Proof of Theorem 1.11. We recall that these solutions of eikonal type are the
solutions which behave like $r^{-\gamma}$ near $0$ or $\infty$. For $c>0$ and
$A\geq 0$ we set $u_{{}_{c,A}}(r)=cr^{-\gamma}+A$ and
$u_{{}_{c}}=u_{{}_{c,0}}$. Then there exist $a,b>0$ depending on $p$ such that

$c^{p}r^{-\gamma p}+aA^{p}+apc^{p-1}Ar^{-\gamma(p-1)}\leq
u^{p}_{{}_{c,A}}(r)\leq c^{p}r^{-\gamma p}+bA^{p}+bpc^{p-1}Ar^{-\gamma(p-1)}.$

(i) Subsolutions. If $u_{{}_{c}}(r)=cr^{-\gamma}$, then

$\begin{array}[]{lll}{\displaystyle\frac{1}{c}}{\mathcal{L}}^{M}_{p,q}u_{{}_{c}}=-\gamma(\gamma+2-N)r^{-\gamma-2}+c^{p-1}r^{-\gamma
p}-Mc^{q-1}\gamma^{q}r^{-(\gamma+1)q}\\\\[0.0pt]
\phantom{{\displaystyle\frac{1}{c}}{\mathcal{L}}^{M}_{p,q}u_{c}}=r^{-\frac{pq}{p-q}}\left(c^{p-1}-c^{q-1}X_{{}_{M}}^{p-q}-\gamma\theta
r^{\frac{\sigma}{p-q}}\right).\end{array}$ (3.84 )

Set $\Phi(c)=c^{p-1}-c^{q-1}X_{{}_{M}}^{p-q}$. Then $\Phi(X_{{}_{M}})=0$ and
$\Phi$ achieves its minimum at
$c_{m}=\left(\frac{q-1}{p-1}X_{{}_{M}}^{p-q}\right)^{\frac{1}{p-q}}$ with
minimal value
$\Phi(c_{m})=-\frac{p-q}{q-1}\left(\frac{q-1}{p-1}X_{{}_{M}}^{p-q}\right)^{\frac{p-1}{p-q}}$.
Notice that if $\theta=0$ i.e. $q=\frac{(N-2)p}{N-1}$, then we find the
explicit solution $u^{*}_{{}_{M,p}}$ defined in $(\ref{U11})$.
(i-a) If $N=1$ or $N=2$, or if $N\geq 3$ and $q>\frac{N-2}{N-1}p$, then
$\theta>0$ and $u_{{}_{c}}$ is a subsolution in
$\mathbb{R}^{N}\setminus\\{0\\}$ provided $c\leq X_{{}_{M}}$.
(i-b) If $q>\frac{2p}{p+1}$ and $\theta<0$ there exists $r_{1}>0$ such that
$u_{{}_{c_{m}}}$ is a subsolution in $B_{r_{1}}\setminus\\{0\\}$ Hence
$\tilde{u}_{{}_{c_{m}}}=c_{m}(r^{-\gamma}-r_{1}^{-\gamma})_{+}$ is a
subsolution in $\mathbb{R}^{N}\setminus\\{0\\}$.

(ii) Supersolutions. We have

$\begin{array}[]{lll}{\displaystyle\frac{1}{c}}{\mathcal{L}}^{M}_{p,q}u_{{}_{c,A}}\geq
r^{-\frac{pq}{p-q}}\left(\Phi(c)-\gamma\theta
r^{\frac{\sigma}{p-q}}\right)+{\displaystyle\frac{aA^{p}}{c}}+apc^{p-2}Ar^{-\gamma(p-1)}\end{array}$
(3.85 )

(ii-a) If $\theta<0$, then for $c\geq X_{{}_{M}}$ and any $A\geq 0$,
$u_{{}_{c,A}}$ is a supersolution in $\mathbb{R}^{N}\setminus\\{0\\}$.
(ii-b) If $q>\frac{2p}{p+1}$ and $\theta>0$, then for any $R>0$ we take
$c>X_{{}_{M}}$ such that $\Phi(c)\geq\gamma\theta R^{\frac{\sigma}{p-q}}$,
hence ${\mathcal{L}}^{M}_{p,q}u_{{}_{c,A}}\geq 0$ in $B_{R}\setminus\\{0\\}$.
Since $-\frac{pq}{p-q}+\frac{\sigma}{p-q}=\frac{q-2p}{p-q}<0$, we take $A>0$
such that $aA^{p}\geq\gamma\theta cR^{-\gamma-2}$, hence
${\mathcal{L}}^{M}_{p,q}u_{{}_{c,A}}\geq 0$ in $B_{R}^{c}$. Consequently
$u_{{}_{c,A}}$ is a supersolution in $\mathbb{R}^{N}\setminus\\{0\\}$.
(ii-c) If $q<\frac{2p}{p+1}$ and $\theta>0$, then we can take $c$ such that
$\Phi(c)\geq\gamma\theta R^{\frac{\sigma}{p-q}}$ and obtain that
$u_{{}_{c,A}}$ is a supersolution in $B_{R}^{c}$.

(iii) Proof of statements 1 and 2.
If $q>\frac{2p}{p+1}$ and whatever is the sign of $\theta$ there exist
$c_{m}\leq c<X_{{}_{M}}<c^{\prime}$ and $A>0$ such that
$u_{{}_{c^{\prime},A}}$ is a supersolution in $\mathbb{R}^{N}\setminus\\{0\\}$
larger than the subsolution $u_{{}_{c}}$. By [25, Theorem 1.4.5] there exists
a radial solution $u$ in $\mathbb{R}^{N}\setminus\\{0\\}$ satisfying
$u_{{}_{c}}\leq u\leq u_{{}_{c^{\prime},A}}$. Its behaviour at infinity is
given by Theorem 1.9. This solution is decreasing by the maximum principle and
it is unique by Theorem 4.6-(3).
The existence of a solution in a bounded domain $\Omega$ containing $0$ and
vanishing on $\partial\Omega$ satisfying $(\ref{Z19})$ follows by Theorem 1.14
which is proved in Section 4. So we deduce statement 1.
If $1<q<\frac{2p}{p+1}$ and $\theta>0$, one has a supersolution
$u_{{}_{c^{\prime},A}}$ in $B_{R}^{c}$ and a subsolution $u_{{}_{c}}$ in
$\mathbb{R}^{N}\setminus\\{0\\}$. Up to increasing the value of $A$ one has
again a supersolution $u_{{}_{c^{\prime},A}}$ larger than the subsolution
$u_{{}_{c}}$. Hence there exists a solution $u$ in between satisfying
$(\ref{Z29})$ which proves statement 2. $\square$

####  3.6.3 Riccati type singular solutions

Proof of Theorem 1.12. We recall that the Riccati equation $(\ref{Z16})$
admits the radial solution $\xi_{{}_{M}}|x|^{-\beta}$ if and only if
$\frac{N}{N-1}<q<2$. This function is a supersolution of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$.

1- Local existence in a neighborhood of $0$. Since $q>\frac{N}{N-1}$ the
system $(\ref{N1})$ in variables $(X,\xi,S)$ admits the equilibria $(0,0,0)$,
$(0,0,N-2)$ and $(0,\xi_{{}_{M}},\beta)$. Our aim is to construct local radial
solutions of $(\ref{W1})$ satisfying $\displaystyle\lim_{r\to
0}r^{\beta}u(r)=\xi_{{}_{M}}$ and $\displaystyle\lim_{r\to
0}r^{\beta+1}|u^{\prime}(r)|=\eta_{{}_{M}}=\beta\xi_{{}_{M}}$, equivalently

$\displaystyle\lim_{t\to-\infty}(X(t),\xi(t),S(t))=(0,\xi_{{}_{M}},\beta).$
(3.86 )

Conversely, any solution $(X,\xi,S)$ satisfying $(\ref{M1})$ corresponds to a
solution $u$ satisfying $\displaystyle\lim_{r\to
0}(r^{\beta}u(r),r^{\beta+1}|u^{\prime}(r)|)=(\xi_{{}_{M}},\beta\xi_{{}_{M}})$.
The system $(\ref{N1})$ may be singular at $\xi_{{}_{M}}=0$ or at $X=0$; hence
we desingularize it by setting $\hat{X}=X^{p-q}$ and $\hat{\xi}=\xi^{q-1}$.
Then $(\hat{X},\hat{\xi},S)$ satisfies

$\begin{array}[]{lll}\hat{X}_{t}=(p-q)\hat{X}(\gamma-S)\\\
\;\hat{\xi}_{t}=(q-1)\hat{\xi}(\beta-S)\\\
\;S_{t}=S(S+2-N)+\hat{\xi}(MS^{q}-\hat{X}).\end{array}$ (3.87 )

So we are led to study solutions in a neighborhood of the equilibrium
$(0,\hat{\xi}_{{}_{M}},\beta)$ where
$\hat{\xi}_{{}_{M}}=\xi^{q-1}_{{}_{M}}=\frac{\kappa}{M\beta^{q-1}}$. We set
$\hat{\xi}=\hat{\xi}_{{}_{M}}+\bar{\xi}$, $\hat{X}=\bar{X}$ and
$S=\beta+\bar{S}$ in order to reduce the study at $(0,0,0)$, and
$(\bar{\xi},\bar{X},\bar{S})$ satisfies the following linearized system

$\begin{array}[]{lll}\bar{X}_{t}=\frac{\sigma}{q-1}\bar{X}\\\
\;\bar{\xi}_{t}=-(q-1)\hat{\xi}_{{}_{M}}\bar{S}\\\
\;\bar{S}_{t}=-\hat{\xi}_{{}_{M}}\bar{X}+M\beta^{q}\bar{\xi}+(\beta+\kappa(q-1))\bar{S}.\end{array}$
(3.88 )

If we denote by ${\mathcal{A}}$ the matrix of this system, then its
charecteristic values are the roots of the polynomial

$\begin{array}[]{lll}det({\mathcal{A}}-\mu
I)=\left(\mu-\mu_{1}\right)\left(\mu-\mu_{2}\right)\left(\mu-\mu_{1}\right),\end{array}$
(3.89 )

with $\mu_{1}={\displaystyle\frac{\sigma}{q-1}}$, $\mu_{2}=\beta$ and
$\mu_{3}=(q-1)\kappa=(N-1)q-N$. Since
$q>\max\left\\{\frac{2p}{p+1},\frac{N}{N-1}\right\\}$ all the eigenvalues are
positive. We find that

${\bf
u}_{2}=\left(0,1,-{\displaystyle\frac{M\beta^{q}}{\kappa(q-1)}}\right)\;\,\text{and
}\;{\bf u}_{3}=\left(0,1,-M\beta^{q-1}\right)$

are eigenvectors corresponding to $\mu_{2}$ and $\mu_{3}$ respectively. If
$\mu_{1}\neq\mu_{2}$ and $\mu_{1}\neq\mu_{3}$, we can take for eigenvector
corresponding to $\mu_{1}$ the vector ${\bf u}_{1}=\left(1,b,c\right)$ for
some real numbers $b$ and $c$. Actually

$b=-{\displaystyle\frac{\mu_{3}}{M\mu_{2}^{q-1}\mu_{1}}}\quad\text{where
}\;c=-{\displaystyle\frac{\kappa\mu_{1}}{M\mu_{2}^{q-1}(\mu_{1}-\mu_{2})(\mu_{1}-\mu_{3})}}.$

Then there exists one trajectory of $(\ref{M2})$ with $X(t)>0$ when
$t\to-\infty$ such that $\xi(t)=\xi_{{}_{M}}+O(e^{\frac{\sigma t}{q-1}})$ when
$t\to-\infty$. Hence there exists at least one solution $u$ of $(\ref{W1})$
such that $u(r)=r^{-\beta}\xi_{{}_{M}}+Cr^{-\beta+(N-1)q-N}(1+o(1))$ when
$r\to 0$.

2- Local existence at infinity. Here we assume
$\frac{N}{N-1}<q<\frac{2p}{p+1}$. Then $\mu_{1}<0$, $\mu_{2}=\beta>0$ and
$\mu_{3}=\kappa(q-1)>0$. Then there exists a unique local trajectory which
converges to $(0,\hat{\xi},\beta)$ when $t\to\infty$, it corresponds to the
stable manifold of this point. If there exists a positive solution in
$\mathbb{R}^{N}\setminus\\{0\\}$, the solution can be extended as a solution
in $\mathbb{R}^{N}$ by [7, Theorem 1.1] since in this range of values of $q$
one has $p>\frac{N}{N-2}$. By Proposition A.1 such a solution is identically
$0$. $\square$

Remark. Note that we have many types of trajectoriess converging to the origin
and their geometry depends in their sign and their relative order. In this
respect we denote

$f(q):=(N-1)q-N+\frac{N-(N-2)q}{2-q},$ (3.90 )

and we have

$\begin{array}[]{lll}(i)&\mu_{3}=\kappa(q-1)>\mu_{2}=\beta\Longleftrightarrow
q>1+{\displaystyle\frac{1}{\sqrt{N-1}}}\\\
(ii)&\mu_{1}>\mu_{2}\Longleftrightarrow
p<{\displaystyle\frac{2(q-1)}{2-q}}\Longleftrightarrow q>\frac{2(p+1)}{p+2}\\\
(iii)&\mu_{1}>\mu_{3}\Longleftrightarrow p<f(q).\end{array}$ (3.91 )

We have that $\mu_{1}=\mu_{2}=\mu_{3}$ only if $p=\frac{2}{\sqrt{N-1}-1}$ and
$q=1+\frac{1}{\sqrt{N-1}}$, a condition which is compatible with $p>1$ only if
$2\leq N\leq 9$.

Global (necessarily singular) solutions in $r^{-\beta}$ are difficult to
construct. We give below a range of exponents in which there exists at least
one.

###### Theorem 3.11

Let $M>0$, $p>1$ and $1<q<2$, $q\neq 1+\frac{1}{\sqrt{N-1}}$. If there holds

$\begin{array}[]{lll}p<\max\left\\{{\displaystyle\frac{2(q-1)}{2-q}},f(q)\right\\},\end{array}$
(3.92 )

in particular if $p<\frac{N}{N-2}$ and $q>\frac{N}{N-1}$, then there exists a
positive radial solution of $(\ref{Z1})$ defined in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying

$\begin{array}[]{lll}\displaystyle\lim_{x\to
0}|x|^{\beta}u(x)=\xi_{{}_{M}}.\end{array}$ (3.93 )

Proof. The function $U(x)=\xi_{{}_{M}}|x|^{-\beta}$ is a supersolution of
$(\ref{Z1})$ in $\mathbb{R}^{N}\setminus\\{0\\}$. We look for a subsolution
under the form $\tilde{\xi}(t)=\xi_{{}_{M}}(1-Ae^{dt})_{+}$ for some $d,A>0$.
Set

$H[\tilde{\xi}](t)=\tilde{\xi}_{tt}+D\tilde{\xi}_{t}-\kappa\beta\tilde{\xi}-e^{\mu_{1}t}\tilde{\xi}^{p}+M|\beta\tilde{\xi}-\tilde{\xi}_{t}|^{q},$

where

$D={\displaystyle\frac{Nq-N-2}{q-1}}=\kappa-\beta.$

Then on the interval $I_{A}:=(-\infty,-\frac{\ln A}{d})$ one has

$\beta\tilde{\xi}-\tilde{\xi}_{t}=\xi_{{}_{M}}\left(\beta-A(\beta-d)e^{dt}\right).$

In order $H[\tilde{\xi}]\geq 0$, one needs

$\begin{array}[]{lll}-A\left(d^{2}+Dd-\kappa\beta\right)e^{dt}+M\xi^{q-1}_{{}_{M}}\beta^{q}\left(1-A{\displaystyle\frac{\beta-d}{\beta}}e^{dt}\right)^{q}-\kappa\beta\\\\[8.53581pt]
\phantom{----------------------}-\xi^{p-1}_{{}_{M}}e^{\mu_{1}t}\left(1-Ae^{dt}\right)^{p}\geq
0.\end{array}$

Since $M\xi_{{}_{M}}^{q-1}\beta^{q}=\kappa\beta$, if we set $Z=Ae^{dt}$, then
$0<Z\leq 1$ on $I_{A}$ and the previous inequality to be verified becomes

$\begin{array}[]{lll}{\displaystyle\frac{\xi^{p-1}_{{}_{M}}}{A}}e^{(\mu_{1}-d)t}Z\left(1-Z\right)^{p}\leq-\left(d^{2}+Dd-\kappa\beta\right)Z+\kappa\beta\left(1-{\displaystyle\frac{\beta-d}{\beta}}Z\right)^{q}-\kappa\beta.\end{array}$

We first impose $d\leq\mu_{1}$, then $\frac{e^{(\mu_{1}-d)t}}{A}\leq
A^{-\frac{\mu_{1}}{d}}$ on $I_{A}$. We set

$\begin{array}[]{lll}Q(Z)=\kappa\beta\left(1-{\displaystyle\frac{\beta-d}{\beta}}Z\right)^{q}-\left(d^{2}+Dd-\kappa\beta+\xi^{p-1}_{{}_{M}}A^{-\frac{\mu_{1}}{d}}\right)Z-\kappa\beta.\end{array}$
(3.94 )

Then

$Q^{\prime}(Z)=-q\kappa(\beta-d)\left(1-{\displaystyle\frac{\beta-d}{\beta}}Z\right)^{q-1}-\left(d^{2}+Dd-\kappa\beta+\xi^{p-1}_{{}_{M}}A^{-\frac{\mu_{1}}{d}}\right),$

and

$Q^{\prime\prime}(Z)={\displaystyle\frac{\kappa(\beta-d)^{2}q(q-1)}{\beta}}\left(1-{\displaystyle\frac{\beta-d}{\beta}}Z\right)^{q-2}.$

Since $\kappa>0$, $Q$ is convex on $[0,1]$. Furthermore $Q(0)=0$. Hence
$H(\tilde{\xi})\geq 0$ if $Q^{\prime}(0)\geq 0$.

