# Liouville type theorems for stable solutions of elliptic system involving
the Grushin operator.

Foued Mtiri<EMAIL_ADDRESS>ANLIG, UR13ES32, University of Tunis El-
Manar, 2092 El Manar II, Tunisia.

###### Abstract

We examine the degenerate elliptic system

$-\Delta_{s}u=v^{p},\quad-\Delta_{s}v=u^{\theta},\quad u,v>0\quad\mbox{in
}\;\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}},\quad\mbox{where
}\;\;\;\;s\geq 0\;\;\mbox{and}\;\;p,\theta>0.$

We prove that the system has no smooth stable solution provided $p,\theta>0$
and $N_{s}<2+\alpha+\beta,$ where

$\alpha=\frac{2(p+1)}{p\theta-1}\quad\mbox{and}\quad\beta=\frac{2(\theta+1)}{p\theta-1}.$

This result is an extension of some result in [15]. In particular, we
establish a new the integral estimate for $u$ and $v$ (see Proposition 1.1),
which is crucial to deal with the case $0<p<1.$

###### keywords:

Stable solutions , Liouville-type theorem , Grushin operator , Critical
exponents, Elliptic system.

##  1 Introduction

We start by noting that throughout this article, $N_{s}:=N_{1}+(1+s)N_{2}$ is
called the homogeneous dimension associated to the Grushin operator:

$\Delta_{s}=\Delta_{x}+|x|^{2s}\Delta_{y},$

where $s\geq 0,$ and

$\Delta_{x}:=\sum_{i=1}^{N_{1}}\frac{\partial^{2}}{\partial
x_{i}^{2}},\quad\mbox{and}\;\;\Delta_{y}:=\sum_{j=1}^{N_{2}}\frac{\partial^{2}}{\partial
y_{j}^{2}},$

are Laplace operators with respect to $x\in\mathbb{R}^{N_{1}},$
$y\in\mathbb{R}^{N_{2}}$ and
$|x|^{2s}=\left(\sum\limits_{i=1}^{N_{1}}x_{i}^{2}\right)^{s}$.

In this paper, we are interested in the classification of stable solutions to
the following degenerate elliptic system

$\displaystyle-\Delta_{s}u=v^{p},\quad-\Delta_{s}v=u^{\theta},\quad
u,v>0\quad\mbox{in
}\;\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}},\quad\mbox{where
}\;p,\theta>0.$ (1.1)

In the case $s=0,$ the system (1.1) becomes

$\displaystyle-\Delta u=v^{p},\quad-\Delta v=u^{\theta},\quad
u,v>0\quad\mbox{in }\;\mathbb{R}^{N},\quad\mbox{where }\;p,\theta>0.$ (1.2)

It is well studied and there are many interesting results on classifying the
solutions of this system for various $p$ and $\theta.$

In the last decade, increased attention has focused on the system (1.1). We
note that the Sobolev critical hyperbola

$\displaystyle\frac{1}{p+1}+\frac{1}{\theta+1}=\frac{N-2}{N},$

which is introduced independently by Mitidieri [13] and Van der Vorst [27],
plays a crucial role in the analysis of (1.1). Among the main interests, the
Lane–Emden conjecture says that there has no positive classical solution if
and only if the pair $(p,\theta)$ lies below the Sobolev critical hyperbola,
i.e.,

Conjecture. Let $p,\theta>0$. If the pair $(p,\theta)$ is subcritical i.e., if
$(p,\theta)$ satisfies

$\displaystyle\frac{1}{p+1}+\frac{1}{\theta+1}>\frac{N-2}{N},$ (1.3)

Then system (1.2) has no positive classical solutions.

The full Lane-Emden conjecture is still open. Only partial results for
nonexistence are known, and many researchers have made contribution in pushing
the progress forward. We shall briefly present some important recent
developments of this conjecture

The case of radial solutions was solved by Mitidieri [12] and Serrin-Zou [20]
constructed positive radial solutions on and above the critical hyperbola, i.e

$\displaystyle\frac{1}{p+1}+\frac{1}{\theta+1}\leq\frac{N-2}{N},$

which means that the nonexistence theorem is optimal for radial solutions. For
non-radial solutions of the Lane-Emden system, certain Liouville theorems are
known. Denote the scaling exponents of system (1.2) by

$\displaystyle\alpha=\frac{2(p+1)}{p\theta-1},\quad\beta=\frac{2(\theta+1)}{p\theta-1},\quad
p\theta>1.$ (1.4)

Then subcritical condition (1.3) is equivalent to

$\displaystyle N<2+\alpha+\beta,\quad\mbox{for }\quad p\theta>1.$ (1.5)

There are no positive classical super-solutions to (1.2) if

$\displaystyle p\theta\leq 1\quad\mbox{or }\quad p\theta>1\quad\mbox{and
}\quad\max\,\left\\{\alpha,\beta\right\\}>N-2,$

see [12, 23, 21]. This implies the conjecture for $N=1,2.$ Also, in [1] the
conjecture is true for

$\displaystyle\min\,\left\\{\alpha,\beta\right\\}>\frac{N-2}{2}\quad\mbox{with
}\quad(\alpha,\beta)\neq(\frac{N-2}{2},\frac{N-2}{2}).$

In dimension $N=3,$ Serrin-Zou [21] gave a proof for the nonexistence of
polynomially bounded solutions, an assumption that was relaxed later by
Polácik, Quittner and Souplet [25]. More recently, in dimension $N=4$, the
conjecture was settled completely by Souplet [24] who provided, in dimensions
$N\geq 5$, a more restrictive new region for the exponents $(p,\theta)$ that
insures that if $(p,\theta)$ satisfies (1.5), along with
$\max\,\left\\{\alpha,\beta\right\\}>N-3,$ then system (1.2) has no positive
classical solutions.

