Short Title]SigVM: Enabling Event-Driven Execution for Autonomous Smart Contracts

Zihan Zhao

University of Toronto

Sidi Mohamed Beillahi

University of Toronto

Ryan Song

University of Toronto

Yuxi Cai

University of Toronto

Andreas Veneris

University of Toronto

Fan Long

University of Toronto

#1     #1#1#1     #1     #1     #1     #1#1          #1#1

authorpersonAave. yearJan.
titleAave Protocol: LendingPoolLiquidationManager.sol.

[æernity dev teamæernity dev
authorpersonæernity dev team.
titleThe Sophia Language.

authorpersonAndreas M. Antonopoulos.
booktitleMastering Bitcoin: Programming the Open
Blockchain (edition2nd ed.).
publisherO'Reilly Media, Inc.

[Bhargavan, Delignat-Lavaud, Fournet,
Gollamudi, Gonthier, Kobeissi, Kulatova, Rastogi, Sibut-Pinote, Swamy, and
Zanella-BéguelinBhargavan et al2016]
authorpersonKarthikeyan Bhargavan,
personAntoine Delignat-Lavaud, personCédric
Fournet, personAnitha Gollamudi, personGeorges
Gonthier, personNadim Kobeissi, personNatalia
Kulatova, personAseem Rastogi, personThomas
Sibut-Pinote, personNikhil Swamy, and
personSantiago Zanella-Béguelin.
Formal Verification of Smart Contracts: Short
Paper. In booktitleProceedings of the 2016 ACM Workshop on
Programming Languages and Analysis for Security. pages91–96.

[Blackshear, Cheng, Dill, Gao, Maurer, Nowacki,
Pott, Qadeer, Rain, Russi, Sezer, Zakian, and ZhouBlackshear
et al2020]
authorpersonSam Blackshear, personEvan
Cheng, personDavid L. Dill, personVictor Gao,
personBen Maurer, personTodd Nowacki,
personAlistair Pott, personShaz Qadeer,
personRain, personDario Russi,
personStephane Sezer, personTim Zakian, and
personRuntian Zhou. year2020.
titleMove: A Language With Programmable Resources.

authorpersonVitalik Buterin.
titleEthereum: A next-generation smart contract and
decentralized application platform.

titleChainLink: Using Price Feeds.

[Coblenz, Oei, Etzel, Koronkevich, Baker,
Bloem, Myers, Sunshine, and AldrichCoblenz et al2019]
authorpersonMichael Coblenz, personReed
Oei, personTyler Etzel, personPaulette Koronkevich,
personMiles Baker, personYannick Bloem,
personBrad A. Myers, personJoshua Sunshine, and
personJonathan Aldrich. year2019.
titleObsidian: Typestate and Assets for Safer Blockchain
[arxiv]1909.03523 [cs.PL]

authorpersonRChain Cooperative.

[Das, Balzer, Hoffmann, Pfenning, and
SanturkarDas et al2021]
authorpersonA. Das, personS. Balzer,
personJ. Hoffmann, personF. Pfenning, and
personI. Santurkar. year2021.
Resource-Aware Session Types for Digital
Contracts. In booktitle2021 IEEE 34th Computer Security
Foundations Symposium (CSF). publisherIEEE Computer Society,
addressLos Alamitos, CA, USA, pages111–126.

authorpersonCorentin Denoeud.
titleWhat Is A Transaction Relayer And How Does It Work?

authorpersonEthereum. year2020.
titleThe Solidity Contract-Oriented Programming Language.

authorpersonPlanet Ethereum.

titleContract: DAIHardFactory.

titleContract: DutchExchange.

titleContract: HistoricalPriceFeed.

titleContract: KyberDxMarketMaker.

titleContract: Metronome.

titleContract: PriceFeed.

titleContract: PriceOracleInterface.

titleThe Ethereum Blockchain Explorer.

[Fabian VogelstellerFabian Vogelsteller2015]
authorpersonVitalik Buterin Fabian Vogelsteller.
title"EIP-20: ERC-20 Token Standard," Ethereum Improvement

authorpersonThe Maker Foundation.
titleThe Maker Protocol: MakerDAO's Multi-Collateral Dai
(MCD) System.

authorpersonThe Maker Foundation.
titleThe Market Collapse of March 12-13, 2020: How It
Impacted MakerDAO.

[Hildenbrandt, Saxena, Rodrigues, Zhu, Daian,
Guth, Moore, Park, Zhang, Stefanescu, and RosuHildenbrandt
et al2018]
authorpersonE. Hildenbrandt, personM.
Saxena, personN. Rodrigues, personX. Zhu,
personP. Daian, personD. Guth, personB.
Moore, personD. Park, personY. Zhang,
personA. Stefanescu, and personG. Rosu.
KEVM: A Complete Formal Semantics of the Ethereum
Virtual Machine. In booktitle2018 IEEE 31st Computer
Security Foundations Symposium (CSF). pages204–217.

[Kalra, Goe, Dhawan, and SharmaKalra
et al2018]
authorpersonSukrit Kalra, personSeep Goe,
personMohan Dhawan, and personSubodh Sharma.
ZEUS: Analyzing Safety of Smart Contracts. In
booktitleProceedings of the 25th Network and Distributed
Systems Security (NDSS) Symposium 2018.

[Leshner and HayesLeshner and Hayes2019]
authorpersonRobert Leshner and
personGeoffrey Hayes. year2019.
titleCompound: The Money Market Protocol.

[Li, Choi, and LongLi et al2020]
authorpersonAo Li, personJemin Andrew Choi,
and personFan Long. year2020.
Securing Smart Contract with Runtime Validation.
In booktitleProceedings of the 41st ACM SIGPLAN Conference
on Programming Language Design and Implementation.

[Li and PalanisamyLi and Palanisamy2020]
authorpersonChao Li and personBalaji
Palanisamy. year2020.
EventWarden: A Decentralized Event-driven Proxy
Service for Outsourcing Arbitrary Transactions in Ethereum-like Blockchains.
In booktitleProceedings of 27rd IEEE International
Conference on Web Services.

[Luu, Chu, Olickel, Saxena, and HoborLuu
et al2016]
authorpersonLoi Luu, personDuc-Hiep Chu,
personHrishi Olickel, personPrateek Saxena, and
personAquinas Hobor. year2016.
Making Smart Contracts Smarter. In
booktitleProceedings of the 2016 ACM SIGSAC Conference on
Computer and Communications Security. pages254–269.

[Nikolić, Kolluri, Sergey, Saxena, and
HoborNikolić et al2018]
authorpersonIvica Nikolić, personAashish
Kolluri, personIlya Sergey, personPrateek Saxena,
and personAquinas Hobor. year2018.
Finding The Greedy, Prodigal, and Suicidal
Contracts at Scale. In booktitleProceedings of the 34th
Annual Computer Security Applications Conference.

authorpersonOCamlPRO. year2020.
titleLiquidity Documentation.

authorpersonRussell O'Connor.
Simplicity: A new language for blockchains. In
booktitleProceedings of the 2017 Workshop on Programming
Languages and Analysis for Security. pages107–120.

[Oved and MositesOved and Mosites2017]
authorpersonMichael Oved and personDon
Mosites. year2017.
titleSwap: A Peer-to-Peer Protocol for Trading Ethereum

[Permenev, Dimitrov, Tsankov,
Drachsler-Cohen, and VechevPermenev et al2020]
authorpersonA. Permenev, personD.
Dimitrov, personP. Tsankov, personD.
Drachsler-Cohen, and personM. Vechev.
VerX: Safety Verification of Smart Contracts. In
booktitle2020 IEEE Symposium on Security and Privacy (SP).

[Peterson, Krug, Zoltu, Williams, and
AlexanderPeterson et al2019]
authorpersonJack Peterson, personJoseph
Krug, personMicah Zoltu, personAustin K. Williams,
and personStephanie Alexander.
titleAugur: a Decentralized Oracle and Prediction Market
Platform (v2.0).

titleProvable: Ethereum Quick Start.

[Sergey, Nagaraj, Johannsen, Kumar, Trunov, and
HaoSergey et al2019]
authorpersonIlya Sergey, personVaivaswatha
Nagaraj, personJacob Johannsen, personAmrit Kumar,
personAnton Trunov, and personKen Chan Guan Hao.
Safer smart contract programming with Scilla.
journalProceedings of the ACM on Programming
Languages volume3, numberOOPSLA
(year2019), pages1–30.

[Sjöberg, Sang, Weng, and ShaoSjöberg
et al2019]
authorpersonVilhelm Sjöberg,
personYuyang Sang, personShu-chun Weng, and
personZhong Shao. year2019.
DeepSEA: A Language for Certified System Software.

Article No: 136.

authorpersonVyper Team.

authorpersonParity Technologies.

authorpersonZeroEx. year2019.

authorpersonZeroEx. yearOct.
titleIDEX Smart Contracts: Exchange.sol.

#1     #1#1#1     #1     #1     #1     #1#1          #1#1

authorpersonAave. yearJan.
titleAave Protocol: LendingPoolLiquidationManager.sol.

[æernity dev teamæernity dev
authorpersonæernity dev team.
titleThe Sophia Language.

authorpersonAndreas M. Antonopoulos.
booktitleMastering Bitcoin: Programming the Open
Blockchain (edition2nd ed.).
publisherO'Reilly Media, Inc.

[Bhargavan, Delignat-Lavaud, Fournet,
Gollamudi, Gonthier, Kobeissi, Kulatova, Rastogi, Sibut-Pinote, Swamy, and
Zanella-BéguelinBhargavan et al2016]
authorpersonKarthikeyan Bhargavan,
personAntoine Delignat-Lavaud, personCédric
Fournet, personAnitha Gollamudi, personGeorges
Gonthier, personNadim Kobeissi, personNatalia
Kulatova, personAseem Rastogi, personThomas
Sibut-Pinote, personNikhil Swamy, and
personSantiago Zanella-Béguelin.
Formal Verification of Smart Contracts: Short
Paper. In booktitleProceedings of the 2016 ACM Workshop on
Programming Languages and Analysis for Security. pages91–96.

[Blackshear, Cheng, Dill, Gao, Maurer, Nowacki,
Pott, Qadeer, Rain, Russi, Sezer, Zakian, and ZhouBlackshear
et al2020]
authorpersonSam Blackshear, personEvan
Cheng, personDavid L. Dill, personVictor Gao,
personBen Maurer, personTodd Nowacki,
personAlistair Pott, personShaz Qadeer,
personRain, personDario Russi,
personStephane Sezer, personTim Zakian, and
personRuntian Zhou. year2020.
titleMove: A Language With Programmable Resources.

authorpersonVitalik Buterin.
titleEthereum: A next-generation smart contract and
decentralized application platform.

titleChainLink: Using Price Feeds.

[Coblenz, Oei, Etzel, Koronkevich, Baker,
Bloem, Myers, Sunshine, and AldrichCoblenz et al2019]
authorpersonMichael Coblenz, personReed
Oei, personTyler Etzel, personPaulette Koronkevich,
personMiles Baker, personYannick Bloem,
personBrad A. Myers, personJoshua Sunshine, and
personJonathan Aldrich. year2019.
titleObsidian: Typestate and Assets for Safer Blockchain
[arxiv]1909.03523 [cs.PL]

authorpersonRChain Cooperative.

[Das, Balzer, Hoffmann, Pfenning, and
SanturkarDas et al2021]
authorpersonA. Das, personS. Balzer,
personJ. Hoffmann, personF. Pfenning, and
personI. Santurkar. year2021.
Resource-Aware Session Types for Digital
Contracts. In booktitle2021 IEEE 34th Computer Security
Foundations Symposium (CSF). publisherIEEE Computer Society,
addressLos Alamitos, CA, USA, pages111–126.

authorpersonCorentin Denoeud.
titleWhat Is A Transaction Relayer And How Does It Work?

authorpersonEthereum. year2020.
titleThe Solidity Contract-Oriented Programming Language.

authorpersonPlanet Ethereum.

titleContract: DAIHardFactory.

titleContract: DutchExchange.

titleContract: HistoricalPriceFeed.

titleContract: KyberDxMarketMaker.

titleContract: Metronome.

titleContract: PriceFeed.

titleContract: PriceOracleInterface.

titleThe Ethereum Blockchain Explorer.

[Fabian VogelstellerFabian Vogelsteller2015]
authorpersonVitalik Buterin Fabian Vogelsteller.
title"EIP-20: ERC-20 Token Standard," Ethereum Improvement

authorpersonThe Maker Foundation.
titleThe Maker Protocol: MakerDAO's Multi-Collateral Dai
(MCD) System.

authorpersonThe Maker Foundation.
titleThe Market Collapse of March 12-13, 2020: How It
Impacted MakerDAO.

[Hildenbrandt, Saxena, Rodrigues, Zhu, Daian,
Guth, Moore, Park, Zhang, Stefanescu, and RosuHildenbrandt
et al2018]
authorpersonE. Hildenbrandt, personM.
Saxena, personN. Rodrigues, personX. Zhu,
personP. Daian, personD. Guth, personB.
Moore, personD. Park, personY. Zhang,
personA. Stefanescu, and personG. Rosu.
KEVM: A Complete Formal Semantics of the Ethereum
Virtual Machine. In booktitle2018 IEEE 31st Computer
Security Foundations Symposium (CSF). pages204–217.

[Kalra, Goe, Dhawan, and SharmaKalra
et al2018]
authorpersonSukrit Kalra, personSeep Goe,
personMohan Dhawan, and personSubodh Sharma.
ZEUS: Analyzing Safety of Smart Contracts. In
booktitleProceedings of the 25th Network and Distributed
Systems Security (NDSS) Symposium 2018.

[Leshner and HayesLeshner and Hayes2019]
authorpersonRobert Leshner and
personGeoffrey Hayes. year2019.
titleCompound: The Money Market Protocol.

[Li, Choi, and LongLi et al2020]
authorpersonAo Li, personJemin Andrew Choi,
and personFan Long. year2020.
Securing Smart Contract with Runtime Validation.
In booktitleProceedings of the 41st ACM SIGPLAN Conference
on Programming Language Design and Implementation.

[Li and PalanisamyLi and Palanisamy2020]
authorpersonChao Li and personBalaji
Palanisamy. year2020.
EventWarden: A Decentralized Event-driven Proxy
Service for Outsourcing Arbitrary Transactions in Ethereum-like Blockchains.
In booktitleProceedings of 27rd IEEE International
Conference on Web Services.

[Luu, Chu, Olickel, Saxena, and HoborLuu
et al2016]
authorpersonLoi Luu, personDuc-Hiep Chu,
personHrishi Olickel, personPrateek Saxena, and
personAquinas Hobor. year2016.
Making Smart Contracts Smarter. In
booktitleProceedings of the 2016 ACM SIGSAC Conference on
Computer and Communications Security. pages254–269.

[Nikolić, Kolluri, Sergey, Saxena, and
HoborNikolić et al2018]
authorpersonIvica Nikolić, personAashish
Kolluri, personIlya Sergey, personPrateek Saxena,
and personAquinas Hobor. year2018.
Finding The Greedy, Prodigal, and Suicidal
Contracts at Scale. In booktitleProceedings of the 34th
Annual Computer Security Applications Conference.

authorpersonOCamlPRO. year2020.
titleLiquidity Documentation.

authorpersonRussell O'Connor.
Simplicity: A new language for blockchains. In
booktitleProceedings of the 2017 Workshop on Programming
Languages and Analysis for Security. pages107–120.