$\begin{array}[]{lll}Q^{\prime}(0)=-q\kappa(\beta-d)-\left(d^{2}+Dd-\kappa\beta+\xi^{p-1}_{{}_{M}}A^{-\frac{\mu_{1}}{d}}\right)\end{array}$
(3.95 )

Replacing $D$ by its value, $(\ref{M11})$ will be achieved, provided $A$ is
large enough, if

$\begin{array}[]{lll}-d^{2}+((q-1)\kappa+\beta)d-(q-1)\kappa\beta=-(d-\mu_{2})(d-\mu_{3})>0.\end{array}$
(3.96 )

The condition is that $\mu_{1}\geq d$ with $d$ satisfying $(\ref{M12})$. It
necessitates $\mu_{2}\neq\mu_{3}$, equivalently $q\neq
1+\frac{1}{\sqrt{N-1}}$, and
(i) either $\mu_{2}<\inf\\{\mu_{1},\mu_{3}\\}$, then we can choose any
$d\in(\mu_{2},\inf\\{\mu_{1},\mu_{3}\\})$,
(ii) or $\mu_{3}<\inf\\{\mu_{1},\mu_{2}\\}$, then we can choose any
$d\in(\mu_{2},\inf\\{\mu_{1},\mu_{2}\\})$.
These conditions are satisfied if $\mu_{1}>\inf\\{\mu_{2},\mu_{3}\\}$ which is
equivalent to $(\ref{M5_*})$. If one of the above conditions is satisfied, it
follows by [25, Corollary 1.4.5] that there exists a radial positive solution
$u$ of $(\ref{Z1})$ in $\mathbb{R}^{N}\setminus\\{0\\}$ which satisfies

$\begin{array}[]{lll}\xi_{{}_{M}}\left(1-A|x|^{d}\right)_{+}|x|^{-\beta}\leq
u(x)\leq\xi_{{}_{M}}|x|^{-\beta}\quad\text{for all
}\,x\in\mathbb{R}^{N}\setminus\\{0\\}.\end{array}$ (3.97 )

$\square$

Remark. Condition $(\ref{M5_*})$ is equivalent to

$\begin{array}[]{lll}(i)&1<p<f(q)&\text{if
}\,1<q<1+{\displaystyle\frac{1}{\sqrt{N-1}}}\\\\[8.53581pt]
(ii)&1<p<{\displaystyle\frac{2(q-1)}{2-q}}&\text{if
}\,1+{\displaystyle\frac{1}{\sqrt{N-1}}}<q<2.\end{array}$ (3.98 )

Condition (ii) is equivalent to

${\displaystyle\frac{2(p+1)}{p+2}}<q<2.$ (3.99 )

Note that the nature of the variations of the function $p=f(q)$ differs
according to the value of $N$.
If $N=3$ or $4$, $f$ is increasing and onto from $(\frac{3}{2},2)$ to
$(3,\infty)$ when $N=3$ and from $(\frac{4}{3},2)$ to $(2,\infty)$ when $N=4$.
If $N\geq 5$, $f$ achieves a maximal value $\tilde{p}$ for $q=\tilde{q}$ with

$\begin{array}[]{lll}\tilde{q}=2-\sqrt{{\displaystyle\frac{N-4}{N-1}}}\,\text{
and }\;\tilde{p}=2\left(N-2-\sqrt{(N-4)(N-1)}\right).\end{array}$ (3.100 )

In particular one has

$f\left(\frac{N}{N-1}\right)=f\left(\frac{N}{N-2}\right)=\frac{N}{N-2}.$

####  3.6.4 Emden-Fowler type singular solutions

Proof of Theorem 1.13-(1). Since $1<p<\frac{N}{N-2}$ the function $x\mapsto
U_{x_{0}}(x)=x_{0}|x|^{-\alpha}$ is a subsolution of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$. In order $x\mapsto
U_{{}_{C}}(x):=C|x|^{-\alpha}$ to be a supersolution, one needs

$C^{p-1}\geq x_{0}^{p-1}+\alpha^{q}C^{q-1}M|x|^{-\frac{\sigma}{p-1}}.$ (3.101
)

The function $C\mapsto C^{p-q}-x_{0}^{p-1}C^{1-q}$ is increasing and onto from
$[x_{0},\infty)$ to $[x_{0},\infty)$. Hence there exists $C>x_{0}$ such that
$\alpha^{q}C^{q-1}M=C^{p-1}-x_{0}^{p-1}$. For such a value we have that

${\mathcal{L}}^{M}_{p,q}U_{{}_{C}}=M\alpha^{q}(1-r^{-\frac{\sigma}{p-1}})r^{-\frac{2p}{p-1}}.$

Since $\sigma<0$ the function $U_{{}_{C}}$ is a supersolution of $(\ref{Z1})$
in $B_{1}\setminus\\{0\\}$. For $A>0$ we set $U_{{}_{C,A}}=U_{{}_{C}}+A$. Then

$\begin{array}[]{lll}{\mathcal{L}}^{M}_{p,q}U_{{}_{C,A}}={\mathcal{L}}^{M}_{p,q}U_{{}_{C}}+U^{p}_{{}_{C,A}}-U^{p}_{{}_{C}}\geq
A^{p}+M\alpha^{q}(1-|x|^{-\frac{\sigma}{p-1}})|x|^{-\frac{2p}{p-1}}.\end{array}$
(3.102 )

Clearly ${\mathcal{L}}^{M}_{p,q}U_{{}_{C,A}}\geq 0$ in
$B_{1}\setminus\\{0\\}$, and for $|x|>1$, one has

${\mathcal{L}}^{M}_{p,q}U_{{}_{C,A}}\geq
A^{p}-M\alpha^{q}|x|^{-\frac{(p+1)q}{p-1}}\geq A^{p}-M\alpha^{q}.$

Therefore, if $A=M\alpha^{\frac{q}{p}}$, the function $U_{{}_{C,A}}$ is a
supersolution in $\mathbb{R}^{N}\setminus\\{0\\}$. Since $U_{x_{0}}\leq
U_{{}_{C,A}}$, it follows by [25, Theorem 1.4.5] that there exists a solution
$u$ of $(\ref{Z1})$ in $\mathbb{R}^{N}\setminus\\{0\\}$ such that
$U_{x_{0}}\leq u\leq U_{{}_{C,A}}$. Then by Theorem 1.8-(1), u satisfies
$(\ref{Z32})$-(i), and by Theorem 1.10-(2), $(\ref{Z32})$-(ii) holds.
Furthermore $r^{\alpha}u(r)\geq x_{0}$ for any $r>0$. Uniqueness (not only for
radial solutions) is a consequence of Theorem 4.6-(2). Obviously $|x|\mapsto
u(x)$ is decreasing. Existence of a positive solution in a bounded domain
$\Omega$ containing $0$ is a consequence of Theorem 1.14, see Section 4.
$\square$

Proof of Theorem 1.13-(2). It is a consequence of Theorem 1.11 and Theorem
1.9. $\square$

####  3.6.5 Solutions behaving like the Newtonian potential

There exist also solutions which behave like the Newtonian kernel at $0$. They
are described in the next result.

###### Theorem 3.12

Let $1<p<\frac{N}{N-2}$ and $1<q<\frac{N}{N-1}$. Then for any $M\geq 0$ and
$k>0$ there exists a minimal positive solution $u_{k}$ of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ such that $(\ref{Z8})$ holds. Furthermore it
is radial and nonincreasing. If we assume $1<q\leq\frac{2p}{p+1}$, this
solution is unique among all the positive solutions.

Proof. Proof of existence. If $M=0$ the result is classical and for $k>0$ we
denote by $v_{k}$ the solution of ${\mathcal{L}}_{p}v=0$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $(\ref{Z11})$. This is a natural
subsolution of $(\ref{Z1})$.
The construction of the supersolution is more involved.
(i) We first assume that $N\geq 3$ and prove that for any $k>0$ there exists
$M_{k}>0$ such that for any $0<M\leq M_{k}$ there exists a supersolution of
$(\ref{Z1})$ satisfying $(\ref{Z8})$. Let $a>0$ set

$w_{k}(x)=k|x|^{2-N}+k^{q}|x|^{2-(N-1)q}+a$

Then there exist $c_{5},c_{6}>0$ depending on $N$ and $q$ such that.

$\begin{array}[]{lll}\\!{\mathcal{L}}_{p,q}^{M}w_{k}=k^{q}((N\\!-\\!1)q-\\!2)(N-(N-\\!1)q))|x|^{(1-N)q}+(k|x|^{2-N}\\!+k^{q}|x|^{2-(N-1)q}\\!+a)^{p}\\\\[2.84526pt]
\phantom{{\mathcal{L}}_{p,q}^{M}w}-M\left((N\\!-\\!2)k|x|^{1-N}\\!+\\!((N\\!-\\!1)q\\!-\\!2)k^{q}|x|^{1-(N-1)q}\right)^{q}\\\\[2.84526pt]
\phantom{{\mathcal{L}}_{p,q}^{M}w_{k}}\geq
c_{5}k^{q}|x|^{(1-N)q}+a^{p}-c_{6}M\left(k^{q}|x|^{(1-N)q}+k^{q^{2}}|x|^{q-(N-1)q^{2}}\right)\\\\[2.84526pt]
\phantom{{\mathcal{L}}_{p,q}^{M}w_{k}}\geq
k^{q}\left(c_{5}-c_{6}M(1+k^{q^{2}-q})\right)|x|^{(1-N)q}+a^{p}-k^{q}\left(c_{6}M+k^{q^{2}-q}\right).\end{array}$
(3.103 )

Note that we have only used inequalities $2\leq(N-1)q\\!\leq N$. Set
$M_{k}=\frac{c_{5}}{c_{6}M(1+k^{q^{2}-q})}$. Then, for $M\leq M_{k}$ we take
$a^{p}=k^{q}\left(c_{6}M+k^{q^{2}-q}\right)$ and we derive that
${\mathcal{L}}_{p,q}^{M}w_{k}\geq 0$. The supersolution $w_{k}$ satisfies
$v_{k}\leq k|x|^{2-N}\leq w_{k}$.
(ii) If $N=2$ and for $b>0$ we denote by $\psi_{k}$ the solution of

$\begin{array}[]{lll}\\!-\Delta\psi+\psi^{p}=|x|^{-q}+2\pi
k\delta_{0},\end{array}$ (3.104 )

and we set $w_{k}=\psi_{k}+b$. Since $1<q<2$, $w_{k}=-k\ln|x|(1+o(1))$ and
$\nabla w_{k}=-k|x|^{-1}(1+o(1))$ as $x\to 0$. Hence

$\begin{array}[]{lll}0\leq\psi_{k}(x)\leq
c_{7}(-k\ln|x|+1)\;{and}\;\,|\nabla\psi_{k}(x)|\leq
c_{7}(k+1)|x|^{-1}\quad\text{for }0<|x|\leq 1.\end{array}$ (3.105 )

Furthermore, by Keller-Osserman technique combined with scaling method, there
holds in $\mathbb{R}^{2}\setminus B_{1}$,

$\begin{array}[]{lll}(i)&0\leq\psi_{k}(x)\leq
c_{8}\max\left\\{|x|^{-\alpha},|x|^{-\frac{q}{p}}\right\\},\\\\[8.53581pt]
(ii)&|\nabla\psi_{k}(x)|\leq
c_{8}\max\left\\{|x|^{-\frac{p+1}{p-1}},|x|^{-\frac{p+q}{p}}\right\\}.\end{array}$
(3.106 )

In the above inequalities, $c_{7}$ and $c_{8}$ are positive constants
depending on $p$ and $q$. Hence

$\begin{array}[]{lll}\\!{\mathcal{L}}_{p,q}^{M}w_{k}=|x|^{-q}+(\psi_{k}+b)^{p}-\psi_{k}^{p}-M|\nabla\psi_{k}|^{q}.\end{array}$
(3.107 )

We infer

$\begin{array}[]{lll}\\!{\mathcal{L}}_{p,q}^{M}w_{k}\geq|x|^{-q}+b^{p}-Mc_{7}^{q}k^{q}|x|^{-q}\quad\text{if
}0<|x|\leq 1,\end{array}$ (3.108 )

and

$\begin{array}[]{lll}\\!{\mathcal{L}}_{p,q}^{M}w_{k}\geq
b^{p}-Mc_{8}^{q}k^{q}\quad\text{if }x|\geq 1.\end{array}$ (3.109 )

If $k$ is fixed, $M\leq M_{k}:=k^{-q}c_{7}^{-q}$ and $b\geq Mc_{8}^{q}k^{q}$
we conclude that $w_{k}$ is a supersolution in
$\mathbb{R}^{2}\setminus\\{0\\}$ larger than $v_{k}$.

We deduce from (i) and (ii) that for any $k>0$ there exists $M_{k}>0$ such
that for any $0<M\leq M_{k}$ there exists a positive radial solution $u_{k}$
of $(\ref{Z1})$ satisfying $(\ref{Z8})$. Furthermore $u_{k}$ satisfies
$(\ref{Y1})$. Therefore $u_{k}$ is necessarily decreasing.

End of the proof of existence. Let $q<\frac{N}{N-1}$ and $q_{1}$ such that
$q<q_{1}<\frac{N}{N-1}$. For $k>0$ let $\epsilon>0$ such that for any
$0<M^{\prime}\leq\epsilon$ there exists a positive radial solution $w_{k}$ to
${\mathcal{L}}_{q_{1},M^{\prime}}w=0$ satisfying $(\ref{Z8})$. If
$M>M^{\prime}$ there holds

$M|X|^{q}\leq M^{\prime}|X|^{q_{1}}+C\qquad\text{for all }X\in\mathbb{R}^{N},$

where
$C=\left(\frac{qM}{q_{1}M^{\prime}}\right)^{\frac{q}{q_{1}-q}}\left(M-\frac{q}{q_{1}}M^{\prime}\right)>0$.
Then

${\mathcal{L}}_{p,q}^{M}w_{k}={\mathcal{L}}_{q_{1},M^{\prime}}w_{k}+(M^{\prime}-M)|\nabla
w_{k}^{q}|\geq-C$

which implies that $w_{k}+C^{\frac{1}{p}}$ is s supersolution of $(\ref{Z1})$
and $v_{k}\leq w_{k}+C^{\frac{1}{p}}$. We conclude as in the first step.

Uniqueness. It is proved in Theorem 4.6, this ends the proof.
When we do not assume $q\leq\frac{2p}{p+1}$ we have only the existence of a
minimal positive solution. This is due to the fact that for two solutions $u$
and $u^{\prime}$ as above, $\min\\{u,u^{\prime}\\}$ is a supersolution larger
that $v_{k}$. The conclusion follows easily. $\square$

In the next statements we prove the existenc of radial solutions defined in
the complement of a ball of $\mathbb{R}^{N}$, $N\geq 3$ which behaves like the
Newtonian potential at infinity. We start with the following lemma dealing
with the positive radial solutions of ${\mathcal{L}}_{p}v=0$ in the complement
of a ball.

###### Lemma 3.13

Assume $N\geq 3$ and $p>\frac{N}{N-2}$. Then for any $c>0$ there exists
$k_{c}>0$ such that the unique solution $v_{c}$ of ${\mathcal{L}}_{p}v=0$ in
$B_{1}^{c}$ verifying $v\lfloor_{\partial B_{1}}=c$ satisfies

$\displaystyle\lim_{|x|\to\infty}|x|^{N-2}v_{c}(x)=k_{c}.$ (3.110 )

Furthermore the mapping $c\mapsto k_{c}$ is continuous and increasing from
$(0,\infty)$ onto $(0,k_{\infty})$ for some $k_{\infty}<\infty$.

Proof. The existence and uniqueness of a solution $v_{c}$ in an exterior
domain and the fact that $(\ref{L3})$ holds is classical (see e.g. [21]).
However the fact that $k_{c}>0$ and the continuity of $c\mapsto k_{c}$ is not
proved there. By the maximum principle $c\mapsto k_{c}$ is nondecreasing. Next
we set $s=\frac{r^{N-2}}{N-2}$ and $v_{c}(r)=r^{2-N}\rho(s)$. Then
$\rho_{c}:=\rho$ satisfies

$\displaystyle s^{2}\rho_{ss}=c_{N,p}s^{\frac{N}{N-2}-p}\rho^{p}\;\text{ on
}\,((N-2)^{-1},\infty)\;\text{ and }\,\rho((N-2)^{-1})=c,$ (3.111 )

where $c_{N,p}=(N-2)^{\frac{4-N}{N-2}-p}$. By the maximum principle
$v_{c}(r)\leq cr^{2-N}$ ($v_{c}$ is the positive harmonic function in
$B_{1}^{c}$ with value $c$ on $\partial B_{1}$), hence $\rho(s)$ is bounded.
Since $\rho$ is convex and bounded, it is decreasing and $\rho^{\prime}s)\to
0$ as $s\to\infty$. Hence

$-\rho^{\prime}(s)=c_{N,p}{\displaystyle\int_{s}^{\infty}}\tau^{\frac{N}{N-2}-p-2}\rho^{p}(\tau)d\tau\leq
c^{\prime}_{N,p}s^{\frac{N}{N-2}-p-1}\rho^{p}(s).$ (3.112 )

Hence by integration the function
$\rho\mapsto\Phi(\rho)=\rho^{1-p}-c^{\prime\prime}\rho^{\frac{N}{N-2}-p}$ is
increasing and bounded. Then it has a finite limit $\ell$ when $\rho\to\infty$
and $\Phi^{1-p}(\rho)$ has the same limit $\ell$. Thus $\ell\neq 0$ and
consequently $k_{c}>0$. Let $\\{c_{n}\\}$ be a decreasing sequence in
$\mathbb{R}_{+}$ converging to $c^{*}>0$. Then the sequence of corresponding
solutions $\\{v_{c_{n}}\\}$ is decreasing to $v_{c^{*}}$ the sequence
$\\{k_{c_{n}}\\}$ is nonincreasing with limit $k^{*}\geq k_{c^{*}}$. From
$(\ref{L5})$ one get

$\displaystyle
c_{n}-k_{c_{n}}=c_{N,p}{\displaystyle\int_{(N-2)^{-1}}^{\infty}}{\displaystyle\int_{s}^{\infty}}\tau^{\frac{N}{N-2}-p-2}\rho_{c_{n}}^{p}(\tau)d\tau
ds,$ (3.113 )

and the same identity holds in $c_{n}$ is replaced by $c^{*}$. By the
dominated convergence theorem, one has that

$\displaystyle
c^{*}-k^{*}=c_{N,p}{\displaystyle\int_{(N-2)^{-1}}^{\infty}}{\displaystyle\int_{s}^{\infty}}\tau^{\frac{N}{N-2}-p-2}\rho_{c^{*}}^{p}(\tau)d\tau
ds,$ (3.114 )

which implies that $k^{*}=k_{c^{*}}$. A similar result holds if $\\{c_{n}\\}$
is an increasing sequence in $\mathbb{R}_{+}$ converging to $c_{*}>0$. Hence
$c\mapsto k_{c}$ is increasing and continuous. When $c\uparrow\infty$ $v_{c}$
increases and converges to the unique positive solution $v_{\infty}$ of
${\mathcal{L}}_{p}v=0$ in $B_{1}^{c}$ such that $\displaystyle\lim_{r\to
1}v(r)=\infty$. Hence $k_{c}\uparrow k_{\infty}$ and $k_{\infty}<\infty$.
$\square$

Remark. Since the equation ${\mathcal{L}}_{p}v=0$ is invariant by the
transformation $T_{\ell}$ defined in $(\ref{Z2})$, the ball $B_{1}$ can be
replaced by $B_{R}$ for any $R>0$. The range of $k_{c}$, that we call
$k_{c,R}$ is modified accordingly and
$\displaystyle\lim_{c\to\infty}k_{c,R}=k_{\infty,R}$. Then one has

$k_{\infty,R}=R^{N-2-\alpha}k_{\infty,1}.$ (3.115 )

###### Theorem 3.14

Let $N\geq 3$, $M>0$, $p>\frac{N}{N-2}$ and $\frac{N}{N-1}<q<p$.

