Liouville-type theorems with finite Morse index for \Delta_{\lambda}-Laplace operator

TL;DR

This paper establishes Liouville-type theorems for solutions of a degenerate elliptic PDE with finite Morse index, showing nonexistence of certain stable solutions under specific conditions.

Contribution

It introduces new Liouville-type theorems for the bla_{\u03bb}-Laplace operator, extending stability analysis to solutions possibly unbounded and sign-changing.

Findings

Proves nonexistence of stable solutions under certain conditions.

Derives integral estimates using stability properties.

Uses Pohozaev identity to extend results to solutions stable outside compact sets.

Abstract

In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_{\lambda} is a strongly degenerate elliptic operator, the functions \lambda=(\lambda_1, ..., \lambda_k) : R^n \rightarrow R^k, satisfies some certain conditions, and |.|_{\lambda} the homogeneous norm associated to the \D_{\lambda}-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of R^n. First, we establish the standard integralestimates via stability property to derive the nonexistence results for stable solutions. Next, by mean of the Pohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compact set.

Equations

$$-{\Delta }_{\lambda }u=|x{|}_{\lambda }^a|u{|}^{p-1}u,\text{mbox}in{\mathbb{R}}^n,n\ge 1,p>1,\text{mbox}anda\ge 0,$$

$$-\Delta u=|u{|}^{p-1}u\text{mbox}in{\mathbb{R}}^n,p>1,$$

$$-L_{\alpha }u=f(u)\text{mbox}in{\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2},$$

$$-{\Delta }_{\lambda }u=|x{|}_{\lambda }^a|u{|}^{p-1}u,\text{mbox}in{\mathbb{R}}^n:={\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times ...\times {\mathbb{R}}^{n_k},$$

$${\Delta }_{\lambda }={\lambda }_1^2{\Delta }_{x^{(1)}}+...+{\lambda }_k^2{\Delta }_{x^{(k)}},|x{|}_{\lambda }:={\left(\sum _{j=1}^k\prod _{i\ne j}{\lambda }_i^2(x){ϵ}_j^2|x^{(j)}{|}^2\right)}^{\frac{1}{2\sigma }},$$

$${\delta }_t:{\mathbb{R}}^n\to {\mathbb{R}}^n,{\delta }_t(x)={\delta }_t(x^{(1)},...,x^{(k)})=(t^{{ϵ}_1}{x}^{(1)},...,t^{{ϵ}_k}{x}^{(k)}),$$

$${\lambda }_i({\delta }_t(x))=t^{{ϵ}_i-1}{\lambda }_i(x),\forall x\in {\mathbb{R}}^n,t>0,i=1,...,k.$$

$${\Delta }_{\lambda }(u({\delta }_t(x)))=t^2({\Delta }_{\lambda }u)({\delta }_t(x)),\forall u\in C^{\infty }({\mathbb{R}}^n).$$

$$Q:={ϵ}_1{n}_1+{ϵ}_2{n}_2+...+{ϵ}_k{n}_k.$$

$$x=(x^{(1)},...,x^{(k)}),x^{(i)}=(x_1^{(i)},...,x_{n_i}^{(i)})\in {\mathbb{R}}^{n_i},$$

$$|x^{(i)}{|}^2=\sum _{j=1}^{n_i}|x_j^{(i)}{|}^2,i=1,2,...,k.$$

$${\Delta }_{x^{(i)}}=\sum _{j=1}^{n_i}{\partial }_{x_j^{(i)}}^2.$$

$${\Delta }_{\lambda }={\Delta }_x+|x{|}^{2\alpha }{\Delta }_y,$$

$${\lambda }_1(x)=1,{\lambda }_2(x)=|x^{(1)}{|}^{\alpha },x\in {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}.$$

$${\delta }_t(x)={\delta }_t(x^{(1)},x^{(2)})=(t{x}^{(1)},t^{\alpha +1}{x}^{(2)}),$$

$${\Delta }_{\alpha }:={\Delta }_{x^{(1)}}+|x^{(1)}{|}^{2{\alpha }_1}{\Delta }_{x^{(2)}}+...+|x^{(k-1)}{|}^{2{\alpha }_{k-1}}{\Delta }_{x^{(k)}}.$$

$${\delta }_t:{\mathbb{R}}^n\to {\mathbb{R}}^n,{\delta }_t(x)={\delta }_t(x^{(1)},...,x^{(k)})=(t^{{ϵ}_1}{x}^{(1)},...,t^{{ϵ}_k}{x}^{(k)}),$$

$${\Delta }_{\alpha }={\Delta }_{x^{(1)}}+|x^{(1)}{|}^2{\Delta }_{x^{(2)}}+...+|x^{(k-1)}{|}^2{\Delta }_{x^{(k)}},$$

