) < ε 0 /2 for all t ∈ [t i , log r 0 ]. Moreover, the function in ( 70) is non-decreasing on [t i , log r 0 ]. Hence, we recover (75) for all t ∈ (t i , log r 0 ]. Since this time w i > 0 on (t i , log r 0 ], instead of (77), we find that (87) By letting t = log r 0 in (89), we conclude (68). This proves the assertions of Step 5.3.

w i (t) [F 0 (w i (t)) -F 0 (w i (t i ))] -1/2 ≥
Step 5.4. Proof of the Claim concluded in Case 1 of Step 5.3: ti = log r 0 .

Proof of Step 5.4. Suppose that w i < 0 on [t i , log r 0 ).

The second inequality in (67) of Step 5.3 reads as follows

(90) log r 0 -t i ≤ 2 n -2 log ε 0 w i (log r 0 ) + c 2 .
The first inequality in (67) and (62

) give that r n-2 2 0 u 0 ≥ c 1 ε q+2 2 0 e λt i 2 .
By applying log to this inequality and to (74) (with ti = log r 0 ), respectively, we find that (91)

λt i /2 ≤ [(n -2)/2] log r 0 + c 4 (log u 0 + log(1/ε 0 ))
for some constant c 4 > 0 independent of ε 0 and i, respectively

(92) log(w i (log r 0 )) ≥ λt i /2 + [(q + 2)/2] log ε 0 + log c 1 .
Using (92) into (90), we deduce that

(93) log r 0 ≤ [1 -λ /(n -2)]t i + c 5 log(1/ε 0 )
for a constant c 5 > 0 independent of ε 0 and i. We have

(94) Θ := 2(q -2 + 2)/(2 -1 -q) > 0 since q ∈ (2 -2, 2 -1).
Plugging into (93) the estimate on t i from (91), we conclude that

(95) -Θ log r 0 ≤ c 6 [log u 0 + log(1/ε 0 )] ,
where c 6 is a positive constant independent of ε 0 and i. Since Θ > 0, we can choose r 0 > 0 small so that the left-hand side of ( 95) is bigger than twice the right-hand side of (95), which is a contradiction with (95). This completes Step 5.4.

Step 5.5. Proof of the Claim in Case 2 of Step 5.3: ti ∈ (t i , log r 0 ).

Proof of Step 5.5. We have w i < 0 on [t i , ti ) and w i > 0 on (t i , log r 0 ]. The first inequality of (67) yields

(96) 2 log w i (t i ) ≥ (q + 2) log ε 0 + λt i + 2 log c 1 .
By adding the second inequality of (67) to that of (68), we get

(97) log r 0 -t i ≤ 2 n -2 [log ε 0 + log w i (log r 0 ) -2 log w i (t i )] +C 1 ,
where C 1 > 0 is a constant independent of ε 0 and i. By (62), we have

(98) log w i (log r 0 ) ≤ log u 0 + [(n -2)/2] log r 0 .
Using ( 96) and ( 98) into (97), we obtain that

(99) [2λ /(n -2) -1]t i ≤ C 2 [log(1/ε 0 ) + log u 0 ] ,
where C 2 > 0 is a a constant independent of ε 0 and i. Since the coefficient of t i in (99) equals 2 -2q, which is negative from the assumption q > 2 -2, using that t i < log r 0 , we infer that

(100) (2 -2 -q) log r 0 ≤ C 2 [log(1/ε 0 ) + log u 0 ] .
By choosing r 0 > 0 small so that the left-hand side of (100) is greater than twice the right-hand side of (100), we reach a contradiction. This proves Step 5.5.

From Steps 5.4 and 5.5 above, we conclude the proof of the Claim.

(CGS) SOLUTIONS

This section is devoted to the proof of part (iii) of Theorem 1.3, restated below. Proposition 6.1. Let q ∈ (1, 2 -1). There exists R 0 > 0 such that for every R ∈ (0, R 0 ) and any positive singular solution U of (2), there exists a unique positive radial (CGS) solution u of (3) with asymptotic profile U near zero.

Proof. Let f be given by ( 56). Let U be a positive singular solution of (2). Then, by defining ϕ(t) = e -(n-2)t/2 U(e -t ) for t ∈ R, we see that ϕ ∈ C ∞ (R) is a positive periodic solution of (101) ϕ (t) = f (ϕ(t)) for all t ∈ R.

Let P denote the set of all positive smooth periodic solutions of (101) to be described in Sect. 6.1. We next show that Proposition 6.1 is equivalent to Lemma 6.2, the proof of which will be given in Sect. 6.2.

Lemma 6.2. Let q ∈ (1, 2 -1). For every ϕ ∈ P, there exists T 0 = T 0 (ϕ) > 0 large for which the nonautonomous first order system

(102) (V ,W ) = (W, f (V ) + µe -λt V q ) in [T 0 , ∞), V > 0 on [T 0 , ∞),
has a unique solution satisfying

(103) (V (t),W (t)) -ϕ(t), ϕ (t) → (0, 0) as t → ∞.
Indeed, assuming that Proposition 6.1 holds, then for every ϕ ∈ P, we use the transformation

(104) ϕ(t) = r n-2 2 U(r), V (t) = r n-2 2 u(r), W (t) = V (t) with t = log(1/r),
where u is the unique positive radial (CGS) solution of (3) satisfying lim r→0 + u(r)/U(r) = 1. Hence, we obtain that (V,W ) is a solution of (102) for any T 0 > log (1/R) and, moreover,

V (t) -ϕ (t) → 0 as t → ∞.
Using (101), we find that W (t)ϕ (t) → 0 as t → ∞. Hence, Wϕ is uniformly continuous on [T 0 , +∞).