1- For any $k>0$ there exist $R:=R_{k}>0$ and a positive radial solution $u$
of $(\ref{Z1})$ in $B_{R}^{c}$ satisfying

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k.\end{array}$
(3.116 )

2- If $\frac{2p}{p+1}<q<p$ there exist $\tilde{k}>0$ and a positive radial
solution, unique among all the positive solutions, $u$ of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $\displaystyle\lim_{r\to
0}r^{\gamma}u(r)=X_{{}_{M}}$ and $(\ref{L8})$ with $k=\tilde{k}$. In the
particular case $q=\frac{(N-2)p}{N-1}$ we have $u=u^{*}_{{}_{M,p}}$ (see
$(\ref{U11})$).

3- If $\frac{2p}{p+1}<q<2$ and the assumption $(\ref{M5_*})$ of Theorem 3.11
is satisfied, there exist $k>0$ and a radial positive solution $u$ of
$(\ref{Z1})$ in $\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $(\ref{L8})$ and
$\displaystyle\lim_{x\to 0}|x|^{\beta}u(x)=\xi_{{}_{M}}$. Furthermore $u$ is
unique among all the positive solutions satisfying $(\ref{L8})$.

Proof. 1- If $w$ is a positive radial and decreasing function such that
${\mathcal{R}}^{M}_{q}w=0$ it satisfies (see $(\ref{T11})$)

$-w^{\prime}(r)=r^{1-N}\left(C+{\displaystyle\frac{M}{\kappa}}r^{N-(N-1)q}\right)^{-\frac{1}{q-1}},$

where, $\kappa=\frac{(N-1)q-N}{q-1}$ and $C\in\mathbb{R}$. If $C>0$, $w$ is
defined on $(0,\infty)$. Hence if $w(r)\to 0$ as $r\to\infty$, one has

$\begin{array}[]{lll}w(r)={\displaystyle\int_{r}^{\infty}}s^{1-N}\left(C+{\displaystyle\frac{M}{\kappa}}s^{N-(N-1)q}\right)^{-\frac{1}{q-1}}ds.\end{array}$
(3.117 )

Then $w(r)=\frac{1}{(N-2)C^{\frac{1}{q-1}}}r^{2-N}(1+o(1))$ as $r\to\infty$.
Hence, if $k>0$ is given, we choose $C>0$ such that
$\frac{1}{(N-2)C^{\frac{1}{q-1}}}=k$. In order that $k$ is in the range of the
application $c\mapsto k_{c,R}$, one takes $R>0$ such that
$k<R^{N-2-\alpha}k_{\infty,1}$. For such an $R$, there exists $c>0$ such that
the solution $v_{c}$ of ${\mathcal{L}}_{p}v=0$ in $B_{R}^{c}$ verifying $v=c$
on $\partial B_{R}$ satisfies $(\ref{L3})$. We then set
$C=\left(\frac{1}{(N-2)k}\right)^{q-1}$. The function $w:=w_{C}$ defined by
$(\ref{L10})$ is a supersolution of $(\ref{Z1})$ in $B_{R}^{c}$, larger than
the subsolution $v_{c}$ and both $v_{c}$ and $w_{C}$ satisfy $(\ref{L8})$.
Then by [25, Theorem 1.4.5] there exists a radial positive solution $u$ of
$(\ref{Z1})$ in $B_{R}^{c}$ such that $v_{c}\leq u\leq w_{C}$, hence
$(\ref{L8})$ follows.
2- The existence of a unique positive and radial solution in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $(\ref{Z19})$ follows from Theorem
1.11. The asymptotic behaviour is a consequence of Theorem 1.9-(2).
3- Under the condition $(\ref{M5_*})$ of Theorem 3.11 there exists a unique
positive solution in $\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $(\ref{M5})$.
From Theorem 1.9, and since $p>\frac{N}{N-2}$, its behaviour at infinity is
given by $(\ref{Z27})$ for some specific $k^{*}>0$. Uniqueness follows from
$Theorem~{}\ref{t:uni}$. $\square$

##  4 Isolated singularities of non-radial solutions

###  4.1 Existence and uniqueness of singular solutions

The results of this paragraph are independent of the description of the radial
singular solutions performed in the previous sections and they provide a
general tool for constructing singular solutions. The existence of singular
solutions is based upon the next variant of [11, Theorem 2.1] proved in [25,
Corollary 1.4.5].

###### Theorem 4.1

Let $G$ be a bounded domain in $\mathbb{R}^{N}$, $B\in
C(G\times\mathbb{R}\times\mathbb{R}^{N})$ a real valued function, $\Gamma\in
C(\mathbb{R}_{+},\mathbb{R}_{+})$ an increasing function such that

$|B(x,r,\xi)|\leq\Gamma(|r|)(1+|\xi|^{2})\quad\text{for all }(x,r,\xi)\in
G\times\mathbb{R}\times\mathbb{R}^{N}.$ (4.1 )

Let ${\mathcal{Q}}$ be the operator defined by

${\mathcal{Q}}(u)=-\Delta u+B(x,u,\nabla u).$ (4.2 )

If there exist a supersolution $\phi\in W^{1,\infty}(G)$ and a subsolution
$\psi\in W^{1,\infty}(G)$ such that $\psi\leq\phi$, then for any $\chi\in
W^{1,\infty}(G)$ satisfying $\psi\leq\chi\leq\phi$ there exists a function
$u\in W^{1,2}(G)$ verifying $\psi\leq u\leq\phi$, solution of
${\mathcal{Q}}(u)=0$ and such that $u-\chi\in W^{1,2}_{0}(G)$.

One of the main application of this result is Theorem 1.14 which is proved
below

Proof of Theorem 1.14. Let $\\{\epsilon_{n}\\}$ be a sequence decreasing to
$0$ and such that $\epsilon_{1}<\mbox{\rm dist}\,(0,\partial\Omega)$ and set
$\displaystyle
m=\max_{z\in\partial\Omega}v(z)+\max_{z\in\partial\Omega}(\phi(z)-v(z))_{+}$.
Then $m\geq\phi$ on $\partial\Omega$ and the function $\overline{v}=v+m$
satisfies ${\mathcal{L}}_{p,q}^{M}\overline{v}\geq 0$ in
$\Omega\setminus\\{0\\}$. The function
$\displaystyle\underline{v}=(v-\max_{z\in\partial\Omega}\phi(z))_{+}$
satisfies ${\mathcal{L}}_{p,q}^{M}\underline{v}\leq 0$. Put
$\chi=\sup\left\\{\underline{v},\inf\left\\{\overline{v},\phi\right\\}\right\\}$.
Then $\chi\in W_{loc}^{1,\infty}(\Omega\setminus\\{0\\})$,
$\underline{v}\leq\chi\leq\overline{v}$ and $\chi=\phi$ on $\partial\Omega$.
By Theorem 4.1 for any $n\in\mathbb{N}^{*}$ there exists a function $u_{n}\in
W^{1,2}(\Omega\setminus\overline{B}_{\epsilon_{n}})$ such that
$(u_{n}-\chi)\lfloor_{\Omega\setminus\overline{B}_{\epsilon_{n}}}\in
W^{1,2}_{0}(\Omega\setminus\overline{B}_{\epsilon_{n}})$ satisfying
${\mathcal{L}}_{p,q}^{M}u_{n}=0$ in
$\Omega\setminus\overline{B}_{\epsilon_{n}}$. Furthermore $u_{n}$ is unique by
the maximum principle. Since $u_{n}=\overline{v}$ on $\partial
B_{\epsilon_{n}}$, $v$, and therefore $\overline{v}$, is radially decreasing
and $u_{n}=\chi$ on $\partial\Omega$ we infer that $u_{n}\leq u_{n^{\prime}}$
in $\Omega\setminus B_{\epsilon_{n}}$ if $n^{\prime}\geq n$. Hence the
sequence $\\{u_{n}\\}$ is increasing and it satisfies

$\displaystyle(v(x)-\max_{z\in\partial\Omega}\phi(z))_{+}\leq u_{n}(x)\leq
v(x)+\max_{z\in\partial\Omega}(\phi(z)-v(z))_{+}\quad\text{for all
}x\in\Omega\setminus B_{\epsilon_{n}}.$ (4.3 )

By standard regularity estimates, $u_{n}$ is relatively compact in
$C_{loc}^{1}(\Omega\setminus\\{0\\})$. Hence it converges to a solution $u$ of
${\mathcal{L}}_{p,q}^{M}u=0$ in $\Omega\setminus\\{0\\}$ which coincides with
$\phi$ on $\partial\Omega$ and satisfies $(\ref{X3})$. $\square$

As a first application we have the following:

###### Corollary 4.2

Let $\Omega$ be any bounded smooth domain containing $0$ and $\phi\in
W^{1,\infty}(\Omega)$ be nonnegative. There exists a positive solution $u$ of
${\mathcal{L}}_{p,\frac{2p}{p+1}}^{M}u=0$ in $\Omega\setminus\\{0\\}$ with
value $\phi$ on $\partial\Omega$ such that $u(x)-a|x|^{-\alpha}$ remains
bounded in $\Omega$ where $a$ is equal to $x_{{}_{M}}$ or $x_{{}_{j,M}}$
(j=1,2) or $x_{m^{*}}$ according to we are in the cases (1)-(2) or (3) or (4)
of Theorem 1.1.

The existence of singular solutions is not restricted to the case
$q=\frac{2p}{p+1}$ where they are explicit. The following easy to prove
corollary shows that existence, and sometimes uniqueness, holds when
$1<q<\frac{N}{N-1}$. This range of exponents is analysed in [7] in connection
with problems with Dirac measure data.

###### Corollary 4.3

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 1$, be any bounded smooth domain
containing $0$. Assume $1<p<\frac{N}{N-2}$ if $N\geq 3$ or any $p>1$ if
$N=1,2$, $1<q<\min\left\\{p,\frac{N}{N-1}\right\\}$ if $N\geq 2$ or any $q>1$
if $N=1$, $M>0$ and $k>0$. Then for any $\phi\in W^{1,\infty}(\Omega)$,
$\phi\geq 0$, there exists a positive solution $u$ of
${\mathcal{L}}_{p,q}^{M}u=0$ in $\Omega\setminus\\{0\\}$ with value $\phi$ on
$\partial\Omega$ satisfying $(\ref{Z8})$.

Proof of Theorem 1.2 and Theorem 1.3. It is a direct consequence of the above
results. $\square$

More general uniqueness results valid for any positive solution, not
necessarily radial, are obtained below. Furthermore the problems involved are
either considered in $\mathbb{R}^{N}\setminus\\{0\\}$ or in a punctured
bounded domain. If $b$ is a positive parameter we define a continuous group of
transformations acting on functions $u$ defined in an open set $G$, $u\mapsto
u_{\ell}^{(b)}$, for $\ell>0$ by the formula

$\displaystyle u_{\ell}^{(b)}(x)=\ell^{b}u(\ell x)\quad\text{ for all
$\,\ell>0$ and $\,x\,$}\in\ell^{-1}G.$ (4.4 )

If $u$ satisfies $(\ref{Z1})$ in $G$, then $u_{\ell}^{(b)}$ satisfies

$-\Delta
u_{\ell}^{(b)}+\ell^{2-b(p-1)}(u_{\ell}^{(b)})^{p}-M\ell^{2-q-b(q-1)}|\nabla{u_{\ell}^{(b)}}|^{q}=0\quad\text{in
}\;\ell^{-1}G.$ (4.5 )

If $\ell>1$, $u_{\ell}^{(b)}$ is a supersolution of $(\ref{Z1})$ if and only
if

$\begin{array}[]{lll}(i)&2-b(p-1)\leq 0&\Longleftrightarrow\alpha\leq
b\phantom{--------------}\\\\[5.69054pt] (ii)&2-q-b(q-1)\geq
0&\Longleftrightarrow\beta\geq b.\end{array}$ (4.6 )

This conditions are compatible if and only if $1<q\leq\frac{2p}{p+1}$.
Similarly, if $\ell<1$, $u_{\ell}^{(b)}$ is a supersolution of $(\ref{Z1})$ if
and only if

$\begin{array}[]{lll}(i)&2-b(p-1)\geq 0&\Longleftrightarrow\alpha\geq
b\phantom{--------------}\\\\[5.69054pt] (ii)&2-q-b(q-1)\leq
0&\Longleftrightarrow\beta\leq b.\end{array}$ (4.7 )

This conditions are compatible if and only if $\frac{2p}{p+1}\leq q<2$.

Proof of Theorem 1.15 First we note that two terms on the right hand-side of
$(\ref{UNX})$ in the statement of the theorem coincide only if
$q=\frac{2p}{p+1}$ since $\alpha\leq\beta$ is equivalent to
$q\leq\frac{2p}{p+1}$ . We first study the problem in
$\mathbb{R}^{N}\setminus\\{0\\}$. We have to consider two cases:
1- Suppose $\alpha\leq\beta$. We choose $b$ such that

$b\in\left(a,\infty\right)\cap\left[\alpha,\beta\right].$ (4.8 )

Let $u$ and $\tilde{u}$ be two positive solutions satisfying $(\ref{UNX1})$.
For $\ell>1$, $u_{\ell}^{(b)}$ is a supersolution. Since

$u_{\ell}^{(b)}(x)=\Lambda\ell^{b-a}|x|^{-a}(-\ln|x|)^{\tilde{a}}(1+o(1))\quad\text{as
}x\to 0$

and $u(x)\to 0$ as $|x|\to\infty$, for any $\epsilon>0$ the function $x\mapsto
u_{\ell}^{(b)}(x)+\epsilon$ which is a supersolution is larger than
$\tilde{u}$ near $0$ and at infinity. Then
$u_{\ell}^{(b)}+\epsilon\geq\tilde{u}$ in $\mathbb{R}^{N}\setminus\\{0\\}$.
Letting $\epsilon\downarrow 0$ and $\ell\downarrow 1$, yields
$u\geq\tilde{u}$. Similarly $\tilde{u}\geq u$.
2- Suppose $\alpha>\beta$. We choose $b$ such that

$b\in\left(0,a\right)\cap\left[\beta,\alpha\right].$ (4.9 )

Then for $\ell<1$, $u_{\ell}^{(b)}+\epsilon$ is a supersolution in
$\mathbb{R}^{N}\setminus\\{0\\}$ which is larger than $\tilde{u}$ at $0$ and
at $\infty$. Hence $\tilde{u}\leq u_{\ell}^{(b)}+\epsilon$ and we conclude as
in the first case.
Next we consider the problem in $\Omega$. Since the solutions are continuous
in $\overline{\Omega}\setminus\\{0\\}$, for $\epsilon>0$ we have that for
$\ell>1$ $u_{\ell}^{(b)}+\epsilon>\tilde{u}$ near $\partial(\ell^{-1}\Omega)$
provided $\ell-1$ is small enough. Hence
$u_{\ell}^{(b)}(x)+\epsilon\geq\tilde{u}$ in
$\ell^{-1}\Omega\setminus\\{0\\}$. This implies that $u\geq\tilde{u}$ by
letting $\ell\uparrow 1$ and then $\epsilon\to 0$. If $\ell<1$ then
$\Omega\subset\ell^{-1}\Omega$, and we compare $u_{\ell}^{(b)}+\epsilon$ and
$\tilde{u}$ in $\Omega$. The proof follows. $\square$

The previous result necessitates to find some $b$ satisfying either
$(\ref{UNX2})$ or $(\ref{UNX3})$ which is not always possible in practice. We
give below a variant of the result which necessitates a slightly sharper blow-
up estimate.

###### Theorem 4.4

Assume $N\geq 1$, $p>1$, $1<q\leq\tfrac{2p}{p+1}$ and $M>0$. Let $a$ such that

$0\leq a\leq\beta$ (4.10 )

There exists at most one positive solution of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying

$u(x)=\Lambda|x|^{-a}+\Lambda^{\prime}|x|^{-a^{\prime}}(1+o(1))\quad\text{as
}x\to 0,$ (4.11 )

or

$u(x)=\Lambda|x|^{-a}(-\ln|x|)^{-a^{\prime\prime}}(1+o(1))\quad\text{as }x\to
0,$ (4.12 )

where $\Lambda,\Lambda^{\prime}$ are some positive constants and
$a>a^{\prime}$ and $a^{\prime\prime}>0$.