On the other direction, the Liouville type theorem for the class of stable
solutions for both system has attracted the attention of many mathematicians.
We refer [3, 10, 9, 15] for Lane–Emden system. The nonexistence of positive
stable classical solutions of (1.2) was examined in [3]. In particular, Cowan
[3] proved that if $N\leq 10$, (1.2) has no stable solution for any $2\leq
p\leq\theta$. This result was then extended in [10] to the Lane–Emden system
with weights. Among other things, we also mention that Hajlaoui et.al. [9]
improved the previous works [3, 10] and mainly obtained a new comparison
property which is key to deal with the case $1<p\leq\frac{4}{3}.$

A new approach which is independently obtained in [15], allows the authors to
prove the following Liouville theorem for classical stable solutions of (1.2)
for any $p,\theta>0,$ satisfying (1.5).

###### Theorem A

If $p,\theta>0$ satisfies (1.3), then (1.2) has no smooth stable solution.

In the case $0<p<1,$ the main tools are based on the relationship between the
stability for the Lane-Emden system (1.2) and the stability for the a fourth
order problem, called the $m$-biharmonic equation

$\Delta^{2}_{m}u:=\Delta(|\Delta u|^{m-2}\Delta u)=u^{\theta}\quad\mbox{in
}\quad\mathbb{R}^{N}.$

The case $p>1$ was handled by the results of [9]. The main motivation was to
provide a proof of the Lane-Emden conjecture dealing with positive stable
solutions.

Coming back to the Lane-Emden system (1.1) for the general case $s\geq 0,$ the
Liouville property is less understood and is more complicated to deal with
than $s=0,$ because the the operator $\Delta_{s}$ no longer has symmetry and
it degenerates on the manifold $\\{0\\}\times\mathbb{R}^{N_{2}}$ which causes
further mathematical difficulties.

In [6] the author extended some of Cowan results [3], in order to prove the
nonexistence of positive stable solutions of (1.1) with
$\frac{4}{3}<p\leq\theta$ and $N_{s}\leq 10,$ or
$1<p\leq\min(\frac{4}{3},\theta)$ with additional assumption. The main idea
used in [6] is a combination of stability inequality, comparison principle and
bootstrap argument. After that, this idea was exploited by Mtiri [16], the
author has obtained a some Liouville theorems for stable solutions of (1.1),
see Theorem B below. This results improves the bound given in [6].

Our main objective is to classify the stable solution of (1.1) for any
$p,\theta>0,$ and the general case $s\geq 0.$ In order to state our results
more accurately, we define the notion of stability where we consider a general
system given by

$\displaystyle-\Delta_{s}u=f(x,v),\quad-\Delta_{s}v=g(x,u)\;\;\mbox{in $K$, a
bounded regular domain }\;\subset\mathbb{R}^{N},$ (1.6)

where $f,g\in C^{1}(K\times\mathbb{R}).$ Following Montenegro [17], a smooth
solution $(u,v)$ of (1.6) is said to be stable in $K$ if the following
eigenvalue problem

$\displaystyle-\Delta\xi=f_{v}(x,v)\zeta+\eta\xi,\quad-\Delta\zeta=g_{u}(x,u)\xi+\eta\zeta\quad\mbox{in
}\,K$

has a nonnegative eigenvalue $\eta$, with a positive smooth eigenfunctions
pair $(\xi,\zeta)$.

The main result in this paper is the following

###### Theorem 1.1

If $p,\theta>0$ and $N_{\alpha}$ satisfying $N_{s}<2+\alpha+\beta$, then (1.1)
has no smooth stable solution.

To prove Theorem 1.1, we borrow crucially the idea in [15] who established
Theorem A. Without loss of generality, we consider only $\theta>p>0$ and
$p\theta>1$. As we will see soon, the $\theta>p\geq 1$ case can be handled by
the results in [16], so our main concern is the case

$\theta>\frac{1}{p}>1>p>0.$

The main difficulty arises from the fact that there has no works in literature
dealing with stable solutions (radial or not) for the $m$-biharmonic–type
Grushin equation. Consequently, it is difficult to use the technique developed
in [15]. We overcome this difficulty, we shall derive new the important
integral estimate for $u$ and $v$ which is crucial to deal with the case
$0<p<1.$ The following Proposition transforms our notion of a stable solution
of (1.1) into an inequality which allows the use of arbitrary test functions.