Since lim t→∞ (Vϕ)(t) = 0, we get that W (t)ϕ (t) → 0 as t → +∞. This proves Lemma 6.2. We prove the reverse implication. If Lemma 6.2 holds, then for every positive singular solution U of (2), by using (104) and Proposition 3.1, we get a unique positive radial (CGS) solution u of (3) satisfying lim r→0 + u(r)/U(r) = 1.

6.1. Description of P. We show that the set P of all positive smooth periodic solutions of ( 101) is given by (107). This is basically standard ODE theory. We state only the essential steps and leave the details to the reader. The function F 0 in ( 27) is increasing on [0, M 0 ] and decreasing on [M 0 , ∞) with M 0 given by (105). Thus F 0 reaches its maximum σ at M 0 , where

M 0 := n -2 2 n-2 2-s and σ := F 0 (M 0 ) = 2 -s n -s n - (105)
.

Note that M 0 is the only positive zero of f (ξ ) = 0. Let ϕ ∈ P. Since F 0 (ξ ) = 2 ξ 0 f (t) dt for all ξ ≥ 0, from (101), there exists a constant σ > 0 such that (106) ϕ (t) 2 = F 0 (ϕ(t))σ for all t ∈ R.

In fact, by taking µ = 0 in (37) for u = U with U given by (104), we precisely obtain that σ = 2P(U)/ω n-1 > 0, where P(U) is the Pohozaev invariant associated to the positive singular solution U of (2). From ( 105) and (106), we must have 0 < σ ≤ σ and ϕ ≡ M 0 on R if σ = σ .

Let σ ∈ (0, σ ) be fixed. Let a σ and b σ denote the two positive solutions of F 0 (ξ ) = σ with 0 < a σ < M 0 < b σ . It follows from standard analysis of the ODE (101) that for any σ ∈ (0, σ ), there is a unique solution ϕ σ to (101

) such that min R ϕ σ = ϕ σ (0) = a σ < b σ = max R ϕ σ .
Moreover, ϕ σ is periodic and we let 2t σ > 0 be its principal period. For every τ ∈ S 1 , let ϕ σ ,τ denote the function whose graph is obtained from that of ϕ σ by a horizontal shift with (t σ /π)Arg τ units, where Arg τ denotes the principal argument of τ. Note that ϕ σ = ϕ σ ,τ 0 with

τ 0 = (1, 0) ∈ S 1 . It follows that (107) P = {ϕ σ } ∪ {ϕ σ ,τ } (σ ,τ)∈(0,σ )×S 1 ,
where ϕ σ ≡ M 0 and ϕ σ ,τ (t) = ϕ σ (t -(t σ /π)Arg τ) for all t ∈ R.

6.2. Proof of Lemma 6.2. We first prove Lemma 6.2 for ϕ ∈ ∪{ϕ σ ,τ } (σ ,τ)∈(0,σ 0 )×S 1 with σ 0 ∈ (0, σ ) and second for ϕ ∈ {ϕ σ } ∪ {ϕ σ ,τ } (σ ,τ)∈[σ 0 ,σ )×S 1 with σ 0 ∈ (0, σ ) close enough to σ .

Step 6.1. For any σ * ∈ (0, σ 0 ) fixed, there exists T 0 > 0 large such that for every ϕ = ϕ σ ,τ with (σ , τ) ∈ (σ * , σ 0 ) × S 1 , the system (102), subject to (103), admits a unique solution (V σ ,τ ,W σ ,τ ).

Proof of Step 6.1. For the existence proof, we make a suitable transformation and use the Fixed Point Theorem for a contraction mapping. Let I 0 be an open interval such that (σ * , σ 0 ) I 0 (0, σ ). The key here is that for every (σ , τ) ∈ I 0 × S 1 , both ϕ σ ,τ and ∂ t ϕ σ ,τ = ϕ σ ,τ are differentiable with respect to σ . This does not hold for σ = σ . By differentiating (106) with respect to σ and using (101), we get

f (ϕ σ ,τ (t)) d [ϕ σ ,τ (t)] dσ -∂ t ϕ σ ,τ (t) d [∂ t ϕ σ ,τ (t)] dσ = 1 2 for all t ∈ R.
We see that there exists C * > 0 such that for every (σ , τ) ∈ I 0 × S 1 , we have

(108) |∂ t ϕ σ ,τ (t)| + d [ϕ σ ,τ (t)] dσ ≤ C * for all t ∈ R.
Moreover, there exists T 0 > 0 such that C * e -λ T 0 /2 < a 0 := inf {a σ : σ ∈ I 0 }, where a σ is the smallest positive root of F 0 (ξ ) = σ . Let X T 0 denote the set of all continuous functions

( f 1 , f 2 ) : [T 0 , ∞) → R 2 with e λt/2 (| f 1 (t)| + | f 2 (t)|) ≤ 1 for all t ≥ T 0 . If we define ( f 1 , f 2 ) := sup t≥T 0 e λt/2 (| f 1 (t)| + | f 2 (t)|) ,
then (X T 0 , • ) is a complete metric space. For ( V , W ) ∈ X T 0 and recalling that ϕ σ ,τ = f (ϕ σ ,τ ) on R, we consider the following transformation:

(109) V (t) -ϕ σ ,τ (t) W (t) -ϕ σ ,τ (t) =   ∂ t ϕ σ ,τ (t) d[ϕ σ ,τ (t)] dσ f (ϕ σ ,τ (t)) d[∂ t ϕ σ ,τ (t)] dσ   V (t) W (t) for t ∈ [T 0 , ∞).
Note that the matrix and its inverse are both uniformly bounded with respect to (σ , τ) ∈ I 0 × S 1 . In particular, (109) yields that (110)

V (t) := ϕ σ ,τ (t) + V (t)∂ t ϕ σ ,τ (t) + W (t) d [ϕ σ ,τ (t)] dσ .
From (108), (110) and our choice of T 0 , we find that (111)

|V (t) -ϕ σ ,τ (t)| ≤ C * e -λt/2 ( V , W ) ≤ C * e -λ T 0 /2 < a 0 for all t ≥ T 0 .
For every t ≥ T 0 and (σ , τ) ∈ I 0 × S 1 , we have ϕ σ ,τ (t) ≥ a σ ≥ a 0 since a σ is increasing in σ . Thus, (111) proves that V in (110) is positive on [T 0 , ∞) for all ( V , W ) ∈ X T 0 . For simplicity of reference, using V in (110) for ( V , W ) ∈ X T 0 , we define

g σ ,τ, V , W (t) := f (V (t)) -f (ϕ σ ,τ (t)) -(V (t) -ϕ σ ,τ (t)) f (ϕ σ ,τ (t)) + µe -λt V q (t).
By (111), there exist positive constants C 0 and C 1 such that for all

( V , W ) ∈ X T 0 , (112) |g σ ,τ, V , W (t)| ≤ C 0 |V (t) -ϕ σ ,τ (t)| 2 + µe -λt V q (t) ≤ C 1 e -λt for every t ∈ [T 0 , ∞) and (σ , τ) ∈ I 0 × S 1 . Remark that (102) is equivalent to the system (113) ( V (t), W (t)) = 2g σ ,τ, V , W (t) d [ϕ σ ,τ (t)] dσ , -2g σ ,τ, V , W (t) ∂ t ϕ σ ,τ (t) on [T 0 , ∞).
For every ( V , W ) ∈ X T 0 and t ≥ T 0 , we define

Φ σ ,τ ( V , W )(t) = -2 ∞ t g σ ,τ, V , W (y) d [ϕ σ ,τ (y)] dσ dy, 2 ∞ t g σ ,τ, V , W (y) ∂ t ϕ σ ,τ (y) dy .
We next prove the existence of T 0 > 0 large such that Φ σ ,τ maps X T 0 into X T 0 and Φ σ ,τ is a contraction mapping on X T 0 for every (σ , τ) ∈ I 0 × S 1 . From (108), (112) and the definition of (X T 0 , • ), we have

(114) Φ σ ,τ ( V , W ) ≤ 2C * sup t≥T 0 e λt/2 ∞ t |g σ ,τ, V , W (y)| dy ≤ 2C * C 1 λ e -λ T 0 /2 .
Thus, for large T 0 > 0, we find that Φ σ ,τ ( V , W ) ∈ X T 0 for every ( V , W ) ∈ X T 0 and all (σ , τ) ∈ I 0 × S 1 . For every ( V 1 , W 1 ) and ( V 2 , W 2 ) in X T 0 , we have

( V 1 , W 1 ) -( V 2 , W 2 ) = sup t≥T 0 e λt 2 S(t) with S(t) := |( V 1 -V 2 )(t)| + |( W 1 -W 2 )(t)|.
Hence, there exist positive constants C 2 and C 3 such that for every (σ , τ)

∈ I 0 × S 1 Φ σ ,τ ( V 1 , W 1 ) -Φ σ ,τ ( V 2 , W 2 ) ≤ 2C * sup t≥T 0 e λt 2 ∞ t g σ ,τ, V 1 , W 1 (y) -g σ ,τ, V 2 , W 2 (y) dy ≤ C 2 sup t≥T 0 e λt 2 ∞ t
( S(y)) 2 + e -λ y S(y) dy

≤ C 3 e -λ T 0 2 ( V 1 , W 1 ) -( V 2 , W 2 ) for all ( V 1 , W 1 ), ( V 2 , W 2 ) ∈ X T 0 . Thus, for T 0 > 0 large, Φ σ ,τ is a contraction on X T 0 for all (σ , τ) ∈ I 0 × S 1 .
Hence, Φ σ ,τ has a unique fixed point in X T 0 , say ( V σ ,τ , W σ ,τ ), which gives a unique solution in X T 0 of (113) such that lim t→∞ ( V σ ,τ , W σ ,τ )(t) = (0, 0). By (108) and ( V , W ) = ( V σ ,τ , W σ ,τ ) in (109), we obtain a solution (V σ ,τ ,W σ ,τ ) of ( 102) satisfying (103) with ϕ = ϕ σ ,τ . Moreover, (V σ ,τ ,W σ ,τ ) is continuous in (σ , τ) since Φ σ ,τ is continuous in (σ , τ).