$${\delta }_t(x)=(t{x}^{(1)},t^2{x}^{(2)},...,t^k{x}^{(k)}).$$

$${\Delta }_{\lambda }={\Delta }_{x^{(1)}}+|x^{(1)}{|}^{2\alpha }{\Delta }_{x^{(2)}}+|x^{(1)}{|}^{2\beta }|x^{(2)}{|}^{2\gamma }{\Delta }_{x^{(3)}},$$

$${\lambda }_1(x)=1,{\lambda }_2(x)=|x^{(1)}{|}^{\alpha },{\lambda }_3(x)=|x^{(1)}{|}^{\beta }|x^{(2)}{|}^{\gamma },x\in {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}\times {\mathbb{R}}^{n_3},$$

$${\delta }_t(x)={\delta }_t(x^{(1)},x^{(2)},x^{(3)})=(t{x}^{(1)},t^{\alpha +1}{x}^{(2)},t^{\beta +(\alpha +1)\gamma +1}{x}^{(3)}).$$

$$Q_u(\psi ):={\int }_{{\mathbb{R}}^n}|{\nabla }_{\lambda }\psi {|}^2-p{\int }_{{\mathbb{R}}^n}|x{|}_{\lambda }^a|u{|}^{p-1}{\psi }^2\ge 0,\forall \psi \in C_c^1({\mathbb{R}}^n),$$

$${\int }_{{\Omega }_R}\left(|x{|}_{\lambda }^a|u{|}^{p+\gamma }+|{\nabla }_{\lambda }(|u{|}^{\frac{\gamma -1}{2}}u){|}^2\right)dx\le C{R}^{Q-\frac{2(p+\gamma )+(\gamma +1)a}{p-1}},\text{mbox}forallR>0.$$

$$p_c(Q,a)=\{\begin{array}{ll}+\infty & \text{if Q≤10+4a},\\ \frac{(Q-2{)}^2-2(a+2)(a+Q)+2\sqrt{(a+2{)}^3(a+2Q-2)}}{(Q-2)(Q-4a-10)}& \text{if Q>10+4a}.\end{array}$$

$$T=\sum _{i=1}^k{ϵ}_i{x}^{(i)}{\nabla }_{x^{(i)}},$$

$${\int }_{{\Omega }_R}\left[\frac{Q-2}{2}|{\nabla }_{\lambda }u{|}^2-\frac{Q+a}{p+1}|x{|}_{\lambda }^a|u{|}^{p+1}\right]\varphi ={\int }_{{\Omega }_R}\left[{\nabla }_{\lambda }u{\nabla }_{\lambda }\varphi T(u)+\left[-\frac{1}{2}|{\nabla }_{\lambda }u{|}^2+\frac{|x{|}_{\lambda }^a}{p+1}|u{|}^{p+1}\right]T(\varphi )\right].$$

$$p_s(Q,a)=\{\begin{array}{ll}+\infty & \text{if Q≤2,}\\ \frac{Q+2+2a}{Q-2}& \text{if Q>2.}\end{array}$$

$${\Delta }_{\lambda }={\lambda }_1^2{\Delta }_{x^{(1)}}+...+{\lambda }_k^2{\Delta }_{x^{(k)}}.$$

$${\lambda }_i(x)=\prod _{j=1}^k|x^{(j)}{|}^{{\alpha }_{ij}},i=1,...,k,$$

Full text

[TABLE]

where $\Delta_{\lambda}$ is a strongly degenerate elliptic operator, the functions $\lambda=(\lambda_{1},...,\lambda_{k}):\;\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$ satisfies some certain conditions, and $|.|_{\lambda}$ the homogeneous norm associated to the $\Delta_{\lambda}$-Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of $\mathbb{R}^{n}$. First, we establish the standard integral estimates via stability property to derive the nonexistence results for stable solutions. Next, by mean of the Pohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compact set.

††journal: arXiv

1 Introduction and main results

The Liouville type theorem is the nonexistence of solutions in the entire space or in half-space. The classical Liouville type theorem stated that a bounded harmonic (or holomorphic) function defined in entire space must be constant. This theorem, known as Liouville theorem, was first announced in 1844 by Liouville [15] for the special case of a doubly-periodic function. Later in the same year, Cauchy [3] published the first proof of the above stated theorem. This classical result has been extended to nonnegative solutions of the semilinear elliptic equation

[TABLE]

in the whole space $\mathbb{R}^{n}$ by Gidas and Spruck [10, 11] see also the paper of Chen and Li [4]. They proved that if $1<p<\frac{n+2}{n-2}$, then the above equation only has the trivial solution $u\equiv 0$ and this result is optimal. In an elegant paper, Farina [7] proved that nontrivial finite Morse index solutions (whether positive or sign changing) to (1.1) exists if and only if $p\geq p_{c}(n)$ and $n\geq 11$, or $p=\frac{n+2}{n-2}$ and $n\geq 3$, where $p_{c}(n)$ is the so-called Joseph-Lundgren exponent. The study of stable solutions in the Hénon type elliptic equation: $-\Delta u=|x|^{a}|u|^{p-1}u,\;\;\mbox{in}\,\,\mathbb{R}^{n},\;p>1\;\mbox{and}\;a>-2$ has been studied recently, Wang and Ye [23] gave a complete classification of stable weak solutions and those of finite Morse index solutions.