###### Proposition 1.1

Let $(u,v)$ denote a stable solution of (1.1) with $0<p<1.$ Then

$\displaystyle\theta
p\int_{\mathbb{R}^{N}}u^{\theta-1}\gamma^{2}dxdy\leq\int_{\mathbb{R}^{N}}v^{1-p}|\Delta_{s}\gamma|^{2}dxdy,$
(1.7)

for all $\gamma\in C_{c}^{\infty}(\mathbb{R}^{N}).$

###### Remark 1.1

  * 1.

If $s=0$, we obtain a similar result in [15] see Theorem A.

  * 2.

Our approach is more easier than those developed in [15], for $p,\theta>0$. To
the best of our knowledge, no general Liouville type result was known for
stable solution of (1.1) with $0<p<1.$

  * 3.

We note also that the method used in the present paper can be applied to study
the weighted systems, and to more general class of degenerate operator, such
as the $\Delta_{s}$ operator (see [11, 8]) of the form

$\Delta_{s}:=\sum_{j=1}^{N}s^{2}_{j}\Delta_{x^{(j)}}\;\;\;s:=(s_{1},\dots,s_{N}):\mathbb{R}^{N}\to\mathbb{R^{N}},$

where $s_{i}:\mathbb{R}^{N}\to\mathbb{R},\quad i=1,\dots,N,$ are nonnegative
continuous functions satisfying some properties such that $\Delta_{s}$
homogeneous of degree two with respect to a group dilation in
$\mathbb{R}^{N}.$

##  2 Liouville Type Theorem.

This section is devoted to the proof of Theorem 1.1. For convenience, we
always denote by $C$ a generic constant whose concrete values may change from
line to line or even in the same line. If this constant depends on an
arbitrary small number $\epsilon$, then we may denote it by $C_{\epsilon}$. We
also use Young inequality in the form $ab\leq\epsilon
a^{q}+C_{\epsilon}b^{q^{\prime}}$ for $q,q^{\prime}>1$ satisfying
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$

As mentioned before, we need only to consider the case $\theta>p$ and
$p\theta>1$. We split the proof into two cases: $\theta>p\geq 1$ and
$\theta>p^{-1}>1>p>0$.

###  2.1 The case $\theta>p^{-1}>1>p>0$.

Here we handle the case $0<p<1.$ We begin with proving the integral estimate
for $u$ and $v$

proof of Proposition 1.1. By the definition of stability, there exist smooth
positive functions $\xi$, $\zeta$ and $\eta\geq 0$ such that

$-\Delta_{s}\xi=pv^{p-1}\zeta+\eta\xi,\;\;-\Delta_{s}\zeta=\theta
u^{\theta-1}\xi+\eta\zeta\quad\mbox{in
}\;\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}.$

Using $(\xi,\zeta)$ as super-solution,
$(\min_{\overline{\Omega}}\xi,\min_{\overline{\Omega}}\zeta)$ as sub-solution,
and the standard monotone iterations, we can claim that there exist positive
smooth functions $\varphi$, $\chi$ verifying

$\displaystyle-\Delta_{s}\varphi=pv^{p-1}\chi,\quad-\Delta_{s}\chi=\theta
u^{\theta-1}\varphi\quad\mbox{in
}\,\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}.$

Therefore, we have

$\displaystyle\theta
u^{\theta-1}\varphi=\Delta_{s}\left(\frac{1}{p}v^{1-p}\Delta_{s}\varphi\right)\quad\mbox{in}\;\;\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}.$

Let $\gamma\in C_{c}^{2}(\mathbb{R}^{N})$. Multiplying the above equation by
$\gamma^{2}\varphi^{-1}$ and integrating by parts, there holds

$\displaystyle\begin{split}&\int_{\mathbb{R}^{N}}\theta
u^{\theta-1}\gamma^{2}dxdy\\\
&=\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}\Delta_{s}\varphi\Delta_{s}(\gamma^{2}\varphi^{-1})dxdy\\\
&=\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}\Delta_{s}\varphi\left(-4\gamma\frac{\nabla_{s}\varphi\cdot\nabla_{s}\gamma}{\varphi^{2}}+\frac{2|\nabla_{s}\gamma|^{2}}{\varphi}+\frac{2\gamma\Delta_{s}\gamma}{\varphi}+\frac{2\gamma^{2}|\nabla_{s}\varphi|^{2}}{\varphi^{3}}-\frac{\gamma^{2}\Delta_{s}\varphi}{\varphi^{2}}\right)dxdy.\end{split}$
(2.1)

Using Cauchy-Schwarz’s inequality and the fact that $-\Delta_{s}\varphi>0,$ we
get

$\displaystyle\begin{split}&\left|-4\int_{\mathbb{R}^{N}}\frac{v^{1-p}}{p}\Delta_{s}\varphi\frac{\nabla_{s}\varphi\cdot\nabla_{s}\gamma}{\varphi^{2}}\gamma
dxdy\right|\\\
&\leq-2\int_{\mathbb{R}^{N}}\frac{v^{1-p}}{p}\Delta_{s}\varphi\frac{|\nabla_{s}\gamma|^{2}}{\varphi}dxdy-2\int_{\mathbb{R}^{N}}\frac{v^{1-p}}{p}\Delta_{s}\varphi\frac{\gamma^{2}|\nabla_{s}\varphi|^{2}}{\varphi^{3}}dxy.\end{split}$
(2.2)