To prove uniqueness, on Ω 0 := I 0 × S 1 × (0, e -T 0 ), we define the functions H, G : Ω 0 → R 3 by H (σ , τ, r) := (V σ ,τ (t (r)) ,W σ ,τ (t (r)) , r) and G (σ , τ, r) := ϕ σ ,τ (t (r)) , ϕ σ ,τ (t (r)) , r for every (σ , τ, r) ∈ Ω 0 , where t (r) := log (1/r). From our construction, H is continuous. Since V σ ,τ is a solution to a second-order ODE and W σ ,τ = V σ ,τ , the uniqueness theorem for ODEs yields that H is one-toone in Ω 0 . Clearly, G is also continuous and one-to-one in Ω 0 . Thus, by applying the Domain Invariance Theorem, we obtain that H and G are open. Moreover, since the functions {ϕ σ ,τ } (σ ,τ)∈I 0 ×S 1 are periodic, we see that G (Ω 0 ) = Σ 0 × (0, e -T 0 ) for some domain Σ 0 in R 2 . Let H 0 : Σ 0 × (-e -T 0 , e -T 0 ) → R 3 be the function defined as

H 0 (ξ 1 , ξ 2 , r) :=      H G -1 (ξ 1 , ξ 2 , r) if r > 0 (ξ 1 , ξ 2 , 0) if r = 0 J H G -1 (ξ 1 , ξ 2 , -r) if r < 0,
where J(ξ 1 , ξ 2 , r) := (ξ 1 , ξ 2 , -r). Since H and G are one-to-one in Ω 0 , we obtain that H 0 is one-to-one in Σ 0 × (-e -T 0 , e -T 0 ). Moreover, since H and G -1 are continuous in Ω 0 , we obtain that H 0 is continuous in Σ 0 × [(-e -T 0 , e -T 0 )\ {0}]. As regards the continuity of H 0 on Σ 0 × {0}, for every

(ζ 1 , ζ 2 ) ∈ Σ 0 and (ξ 1 , ξ 2 , r) ∈ Σ 0 × [(-e -T 0 , e -T 0 )\ {0}], we write (115) |H 0 (ξ 1 , ξ 2 , r) -H 0 (ζ 1 , ζ 2 , 0)| ≤ |H 0 (ξ 1 , ξ 2 , r) -(ξ 1 , ξ 2 , r)| + |(ξ 1 , ξ 2 , r) -(ζ 1 , ζ 2 , 0)| ≤ V σ ,τ (t (|r|)) -ϕ σ ,τ (t (|r|)) ,W σ ,τ (t (|r|)) -ϕ σ ,τ (t (|r|)) + |(ξ 1 , ξ 2 , r) -(ζ 1 , ζ 2 , 0)| ,
where (σ , τ, |r|) = G -1 (ξ 1 , ξ 2 , |r|). Since ( V σ ,τ , W σ ,τ ) ∈ X T 0 , it follows from (108), ( 109) and (115) that for every

(ζ 1 , ζ 2 , 0) ∈ Σ 0 × {0}, H 0 (ξ 1 , ξ 2 , r) → H 0 (ζ 1 , ζ 2 , 0) as (ξ 1 , ξ 2 , r) → (ζ 1 , ζ 2 , 0) i.e., H 0 is continuous at (ζ 1 , ζ 2 , 0)
. By another application of the Domain Invariance Theorem, we obtain that H 0 is open. We let Σ * be the domain such that Σ * × {r} = G((σ * , σ 0 ) × S 1 , r) for every r > 0. In particular, since Σ * is open, we obtain that for every (σ , τ) ∈ (σ * , σ 0 ) × S 1 , every solution (V (t) ,W (t)) of ( 102) satisfying

(116) (V (t),W (t)) -ϕ σ ,τ (t), ϕ σ ,τ (t) → (0, 0) as t → ∞
must satisfy (X(t (r)),Y (t (r))) ∈ Σ * for small r > 0. Since Σ * × {0} H 0 (Σ 0 × (-e -T 0 , e -T 0 )) and H 0 is open, we obtain that there exists R 0 ∈ (0, e -T 0 ) such that Σ * × [-R 0 , R 0 ] ⊂ H 0 (Σ 0 × (-e -T 0 , e -T 0 )). It then follows from the definitions of H 0 and Σ * that Σ * × (0, R 0 ] ⊂ H(I 0 × S 1 × (0, R 0 ]). Hence, for every solution (X (t) ,Y (t)) of ( 102) satisfying (116) for some (σ , τ) ∈ (σ * , σ 0 ) × S 1 , we obtain (X(t (r)),Y (t (r)), r) ∈ H(I 0 × S 1 × (0, R 0 ]) and so (X(t (r)),Y (t (r))) = (V σ ,τ (t (r)),W σ ,τ (t (r))) for small r > 0. Hence, for every ϕ = ϕ σ ,τ with (σ , τ) ∈ (σ * , σ 0 ) × S 1 , we conclude that (V σ ,τ ,W σ ,τ ) is the unique solution of (102) satisfying (103). This ends Step 6.1.

Step 6.2. Proof of Lemma 6.

2 if ϕ ∈ ∪{ϕ σ ,τ } (σ ,τ)∈[σ 0 ,σ ]×S 1 for σ 0 ∈ (0, σ ) close enough to σ . Proof of Step 6.2. For (σ , τ) ∈ (0, σ ] × S 1 , we search for T 0 ∈ R and V,W ∈ C 1 ([T 0 , +∞)) such that (102) holds and (V (t),W (t)) -(ϕ σ ,τ (t), ϕ σ ,τ (t)) → (0, 0) as t → ∞. Writing Ṽ = V -ϕ σ ,τ and W = W -ϕ σ ,τ , this is equivalent to finding T 0 ∈ R and Ṽ , W ∈ C 1 ([T 0 , +∞)) such that (117) ( Ṽ , W ) = ( W , f (ϕ σ ,τ + Ṽ ) -f (ϕ σ ,τ ) + µe -λt (ϕ σ ,τ + Ṽ ) q ) in [T 0 , ∞),
( Ṽ (t), W (t)) → (0, 0) as t → +∞.