[Oved and MositesOved and Mosites2017]
authorpersonMichael Oved and personDon
Mosites. year2017.
titleSwap: A Peer-to-Peer Protocol for Trading Ethereum

[Permenev, Dimitrov, Tsankov,
Drachsler-Cohen, and VechevPermenev et al2020]
authorpersonA. Permenev, personD.
Dimitrov, personP. Tsankov, personD.
Drachsler-Cohen, and personM. Vechev.
VerX: Safety Verification of Smart Contracts. In
booktitle2020 IEEE Symposium on Security and Privacy (SP).

[Peterson, Krug, Zoltu, Williams, and
AlexanderPeterson et al2019]
authorpersonJack Peterson, personJoseph
Krug, personMicah Zoltu, personAustin K. Williams,
and personStephanie Alexander.
titleAugur: a Decentralized Oracle and Prediction Market
Platform (v2.0).

titleProvable: Ethereum Quick Start.

[Sergey, Nagaraj, Johannsen, Kumar, Trunov, and
HaoSergey et al2019]
authorpersonIlya Sergey, personVaivaswatha
Nagaraj, personJacob Johannsen, personAmrit Kumar,
personAnton Trunov, and personKen Chan Guan Hao.
Safer smart contract programming with Scilla.
journalProceedings of the ACM on Programming
Languages volume3, numberOOPSLA
(year2019), pages1–30.

[Sjöberg, Sang, Weng, and ShaoSjöberg
et al2019]
authorpersonVilhelm Sjöberg,
personYuyang Sang, personShu-chun Weng, and
personZhong Shao. year2019.
DeepSEA: A Language for Certified System Software.

Article No: 136.

authorpersonVyper Team.

authorpersonParity Technologies.

authorpersonZeroEx. year2019.

authorpersonZeroEx. yearOct.
titleIDEX Smart Contracts: Exchange.sol.


# Classification of radial solutions to
$-\Delta_{g}u=e^{u}$ on Riemannian models

Elvise BERCHIO  Dipartimento di Scienze Matematiche,
Politecnico di Torino,
Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
E-mail address<EMAIL_ADDRESS>,  Alberto FERRERO  Dipartimento di
Scienze e Innovazione Tecnologica,
Universitá del Piemonte Orientale,
Viale Teresa Michel 11, Alessandria, 15121, Italy.
E-mail address<EMAIL_ADDRESS>,  Debdip Ganguly  Department of
Mathematics,
Indian Institute of Technology Delhi,
IIT Campus, Hauz Khas, Delhi,
New Delhi 110016, India.
E-mail address<EMAIL_ADDRESS>and  Prasun Roychowdhury  Mathematics
Division,
National Center for Theoretical Sciences,
NTU, Cosmology Building, No. 1, Sec. 4
Roosevelt RD, Taipei City 106, Taiwan.
E-mail address<EMAIL_ADDRESS>
###### Abstract.

We provide a complete classification with respect to asymptotic behaviour,
stability and intersections properties of radial smooth solutions to the
equation $-\Delta_{g}u=e^{u}$ on Riemannian model manifolds $(M,g)$ in
dimension $N\geq 2$. Our assumptions include Riemannian manifolds with
sectional curvatures bounded or unbounded from below. Intersection and
stability properties of radial solutions are influenced by the dimension $N$
in the sense that two different kinds of behaviour occur when $2\leq N\leq 9$
or $N\geq 10$, respectively. The crucial role of these dimensions in
classifying solutions is well-known in Euclidean space; here the analysis
highlights new properties of solutions that cannot be observed in the flat
case.

###### Key words and phrases:

Riemannian models, hyperbolic space, radial solutions, stability of solutions,
asymptotics of solutions

###### 2010 Mathematics Subject Classification:

58J60, 35B35, 35J61, 35R01

##  1\. Introduction

Let $N\geq 2$ and let $M$ be an $N$-dimensional Riemannian model $(M,g)$,
namely a manifold admitting a pole $o$ and whose metric is given, in polar or
spherical coordinates around $o$, by

(1.1) $g={\rm d}r^{2}+(\psi(r))^{2}\,{\rm d}\omega^{2},\qquad
r>0,\,\omega\in\mathbb{S}^{N-1}\,,$

for some function $\psi$ satisfying suitable assumptions. Here ${\rm
d}\omega^{2}$ denotes the canonical metric on the unit sphere
$\mathbb{S}^{N-1}$ and $r$ is by construction the distance between a point of
spherical coordinates $(r,\omega)$ and the pole $o$.

In this article, we are concerned with radial smooth solutions of

(1.2) $\displaystyle-\Delta_{g}u\,=\,e^{u}\quad\text{ in }M\,,$

namely smooth solutions of (1.2) depending only on the geodesic distance from
the pole. Here $\Delta_{g}$ denotes the Laplace-Beltrami operator in $(M,g)$.

The great interest for equation (1.2) is motivated by its applications to
geometry and physics see e.g., [9, 14, 18]. For instance, in fluid dynamics, a
variant of (1.2) appears when studying “Stuart vortices” on a unit sphere
$\mathbb{S}^{2}$ in a stationary regime, i.e.

$-\Delta_{\mathbb{S}^{2}}\,\psi=b+c\,e^{d\psi}\qquad\text{on }\mathbb{S}^{2}$

where $\psi$ represents the stream function, $b,c,d$ are real constants and
the right hand side of the equation represents a vorticity $\omega$ depending
exponentially on the streamfunction $\psi$, see [11]. Related semilinear
equations on rotating spheres also arise in the study of stratospheric
planetary stationary flows, see [10] and the discussion therein. When looking
for stationary stream functions, the authors of [10] are led to consider the
semilinear elliptic equation

$\Delta_{\mathbb{S}^{2}}\,\psi=F(\psi)-2\omega\sin\theta\qquad\text{on
}\mathbb{S}^{2}$

where now $\omega$ represents the rotation speed around the polar axis, $F$ is
a smooth function and $\theta$ is the angle of latitude.

The behaviour of solutions to equation (1.2), when posed in the Euclidean
space, has been fully understood from different points of view: asymptotic
behaviour, stability and intersections properties. In the seminal paper [21]
existence and asymptotic behaviour of radial solutions were established by
means of a dynamical system analysis based on the so-called Emden
transformation of the phase plane. We refer to [25, Theorem 1.1] for the
intersection properties of solutions, while the classification of solutions
with respect to stability is given in [12, 15]. See also [13] and references
therein for more recent results in the case of general nonlinearities.
Finally, related results in the weighted case: $-\Delta u=k(x)e^{u}$ in
$\mathbb{R}^{n}$ can be found in [1, 7, 8], extensions to the higher order
case are instead given in [3] and in [16].

As already pointed out in [21], equation (1.2) can be regarded as the limit
case as $p\rightarrow+\infty$ of the Lane-Emden-Fowler equation:
$-\Delta_{g}u=|u|^{p-1}u$ in $M$ ($p>1$). In the last ten years, there has
been an intense study of this equation in non-euclidean frameworks including
the hyperbolic space and more general Riemannian models, see [2, 4, 5, 17, 20,
23, 22] and references therein. In these papers existence, multiplicity,
asymptotic and stability results were provided. The analysis settled on
general manifolds highlights deep relationships between the qualitative
behaviour of solutions and intrinsic properties of the manifold itself and,
sometimes, it reveals a number of unexpected phenomena, if compared with the
euclidean case, see e.g, the introductions of [4] and [5]. Another reason for
the interest in this kind of research is the fact that some classical tools in
the Euclidean case do not work in non-euclidean frameworks, therefore the
analysis requires new ideas and alternative approaches that could be useful
also in other contexts. As a matter of example, an analogous of the above-
mentioned Emden transformation seems not known in non-euclidean settings,
therefore different arguments, such as the exploitation of ad hoc Lyapunov
functionals, fine asymptotic analysis and blow-up methods, must be exploited,
see e.g., Sections 3-5 below.

Coming back to (1.2), its investigation in non-euclidean settings turns out to
be the natural subsequent step in order to complete the scenario of results
for this equation; this motivates the present paper. More precisely, we
provide a complete classification with respect to asymptotic behaviour,
stability and intersections properties of radial smooth solutions to (1.2) for
manifolds $M$ having strictly negative sectional curvatures, possibly
unbounded, see assumptions (A1)-(A3) in Section 2. It is worth mentioning
that, even if in our proofs we take advantage of some arguments already
employed in the study of the Lane-Emden-Fowler equation on Riemannian models,
the fact of dealing with an exponential nonlinearity brought a number of
considerable technical difficulties related, for instance, to the different
sign and decay behaviour of solutions in the two cases. Moreover, our analysis
made it possible to highlight a number of properties of solutions that cannot
be observed in the flat case, see e.g., Remarks 2.3, 2.8 and 2.10 in Section
2.

The paper is organised as follows. In Section 2 we give the precise
formulation of the problem and we state our main results about continuation
and asymptotic behaviour of solutions (Proposition 2.1 and Theorem 2.2),
stability properties (Theorems 2.6 and 2.7 ), and intersection properties
(Theorem 2.5 and Theorem 2.9); the remaining sections of the paper are devoted
to the proofs. More precisely, in Section 3 we prove Proposition 2.1 while
Section 4 is devoted to the proof of Theorem 2.2. Section 5 contains a number
of technical lemmas that will be exploited to prove Theorems 2.5, 2.6, 2.7 and
2.9. At last, for the sake of the reader, in the Appendix, we briefly recall
some well-known facts in the Euclidean case that highlight the role of the
dimension $N$ in the stability and intersection properties analysis.

##  2\. Statement of the problem and main results

###  2.1. Notations

Let $\psi$ be the function introduced in (1.1). We assume that $\psi$
satisfies

(A1) $\psi\in C^{2}([0,\infty))$, $\psi>0$ in $(0,+\infty)$,
$\psi(0)=\psi^{\prime\prime}(0)=0$ and $\psi^{\prime}(0)=1$;

and

(A2) $\psi^{\prime}(r)>0$ for any $r>0$ .

We recall that the Riemannian model associated with the choice $\psi(r)=\sinh
r$ is a well-known representation of the hyperbolic space $\mathbb{H}^{N}$,
see e.g., [19] and the references therein, while the Euclidean space
$\mathbb{R}^{N}$ corresponds to $\psi(r)=r$.

The following list summarizes some notations we shall use throughout this
paper.

  * -

For any $P\in M$ we denote by $T_{P}M$ the tangent space to $M$ at the point
$P$.

  * -

For any $P\in M$ and $U_{1},U_{2}\in T_{P}M$ we denote by $\langle
U_{1},U_{2}\rangle_{g}$ the scalar product on $T_{P}M$ associated with the
metric $g$.

  * -

For any $P\in M$ and $U\in T_{P}M$ we denote by $|U|_{g}:=\sqrt{\langle
U,U\rangle_{g}}$ the norm of the vector $U$.

  * -

${\rm d}V_{g}$ denotes the volume measure in $(M,g)$.

  * -

$\nabla_{g}$ denotes the Riemannian gradient in $(M,g)$ and in spherical
coordinates it is given by $\nabla_{g}u(r,\omega)=\frac{\partial u}{\partial
r}(r,\omega)\,\frac{\partial}{\partial
r}+\frac{r}{(\psi(r))^{2}}\,\nabla_{\omega}u(r,\omega)$ for any $u\in
C^{1}(M)$, where we denoted by $\nabla_{\omega}$ the Riemannian gradient in
the unit sphere $\mathbb{S}^{N-1}$.

  * -

$\Delta_{g}$ denotes the Laplace-Beltrami operator in $(M,g)$ and in spherical
coordinates it is given by
$\Delta_{g}u(r,\omega)=\frac{\partial^{2}u}{\partial
r^{2}}(r,\omega)+(N-1)\frac{\partial u}{\partial
r}(r,\omega)+\frac{1}{(\psi(r))^{2}}\,\Delta_{\omega}u(r,\omega)$ for any
$u\in C^{2}(M)$, where we denoted by $\Delta_{\omega}$ the Laplace-Beltrami
operator in the unit sphere $\mathbb{S}^{N-1}$.

  * -

$C^{\infty}_{c}(M)$ denotes the space of $C^{\infty}(M)$ functions compactly
supported in $M$.

From the above notations, we deduce that if $u\in C^{2}(M)$ is a radial
function, then

(2.1)
$\displaystyle\Delta_{g}u(r)=u^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,u^{\prime}(r)=\frac{1}{(\psi(r))^{N-1}}[(\psi(r))^{N-1}u^{\prime}(r)]^{\prime}.$

Since our aim is to study _smooth_ radial solutions to (1.2), for any
$\alpha\in\mathbb{R}$, we focus our attention on the following initial value
problem

(2.2)
$\begin{dcases}-u^{\prime\prime}(r)-(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}u^{\prime}(r)=e^{u(r)}&(r>0)\\\
u(0)=\alpha&\\\ u^{\prime}(0)=0.&\\\ \end{dcases}$

The existence and uniqueness of a local solution $u(r)$ to (2.2) in $0\leq
r<R$ (here $R$ denotes the maximal interval of existence) follows by arguing
as in Proposition 1 in the Appendix of [24]. A classical argument allows to
prove that actually $R=+\infty$, thus showing that the solution $u=u(r)$ to
(2.2) is globally define in $[0,+\infty)$:

######  Proposition 2.1.

Let $N\geq 2$. Suppose that $\psi$ satisfies assumptions (A1)-(A2). For any
$\alpha\in\mathbb{R}$ the local solution to (2.2) may be continued to the
whole interval $[0,+\infty)$. Moreover, the functions $r\mapsto u^{\prime}(r)$
and $r\mapsto e^{u(r)}$ are bounded in $[0,+\infty)$, $u^{\prime}(r)<0$ for
any $r>0$ and in particular $u$ is decreasing in $[0,+\infty)$.

###  2.2. Asymptotic behaviour

In order to study the asymptotic behaviour of global solutions to (2.2) we
require the additional condition:

(A3)
$\lim_{r\rightarrow+\infty}\frac{\psi^{\prime}(r)}{\psi(r)}=:\Lambda\in(0,\infty].$

Clearly, the hyperbolic space satisfies condition (A3) and Riemannian models
which are asymptotically hyperbolic satisfy it as well. Furthermore, such a
condition allows for unbounded negative sectional curvatures: a typical
example in which this can hold corresponds to the choice $\psi(r)=e^{r^{a}}$
for a given $a>1$ and $r$ large, a case for which (see [4, Section 1.1])
sectional curvatures in the radial direction diverge like $-a^{2}r^{2(a-1)}$
as $r\to+\infty$. In addition, we remark that under assumptions (A1)-(A3), the
$L^{2}$ spectrum of $-\Delta_{g}$ is bounded away from zero whereas if
$\lim_{r\rightarrow+\infty}\frac{\psi^{\prime}(r)}{\psi(r)}=0$ then there is
no gap in the $L^{2}$ spectrum of $-\Delta_{g}$, see e.g., [4, Lemma 4.1].
Moreover, it can be proved that if the radial sectional curvature goes to zero
as $r\to+\infty$ then necessarily
$\lim_{r\rightarrow+\infty}\frac{\psi^{\prime}(r)}{\psi(r)}=0$, see again [4,
Lemma 4.1], therefore no spectral gap is present and the expected picture is
of Euclidean type.