Combining (2.1) and (2.2), one obtains, using again the Cauchy-Schwartz
inequality,

$\displaystyle\int_{\mathbb{R}^{N}}\theta u^{\theta-1}\gamma^{2}dxdy$
$\displaystyle\leq\frac{2}{p}\int_{\mathbb{R}^{N}}v^{1-p}\Delta_{s}\varphi\frac{\gamma\Delta_{s}\gamma}{\varphi}dxdy-\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}\frac{(\Delta_{s}\varphi)^{2}}{\varphi^{2}}\gamma^{2}dxdy$
$\displaystyle\leq\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}\frac{(\Delta_{s}\varphi)^{2}}{\varphi^{2}}\gamma^{2}dxdy+\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}(\Delta_{s}\gamma)^{2}dxdy-\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}\frac{(\Delta_{s}\varphi)^{2}}{\varphi^{2}}\gamma^{2}dxdy$
$\displaystyle=\frac{1}{p}\int_{\mathbb{R}^{N}}v^{1-p}(\Delta_{s}\gamma)^{2}dxdy.$

The proof is completed.

As a consequence of Proposition 1.1, we derive immediately the following
integral estimate for stable solution $(u,v)$ of system (1.1), in the case
$p\in(0,1)$ and $p\theta>1,$ which is a crucial tool in our approach.

###### Lemma 2.1

Let $(u,v)$ be a stable solution of (1.1), with $p\in(0,1)$. Then, for any
integer

$k\geq\max\left(\frac{1+p}{2p},\frac{\beta(p+1)}{4},\frac{\alpha(\theta+1)}{4}\right),$

there exists a positive constant $C=C(N,\epsilon,p,k)$ such that for any
$\zeta\in C_{c}^{2}(\mathbb{R}^{N})$ satisfying $0\leq\zeta\leq 1$,

$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy\leq
C\left[\int_{\mathbb{R}^{N}}\Big{(}|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\Big{)}^{\frac{p\beta}{2}}dxdy\right].\end{split}$
(2.3)

Here $\alpha$ and $\beta$ are defined in (1.4).

Proof. First, for any $\epsilon\in(0,1)$ and $\eta\in C^{2}(\mathbb{R}^{N})$,
there holds

$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}v^{1-p}[\Delta_{s}(u\eta)]^{2}dxdy&=\int_{\mathbb{R}^{N}}v^{1-p}\left(u\Delta_{s}\eta+2\nabla_{s}u\nabla_{s}\eta+\eta\Delta_{s}u\right)^{2}dxdy\\\
&\leq\left(1+C\epsilon\right)\int_{\mathbb{R}^{N}}v^{p+1}{\eta}^{2}dxdy+\frac{C}{\epsilon}\int_{\mathbb{R}^{N}}v^{1-p}u^{2}|\Delta_{s}\eta|^{2}dxdy\\\
&+\frac{C}{\epsilon}\int_{\mathbb{R}^{N}}v^{1-p}|\nabla_{s}u|^{2}|\nabla_{s}\eta|^{2}dxdy.\end{split}$
(2.4)

Take $\eta=\zeta^{2k}$ with $k\geq\frac{1}{p}+1>2$ and $0\leq\zeta\leq 1.$
Apply Young’s inequality, we get

$\displaystyle\int_{\mathbb{R}^{N}}v^{1-p}u^{2}|\Delta_{s}(\zeta^{2k})|^{2}dxdy$
$\displaystyle\leq
C_{k}\int_{\mathbb{R}^{N}}v^{1-p}u^{2}\left(|\Delta_{s}\zeta|^{2}+|\nabla_{s}\zeta|^{4}\right)\zeta^{\frac{4k(1-p)}{1+p}+\frac{2(4pk-2(1+p)}{1+p}}dxdy$
$\displaystyle\leq
C_{p,k}\epsilon^{2}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+C_{\epsilon,k,p}\int_{\mathbb{R}^{N}}u^{\frac{1+p}{p}}\left(|\Delta_{s}\zeta|^{2}+|\nabla_{s}\zeta|^{4}\right)^{\frac{p+1}{2p}}\zeta^{4k-2(\frac{1}{p}+1)}dxdy,$

and

$\displaystyle\int_{\mathbb{R}^{N}}v^{1-p}|\nabla_{s}u|^{2}|\nabla_{s}(\zeta^{2k})|^{2}dxdy$
$\displaystyle=4k^{2}\int_{\mathbb{R}^{N}}v^{1-p}|\nabla_{s}u|^{2}|\nabla_{s}\zeta|^{2}\zeta^{\frac{4k(1-p)}{1+p}+\frac{2(4pk-(1+p)}{1+p}}dxdydy$
$\displaystyle\leq
C_{p,k}\epsilon^{2}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\frac{C_{p,k}}{\epsilon^{2}}\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy.$