We define L(ϕ σ ,τ , ϕ σ )(t) := f (ϕ σ ,τ (t))f (ϕ σ (t)) for t ∈ R and

A := 0 1 f (M 0 ) 0 .
Since ϕ σ ≡ M 0 and ϕ σ ,τ → ϕ σ as σ → σ uniformly with respect to τ ∈ S 1 , we get that (118) lim

σ →σ sup τ∈S 1 ,t∈R |L(ϕ σ ,τ , ϕ σ )(t)| = 0.
With a Taylor expansion, we write

f (ϕ σ ,τ + Ṽ ) -f (ϕ σ ,τ ) = f (ϕ σ ) Ṽ + L(ϕ σ ,τ , ϕ σ ) Ṽ + Q(ϕ σ ,τ , Ṽ ) with |Q(ϕ σ ,τ , Ṽ )| ≤ C| Ṽ | 2
. Therefore, the system in (117) rewrites as follows

(119) Ṽ W = A Ṽ W + 0 L(ϕ σ ,τ , ϕ σ ) Ṽ + Q(ϕ σ ,τ , Ṽ ) + µe -λt (ϕ σ ,τ + Ṽ ) q .
Since f (M 0 ) < 0, we get that A has two conjugate pure imaginary eigenvalues. Therefore, there exists C > 0 such that e tA + e -tA ≤ C for all t ∈ R, where • is any operator norm on R 2 . For all t ≥ T 0 , we define X(t) := e -tA Ṽ (t) W (t) and Φ ϕ σ ,τ (t, X) := e -tA 0 L(ϕ σ ,τ , ϕ σ )(e tA X) 1 + Q(ϕ σ ,τ , (e tA X) 1 ) + µe -λt (ϕ σ ,τ + (e tA X) 1 ) q , where (e tA X) 1 denotes the first coordinate of e tA X ∈ R 2 . Then getting a solution to (117) amounts to finding a solution X ∈ C 1 ([T 0 , +∞), R 2 ) to (120) X (t) = Φ ϕ σ ,τ (t, X(t)) for t ≥ T 0 and lim t→+∞ X(t) = 0.

As in Step 6.1, we find a solution to (120) via the Fixed Point Theorem for contracting maps on a complete metric space. Since Q(ϕ σ ,τ , Ṽ ) is quadratic in Ṽ , the last two terms of the second coordinate of Φ ϕ σ ,τ (t, X) are tackled as in Step 6.1. The first term is linear in Ṽ and controled by L(ϕ σ ,τ , ϕ σ ): with (118), this term is contracting for σ close enough to σ . Mimicking the existence proof of Step 6.1, we get the following:

There exist ε > 0 and T 0 > 0 such that for every (σ , τ) ∈ [σ -3ε, σ ]×S 1 , there exists a solution (V σ ,τ ,W σ ,τ ) ∈ C 1 ([T 0 , +∞), R 2 ) to (102) such that (103) holds for ϕ = ϕ σ ,τ . Moreover, since (σ , τ) -→ (ϕ σ ,τ , ϕ σ ,τ ) is continuous on (0, σ ] × S 1 (despite the issue for σ ), the continuity of the fixed points depending on a parameter yields that

(σ , τ) -→ (V σ ,τ ,W σ ,τ ) is continuous on [σ -3ε, σ ]×S 1 .
Here we have taken the supremum norm on C 0 ([T 0 , +∞), R 2 ): via the fixed point construction, we also get that this holds with a weighted norm.

We only sketch the uniqueness proof. For τ 0 = (1, 0) ∈ S 1 and every ξ ∈ B(0, 2ε) ⊂ R 2 , we define

σ (ξ ) := σ -|ξ | and {τ(ξ ) := ξ /|ξ | if ξ = 0 and τ(0) = τ 0 } .
Due to the uniqueness of solution for σ = σ , as one checks, we have the continuity of the mappings ξ -→ (ϕ σ (ξ ),τ(ξ ) , ϕ σ (ξ ),τ(ξ ) ) and ξ -→ (V σ (ξ ),τ(ξ ) ,W σ (ξ ),τ(ξ ) ) on B(0, 2ε). We introduce the domain Ω 0 := B(0, 2ε) × (0, e -T 0 ) and the functions H, G : Ω 0 → R 3 defined as H (ξ , r) := (V σ ,τ (t) ,W σ ,τ (t) , r) and G (ξ , r) := ϕ σ ,τ (t) , ϕ σ ,τ (t) , r for every (ξ , r) ∈ Ω 0 , where t (r) := log (1/r), σ = σ (ξ ) and τ = τ(ξ ). Arguing as in Step 6.1 and with some extra care for the case ξ = 0, we get the uniqueness of the solution of (102) satisfying (103) for ϕ = ϕ σ ,τ . This ends the proof of Step 6.2.

This completes the proof of Lemma 6.2 and thus of Proposition 6.1.

