Some results for semi-stable radial solutions of k-Hessian equations with weight on ℝⁿ

Abstract

We devote this paper to study semi-stable nonconstant radial solutions of $S_k(D^2u)=w(\left \vert x \right \vert )g(u)$ on the Euclidean space $\mathbb {R}^n$. We establish pointwise estimates and necessary conditions for the existence of such solutions (not necessarily bounded) for this equation. For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behaviour at infinity of the growth rate function of $w$.

1. Introduction and main results

This paper deals with the semi-stability of nonconstant radial solutions of

(1.1)\begin{equation} S_k(D^2u)=w(\left\vert x\right\vert)g(u), \end{equation}

posed in $\mathbb {R}^n$, where $n\geq 1$, $k\in \{1,\,\ldots,\,n\}$, the function $g\in C^1(\mathbb {R})$ is nonnegative and nonincreasing, and $w$ is a nonnegative radial function that satisfies some suitable conditions. The operator $S_k(D^2u)$ is the $k$-Hessian of $u$, which is defined by the sum of all $k$-th principal minors of the Hessian matrix. Alternatively, $S_k(D^2u)$ is the $k$-th elementary symmetric polynomial of the eigenvalues of the Hessian matrix $D^2 u$. According to [1], to ensure ellipticity of equation (1.1), we consider $k$-admissible or $k$-convex solutions, i.e., functions that belong to

\[ \Phi^k:=\left\lbrace u\in C^2(\mathbb{R}^n): S_l(D^2 u)\geq 0,\quad l=1,2,\ldots,k\right\rbrace. \]

We point out that there are no previous works concerning semi-stable solutions to the $k$-Hessian operator in the whole space $\mathbb {R}^n$ for general nonlinearities and $k\neq 1$. In fact, to the best of our knowledge, the only results devoted to stable solutions to (1.1) are contained in the work [10] and they concern only the particular cases $w(r)\equiv 1$ and $g(u)$ a power nonlinearity. More precisely, in [10], the authors gave a definition of stable radial solutions of the $k$-Hessian equation $F_{k}(D^2V)=(-V)^p$ in $\mathbb {R}^n$, where $F_{k}(D^2V)=S_k(D^2\,V)$. They stablished connections between stability and certain critical exponents of Joseph–Lundgren type available for $k$-Hessian operators. Their tools also include Wolff potentials.

For existence and non-existence results for equations of the form (1.1) we refer to [3], where a special emphasis was put in the model equation

\[ ({-}1)^kH_k u=a(\left\vert x\right\vert)\left\vert u\right\vert^{q-1}u, \]

being

\[ H_k=\frac{k}{\binom{n-1}{k-1}}T_k,\mbox{ for }1\leq k \leq n\mbox{ (integer)}\mbox{ and }T_k=S_k(\nabla^2). \]

Some existence and non-existence results for radial solutions are given in terms of an integral condition involving the function $a$. In [6], among other results, the author constructs explicit negative solutions of the equation $F_{k}(D^2V)=R(x)(-V)^q$ in $\mathbb {R}^n$, where $F_{k}(D^2V)=S_k(D^2\,V)$ and $R(x)$ is a radial function that satisfies $C^{-1}\leq R(x)\leq C$ for some constant $C>1$. See [6, theorem 4.2].

Throughout this work, we identify a radial solution $u$ by their one variable representant, that is, $u(x) = u(r),\, \left \vert x \right \vert =r$. At the point $x=(r,\,0,\,\ldots,\,0)$, the eigenvalues of $D^2 u$ are $\lambda _1=u''$, which is simple, and $\lambda _2=\frac {u'}{r}$, which has multiplicity $n-1$, where by abuse of notation, we write $u'$ or $\partial _r u$ as the radial derivative of a radial function $u$. Thus the $k$-Hessian operator acting on radially symmetric $C^2$ functions can be written as

(1.2)\begin{equation} S_k(D^2 u)=c_{n,k}\lambda_2^{k-1}\left(n\lambda_2+k\left(\lambda_1-\lambda_2\right)\right)=c_{n,k}r^{1-n}\partial_r\left (r^n\lambda_2^k\right ),\quad r>0, \end{equation}

where $c_{n,k}$ is defined by $c_{n,k}=\binom {n}{k}/n$.