In the following statement we show that the asymptotic behaviour of solutions
of (2.2) is related to the behaviour at infinity of the ratio
$\frac{\psi^{\prime}(r)}{\psi(r)}$ and hence, for what remarked above, to the
curvatures of the manifold:

######  Theorem 2.2.

Let $N\geq 2$. Suppose that $\psi$ satisfies assumptions (A1)-(A3). Finally,
in the case $\Lambda=+\infty$ we also assume that

(A4)
$\displaystyle\left[\log\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)\right]^{\prime}=O(1)\quad\text{as
}r\rightarrow+\infty\,.$

Let $u$ be a solution to (2.2). Then two cases may occur:

  * $(i)$

if $\frac{\psi}{\psi^{\prime}}\in L^{1}(0,\infty)$, then

$\lim_{r\rightarrow+\infty}u(r)\in(-\infty,\alpha)\,;$

  * $(ii)$

if $\frac{\psi}{\psi^{\prime}}\not\in L^{1}(0,\infty)$, then $u$ goes to
$-\infty$ with the following rate:

$\displaystyle\lim_{r\rightarrow+\infty}\frac{u(r)}{\log\big{(}\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}{\rm
d}s\big{)}}=-1\,;$

in particular, when $\Lambda\in(0,+\infty)$ we have

$\displaystyle\lim_{r\rightarrow+\infty}\frac{u(r)}{\log r}=-1.$

######  Remark 2.3.

As a prototype of function $\psi$ satisfying the assumptions of Theorem 2.2
when $\Lambda=+\infty$ consider the function $\psi(r)=re^{r^{2\gamma}}$ with
$\gamma>\frac{1}{2}$. If $\frac{1}{2}<\gamma\leq 1$ case $(ii)$ occurs while
if $\gamma>1$ assumption $(i)$ holds. Clearly, if $M=\mathbb{H}^{N}$ we have
that $\Lambda=1$ and case $(ii)$ occurs. From [21] we recall that in the flat
case solutions diverge to $-\infty$ being asymptotically equivalent to $-2\log
r$, therefore the effect of curvatures, in general, results in a slower decay
of solutions, indeed they may even remain bounded if case $(i)$ occurs.

###  2.3. Stability results and intersection properties of solutions

Let us start with the definition of stability.

######  Definition 2.4.

A solution $u\in C^{2}(M)$ to (1.2) is stable if

(2.3) $\displaystyle\int_{M}|\nabla_{g}v|_{g}^{2}\,{\rm
d}V_{g}-\int_{M}e^{u}\,v^{2}\,{\rm d}V_{g}\geq 0\quad\forall v\in
C_{c}^{\infty}(M).$

If $u$ does not satisfy (2.3), we say that it is unstable.

It is well known that stability plays an important role in the classification
of solutions of elliptic partial differential equations and the analysis of
qualitative properties of solutions, see e.g., the seminal paper [6]. In this
section, we provide a complete classification of smooth radial solutions of
(1.2) with respect to stability and we show that the same conditions determine
either the stability or the intersection properties. The relationship between
stability properties and intersection properties is clarified by the following
result.

######  Theorem 2.5.

Let $N\geq 2$ and let $\psi$ satisfy (A1)-(A3). Then the following statements
hold:

  * $(i)$

let $\alpha$ and $\beta$ be two distinct real numbers and let $u_{\alpha}$ and
$u_{\beta}$ be the corresponding solutions of (2.2), i.e.
$u_{\alpha}(0)=\alpha$ and $u_{\beta}(0)=\beta$. If $u_{\alpha}$ and
$u_{\beta}$ are stable then they do not intersect;

  * $(ii)$

if $u_{\alpha}$ is a unstable solution of (2.2) for some
$\alpha\in\mathbb{R}$, then for any $\beta>\alpha$ we have that $u_{\beta}$
intersects $u_{\alpha}$ at least once.

We observe that in the proof Theorem 2.5 -$(ii)$, we actually show the
validity of a more general result involving also non-radial smooth solutions
of (1.2). Indeed, in Lemma 6.2 we prove that if $u$ is a smooth unstable
solution of (1.2) then (1.2) does not admit any smooth solution $v$ satisfying
$v>u$ in $M$.

We now state the two main results about the stability of radial smooth
solutions of (1.2) characterized by dimensions $2\leq N\leq 9$ and $N\geq 10$,
respectively.

######  Theorem 2.6.

Let $2\leq N\leq 9$ and let $\psi$ satisfy (A1)-(A3). For any
$\alpha\in\mathbb{R}$ denote by $u_{\alpha}$ the unique solution to (2.2).
Then there exist $\eta\in\mathbb{R}$ such that

  * $(i)$

if $\alpha\in(-\infty,\eta]$ then $u_{\alpha}$ is stable;

  * $(ii)$

if $\alpha>\eta$ then $u_{\alpha}$ is unstable.

Furthermore, we have that $\eta\geq\log(\lambda_{1}(M))$ with the strict
inequality if $\frac{\psi}{\psi^{\prime}}\notin L^{1}(0,+\infty)$ where
$\lambda_{1}(M)$ denotes the bottom of the spectrum of $-\Delta_{g}$ in $M$.

######  Theorem 2.7.

Let $N\geq 10$ and let $\psi$ satisfy (A1)-(A3) and the additional condition

(A5) $\psi\in
C^{3}([0,+\infty)),\quad[\log(\psi^{\prime}(r))]^{\prime\prime}>0\quad\text{for
}r>0\,.$

Then all solutions to (2.2) are stable.

######  Remark 2.8.

The fact that $N=10$ is a critical threshold for stability is well-known in
the Euclidean case, see [15] and the Appendix. Here a new critical value
$\eta$ arises which has no analogous in the flat case where solutions are
always unstable if $2\leq N\leq 9$ and Theorem 2.6-$(i)$ never occurs. In
other words, we can say that, by the effect of assumption (A3), the critical
dimension does not exist if $\alpha\leq\eta$ since solutions are stable for
all $N\geq 2$.
About assumption (A5), it is technical but it includes the most interesting
examples. Indeed, it holds in the relevant case of the hyperbolic space, since
$[\log(\psi^{\prime}(r))]^{\prime\prime}=(\cosh r)^{-2}$. Models with
unbounded curvatures satisfying (A5) can be build as well, e.g., by taking
$\psi(r)=e^{r^{a}}$ with $a>1$ for $r\geq 1$, indeed
$[\log(\psi^{\prime}(r))]^{\prime\prime}=\frac{a-1}{r^{2}}(ar^{a}-1)>0$ for
$r\geq 1$.

At last, concerning intersection properties, we have:

######  Theorem 2.9.

Let $N\geq 2$, let $\psi$ satisfy (A1)-(A3) and let $\eta$ be as in the
statement of Theorem 2.6. We have that:

  * $(i)$

if $2\leq N\leq 9$ and $u_{\alpha}$, $u_{\beta}$ are two solutions of (2.2)
with $\alpha,\beta\in(-\infty,\eta]$ then $u_{\alpha}$ and $u_{\beta}$ do not
intersect;

  * $(ii)$

if $2\leq N\leq 9$ and $u_{\alpha}$, $u_{\beta}$ are two solutions of (2.2)
with $\alpha,\beta>\eta$ then $u_{\alpha}$ and $u_{\beta}$ intersect at least
once;

  * $(iii)$

if $N\geq 10$ and the additional condition (A5) holds true, then for any
$\alpha,\beta\in\mathbb{R}$ the corresponding solutions $u_{\alpha}$ and
$u_{\beta}$ of (2.2) do not intersect.

######  Remark 2.10.

In $\mathbb{R}^{N}$ it is known, see e.g., [25, Theorem 1.1], that every two
smooth radial solutions intersect each other once if $N=2$ and infinitely many
times if $3\leq N\leq 9$ while for $N\geq 10$ smooth radial solutions do not
intersect. Therefore, the behaviour stated in Theorem 2.9-$(i)$ (with its
counterpart Theorem 2.6-$(i)$) is, as a matter of fact, an effect of non-
vanishing curvatures.

##  3\. Proof of Proposition 2.1

Let $(0,R)$ be the maximum interval where the local solution $u$ of (2.2) is
defined and consider the Lyapunov functional

(3.1) $\displaystyle
F(r):=\frac{1}{2}(u^{\prime}(r))^{2}+e^{u(r)}\quad\text{for any }r\in(0,R)\,.$

It is readily seen that

$\displaystyle
F^{\prime}(r)=u^{\prime}(r)[u^{\prime\prime}(r)+e^{u(r)}]=-(N-1)u^{\prime}(r)^{2}\frac{\psi^{\prime}(r)}{\psi(r)}\leq
0\quad\text{for any }r\in(0,R)\,.$

This proves that $F$ is decreasing and $0<F(r)\leq e^{\alpha}$ for any
$r\in(0,R)$, therefore both $u^{\prime}(r)$ and $e^{u(r)}$ are bounded in
$(0,R)$. On the other hand, integrating (2.1) from $0$ to $r$ we deduce

$\displaystyle(\psi(r))^{N-1}u^{\prime}(r)=-\int_{0}^{r}(\psi(s))^{N-1}\,e^{u(s)}\>{\rm
d}s<0\quad\text{for any }r\in(0,R)\,.$

Therefore, $u^{\prime}$ is a negative function in $(0,R)$ and hence $u$ is
decreasing in $(0,R)$ as claimed.

Finally, we show that $R=+\infty$. Assume by contradiction that $R$ is finite
and let

$\displaystyle l:=\lim_{r\rightarrow R^{-}}u(r)\in[-\infty,\alpha)\,.$

If $l=-\infty$ we immediately get a contradiction since

$\displaystyle u(r)=\alpha+\int_{0}^{r}u^{\prime}(s)\>{\rm d}s\qquad\text{for
any }r\in(0,R)$

and taking the limit as $r\rightarrow R^{-}$ we have that the left hand side
goes to $-\infty$ while the right hand side remains bounded since
$u^{\prime}(r)$ is bounded. Being $l$ finite, from local existence for
solutions of ordinary differential equations with initial value conditions,
the solution $u$ may be extended in a right neighbourhood of $R$ thus
contradicting the maximality of $R$. This shows that $R=+\infty$ completes the
proof of the proposition. $\Box$

##  4\. Proof of Theorem 2.2

Throughout this section, we will always assume the validity of conditions
(A1)-(A3) and when $\Lambda=+\infty$ in (A3) we also assume the validity of
the additional condition (A4). We start by showing that $u^{\prime}$ admits a
limit at infinity and this limit is zero:

######  Lemma 4.1.

Let $u$ be the unique solution of (2.2). Then

$\lim_{r\to+\infty}u^{\prime}(r)=0\,.$

###### Proof.

From Proposition 2.1 we know that $u$ is decreasing in $[0,+\infty)$ and hence
it admits a limit as $r\to+\infty$. We put

(4.1) $l:=\lim_{r\to+\infty}u(r)\in[-\infty,\alpha)\,,$

where $\alpha=u(0)$. Let $F$ be the function defined in (3.1). Since $F$ is
non-increasing, then $0<F(r)\leq F(0)=e^{\alpha}$ for any $r\geq 0$. Hence,
there exists $c\in[0,+\infty)$ such that $c=\lim_{r\rightarrow+\infty}F(r)$.

By (4.1) we deduce that $\bar{l}:=\lim_{r\to+\infty}e^{u(r)}\in[0,+\infty)$ so
that by (3.1) we obtain

(4.2)
$\lim_{r\to+\infty}u^{\prime}(r)=\lim_{r\to+\infty}-\sqrt{2F(r)-e^{u(r)}}=-\sqrt{2c-\bar{l}}=\gamma\in(-\infty,0]\,.$

It remains to prove that $\gamma=0$. Suppose by contradiction that $\gamma<0$.
This implies

(4.3)
$\lim_{r\to+\infty}u(r)=\lim_{r\to+\infty}\left[\alpha+\int_{0}^{r}u^{\prime}(s)\,{\rm
d}s\right]=-\infty\,.$

Letting $\Lambda\in(0,+\infty]$ be as in (A3), by (2.2), (4.2) and (4.3) we
obtain

(4.4)
$\lim_{r\to+\infty}u^{\prime\prime}(r)=\lim_{r\to+\infty}\left[-(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,u^{\prime}(r)-e^{u(r)}\right]=-(N-1)\Lambda\gamma\in(0,+\infty]\,.$

In turn, by (4.4) and (2.2), we infer that

$\lim_{r\to+\infty}u^{\prime}(r)=\lim_{r\to+\infty}\int_{0}^{r}u^{\prime\prime}(s)\,{\rm
d}s=+\infty\,,$

in contradiction with (4.2). This completes the proof of the lemma. $\Box$

By exploiting Lemma 4.1 we prove:

######  Lemma 4.2.

Let $u$ be the unique solution of (2.2). Then the following limit exists

$\displaystyle\lim_{r\rightarrow+\infty}\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}\,.$

###### Proof.

We divide the proof into two parts depending on whether $\Lambda$ is finite or
not.

The case $\Lambda\in(0,+\infty)$. From (2.2), we have that

(4.5)
$\displaystyle-\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}-(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}=\frac{e^{u(r)}}{u^{\prime}(r)}.$

The above identity suggests that if $\chi(r):=\frac{e^{u(r)}}{u^{\prime}(r)}$
admits a limit as $r\to+\infty$, then the same does the function
$\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}$. This means that the proof of
Lemma 4.2 follows once we prove that $\chi$ admits a limit as $r\to+\infty$.

We proceed by contradiction assuming that $\chi$ does not admit a limit as
$r\to+\infty$. By direct computation, we see that

(4.6)
$\displaystyle\chi^{\prime}(r)=\frac{e^{u(r)}[({u^{\prime}}(r))^{2}-u^{\prime\prime}(r)]}{({u^{\prime}}(r))^{2}}\quad\text{for
any }r>0.$

We may assume that $\chi$ admits infinitely many local maxima and minima at
some points $r_{m}$, with $r_{m}\rightarrow+\infty$ as $m\rightarrow+\infty$,
and $\chi(r_{m})$ does not admit a limit as $m\to+\infty$.

In particular by (4.6) we have
$(u^{\prime}(r_{m}))^{2}-u^{\prime\prime}(r_{m})=0$ for any $m$. Hence,
evaluating (4.5) at $r_{m}$, we obtain

$\displaystyle-u^{\prime}(r_{m})-(N-1)\frac{\psi^{\prime}(r_{m})}{\psi(r_{m})}=\chi(r_{m})\,,$

for any $m$. Now, by (A3) and Lemma 4.1, we find that
$\chi(r_{m})\rightarrow-(N-1)\Lambda$ as $m\rightarrow+\infty$, a
contradiction. This completes the proof of the lemma in the case
$\Lambda\in(0,+\infty)$.

The case $\Lambda=+\infty$. We proceed similarly to the previous case by
writing (2.2) in the form

(4.7)
$\displaystyle-\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}-(N-1)=\frac{e^{u(r)}}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}$

and by defining this time
$\chi(r):=\frac{e^{u(r)}}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}$ in
such a way that if $\chi$ admits a limit as $r\rightarrow+\infty$ then the
same conclusion occurs for
$\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}$
and we complete the proof of the lemma also in this case. By contradiction,
assume that $\chi$ does not admit a limit as $r\to+\infty$. A simple
computation gives

(4.8)
$\displaystyle\chi^{\prime}(r)=\chi(r)\left\\{u^{\prime}(r)-\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}-\left[\log\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)\right]^{\prime}\right\\}\qquad\text{for
any }r>0.$

We may assume that $\chi$ admits infinitely many local maxima and minima at
some points $r_{m}$ with $r_{m}\rightarrow+\infty$ as $m\rightarrow+\infty$,
and $\chi(r_{m})$ does not admit a limit as $m\to+\infty$. In particular, by
(4.8), we have that

(4.9) $\displaystyle
u^{\prime}(r_{m})-\frac{u^{\prime\prime}(r_{m})}{u^{\prime}(r_{m})}-\left[\log\left(\frac{\psi^{\prime}(r_{m})}{\psi(r_{m})}\right)\right]^{\prime}=0\,.$

Hence, evaluating (4.7) at $r_{m}$ and using (4.9), we obtain

$\displaystyle-(N-1)-\frac{\psi(r_{m})u^{\prime}(r_{m})}{\psi^{\prime}(r_{m})}+\frac{\psi(r_{m})}{\psi^{\prime}(r_{m})}\left[\log\left(\frac{\psi^{\prime}(r_{m})}{\psi(r_{m})}\right)\right]^{\prime}=\chi(r_{m})\,,$

for any $m$.