Combining all these inequalities with (2.4), we get the following estimates

$\displaystyle\begin{split}&\int_{\mathbb{R}^{N}}v^{1-p}[\Delta_{s}(u\zeta^{2k})]^{2}dxdy\\\
&\leq\left(1+C_{p,k}\epsilon\right)\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\frac{C_{p,k}}{\epsilon^{3}}\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
&+C_{\epsilon,k,p}\int_{\mathbb{R}^{N}}u^{\frac{1+p}{p}}\left(|\Delta_{s}\zeta|^{2}+|\nabla_{s}\zeta|^{4}\right)^{\frac{p+1}{2p}}\zeta^{4k-2(\frac{1}{p}+1)}dxdy.\end{split}$
(2.5)

We will use also the following lemma

###### Lemma 2.2

Let $k\geq\frac{1+p}{2p}>1,$ there exists a positive constant
$C=C_{N,\epsilon,p,k}>0$ such that for any $\epsilon>0,$ and $\zeta\in
C_{c}^{\infty}(\mathbb{R}^{N})$ with $0\leq\zeta\leq 1$, there holds

$\displaystyle\;\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle\;\leq\epsilon^{4}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.$

Proof. Since the simple calculation implies that

$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy=&\;\int_{\mathbb{R}^{N}}\nabla_{s}u\cdot\nabla_{s}u|\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
=&\;-\int_{\mathbb{R}^{N}}\mathrm{div}_{s}\left(\nabla_{s}u|\nabla_{s}u|^{\frac{p+1}{p}-2}\right)u|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
&\;-\int_{\mathbb{R}^{N}}u|\nabla_{s}u|^{\frac{p+1}{p}-2}\nabla_{s}u\cdot\nabla_{s}\left(|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}\right)dxdy\\\
:=&\;Q_{1}+Q_{2},\end{split}$ (2.6)

where $\mathrm{div}_{s}=\mathrm{div}_{x}+|x|^{s}\mathrm{div}_{y}.$ hence the
first term on the right hand side of (2.6), can be estimated as

$\displaystyle Q_{1}$
$\displaystyle=-\left(\frac{p+1}{p}-2\right)\int_{\mathbb{R}^{N}}u|\nabla_{s}u|^{\frac{p+1}{p}-4}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\nabla_{s}^{2}u(\nabla_{s}u,\nabla_{s}u)\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle\;+\int_{\mathbb{R}^{N}}uv^{p}|\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle\leq
C_{p}\int_{\mathbb{R}^{N}}u|\nabla_{s}^{2}u||\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy+\int_{\mathbb{R}^{N}}uv^{p}|\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy.$

Observe that

$4k-\frac{p+1}{p}=\left(\frac{4k-\frac{p+1}{p}}{\frac{p+1}{p}}\right)\left(\frac{p+1}{p}-2\right)+2\left(\frac{4k-\frac{p+1}{p}}{\frac{p+1}{p}}\right),$

and applying Young’s inequality, there holds, for any $\epsilon>0$,

$\displaystyle\begin{split}&\int_{\mathbb{R}^{N}}uv^{p}|\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}-2+2}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
\leq&\;C_{\epsilon,p}\int_{\mathbb{R}^{N}}\left(uv^{p}\right)^{\frac{p+1}{2p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy+C_{p}\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
\leq&\;C_{\epsilon,p}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}\zeta^{4k-\frac{2(p+1)}{p}}dxdy+C_{p}\epsilon^{4}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy\\\
&+C_{p}\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy.\end{split}$
(2.7)

and

$\displaystyle\begin{split}&\;C_{p}\int_{\mathbb{R}^{N}}u|\nabla_{s}^{2}u||\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}-2+2}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
\leq&\;C_{\epsilon,p}\int_{\mathbb{R}^{N}}\left(u|\nabla_{s}^{2}u|\right)^{\frac{p+1}{2p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy++C_{p}\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
\leq&\;C_{\epsilon,p}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}\zeta^{4k-\frac{2(p+1)}{p}}dxdy+C_{p}\epsilon^{4}\int_{\mathbb{R}^{N}}|\nabla_{s}^{2}u|^{\frac{p+1}{p}}\zeta^{4k}dxdy\\\
&+C_{p}\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy.\end{split}$
(2.8)

Now we shall estimate the integral

$\int_{\mathbb{R}^{N}}|\nabla_{s}^{2}u|^{\frac{p+1}{p}}\zeta^{4k}dxdy.$

For the last inequality, we have used the following estimates from [[18],
Proof of Theorem 2.6]

$\|\nabla_{x_{i}x_{j}}u\|_{L^{2}(\mathbb{R}^{N_{1}})}\leq
C(N_{1})\|\Delta_{x}u\|_{L^{2}(\mathbb{R}^{N_{1}})},\quad\mbox{for}\quad 1\leq
i,j\leq N_{1},$

and

$\|\nabla_{y_{h}y_{k}}u\|_{L^{2}(\mathbb{R}^{N_{2}})}\leq
C(N_{2})\|\Delta_{y}u\|_{L^{2}(\mathbb{R}^{N_{2}})},\quad\mbox{for}\quad 1\leq
h,k\leq N_{2}.$