In the past years, the Liouville property has been refined considerably and emerged as one of the most powerful tools in the study of initial and boundary value problems for nonlinear PDEs. It turns out that one can obtain from Liouville-type theorems a variety of results on qualitative properties of solutions such as universal, pointwise, a priori estimates of local solutions; universal and singularity estimates; decay estimates; blow-up rate of solutions of nonstationary problems, etc., see [19, 21] and references therein.

Liouville-type theorems for degenerate elliptic equations have been attracted the interest of many mathematicians. The classical Liouville theorem was generalized to $p$-harmonic functions on the whole space $\mathbb{R}^{n}$ and on exterior domains by Serrin and Zou [22], see also [5] for related results. The Liouville theorems for some linear degenerate elliptic operators such as $X$-elliptic operators, Kohn-Laplacian (and more general sublaplacian on Carnot groups) and degenerate Ornstein-Uhlenbeck operators were proved in [14, 13].

More recently, Yu [24] studied the equation

[TABLE]

where $L_{\alpha}=\Delta_{x}+(1+\alpha)^{2}\Delta_{y}$, $\alpha>0$ and $Q=n_{1}+(1+\alpha)n_{2}$ is the homogeneous dimension of the space. Under some assumptions on the nonlinear term $f$ , he showed that the above equation possesses no positive solutions and the main technique used is the moving plane method in the integral form.

In this paper, we are concerned with the Liouville-type theorems for the following problem

[TABLE]

where $n\geq 1$, $a\geq 0$, $p>1$,

[TABLE]

$\sigma=1+\sum_{i=1}^{k}(\epsilon_{i}-1)$, $1\leq\epsilon_{1}\leq...\leq\epsilon_{k}$, $x=(x^{(1)},...,x^{(k)})\in\mathbb{R}^{n}$. Here the functions $\lambda_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ are continuous, strictly positive and of class $C^{1}$ outside the coordinate hyperplanes, i.e. $\lambda_{i}>0$, $i=1,...,k$ in $\mathbb{R}^{n}\backslash\prod$, where $\prod=\{x=(x_{1},...,x_{n})\in\mathbb{R}^{n}:\prod_{i=1}^{n}x_{i}=0\}$, and $\Delta_{x^{(i)}}$ denotes the classical Laplacian in $\mathbb{R}^{n_{i}}$, $i=1,...,k$. As in [12] we assume that $\lambda_{i}$ satisfy the following properties:

$(H_{1})$ $\lambda_{1}(x)=1$, $\lambda_{i}(x)=\lambda_{i}(x^{(1)},...,x^{(i-1)})$, $i=2,...,k$.

$(H_{2})$ For every $x\in\mathbb{R}^{n}$, $\lambda_{i}(x)=\lambda_{i}(x^{*})$, $i=1,...,k$, where $x^{*}=(|x^{(1)}|,...,|x^{(k)}|)$ if $x=(x^{(1)},...,x^{(k)})$.

$(H_{3})$ There exists a group of dilations $\{\delta_{t}\}_{t>0}$

[TABLE]

where $1\leq\epsilon_{1}\leq\epsilon_{2}\leq...\leq\epsilon_{k}$, such that $\lambda_{i}$ is $\delta_{t}$-homogeneous of degree $\epsilon_{i}-1$, i.e.

[TABLE]

This implies that the operator $\Delta_{\lambda}$ is $\delta_{t}$-homogeneous of degree two, i.e.

[TABLE]

We denote by $Q$ the *homogeneous dimension * of $\mathbb{R}^{n}$ with respect to the group of dilations $\{\delta_{t}\}_{t>0}$, i.e.

[TABLE]

The $\Delta_{\lambda}$-Laplace operator was first introduced by Franchi and Lanconelli [8], and recently reconsidered in [12] under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation in $\mathbb{R}^{n}$. It was proved in [1], that the autonomous case, i.e. $a=0$, (1.2) has no positive classical solution if $1<p\leq\frac{Q}{Q-2}$, with $Q=\epsilon_{1}+\epsilon_{2}+...+\epsilon_{n}$, ($n_{i}=1$, $i=1,...,n$).