Remark 1.1 Note that, if $u$ is a radial solution of (1.1) then, in particular, $S_1(D^2 u)=r^{1-n}(r^{n-1}u')'\geq 0$. Thus $G(r)=r^{n-1}u'$ is nondecreasing, since ${G(0)=0}$, we deduce that $G\geq 0$ and hence $u$ is nondecreasing. As a consequence, $\lambda _2\geq 0$.

Definition 1.2 We say that a radial solution $u\in \Phi ^k$ of (1.1) is semi-stable if

(1.3)\begin{equation} \mathcal{Q}_u(\xi):=\int_{\mathbb{R}^n}{kc_{n,k}\left\vert x\right\vert^{1-k}\left\vert\nabla u\right\vert^{k-1}\left\vert\nabla\xi\right\vert^2+w(\left\vert x\right\vert)g'(u)\xi^2}\geq 0, \end{equation}

for every radially symmetric function $\xi \in C_c^{1}(\mathbb {R}^n)$.

In this paper, we establish pointwise estimates and necessary conditions for the existence of semi-stable solutions (not necessarily bounded) of (1.1). For bounded solutions we estimate their asymptotic behaviour at infinity. All the estimates are given in terms of the spatial dimension $n$, the values of $k$ and the behaviour at infinity of the growth rate function of $w$, that is, the function $W(r):=\frac {rw'(r)}{w(r)}$.

We now establish our precise assumptions on the weight $w(r)$ that we will assume throughout the paper:

(1.4)\begin{equation} \begin{cases} \mbox{The functions } w \mbox{ and } W \mbox{ belong to }C^1\left(\mathbb{R}^n{\setminus}\{0\}\right) \mbox{ and } W(r) \mbox{ is}\\ \mbox{nonincreasing on }(0,\infty). \mbox{ Further, }\lim\limits_{r\to +\infty}{rW'(r)}=0.\\ \mbox{Set }\Gamma=\lim\limits_{r\to 0}W(r),~\gamma=\lim\limits_{r\to +\infty}W(r)\mbox{, and assume that }\Gamma\mbox{ and}\\ \gamma\mbox{ are finite with }2k+\gamma>0. \end{cases} \end{equation}

In order to state our main results, we need the following notation:

Let

(1.5)\begin{align} & \alpha(r)=\frac{\displaystyle\int_1^r{\left\lbrace \frac{v(s)+\sqrt{v(s)^2+\frac{n}{k}v(s)+\frac{n}{k}-1 -sv'(s)}}{s}\right\rbrace ds}}{\log r}, \end{align}

(1.6)\begin{align} & \delta(r)=\frac{-n+2\alpha(r)+2k+2}{k+1},\,\forall r\geq 1\mbox{ and } \delta_{\infty}(\gamma)=\lim\limits_{r\to +\infty}{\delta(r)}, \end{align}

where

(1.7)\begin{equation} v(s)=\frac{k-1+W(s)}{k+1},\quad\forall s\geq 0. \end{equation}

Note that when $k=1$ and $w\equiv 1$, $\delta (r)$ takes the constant value $-\frac {n}{2}+\sqrt {n-1}+2$, which plays a crucial role in pointwise estimates. We refer to [9] by Villegas for details. A great difference here is that $\delta (r)$ is a variable exponent, this produces additional technical difficulties.

Our main results are

Proposition 1.3 Let $n>k(k+1)/(2k+\gamma )$ and $u$ be a semi-stable nonconstant radial solution of (1.1). Then $\left \vert \nabla u(x)\right \vert >0$ for all $\left \vert x \right \vert >0$.

We have the following pointwise estimate for not necessarily bounded solutions of (1.1).

Theorem 1.4 Let $w\in C^1(\mathbb {R}^n{\setminus} \{0\})$ be a radial function that satisfies (1.4) , $n>k(k+1)/(2k+\gamma )$ and $u$ be a semi-stable nonconstant radial solution of (1.1) (not necessarily bounded). Then, there exist $M>0$ and $r_0\geq 1$ depending on $u$ and $w$ such that for all $r\geq r_0$,

(1.8)\begin{equation} \left\vert u(r)\right\vert\geq M\begin{cases} r^{\delta(r)} & \mbox{if }n\neq 2\left(k+\frac{2\gamma}{k}+4\right),\\ \log r & \mbox{if }n=2\left(k+\frac{2\gamma}{k}+4\right). \end{cases} \end{equation}