APPENDIX

Here, we establish Theorem 7.1, a critical result that was used in the proof of Lemma 2.2. The proof of Theorem 7.1 is strongly inspired by Kelley's paper [START_REF] Kelley | The stable, center-stable, center, center-unstable, unstable manifolds[END_REF]. We denote by B δ (0) ⊂ R 3 the ball centered at 0 with radius δ > 0. For any r 0 > 0, we set D r 0 := [0, r 0 ] × [-r 0 , r 0 ]. Theorem 7.1. Let h j ∈ C 1 (B δ (0)) for some δ > 0 with j = 1, 2, 3. Suppose there exist constants C 1 > 0 and p > 1 such that for all ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ B δ (0), we have

(121)        3 ∑ j=1 |h j ( ξ )| ≤ C 1 3 ∑ j=1 ξ 2 j and 3 ∑ j=1 |∇h j ( ξ )| ≤ C 1 3 ∑ j=1 |ξ j |, h 2 ( ξ ) ≤ -C 1 |ξ 2 | p and h 2 (ξ 1 , 0, ξ 3 ) = 0.
For fixed a > 0 and c < 0, we consider the system

(122) Z (t) = (aZ 1 (t) + h 1 ( Z (t)), h 2 ( Z (t)), cZ 3 (t) + h 3 ( Z (t))) for t ≥ 0, Z (0) = (x 0 , y 0 , z 0 ).
Then there exist r 0 ∈ (0, δ /2) and a Lipschitz function w : D r 0 → [-r 0 , r 0 ] such that for all (y 0 , z 0 ) ∈ D r 0 and x 0 = w(y 0 , z 0 ), the initial value system (122) has a solution Z on [0, ∞) and

(123) lim t→+∞ Z (t) = (0, 0, 0).

Moreover, we have that the parametrized surface

(Z 2 , Z 3 ) -→ (w(Z 2 , Z 3 ), Z 2 , Z 3 ) is stable in the sense that Z 1 (t) = w(Z 2 (t), Z 3 (t)
) for all t ≥ 0.

Proof. Since h j ∈ C 1 (B δ (0)) for 1 ≤ j ≤ 3, the Cauchy-Lipschitz theory applies to the system. For r 0 ∈ (0, δ /2) and C 2 > 0, we define X as the set of all continuous functions w : D r 0 → [-r 0 , r 0 ] such that w(0, 0) = 0 and w is C 2 -Lipschitz. Note that (X , • ∞ ) is a complete metric space. For any w ∈ X , we consider the system (S w ) (y , z ) = (h 2 (w(y, z), y, z), cz + h 3 (w(y, z), y, z)) on [0, ∞), (y(0), z(0)) = (y 0 , z 0 ).

We now divide the proof of Theorem 7.1 in five Steps.

Step 7.1. Let r 0 ∈ (0, δ /2) be such that 4C Proof of (i). We write h 2 (w(y(t), z(t)), y(t), z(t)) = h 2 (t, y Proof of (ii). Since c < 0, using the system (S w ), we find that

1 (1 + C 2 2 ) r 0 ≤ |c|. If (y 0 , z 0 ) ∈ D r 0 , then the flow Φ w t (y 0 , z 0 ) associated to (S w ) is defined for all t ∈ [0, +∞). If we set (124) (y(t), z(t)) := Φ w t (y 0 , z 0 ) for all t ∈ [0, ∞), then 0 ≤ y(t) ≤
(126) (z 2 ) = 2 -|c|z 2 + z h 3 (w(y, z), y, z) on [0, b).
Since w is a C 2 -Lipschitz function, using the hypothesis on h 3 in (121), we have

(127) |z h 3 (w(y, z), y, z)| ≤ C 1 |z| w 2 (y, z) + y 2 + z 2 ≤ C 1 1 +C 2 2 |z| y 2 + z 2 .
Using |z| ≤ r 0 and the choice of r 0 > 0, from (127) we obtain that We now prove (ii) when y 0 > 0. If there exists t 0 ∈ [0, b) such that |z(t 0 )| = y 0 , then using (i) and (128), we find that |z h 3 (w(y, z), y, z)| (t 0 ) < |c|z 2 (t 0 ). Thus, (126) yields that (z 2 ) (t 0 ) < 0. This means that |z(t)| = y 0 has at most a solution in [0, b). Hence, one of the following holds: By (i), (ii) and the finite-time blow-up of solutions of ODEs, the flow Φ w t (y 0 , z 0 ) associated to (S w ) is defined for all t ∈ [0, +∞). Let (y(t), z(t)) be as in (124).

Proof of (125). If y 0 = 0, then y ≡ 0 on [0, ∞). Assuming y 0 > 0, then y > 0 on [0, ∞). The hypothesis on h 2 in (121) implies that (y 1-p ) (t) ≥ (p -1)C 1 for all t ≥ 0. By integration, we get that lim t→+∞ y(t) = 0. Hence, for every ε > 0, there exists t ε > 0 large such that 0 ≤ y ≤ ε on [t ε , ∞). To prove that lim t→+∞ z(t) = 0, we show that there exists t ε ≥ t ε such that |z(t)| ≤ ε for all t ≥ t ε . Indeed, with a similar argument to the proof of (ii), it can be shown that |z(t)| = ε has at most one zero on [t ε , ∞). The option |z| ≥ ε on [t ε , ∞) is not viable here. Indeed, if |z| ≥ ε on [t ε , ∞), then again from (126) and (128), we would have (z 2 ) ≤ -|c|z 2 ≤ -|c|ε 2 on [t ε , ∞), leading to a contradiction. Hence, either |z| ≤ ε on [t ε , ∞) or there exists t ε ∈ (t ε , ∞) such that |z| > ε on [t ε , t ε ) and |z| < ε on ( t ε , ∞). In either of these cases, the conclusion lim t→+∞ z(t) = 0 follows. This proves (125).

The proof of Step 7.1 is now complete.