Let $\psi^{r}\in C_{0}^{\infty}(\mathbb{R}^{N}),$ with $r>2.$ By direct
calculations, we get

$\displaystyle|\nabla_{s}^{2}(u)|\psi^{r}\leq$
$\displaystyle\;C_{r}\Big{[}u\left(|\nabla_{s}\psi|^{2}\psi^{r-2}+|\nabla_{s}^{2}\psi|\psi^{r-1}\right)+|\nabla_{s}u||\nabla_{s}\psi|\psi^{r-1}+|\nabla_{s}^{2}(u\psi^{r})|\Big{]}.$

Consider $\psi=\zeta,$ and $r=\frac{4kp}{p+1}\geq 2,$ so that
$k\geq\frac{p+1}{2p}.$ We can claim, for any $0\leq\zeta\leq 1,$ there exists
$C_{p,k}>0$ such that

$\displaystyle\begin{split}&\;\int_{\mathbb{R}^{N}}|\nabla_{s}^{2}u|^{\frac{p+1}{p}}\zeta^{4k}dxdy\\\
&\;\leq
C_{p,k}\int_{\mathbb{R}^{N}}|\nabla_{s}^{2}(u\zeta^{\frac{4kp}{p+1}})|^{\frac{p+1}{p}}dxdy+C_{p,k}\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy\\\
&\;+C_{p,k}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.\end{split}$
(2.9)

Clearly $u\zeta\in H_{0}^{2}(\mathbb{R}^{N}),$ so from a standard
approximation and scaling argument imply then

$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}|\nabla_{s}^{2}(u\zeta^{\frac{4kp}{p+1}})|^{\frac{p+1}{p}}dxdy&\leq\;C_{N,p}\int_{\mathbb{R}^{N}}|\Delta_{s}(u\zeta^{\frac{4kp}{p+1}})|^{\frac{p+1}{p}}dxdy\\\
&\leq\;C_{N,p,k}\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy+C_{N,p}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy\\\
&+\;C_{N,p,k}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.\end{split}$
(2.10)

Combining (2.7)–(2.10), we obtain the estimate for the first left term in
(2.6):

$\displaystyle Q_{1}\leq$
$\displaystyle\;C_{N,p,k}\epsilon^{4}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+C_{N,p,k}\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle+\;C_{N,p,k}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.$

Furthermore, by Young’s inequality,

$\displaystyle Q_{2}=$
$\displaystyle-\frac{p+1}{p}\int_{\mathbb{R}^{N}}u\Big{(}|\nabla_{s}u||\nabla_{s}\zeta|\Big{)}^{\frac{p+1}{p}-2}\nabla_{s}^{2}\zeta(\nabla_{s}\zeta,\nabla_{s}u)\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle-(4k-\frac{p+1}{p})\int_{\mathbb{R}^{N}}u|\nabla_{s}u|^{\frac{p+1}{p}-2}|\nabla_{s}\zeta|^{\frac{p+1}{p}}(\nabla_{s}u\cdot\nabla_{s}\zeta)\zeta^{4k-\frac{p+1}{p}-1}dxdy$
$\displaystyle\leq$
$\displaystyle\;C_{p,k}\int_{\mathbb{R}^{N}}u\Big{(}|\nabla_{s}u||\nabla_{s}\zeta|\Big{)}^{\frac{p+1}{p}-1}\left(|\nabla_{s}\zeta|^{2}+|\nabla_{s}^{2}\zeta|\right)\zeta^{4k-\frac{p+1}{p}-1}dxdy$
$\displaystyle\leq$
$\displaystyle\;C_{\epsilon,p,k}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy+\epsilon\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy.$

Combining the last tow inequality with (2.6), we get readily

$\displaystyle\;(1-C_{N,p,k}\epsilon)\int_{\mathbb{R}^{N}}|\nabla_{s}u|^{\frac{p+1}{p}}|\nabla_{s}\zeta|^{\frac{p+1}{p}}\zeta^{4k-\frac{p+1}{p}}dxdy$
$\displaystyle\leq
C_{\epsilon,p,N,k}\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy$
$\displaystyle\;+C_{N,p,k}\epsilon^{4}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy.$

Take $\epsilon$ small enough, the lemma follows.