The $\Delta_{\lambda}$-operator contains many degenerate elliptic operators. We now give some examples of $\Delta_{\lambda}$-Laplace operators (see also [12]). We use the following notation: we split $\mathbb{R}^{n}$ as follows $\mathbb{R}^{n}=\mathbb{R}^{n_{1}}\times...\times\mathbb{R}^{n_{k}}$ and write

[TABLE]

[TABLE]

We denote the classical Laplace operator in $\in\mathbb{R}^{n_{i}}$ by

[TABLE]

Example 1. Let $\alpha$ be a real positive constant and $k=2$. We consider the Grushin-type operator

[TABLE]

where $\lambda=(\lambda_{1},\lambda_{2})$ with

[TABLE]

Our group of dilations is

[TABLE]

and the homogenous dimension with respect to $(\delta_{t})_{t>0}$ is $Q=n_{1}+(\alpha+1)n_{2}$.

Example 2. Given a multi-index $\alpha=(\alpha_{1},...,\alpha_{k-1})$, $\alpha_{j}\geq 1$, $j=1,...,k-1$, define

[TABLE]

Then $\Delta_{\alpha}=\Delta_{\lambda}$ with $\lambda=(\lambda_{1},...,\lambda_{k})$ and $\lambda_{i}=|x^{(i-1)}|^{\alpha_{i-1}}$, $i=1,...,k$. Here we agree to let $|x^{(0)}|^{\alpha_{0}}=1$. A group of dilations for which $\lambda$ satisfies $(H_{3})$ is given by

[TABLE]

with $\epsilon_{1}=1$ and $\epsilon_{i}=\alpha_{i-1}\epsilon_{i-1}+1$, $i=2,...,k$. In particular, if $\alpha_{1}=...=\alpha_{k-1}=1$, the operator $\Delta_{\alpha}$ and the dilation $\delta_{t}$ becomes, respectively

[TABLE]

and

[TABLE]

Example 3. Let $\alpha$, $\beta$ and $\gamma$ be positive real constants. For the operator

[TABLE]

where $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})$ with

[TABLE]

we find the group of dilations

[TABLE]

The aim of the present paper was to establish the Liouville-type theorems with finite Morse index for the equation (1.2). In order to state our results we need the following:

Definition 1.1

*We say that a solution $u$ of (1.2) belonging to $C^{2}(\mathbb{R}^{n})$

$\bullet$ is stable, if*

[TABLE]

*where $\nabla_{\lambda}=(\lambda_{1}\nabla_{x^{(1)}},...,\lambda_{k}\nabla_{x^{(k)}})$.

$\bullet$ has Morse index equal to $K\geq 1$ if $K$ is the maximal dimension of a subspace $X_{K}$ of $C^{1}_{c}(\mathbb{R}^{n})$ such that $Q_{u}(\psi)<0$ for any $\psi\in X_{K}\backslash\{0\}$.

$\bullet$ is stable outside a compact set $\mathcal{K}\subset\mathbb{R}^{n}$ if $Q_{u}(\psi)\geq 0$ for any $\psi\in C^{1}_{c}(\mathbb{R}^{n}\backslash\mathcal{K})$.*

Remark 1.1

**a) Clearly, a solution stable if and only if its Morse index is equal to zero.

b) It is well know that any finite Morse index solution $u$ is stable outside a compact set $\mathcal{K}\subset\mathbb{R}^{n}$. Indeed, there exists $m_{0}\geq 1$ and $X_{m_{0}}:=\mbox{Span}\{\phi_{1},...,\phi_{m_{0}}\}\subset C^{1}_{c}(\mathbb{R}^{n})$ such that $Q_{u}(\phi)<0$ for any $\phi\in X_{m_{0}}\backslash\{0\}$. Hence, $Q_{u}(\psi)\geq 0$ for every $\psi\in C^{1}_{c}(\mathbb{R}^{n}\backslash\mathcal{K})$, where $\mathcal{K}:=\displaystyle{\cup_{j=1}^{m_{0}}}supp(\phi_{j})$.**

In the following, we state Liouville-type results for solutions $u\in C^{2}(\mathbb{R}^{n})$ of (1.2). In what follows, we divide our study to stable solutions and solutions which are stable outside a compact set.

1.1 Stable solutions

To state the following result we need to introduce some notation. We set $\Gamma_{M}(p)=2p-1+2\sqrt{p(p-1)}$ and denote by $\Omega_{R}=B_{1}(0,R^{\epsilon_{1}})\times B_{2}(0,R^{\epsilon_{2}})\times...\times B_{k}(0,R^{\epsilon_{k}})$, where $B_{i}(0,R^{\epsilon_{i}})\subset\mathbb{R}^{n_{i}}$, $i=1,...,k$, the balls of center [math] and radius $R^{\epsilon_{i}}$.

Proposition 1.1

Let $u\in C^{2}(\mathbb{R}^{n})$ be a stable solution of (1.2). Then, for any $\gamma\in\left[1,\Gamma_{M}(p)\right)$, there exists a positive constant $C$ independent of $R$, such that

[TABLE]

Proposition 1.1 provides an important estimate on the integrability of $u$ and $\nabla_{\lambda}u$. As we will see, our nonexistence results will follow by showing that the right-hand side of (1.3) vanishes under the right assumptions on $p$ when $R\rightarrow+\infty$. More precisely, as a corollary of Proposition 1.1, we can state our first Liouville type theorem.