Proof. The principle of the proof is to replace $(\ref{UNX2})$ by

$b\in\left[a,\infty\right)\cap\left[\alpha,\beta\right].$ (4.13 )

when $\alpha\leq\beta$. Then, for $\ell>1$, $u_{\ell}^{(b)}$ is a
supersolution. If $u$ satisfies $(\ref{UNX'1})$ then, as $x\to 0$,

$u_{\ell}^{(b)}(x)=\Lambda\ell^{b-a}|x|^{-a}+\Lambda^{\prime}\ell^{b-a^{\prime}}|x|^{-a}|x|^{-a^{\prime}}(1+o(1))$

Since $b\geq a>a^{\prime}$, $u_{\ell}^{(b)}$ is larger than another solution
$\tilde{u}$ near $0$. Thus $u_{\ell}^{(b)}+\epsilon\geq\tilde{u}$ for any
$\epsilon>0$, which implies the claim.
If $u$ satisfies $(\ref{UNX''1})$, then

$\begin{array}[]{lll}u_{\ell}^{(b)}(x)=\Lambda\ell^{b-a}|x|^{-a}(-\ln|x|-\ln\ell)^{-a^{\prime\prime}}(1+o(1))\\\\[5.69054pt]
\phantom{u_{\ell}^{(b)}(x)}=\Lambda\ell^{b-a}|x|^{-a}(-\ln|x|)^{-a^{\prime\prime}}\left(1+a^{\prime\prime}{\displaystyle\frac{\ln\ell}{-\ln|x|}}\right)(1+o(1)).\end{array}$

Again $u_{\ell}^{(b)}$ is larger than another solution $\tilde{u}$ in a
neighborhood of $0$ and we end the proof as in the first case. $\square$

Remark. The method developed above allows to give uniqueness result for large
solutions under some starshapedness assumption. Let $G\subset\mathbb{R}^{N}$
be a domain with compact boundary and $\rho_{G}(x)=\mbox{\rm
dist}\,(x,\partial G)$, we consider the problem

$\begin{array}[]{lll}-\Delta u+u^{p}-M|\nabla u|^{q}=0\quad\text{in }G\\\
\phantom{---,;;}\displaystyle\\!\lim_{\rho_{G}(x)\to
0}u(x)=\infty.\end{array}$ (4.14 )

Such a solution, if it exists is called a large solution.

###### Theorem 4.5

Assume $N\geq 1$, $M>0$ and $p,q>1$ and $\Omega$ is a bounded domain
starshaped with respect to $0$. There exists at most one positive function
satisfying $(\ref{UNX4})$ in one of the following case:

1- $\frac{2p}{p+1}\leq q<2$ and $G=\Omega$.

2- $1<q\leq\frac{2p}{p+1}$ and $G=\overline{\Omega}^{c}$.

Proof. Let $u$ and $\tilde{u}$ be two positive solutions of $(\ref{UNX4})$. In
the first case with $G=\Omega$. Then for $\alpha\leq b\leq\beta$ and $\ell>1$,
$u^{b}_{\ell}$ is a supersolution of $(\ref{UNX4})$ in
$\Omega_{\ell}:=\ell^{-1}\Omega$. Since
$\overline{\Omega}_{\ell}\subset\Omega$, $u^{b}_{\ell}\geq\tilde{u}$, it
follows that $u\geq\tilde{u}$.
In the second case with $G=\overline{\Omega}^{c}$, then for $0<\ell<1$ and
$\beta\leq b\leq\alpha$. Then $u^{b}_{\ell}$ is a supersolution in
$\ell^{-1}\overline{\Omega}^{c}\subset\overline{\Omega}^{c}$. Then for
$\epsilon>0$, $u^{b}_{\ell}+\epsilon\geq\tilde{u}$ in
$\ell^{-1}\overline{\Omega}^{c}$. Letting $\epsilon\to 0$ and $\ell\uparrow 1$
yields $u\geq\tilde{u}$. This ends the proof. $\square$

If we combine the results of existence of radial singular solutions in
$\mathbb{R}^{N}\setminus\\{0\\}$ with the uniqueness results of Theorem 1.15
and Theorem 4.4 we have the following:

###### Theorem 4.6

Assume $N\geq 3$, $p,q>1$ and $M>0$. There exists one and only one positive
solution $u$ of $(\ref{Z1})$ in $\mathbb{R}^{N}\setminus\\{0\\}$, if one of
the following conditions holds:

1- $1<p<\tfrac{N}{N-2}$, $q=\frac{2p}{p+1}$, $M>0$ and $u$ satisfies
$(\ref{Z8})$-(i) for some $k>0$.
2- $\frac{2p}{p+1}<q<p$ and $\displaystyle\lim_{x\to
0}|x|^{\gamma}u(x)=X_{{}_{M}}$.
3- $1<p<\tfrac{N}{N-2}$, $1<q<\frac{2p}{p+1}$, $M>0$ and either
$\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{0}$, or $u$ satisfies
$(\ref{Z8})$-(i) for some $k>0$.
4- $p=\frac{N}{N-2}$, $q=\frac{2p}{p+1}$, $M>0$ and $u$ satisfies
$(\ref{Z11})$-(i).

Furthermore, existence and uniqueness of a solution holds if the equation
$(\ref{Z1})$ is considered in $\Omega\setminus\\{0\\}$ where $\Omega$ is a
bounded smooth domain starshaped with respect to $0$ and is the function $u$
is equal to some $\phi$ on $\partial\Omega$ where $\phi\in
C^{1}(\partial\Omega)$ is nonnegative.

Proof. By applying Theorem 1.14 and Theorem 1.15 the proof is reduced to use
results of existence of radial positive singular solutions in
$\mathbb{R}^{N}\setminus\\{0\\}$ and to check that the parameters fulfill the
conditions of Theorem 1.15.
Case 1- If $q=\tfrac{2p}{p+1}$, $\alpha=\beta$ and $N-2<\beta$. Existence of
radial positive solutions satisfying $(\ref{Z8})$-(i) is proved in Theorem
1.3.
Case 2- Then $\gamma>\beta$. Existence of a radial positive solution
satisfying $\displaystyle\lim_{x\to 0}|x|^{\gamma}u(x)=X_{{}_{M}}$ is proved
in Theorem 1.11-1.
Case 3- If $1<q<\frac{2p}{p+1}$, then $\alpha<\beta$, and since
$p<\frac{N}{N-2}$, $N-2<\alpha$. Existence of a a radial positive solution
satisfying $\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{0}$ is proved in
Theorem 1.13-1. Furthermore the assumptions on $p$ and $q$ imply that
$q<\frac{N}{N-1}$. The existence of a positive solution satisfying
$(\ref{Z8})$-(i) for any $k>0$ is proved in Theorem 3.12.
Case 4- When $p=\frac{N}{N-2}$, $q=\frac{2p}{p+1}$ there exists a radial
global solution satisfying $(\ref{Z11})$-(i) by Theorem 1.4. We apply estimate
$(\ref{UNX''1})$ in Theorem 4.4 with $a=N-2=\alpha=\beta$ and
$a^{\prime\prime}=N-1$. The result follows. $\square$

Remark. In the case $p=\frac{N}{N-2}$, $q=\frac{N}{N-1}$ we conjecture that
the function $u_{x_{M}}$ is the only positive solution of $(\ref{Z1})$ defined
in $\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $\displaystyle\lim_{x\to
0}|x|^{N-2}u(x)=x_{{}_{M}}$.

###  4.2 Characterization of singular solutions

In this section we give some results showing how the characterization of
singularities of radial solutions can be extended to nonradial solutions. An
important tool for studying positive isolated singularities is Harnack
inequality.

###### Proposition 4.7

Assume $M>0$, $p>1$ and $1<q\leq\frac{2p}{p+1}$. If $u$ is a positive solution
of $(\ref{Z1})$ in $B_{R_{0}}\setminus\\{0\\}$, there exists
$c_{9}=c_{9}(N,p,q,R_{0},M)>0$ such that for any $R\in(0,\frac{R_{0}}{2}]$
there holds

$\begin{array}[]{lll}\displaystyle\max_{|x|=R}u(x)\leq
c_{9}\displaystyle\min_{|x|=R}u(x).\end{array}$ (4.15 )

Proof. We write $(\ref{Z1})$ under the form

$\begin{array}[]{lll}-\Delta u+C(x)u+V(x)|\nabla u|=0,\end{array}$ (4.16 )

where $C(x)=|u(x)|^{p-1}$ and $V(x)=M|\nabla u(x)|^{q-1}$. By Proposition A.1,

$C(x)\leq
c_{1}^{p-1}\max\left\\{M^{\frac{p-1}{p-q}}|x|^{-\frac{q(p-1)}{p-q}},|x|^{-2}\right\\}$

and

$V(x)\leq
c_{4}^{q-1}\max\left\\{M^{\frac{q-1}{p-q}}|x|^{-\frac{p(q-1)}{p-q}},|x|^{-\frac{(q-1)(p+1)}{(p-1)}}\right\\}$

Under the assumptions $1<q\leq\frac{2p}{p+1}$, the terms $|x|^{2}C(x)$ and
$|x|V(x)$ are uniformly bounded in $B_{R_{0}}\setminus\\{0\\}$. The result
follows by [13, Chapter 8]. $\square$

###  4.3 The case $1<p<\frac{N}{N-2}$ and $1<q<\frac{N}{N-1}$

In this section, the results are obtained by a combination of Theorem 3.12 for
existence of solutions and Theorem 4.6 for their uniqueness.

###### Theorem 4.8

Let $N\geq 3$, $M>0$, $1<p<\frac{N}{N-2}$, $1<q<\frac{N}{N-1}$ and $\Omega$ be
a bounded domain containing $0$. For any $k>0$ there exists a unique positive
function $u:=u_{k}$ solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$,
vanishing on $\partial\Omega$ and satisfying

$\displaystyle\lim_{x\to 0}|x|^{N-2}u(x)=k.$ (4.17 )

Furthermore $k\mapsto u_{k}$ is increasing by the maximum principle and
converges to a solution $u_{\infty}$ of $(\ref{Z1})$ in
$\Omega\setminus\\{0\\}$, vanishing on $\partial\Omega$ and satisfying

$\begin{array}[]{lll}\displaystyle\lim_{x\to
0}|x|^{\gamma}u(x)=X_{M}\qquad\text{if
}\;\frac{2p}{p+1}<q<\frac{N}{N-1},\end{array}$ (4.18 )

where $X_{{}_{M}}$ is defined at $(\ref{Z19})$, or

$\displaystyle\lim_{x\to
0}|x|^{\alpha}u(x)=\left\\{\begin{array}[]{lll}x_{0}&\text{if
}\;1<q<\frac{2p}{p+1}\\\ x_{{}_{M}}&\text{if
}\;q=\frac{2p}{p+1},\end{array}\right.$ (4.19 )

where $x_{0}$, $x_{{}_{M}}$ are the unique positive root of equation
$(\ref{Z3})$ with $M=0$ and $M>0$ respectively.

Proof. Let $0<R_{1}<R_{2}$ be such that
$\overline{B}_{R_{1}}\subset\Omega\subset\overline{\Omega}\subset B_{R_{2}}$.
By Theorem 1.13-1 and Theorem 4.6, for $k>0$ there exists a unique solution
$u_{1,k}$ (resp. $u_{2,k}$) of $(\ref{Z1})$ in $B_{R_{1}}\setminus\\{0\\}$
(resp. $B_{R_{2}}\setminus\\{0\\}$) satisfying $(\ref{Z8})$-(i) and vanishing
on $\partial B_{R_{1}}$ (resp. $\partial B_{R_{2}}$). If we extend $u_{1,k}$
by $0$ in $B^{c}_{R_{1}}$, we have,

$u_{1,k}(|x|)\leq u_{k}(x)\leq u_{2,k}(|x|)\quad\text{in
}\,\Omega\setminus\\{0\\}.$

All the above functions are locally bounded in
$\overline{\Omega}\setminus\\{0\\}$ and $\overline{B}_{R_{2}}\setminus\\{0\\}$
by Proposition A.1. Since the mappings $k\mapsto u_{k}$ and $k\mapsto u_{j,k}$
are increasing, we have, by letting $k\to\infty$,

$u_{1,\infty}(|x|)\leq u_{\infty}(x)\leq u_{2,\infty}(|x|)\quad\text{in
}\,\Omega\setminus\\{0\\}.$

Then we obtain $(\ref{K3})$ by Theorem 1.7-(1) and $(\ref{K4})$ by Theorem
1.8-(1) and Theorem 1.3. $\square$

The main characterization of isolated singularities is the next result.

###### Theorem 4.9

Let $N\geq 3$, $\Omega$ be an open subset containing $0$, $M>0$,
$1<p<\frac{N}{N-2}$ and $1<q\leq\frac{2p}{p+1}$. If $u$ is a positive solution
of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$, then either its behaviour at
$x=0$ is given by $(\ref{K3})$ or $(\ref{K4})$, or there exists $k\geq 0$ such
that $(\ref{K5})$ holds. If $k=0$ the singularity at $0$ is removable.

The proof needs a few intermediate steps.

###### Lemma 4.10

Let $N\geq 3$, $\Omega$ be an open subset containing $0$, $M>0$,
$1<p<\frac{N}{N-2}$ and $1<q<\frac{N}{N-1}$. If $u$ is a positive solution of
$(\ref{Z1})$ in $\Omega\setminus\\{0\\}$ vanishing on $\partial\Omega$ and
such that

$\displaystyle\limsup_{x\to 0}|x|^{N-2}u(x)<\infty.$

Then there exists $k\geq 0$ such that $(\ref{K5})$ holds. If $k=0$, then $u$
coincides in $\Omega\setminus\\{0\\}$ with a $C^{2}(\Omega)$ solution of
$(\ref{Z1})$ in $\Omega$.

Proof. By assumption $u(x)\leq c|x|^{2-N}$ and by [4, Lemma 3.10] we have the
following: if $u$ is a solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$
(not necessarily positive) such that $|x|^{m}|u(x)|$ is bounded near $x=0$ for
some $m<\inf\\{\alpha,\frac{2-q}{q-1}\\}$, then $|x|^{m+1}|\nabla u(x)|$ is
also bounded near $x=0$. Actually, in the reference the result is proved for a
more general operator, without the absorption $u^{p}$, but the adaptation is
straightforward. The result applies there with $m=N-2$ and in particular
$|\nabla u(x)|\leq c^{\prime}|x|^{1-N}$. We write $(\ref{Z1})$ under the form
$(\ref{K2})$. Since $p<\frac{N}{N-2}$ and $q<\frac{N}{N-1}$ we have

$|C(x)|\leq c|x|^{(2-N)(p-1)}\leq c|x|^{-2+\epsilon_{1}}\,\text{ and
}\;|V(x)|\leq c|x|^{(1-N)(q-1)}\leq c|x|^{-1+\epsilon_{2}},$

for some $\epsilon_{1},\epsilon_{2}>0$. It follows by Serrin’s result that
either the singularity at $0$ is removable, or there exist $c_{1}>c_{2}>0$
such that

$c_{2}|x|^{2-N}\leq u(x)\leq c_{1}|x|^{2-N}\quad\text{for all }0<|x|\leq 1.$

In order to make the convergence precise, we denote by $u_{2}$ the solution of

$\begin{array}[]{lll}-\Delta u_{2}=M|\nabla u|^{q}-u^{p}&\text{in }B_{1}\\\
\phantom{-\Delta}u_{2}=0&\text{on }\partial B_{1}.\end{array}$

Then $-v_{2}\leq u_{2}\leq v^{\prime}_{2}$ where

$\begin{array}[]{lll}-\Delta v^{\prime}_{2}=Mc^{\prime q}|x|^{(1-N)q}&\text{in
}B_{1}\\\ \phantom{-\Delta}v^{\prime}_{2}=0&\text{on }\partial
B_{1},\end{array}$

and

$\begin{array}[]{lll}-\Delta v_{2}=c^{p}|x|^{(2-N)p}&\text{in }B_{1}\\\
\phantom{-\Delta}v_{2}=0&\text{on }\partial B_{1},\end{array}$

Because $(N-1)q<N$ and $(N-2)p<N$, $v_{2}$ and $v^{\prime}_{2}$ satisfy

$0\leq v_{2}(r)\leq c_{10}r^{2-N+\delta}\,\text{ and }\;0\leq
v^{\prime}_{2}(r)\leq c_{10}r^{2-N+\delta^{\prime}},$

for some $c_{10}>0$ and $\delta=N-(N-2)p>0$, $\delta^{\prime}=N-(N-1)q>0$.
Then $u_{2}$ satisfies

$\displaystyle\lim_{x\to 0}|x|^{N-2}u_{2}(x)=0.$

The function $u_{1}=u-u_{2}$ is harmonic in $B_{1}\setminus\\{0\\}$ and is
bounded from below by $-v^{\prime}_{2}$ which satisfies
$\displaystyle\lim_{x\to 0}|x|^{N-2}v^{\prime}_{2}(x)=0$. Hence by standard
result on singularities of harmonic functions, $|x|^{N-2}u_{1}(x)$ admits a
limit $k\geq 0$ when $x\to 0$. Combined with Serrin’s estimates it follows
that either $k=0$ and the singularity is removable, or $k>0$. Note that if
$k=0$, then $u$ is a $C^{2}$ solution in $\Omega$.
Another proof based on a perturbation is the following: let
$u(x)=u(r,s)=|x|^{2-N}\phi(t,s)$ with $r=|x|$ and $t=\ln r$. Then

$\begin{array}[]{lll}\phi_{tt}+(N-2)\phi_{t}-e^{(N-p(N-2))t}\phi^{p}+\Delta^{\prime}\phi\\\\[5.69054pt]
\phantom{---------}+Me^{(N-q(N-1))t}\left(((N-2)\phi-\phi_{t})^{2}+|\nabla^{\prime}\phi|^{2}\right)^{\frac{q}{2}}=0\end{array}$

Since $u=O(|x|^{2-N})$, we can write

$\phi_{tt}+(N-2)\phi_{t}+\Delta^{\prime}\phi=-e^{at}\psi,$

where $a=\min\\{(N-p(N-2),(N-q(N-1)\\}>0$ and $\psi$ is bounded. Then the
result follows by [10, Proposition 4.1]. $\square$

We give below another application of the perturbation method and specific to
the case $q<\frac{2p}{p+1}$.

###### Proposition 4.11

Assume $\Omega$ is an open subset containig $0$, $M>0$, $1<p<\frac{N}{N-2}$
and $1<q<\frac{2p}{p+1}$. If $u$ is a solution of

$-\Delta u+|u|^{p-1}u-M|\nabla u|^{q}=0$ (4.20 )

not necessarily nonnegative in $\Omega\setminus\\{0\\}$, then
$r^{\alpha}u(r,s)$ converges uniformly with respect to $s\in S^{N-1}$ when
$r\to 0$ to a non-empty compact and connected subset of the set of solutions
$\omega$ of

$-\Delta^{\prime}\omega+\alpha(N-2-\alpha)\omega+|\omega|^{p-1}\omega=0\qquad\text{on
}\,S^{N-1}.$ (4.21 )

If $u\geq 0$, $\omega$ is either $x_{0}$ or $0$.