Now, using Lemma 2.2 and (2.5) , we obtain also

$\displaystyle\begin{split}\int_{\mathbb{R}^{N}}v^{1-p}[\Delta_{s}(u\zeta^{2k})]^{2}dxdy\leq&\;C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy\\\
&\;+\left(1+C\epsilon\right)\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy.\end{split}$
(2.11)

Thanks to the approximation argument, the stability property (1.7), with
$\gamma=u\zeta^{2k}$ where $\zeta\in C_{0}^{\infty}(\mathbb{R}^{N}),$ we get

$\displaystyle\begin{split}&\;\theta
p\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy-\left(1+C\epsilon\right)\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy\\\
\leq&\;C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.\end{split}$
(2.12)

Multiplying the equation $-\Delta_{s}v=u^{\theta}$ by $u\zeta^{4k}$ and
integrating by parts, there holds

$\displaystyle\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy-\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy\leq\int_{\mathbb{R}^{N}}uv|\Delta(\zeta^{4k})|dxdy+\int_{\mathbb{R}^{N}}v|\nabla_{s}u||\nabla_{s}(\zeta^{4k})|dxdy.$

Using Young’s inequality and applying again Lemma 2.2, we can conclude that
for any $\epsilon>0$, there exists $C=C_{N,\epsilon,p,k}>0$ such that

$\displaystyle\begin{split}&\;\left(1-C\epsilon\right)\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy-\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy\\\
\leq&\;C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.\end{split}$
(2.13)

Now, multiplying (2.13) by $\frac{1+2C\epsilon}{1-C\epsilon}$, adding it with
(2.12), we get

$\displaystyle
C\epsilon\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\left(p\theta-\frac{1+2C\epsilon}{1-C\epsilon}\right)\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy$
$\displaystyle\leq$
$\displaystyle\;C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.$

As $\theta>p^{-1}>1.$ Fix $0<\epsilon<\frac{p\theta-1}{C(p\theta+2)}$, there
holds

$\displaystyle\begin{split}&\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy\\\
\leq&\;C\int_{\mathbb{R}^{N}}u^{\frac{p+1}{p}}\left(|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\right)\zeta^{4k-\frac{2(p+1)}{p}}dxdy.\end{split}$
(2.14)

Apply Young’s inequality, we deduce then

$\displaystyle\;\int_{\mathbb{R}^{N}}v^{p+1}\zeta^{4k}dxdy+\int_{\mathbb{R}^{N}}u^{\theta+1}\zeta^{4k}dxdy$
$\displaystyle\leq$
$\displaystyle\;C_{\epsilon}\int_{\mathbb{R}^{N}}\Big{(}|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\Big{)}^{\frac{p\beta}{2}}dxdy+\epsilon^{\prime}\int_{\mathbb{R}^{N}}|u|^{\theta+1}\zeta^{\frac{4kp(\theta+1)}{p+1}-2(\theta+1)}dxdy$
$\displaystyle\leq$
$\displaystyle\;C_{\epsilon}\int_{\mathbb{R}^{N}}\Big{(}|\Delta_{s}\zeta|^{\frac{p+1}{p}}+|\nabla_{s}\zeta|^{\frac{2(p+1)}{p}}+|\nabla_{s}^{2}\zeta|^{\frac{p+1}{p}}\Big{)}^{\frac{p\beta}{2}}dxdy+\epsilon^{\prime}\int_{\mathbb{R}^{N}}|u|^{\theta+1}\zeta^{4k}dxdy.$

Choose $k\geq\frac{\beta(p+1)}{4}=\frac{\alpha(\theta+1)}{4}$ so that
$4k\leq\frac{4kp(\theta+1)}{p+1}-2(\theta+1),$ for the last line. Take
$\epsilon^{\prime}$ small enough, the estimate (2.3) is proved.

We are now in position to conclude. Choose $\chi$ a cut-off function in
$C_{c}^{\infty}\left(\mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}},[o,1]\right),$
such that

$\chi=1\quad\mbox{on}\quad B_{1}\times
B_{2},\quad\mbox{and}\quad\chi=0\quad\mbox{outside }\quad B_{2}\times
B_{2^{1+s}}.$

For $R>0,$ put $\eta_{R}(x,y)=\chi(\frac{x}{R},\frac{y}{R^{1+s}}),$ it is easy
to verify that there exists $C>0$ independent of $R$ such that

$|\nabla_{x}\eta_{R}|\leq\frac{C}{R}\quad\mbox{and}\quad|\nabla_{y}\eta_{R}|\leq\frac{C}{R^{1+s}},$

$|\nabla^{2}_{x}\eta_{R}|+|\Delta_{x}\eta_{R}|\leq\frac{C}{R^{2}}\quad\mbox{and}\quad|\nabla^{2}_{y}\eta_{R}|+|\Delta_{y}\eta_{R}|\leq\frac{C}{R^{2(1+s)}}.$

Applying (2.3) with $\zeta=\eta_{R}(x,y),$ we get

$\displaystyle\int_{B_{R}\times
B_{R^{1+s}}}u^{\theta+1}dxdy\leq\int_{\mathbb{R}^{N}}u^{\theta+1}\eta_{R}^{4k}dxdy\leq
CR^{\frac{2N_{s}}{p\beta}-\frac{2(p+1)}{p}}.$

Since under our assumptions
$N_{s}<\frac{2(p+1)(\theta+1)}{p\theta-1}:=2+\alpha+\beta,$ the desired claim
follows by letting $R\to\infty.$

###  2.2  The case $\theta>p\geq 1$.

Let us recall a consequence of Theorem 1.1 (with $\alpha=0$ there) in [16].

###### Theorem B

Let $x_{0}$ be the largest root of the polynomial

$\displaystyle
H(x)=x^{4}-p\theta\alpha\beta\left[4x^{2}-2(\alpha+\beta)x+1\right].$ (2.15)

  1. 1.