Finally, by Lemma 4.1, (A3) and (A4), we find that
$\chi(r_{m})\rightarrow-(N-1)$ for $m\rightarrow+\infty$, a contradiction. The
proof of the lemma is complete also in this case. $\Box$

Our next purpose is to show that the limit in Lemma 4.2 must be $0$ under the
additional assumption:

(4.10) $\lim_{r\to+\infty}u(r)=-\infty\,.$

In Lemma 4.6 below we discuss the occurrence of (4.10) and we provide a
sufficient condition for (4.10) in terms of the integrability properties of
the ratio $\psi/\psi^{\prime}$.

Before proving in Lemma 4.4 that the limit in Lemma 4.2 is zero when (4.10)
holds, we state the following result which deals with the behaviour of $\psi$
at infinity.

######  Lemma 4.3.

Let $\Lambda$ be as in (A3), the following statements hold true:

  * $(i)$

for any $\Lambda\in(0,+\infty]$ we have that

(4.11) $\displaystyle\lim_{r\rightarrow+\infty}\psi(r)=+\infty$

and moreover for any $M>0$ and $0<\delta<N-1$ we have that

(4.12) $\displaystyle\psi^{-(N-1)+\delta}\in L^{1}(M,\infty)\,;$

  * $(ii)$

if $\Lambda\in(0,+\infty)$ then there exist $C>0$ and $M>0$ such that

(4.13) $\psi(r)>Ce^{\frac{\Lambda}{2}\,r}\qquad\text{for any }r>M\,.$

###### Proof.

Let us start with the proof of $(i)$. First of all, we observe that (A3) also
reads
$\lim_{r\rightarrow+\infty}\left[\log(\psi(r))\right]^{\prime}=\Lambda>0$ and
(4.11) follows by writing

$\log(\psi(r))=\int_{R}^{r}\left[\log(\psi(s))\right]^{\prime}\,ds+\log(\psi(R))$

for some $R>0$ and letting $r\to+\infty$.

Let us proceed with the proof of (4.12) when $\Lambda=+\infty$. By (A3) it
follows that there exists $R>0$ such that $\psi^{\prime}(r)>M\psi(r)$ for any
$r>R$, for some positive constant $M>0$. This implies

$\displaystyle\int_{R}^{+\infty}\frac{1}{(\psi(r))^{N-1-\delta}}\,{\rm
d}r=\int_{\psi(R)}^{+\infty}\frac{1}{s^{N-1-\delta}\,\psi^{\prime}(\psi^{-1}(s))}\,{\rm
d}s$
$\displaystyle\qquad<\frac{1}{M}\int_{\psi(R)}^{+\infty}\frac{1}{s^{N-1-\delta}\,\psi(\psi^{-1}(s))}\,{\rm
d}s=\frac{1}{M}\int_{\psi(R)}^{+\infty}\frac{1}{s^{N-\delta}}\,{\rm
d}s<+\infty\,.$

This proves (4.12) and completes the proof of $(i)$ when $\Lambda=+\infty$.
The validity of (4.12) when $\Lambda<+\infty$ is an easy consequence of
statement $(ii)$.

Let us proceed with the proof of $(ii)$. By (A3) we have that for any
$\varepsilon>0$ there exists $r_{\varepsilon}>0$ such that

$\Lambda-\varepsilon<[\log(\psi(r))]^{\prime}<\Lambda+\varepsilon\qquad\text{for
any }r>r_{\varepsilon}\,.$

After integration we get

$\psi(r_{\varepsilon})e^{(\Lambda-\varepsilon)(r-r_{\varepsilon})}<\psi(r)<\psi(r_{\varepsilon})e^{(\Lambda+\varepsilon)(r-r_{\varepsilon})}\qquad\text{for
any }r>r_{\varepsilon}\,.$

The proof of (4.13) now follows choosing $\varepsilon=\frac{\Lambda}{2}$ and
$C=\psi(r_{\varepsilon})e^{-\frac{\Lambda}{2}\,r_{\varepsilon}}$ in the left
inequality above. $\Box$

######  Lemma 4.4.

Let $u$ be the unique solution of (2.2) and suppose that $u$ satisfies (4.10).
Then we have

$\displaystyle\lim_{r\rightarrow+\infty}\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}=0\,.$

###### Proof.

If $u^{\prime\prime}$ vanishes infinitely many times at infinity, then by
Lemma 4.2 we are done. If this does not occur, using again Lemma 4.2 and,
recalling Proposition 2.1 and assumptions (A1)-(A2), we infer that the
following limit exists

$\lim_{r\rightarrow+\infty}\frac{u^{\prime}(r)\psi^{\prime}(r)}{u^{\prime\prime}(r)\psi(r)}=:L\in[-\infty,+\infty]\,.$

The proof of the lemma now follows if we prove that $|L|=+\infty$.

We divide the remaining part of the proof into two steps.

Step 1. We show that $L\neq 0$. By contradiction, assume $L=0$. Notice that by
Lemma 4.1, (4.10), de l’Hôpital’s rule and (A3)

$\displaystyle\lim_{r\rightarrow+\infty}\frac{e^{u(r)}}{u^{\prime}(r)}=\lim_{r\rightarrow+\infty}\frac{u^{\prime}(r)e^{u(r)}}{u^{\prime\prime}(r)}=\lim_{r\rightarrow+\infty}\frac{u^{\prime}(r)\psi^{\prime}(r)}{u^{\prime\prime}(r)\psi(r)}\,e^{u(r)}\,\frac{\psi(r)}{\psi^{\prime}(r)}=0$

by which, from (A3), we readily get

(4.14)
$\displaystyle\lim_{r\rightarrow+\infty}\frac{e^{u(r)}\psi(r)}{u^{\prime}(r)\psi^{\prime}(r)}=0\,.$

Recalling (4.7), by (4.14) we deduce that

$\displaystyle\lim_{r\rightarrow+\infty}\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}=-(N-1).$

This yields
$\lim_{r\rightarrow+\infty}\frac{u^{\prime}(r)\psi^{\prime}(r)}{u^{\prime\prime}(r)\psi(r)}=-\frac{1}{N-1}$,
a contradiction.

This completes the proof of Step 1.

Step 2. We now prove that $L$ cannot be finite. Thanks to Step 1, we may
assume by contradiction that $L\in\mathbb{R}\setminus\\{0\\}$. Arguing as in
Step 1 we get that (4.14) also holds in this case and, in turn, we obtain that
$L=-\frac{1}{N-1}$. Therefore, for any $\varepsilon>0$ there exists
$r_{\varepsilon}>$ such that for any $r>r_{\varepsilon}$

$-(N-1)-\varepsilon\leq\frac{u^{\prime\prime}(r)\psi(r)}{u^{\prime}(r)\psi^{\prime}(r)}\leq-(N-1)+\varepsilon\,,$

whence

$[-(N-1)-\varepsilon]\,[\log(\psi(r))]^{\prime}\leq[\log(-u^{\prime}(r))]^{\prime}\leq[-(N-1)+\varepsilon]\,[\log(\psi(r))]^{\prime}\,.$

Integrating from $r_{\varepsilon}$ to $r$, with $r>r_{\varepsilon}$, we deduce
that

$\log\bigg{(}\frac{-u^{\prime}(r)}{-u^{\prime}(r_{\varepsilon})}\bigg{)}\leq[-(N-1)+\varepsilon]\log\bigg{(}\frac{\psi(r)}{\psi(r_{\varepsilon})}\bigg{)}$

and, in turn, that

$-u^{\prime}(r)\leq A_{\varepsilon}(\psi(r))^{-(N-1)+\varepsilon}\,,$

where
$A_{\varepsilon}=\frac{-u^{\prime}(r_{\varepsilon})}{(\psi(r_{\varepsilon}))^{-(N-1)+\varepsilon}}$
is a positive constant. By a further integration, for any $r>r_{\varepsilon}$,
we obtain

(4.15) $\displaystyle-u(r)+u(r_{\varepsilon})\leq
A_{\varepsilon}\int_{r_{\varepsilon}}^{r}(\psi(s))^{-(N-1)+\varepsilon}\,{\rm
d}s\,.$

By (4.12) we deduce that in both cases $\Lambda=+\infty$ and
$\Lambda\in(0,+\infty)$, the function

$(\psi(s))^{-(N-1)+\varepsilon}\in L^{1}(r_{\varepsilon},+\infty)$

provided that $\varepsilon<N-1$. This means that the right-hand side of (4.15)
admits a finite limit as $r\to+\infty$ and this is absurd since the left-hand
side of (4.15) blows up to $+\infty$ in view of (4.10). $\Box$

Finally, we determine the exact asymptotic behaviour at infinity for solutions
of (2.2) satisfying (4.10).

######  Lemma 4.5.

Let $u$ be the unique solution of (2.2) and suppose that $u$ satisfies (4.10).
Then

(4.16)
$\displaystyle\lim_{r\rightarrow+\infty}\frac{e^{-u(r)}}{\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}{\rm
d}s}=\frac{1}{N-1}.$

Consequently, the following decay estimate holds

(4.17)
$\displaystyle\lim_{r\rightarrow+\infty}\frac{u(r)}{\log\big{(}\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}{\rm
d}s\big{)}}=-1.$

In particular if $\Lambda\in(0,+\infty)$ then

(4.18) $\lim_{r\rightarrow+\infty}\frac{u(r)}{\log r}=-1$

###### Proof.

By (2.2), Lemma 4.2 and Lemma 4.4, we get

$\displaystyle\lim_{r\rightarrow+\infty}-\frac{e^{u(r)}\psi(r)}{u^{\prime}(r)\psi^{\prime}(r)}=N-1\,.$

Therefore, for any $\varepsilon>0$ there exists $r_{\varepsilon}>0$ such that

$\bigg{(}\frac{1}{N-1}-\varepsilon\bigg{)}\frac{\psi(r)}{\psi^{\prime}(r)}<\big{(}e^{-u(r)}\big{)}^{\prime}<\bigg{(}\frac{1}{N-1}+\varepsilon\bigg{)}\frac{\psi(r)}{\psi^{\prime}(r)}\,.$

Integrating from $r_{\varepsilon}$ to $r$, for any $r>r_{\varepsilon}$ we get

(4.19) $\displaystyle
e^{-u(r_{\varepsilon})}+\bigg{(}\frac{1}{N-1}-\varepsilon\bigg{)}\int_{r_{\varepsilon}}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s<e^{-u(r)}<e^{-u(r_{\varepsilon})}+\bigg{(}\frac{1}{N-1}+\varepsilon\bigg{)}\int_{r_{\varepsilon}}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s.$

Now, if $\frac{\psi}{\psi^{\prime}}$ is integrable in a neighbourhood of
infinity, we reach a contradiction with (4.10), therefore
$\frac{\psi}{\psi^{\prime}}$ has to be not integrable in a neighbourhood of
infinity and we obtain

$\displaystyle\lim_{r\rightarrow+\infty}\frac{e^{-u(r)}}{\int_{r_{\varepsilon}}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s}=\frac{1}{N-1}.$

Then, (4.16) readily follows from the above limit by recalling (A1). The limit
(4.17) follows from (4.19) with a similar argument.

It remains to prove (4.18) when $\Lambda\in(0,+\infty)$. We proceed by
considering the limit:

$\displaystyle\lim_{r\to+\infty}\frac{\log\left(\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s\right)}{\log
r}=\lim_{r\to+\infty}\frac{\psi(r)}{\psi^{\prime}(r)}\frac{r}{\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s}=\frac{1}{\Lambda}\,\lim_{r\to+\infty}\frac{r}{\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s}$
$\displaystyle\qquad=\frac{1}{\Lambda}\,\lim_{r\to+\infty}\frac{\psi^{\prime}(r)}{\psi(r)}=\frac{1}{\Lambda}\cdot\Lambda=1\,,$

where we used twice de l’Hôpital rule.

This proves that $\log\left(\int_{0}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s\right)\sim\log r$ as $r\to+\infty$ and the proof of (4.18) follows from
(4.17).

$\Box$

At last, we provide a sufficient condition for (4.10) in terms of
integrability properties of the ratio $\psi/\psi^{\prime}$.

######  Lemma 4.6.

Let $u$ be the unique solution of (2.2). Then the two alternatives hold:

  * $(i)$

if $\frac{\psi}{\psi^{\prime}}\in L^{1}(0,\infty)$ then

$\lim_{r\to+\infty}u(r)\in(-\infty,\alpha)\,;$

  * $(ii)$

if $\frac{\psi}{\psi^{\prime}}\not\in L^{1}(0,\infty)$ then

$\lim_{r\to+\infty}u(r)=-\infty\,.$

###### Proof.

The existence of the limit of $u$ as $r\to+\infty$ is known from Proposition
2.1 as well as the fact that this limit is less than $\alpha$. It remains to
prove that the limit is finite in case $(i)$ and $-\infty$ in case $(ii)$.

Let us start with the proof of $(i)$. Suppose by contradiction that the limit
is $-\infty$ so that (4.10) holds true. Then we can apply Lemma 4.4 and
proceed as in the proof of Lemma 4.5 to obtain (4.19). The integrability of
$\psi/\psi^{\prime}$ shows that $u$ remains bounded as $r\to+\infty$, a
contradiction.

Let us proceed with the proof of $(ii)$. Set $l_{1}:=\lim_{r\to+\infty}u(r)$.
Suppose by contradiction that $l_{1}$ is finite. We claim that

(4.20)
$\lim_{r\to+\infty}\frac{\psi(r)}{u^{\prime}(r)\psi^{\prime}(r)}=-\infty\,.$

From the proof of Lemma 4.2 we know that the function
$\frac{e^{u(r)}}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}$ admits a
limit as $r\to+\infty$, hence, since $e^{-u(r)}\to e^{-l_{1}}\in(0,+\infty)$
as $r\to+\infty$, the limit in (4.20) exists and it belongs to $[-\infty,0]$,
thanks to (A1), (A2) and Proposition 2.1.

Let us denote by $l_{2}\leq 0$ the limit in (4.20) and suppose by
contradiction that it is finite.

Then for any $\varepsilon>0$ there exists $r_{\varepsilon}>0$ such that

$l_{2}-\varepsilon<\frac{\psi(r)}{u^{\prime}(r)\psi^{\prime}(r)}<l_{2}+\varepsilon\qquad\text{for
any }r>r_{\varepsilon}$

and from the left inequality, it follows that

$0<\frac{\psi(r)}{\psi^{\prime}(r)}<(l_{2}-\varepsilon)u^{\prime}(r)\qquad\text{for
any }r>r_{\varepsilon}\,.$

Integrating the inequality above we obtain

$0<\int_{r_{\varepsilon}}^{r}\frac{\psi(s)}{\psi^{\prime}(s)}\,{\rm
d}s<(l_{2}-\varepsilon)[u(r)-u(r_{\varepsilon})]\qquad\text{for any
}r>r_{\varepsilon}\,.$

Passing to the limit as $r\to+\infty$ and recalling that $l_{1}$ is finite we
infer that $\psi/\psi^{\prime}$ is integrable at infinity, in contradiction
with the assumption in $(ii)$. This completes the proof of (4.20).