Remark 1.5 This theorem is sharp for some $\beta \in \mathbb {R},\, \mu =\frac {\sigma }{k}+2$ and

\[ u_{\beta} (r):=\begin{cases} {\rm sgn}\left(\beta\right)\left(1+r^{\mu}\right )^{\frac{\beta}{\mu}} & \mbox{if } \beta \neq 0,\forall r\geq 0,\\ \left(\frac{1}{\mu}\right)\log{\left (1+r^{\mu}\right )} & \mbox{if } \beta=0,\forall r\geq 0, \end{cases} \]

where $\sigma >-k$ and $w(r)=r^{\sigma }$, as shown in the appendix.

Theorem 1.6 Let $w\in C^1(\mathbb {R}^n{\setminus} \{0\})$ be a radial function that satisfies (1.4), $n\geq 2$ and $u$ be a semi-stable nonconstant bounded radial solution of (1.1). Then,

1. (i) $n>2(k+\frac {2\gamma }{k}+4)$.

2. (ii) There exists $u_\infty =\lim \limits _{r\rightarrow +\infty }{u(r)}\in \mathbb {R}$ and $M>0$ depending on $u$ and $w$ such that for all $r\geq 1,$

   (1.9)\begin{equation} \left\vert u(r)-u_\infty\right\vert \geq M r^{\delta(r)}. \end{equation}

Remark 1.7 Theorem 1.6 is sharp (see example A.1 in the appendix).

Remark 1.8 See [9] to compare our results with the semilinear case ($k=1$) and for related equations involving the $p$-Laplacian operator, see [2, 5, 8].

This paper is organized as follows. In § 2 we prove our main results, proposition 1.3, theorem 1.4 and theorem 1.6. We conclude the paper by presenting in the appendix some examples of functions $w$ and $g$ for which our theorems are sharp.

2. Proof of the main results

We claim that if $u$ is a $C^2$ radial solution, then $u'\in C_{loc}^{0,1}(\mathbb {R}^n)$. To prove our claim, we first observe that $u'\in C^0(\mathbb {R}^n)\cap C^1(\mathbb {R}^n{\setminus} \{0\})$ with $u'(0)=0$ and $\left \vert u '(x)\right \vert \leq C_R\left \vert x \right \vert$ in any open ball $B_R$ of radius $R>0$. Now from (1.1) and (1.2) we obtain, for $r>0$,

\[ kc_{n,k}\left(\frac{u'}{r}\right)^{k-1}\left(\left(\frac{n-k}{k}\right)\left(\frac{u'}{r}\right)+u''\right)=wg(u), \]

from which we deduce that

\[ u''=\left(\frac{wg(u)}{kc_{n,k}}\right) \left(\frac{u'}{r}\right)^{1-k}-\left(\frac{n-k}{k}\right)\left(\frac{u'}{r}\right),\quad r>0. \]

Thus, for any $R>0$, the function $u''$ is bounded on $B_R{\setminus} \{0\}$. Also for any $x,\,y\in \mathbb {R}^n$ such that $R>\left \vert y \right \vert >\left \vert x \right \vert >0$, we have

\begin{align*} \left\vert u'(\left\vert y\right\vert)-u'(\left\vert x\right\vert)\right\vert& \leq\int_{\left\vert x\right\vert}^{\left\vert y\right\vert}\left\vert u''(t)\right\vert\,{\rm d}t\\ & \leq\sup_{\xi\in [\left\vert x\right\vert,\left\vert y\right\vert]}\left\vert u''(\xi)\right\vert(\left\vert y\right\vert-\left\vert x\right\vert)\\ & \leq\sup_{z\in B_{R}{\setminus}\{0\}}\left\vert u''(\left\vert z\right\vert)\right\vert\left\vert y-x\right\vert. \end{align*}

We conclude that $u'\in C_{loc}^{0,1}(\mathbb {R}^n)$. This will be used in the proof of lemma 2.1 below.