Step 7.2. For any ρ > 0, let r 0 ∈ (0, δ /2) be as in Step 7.1 and 3C 1 (3 + 2C 2 )r 0 < ρ. Then for any w j ∈ X and y ∞ + (Y 2 + Z 2 )(0) for all t ≥ 0. When clear, we drop the dependence on t in notation. For j = 1, 2, we set (131) P j := (w j (y j , z j ), y j , z j ) and L := Y [h 2 (P 1 )h 2 (P 2 )] + Z [h 3 (P 1 )h 3 (P 2 )] .

By a simple calculation, we see that

(132) e -2ρt (Y 2 + Z 2 ) = 2e -2ρt -ρ Y 2 + Z 2 -|c| Z 2 + L .
We show that L in (131) satisfies Plugging (136) into (135), then using the inequality 2a 1 a 2 ≤ a 2 1 + a 2 2 , we conclude (133).

Using (133) into (132), we get that

(137) e -2ρt Y 2 + Z 2 (t) ≤ 2e -2ρt α 0 (Y 2 + Z 2 )(t) + 3C 1 r 0 w 1 -w 2 2
∞ , where α 0 := 3C 1 (3 + 2C 2 )r 0ρ is negative from our choice of r 0 . Hence, from (137), for every t ∈ (0, ∞), we deduce that (138) e -2ρt Y 2 + Z 2 (t) ≤ 2ρ e -2ρt w 1w 2 2 ∞ . By integrating (138), we obtain (130), which completes Step 7.2.

Step 7.3. Let r 0 ∈ (0, δ /2) be as in Step 7.2 with ρ = a/2 and 6C 1 (1 + C 2 )r 0 < aC 2 . Then, the map T : X → X is well-defined, where for every w ∈ X , we put Tw (y 0 , z 0 ) := -∞ 0 e -at h 1 (w (Φ w t (y 0 , z 0 )) , Φ w t (y 0 , z 0 )) dt for all (y 0 , z 0 ) ∈ D r 0 .

Proof of Step 7.3. For all (y 0 , z 0 ) ∈ D r 0 , we define (y(t), z(t)) := Φ w t (y 0 , z 0 ). We now observe that for all t ≥ 0, h 1 (w(y(t), z(t)), y(t), z(t)) stays bounded since max{|w(y(t), z(t))|, |y(t)|, |z(t)|} ≤ r 0 . Then, Tw (y 0 , z 0 ) is well-defined since a > 0. From w(0, 0) = 0, we have Φ w t (0, 0) = (0, 0) for all t ≥ 0, which yields Tw (0, 0) = 0. To prove that Tw ∈ X , it remains to show that Tw ranges in [-r 0 , r 0 ] and Tw is C 2 -Lipschitz. Indeed, using (121), for every (y 0 , z 0 ) ∈ D r 0 , we find that Since 3C 1 r 0 < a, we have |Tw (y 0 , z 0 ) | ≤ r 0 so that Tw ranges in [-r 0 , r 0 ]. We prove that Tw is C 2 -Lipschitz. We fix (y ( j) 0 , z ( j) 0 ) ∈ D r 0 for j = 1, 2, then define (y j (t), z j (t)) := Φ w t y ( j) 0 , z ( j) 0 and P j (t) := (w(y j , z j ), y j , z j )(t) for all t ≥ 0. By the definition of Tw, we see that (140) Tw y

(1)

0 , z (1) 0
-Tw y

(2)

0 , z (2) 0 ≤
.

From (140) and (141), we see that Tw is C 2 -Lipschitz, completing Step 7.3.

Step 7.4. If also 12C 1 r 0 (2 +C 2 ) < a in Step 7.3, then T is a contraction on X .

5 )

 5 admits no positive smooth solutions of non-(ND) type.

(a) Assumption ( 7 )

 7 holds for ≥ (n -2)/2 (see [8, Theorems 1.1 and 1.2]); (b) If (7) holds with ∈ (0, (n -2)/2), together with extra conditions, see [17, Theorem 1.2].

  (a) y and y are absolutely continuous on [0, T ]; (b) y satisfies the ODE in (44) a.e. on [0, T ];

  y 0 and |z(t)| ≤ max{y 0 , |z 0 |} on [0, ∞). Moreover, we have (125) lim t→∞ (y(t), z(t)) = (0, 0). Proof of Step 7.1. Let (y 0 , z 0 ) ∈ D r 0 be arbitrary. Since the Cauchy-Lipschitz theory applies, the initial value problem (S w ) has a unique solution (y, z) on an interval [0, b) with b > 0. We prove the following: (i) y ≡ 0 if y 0 = 0 and 0 < y(t) ≤ y 0 for all t ∈ [0, b) if y 0 ∈ (0, r 0 ]; (ii) |z(t)| ≤ max{y 0 , |z 0 |} for every t ∈ [0, b).

  (t)) for t ∈ [0, b), where h 2 (t, y) is continuous in t ∈ [0, b) and Lipschitz with respect to y ∈ [0, r 0 ]. The assumption (121) yields h 2 (•, 0) = 0 on [0, b) and h 2 (t, y(t)) ≤ 0 for all t ∈ [0, b). The claim of (i) holds since y (t) ≤ 0 on [0, b) and y is the unique solution of y (t) = h 2 (t, y(t)) for t ∈ [0, b), subject to y(0) = y 0 .

|z h 3

 3 (w(y, z), y, z)| ≤ |c| max{y 2 , z 2 }/2 on [0, b). If y 0 = 0, then y ≡ 0 on [0, b) by (i). From (128) and (126), we have (z 2 ) ≤ 0 on [0, b), which yields |z(t)| ≤ |z 0 | for all t ∈ [0, b), proving (ii) if y 0 = 0.