Theorem 1.1

Let $u\in C^{2}(\mathbb{R}^{n})$ be a stable solution of (1.2) with,

[TABLE]

Then $u\equiv 0$.

1.2 Solutions which are stable outside a compact set

In this subsection we prove some integral identities extending to the $\Delta_{\lambda}$ setting the classical Pohozaev identity for semilinear Poisson equation [18]. Pohozaev identity has been extended by several authors to general elliptic equations and systems, both in Riemannian and sub-Riemannian context, see, e.g., [2, 9, 20] and the references therein. To prove our identities we closely follow the original procedure of Pohozaev, just replacing the vector field $P=\sum_{i=1}^{n}x_{i}\partial_{x_{i}}$ in [18], page $1410$], by

[TABLE]

the generator of the group of dilation $(\delta_{t})_{t\geq 0}$ in $(H_{3})$(we say that $T$ generates $(\delta_{t})_{t\geq 0}$ since a function $u$ is $\delta_{t}$-homogeneous of degree $m$ if and only if $Tu=mu$).

Proposition 1.2

Let $u\in C^{2}(\mathbb{R}^{n})$ be a solution of (1.2) and $\phi\in C^{1}_{c}(\Omega_{R})$. If $T\left(|x|_{\lambda}\right)=|x|_{\lambda}$, then

[TABLE]

Thanks to Proposition 1.2, we derive

Theorem 1.2

Let $u\in C^{2}(\mathbb{R}^{n})$ be a solution of (1.2) which is stable outside a compact set of $\mathbb{R}^{n}$, with

[TABLE]

If $T\left(|x|_{\lambda}\right)=|x|_{\lambda}$, then $u\equiv 0$.

2 Example which satisfies $T\left(|x|_{\lambda}\right)=|x|_{\lambda}$

The degenerate elliptic operators we consider are of the form

[TABLE]

We denote by $|x^{(j)}|$ the euclidean norm of $x^{(j)}\in\mathbb{R}^{n_{j}}$ and assume the functions $\lambda_{i}$ are of the form

[TABLE]

such that

1. $\alpha_{ij}\geq 0$ for $i=2,...,k$, $j=1,...,i-1$.

2. $\alpha_{ij}=0$ for $j\geq i$.

3. $\sum_{l=1}^{k}\epsilon_{l}\alpha_{jl}=\epsilon_{j}-1$, $j=1,...,k$ with $1=\epsilon_{1}\leq\epsilon_{2}\leq...\leq\epsilon_{k}$.

Clearly, $\lambda_{i}$ is $\delta_{t}$-homogeneous of degree $\epsilon_{i}-1$ with respect to a group of dilations $\{\delta_{t}\}_{t>0}$

[TABLE]

Now, using the relation $\sum_{l=1}^{k}\epsilon_{l}\alpha_{jl}=\epsilon_{j}-1$, we get $T\left(|x|_{\lambda}\right)=|x|_{\lambda}$ is satisfied.

This paper is organized as follows. In section $3$, we give the proof of Proposition 1.1 and Theorem 1.1. Section $4$ is devoted to the proof of Proposition 1.2 and Theorem 1.2.

3 The Liouville theorem for stable solutions: proof of Theorem 1.1

In this section we prove all the results concerning the classification of stable solutions, i.e., Proposition 1.1 and Theorem 1.1. First, to prove Proposition 1.1, we need the following technical Lemma.

Let $R>0$, $\Omega_{2R}=B_{1}(0,2R^{\epsilon_{1}})\times B_{2}(0,2R^{\epsilon_{2}})\times...\times B_{k}(0,2R^{\epsilon_{k}})$, where $B_{i}(0,2R^{\epsilon_{i}})\subset\mathbb{R}^{n_{i}}$, $i=1,...,k$, and consider $k$ functions $\psi_{1,R}$,…, $\psi_{k,R}$ such that

[TABLE]

with $\psi_{1,R},\;...\;\psi_{k,R}\in C^{\infty}_{c}([0,+\infty)),\;0\leq\psi_{1,R},\;...\;\psi_{k,R}\leq 1,$

[TABLE]

and for some constant $C>0$ and $\psi_{1,R},\;...\;\psi_{k,R}$ satisfy

[TABLE]

[TABLE]

where $r^{(i)}=|x^{(i)}|$, $i=1,...,k$.

Lemma 3.1

(1) There exists a constant $C>0$ independent of $R$ such that

a) $|\lambda_{i}(x)|\leq CR^{\epsilon_{i}-1},\;\forall\;x\in\Omega_{2R},\;i=1,...,k.$

**b) $|\nabla_{\lambda}\psi_{R}|^{2}+|\Delta_{\lambda}\psi_{R}|\leq CR^{-2}$, where $\psi_{R}=\prod_{i=1}^{k}\psi_{i,R}$.