Proof. We can assume that $\overline{B}_{1}\subset\Omega$ and set
$\phi(t,s)=r^{\alpha}u(r,s)$ with $t=\ln r$. Then $\phi$ satisfies

$\begin{array}[]{lll}\phi_{tt}+L\phi_{t}+\alpha
K\phi+\Delta^{\prime}\phi-|\phi|^{p-1}\phi+Me^{\frac{\sigma
t}{p-1}}\left((\alpha\phi-\phi_{t})^{2}+|\nabla^{\prime}\phi|^{2}\right)^{\frac{q}{2}}=0.\end{array}$
(4.22 )

in $\mathbb{R}_{-}\times S^{N-1}$ where $K=N-2-\alpha$ and $L=K-\alpha$. By
assumption $\sigma<0$, hence $(\ref{K7})$ is an exponentially small
perturbation of the autonomous equation associated to the Emden-Fowler
equation by the same change of variables. The result follows from [10, Theorem
4.1] but for the sake of comprehension, we recall its proof. By Proposition
A.2 the function $\phi$ is uniformly bounded, and by $(\ref{Y3''})$ $\phi_{t}$
and $\nabla^{\prime}\phi$ are also uniformly bounded. By standard local
regularity theory for elliptic equations, there holds

$\left\|\partial_{t^{i}}\nabla^{\prime j}\phi\right\|_{C[T-1,T+1]\times
S^{N-1}}\leq c_{11}\quad\text{for all
}(i,j)\in\mathbb{N}\times\mathbb{N},i+j\leq 3\quad\text{and }T\leq-2,$

where $\nabla^{\prime j}$ is the covariant derivative of order $j$ on
$S^{N-1}$. Then the omega-limit at $-\infty$ of the trajectory
$\\{\phi(t,.)\\}_{t\in\mathbb{R}_{-}}$ in $C^{2}(S^{N-1})$ is a non-empty
compact connected denoted by $\Gamma_{\phi}\subset C^{2}(S^{N-1})$. From
$(\ref{K7})$ we have that

${\displaystyle\frac{d}{dt}}E=L{\displaystyle\int_{S^{N-1}}}\phi^{2}_{t}dS(s)+Me^{\frac{\sigma
t}{p-1}}{\displaystyle\int_{S^{N-1}}}\left((\alpha\phi-\phi_{t})^{2}+|\nabla^{\prime}\phi|^{2}\right)^{\frac{q}{2}}\phi_{t}dS(s),$

where

$J(t)={\displaystyle\frac{1}{2}}{\displaystyle\int_{S^{N-1}}}\left(|\nabla^{\prime}\phi|^{2}+{\displaystyle\frac{2}{p+1}}|\phi|^{p+1}-\phi^{2}_{t}\right)dS(s).$

Because $L\neq 0$ and $J$ is uniformly bounded, there holds

${\displaystyle\int_{-\infty}^{1}}{\displaystyle\int_{S^{N-1}}}\phi^{2}_{t}dS(s)dt<\infty.$

Multiplying $(\ref{K7})$ by $w_{tt}$ and using the previous estimate, we
obtain

${\displaystyle\int_{-\infty}^{1}}{\displaystyle\int_{S^{N-1}}}\phi^{2}_{tt}dS(s)dt<\infty.$

As $\phi_{t}$ and $\phi_{tt}$ are uniformly continuous in $(-\infty,1]\times
S^{N-1}$, this implies that

$\displaystyle\lim_{t\to-\infty}{\displaystyle\int_{S^{N-1}}}\left(\phi^{2}_{t}(t,.)+\phi^{2}_{tt}(t,.)\right)dS(s)=0.$

Therefore $\Gamma_{\phi}\subset\left\\{\omega\in C^{2}(S^{N-1}):\alpha
K\omega+\Delta^{\prime}\omega-|\omega|^{p-1}\omega=0\text{ in
}S^{N-1}\right\\}$. If $u\geq 0$, then $\omega\geq 0$ for any
$\omega\in\Gamma_{\phi}$ and the result follows by the maximum principle.
$\square$

###### Lemma 4.12

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded open subset
containing $0$, $M>0$, $1<p<\frac{N}{N-2}$ and $1<q<\frac{2p}{p+1}$. If $u$ is
a nonnegative solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$ such that

$\displaystyle\limsup_{x\to 0}|x|^{N-2}u(x)=\infty,$ (4.23 )

then

$\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{0}.$ (4.24 )

Proof. Without loss of generality we can assume that
$\overline{B}_{1}\subset\Omega$. It follows from Proposition 4.7 that
$\displaystyle\liminf_{x\to 0}|x|^{N-2}u(x)=\infty,$ Hence there exists a
decreasing sequence $\\{r_{n}\\}$ converging to $0$ such that

$\displaystyle\alpha_{n}:=\inf_{|x|=r_{n}}|x|^{N-2}u(x)\uparrow\infty\quad\text{as
}n\to\infty.$

Let $v_{n}$ be the solution of

$\begin{array}[]{lll}-\Delta v+v^{p}=c_{N}\alpha_{n}\delta_{0}&\text{in
}{\mathcal{D}}^{\prime}(B_{1})\\\ \phantom{-\Delta+v^{p}}v=0&\text{on
}\partial B_{1},\\\ \end{array}$ (4.25 )

where $c_{N}$ is the explicit constant such that
$\mathbb{G}[c_{N}\delta_{0}](x)=|x|^{2-N}$, where $\mathbb{G}$ is the
Newtonian potential in $\mathbb{R}^{N}$ ; note that $v_{n}$ is radial because
of uniqueness. Then the sequence $\\{v_{n}\\}$ is increasing and converges to
the function $v_{\infty}$ which satisfies (see [22])

$\begin{array}[]{lll}\phantom{i}-\Delta v+v^{p}=0&\text{in
}B_{1}\setminus\\{0\\}\\\ \phantom{-\Delta i+iv^{p}i}v=0&\text{on }\partial
B_{1}\\\ \displaystyle\lim_{x\to 0}|x|^{\alpha}v(x)=x_{0}.\end{array}$ (4.26 )

Moreover, $v_{n}(x)\leq\alpha_{n}|x|^{2-N}$ and since $v_{n}$ is a subsolution
of $(\ref{Z1})$, we have that $v_{n}\leq u$ in $\overline{B}_{1}\cap
B^{c}_{r_{n}}$. Letting $n\to\infty$ implies that $v_{\infty}(x)\leq u(x)$ in
$B_{1}\setminus\\{0\\}$. Therefore

$\displaystyle\liminf_{x\to 0}|x|^{\alpha}u(x)\geq x_{0}.$

Combined with Proposition 4.11, this inequality implies $(\ref{K7})$.
$\square$

This result admits an extension to the case $q=\frac{2p}{p+1}$.

###### Lemma 4.13

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded open subset
containing $0$, $M>0$, $1<p<\frac{N}{N-2}$ and $q=\frac{2p}{p+1}$. If $u$ is a
nonnegative solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$ such that

$\displaystyle\limsup_{x\to 0}|x|^{N-2}u(x)=\infty,$ (4.27 )

then

$\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{{}_{M}}.$ (4.28 )

Proof. Assuming for simplicity that $\overline{B}_{1}\subset\Omega$ and using
Proposition 4.7 we obtain that for some decreasing sequence $\\{r_{n}\\}$
converging to $0$ we have

$\displaystyle\alpha_{n}:=\inf_{|x|=r_{n}}|x|^{N-2}u(x)\uparrow\infty\quad\text{as
}n\to\infty.$

Therefore, $u$ is bounded from below in $B_{1}\setminus B_{r_{n}}$ by the
(radial) solution $u_{n}$ of

$\begin{array}[]{lll}-\Delta u+u^{p}-M|\nabla u|^{q}=0&\text{in
}B_{1}\setminus\overline{B}_{r_{n}}\\\ \phantom{-\Delta+u^{p}-M|\nabla
u|^{q}}v=0&\text{on }\partial B_{1}\\\ \phantom{-\Delta+u^{p}-M|\nabla
u|^{q}}u=\alpha_{n}r_{n}^{2-N}&\text{on }\partial B_{r_{n}}.\end{array}$ (4.29
)

The sequence $\\{u_{n}\\}$ may not be monotone, but $u_{n}\geq v_{n}$ where
$v_{n}$ has been defined in $(\ref{K10})$. Since $\\{u_{n}\\}$ is eventually
locally bounded in $\overline{B}_{1}\setminus\\{0\\}$ by Proposition A.1 and
standard regularity results (see e.g. [13]), up to a subsquence, it converges
locally uniformly in $\overline{B}_{1}\setminus\\{0\\}$ to a radial function
$u_{\infty}$ which satisfies

$\begin{array}[]{lll}-\Delta u+u^{p}-M|\nabla u|^{q}=0&\text{in
}B_{1}\setminus\\{0\\}\\\ \phantom{-\Delta+u^{p}-M|\nabla u|^{q}}v=0&\text{on
}\partial B_{1},\end{array}$ (4.30 )

and

$\begin{array}[]{lll}\displaystyle\liminf_{x\to
0}|x|^{\alpha}u_{\infty}(x)\geq\liminf_{x\to
0}|x|^{\alpha}v_{\infty}(x)=x_{0}.\end{array}$ (4.31 )

By Theorem 2.4-(4) we have that

$\begin{array}[]{lll}\displaystyle\liminf_{x\to
0}|x|^{\alpha}u(x)\geq\displaystyle\lim_{x\to
0}|x|^{\alpha}u_{\infty}(x)=x_{{}_{M}}.\end{array}$ (4.32 )

The upper estimate is obtained as follows. By Proposition 4.7 the function $u$
is bounded from above in $B_{1}\setminus B_{r_{n}}$ by $\displaystyle
u^{\prime}_{n}+\max_{|y|=1}u(y)$ where $u^{\prime}_{n}$ is the solution of

$\begin{array}[]{lll}-\Delta u+u^{p}-M|\nabla u|^{q}=0&\text{in
}B_{1}\setminus\overline{B}_{r_{n}}\\\ \phantom{-\Delta+u^{p}-M|\nabla
u|^{q}}v=0&\text{on }\partial B_{1}\\\ \phantom{-\Delta+u^{p}-M|\nabla
u|^{q}}u=c_{9}\alpha_{n}r_{n}^{2-N}&\text{on }\partial B_{r_{n}}.\end{array}$
(4.33 )

For the same reason as above there exists a subsequence,
$\\{u^{\prime}_{n_{\ell}}\\}$ which converges locally uniformly in
$\overline{B}_{1}\setminus\\{0\\}$ to a radial solution $u^{\prime}_{\infty}$
of $(\ref{K15bis})$. By Theorem 2.4-(4) we have that
$u^{\prime}_{\infty}=u_{\infty}$. Then $\displaystyle\limsup_{x\to
0}|x|^{\alpha}u(x)\leq\displaystyle\lim_{x\to
0}|x|^{\alpha}u^{\prime}_{\infty}(x)=x_{{}_{M},}$ which ends the proof.
$\square$

Proof of Theorem 4.9 It is a direct consequence of Lemma 4.10, Lemma 4.12 and
Lemma 4.13. $\square$

When $q>\frac{2p}{p+1}$ and $|x|^{N-2}u(x)$ is not bounded, Harnack inequality
may not hold. However we still have a dichotomy for the possible behaviour
which extends Theorem 4.9.

###### Theorem 4.14

Let $\Omega\subset\mathbb{R}^{N}$ ($N\geq 3$) be an open set containing $0$,
$M>0$, $1<p<\frac{N}{N-2}$ and $\frac{2p}{p+1}<q<\frac{N}{N-1}$. If $u$ is a
nonnegative solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$, then the
following dichotomy holds:
1- either there exists $k\geq 0$ such that $r^{N-2}u(r,.)$ converges to $k$ in
measure on $S^{N-1}$ as $r\to 0$,
2- or $\displaystyle\lim_{x\to 0}|x|^{\gamma}u(x)=X_{{}_{M}}$.

Proof. We recall Richard-Véron’s isotropy theorem [20, Theorem 1.1]:
Let $g:\mathbb{R}_{+}\mapsto\mathbb{R}_{+}$ be a continuous nondecreasing
function satisfying

$\begin{array}[]{lll}{\displaystyle\int_{0}^{1}}g(r^{2-N})r^{N-1}dr<\infty.\end{array}$
(4.34 )

If $u\in C^{1}(\Omega\setminus\\{0\\})$ is a nonnegative function satisfying

$\begin{array}[]{lll}\Delta u\leq g(u)+f\quad\text{in
}\,\Omega\setminus\\{0\\},\end{array}$ (4.35 )

where $f\in L^{1}_{loc}(\Omega\setminus\\{0\\})$ is a nonnegative radial
function, then we have the following:

(i) either $r^{N-2}u(r,.)$ converges in measure on $S^{N-1}$ to some $k\geq 0$
as $r\to 0$,
(ii) or

$\begin{array}[]{lll}\displaystyle\lim_{x\to
0}|x|^{N-2}u(x)=\infty.\end{array}$ (4.36 )

Since $p<\frac{N}{N-2}$, assumption $(\ref{K15})$ is satisfied with
$g(r)=r^{p}$ and equation $(\ref{K16bis})$ with $f=0$. Then either the
statement 1 holds, or $(\ref{K17bis})$ holds. If it is the case, then for any
$k>0$, $u$ is bounded from below in $B_{1}\setminus\\{0\\}$ by the solution
$u_{k}$ of $(\ref{Z1})$ in $B_{1}\setminus\\{0\\}$ vanishing on $\partial
B_{1}$ and satisfying $(\ref{K5})$. Such a solution exists and is unique by
Theorem 4.8. Letting $k\to\infty$, this implies

$\begin{array}[]{lll}\displaystyle\liminf_{x\to 0}|x|^{\gamma}u(x)\geq
X_{{}_{M}}.\end{array}$ (4.37 )

Next, we denote by $\upsilon$ the solution of $(\ref{Z1})$ on $(0,\infty)$
(hence $\upsilon$ is 1-dimensional) satisfying

$\displaystyle\lim_{r\to 0}r^{\gamma}\upsilon(r)=X_{{}_{M}}\quad\text{and
}\,\lim_{r\to\infty}r^{\alpha}\upsilon(r)=x_{0}.$

Its existence is proved in Theorem 1.10, Theorem 1.11. It is decreasing. For
$\epsilon>0$ the function
$r\mapsto\upsilon_{\epsilon}(x)=\upsilon(|x|-\epsilon)$ satisfies
${\mathcal{L}}^{M}_{p,q}(\upsilon_{\epsilon})\geq 0$ in
$\overline{B}_{\epsilon}^{c}$ and $\upsilon_{\epsilon}(x)\to\infty$ when
$|x|\downarrow\epsilon$. If $c=\max u\lfloor_{\partial B_{\delta}}$ for some
$\delta>\epsilon>0$ such that $\overline{B}_{\delta}\subset\Omega$, then
$\upsilon_{\epsilon}+c$ is a supersolution of $(\ref{Z1})$ in
$B_{\delta}\setminus\overline{B}_{\epsilon}$ which is larger than $u$ for
$|x|=\epsilon$ and $|x|=\delta$. Hence $u\leq gu_{\epsilon}+c$ in
$B_{\delta}\setminus\overline{B}_{\epsilon}$. Letting $\epsilon\to 0$ yields
$u(x)\leq\upsilon(x)+c$ for $0<|x|\leq\delta$ and finally

$\begin{array}[]{lll}\displaystyle\limsup_{x\to 0}|x|^{\gamma}u(x)\leq
X_{{}_{M}}.\end{array}$ (4.38 )

Combining $(\ref{K18})$ and $(\ref{K19})$ we obtain $\displaystyle\lim_{x\to
0}|x|^{\gamma}u(x)=X_{{}_{M}}$. $\square$

Remark. We conjecture that the convergence in Theorem 4.14-1 holds in the
strong sense.

Remark. Most of the results of this section can be extended to the case $N=2$.
The subcritical case corresponds then to $p>1$ and $1<q<2$. The main change is
that $|x|^{2-N}$ has to be replaced by $-\ln|x|$.

###  4.4 The case $p\geq\frac{N}{N-2}$ and $q=\frac{2p}{p+1}$

The cases that we consider are $q=\frac{2p}{p+1}$, $p\geq\frac{N}{N-2}$ and
$M>0$. We recall that the stationary equation $(\ref{Z3})$ admits two positive
constant solutions $x_{{}_{1,M}}<x_{{}_{2,M}}$ if $p>\frac{N}{N-2}$ and
$M>m^{*}$, and only one denoted by $x_{{}_{M}}$ if $p=\frac{N}{N-2}$ and $M>0$
or if $p>\frac{N}{N-2}$ and $M=m^{*}$. The following result is an improvement
of Proposition A.1

###### Lemma 4.15

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded domain containing
$0$ such that $\overline{B}_{R}\subset\Omega$ and $p\geq\frac{N}{N-2}$.
1- If $p>\frac{N}{N-2}$ and $M>m^{*}$, then any positive solution $u$ of
$(\ref{Z7})$ in $\Omega\setminus\\{0\\}$ satisfies

$\begin{array}[]{lll}u(x)\leq
x_{{}_{2,M}}|x|^{-\alpha}+\displaystyle\sup_{|z|=R}u(z),\end{array}$ (4.39 )

2- If $p>\frac{N}{N-2}$ and $M=m*$, or $p=\frac{N}{N-2}$ and $M>0$, the same
inequality holds with $x_{{}_{2,M}}$ replaced by $x_{m^{*}}$ and $x_{{}_{M}}$
respectively.