  (a) |z(t)| ≤ y 0 for all t ∈ [0, b), which immediately yields (ii); (b) |z(t)| ≥ y 0 for all t ∈ [0, b);(c) For some t 0 ∈ (0, b), we have |z| > y 0 on t ∈ [0,t 0 ) and |z| < y 0 on (t 0 , b). Using (128) into (126), we get (z 2 ) < 0 on [0, b) in case (b) and on [0,t 0 ) in case (c) since max{y 2 , z 2 } = z 2 . Thus in case (b) and (c) respectively, we find that |z| ≤ |z 0 | on [0, b) and [0,t 0 ), respectively. This proves (ii) when y 0 > 0.

∈ 0 )

 0 D r 0 with j = 1, 2, we have(129) |(y 1 , z 1 ) -(y 2 , z 2 )| (t) ≤ e ρt w 1w 2 ∞ ∈ [0, ∞), where we denote (y j (t), z j (t)) := Φ for j = 1, 2. Proof of Step 7.2. We denote Y := y 1y 2 and Z := z 1z 2 . It suffices to prove that (130) e -2ρt (Y 2 + Z 2 )(t) ≤ w 1w 2 2

3C 1 r 0 ( 3 + 2 ∞ 3 √ 2C 1 r 0 |P 1 -P 2 | Y 2 + Z 2 .√ Y 2 + 2 ≤ |w 1 (y 1 , z 1 )

 323122222111 2C 2 )(Y 2 + Z 2 ) + w 1w 2 .Proof of (133). Since max{|y j |, |z j |, |w j (y j , z j )|} ≤ r 0 for j = 1, 2, by the assumption on |∇h 2 | and |∇h 3 | in (121), we infer that (134) supξ ∈[0,1] |(∇φ )(ξ P 1 + (1ξ )P 2 )| ≤ 3C 1 r 0 with φ = h 2 and φ = h 3 . Therefore, we get that (135) |L| ≤ 3C 1 r 0 |P 1 -P 2 | (|Y | + |Z|) ≤ Set a 1 = w 1w 2 ∞ and a 2 = Z 2 .Using (131) and that w 1 is C 2 -Lipschitz, we get (136) |P 1 -P 2 | ≤ |w 1 (y 1 , z 1 )w 2 (y 2 , z 2 )| + aw 1 (y 2 , z 2 )| + a 1 + a 2 ≤ (1 +C 2 )a 2 + a 1 .

1 ∞ 0 e

 10 (139) |Tw (y 0 , z 0 ) | ≤ C -at (w 2 (y, z) + y 2 + z 2 ) dt ≤ 3C 1 r 2 0 a .

∞ 0 e

 0 -at |h 1 (P 1 )h 1 (P 2 )| dt.Since (134) holds for φ = h 1 , using w 1 = w 2 = w in (136), we get that|h 1 (P 1 )h 2 (P 2 )| ≤ 3C 1 (1 +C 2 )r 0 |(y 1 , z 1 ) -(y 2 , z 2 )| .Using that 6C 1 (1 +C 2 )r 0 < aC 2 and taking ρ = a/2 in Step 7.2, we arrive at (141) |h 1 (P 1 )h 1 (P 2 )

Acknowledgements: Work on this project started during first author's visits to McGill University (July-August 2017) and University of Lorraine (September-October 2017). She is very grateful for the support and hospitality of her co-authors while carrying out research at their institutions.

Proof of Step 7.4. For w 1 , w 2 ∈ X and (y 0 , z 0 ) ∈ D r 0 , we define (y j (t), z j (t)) := Φ w j t (y 0 , z 0 ) and P j (t) := (w j (y j (t), z j (t)), y j (t), z j (t)) for all t ≥ 0 and j = 1, 2. As in Step 7.2 with ρ = a/2, we obtain that (142)

Then, using (142) and our choice of r 0 , we see that

Therefore, T is 1/2-Lipschitz, so it is a contraction mapping. This ends Step 7.4.

Step 7.5. Let r 0 ∈ (0, δ /2) be as in Step 7.4. Then, there exists w ∈ X such that for every (y 0 , z 0 ) ∈ D r 0 and x 0 = w(y 0 , z 0 ), the initial value system (122) has a solution Z = (x, y, z) defined on [0, ∞) and satisfying (123).

Proof of Step 7.5. The choice of r 0 in Step 7.4 depends only on a, |c|,C 1 ,C 2 . Picard's fixed point theorem yields the existence of w ∈ X such that Tw = w, where T is given by Step 7.3, that is,

for all (y, z) ∈ D r 0 . We fix (y 0 , z 0 ) ∈ D r 0 arbitrarily. We show that Z (t) = (x(t), y(t), z(t)) is a solution to the initial system (122), subject to (123), where we define (144) (y(t), z(t)) := Φ w t (y 0 , z 0 ) and x(t) := w(y(t), z(t)) for all t ≥ 0.

Indeed, Step 7.1 yields that (y , z ) = (h 2 (x, y, z), cz + h 3 (x, y, z)) on [0, ∞). In view of w(0, 0) = 0 and lim t→+∞ (y(t), z(t)) = (0, 0), we get lim t→+∞ x(t) = 0, proving (123). Since

for all ξ ,t ≥ 0, from (143) and (144), we obtain that (145)

for all t ≥ 0. This ends Step 7.5 since x ∈ C 1 [0, +∞) and x = ax + h 1 (x, y, z) on [0, ∞).

Using the definition of X and Step 7.5, we finish the proof of Theorem 7.1.