Lemma 2.1 Let $n\geq 2$ and $u$ be any radial solution of (1.1). Then

(2.1)\begin{equation} \mathcal{Q}_u(u'\eta)=kc_{n,k}\int_{\mathbb{R}^n}{\!\lambda_2^{k+1}\left\lbrace\left\vert\left\vert x\right\vert\nabla\eta+v\frac{x\eta}{\left\vert x\right\vert}\right\vert^2\!\!\!-\left(v^2+\frac{n}{k}v+\frac{n}{k}-1-\left(x,\nabla v\right)\right)\eta^2\right\rbrace }, \end{equation}

for every radially symmetric function $\eta \in (H_c^1\cap L_{loc}^{\infty })(\mathbb {R}^n)$, where $v$ is defined as in (1.7). Here $(\cdot,\, \cdot )$ denotes the standard scalar product in $\mathbb {R}^n$.

Proof. Let $\eta \in H_c^1(\mathbb {R}^n)\cap L_{loc}^\infty (\mathbb {R}^n)$ and $\zeta \in C_{loc}^{0,1}(\mathbb {R}^n)$ be radial functions. Then, by a standard density argument, we can take $\xi =\zeta \eta \in H_c^1(\mathbb {R}^n)\cap L_{loc}^\infty (\mathbb {R}^n)$ in (1.3) to obtain

(2.2)\begin{equation} \mathcal{Q}_u(\zeta\eta)=\int_{\mathbb{R}^n}{kc_{n,k}\left\vert\frac{u'}{\left\vert x\right\vert}\right\vert^{k-1}\left\vert\nabla(\zeta\eta)\right\vert^2+w(\left\vert x\right\vert)g'(u)(\zeta\eta)^2}. \end{equation}

Thus, as $u$ and $\zeta$ are radial functions, we get

(2.3)\begin{equation} \left\vert\frac{u'}{\left\vert x\right\vert}\right\vert^{k-1}\left\vert\nabla(\zeta\eta)\right\vert^2=\lambda_2^{k-1}\left(\zeta^2\left\vert\nabla\eta\right\vert^2+\left(2\zeta\eta(\nabla\zeta,\nabla\eta)+\eta^2\left\vert\nabla\zeta\right\vert^2\right)\right), \end{equation}

where $\lambda _2=u'/{\left \vert x \right \vert }$.

From (1.2) and differentiating (1.1) with respect to $r$, we obtain

\begin{align*} c_{n,k}^{{-}1}wg'(u)u'& ={-}c_{n,k}^{{-}1}w'g(u)-\frac{n-1}{r^2}\left(\frac{u'^k}{r^{k-1}}\right)+\Delta\left(\frac{u'^k}{r^{k-1}}\right)\\ & ={-}c_{n,k}^{{-}1}wg(u)\frac{W}{r}-\frac{(n-1)\lambda_2^k}{r}+\Delta\left(\lambda_2^{k-1}u'\right),\quad r>0. \end{align*}

Then, multiplying the latter equation by $u'\eta ^2=r\lambda _2\eta ^2$, we have

(2.4)\begin{equation} \begin{aligned} \frac{1}{c_{n,k}}\int_{\mathbb{R}^n}{wg'(u)\left(u'\eta\right)^2} & ={-}\frac{1}{c_{n,k}}\int_{\mathbb{R}^n}{wg(u)\left(\lambda_2W\eta^2\right)}-(n-1)\int_{\mathbb{R}^n}{\lambda_2^{k+1}\eta^2}\\ & \quad+\int_{\mathbb{R}^n}{\left(\left\vert x\right\vert\lambda_2\eta^2\right) \Delta\left(\left\vert x\right\vert\lambda_2^{k}\right)}. \end{aligned} \end{equation}

Thus, integrating by parts, and taking into account that $\lambda _2\in L_{loc}^\infty (\mathbb {R}^n)$, we obtain

(2.5)\begin{equation} \begin{aligned} \frac{1}{c_{n,k}}\int_{\mathbb{R}^n}{wg(u)\left(\lambda_2W\eta^2\right)} & =\int_{\mathbb{R}^n}{\left(x,\nabla\left(\left\vert x\right\vert^{n}\lambda_2^k\right) \right)\left(\left\vert x\right\vert^{{-}n}\lambda_2W\eta^2\right)}\\ & ={-}\int_{\mathbb{R}^n}{\left(\left\vert x\right\vert^{n}\lambda_2^k\right)\left(x,\nabla\left(\left\vert x\right\vert^{{-}n}\lambda_2W\eta^2\right)\right)}\\ & =\frac{k}{k+1}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left(\frac{n}{k}W\eta^2-\left(x,\nabla\left(W\eta^2\right)\right)\right)} \end{aligned} \end{equation}