Combining (4.20) and (4.7) with the fact that $l_{1}$ is finite we deduce that

$\displaystyle\lim_{r\to+\infty}\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}=-(N-1)-\lim_{r\to+\infty}\frac{e^{u(r)}}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}=+\infty\,.$

This implies that for any $M>0$ there exists $r_{M}>0$ such that

$\frac{u^{\prime\prime}(r)}{u^{\prime}(r)}\frac{\psi(r)}{\psi^{\prime}(r)}>M\qquad\text{for
any }r>r_{M}\,.$

Multiplying both sides of the above inequality by $\psi^{\prime}(r)/\psi(r)$
and integrating we obtain

$\log\left(\frac{|u^{\prime}(r)|}{|u^{\prime}(r_{M})|}\right)>M\log\left(\frac{\psi(r)}{\psi(r_{M})}\right)\qquad\text{for
any }r>r_{M}$

from which it follows that

$|u^{\prime}(r)|>\frac{|u^{\prime}(r_{M})|}{(\psi(r_{M}))^{M}}\,(\psi(r))^{M}\qquad\text{for
any }r>r_{M}\,.$

Passing to the limit as $r\to+\infty$ and recalling (4.11), we conclude that
$|u^{\prime}(r)|\to+\infty$ as $r\to+\infty$ in contradiction with Lemma 4.1.
This completes the proof of $(ii)$. $\Box$

End of the proof of Theorem 2.2. The proof of $(i)$ in the case
$\frac{\psi}{\psi^{\prime}}\in L^{1}(0,\infty)$ is an immediate consequence of
Lemma 4.6 -$(i)$.

Let us proceed with the proof of $(ii)$. First of all by Lemma 4.6-$(ii)$ we
have that $u$ diverges to $-\infty$ as $r\to+\infty$. In this way (4.10) is
satisfied, hence Lemma 4.5 completes the proof of Theorem 2.2. $\Box$

##  5\. Preliminary results about stability and intersection properties

We start by stating an equivalent characterization of stability in the case of
radial solutions of (1.2). Since the proof can be obtained by following the
proof of [4, Lemma 5.1] with obvious changes we omit it.

######  Lemma 5.1.

Let $\psi$ satisfy (A1)-(A3) and let $u$ be a radial solution of (1.2). Then
$u$ is stable if and only if

(5.1)
$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r-\int_{0}^{+\infty}e^{u(r)}(\chi(r))^{2}\,(\psi(r))^{N-1}\>{\rm d}r\geq 0,$

for every radial function $\chi\in C_{c}^{\infty}(M)$.

We now give the statement of a series of lemmas that will be exploited in the
proofs of the main results about stability and intersection properties.

In the sequel, we denote by $u_{\alpha}$ the unique solution of (2.2) with
$\alpha=u_{\alpha}(0)$ and we consider the set

(5.2) $\mathcal{S}:=\\{\alpha\in\mathbb{R}:u_{\alpha}\text{ is stable}\\}\,.$

Before proceeding, we recall the variational characterization of the bottom of
the $L^{2}$ spectrum of $-\Delta_{g}$ in $M$:

(5.3) $\lambda_{1}(M):=\inf_{\varphi\in
C^{\infty}_{c}(M)\setminus\\{0\\}}\frac{\int_{M}|\nabla_{g}\varphi|_{g}^{2}\,dV_{g}}{\int_{M}\varphi^{2}\,dV_{g}}\,.$

It is well known that under assumptions (A1)-(A3) we have that
$\lambda_{1}(M)>0$, see e.g., [4, Lemma 4.1]. The bottom of the spectrum is
involved in the stability of solutions $u_{\alpha}$ of (2.2) for sufficiently
small values of $\alpha$. Indeed, we prove:

######  Lemma 5.2.

Let $\psi$ satisfy (A1)-(A3) and let $u_{\alpha}$ be a solution of (2.2) with
$\alpha\leq\log(\lambda_{1}(M))$. Then $u_{\alpha}$ is stable and in
particular, the set $\mathcal{S}$ is not empty.

###### Proof.

Using (5.3) and Proposition 2.1 we infer

$\displaystyle\int_{M}|\nabla_{g}v|_{g}^{2}\,{\rm
d}V_{g}\geq\frac{\lambda_{1}(M)}{e^{\alpha}}\int_{M}e^{\alpha}v^{2}\,{\rm
d}V_{g}\geq\frac{\lambda_{1}(M)}{e^{\alpha}}\int_{M}e^{u_{\alpha}(r)}v^{2}\,{\rm
d}V_{g}\qquad\text{for any }v\in C^{\infty}_{c}(M)$

which gives the stability of $u_{\alpha}$ if $\alpha\leq\log(\lambda_{1}(M))$.
$\Box$

The next step is to prove that the set $\mathcal{S}$ is an interval. First, we
state three preliminary lemmas; the fact that $\mathcal{S}$ is an interval
will be proved in the fourth one (Lemma 5.6 below).

######  Lemma 5.3.

Let $\psi$ satisfy (A1)-(A3) and let $a,\,b,\,R\in\mathbb{R}$ be such that
$b>a$. If $u_{\alpha}(r)$ is the solution of (2.2) with $\alpha\in[a,b]$, then
for any $r\in[0,R]$, the map $\alpha\mapsto u(\alpha,r):=u_{\alpha}(r)$ is
differentiable in $[a,b]$ and for any $\alpha_{0}\in[a,b]$

(5.4)
$\displaystyle\lim_{\alpha\rightarrow\alpha_{0}}\,\sup_{r\in[0,R]}\bigg{|}\frac{\partial
u}{\partial\alpha}(\alpha,r)-\frac{\partial
u}{\partial\alpha}(\alpha_{0},r)\bigg{|}=0.$

Moreover for any $\alpha\in[a,b]$ and $r\in[0,R]$, the function
$v_{\alpha}(r):=\frac{\partial u}{\partial\alpha}(\alpha,r)$ is the solution
of the following problem

(5.5)
$\begin{dcases}-v^{\prime\prime}(r)-(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}v^{\prime}(r)=e^{u_{\alpha}(r)}v(r)&\\\
v(0)=1&\\\ v^{\prime}(0)=0.&\\\ \end{dcases}$

###### Proof.

The proof can be obtained by proceeding along the lines of [4, Lemma 5.6] with
suitable changes concerning essentially the fact that the power nonlinearity
is replaced here by the exponential nonlinearity. For the sake of
completeness, we recall the main steps here below.

For any $r\in[0,R]$ and $\alpha\in[a,b]$, let us define

$\displaystyle
w(r)=\frac{u_{\alpha}(r)-u_{\alpha_{0}}(r)}{\alpha-\alpha_{0}}-v_{\alpha_{0}}(r)\text{
and }z(r)=w^{\prime}(r)$

where $v_{\alpha_{0}}$ is the solution of problem (5.5) with
$\alpha=\alpha_{0}$.

Then

(5.6) $\displaystyle
z^{\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}z(r)=-\bigg{(}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-e^{u_{\alpha_{0}}(r)}v_{\alpha_{0}}(r)\bigg{)}\,.$

For any $\delta>0$, $\alpha\in(\alpha_{0}-\delta,\alpha_{0}+\delta)\cap[a,b]$
and $r\in[0,R]$, we have

$\displaystyle\bigg{|}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-e^{u_{\alpha_{0}}(r)}v_{\alpha_{0}}(r)\bigg{|}$
$\displaystyle\leq\bigg{|}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-\frac{u_{\alpha}(r)-u_{\alpha_{0}}(r)}{\alpha-\alpha_{0}}\,e^{u_{\alpha_{0}}(r)}+e^{u_{\alpha_{0}}(r)}w(r)\bigg{|}$
$\displaystyle\leq\frac{|u_{\alpha}(r)-u_{\alpha_{0}}(r)|}{|\alpha-\alpha_{0}|}|e^{\xi(r)}-e^{u_{\alpha_{0}}(r)}|+e^{u_{\alpha_{0}}(r)}|w(r)|$

where, by Lagrange Theorem,
$\min\\{u_{\alpha}(r),u_{\alpha_{0}}(r)\\}<\xi(r)<\max\\{u_{\alpha}(r),u_{\alpha_{0}}(r)\\}$
any $r\in[0,R]$. Recalling that for any $\alpha$ the functions $u_{\alpha}$
are decreasing and using again Lagrange Theorem, we obtain

$\displaystyle\bigg{|}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-e^{u_{\alpha_{0}}(r)}v_{\alpha_{0}}(r)\bigg{|}\leq
e^{\alpha_{0}+\delta}\,\frac{|u_{\alpha}(r)-u_{\alpha_{0}}(r)|^{2}}{|\alpha-\alpha_{0}|}+e^{\alpha_{0}}|w(r)|$
$\displaystyle\qquad\leq|w(r)|\left\\{e^{\alpha_{0}+\delta}\,|u_{\alpha}(r)-u_{\alpha_{0}}(r)|+e^{\alpha_{0}}\right\\}+e^{\alpha_{0}+\delta}\,|v_{\alpha_{0}}(r)|\,|u_{\alpha}(r)-u_{\alpha_{0}}(r)|.$

Now, by continuous dependence, for any $\varepsilon>0$ we may choose $\delta$
small enough in such a way that
$\sup_{r\in[0,R]}|u_{\alpha}(r)-u_{\alpha_{0}}(r)|<\varepsilon$ and we obtain
for $r\in[0,R]$,

$\displaystyle\bigg{|}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-e^{u_{\alpha_{0}}(r)}v_{\alpha_{0}}(r)\bigg{|}\leq(1+\varepsilon)e^{\alpha_{0}+\delta}|w(r)|+C\varepsilon$

where we put $C=e^{\alpha_{0}+\delta}\sup_{r\in[0,R]}|v_{\alpha_{0}}(r)|$.
Furthermore, observing that $w(0)=0$, we infer

(5.7)
$\displaystyle\bigg{|}\frac{e^{u_{\alpha}(r)}-e^{u_{\alpha_{0}}(r)}}{\alpha-\alpha_{0}}-e^{u_{\alpha_{0}}(r)}v_{\alpha_{0}}(r)\bigg{|}\leq(1+\varepsilon)e^{\alpha_{0}+\delta}\int_{0}^{r}|z(s)|{\rm
d}s+C\varepsilon.$

Combining (5.6) and (5.7) and observing that $z(0)=0$, we obtain

$\displaystyle|z(r)|\leq
K(1+\varepsilon)e^{\alpha_{0}+\delta}\int_{0}^{r}|z(s)|{\rm
d}s+KC\varepsilon,$

for any $r\in[0,R]$ and
$\alpha\in(\alpha_{0}-\delta,\alpha_{0}+\delta)\cap[a,b]$, where we put
$K:=\sup_{r\in[0,R]}\frac{\int_{0}^{r}(\psi(s))^{N-1}{\rm
d}s}{(\psi(r))^{N-1}}$.

Finally, exploiting standard Gronwall-type estimates we obtain

$\displaystyle\lim_{\alpha\rightarrow\alpha_{0}}\sup_{r\in[0,R]}|z(r)|=0\implies\lim_{\alpha\rightarrow\alpha_{0}}\sup_{r\in[0,R]}|w(r)|=0.$

This proves the differentiability with respect to $\alpha$ of the map
$\alpha\mapsto u(\alpha,r)$ and shows that the derivative with respect to
$\alpha$ is a solution of (5.5). The proof of (5.4) is a consequence of a
standard continuous dependence result for the Cauchy problem (5.5). $\Box$

######  Lemma 5.4.

Let $\psi$ satisfy (A1)-(A3) and let
$\alpha_{1}>\alpha_{2}\geq\alpha_{3}>\alpha_{4}$. Then the first intersection
between $u_{\alpha_{1}}$ and $u_{\alpha_{2}}$ cannot take place after the
first intersection between $u_{\alpha_{3}}$ and $u_{\alpha_{4}}$.

###### Proof.

We follow closely the proof of [5, Lemma 7.3]. We divide the proof into two
steps.

Step 1. We first prove the lemma when only three functions
$u_{\alpha_{1}},u_{\alpha_{2}},u_{\alpha_{3}}$ with
$\alpha_{1}>\alpha_{2}>\alpha_{3}$, are involved. In other words, we prove
that the first intersection between $u_{\alpha_{1}}$ and $u_{\alpha_{2}}$
cannot take place after the first intersection between $u_{\alpha_{2}}$ and
$u_{\alpha_{3}}$.

Let $w_{1}=u_{\alpha_{1}}-u_{\alpha_{2}}$ and
$w_{2}=u_{\alpha_{2}}-u_{\alpha_{3}}$. Then $w_{i}(0)>0$ and
$w_{i}^{\prime}(0)=0$ for $i=1,2$. Let $r_{1}>0$ be such that $w_{i}$ has no
zero in $[0,r_{1}]$. Then for $i=1,2$, the functions $w_{i}$ satisfy

$\displaystyle
w_{i}^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,w_{i}^{\prime}(r)+b_{i}(r)w_{i}(r)=0,$

where the functions $b_{i}$ are positive in $[0,r_{1}]$ and they satisfy

$\displaystyle
u_{\alpha_{2}}(r)<\log(b_{1}(r))<u_{\alpha_{1}}(r)\quad\text{and}\quad
u_{\alpha_{3}}(r)<\log(b_{2}(r))<u_{\alpha_{2}}(r)\qquad\text{for any
}r\in[0,r_{1}]\,,$

thanks to Lagrange Theorem. In particular this gives $b_{1}(r)>b_{2}(r)$ in
$[0,r_{1}]$. Putting $z=w_{1}/w_{2}$ we have that

$\displaystyle
w_{2}(r)z^{\prime\prime}(r)+\left[2w_{2}^{\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,w_{2}(r)\right]z^{\prime}(r)=-z(r)(b_{1}(r)-b_{2}(r))w_{2}(r)<0$

for any $r\in[0,r_{1}]$ and moreover $z(0)>0$ and $z^{\prime}(0)=0$. If we set
$a(r):=2\frac{w_{2}^{\prime}(r)}{w_{2}(r)}+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}$,
the above inequality can be written as
$z^{\prime\prime}(r)+a(r)z^{\prime}(r)<0$ in $[0,r_{1}]$, being $w_{2}>0$ in
$[0,r_{1}]$. Then, for $\varepsilon>0$, multiplying both sides by
$e^{\int_{\varepsilon}^{r}a(t)\,dt}$ and integrating in $[\varepsilon,r]$, we
get

$z^{\prime}(r)\leq
z^{\prime}(\varepsilon)\,e^{-\int_{\varepsilon}^{r}a(t)\,dt}\quad\text{for all
}r\in(\varepsilon,r_{1}]\,.$

By (A1) and the fact that $w_{2}(0)>0$ and $w_{2}^{\prime}(0)=0$ we deduce
that $a(r)\sim\frac{N-1}{r}$ as $r\rightarrow 0^{+}$. Then, letting
$\varepsilon\rightarrow 0^{+}$ in the above inequality and recalling that
$z^{\prime}(\varepsilon)\to z^{\prime}(0)=0$, we conclude that $z^{\prime}\leq
0$ in $(0,r_{1}]$ so that $z$ is non increasing in the same interval and in
particular $z(r)\leq z(0)$ for any $r\in(0,r_{1}]$. This completes the proof
of Step 1. Indeed, if $\zeta_{i}$ is the first zero of $w_{i}$, $i=1,2$ and if
we assume by contradiction that $\zeta_{1}>\zeta_{2}$ then we may apply the
previous estimate for any $0<r_{1}<\zeta_{2}$ and obtain

$w_{1}(r)\leq z(0)w_{2}(r)\qquad\text{for any }r\in(0,\zeta_{2})\,.$

Then, letting $r\to\zeta_{2}^{-}$ we conclude that $w_{1}(\zeta_{2})\leq 0$ in
contradiction with $\zeta_{1}>\zeta_{2}$.