Proof. We assume first that $M>m^{*}$.
1- Construction of the maximal solution. We claim that $x\mapsto
x_{{}_{2,M}}|x|^{-\alpha}$ is the maximal solution of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$. For $a>0$ we set $\phi_{a}(s)=as^{\alpha}$.
Then

$\begin{array}[]{lll}\tilde{L}\phi_{a}(s):=-\phi^{\prime\prime}_{a}+\phi^{p}_{a}-M|\phi^{\prime}_{a}|^{\frac{2p}{p+1}}=as^{-\alpha
p}\left(a^{p-1}-\alpha^{q}Ma^{q-1}-\alpha(\alpha+1)\right).\end{array}$

Taking $a$ large enough we obtain that $\phi_{a}$ is a supersolution in
$(0,\infty)$. We set $\Phi_{a,\epsilon}(x)=\phi_{a}(x_{1}-\epsilon)$ for
$x_{1}>\epsilon>0$ and as in the proof of Theorem 4.14 we deduce that the
function $\tilde{\Phi}_{a,\epsilon}$ defined by

$\tilde{\Phi}_{a,\epsilon}=\inf\left\\{{\mathcal{R}}[\Phi_{a,\epsilon}]:{\mathcal{R}}\in
SO(N)\right\\}\qquad\text{($SO(N)$ is the group of rotations in
$\mathbb{R}^{N}$)},$

is a positive and radial supersolution of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}$ which tends to infinity on
$\partial B_{\epsilon}$. It dominates any positive solution of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}$. Next we set

$\tilde{\Psi}_{\epsilon}(x)=\sup\left\\{x_{{}_{2,M}}|x-z|^{-\alpha}:|z|<\epsilon\right\\}.$

It is a subsolution of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}$ dominated by
$\tilde{\Phi}_{a,\epsilon}$. Since the supremum is achieved for
$z=\epsilon\frac{x}{|x|}$, the function $\tilde{\Psi}_{\epsilon}$ is radial
and positive. By [25, Theorem 1.4.5] there exists a solution $U_{\epsilon}$ of
$(\ref{Z7})$ in $\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}$ such that

$\tilde{\Psi}_{\epsilon}\leq
U_{\epsilon}\leq\tilde{\Phi}_{a,\epsilon}\quad\text{in
}\,\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}.$

The function $U_{\epsilon}$ is positive and radial. Since any positive
solution $u$ in $\mathbb{R}^{N}\setminus\\{0\\}$ is dominated by
$\tilde{\Psi}_{\epsilon}$, the function $U_{\epsilon}$ is larger than $u$ in
$\mathbb{R}^{N}\setminus\overline{B}_{\epsilon}$. This implies the relation,
valid for any $\ell>0$,

$\begin{array}[]{lll}T_{\ell}[U_{\epsilon}](x):=\ell^{\alpha}U_{\epsilon}(\ell
x)=U_{\ell^{-1}\epsilon}.\end{array}$ (4.40 )

When $\epsilon\downarrow 0$ the sequence $\\{U_{\epsilon}\\}$ decreases and
converges to a positive radial solution $U_{0}$ of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ which dominates any other positive solution.
Hence $U_{0}$ is the maximal solution in $\mathbb{R}^{N}\setminus\\{0\\}$.
Letting $\epsilon\downarrow 0$ in $(\ref{K29})$ we infer that
$T_{\ell}[U_{0}]=U_{0}$ for any $\ell>0$. Hence $U_{0}$ is self-similar. Since
it is radial and larger than any other positive solution, we deduce that

$\begin{array}[]{lll}U_{0}(x)=x_{{}_{2,M}}|x|^{-\alpha}\quad\text{for all
}|x|>0.\end{array}$ (4.41 )

2- End of the proof. If $u$ is any positive solution in
$\Omega\setminus\\{0\\}$, then $U_{\epsilon}+\displaystyle\sup_{|z|=R}u(z)$ is
a supersolution larger than $u$ in $B_{R}\setminus B_{\epsilon}$. Letting
$\epsilon\downarrow 0$ yields the result. The proof in the other case is
similar. $\square$

###### Theorem 4.16

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded domain containing
$0$, $p>\frac{N}{N-2}$ and $M>m^{*}$. If $u$ is a positive solution of
$(\ref{Z7})$ in $\Omega\setminus\\{0\\}$, there holds

1- If

$\begin{array}[]{lll}\displaystyle\liminf_{x\to
0}|x|^{\alpha}u(x)=0,\end{array}$ (4.42 )

then $u$ can be extended as a $C^{2}$ solution of $(\ref{Z7})$ in $\Omega$.

2- If

$\begin{array}[]{lll}\displaystyle\limsup_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{2,M}},\end{array}$ (4.43 )

then

$\begin{array}[]{lll}\displaystyle\lim_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{2,M}},\end{array}$ (4.44 )

3- If

$\begin{array}[]{lll}\displaystyle\liminf_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{1,M}}\quad\text{or }\;\limsup_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{1,M}},\end{array}$ (4.45 )

then there exists a sequence $\\{r_{n}\\}\subset\mathbb{R}^{+}_{*}$ converging
to $0$ such that

$\begin{array}[]{lll}\displaystyle\lim_{r_{n}\to
0}r_{n}^{\alpha}u(r_{n},s)=x_{{}_{1,M}}\quad\text{uniformly on
}S^{N-1}.\end{array}$ (4.46 )

4- Any one of the two following situations never occur

$\begin{array}[]{lll}(i)\displaystyle&\displaystyle 0<\limsup_{x\to
0}|x|^{\alpha}u(x)<x_{{}_{1,M}}\\\\[5.69054pt] (ii)\displaystyle&\displaystyle
x_{{}_{1,M}}<\liminf_{x\to 0}|x|^{\alpha}u(x)<x_{{}_{2,M}}.\end{array}$ (4.47
)

Proof. 1- If relation $(\ref{K23})$ holds, it follows by Harnack inequality
proved in Proposition 4.7, that there exists a sequence $\\{r_{n}\\}$
converging to $0$ as $n\to\infty$ such that

$\begin{array}[]{lll}\displaystyle\lim_{r_{n}\to
0}r_{n}^{\alpha}u(r_{n},s)=0\quad\text{uniformly on }S^{N-1}.\end{array}$
(4.48 )

For any $\epsilon>0$ and $m=\sup\\{u(z):|z|=R\\}$, the function
$x\mapsto\epsilon|x|^{\alpha}+m$ is a supersolution of $(\ref{Z7})$ in
$B_{R}\setminus\\{0\\}$ which is larger than $u$ near $x=0$ and on $\partial
B_{R}$. Hence $u(x)\leq\epsilon|x|^{\alpha}+m$. Letting $\epsilon\to 0$
implies $u\leq m$, and the result follows by standard regularity.
2- If $(\ref{K24})$ holds there exists a sequence $\\{r_{n}\\}$ converging to
$0$ such that

$r_{n}^{\alpha}\max\left\\{u(r_{n},s):s\in
S^{N-1}\right\\}=r_{n}^{\alpha}u(r_{n},s_{n})\to x_{{}_{2,M}}\quad\text{as
}r_{n}\to 0.$

Furthermore, we can assume that $s_{n}\to s^{*}$ when $n\to\infty$. Using
Lemma 4.15 there exist a nondecreasing sequence $\\{w_{1,n}\\}$ converging to
$x_{{}_{2,M}}$ and a bounded sequence $\\{w_{2,n}\\}$ such that
$r_{n}^{\alpha}u(r_{n},s_{n})=w_{1,n}+r_{n}^{\alpha}w_{2,n}$. We set
$w(t,s)=r^{\alpha}u(r,s)$ with $t=\ln r$, then there holds,

$\begin{array}[]{lll}w_{tt}+Lw_{t}-\alpha
Kw+\Delta^{\prime}w-w^{p}+M\left(\left(w_{t}-\alpha
w\right)^{2}+|\nabla^{\prime}w|^{2}\right)^{\frac{p}{p+1}}=0,\end{array}$
(4.49 )

on $\mathbb{R}_{-}\times S^{N-1}$. By standard regularity estimates and
Ascoli-Arzela theorem there exist a subsequence $\\{t_{n_{j}}\\}$ of
$\\{t_{n}\\}=\\{\ln r_{n}\\}$ and a nonnegative $C^{2}$ function $W$ such that
$w(t_{n}+t,s)$ converges to $W$ in the $C^{2}$ topology of $[-a,a]\times
S^{N-1}$, for any $a>0$, and $W$ is a solution of $(\ref{K33})$ in
$\mathbb{R}\times S^{N-1}$. Furthermore $W(0,s^{*})=x_{{}_{2,M}}$. By Lemma
4.15, $x_{{}_{2,M}}$ is the maximal solution of $(\ref{K33})$ in
$\mathbb{R}\times S^{N-1}$, it then follows from the strong maximum principle
that $W=x_{{}_{2,M}}$ and $w(t_{n}+t,s)\to x_{{}_{2,M}}$ uniformly in
$[-a,a]\times S^{N-1}$. Let $\epsilon>0$, then there exists
$n_{\epsilon}\in\mathbb{N}$ such that for any $n\geq n_{\epsilon}$, we have
that $u(r_{n},s)\geq r_{n}^{-\alpha}(x_{{}_{2,M}}-\epsilon)$ for any $s\in
S^{N-1}$. Since $r\mapsto r^{-\alpha}(x_{{}_{2,M}}-\epsilon)$ is a subsolution
of $(\ref{Z7})$, it follows that for $m>n\geq n_{\epsilon}$, one has

$u(r,s)\geq r^{-\alpha}(x_{{}_{2,M}}-\epsilon)\quad\text{for all
}(r,s)\in[r_{m},r_{n}]\times S^{N-1}.$

Letting $r_{m}\to 0$ yields

$\begin{array}[]{lll}u(x)\geq|x|^{-\alpha}(x_{{}_{2,M}}-\epsilon)\qquad\text{for
all }x\in B_{r_{n_{\epsilon}}}\setminus\\{0\\}.\end{array}$ (4.50 )

Since $\epsilon$ is arbitrary we infer that

$\begin{array}[]{lll}\displaystyle\liminf_{x\to 0}|x|^{\alpha}u(x)\geq
x_{{}_{2,M}}.\end{array}$ (4.51 )

By assumption $\displaystyle\limsup_{x\to 0}|x|^{\alpha}u(x)\geq
x_{{}_{2,M}}$; then $(\ref{K25})$ holds.
3- Let us assume that the first condition in $(\ref{K26})$ holds. If the
function

$\tau\mapsto\displaystyle w(\tau,s_{\tau}):=\min_{s\in S^{N-1}}w(\tau,s)\\}$
(4.52 )

is asymptotically monotone, nonincreasing or nondecreasing, then either
$w(\tau,s_{\tau})\uparrow x_{{}_{1,M}}$ in the first case, or
$w(\tau,s_{\tau})\downarrow x_{{}_{1,M}}$ in the second case. Using again the
uniform $C^{2,\alpha}$ estimate and Ascoli-Arzela theorem we have that there
exists a sequence $\\{\tau_{n}\\}$ converging to $-\infty$ such that
$w(t+\tau_{n},s)$ converges in the $C^{2}$-topology of $[-a,a]\times S^{N-1}$,
for any $a>0$, to a positive solution $W$ of $(\ref{K33})$ in
$\mathbb{R}\times S^{N-1}$ such that $W(t,s)\geq x_{{}_{1,M}}$ and
$\displaystyle\min_{s\in S^{N-1}}W(0,s)=W(0,s^{*})=x_{{}_{1,M}}$ for some
$s^{*}\in S^{N-1}$ in the first case. By the strong maximum principle to $w$
and to $x_{{}_{1,M}}$ which are ordered solutions of $(\ref{K33})$ in
$\mathbb{R}\times S^{N-1}$ we infer that $W\equiv x_{{}_{1,M}}$, hence
$w(t,s)$ converges to $x_{{}_{1,M}}$ uniformly on $S^{N-1}$ when
$t\to-\infty$. In the second case we obtain that the limit function $W$
satisfies $W(t,s)\geq x_{{}_{1,M}}$ and $\displaystyle\min_{s\in
S^{N-1}}W(0,s)=W(0,s^{*})=x_{{}_{1,M}}$. This implies again $W\equiv
x_{{}_{1,M}}$ by the strong maximum principle. Finally we do not suppose that
the function $w(\tau,s_{\tau})$ defined in $(\ref{K35-1})$ is monotone. By the
definition of the liminf, there exist sequences $\\{t_{n}\\}$ tending to
$-\infty$ and $\\{s_{n}\\}\subset S^{N-1}$ such that

$w(t_{n},s_{n})=\inf\\{w(t,s):t\leq t_{n},s\in S^{N-1}\\}\uparrow
x_{{}_{1,M}}\,\text{ as }\,n\to\infty.$

Using again Ascoli-Arzela theorem we deduce that, up to a subsequence
$\\{t_{n_{k}}\\}$, $w(t+t_{n_{k}},s)$ converges in the $C^{2}$-topology of
$[-a,a]\times S^{N-1}$, for any $a>0$, to a positive solution $W$ of
$(\ref{K33})$ in $\mathbb{R}\times S^{N-1}$ and $W\geq x_{{}_{1,M}}$ and
$W(0,s^{*})=x_{{}_{1,M}}$. Hence $W\equiv x_{{}_{1,M}}$.
The proof of $(\ref{K27})$ under the second condition of $(\ref{K26})$ is
similar.

4- Let $(\ref{K31})$-(i) be satisfied and $w$ be defined as in the previous
steps. Then as in Step 3, $w(t_{n},s)$ converges locally uniformly to a
positive solution $W$ of $(\ref{K33})$ defined on $\mathbb{R}_{-}\times
S^{N-1}$, $w(t_{n},s_{n})\to W(0,s^{*})$, $W(0,s^{*})$ is a local maximum of
$W$ and it is smaller than $x_{{}_{1,M}}$. Hence
$W_{t}(0,s^{*})=|\nabla^{\prime}W(0,s^{*})|=0$ and $W_{tt}(0,s^{*})\leq 0$ and
$\Delta^{\prime}W(0,s^{*})\leq 0$. Then

$W^{p}(0,s^{*})-M\alpha^{q}W^{\frac{2p}{p+1}}(0,s^{*})+\alpha KW(0,s^{*})\leq
0.$

This contradicts the fact that ${\mathcal{P}}_{{}_{M}}$ defined in
$(\ref{Z5})$ is positive on the interval $(0,x_{{}_{1,M}})$.
Similarly, if $(\ref{K31})$-(ii), we obtain that the limit function $W$ and
the limit point $s^{*}$ where $W(0,s^{*})$ is a local minimum of $W$ satisfies
$x_{{}_{1,M}}<W(0,s^{*})<x_{{}_{2,M}}$ and

$W^{p}(0,s^{*})-M\alpha^{q}W^{\frac{2p}{p+1}}(0,s^{*})+\alpha KW(0,s^{*})\geq
0.$

which is not compatible with the fact that ${\mathcal{P}}_{{}_{M}}$ is
negative on $(x_{{}_{1,M}},x_{{}_{2,M}})$. $\square$

Remark. We conjecture that the following stronger form of Theorem 4.16 holds:

1- Either $u$ can be extended as a $C^{2}$ solution of $(\ref{Z7})$ in
$\Omega$,

2- or $\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{{}_{2,M}}$,

3- or $\displaystyle\lim_{x\to 0}|x|^{\alpha}u(x)=x_{{}_{1,M}}$.

In the case $M=m^{*}$, we prove the following.

###### Theorem 4.17

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded domain containing
$0$, $p>\frac{N}{N-2}$ and $M=m^{*}$. If $u$ is a positive solution of
$(\ref{Z7})$ in $\Omega\setminus\\{0\\}$, there we have the following:

1- either $u$ can be extended as a $C^{2}$ solution in $\Omega$,

2- or there exists a sequence $\\{r_{n}\\}$ converging to $0$ such that
$r_{n}^{\alpha}u(r_{n},s)$ converges to $x_{m^{*}}$ uniformly on $S^{N-1}$.

Proof. If $u$ satisfies $(\ref{K23})$, then the singularity of $u$ at zero is
removable since the function $P_{{}_{M}}$ is positive on $(0,m^{*})$ and on
$(m^{*},\infty)$, see the argument in the proof of Theorem 4.16-(1). Thus we
are left with the case

$\begin{array}[]{lll}\displaystyle\liminf_{x\to
0}|x|^{\alpha}u(x)>0.\end{array}$ (4.53 )

If

$\begin{array}[]{lll}\displaystyle\limsup_{x\to
0}|x|^{\alpha}u(x)=x_{m^{*}},\end{array}$ (4.54 )

then, as in the proof of Theorem 4.16-(3) we deduce that there exists a
sequence $\\{r_{n}\\}$ converging to $0$ such that $r_{n}^{\alpha}u(r_{n},s)$
converges to $x_{m^{*}}$ uniformly on $S^{N-1}$. If there exists
$m\neq{m^{*}}$ such that

$\begin{array}[]{lll}\displaystyle\limsup_{x\to
0}|x|^{\alpha}u(x)=m,\end{array}$ (4.55 )

then there exists a sequence $\\{t_{n}\\}$ converging to $-\infty$ and
$\\{s_{n}\\}\subset S^{N-1}$ such that $w(t+t_{n},.)$ converges in the $C^{2}$
topology of $[-a,a]\times S^{N-1}$ for any $a>$ to a solution $W$ of
$(\ref{K33})$. Furthermore $w_{t}(t_{n},s_{n})\to 0$ and
$\displaystyle\liminf_{t_{n}\to-\infty}w_{tt}(t_{n},s_{n})\leq 0$. Since
$\nabla^{\prime}w(t_{n},s_{n})=0$ and $\Delta^{\prime}w(t_{n},s_{n})\leq 0$,
one has that

$-\alpha Km-m^{p}+M(\alpha
m)^{\frac{2p}{p+1}}=-m\tilde{P}_{m^{*}}(m^{\frac{p-1}{p+1}})=-\Delta^{\prime}W(0,s^{*})-w_{tt}(0,s^{*})\geq
0.$ (4.56 )

Since $\tilde{\mathcal{P}}_{m^{*}}\geq 0$ and vanishes only at $m^{*}$, it
implies $m=m^{*}$, contradiction. The proof of the uniform convergence of
$w(t_{n},.)$ to $m^{*}$ follows from the strong maximum principle since $W$ is
a positive solution $(\ref{K33})$ as inTheorem 4.16. $\square$

Remark. We conjecture that assertion (2) holds under the form

$\lim_{x\to 0}|x|^{\alpha}u(x)=x_{m^{*}}.$

Finally we have the following result dealing with the case $p=\frac{N}{N-2}$
and $M>0$ where there exists a unique and explicit positive constant solution
$x_{{}_{M}}$ to $(\ref{Z3})$.

###### Theorem 4.18

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq 3$, be a bounded domain containing
$0$, $p=\frac{N}{N-2}$ and $M>0$. If $u$ is a positive solution of
$(\ref{Z7})$ in $\Omega\setminus\\{0\\}$ which satisfies

$\begin{array}[]{lll}\displaystyle\limsup_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{M}},\end{array}$ (4.57 )

then

$\begin{array}[]{lll}\displaystyle\lim_{x\to
0}|x|^{\alpha}u(x)=x_{{}_{M}},\end{array}$ (4.58 )

Proof. Since the function ${\mathcal{P}}_{{}_{M}}$ is negative on
$(0,x_{{}_{M}})$ and positive on $(x_{{}_{M}},\infty)$, for any $\epsilon>0$
the function $x\mapsto(x_{{}_{M}}-\epsilon)|x|^{-\alpha}$ is a subsolution of
$(\ref{Z7})$. The proof follows as in the proof of Theorem 4.16-(2). $\square$

Remark. We conjecture that the following dichotomy occurs: if $u$ is a
positive solution of $(\ref{Z7})$ in $\Omega\setminus\\{0\\}$ unbounded near
$0$, then,

1- either $(\ref{K39})$ holds,

2- or $(\ref{Z11})$-(i) holds.