(2) The homogeneous norm, $|.|_{\lambda}$, is $\delta_{t}$-homogeneous of degree one, i.e.**

[TABLE]

(3) There exists a constant $C>0$ independent of $R$ such that

[TABLE]

**Proof.

**Proof of (1) a). For any $x=(x^{(1)},...,x^{(k)})\in\Omega_{2R}$, we have $x^{(i)}\in B_{i}(0,2R^{\epsilon_{i}})$, $i=1,...,k$, this implies $\frac{|x^{(i)}|}{R^{\epsilon_{i}}}\leq 2$, $i=1,...,k$. Therefore, if we write

[TABLE]

and let $y=(y^{(1)},...,y^{(k)})=\left(\frac{x^{(1)}}{R^{\epsilon_{1}}},...,\frac{x^{(k)}}{R^{\epsilon_{k}}}\right)$, then $y\in\overline{\Omega_{2}}$. Hence by assumption $(H_{3})$ made on functions $\lambda_{i}$, we get

[TABLE]

Moreover, since $\lambda_{i}$, $i=1,...,k$ are continuous, then

[TABLE]

Therefore, from (3.6) and (3.7), we obtain

[TABLE]

Proof of (1) b). Using assumption $(H_{2})$ made on functions $\lambda_{i}$, $i=1,...,k$, with $r=(r^{(1)},...,r^{(k)})=(|x^{(1)}|,...,|x^{(k)}|)$, we have

[TABLE]

If we denote by $\psi_{R}=\prod_{i=1}^{k}\psi_{i,R}$, we get

[TABLE]

and

[TABLE]

Since $|\lambda_{i}(r)|=|\lambda_{i}(x)|\leq CR^{\epsilon_{i}-1}$, $\forall\;x\in\Omega_{2R}$, $i=1,...,k$, then there exists a constant $C>0$ independent of $R$ such that

[TABLE]

Proof of (2). Let $x\in\mathbb{R}^{n}$. The homogeneity of the functions $\lambda_{i}$ implies that

[TABLE]

Proof of (3). For any $x=(x^{(1)},...,x^{(k)})\in\Omega_{2R}$, we have $x^{(i)}\in B_{i}(0,2R^{\epsilon_{i}})$, $i=1,...,k$, this implies $\frac{|x^{(i)}|}{R^{\epsilon_{i}}}\leq 2$, $i=1,...,k$. Therefore, if we write

[TABLE]

and let $y=(y^{(1)},...,y^{(k)})=\left(\frac{x^{(1)}}{R^{\epsilon_{1}}},...,\frac{x^{(k)}}{R^{\epsilon_{k}}}\right)$, then $y\in\overline{\Omega_{2}(0)}$.

Using (3.8), we get

[TABLE]

Since $|\lambda_{i}(y)|\leq C$, $\forall\;y\in\overline{\Omega_{2}}$, $i=1,...,k$, then there exists a constant $C>0$ independent of $R$ such that

[TABLE]

This completes the proof of Lemma 3.1. \qed

Proof of Proposition 1.1. The proof follows the main lines of the demonstration of proposition $4$ in [7], with more modifications. We split the proof into four steps:

Step 1. For any $\phi\in C^{2}_{c}(\mathbb{R}^{n})$ we have

[TABLE]

Multiply equation (1.2) by $|u|^{\gamma-1}u\phi^{2}$ and integrate by parts to find

[TABLE]

therefore

[TABLE]

Identity (3.9) then follows by multiplying the latter identity by the factor $\frac{(\gamma+1)^{2}}{4\gamma}$.

Step 2. For any $\phi\in C^{2}_{c}(\mathbb{R}^{n})$ we have

[TABLE]

The function $|u|^{\frac{\gamma-1}{2}}u\phi$ belongs to $C^{1}_{c}(\mathbb{R}^{n})$, and thus it can be used as a test function in the quadratic form $Q_{u}$. Hence, the stability assumption on $u$ gives

[TABLE]

A direct calculation shows that the right hand side of (3.11) equals to

[TABLE]

From (3.11) and (3.12), we obtain that

[TABLE]

Putting this back into (3.9) gives

[TABLE]

Step 3. For any $\gamma\in\left[1,\,\Gamma_{M}(p)\right)$ and any integer $m\geq\max\{\frac{p+\gamma}{p-1},\;2\}$ there exists a constant $C(p,m,\gamma)>0$ depending only on $p$, $m$ and $\gamma$

[TABLE]

[TABLE]

where $\psi_{R}=\prod_{i=1}^{k}\psi_{i,R}$. Moreover, the constant $C(p,m,\gamma)$ can be explicitly computed.

From (3.10), we obtain that

[TABLE]

where we have set $\alpha=\left(p-\frac{(\gamma+1)^{2}}{4\gamma}\right)$ and $\beta=\frac{1-\gamma}{4\gamma}$. Notice that $\alpha>0$ and $\beta<0$, since $p>1$ and $\gamma\in\left[1,\,\Gamma_{M}(p)\right)$.