If $\frac{4}{3}<p\leq\theta$ then (1.1) has no stable solution if
$N_{s}<2+2x_{0}.$

  2. 2.

If $1<p\leq\min(\frac{4}{3},\theta)$, then (1.1) has no bounded stable
solution, if

$N_{s}<2+2x_{0}\left[\frac{p}{2}+\frac{(2-p)(p\theta-1)}{(\theta+p-2)(\theta+1)}\right].$

For $\theta>p\geq 1,$ we can proceed similarly as the proof of Proposition in
[15], we sketch a proof here for the reader’s convenience. Performing the
change of variables $x=\frac{\beta}{2}z$ in (2.15), a direct computation shows
that $H(x)=\left(\frac{\beta}{2}\right)^{4}L(z)$ where

$L(z):=z^{4}-\frac{16p\theta(p+1)}{\theta+1}z^{2}+\frac{16p\theta(p+1)(p+\theta+2)}{(\theta+1)^{2}}z-\frac{16p\theta(p+1)^{2}}{(\theta+1)^{2}}.$

Denote by $z_{0}$ the largest root of $L,$ hence $x_{0}=\frac{\beta}{2}z_{0}$
and $H(x)<0$ if and only if $L(z)<0$. For $\theta>p\geq 1$, there holds

$\displaystyle L(p+1)$
$\displaystyle=(p+1)^{4}-\frac{16p\theta(p+1)^{3}}{(\theta+1)}+\frac{16p\theta(p+1)^{2}(p+\theta+2)}{(\theta+1)^{2}}-\frac{16p\theta(p+1)^{2}}{(\theta+1)^{2}}$
$\displaystyle=(p+1)^{4}-\frac{16p\theta(p+1)^{3}}{(\theta+1)}+\frac{16p\theta(p+1)^{2}}{(\theta+1)}+\frac{16p\theta(p+1)^{3}}{(\theta+1)^{2}}-\frac{16p\theta(p+1)^{2}}{(\theta+1)^{2}}$
$\displaystyle=(p+1)^{2}\left[(p+1)^{2}-\frac{16p^{2}\theta}{(\theta+1)}+\frac{16p^{2}\theta}{(\theta+1)^{2}}\right]$
$\displaystyle=\left(\frac{p+1}{\theta+1}\right)^{2}\left[(p+1)^{2}(\theta+1)^{2}-16p^{2}\theta^{2}\right]<0.$

The last inequality holds true since

$4p\theta-(p+1)(\theta+1)>4p^{2}-(p+1)^{2}\geq 0,\quad\forall\;\theta>p\geq
1.$

As $\lim_{z\rightarrow\infty}L(z)=\infty,$ it follows that $z_{0}>p+1.$ We get
then

$\displaystyle 2x_{0}>(p+1)\beta=2+\alpha+\beta,\quad\forall\;\theta>p\geq 1.$

If $p>\frac{4}{3}$, by $(i)$ of Theorem B, the system (1.1) has no classical
stable solution if $N_{s}<2+\alpha+\beta$. Suppose now $1\leq
p\leq\min(\frac{4}{3},\theta)$. Observe that for all $\theta\geq p\geq 1$,

$\displaystyle\left[p+\frac{2(2-p)(p\theta-1)}{(\theta+p-2)(\theta+1)}\right]\beta\geq\alpha+\beta$
$\displaystyle\Leftrightarrow\left[p+\frac{2(2-p)(p\theta-1)}{(\theta+p-2)(\theta+1)}\right](\theta+1)\geq
p+\theta+2$ $\displaystyle\Leftrightarrow
p\theta-1+\frac{2(2-p)(p\theta-1)}{\theta+p-2}\geq\theta+1$
$\displaystyle\Leftrightarrow(p\theta-1)\left[1+\frac{2(2-p)}{\theta+p-2}\right]\geq\theta+1$
$\displaystyle\Leftrightarrow(p\theta-1)(\theta+2-p)\geq(\theta+p-2)(\theta+1)$
$\displaystyle\Leftrightarrow
p\theta^{2}-\theta+(2-p)p\theta\geq\theta^{2}+(p-1)\theta$
$\displaystyle\Leftrightarrow(p-1)(\theta-p)\geq 0.$

As $z_{0}>p+1\geq 2,$ we have $x_{0}=\frac{\beta z_{0}}{2}\geq\beta$ and

$2+\alpha+\beta\leq
2+\beta\left[p+\frac{2(2-p)(p\theta-1)}{(\theta+p-2)(\theta+1)}\right]\leq
2+x_{0}\left[p+\frac{2(2-p)(p\theta-1)}{(\theta+p-2)(\theta+1)}\right].$

If $N_{s}<2+\alpha+\beta,$ using $(ii)$ of Theorem B, we are done. To conclude

$\displaystyle\mbox{ for all}\quad\theta>p\geq 1,\mbox{ and}\quad
N_{s}<2+\alpha+\beta,\quad\eqref{1.1}\,\mbox{has no smooth stable solution}.$

The proof is finished.

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