##  5 Behaviour at infinity of non-radial solutions

In this section we present some results dealing with the asymptotic behaviour
of solutions which extend to the non-radial case what has already been proved
in the radial one. The results are more complete if there exists only one
possible behaviour for radial positive solutions; they have to be compared
with what was obtained in [21] when $M=0$.

###  5.1 The case $q=\frac{2p}{p+1}$

###### Theorem 5.1

Let $p>1$, $M>0$ and $u$ be a positive solution of $(\ref{Z7})$ in
$\mathbb{R}^{N}\setminus B_{R}$ with $N\geq 1$.

1- If $N=1,2$ and $p>1$, or $N\geq 3$ and $1<p<\frac{N}{N-2}$, then

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{\alpha}u(x)=x_{{}_{M}}.\end{array}$
(5.1 )

2- If $N\geq 3$, $p=\frac{N}{N-2}$, then

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}\left(\ln|x|\right)^{\frac{N-2}{2}}u(x)=\left({\displaystyle\frac{N-2}{\sqrt{2}}}\right)^{N-2}.\end{array}$
(5.2 )

3- If $N\geq 3$, $p>\frac{N}{N-2}$ and $M<m^{*}$, then there exists $k>0$ such
that

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k.\end{array}$
(5.3 )

Proof. The method of the proof is firstly to construct two positive radial
solutions $u_{j}$, $j=1,2$ of $(\ref{Z7})$ in $B_{R}^{c}$ such that $u_{1}\leq
u\leq u_{2}$, and to use Proposition 1.6. The solution $v$ of
${\mathcal{L}}_{p}v=0$ in $B_{R}^{c}$ with value $\min u\lfloor_{\partial
B_{R}}$ for $|x|=R$ is a subsolution smaller than $u$. For cases 1 and 2, we
can take for supersolution the function $u_{X_{{}_{M}}}+\max
u\lfloor_{\partial B_{R}}$. Therefore there exist two positive radial
solutions $u_{1}$ and $u_{2}$ of $(\ref{Z7})$ in $B_{R}^{c}$ with respective
value $\min u\lfloor_{\partial B_{R}}$ and $\max u\lfloor_{\partial B_{R}}$ on
$\partial B_{R}$ and such that

$\begin{array}[]{lll}v(x)\leq u_{1}(x)\leq u(x)\leq u_{2}(x)\leq
u_{X_{{}_{M}}}(x)+\max u\lfloor_{\partial B_{R}}\quad\text{for }|x|\geq
R,\end{array}$ (5.4 )

by Theorem 2.4 and Theorem 2.5. Since $u_{1}$ and $u_{2}$ satisfy either
$(\ref{K40})$ or $(\ref{K41})$ in cases 1 and 2 respectively, $u$ shares this
behaviour.

In case 3 with $p>\frac{N}{N-2}$ the function $v$ satisfies the same behaviour
$(\ref{K41+})$ up to the constant $c>0$ which is not fixed. By Theorem
2.6-(3), $u_{1}$ and $u_{2}$ satisfy $(\ref{K41+})$ with two different
constants $0<c_{1}\leq c_{2}$. In order to prove that $(\ref{K41+})$ holds for
some $c\in[c_{1},c_{2}]$ we use the method introduced in [21]. We set
$u(r,s)=r^{-\nu}w(t,s)$ with $\nu=N-2$ and $t=\ln r$, then $w$ satisfies

$\begin{array}[]{lll}w_{tt}-\nu
w_{t}+\Delta^{\prime}w+e^{(N-p(N-2))t}\left(\nu^{q}M\left((w_{t}-\nu
w)^{2}+|\nabla^{\prime}w|^{2}\right)^{\frac{q}{2}}-w^{p}\right)=0\end{array}$
(5.5 )

in $[0,\infty)\times S^{N-1}$. Since $w$ and $\left(\nu^{q}M\left((w_{t}-\nu
w)^{2}+|\nabla^{\prime}w|^{2}\right)^{\frac{q}{2}}-w^{p}\right)$ are bounded,
it follows from [10, Proposition 4.1] that there exists $c\geq 0$ such that
$w(t,.)\to c$ uniformly on $S^{N-1}$ when $t\to\infty$. This ends the proof.
$\square$

###### Theorem 5.2

Let $N\geq 3$, $p>\frac{N}{N-2}$ and $M=m^{*}$. If $u$ is a positive solution
of $(\ref{Z7})$ in $\mathbb{R}^{N}\setminus B_{R}$, we have the following
alternative,

1- either there exists a sequence $\\{r_{n}\\}$ converging to infinity such
that

$\begin{array}[]{lll}\displaystyle\lim_{r_{n}\to\infty}r_{n}^{\alpha}u(r_{n},s)=x_{m^{*}},\end{array}$
(5.6 )

uniformly on $S^{N-1}$.

2- or there exists $k>0$ such that

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k.\end{array}$
(5.7 )

Proof. We can assume that $u$ is continuous in $B_{R}^{c}$. By Theorem
2.6-(2), $u$ is bounded from above by the function $u_{R}$ where $u_{R}$ is a
radial soluion of $(\ref{Z7})$ in $\overline{B}_{R}^{c}$ which tends to
$\infty$ when $r\downarrow R$ and satisfies $r^{\alpha}u_{R}(r)\downarrow
x_{m^{*}}$ when $r\to\infty$. Hence

$m:=\limsup_{|x|\to\infty}|x|^{\alpha}u(x)\leq\sup_{|x|\geq
R}|x|^{\alpha}u(x)\leq x_{m^{*}}.$ (5.8 )

We claim that either $m=0$ or $m=x_{m^{*}}$ As in the proof of Theorem 4.17,
there exists a sequence $\\{t_{n}\\}$ tending to $\infty$ and
$\\{s_{n}\\}\subset S^{N-1}$ converging to $s^{*}$ such that
$w_{n}(t,.):=w(t+t_{n},.)$ converges in the $C^{2}$-topology of $[-a,a]\times
S^{N-1}$ for any $a>0$ to a solution $W$ of $(\ref{K33})$ in $\mathbb{R}\times
S^{N-1}$. Furthermore $W$ achieves its maximal value $m$ at $(0,s^{*})$, hence
$W_{t}(0,s^{*})=0$, $\nabla^{\prime}W(0,s^{*})=0$,
$\Delta^{\prime}W(0,s^{*})\leq 0$ and $W_{tt}(0,s^{*})\leq 0$. Therefore

$-m\tilde{\mathcal{P}}_{m^{*}}(m^{\frac{p-1}{p+1}})=-\alpha Km-m^{p}+M(\alpha
m)^{\frac{2p}{p+1}}\geq 0.$

Since $(\ref{K37-1})$ holds this implies that either $m=0$ or $m=x_{m^{*}}$.
If $m=x_{m^{*}}$ it follows by the strong maximum principle, as in the proof
of Theorem 4.16, that $w(t_{n},s)\to x_{m^{*}}$ as $n\to\infty$, uniformly on
$S^{N-1}$.

If $m=0$, then

$\displaystyle\lim_{|x|\to\infty}|x|^{\alpha}u(x)=0.$

For any $a<x_{m^{*}}$ and $\rho>R$ such that $u\lfloor_{\partial B_{\rho}}\leq
a$, we consider the problem

$\begin{array}[]{lll}\displaystyle-
v_{rr}-\frac{N-1}{r}v_{r}+v^{p}-M|v_{r}|^{\frac{2p}{p+1}}=0\quad\text{in
}\,(\rho,\infty)\\\
\phantom{--;,\frac{N-1}{r}v_{r}+v^{p}-M|v_{r}|^{\frac{2p}{p+1}}}v(\rho)=\rho^{-\alpha}a\end{array}$
(5.9 )

Since the solution of

$\begin{array}[]{lll}\displaystyle-\upsilon_{rr}-\frac{N-1}{r}\upsilon_{r}+\upsilon^{p}=0\quad\text{in
}\,(\rho,\infty)\\\
\phantom{--;,\frac{N-1}{r}\upsilon_{r}+\upsilon^{p}}\upsilon(\rho)=\rho^{-\alpha}a\end{array}$
(5.10 )

is a subsolution and $x_{m^{*}}|x|^{-\alpha}$ a supersolution, the solution
$v$ exists and it is unique. By the phase plane analysis of Figure 4, the
function $\tilde{\upsilon}(t)=e^{\alpha t}\tilde{u}(r)$ which initial value
belongs to the region (F) converges to $0$ when $t\to\infty$. Since $(0,0)$ is
a saddle point for the system $(\ref{W4})$ it follows that the corresponding
trajectory is is the unstable one of this point. The initial slope of this
curve is $N-2$. By Lemma A.4 it follows that there exists $\ell>0$ such that

$\begin{array}[]{lll}\displaystyle\lim_{r\to\infty}r^{N-2}v(r)=\ell.\end{array}$
(5.11 )

Consequently $|x|^{N-2}u(x)$ is bounded, the proof of Theorem 5.1-3 applies
and deduce from $(\ref{K41-11})$ there exists $c>0$ such that
$|x|^{N-2}u(x)\to c$ when $|x|\to\infty$. $\square$

In the case $M>m^{*}$ the situation is even more complicated and the results
are still incomplete.

###### Theorem 5.3

Let $N\geq 3$, $p>\frac{N}{N-2}$ and $M>m^{*}$ If $u$ is a positive solution
of $(\ref{Z7})$ in $\mathbb{R}^{N}\setminus B_{R}$, we have the following,

1-

$\begin{array}[]{lll}\displaystyle\limsup_{|x|\to\infty}|x|^{\alpha}u(x)=x_{{}_{2,M}}\Longrightarrow\lim_{|x|\to\infty}|x|^{\alpha}u(x)=x_{{}_{2,M}},\end{array}$
(5.12 )

2- If $\displaystyle\liminf_{|x|\to\infty}|x|^{\alpha}u(x)=x_{{}_{1,M}}$,
there exists a sequence $\\{r_{n}\\}$ tending to $\infty$ such that

$\begin{array}[]{lll}\displaystyle\lim_{r_{n}\to\infty}r_{n}^{\alpha}u(r_{n},s)=x_{{}_{1,M}},\end{array}$
(5.13 )

uniformly on $S^{N-1}$.

3- If $\displaystyle\liminf_{|x|\to\infty}|x|^{\alpha}u(x)=0$, there exists
$k>0$ such that

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k.\end{array}$
(5.14 )

Proof. By Theorem 2.6-(1) $u$ is bounded from above by the solution $u_{R}$ of
$(\ref{Z7})$ in $\mathbb{R}^{N}\setminus B_{R}$ which tends to infinity as
$|x|\downarrow R$ and satisfies

$\displaystyle\lim_{|x|\to\infty}|x|^{\alpha}u_{R}(x)=x_{{}_{2,M}}.$

Hence

$\displaystyle\tilde{m}:=\liminf_{|x|\to\infty}|x|^{\alpha}u(x)\leq
m:=\limsup_{|x|\to\infty}|x|^{\alpha}u(x)\leq x_{{}_{2,M}}$

1- If $m=x_{{}_{2,M}}$ there exists a sequence $\\{t_{n}\\}$ tending to
$\infty$ and $\\{s_{n}\\}\subset S^{N-1}$ converging to $s^{*}$ such that
$w_{n}(t,.):=w(t+t_{n},.)$ converges in the $C^{2}$-topology of $[-a,a]\times
S^{N-1}$ for any $a>0$ to a solution $W$ of $(\ref{K33})$ in $\mathbb{R}\times
S^{N-1}$. Furthermore $W$ achieves its maximal value $m$ at $(0,s^{*})$, hence
$W_{t}(0,s^{*})=0$, $\nabla^{\prime}W(0,s^{*})=0$,
$\Delta^{\prime}W(0,s^{*})\leq 0$ and $W_{tt}(0,s^{*})\leq 0$. Therefore

$-m\tilde{\mathcal{P}}_{{}_{M}}(m^{\frac{p-1}{p+1}})=-\alpha Km-m^{p}+M(\alpha
m)^{\frac{2p}{p+1}}\geq 0.$

This implies that either $x_{{}_{1,M}}\leq m\leq x_{{}_{2,M}}$ or $m=0$. For
the liminf the same analysis yields that

$-\tilde{m}\tilde{\mathcal{P}}_{{}_{M}}(\tilde{m}^{\frac{p-1}{p+1}})=-\alpha
K\tilde{m}-\tilde{m}^{p}+M(\alpha\tilde{m})^{\frac{2p}{p+1}}\leq 0,$

hence either $0\leq\tilde{m}\leq x_{{}_{1,M}}$ or $\tilde{m}\geq
x_{{}_{2,M}}$. Note that in the latter case $(\ref{K40-2})$ holds.
2- If $m=x_{{}_{2,M}}$, then using the function $W$ as in the proof of Theorem
5.2, we infer by the strong maximum principle that there exists a sequence
$\\{r_{n}\\}$ tending to infinity such that $r_{n}^{\alpha}u(r_{n},s)$
converges to $x_{{}_{2,M}}$. For any $\epsilon>0$ there exists
$n_{\epsilon}>0$ such that for $n\geq n_{\epsilon}$ we have
$r_{n}^{\alpha}u(r_{n},s)\geq x_{{}_{2,M}}-\epsilon$ for all $s\in S^{N-1}$.
Since ${\mathcal{P}}_{{}_{M}}(x_{{}_{2,M}}-\epsilon)\leq 0$ the function
$x\mapsto(x_{{}_{2,M}}-\epsilon)|x|^{-\alpha}$ is a subsolution of
$(\ref{Z7})$. Then, for any $r_{n}>r_{n_{\epsilon}}$,
$u(x)\geq(x_{{}_{2,M}}-\epsilon)|x|^{-\alpha}$ in
$\\{x:r_{n_{\epsilon}}\leq|x|\leq r_{n}\\}$. This implies

$\displaystyle\liminf_{|x|\to\infty}|x|^{\alpha}u(x)\geq
x_{{}_{2,M}}-\epsilon.$

Since $\epsilon$ is arbitrary, this yields $(\ref{K40-2})$.
If $\tilde{m}=x_{1,M}$ then we proceed as in the case above and deduce that
there exist a function $W\geq 0$ satisfying $(\ref{K33})$ in $\mathbb{R}\times
S^{N-1}$ and a sequence $\\{t_{n}\\}$ tending to infinity such that
$w(t_{n}+t,s)$ converges in the $C^{2}$-topology of $[-a,a]\times S^{N-1}$ to
$W$ for any $a>0$. The function $W$ is larger or equal to $x_{{}_{1,M}}$ and
coincides with $x_{{}_{1,M}}$ at $(0,s^{*})$ for some $s^{*}\in S^{N-1}$. By
the strong maximum principle we have that $W\equiv x_{{}_{1,M}}$. This implies
assertion 2.

3- If $\displaystyle\liminf_{|x|\to\infty}|x|^{\alpha}u(x)=0$, then we deduce
by Harnack inequality that there exists a sequence $\\{r_{n}\\}$ tending to
$\infty$ such that

$\lim_{r_{n}\to\infty}r_{n}^{\alpha}u(r_{n},s)=0\quad\text{uniformly on
}S^{N-1}.$

Then for any $\epsilon>0$ the function there exists
$n_{\epsilon}\in\mathbb{N}$ such that for any $n\geq n_{\epsilon}$,
$u(r_{n},s)\leq\epsilon|r_{n}|^{-\alpha}$ for all $s\in S^{N-1}$. The function
$x\mapsto\epsilon|x|^{-\alpha}$ is a supersolution of $(\ref{Z7})$. Since
there exists a sequence $\\{r_{n}\\}$ tending to infinity such that
$u(r_{n},s)\leq\epsilon r_{n}^{-\alpha}$ for all $s\in S^{N-1}$ for $n\geq
n_{\epsilon}$ it follows by the comparison principle applied to the sequence
of annuli $\\{x:r_{n_{\epsilon}}\leq|x|\leq r_{n}\\}$, that
$x(x)\leq\epsilon|x|^{-\alpha}$. Since the function
$x\mapsto\epsilon|x|^{-\alpha}$ is a supersolution of $(\ref{Z7})$, it follows
by the comparison principle that $u(x)\leq\epsilon|x|^{-\alpha}$ in the annuli
$\\{x:r_{n_{\epsilon}}\leq|x|\leq r_{n}\\}$. Letting $n\to\infty$ yields

$u(x)\leq\epsilon|x|^{-\alpha}\quad\forall x\in
B_{r_{n_{\epsilon}}}\Longrightarrow\limsup_{|x|\to\infty}|x|^{\alpha}u(x)\leq\epsilon.$

Since $\epsilon$ is arbitrary we infer that $|x|^{\alpha}u(x)$ converges to
$0$ when $|x|\to\infty$. By the phase plane analysis of section 2.4
(seeTheorem 2.6), as in the proof of Theorem 5.2-2, we have that
$|x|^{N-2}u(x)$ is bounded. Hence $(\ref{K41-2})$ follows as in the previous
proof. $\square$

Remark. We conjecture that the results of Theorem 5.2 and Theorem 5.3 hold
under the following forms:

For Theorem 5.2

1- either $\displaystyle\lim_{x\to\infty}|x|^{\alpha}u(x)=x_{m^{*}},$

2- or there exists $k>0$ such that
$\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k$

For Theorem 5.3

1- either $\displaystyle\lim_{x\to\infty}|x|^{\alpha}u(x)=x_{{}_{2,M}}$,

2- or $\displaystyle\lim_{x\to\infty}|x|^{\alpha}u(x)=x_{{}_{1,M}}$,

3- or there exists $k>0$ such that
$\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k$.

###  5.2 The case $q\neq\frac{2p}{p+1}$

The next results extend the asymptotic behaviour described in Theorem 1.9 and
Theorem 1.10 to non radial solutions. The following statement shows that
equation $(\ref{Z1})$ inherits the properties of the Emden-Fowler equation
${\mathcal{L}}_{p}u=0$ if $\frac{2p}{p+1}<q<p$.