Step 2. We complete here the proof of the lemma. We denote by $\zeta_{ij}$ the
first intersection between $u_{\alpha_{i}}$ and $u_{\alpha_{j}}$ with $i\leq
j$. We have to prove that $\zeta_{12}\leq\zeta_{34}$. We apply Step 1 twice,
first to prove that $\zeta_{12}\leq\zeta_{23}$ and then to prove that
$\zeta_{23}\leq\zeta_{34}$. The combination of the two inequalities readily
completes the proof of Step 2. $\Box$

######  Lemma 5.5.

[4, Lemma 5.8] Let $\psi$ satisfy (A1)-(A3). Then $\lim_{r\rightarrow
0^{+}}\,\lambda_{1}(B_{r})=+\infty$.

By combining the above lemmas one gets:

######  Lemma 5.6.

Let $\psi$ satisfy (A1)-(A3) and assume $\alpha>\beta$. If $u_{\beta}$ is
unstable then $u_{\alpha}$ is also the unstable solution. In particular, the
set $\mathcal{S}$ defined in (5.2) is an interval.

###### Proof.

Suppose that $u_{\alpha}$ and $u_{\beta}$ have no intersections. Then
$u_{\alpha}(r)>u_{\beta}(r)$ for any $r\geq 0$ and
$e^{u_{\alpha}(r)}>e^{u_{\beta}(r)}$ for any $r\geq 0$. Suppose by
contradiction that $u_{\alpha}$ is stable, then (5.1) implies

$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r\geq\int_{0}^{+\infty}e^{u_{\alpha}(r)}(\chi(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r$
$\displaystyle\geq\int_{0}^{+\infty}e^{u_{\beta}(r)}(\chi(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r,$

for every radial function $\chi\in C_{c}^{\infty}(M)$. This implies the
stability of $u_{\beta}$, a contradiction.

We may suppose now that $u_{\alpha}$ and $u_{\beta}$ have at least one
intersection. Exploiting Lemma 5.3, Lemma 5.4 and Lemma 5.5, we may follow the
part of the proof of [4, Lemma 5.9] in which $u_{\alpha}$ and $u_{\beta}$ have
at least one intersection, and prove the instability of $u_{\alpha}$. $\Box$

###  5.1. Low dimensions

In the next lemma we prove that if $2\leq N\leq 9$, the set $\mathcal{S}$ is
bounded above. To this aim we first set

(5.8) $\displaystyle\eta:=\sup\mathcal{S}\,.$

######  Lemma 5.7.

Let $\psi$ satisfy (A1)-(A3) and let $2\leq N\leq 9$. Then there exists
$\alpha_{0}>0$ such that for any $\alpha>\alpha_{0}$, the solution
$u_{\alpha}$ of (2.2) is unstable. In particular, the number $\eta$ defined in
(5.8) is finite.

###### Proof.

We adapt to our framework the blow-up arguments of [4, Lemmas 4.9 and 5.5] and
[5, Lemma 7.1]. Since we know from Lemma 5.6 that the set $\mathcal{S}$ is an
interval, we proceed by contradiction assuming that $u_{\alpha}$ is stable for
any $\alpha\in\mathbb{R}$.

Let us define $u_{\lambda}$ as the solution of (2.2) with initial condition
$\alpha=\log(\frac{e}{\lambda^{2}})$ and define

$\displaystyle v_{\lambda}(s)=u_{\lambda}(\lambda s)+2\log(\lambda).$

Now, one can check that $v_{\lambda}(0)=1$ and it satisfies

(5.9) $\displaystyle
v_{\lambda}^{\prime\prime}(s)+\frac{N-1}{s}\frac{\psi^{\prime}(\lambda
s)}{\psi(\lambda s)}\,\lambda
s\,v^{\prime}_{\lambda}(s)+e^{v_{\lambda}(s)}=0.$

Furthermore, using the assumptions on $\psi$, we can find that for any fixed
$S>0$,

(5.10) $\displaystyle\frac{\psi^{\prime}(\lambda s)}{\psi(\lambda s)}\,\lambda
s\to 1\qquad\text{as $\lambda\to 0^{+}$, uniformly in $(0,S]$. }$

If we define
$F_{\lambda}(r)=\frac{1}{2}(u_{\lambda}^{\prime}(r))^{2}+e^{u_{\lambda}(r)}$
then we know from the proof of Proposition 2.1 that $F_{\lambda}$ is
decreasing and hence

$\displaystyle|u_{\lambda}^{\prime}(r)|^{2}=2\Big{[}F_{\lambda}(r)-e^{u_{\lambda}(r)}\Big{]}\leq
2\Big{[}e^{u_{\lambda}(0)}-e^{u_{\lambda}(r)}\Big{]}\leq
2e^{u_{\lambda}(0)}\Big{[}u_{\lambda}(0)-u_{\lambda}(r)\Big{]}$

from which we obtain

$\displaystyle|u_{\lambda}^{\prime}(r)|\leq\frac{\sqrt{2e}}{\lambda}\,\left(\int_{0}^{r}|u_{\lambda}^{\prime}(t)|\,{\rm
d}t\right)^{1/2}\qquad\text{for any }r>0$

where we recall that $e^{u_{\lambda}(0)}=\frac{e}{\lambda^{2}}$.

By Gronwall-type estimates we obtain

$|u_{\lambda}^{\prime}(r)|\leq\frac{e}{\lambda^{2}}\,r\qquad\text{for any
}r>0$

and from the definition of $v_{\lambda}$ we finally obtain

(5.11) $|v_{\lambda}^{\prime}(s)|\leq es\qquad\text{for any }s>0\,.$

Combining (5.9), (5.10), (5.11) we deduce that for any fixed $S>0$ there
exists $\bar{\lambda}(S)>0$ such that

$|v_{\lambda}^{\prime\prime}(s)|\leq
2(N-1)e+e^{v_{\lambda}(0)}=(2N-1)e\quad\text{for any $s\in[0,S]$ and
$0<\lambda<\bar{\lambda}(S)$}\,.$

Hence, using Ascoli-Arzelá Theorem on $[0,S]$, we deduce that there exists
$\bar{v}\in C^{1}([0,S])$ such that $v_{\lambda}\rightarrow\bar{v}$ in
$C^{1}([0,S])$ as $\lambda\rightarrow 0^{+}$ and $\bar{v}$ satisfies

$\displaystyle\bar{v}^{\prime\prime}(s)+\frac{N-1}{s}\,\bar{v}^{\prime}(s)+e^{\bar{v}(s)}=0\text{
and }\bar{v}(0)=1.$

Since $u_{\lambda}$ is a stable radial solution, using (5.1) we have that

$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r-\int_{0}^{+\infty}e^{u_{\lambda}(r)}(\chi(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r\geq 0\,,$

for every radial function $\chi\in C_{c}^{\infty}(M)$. In terms of
$v_{\lambda}$ the above inequality reads

$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}(\psi(r))^{N-1}\>{\rm
d}r-\frac{1}{\lambda^{2}}\int_{0}^{+\infty}e^{v_{\lambda}(\frac{r}{\lambda})}(\chi(r))^{2}\,(\psi(r))^{N-1}\>{\rm
d}r\geq 0\,,$

for every radial function $\chi\in C_{c}^{\infty}(M)$. Now, choosing
$\eta_{\lambda}(r):=\eta(\frac{r}{\lambda})$ as test function for some radial
function $\eta\in C_{c}^{\infty}(M)$ and then using the change of variable
$s=\frac{r}{\lambda}$ we infer

$\displaystyle\int_{0}^{+\infty}(\eta^{\prime}(s))^{2}(\psi(s\lambda))^{N-1}\>{\rm
d}s-\int_{0}^{+\infty}e^{v_{\lambda}(s)}(\eta(s))^{2}\,(\psi(s\lambda))^{N-1}\>{\rm
d}s\geq 0,$

for every radial function $\eta\in C_{c}^{\infty}(M)$. Now fix $S>0$ and then
choose $\eta$ in such a way that supp$(\eta)\subset B_{S}$ where $B_{S}$
denotes the geodesic ball of radius $S$ centered at the pole $o$. By Lagrange
Theorem for any $s\in[0,S],$ there exist $0<\zeta<s\lambda$ and
$0<|t|<\frac{|\psi^{\prime\prime}(\zeta)|}{2}(s\lambda)$ such that for
$\lambda\rightarrow 0^{+}$

$\displaystyle(\psi(s\lambda))^{N-1}=(s\lambda)^{N-1}+g(\zeta,t)(s\lambda)^{N},$

where $g(\zeta,t)=(N-1)(1+t)^{N-2}\,\frac{\psi^{\prime\prime}(\zeta)}{2}$.
This gives, after cancelling $\lambda^{N-1}$, that

$\displaystyle\int_{0}^{+\infty}(\eta^{\prime}(s))^{2}s^{N-1}\>{\rm
d}s+\int_{0}^{+\infty}(\eta^{\prime}(s))^{2}\,s^{N}g(\zeta,t)\lambda\>{\rm
d}s$
$\displaystyle-\int_{0}^{+\infty}e^{v_{\lambda}(s)}(\eta(s))^{2}s^{N-1}\>{\rm
d}s-\int_{0}^{+\infty}e^{v_{\lambda}(s)}(\eta(s))^{2}\,s^{N}g(\zeta,t)\lambda\>{\rm
d}s\geq 0.$

Therefore taking the limit as $\lambda\rightarrow 0^{+}$, we finally obtain

$\displaystyle\int_{0}^{+\infty}(\eta^{\prime}(s))^{2}s^{N-1}\>{\rm
d}s-\int_{0}^{+\infty}e^{\bar{v}(s)}(\eta(s))^{2}s^{N-1}\>{\rm d}s\geq 0,$

for every radial function $\eta\in C_{c}^{\infty}(M)$. Therefore $\bar{v}$ is
a stable solution of the equation $-\Delta u=e^{u}$ on $\mathbb{R}^{N}$ for
$2\leq N\leq 9$. This contradicts the results of [15] and concludes the proof.
$\Box$

We complete the stability picture for $2\leq N\leq 9$ by showing that
$\mathcal{S}$ is a closed interval.

######  Lemma 5.8.

Let $\psi$ satisfy (A1)-(A3) and let $2\leq N\leq 9$, then
$\mathcal{S}=(-\infty,\eta]$.

###### Proof.

Suppose $\eta\notin\mathcal{S}$ and so $u_{\eta}$ is unstable solution. Then
for each $n\in\mathbb{N}$ there exist $\eta_{n}\in\mathcal{S}$ such that
$\eta_{n}\rightarrow\eta$ for $n\rightarrow+\infty$. The definition of
$\eta_{n}$ gives that each $u_{\eta_{n}}$ is a stable solution. Hence for each
radial function $\chi\in C_{c}^{\infty}(M)$ there holds

$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}(\psi(r))^{N-1}\>{\rm
d}r-\int_{0}^{+\infty}e^{u_{\eta_{n}}(r)}(\chi(r))^{2}(\psi(r))^{N-1}\>{\rm
d}r\geq 0.$

Now, for any function $\chi\in C_{c}^{\infty}(M)$, we have that supp($\chi$)
is a compact set and continuous dependence on initial data gives
$u_{\eta_{n}}\rightarrow u_{\eta}$ uniformly. This implies

$\displaystyle\int_{0}^{+\infty}(\chi^{\prime}(r))^{2}(\psi(r))^{N-1}\>{\rm
d}r-\int_{0}^{+\infty}e^{u_{\eta}(r)}(\chi(r))^{2}(\psi(r))^{N-1}\>{\rm
d}r\geq 0.$

contradicting the fact that $u_{\eta}$ is an unstable solution. $\Box$

By Lemma 5.2 we know that $\eta\geq\log(\lambda_{1}(M))$; the next two lemmas
allow us to improve these bounds when $\psi/\psi^{\prime}$ is not integrable.

######  Lemma 5.9.

Let $\psi$ satisfy (A1)-(A3). Then for any $\alpha$,

$\displaystyle\Lambda(M,\alpha):=\,\,\inf_{v\in
H^{1}(M)\setminus\\{0\\}}\frac{\int_{M}|\nabla_{g}v|_{g}^{2}\,{\rm
d}V_{g}}{\int_{M}e^{u_{\alpha}}v^{2}\,{\rm d}V_{g}}$

admits a minimizer.

###### Proof.

The proof is a straightforward adaptation of [4, Lemma 5.11] to this setting,
therefore we omit it. $\Box$

######  Lemma 5.10.

Let $\psi$ satisfy (A1)-(A3) and assume that $\frac{\psi}{\psi^{\prime}}\notin
L^{1}(0,+\infty)$. Then $\eta>\log(\lambda_{1}(M))$.

###### Proof.

Set $\bar{\alpha}:=\log(\lambda_{1}(M))$ and let $\Lambda(M,\alpha)$ be as in
the statement of Lemma 5.9, then $\Lambda(M,\bar{\alpha})>1$. Indeed, if $w\in
H^{1}(M)$ is the minimizer of $\Lambda(M,\bar{\alpha})$, then

$\displaystyle\Lambda(M,\bar{\alpha})=\frac{\int_{M}|\nabla_{g}w|_{g}^{2}\,{\rm
d}V_{g}}{\int_{M}e^{u_{\bar{\alpha}}}|w|^{2}\,{\rm
d}V_{g}}\geq\frac{\int_{M}|\nabla_{g}w|_{g}^{2}\,{\rm
d}V_{g}}{e^{\bar{\alpha}}\int_{M}|w|^{2}\,{\rm d}V_{g}}\geq 1.$

Therefore, if $\Lambda(M,\bar{\alpha})=1$, $w$ solves

$\displaystyle-\Delta_{g}w=\lambda_{1}(M)w\,\,\text{ in }M\,\,\text{ and
}-\Delta_{g}w=e^{u_{\bar{\alpha}}}w\,\,\text{ in }M.$

This implies that $u_{\bar{\alpha}}$ is constant, which is a contradiction.
Hence, $\Lambda(M,\bar{\alpha})>1$.

Next we consider a decreasing sequence $\\{\alpha_{k}\\}$ such that
$\alpha_{k}\rightarrow\bar{\alpha}$ as $k\rightarrow+\infty$. We will show
that $\Lambda(M,\alpha_{k})>1$ definitively, hence $\eta>\log(\lambda_{1}(M))$
which is the thesis.

By contradiction, assume that there exists $K>0$ such that
$\Lambda(M,\alpha_{k})\leq 1$ for $k>K$. For any $k$ let $w_{k}$ be a
minimizer of $\Lambda(M,\alpha_{k})$ satisfying
$\int_{M}e^{u_{\alpha_{k}}}w_{k}^{2}\,{\rm d}V_{g}=1$. The sequence
$\\{w_{k}\\}$ is bounded in $H^{1}(M)$ hence, up to a subsequence,
$w_{k}\rightharpoonup w$ in $H^{1}(M)$ as $k\rightarrow+\infty$ and $w_{k}\to
w$ strongly in $L^{2}(B_{R})$ for any $R>0$, where we recall that $B_{R}$
denotes the geodesic ball of radius $R$ centered at the pole $o$.