Now, we set $\phi=\psi_{R}^{m}$. The function $\phi$ belongs to $C^{2}_{c}(\mathbb{R}^{n})$, since $m\geq 2$ and $m$ is an integer, hence it can be used in (3.16). A direct computation gives

[TABLE]

hence

[TABLE]

with $C_{1}=\frac{m^{2}+\beta m(m-1)}{\alpha}>-\frac{\beta m}{\alpha}\geq 0$.

An application of Young’s inequality yields

[TABLE]

At this point we notice that $m\geq\max\{\frac{p+\gamma}{p-1},\;2\}$ implies $(2m-2)\frac{p+\gamma}{p-1}\geq 2m$ and thus $\psi_{R}^{(2m-2)\frac{p+\gamma}{\gamma+1}}\leq\psi_{R}^{2m}$ in $\mathbb{R}^{n}$, since $0\leq\psi_{R}\leq 1$ everywhere in $\mathbb{R}^{n}$.

Therefore, we obtain

[TABLE]

The latter immediately implies

[TABLE]

which proves inequality (3.14) with $C(p,m,\gamma)=C_{1}$.

To prove (3.15), we combine (3.9) and (3.10). This leads to

[TABLE]

where $A=\frac{(\gamma+1)^{2}}{4\gamma\alpha}+\frac{(\gamma+1)}{2\gamma}>0$ and $B=\frac{\beta(\gamma+1)^{2}}{4\gamma\alpha}+\frac{(\gamma+1)}{2\gamma}\in\mathbb{R}$.

Now, we insert the test function $\phi=\psi_{R}^{m}$ in the latter inequality to find,

[TABLE]

and hence

[TABLE]

with $C_{2}=\max\{Am^{2}+Bm(m-1),\;|B|m\}>0$.

Using Hölder’s inequality in (3.21) yields

[TABLE]

Finally, inserting (3.20) into the latter we obtain

[TABLE]

which gives the desired inequality (3.15).

Step 4. For any $\gamma\in\left[1,\,\Gamma_{M}(p)\right)$, there exists a constant $C>0$ independent of $R$ such that

[TABLE]

The proof of (3.22) follows immediately by adding inequality (3.14) to inequality (3.15) and using Lemma 3.1. \qed

Proof of Theorem 1.1. By Proposition 1.1, there exists a positive constant $C$ independent of $R$ such that

[TABLE]

Then it suffices to show that we can always choose a $\gamma\in\left[1,\,\Gamma_{M}(p)\right)$, such that $Q-\frac{2(p+\gamma)+a(\gamma+1)}{p-1}<0$. Therefore, by letting $R\rightarrow+\infty$ in (3.23), we deduce that

[TABLE]

which implies that $u\equiv 0$ in $\mathbb{R}^{n}$.

Next, we claim that, under the assumptions on the exponent $p$ assumed in Theorem 1.1, we can always choose $\gamma\in[1,\,\Gamma_{M}(p))$ such that

[TABLE]

As in [7], we consider separately the case $Q\leq 10+4a$ and the case $Q>10+4a$.

Case 1. $Q\leq 10+4a$ and $p>1$. In this case we have

[TABLE]

and therefore

[TABLE]

The latter inequality and the continuity of the function $x\mapsto Q-\frac{2(p+x)+a(x+1)}{p-1}$ immediately imply the existence of $\gamma\in[1,\,\Gamma_{M}(p))$ satisfying (3.24).

Case 2. $Q>10+4a$ and $1<p<p_{c}(Q,a)$. In this case we consider the real-valued function $x\mapsto g(x):=\frac{2(x+\Gamma_{M}(x))+a(\Gamma(x)+1)}{x-1}$ on $(1,\,+\infty)$. Since $g$ is strictly decreasing function satisfying $\lim_{x\rightarrow 1^{+}}g(x)=+\infty$ and $\lim_{x\rightarrow+\infty}g(x)=10+4a$, there exists a unique $p_{0}>1$ such that $Q=g(p_{0})$. We claim that $p_{0}=p_{c}(Q,a)$. Indeed,

[TABLE]

[TABLE]

which implies that

[TABLE]

and

[TABLE]

The roots of (3.26)

[TABLE]

[TABLE]

while (3.27) easily implies $p_{0}>\frac{Q-6-2a}{Q-4a-10}>p_{2}$. This proves that $p_{0}=p_{1}$. Hence

[TABLE]

as claimed. Since we have just proven that $g(p_{c}(Q,a))=Q$ and $g$ is a strictly decreasing function, it follows that

[TABLE]

Now we can conclude as in the first case, i.e, the continuity of $x\mapsto Q-\frac{2(p+x)+a(x+1)}{p-1}$ immediately implies the existence of $\gamma\in[1,\,\Gamma_{M}(p))$ satisfying (3.24). \qed

4 The Liouville theorem for solutions which are stable outside a compact set of $\mathbb{R}^{n}$: proof of Theorem 1.2

In this section, we prove Proposition 1.2 and Theorem 1.2.