###### Theorem 5.4

Let $N\geq 1$, $M>0$, $\frac{2p}{p+1}<q<p$ and $u$ be a positive solution of
$(\ref{Z1})$ in $B_{R}^{c}$. Then

1- If $N=1,2$ and $p>1$, or $N\geq 3$ and $1<p<\frac{N}{N-2}$, then

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{\alpha}u(x)=x_{0}.\end{array}$
(5.15 )

2- If $N\geq 3$ and $p>\frac{N}{N-2}$, then

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{N-2}u(x)=k>0.\end{array}$
(5.16 )

3- If $N\geq 3$ and $p=\frac{N}{N-2}$, then

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}\left(\ln|x|\right)^{\frac{N-2}{2}}|x|^{N-2}u(x)=\left({\displaystyle\frac{N-2}{\sqrt{2}}}\right)^{N-2}.\end{array}$
(5.17 )

Proof. In the first case, the solution $v$ of ${\mathcal{L}}_{p}v=0$ in
$B_{R}^{c}$ with value $\min u\lfloor_{\partial B_{R}}$ on $\partial B_{R}$ is
a subsolution smaller than $u$ (it is obtained by minimization), and it has
the behaviour expressed by $(\ref{K44})$. By Theorem 1.13 there exists a
global positive solution $\tilde{u}$ of $(\ref{Z1})$ in
$\mathbb{R}^{N}\setminus\\{0\\}$ satisfying $(\ref{K44})$. The difficulty is
that this solution may not be larger than $u$ for $|x|=R$. In such a case, for
$a>\max u\lfloor_{\partial B_{R}}$, the function $\tilde{u}_{a}:=\tilde{u}+a$
is a supersolution of $(\ref{Z1})$ in $B_{R}^{c}$. The solution $\tilde{v}$
which satisfies ${\mathcal{L}}_{p}\tilde{v}=0$ in $B_{R}^{c}$ with value $a$
for $|x|=R$ is a subsolution smaller than $\tilde{u}_{a}$. Hence there exists
a radial solution $u_{a}$ of $(\ref{Z1})$ in $B_{R}^{c}$ such that
$u_{a}(R)=a+\tilde{u}(R)$ and it dominates $u$ in $B_{R}^{c}$. By Theorem
1.9-(1), the function $u_{a}$, and therefore $u$, satisfies $(\ref{K44})$.

In the second case, We proceed as in the proof of Theorem 5.1-(2), with the
help of Theorem 1.9-(2). The function $u$ satisfies $u_{1}\leq u\leq u_{2}$
where $u_{1}$ and $u_{2}$ are radial solutions of $(\ref{Z1})$ in $B_{R}^{c}$,
hence $u(x)\leq C|x|^{2-N}$. If we set $w(t,s)=r^{\nu}u(r,s)$ with $t=\ln r$
and $\nu=N-2$, then $w$ satisfies

$\begin{array}[]{lll}w_{tt}-\nu
w_{t}+\Delta^{\prime}w+\nu^{q}Me^{(N-q(N-1))t}\left((w_{t}-\nu
w)^{2}+|\nabla^{\prime}w|^{2}\right)^{\frac{q}{2}}\\\\[5.69054pt]
\phantom{------------}-e^{(N-p(N-2))t}w^{p}=0\qquad\text{in }[0,\infty)\times
S^{N-1}.\end{array}$ (5.18 )

Since $w$, $w_{t}$ and $|\nabla^{\prime}w|$ are bounded, it follows by the
same argument [10, Proposition 4.1] that $w(t,.)\to c\geq 0$ when $t\to\infty$
and $c>0$ since $u$ is bounded from below by the solution of
${\mathcal{L}}^{p}v=0$ in $B_{R}^{c}$ which satisfies the same type of
asymptotic behaviour with a positive limit of $r^{N-2}v(r)$ when $r\to\infty$.

In the third case, it is proved in [21, Théorème 3.1] that the solution $v$ of
${\mathcal{L}}_{p}v=0$ in $B_{R}^{c}$ which coincide with $\min
u\lfloor_{\partial B_{R}}$ for $|x|=R$ verifies the relation $(\ref{K46})$. In
order to have the estimate from above, for $a>0$ the function
$h_{a}(x)=a|x|^{2-N}$ satisfies

${\mathcal{L}}^{M}_{p,q}h_{a}(x)=a^{p}|x|^{-p(N-2)}-Ma^{q}(N-2)^{q}|x|^{-q(N-1)}=a^{p}|x|^{-N}-Ma^{q}(N-2)^{q}|x|^{-q(N-1)}.$

Since $q>\frac{2p}{p+1}=\frac{N}{N-1}$, we obtain for $|x|\geq R$

$\begin{array}[]{lll}{\mathcal{L}}^{M}_{p,q}h_{a}(x)=a^{q}|x|^{(1-N)q}\left(a^{p-a}|x|^{(N-1)q-N}-M(N-2)^{q}\right)\\\\[5.69054pt]
\phantom{{\mathcal{L}}^{M}_{p,q}h_{a}(x)}\geq
a^{q}|x|^{(1-N)q}\left(a^{p-a}R^{(N-1)q-N}-M(N-2)^{q}\right),\end{array}$

Therefore, for $a>0$ large enough ${\mathcal{L}}^{M}_{p,q}h_{a}\geq 0$ in
$B^{c}_{R}$. Since the solution $v_{a}$ of ${\mathcal{L}}_{p}v=0$ in
$B^{c}_{R}$ with value $aR^{2-N}$ for $|x|=R$ is a subsolution of $(\ref{Z1})$
smaller than $h_{a}$ it follows from [25, Theorem 1.4.5] that there exists a
radial solution $u_{a}$ of $(\ref{Z1})$ in $B_{R}^{c}$ such that $v_{a}\leq
u_{a}\leq h_{a}$. If we choose $a$ large enough so that $aR^{2-N}\geq\max
u\lfloor_{\partial B_{R}}$, then $u_{a}$ is larger than $u$ in $B_{R}^{c}$.
Finally, by Theorem 1.9-(3) the function $u_{a}$ satisfies also $(\ref{K46})$,
which ends the proof. $\square$

In the following result we extend Theorem 1.10-(2) to the non-radial case.

###### Theorem 5.5

Let $N\geq 3$, $M>0$, $p>1$ and $1<q<\frac{2p}{p+1}$ with
$q\leq\frac{N}{N-1}$. If $u$ is a positive solution of $(\ref{Z1})$ in
$\overline{B}_{R}^{c}$ then there holds

$\begin{array}[]{lll}\displaystyle\lim_{|x|\to\infty}|x|^{\gamma}u(x)=X_{{}_{M}}.\end{array}$
(5.19 )

Proof. We can assume that $u$ is continuous in $B_{\rho}^{c}$ for any
$\rho>R$.
For constructing a supersolution we proceed as in Theorem 4.14 using the
solution $\upsilon$ of $(\ref{Z1})$ on $(0,\infty)$. The function
$x\mapsto\upsilon(|x|-\rho)+\delta$ is a supersolution 1.1 in $B_{\rho}^{c}$
which is larger than $u$ for $|x|=\rho$ and at infinity. Hence it is larger
than $u$ in $\overline{B}_{\rho}^{c}$. Letting $\delta\to 0$ yields
$u(x)\leq\upsilon(|x|-\rho)$ for all $x>\rho$.

Next, we construct a subsolution: we set $c=\min u\lfloor_{\partial
B_{\rho}}$. For $n>\rho$ we denote by $w_{n}$ the solution of

$\begin{array}[]{lll}-\Delta w+w^{p}=0\quad\text{in
}\,\Gamma_{\rho,n}:=B_{n}\setminus\overline{B}_{\rho}\\\ w_{n}=c\quad\text{on
}\partial B_{\rho}\,,\;w_{n}=0\quad\text{on }\partial B_{n}.\end{array}$

The function $w_{n}$ which is unique is a subsolution of 1.1 in
$\Gamma_{\rho,n}$ where it satisfies $w_{n}\leq\upsilon(|x|-\rho)$. By Theorem
4.1 there exists a solution $u_{n}$ of $(\ref{Z1})$ in $\Gamma_{\rho,n}$ which
coincides with $w_{n}$ on $\partial\Gamma_{\rho,n}$ and is radial as $w_{n}$
and $\upsilon(|.|-\rho)$ are (or by uniqueness), and $u_{n}\leq u$ in
$\Gamma_{\rho,n}$. When $n\to\infty$, $u_{n}\uparrow u_{\infty}$. The function
$u_{\infty}$ is a radial positive solution of $(\ref{Z1})$ in
$\overline{B}_{\rho}^{c}$ and it satisfies

$u_{\infty}(x)\leq u(x)\leq\upsilon(|x|-\rho)\quad\text{for all }\,|x|>\rho.$

It follows from Theorem 1.10-(2) that $u_{\infty}$ satisfies $(\ref{K48})$, as
$\upsilon$ does it too. This ends the proof. $\square$

##  Appendix A Appendix

###  A.1 The a priori estimates

The following a priori estimates proved in [7], [8] are fundamental throughout
the paper. They do not depend on the sign of $q-\frac{2p}{p+1}$.

###### Proposition A.1

Let $\Omega\subset\mathbb{R}^{N}$ be a domain containing $0$, $1<q<p$ and
$M\in\mathbb{R}$. If $u\in C^{1}(\Omega\setminus\\{0\\})$ is a nonnegative
solution of $(\ref{Z1})$ in $\Omega\setminus\\{0\\}$, then there holds for any
$0<R\leq\frac{1}{2}\mbox{\rm dist}\,(x,\partial\Omega)$:
1- If $M>0$,

$u(x)\leq
c_{1}\max\left\\{M^{\frac{1}{p-q}}|x|^{-\gamma},|x|^{-\alpha}\right\\}\quad\text{for
all }x\in\overline{B}_{R}\setminus\\{0\\},$ (A.1 )

where $c_{1}=c_{1}(N,p,q)>0$.
2- If $M\leq 0$ and $q<2$ if $M<0$,

$\displaystyle
u(x)\leq\min\left\\{c_{2}|x|^{-\alpha},c_{3}|M|^{-\frac{1}{q-1}}|x|^{-\beta}+\max_{|y|=R}u(y)\right\\}\quad\text{for
all }x\in\overline{B}_{R}\setminus\\{0\\},$ (A.2 )

where $c_{2}=c_{2}(N,p)>0$ and $c_{3}=c_{3}(N,q)>0$.

When $u$ is a signed solution the following estimate holds [8, Corollary 2.2].

###### Proposition A.2

Under the assumptions on $\Omega$, $p$ and $q$ of Proposition A.1 and assuming
that $M>0$, any signed solution $u$ of $(\ref{K6_})$ in
$\Omega\setminus\\{0\\}$ satisfies for any $0<R\leq\frac{1}{2}\mbox{\rm
dist}\,(x,\partial\Omega)$,

$\begin{array}[]{lll}-\min\left\\{c_{4}M^{\frac{1}{q-1}}|x|^{-\beta},c_{2}|x|^{-\alpha}\right\\}\leq-
u_{-}(x)\leq 0\\\\[8.53581pt] \phantom{--------------}\leq u_{+}(x)\leq
c_{1}\max\left\\{M^{\frac{1}{p-q}}|x|^{-\frac{q}{p-q}},|x|^{-\alpha}\right\\},\end{array}$
(A.3 )

for all $x\in\overline{B}_{R}\setminus\\{0\\}$, where $c_{1}=c_{1}(N,p,q)>0$,
$c_{2}=c_{2}(N,p)>0$ and $c_{4}=c_{2}(N,p,q)>0$.

Using scaling method when $1<q\leq 2$ and the Bernstein method when $1<q<p$,
it is proved in [7, Proposition 2.3, Corollary 2.5] a gradient estimate that
we recall.

###### Proposition A.3

Let $\Omega\subset\mathbb{R}^{N}$ be a domain containing $0$, $1<q<p$ and
$M>0$. If $u\in C^{1}(\Omega\setminus\\{0\\})$ is a nonnegative solution of
$(\ref{Z1})$ in $\Omega\setminus\\{0\\}$, there holds for any $0<R\leq
R_{0}:=\frac{1}{2}\mbox{\rm dist}\,(x,\partial\Omega)$ and some constants
$c_{4}=c_{4}(N,p,q,R_{0})>0$:
1- When $1<q\leq\frac{2p}{p+1}$

$|\nabla u(x)|\leq
c_{4}\max\left\\{M^{\frac{1}{p-q}}|x|^{-\frac{p}{p-q}},|x|^{-\frac{p+1}{p-1}}\right\\},$
(A.4 )

for all $x\in\overline{B}_{R}\setminus\\{0\\}$. If $q=\frac{2p}{p+1}$, $c_{4}$
is independent of $R_{0}$.
2- When $1<q<p$

$|\nabla u(x)|\leq
c_{5}\left(|x|^{-\frac{1}{q-1}}+\max\left\\{M^{\frac{p}{q(p-q)}}|x|^{-\frac{p}{p-q}},|x|^{-\frac{2p}{q(p-1)}}\right\\}\right),$
(A.5 )

for all $x\in\overline{B}_{R}\setminus\\{0\\}$.

Remark. When $\frac{2p}{p+1}\leq q\leq 2$ any nonnegative solution of
$(\ref{Z1})$ in $\mathbb{R}^{N}\setminus B_{\frac{R}{2}}$ satisfies
$(\ref{Y3})$ in $\mathbb{R}^{N}\setminus B_{R}$.

Remark. If $u$ is a signed solution, the Bernstein method that we developed in
[7], [8] cannot be applied, however the scaling method can be used if $1<q<p$.
In the particular case $1<q\leq\frac{2p}{p+1}$ there holds in a neighborhood
of $x=0$,

$|\nabla u(x)|\leq
c_{5}\max\left\\{M^{\frac{1}{p-q}}|x|^{-\gamma-1},|x|^{-\alpha-1}\right\\}.$
(A.6 )

###  A.2 Equilibrium with a simple eigenvalue

Consider the system in $\mathbb{R}^{2}$

$\begin{array}[]{lll}x_{t}=ax+by+f(x,y)\\\ y_{t}=cx+dy+g(x,y)\end{array}$ (A.7
)

where $a,b,c,d$ are real numbers and $f$ and $g$ two $C^{1}$ real functions
satisfying

$\begin{array}[]{lll}|f(x,y)+g(x,y)|\leq
c\left(|x|^{s}+|y|^{s}\right)\qquad\text{for all }(x,y)\in B_{1}.\end{array}$
(A.8 )

Suppose that the matrix $A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ of the
system admits two eigenvalues $\mu_{1}\neq\mu_{2}$ and $\mu_{2}>0$. By
reduction to the diagonal form $\begin{pmatrix}x\\\
y\end{pmatrix}=P\begin{pmatrix}\tilde{x}\\\ \tilde{y}\end{pmatrix}$ the system
becomes

$\begin{array}[]{lll}\tilde{x}_{t}=\mu_{1}\tilde{x}+\tilde{f}(\tilde{x},\tilde{y})\\\
\tilde{y}_{t}=\mu_{2}\tilde{y}+\tilde{g}(\tilde{x},\tilde{y}).\end{array}$
(A.9 )

###### Lemma A.4

Under the above assumptions there exist at least two trajectories
$\tilde{\mathcal{T}}_{1}=(\tilde{x}_{1},\tilde{y}_{1})$ and
$\tilde{\mathcal{T}}_{2}=(\tilde{x}_{2},\tilde{y}_{2})$ tangent to the axis
$0\tilde{y}$ converging to $(0,0)$ when $t\to-\infty$, one with
$\tilde{y}_{1}(t)>0$, the other with $\tilde{y}_{2}(t)<0$ for $t\leq-T$. Any
trajectory $\tilde{\mathcal{T}}=\\{(\tilde{x},\tilde{y})\\}_{t\leq T}$
converging to $(0,0)$ when $t\to-\infty$ and tangent at $(0,0)$ to the axis
$0\tilde{y}$ satisfies for some $\ell\neq 0$

$\begin{array}[]{lll}\displaystyle\lim_{t\to-\infty}e^{-\mu_{2}t}\tilde{y}(t)=\ell.\end{array}$
(A.10 )

Proof. The existence of the solutions tangent to the axis $0\tilde{y}$ is
classical. Consider a $(\tilde{x},\tilde{y})$ converging to $(0,0)$
tangentialy to $0\tilde{y}$ and such that $\tilde{y}(t)>0$ for $t\leq-T$. Then

$|\tilde{g}(\tilde{x}(t),\tilde{y}(t))|\leq
c\left(|\tilde{x}(t)|^{s}+|\tilde{y}(t)|^{s}\right)\leq
c^{\prime}|\tilde{y}(t)|^{s}$

with $c^{\prime}>c$, since $\frac{\tilde{x}(t)}{\tilde{y}(t)}\to 0$ when
$t\to-\infty$. Put $\upsilon(t)=e^{-\mu_{2}t}\tilde{y}(t)$. Then

$|\upsilon_{t}(t)|=|\tilde{g}(\tilde{x}(t),\tilde{y}(t))|e^{-\mu_{2}t}\leq
ce^{-\mu_{2}t+s\mu_{2}t}\upsilon^{s}(t).$

Therefore

$\left|\left(\frac{\upsilon^{1-s}}{s-1}\right)_{t}\right|\leq
ce^{\mu_{2}(s-1)t}\Longrightarrow\left(\upsilon^{1-s}\right)_{t}\in
L^{1}(-\infty,T)$ (A.11 )

Then $\upsilon^{1-s}(t)$ admits a nonnegative limit $\ell$ when $t\to-\infty$.
If $\ell=0$, it would follow from $(\ref{Ap5})$ that

$\left(e^{\mu_{2}t}\upsilon(t)\right)^{s-1}\geq\tilde{c}\Longrightarrow\tilde{y}(t)\geq
C>0,$

which contradict the fact that $\tilde{y}(t)\to 0$. $\square$

Remark. This result is easily extendable to higher dimension, where $A$ is a
$N\times N$ matrix with a simple eigenvalue $\mu>0$ and such that
$\mathbb{R}^{N}=ker(A-\mu I)\oplus E$, where $E$ is A-invariant. Consider the
system

$X^{\prime}=AX+F(X)$ (A.12 )

where $|F(X)|\leq c|X|^{s}$ in $B_{1}$ for some $s>1$. If $X=X_{1}+X^{\prime}$
where $X_{1}\in ker(A-\mu I)$ and $X^{\prime}\in E$, then there exist two
tajectories $X_{j}(t)$ of $(\ref{Ap6})$ admitting a limit direction $\tau\in
ker(A-\mu I)\setminus\\{0\\}$ and , j=1,2, and they satisfy for some $a\neq
0$,

$\displaystyle\lim_{t\to-\infty}e^{-\mu t}X(t)=a\tau.$ (A.13 )

## References

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