For any $\alpha\in\mathbb{R}$ consider the Lyapunov functional (3.1):
$F_{\alpha}(r)=\frac{1}{2}(u_{\alpha}^{\prime}(r))^{2}+e^{u_{\alpha}(r)}$ for
$r>0$. From the proof of Theorem 2.2, the stated assumptions imply that
$\lim_{r\rightarrow+\infty}u_{{\alpha}}(r)=-\infty$ and
$\lim_{r\rightarrow+\infty}u^{\prime}_{{\alpha}}(r)=0$. Therefore, for every
$\varepsilon>0$ there exists $R_{\varepsilon}>0$ such that
$F_{\bar{\alpha}}(R_{\varepsilon})<\varepsilon$. Since for any $r>0$,
$u_{\alpha_{k}}(r)\rightarrow u_{\bar{\alpha}}(r)$ and
$u^{\prime}_{\alpha_{k}}(r)\rightarrow u^{\prime}_{\bar{\alpha}}(r)$ as
$k\rightarrow+\infty$, then there exists $\bar{K}=\bar{K}(\varepsilon)>0$ such
that $F_{\alpha_{k}}(R_{\varepsilon})<\varepsilon$ for $k>\bar{K}$. Since the
functions $F_{\alpha_{k}}$ are nonincreasing, it follows that
$F_{\alpha_{k}}(r)<\varepsilon$ and $e^{u_{\alpha_{k}}(r)}<\varepsilon$ for
any $r\geq R_{\varepsilon}$ and $k>\bar{K}$. Therefore,

$\displaystyle\bigg{|}\int_{M}e^{u_{\alpha_{k}}}w_{k}^{2}\,{\rm
d}V_{g}-\int_{M}e^{u_{\bar{\alpha}}}w^{2}\,{\rm d}V_{g}\bigg{|}$
$\displaystyle\leq\bigg{|}\int_{B_{R_{\varepsilon}}}e^{u_{\alpha_{k}}}w_{k}^{2}\,{\rm
d}V_{g}-\int_{B_{R_{\varepsilon}}}e^{u_{\bar{\alpha}}}w_{k}^{2}\,{\rm
d}V_{g}\bigg{|}+\bigg{|}\int_{B_{R_{\varepsilon}}}e^{u_{\bar{\alpha}}}w_{k}^{2}\,{\rm
d}V_{g}-\int_{B_{R_{\varepsilon}}}e^{u_{\bar{\alpha}}}w^{2}\,{\rm
d}V_{g}\bigg{|}$
$\displaystyle+\bigg{|}\int_{M\setminus{B_{R_{\varepsilon}}}}e^{u_{\alpha_{k}}}w_{k}^{2}\,{\rm
d}V_{g}-\int_{M\setminus{B_{R_{\varepsilon}}}}e^{u_{\bar{\alpha}}}w^{2}\,{\rm
d}V_{g}\bigg{|}$
$\displaystyle\leq\sup_{{B_{R_{\varepsilon}}}}|e^{u_{\alpha_{k}}}-e^{u_{\bar{\alpha}}}|\int_{B_{R_{\varepsilon}}}w_{k}^{2}\,{\rm
d}V_{g}+\bigg{|}\int_{B_{R_{\varepsilon}}}e^{u_{\bar{\alpha}}}w_{k}^{2}\,{\rm
d}V_{g}-\int_{B_{R_{\varepsilon}}}e^{u_{\bar{\alpha}}}w^{2}\,{\rm
d}V_{g}\bigg{|}$
$\displaystyle+\varepsilon\int_{M\setminus{B_{R_{\varepsilon}}}}w_{k}^{2}\,{\rm
d}V_{g}+\varepsilon\int_{M\setminus{B_{R_{\varepsilon}}}}w^{2}\,{\rm d}V_{g}$
$\displaystyle\leq\sup_{{B_{R_{\varepsilon}}}}|e^{u_{\alpha_{k}}}-e^{u_{\bar{\alpha}}}|\int_{B_{R_{\varepsilon}}}\\!\\!w_{k}^{2}\,{\rm
d}V_{g}+o(1)+\frac{\varepsilon}{\lambda_{1}(M)}\bigg{(}\int_{M}|\nabla_{g}w_{k}|_{g}^{2}{\rm
d}V_{g}+\liminf_{k\rightarrow+\infty}\int_{M}|\nabla_{g}w_{k}|_{g}^{2}{\rm
d}V_{g}\bigg{)}$ $\displaystyle\leq
C\varepsilon+\frac{\varepsilon}{\lambda_{1}(M)}\bigg{(}\Lambda(M,\alpha_{k})+\liminf_{k\rightarrow+\infty}\Lambda(M,\alpha_{k})\bigg{)}+o(1)\leq
C\varepsilon+\frac{2\varepsilon}{\lambda_{1}(M)}+o(1)\,.$

In the above estimate we used the following facts: $w_{k}\rightarrow w$ in
$L^{2}(B_{R_{\varepsilon}})$ strongly, $e^{u_{\alpha_{k}}}\rightarrow
e^{u_{\bar{\alpha}}}$ uniformly, the lower semicontinuity of the
$H^{1}(M)$-norm and the inequality $\Lambda(M,\alpha_{k})\leq 1$. Letting
$k\to+\infty$, since $\varepsilon$ was chosen arbitrarily, we conclude that

$\displaystyle\lim_{k\rightarrow+\infty}\int_{M}e^{u_{\alpha_{k}}}w_{k}^{2}\,{\rm
d}V_{g}=\int_{M}e^{u_{\bar{\alpha}}}w^{2}{\rm d}V_{g}.$

Using again the lower semicontinuity of the $H^{1}(M)$-norm, we finally have

$\displaystyle
1<\Lambda(M,\bar{\alpha})\leq\frac{\int_{M}|\nabla_{g}w|_{g}^{2}\,{\rm
d}V_{g}}{\int_{M}e^{u_{\bar{\alpha}}}|w|^{2}\,{\rm
d}V_{g}}\leq\liminf_{k\rightarrow+\infty}\frac{\int_{M}|\nabla_{g}w_{k}|_{g}^{2}\,{\rm
d}V_{g}}{\int_{M}e^{u_{\alpha_{k}}}|w_{k}|^{2}\,{\rm
d}V_{g}}=\liminf_{k\to+\infty}\Lambda(M,\alpha_{k}),$

a contradiction. This ensures that $\Lambda(M,\alpha_{k})>1$ definitively and
concludes the proof. $\Box$

###  5.2. High dimensions

We are going to state the key ingredients in the proofs of Theorem 2.7 and
Theorem 2.9-$(iii)$. Inspired by [20] and [21] (see the Appendix), given a
radial regular solution $u$ of (2.2), we define the function

(5.12) $v(r)=u(r)+2\log(\psi(r))-\log[2(N-2]\,.$

Then $v$ solves the equation

(5.13)
$v^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,v^{\prime}(r)+2(N-2)\left[\frac{e^{v(r)}}{(\psi(r))^{2}}-\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}\right]-\frac{2\psi^{\prime\prime}(r)}{\psi(r)}=0$

The linearized equation at a function $v=v(r)$ becomes

$\varphi^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,\varphi^{\prime}(r)+\frac{2(N-2)}{(\psi(r))^{2}}\,e^{v(r)}\,\varphi(r)=0\,.$

In particular, linearizing at $v(r)=2\log(\psi^{\prime}(r))$, we obtain the
equation

(5.14)
$\varphi^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,\varphi^{\prime}(r)+2(N-2)\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}\,\varphi(r)=0\,.$

Next we define the operator $\mathcal{L}$ as the left hand sight of (5.14),
i.e.

(5.15)
$\mathcal{L}\varphi(r)=\varphi^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,\varphi^{\prime}(r)+2(N-2)\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}\,\varphi(r)$

and we consider the polynomial:

(5.16) $P(\lambda):=\lambda^{2}+(N-2)\lambda+2(N-2)\,.$

It is readily seen that $P$ admits the negative root:

(5.17) $\lambda_{1}:=\frac{-(N-2)-\sqrt{(N-2)(N-10)}}{2}\quad(N\geq 10)\,.$

Adapting in a non trivial way the proof of [20, Lemma 3.1] to our setting, we
prove the following property of the operator $\mathcal{L}$:

######  Lemma 5.11.

Suppose that $\psi$ satisfies (A1), (A2) and (A5). Let $N\geq 10$ and let
$0<R<+\infty$, then there exists no function $z\in C^{2}((0,R])$ such that

  * (i)

$\mathcal{L}z>0$ in $(0,R)$;

  * (ii)

$z>0$ in $(0,R)$ and $z(R)=0$;

  * (iii)

$\psi(r)z^{\prime}(r)=O(1)$ and
$z(r)=o\left((\psi(r))^{-(N-2+\lambda_{1})}\right)$ as $r\to 0^{+}$,

with $\lambda_{1}$ as in (5.17).

###### Proof.

Suppose by contradiction that there exists $z\in C^{2}((0,R])$ satisfying
$(i)$-$(iii)$. Let us define

$Z(r):=(\psi(r))^{\lambda_{1}}\qquad\text{for any }r>0$

with $\lambda_{1}$ as in (5.17). Differentiating we have that

$Z^{\prime}(r)=\lambda_{1}(\psi(r))^{\lambda_{1}-1}\,\psi^{\prime}(r)\,,\qquad
Z^{\prime\prime}(r)=\lambda_{1}(\lambda_{1}-1)(\psi(r))^{\lambda_{1}-2}\,(\psi^{\prime}(r))^{2}+\lambda_{1}(\psi(r))^{\lambda_{1}-1}\,\psi^{\prime\prime}(r)$

and hence

(5.18)
$\displaystyle\mathcal{L}Z(r)=\lambda_{1}(\psi(r))^{\lambda_{1}-1}\,\psi^{\prime\prime}(r)+(\psi(r))^{\lambda_{1}-2}(\psi^{\prime}(r))^{2}\,P(\lambda_{1})$
$\displaystyle\qquad=\lambda_{1}(\psi(r))^{\lambda_{1}-1}\,\psi^{\prime\prime}(r)<0$

since $\lambda_{1}$ is a negative root of the polynomial $P$ and by (A1),
(A2), (A5) we have

(5.19)
$\frac{\psi^{\prime\prime\prime}(r)}{\psi^{\prime}(r)}>\left(\frac{\psi^{\prime\prime}(r)}{\psi^{\prime}(r)}\right)^{2}\,,\quad\psi^{\prime\prime}(r)>0\qquad\text{for
any }r>0\,.$

Indeed, the first inequality in (5.19) is equivalent to (A5) and from it we
also have that $\psi^{\prime\prime\prime}>0$ which implies
$\psi^{\prime\prime}$ increasing and, in turn, by (A1) we finally obtain
$\psi^{\prime\prime}>0$ in $(0,+\infty)$.

Now, combining $(i)$, $(ii)$ and (5.18), we obtain for any $r\in(0,R)$

$\displaystyle
0<\mathcal{(}\mathcal{L}z(r))Z(r)-\mathcal{(}\mathcal{L}Z(r))z(r)$
$\displaystyle\quad=\left(z^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,z^{\prime}(r)\right)Z(r)-\left(Z^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,Z^{\prime}(r)\right)z(r)$

which is equivalent to

$\left[(\psi(r))^{N-1}z^{\prime}(r)Z(r)\right]^{\prime}-\left[(\psi(r))^{N-1}Z^{\prime}(r)z(r)\right]^{\prime}>0\,.$

In particular, we have the function

$r\mapsto(\psi(r))^{N-1}\left[z^{\prime}(r)Z(r)-Z^{\prime}(r)z(r)\right]$

is increasing in $(0,R)$ and this implies

$\displaystyle(\psi(R))^{N-1}z^{\prime}(R)Z(R)>\lim_{r\to
0^{+}}(\psi(r))^{N-1}\left[z^{\prime}(r)Z(r)-Z^{\prime}(r)z(r)\right]$
$\displaystyle\quad=\lim_{r\to
0^{+}}\left[(\psi(r))^{N-2+\lambda_{1}}\,z^{\prime}(r)\psi(r)-\lambda_{1}(\psi(r))^{N-2+\lambda_{1}}z(r)\psi^{\prime}(r)\right]=0$

thanks to $(ii)$, $(iii)$, (A1) and the fact that $N-2+\lambda_{1}>0$.

We may conclude that $z^{\prime}(R)>0$ in contradiction with $(ii)$. $\Box$

Thanks to Lemma 5.11, we now prove a uniform estimate for functions defined in
(5.12).

######  Lemma 5.12.

Suppose that $\psi$ satisfies (A1), (A2) and (A5) and let $N\geq 10$. Then any
function $v$ defined by (5.12) satisfies

$v(r)<2\log(\psi^{\prime}(r))\qquad\text{for any }r>0\,.$

###### Proof.

Let us define

(5.20) $V(r)=2\log(\psi^{\prime}(r))\qquad\text{for any }r>0\,.$

If we define $W(r)=V(r)-v(r)$, the statement of the lemma is equivalent to say
that $W(r)>0$ for any $r>0$.

We first observe that by (5.12), (5.20), (A1) and the fact that $u$ is a
radial smooth function in $M$, we have

(5.21) $\displaystyle
W(r)=2\log(\psi^{\prime}(r))-u(r)-2\log(\psi(r))+\log[2(N-2)]\sim-2\log(\psi(r))\to+\infty$

as $r\to 0^{+}$.

Then, recalling the definition of $\lambda_{1}$ given in (5.17), by (5.21) and
de l’Hôpital’s rule, we infer

$\displaystyle\lim_{r\to 0^{+}}(\psi(r))^{N-2+\lambda_{1}}\,W(r)=\lim_{r\to
0^{+}}\frac{-2\log(\psi(r))}{(\psi(r))^{-(N-2+\lambda_{1})}}=\lim_{r\to
0^{+}}\,\frac{2}{N-2+\lambda_{1}}\,(\psi(r))^{N-2+\lambda_{1}}=0$

being $\psi(0)=0$ and $\psi\in C^{0}([0,+\infty))$ by (A1) and
$N-2+\lambda_{1}>0$ as already observed in the proof of Lemma 5.11. In
particular, we have that

(5.22) $W(r)=o\left((\psi(r))^{-(N-2+\lambda_{1})}\right)\qquad\text{as }r\to
0^{+}\,.$

Moreover, differentiating in (5.21), using (A1) and exploiting the fact that
$u^{\prime}(0)=0$, we also obtain

$\displaystyle
W^{\prime}(r)=\frac{2\psi^{\prime\prime}(r)}{\psi^{\prime}(r)}-u^{\prime}(r)-\frac{2\psi^{\prime}(r)}{\psi(r)}\sim-\frac{2}{\psi(r)}\qquad\text{as
}r\to 0^{+}$

so that in particular

(5.23) $\psi(r)W^{\prime}(r)=O(1)\qquad\text{as }r\to 0^{+}\,.$

By (5.21) we know that $W(r)>0$ for any $r$ small enough. We have to prove
that actually $W$ is positive in $(0,+\infty)$ and hence we may proceed by
contradiction assuming that there exists $R>0$ such that

(5.24) $W(r)>0\qquad\text{for any }r\in(0,R)\qquad\text{and}\qquad W(R)=0\,.$

We claim that $\mathcal{L}W>0$ in $(0,R)$ with $\mathcal{L}$ as in (5.15).
First of all we observe that by (5.19), $V$ satisfies

(5.25) $\displaystyle
V^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,V^{\prime}(r)+2(N-2)\left[\frac{e^{V(r)}}{(\psi(r))^{2}}-\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}\right]-\frac{2\psi^{\prime\prime}(r)}{\psi(r)}$
$\displaystyle=2\left[\frac{\psi^{\prime\prime\prime}(r)}{\psi^{\prime}(r)}-\left(\frac{\psi^{\prime\prime}(r)}{\psi^{\prime}(r)}\right)^{2}\right]+2(N-2)\frac{\psi^{\prime\prime}(r)}{\psi(r)}>0\qquad\text{for
any }r>0\,.$

Subtracting (5.13) to (5.25), applying Lagrange Theorem to the exponential
function and exploiting (5.24), we infer

(5.26) $\displaystyle
0<W^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,W^{\prime}(r)+\frac{2(n-2)}{(\psi(r))^{2}}\left[e^{V(r)}-e^{v(r)}\right]$
$\displaystyle\quad<W^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,W^{\prime}(r)+\frac{2(N-2)}{(\psi(r))^{2}}\,e^{V(r)}[V(r)-v(r)]$
$\displaystyle\quad=W^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,W^{\prime}(r)+2(N-2)\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}W(r)=\mathcal{L}W(r)\,$

for any $r\in(0,R)$, thus completing the proof of the claim.