Proof of Proposition 1.2. Let $u\in C^{2}(\mathbb{R}^{n})$ be a solution of (1.2) and $\phi\in C^{1}_{c}(\Omega_{R})$. Multiplying equation (1.2) by $T(u)\phi$ and integrating by parts in $\Omega_{R}$, we obtain

[TABLE]

Here and in the sequel, we use the Einstein summation convention: an index occurring twice in a product is to be summed from $1$ up to the space dimension.

Obviously

[TABLE]

Moreover, an integration by parts in $I_{2}$ gives

[TABLE]

Since $\lambda_{i}$ is $\delta_{t}$-homogeneous of degree $\epsilon_{i}-1$, then $T(\lambda_{i})=(\epsilon_{i}-1)\lambda_{i}$. Hence

[TABLE]

Then

[TABLE]

It is easily seen that

[TABLE]

Hence, by (4.31),

[TABLE]

On the other hand, an integration by parts gives

[TABLE]

If $T(|x|_{\lambda})=|x|_{\lambda}$, then

[TABLE]

Clearly (1.4) follows directly from (4) and (4.36). \qed

Proof of Theorem 1.2. Let $u$ be a solution of (1.2) which is stable outside a compact set. We begin defining some smooth compactly supported functions which will be used several times in the sequel. More precisely, for $R_{*}>0$, we choose a function $\zeta_{i,R}\in C^{2}_{c}(\mathbb{R}^{n_{i}})$, $i=1,...,k$, $0\leq\zeta_{i,R}\leq 1$, everywhere on $\mathbb{R}^{n_{i}}$ and

[TABLE]

The rest of the proof splits into several steps.

Step 1. Let $p>1$. There exists $R_{*}>0$ such that for every $\gamma\in\left[1,\,\Gamma_{M}(p)\right)$ and every $R^{\epsilon_{i}}>R_{*}+2$, we have

[TABLE]

where $\Sigma_{0}(R)=\Omega_{R}\backslash B_{1}(0,R_{*}+2)\times...\times B_{k}(0,R_{*}+2)$, $C_{R_{*}}$ and $C$ are positive constants depending on $p$, $\gamma$, $R_{*}$ but not on $R$.

Since $u$ is stable outside a compact set of $\mathbb{R}^{n}$, there exists $R_{*}>0$ such that, similar to that of Proposition 1.1 we derive

[TABLE]

where $\zeta_{R}=\prod_{i=1}^{n}\zeta_{i,R}$. Hence, the desired integral estimate (4.37) follows.

Step 2. If $Q=2$ and $1<p<+\infty$ or $Q\geq 3$ and $1<p<\frac{Q+2+2a}{Q-2}$, then $u\equiv 0$.

By choosing $\gamma=1$ and using Step $1$, we get $|x|^{\frac{a}{p+1}}_{\lambda}u\in L^{p+1}(\mathbb{R}^{n})$ and $|\nabla_{\lambda}u|\in L^{2}(\mathbb{R}^{n})$ for $1<p<p_{s}(Q,a)$.

Take $\phi=\psi_{R}=\prod_{i=1}^{k}\psi_{i,R}$ in (1.4) where $\psi_{i,R}$ defined as above. Since $|x|^{\frac{a}{p+1}}_{\lambda}u\in L^{p+1}(\mathbb{R}^{n})$ and $|\nabla_{\lambda}u|\in L^{2}(\mathbb{R}^{n})$, then

[TABLE]

where $\Sigma_{R}=\Omega_{2R}\backslash\Omega_{R}$.

Recalling that $\lambda_{i}$ and $\lambda_{i}\nabla_{x^{(i)}}u$ are $\delta_{t}$-homogeneous of degree $\epsilon_{i}-1$ and one respectively. Then, since $T$ generates $(\delta_{t})_{t\geq 0}$, we have

[TABLE]

Integrating by parts and using (4.39), we derive

[TABLE]

By Lemma 3.1, (4.40) and using Hölder’s inequality, we obtain

[TABLE]

Similarly, we get

[TABLE]

From (4.38), (4.41) and (4.42), we obtain

[TABLE]

As a consequence, (1.4) becomes

[TABLE]

On the other hand, multiplying equation (1.2) by $u\psi_{R}$ and integrating by parts yields

[TABLE]

Since $1<p<p_{s}(Q,a)$, we get

[TABLE]

Then

[TABLE]

To complete the proof we combine (4.43) and (4.44) to get

[TABLE]

but $\frac{Q-2}{2}-\frac{Q+a}{p+1}\neq 0$, since $p$ is subcritical, hence $u$ must be identically zero, as claimed. \qed

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