By (5.22), (5.23), (5.24), (5.26), we see that $W$ satisfies conditions $(i)$,
$(ii)$, $(iii)$ of Lemma 5.11 and the same lemma states that this is
impossible.

We reached a contradiction and this means that $W>0$ in $(0,+\infty)$. The
proof of the lemma now follows immediately from the definition of $V$ and $W$.
$\Box$

The estimate stated in Lemma 5.12 allows us to prove that for $N\geq 10$
solutions of (2.4) are ordered in the following sense:

######  Lemma 5.13.

Suppose that $\psi$ satisfies (A1), (A2) and (A5) and let $N\geq 10$. Let
$\alpha>\beta$ and let $u_{\alpha}$ and $u_{\beta}$ two solutions of (2.2)
satisfying $u_{\alpha}(0)=\alpha$ and $u_{\beta}(0)=\beta$. Then
$u_{\alpha}(r)>u_{\beta}(r)$ for any $r\geq 0$.

###### Proof.

Since $\alpha>\beta$ we have that $u_{\alpha}(r)>u_{\beta}(r)$ for any $r\geq
0$ small enough. We proceed by contradiction assuming that there exists $R>0$
such that

(5.27) $u_{\alpha}(r)>u_{\beta}(r)\quad\text{for any
}r\in[0,R)\quad\text{and}\quad u_{\alpha}(R)=u_{\beta}(R)\,.$

Recalling (5.12) we may define two functions $v_{\alpha}$ and $v_{\beta}$
corresponding to $u_{\alpha}$ and $u_{\beta}$ respectively.

By (5.27) we obtain

(5.28) $v_{\alpha}(r)>v_{\beta}(r)\quad\text{for any
}r\in[0,R)\quad\text{and}\quad v_{\alpha}(R)=v_{\beta}(R)\,.$

By Lemma 5.12 we know that

(5.29) $v_{\alpha}(r)<2\log(\psi^{\prime}(r))\quad\text{and}\quad
v_{\beta}(r)<2\log(\psi^{\prime}(r))\qquad\text{for any }r>0\,.$

Let us define $w(r)=v_{\alpha}(r)-v_{\beta}(r)=u_{\alpha}(r)-u_{\beta}(r)$ for
any $r\geq 0$. Since both $v_{\alpha}$ and $v_{\beta}$ solve (5.13), by (5.29)
and Lagrange Theorem we have that

(5.30) $\displaystyle
0=w^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,w^{\prime}(r)+\frac{2(N-2)}{(\psi(r))^{2}}\left[e^{v_{\alpha}(r)}-e^{v_{\beta}(r)}\right]$
$\displaystyle\quad<w^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,w^{\prime}(r)+\frac{2(N-2)}{(\psi(r))^{2}}\,e^{v_{\alpha}(r)}\,[v_{\alpha}(r)-v_{\beta}(r)]$
$\displaystyle\quad<w^{\prime\prime}(r)+(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}\,w^{\prime}(r)+2(N-2)\left(\frac{\psi^{\prime}(r)}{\psi(r)}\right)^{2}\,w(r)=\mathcal{L}w(r)$

for any $r\in(0,R)$.

We observe that, by (A1), $w$ trivially satisfies the following two conditions

(5.31) $\psi(r)w^{\prime}(r)=O(1)\,,\quad
w(r)=o\left((\psi(r))^{-(N-2+\lambda_{1})}\right)\qquad\text{as }r\to
0^{+}\,,$

being $w=u_{\alpha}-u_{\beta}\in C^{2}([0,+\infty))$.

By (5.28), (5.30) and (5.31), we see that $w$ satisfies conditions $(i)$,
$(ii)$ and $(iii)$ of Lemma 5.11 and the lemma itself says that this is
impossible. We proved that $w>0$ in $[0,+\infty)$ and now the proof of the
lemma follows from the definition of $w$. $\Box$

##  6\. Proofs of Theorems 2.5, 2.6, 2.7 and 2.9.

###  6.1. Proof of Theorem 2.5.

The proof of statements $(i)$ and $(ii)$ follow, respectively, from Lemma 6.1
and Lemma 6.2 below.

######  Lemma 6.1.

Let $N\geq 2$ and let $\psi$ satisfy (A1)-(A2). Let $\alpha$ and $\beta$ be
two distinct real numbers and let $u_{\alpha}$ and $u_{\beta}$ be the
corresponding solutions of (2.2), i.e. $u_{\alpha}(0)=\alpha$ and
$u_{\beta}(0)=\beta$. If $u_{\alpha}$ and $u_{\beta}$ are stable then they do
not intersect.

###### Proof.

Without loss of generality we may assume $\alpha>\beta$; we establish that
$u_{\alpha}(r)>u_{\beta}(r)$ for any $r>0$. If this is not true then there
exists $r_{0}>0$ such that $u_{\alpha}(r_{0})\leq u_{\beta}(r_{0})$. Now using
Lagrange Theorem and Lemma 5.3 we deduce that for some
$\sigma\in(\beta,\alpha)$ we have

$\displaystyle v_{\sigma}(r_{0})=\frac{\partial
u}{\partial\alpha}(\sigma,r_{0})=\frac{u_{\alpha}(r_{0})-u_{\beta}(r_{0})}{\alpha-\beta}\leq
0$

and $v_{\sigma}(0)=1$. Hence there exists $R\in(0,r_{0}]$ such that
$v_{\sigma}(R)=0$. Then $v_{\sigma}$ satisfies

$\displaystyle-\Delta_{g}v_{\sigma}=e^{u_{\sigma}}v_{\sigma}\text{ in
}B_{R}\quad\text{ with }v_{\sigma}=0\text{ on }\partial B_{R}.$

Multiplying this equation by $v_{\sigma}$ and integrating by parts we obtain

$\displaystyle\int_{B_{R}}|\nabla_{g}v_{\sigma}|_{g}^{2}\,{\rm
d}V_{g}-\int_{B_{R}}e^{u_{\sigma}}v_{\sigma}^{2}\,{\rm d}V_{g}=0.$

Let $w_{\sigma}$ be the trivial extension of $v_{\sigma}$ to the whole $M$ in
such a way that $w_{\sigma}\in H^{1}(M)$ and

$\displaystyle\int_{M}|\nabla_{g}w_{\sigma}|_{g}^{2}\,{\rm
d}V_{g}-\int_{M}e^{u_{\sigma}}w_{\sigma}^{2}\,{\rm d}V_{g}=0.$

Since $u_{\alpha}$ is a stable solution and $\sigma<\alpha$, by Lemma 5.6 we
deduce that $u_{\sigma}$ is a stable solution too. This implies that

$\displaystyle\inf_{v\in
H^{1}(M)\setminus\\{0\\}}\frac{\int_{M}|\nabla_{g}v|_{g}^{2}\,{\rm
d}V_{g}}{\int_{M}e^{u_{\sigma}}|v|^{2}\,{\rm d}V_{g}}\geq 1,$

and $w_{\sigma}$ attains the infimum. In particular, $w_{\sigma}$ satisfies
the following equation

$\displaystyle-\Delta_{g}w_{\sigma}=e^{u\sigma}w_{\sigma}\text{ in }M.$

Now by standard regularity theory $w_{\sigma}\in C^{2}(M)$ and satisfies the
following equation

$\displaystyle-
w_{\sigma}^{\prime\prime}(r)-(N-1)\frac{\psi^{\prime}(r)}{\psi(r)}w_{\sigma}^{\prime}(r)=e^{u_{\sigma}(r)}w_{\sigma}(r)\text{
for any }r>0.$

Moreover, by construction $w_{\sigma}(r)=0$ for $r>R$ and so, by unique
continuation, we conclude that $w_{\sigma}\equiv 0$ in $M$. This is a
contradiction. $\Box$

######  Lemma 6.2.

Let $N\geq 2$ and let $\psi$ satisfy (A1)-(A3). If $u\in C^{2}(M)$ is an
unstable solution of (2.2), then there exists no solution $v\in C^{2}(M)$ of
(2.2) such that $v>u$ in M.

###### Proof.

Suppose that there exists a solution $v\in C^{2}(M)$ of (2.2) such that $v>u$.
Let $w=v-u$, so $w>0$ in $M$. Using the Lagrange Theorem we have

$\displaystyle-\Delta_{g}w=e^{v}-e^{u}\geq e^{u}w\text{ in }M\,.$

Now taking $\varphi\in C^{\infty}_{c}(M)$ and multiplying in the above
inequality by $\frac{\varphi^{2}}{w}$, integrating by parts and using Cauchy
Schwarz and Young inequalities, we infer

$\displaystyle\int_{M}e^{u}\varphi^{2}{\rm d}V_{g}$
$\displaystyle\leq-\int_{M}\left\langle\nabla_{g}w,\nabla_{g}\left(\frac{\varphi^{2}}{w}\right)\right\rangle_{g}\,{\rm
d}V_{g}$
$\displaystyle=2\int_{M}\left\langle\nabla_{g}w,\frac{\varphi}{w}\nabla_{g}\varphi\right\rangle_{g}{\rm
d}V_{g}-\int_{M}\frac{\varphi^{2}}{w^{2}}|\nabla_{g}w|_{g}^{2}\,{\rm
d}V_{g}\leq\int_{M}|\nabla_{g}\varphi|_{g}^{2}\,{\rm d}V_{g}.$

This implies that $u$ is a stable solution and gives a contradiction. $\Box$

###  6.2. Proofs of Theorems 2.6 and 2.7.

The proofs of Theorem 2.6 \- $(i)$ and $(ii)$ follow by Lemma 5.7 and Lemma
5.8 which give that $\mathcal{S}=(-\infty,\eta]$. By Lemma 5.2 we know that
$\eta\geq\log(\lambda_{1}(M))$ while Lemma 5.10 allows improving this bound
when $\psi/\psi^{\prime}$ is not integrable.

The proof of Theorem 2.7 follows arguing by contradiction. Indeed, suppose
that there exists a radial solution $u\in C^{2}(M)$ of (2.2) with $N\geq 10$
which is unstable. Let $v\in C^{2}(M)$ be a radial solution of (2.2) such that
$v(0)>u(0)$. Then, by Lemma 5.13 we have that $v>u$ on $M$, in contradiction
with Lemma 6.2.

###  6.3. Proof of Theorem 2.9.

The proofs of Theorem 2.9 -$(i)$ and $(iii)$ follow by combining Theorem
2.6-$(i)$ and Theorem 2.7 with Lemma 6.1 . Instead, the proof of Theorem
2.9-$(ii)$ follows by combining Theorem 2.6-$(ii)$ with Lemma 6.2.

## Appendix: some (well) known facts in the Euclidean case

Consider the equation

(6.1) $-\Delta u=e^{u}\qquad\text{in }\mathbb{R}^{N}$

and let $u$ be a radial regular solution of (6.1). Letting $\alpha=u(0)$, then
$u$ solves (2.2) with $\psi(r)=r$. Following [21], we consider the function
$v(r)=u(r)+2\log r-\log[2(N-2)]$ (i.e., (5.12) with $\psi(r)=r$) which
satisfies the equation

$-v^{\prime\prime}(r)-\frac{N-1}{r}\,v^{\prime}(r)=\frac{2(N-2)}{r^{2}}[e^{v(r)}-1]\quad(r>0)\,.$

Now, let $w(t)=v(e^{t})$ so that $w$ solves the autonomous equation

(6.2)
$w^{\prime\prime}(t)+(N-2)w^{\prime}(t)+2(N-2)[e^{w(t)}-1]=0\quad(t\in\mathbb{R})\,.$

Following [21] one can reduce equation (6.2) into an _autonomous_ system in
the plane $(y,z)$ admitting the unique stationary point $(0,0)$ where we put
$z(t)=w(t)$ and $y(t)=w^{\prime}(t)$. Clearly, the system is given by

(6.3) $\begin{cases}y^{\prime}(t)=-(N-2)y(t)-2(N-2)[e^{z(t)}-1]\,,\\\\[5.0pt]
z^{\prime}(t)=y(t)\,.\end{cases}$

The behavior and, in turn, the stability of radial solutions to (6.1) depend
on the nature of the stationary point $(0,0)$ of (6.3) and, in particular,
after linearization at $(0,0)$, on the nature of the eigenvalues of the matrix

(6.4) $\begin{pmatrix}-(N-2)&-2(N-2)\\\ 1&0\end{pmatrix}\,.$

We observe that the characteristic polynomial of the matrix (6.4) is exactly
the polynomial $P$ given in (5.16). For $3\leq N\leq 9$ it admits two complex
conjugate eigenvalues while for $N\geq 10$ it admits two negative eigenvalues
which are coinciding if $N=10$ and distinct if $N\geq 11$:

$\lambda_{1}=\frac{-(N-2)-\sqrt{(N-2)(N-10)}}{2}\,,\ \ \
\lambda_{2}=\frac{-(N-2)+\sqrt{(N-2)(N-10)}}{2}\quad(N\geq 10)\,.$

The above computations highlight a change in the nature of solutions when
passing from low ($N\leq 9$) to high dimensions ($N\geq 10$). Starting from
this observation, see e.g., [25, Theorem 1.1], it can be proved that radial
smooth solutions intersect themselves infinitely many times if $3\leq N\leq 9$
and do not intersect if $N\geq 10$. Furthermore, for $N\leq 9$ all regular
solutions of (6.1) are unstable while for $N\geq 10$ radial regular solutions
are stable, see [15] and reference therein.

Acknowledgments. The first two authors are members of the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the
Instituto Nazionale di Alta Matematica (INdAM) and are partially supported by
the PRIN project 201758MTR2: “Direct and inverse problems for partial
differential equations: theoretical aspects and applications” (Italy). The
first two authors acknowledge partial financial support from the INdAM -
GNAMPA project 2022 “Modelli del 4 ordine per la dinamica di strutture
ingegneristiche: aspetti analitici e applicazioni”.

The second author acknowledges partial financial support from the research
project “Metodi e modelli per la matematica e le sue applicazioni alle
scienze, alla tecnologia e alla formazione” Progetto di Ateneo 2019 of the
University of Piemonte Orientale “Amedeo Avogadro”. The third author is
partially supported by the INSPIRE faculty fellowship (IFA17-MA98). The fourth
author is supported in part by National Theoretical Science Research Center
Operational Plan V-Mathematics Field (2/5) (Project number
111-2124-M-002-014-G58-01).

The authors are grateful to the anonymous referee for bringing to their
attention two articles containing interesting applications to fluid dynamics
of equation (1.2) and related semilinear equations.

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