and

(2.6)\begin{equation} \begin{aligned} \int_{\mathbb{R}^n}{\left(\left\vert x\right\vert\lambda_2\eta^2\right) \Delta\left(\left\vert x\right\vert\lambda_2^{k}\right)} & ={-}\int_{\mathbb{R}^n}{\left(\nabla\left(\left\vert x\right\vert\lambda_2\eta^2\right),\nabla\left(\left\vert x\right\vert\lambda_2^{k}\right)\right)}\\ & =\frac{k(k-1)}{k+1}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left(\eta^2+\left(x,\nabla\eta^2\right)\right)}\\ & \quad-k\int_{\mathbb{R}^n}{\lambda_2^{k-1}\frac{(\nabla\left(\left\vert x\right\vert\lambda_2\right)^2,\nabla\eta^2)}{2}}\\ & \quad-k\int_{\mathbb{R}^n}{\lambda_2^{k-1}\left\vert\nabla\left(\left\vert x\right\vert\lambda_2\right)\right\vert^2\eta^2}. \end{aligned} \end{equation}

Next, using $\zeta =u'$ in (2.2), (2.3), we have

(2.7)\begin{equation} \begin{aligned} \int_{\mathbb{R}^n}{\lambda_2^{k-1}\left\vert\nabla(u'\eta)\right\vert^2} & =\int_{\mathbb{R}^n}{\lambda_2^{k-1}\left(u'^2\left\vert\nabla\eta\right\vert^2+2u'\eta(\nabla u',\nabla\eta)+\left\vert\nabla u'\right\vert^2\eta^2\right)}\\ & =\int_{\mathbb{R}^n}{\left\lbrace \left\vert x\right\vert^2\lambda_2^{k+1}\left\vert\nabla\eta\right\vert^2+\lambda_2^{k-1}\frac{(\nabla\left(\left\vert x\right\vert\lambda_2\right)^2,\nabla\eta^2)}{2}\right\rbrace }\\ & \quad+\int_{\mathbb{R}^n}{\lambda_2^{k-1}\left\vert\nabla\left(\left\vert x\right\vert\lambda_2\right)\right\vert^2\eta^2}\\ & =\int_{\mathbb{R}^n}{\left\vert x\right\vert^2\lambda_2^{k+1}\left\vert\nabla\eta\right\vert^2}\\ & \quad+\frac{(k-1)}{k+1}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left(\eta^2+\left(x,\nabla\eta^2\right)\right)}\\ & \quad-\frac{1}{k}\int_{\mathbb{R}^n}{\left(\left\vert x\right\vert\lambda_2\eta^2\right) \Delta\left(\left\vert x\right\vert\lambda_2^{k}\right)}. \end{aligned} \end{equation}

Then, from (2.2), (2.4), (2.5), (2.6) and (2.7), we obtain

\begin{align*} \mathcal{Q}_u(u'\eta)& =\int_{\mathbb{R}^n}{\lambda_2^{k-1}\left\vert\nabla\left(u'\eta\right)\right\vert^2}+\frac{1}{kc_{n,k}}\int_{\mathbb{R}^n}{wg'(u)(u'\eta)^2}\\ & ={-}\frac{1}{k+1}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left(\frac{n}{k}W\eta^2-\left(x,\nabla\left(W\eta^2\right)\right)\right)}\\ & \quad-\frac{(n-1)}{k}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\eta^2}+\int_{\mathbb{R}^n}{\left\vert x\right\vert^2\lambda_2^{k+1}\left\vert\nabla\eta\right\vert^2}\\ & \quad+\frac{(k-1)}{k+1}\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left(\eta^2+\left(x,\nabla\eta^2\right)\right)}. \end{align*}

Since

\[ \left\vert x\right\vert^2\left\vert\nabla\eta\right\vert^2+v\left(x,\nabla\eta^2\right)=\left\vert\left\vert x\right\vert\nabla\eta+v\frac{x\eta}{\left\vert x\right\vert}\right\vert^2-v^2\eta^2, \]

from (1.7), we have

\[ W=(k+1)v+1-k\mbox{ and }\left(x,\nabla W\right)=(k+1)\left(x,\nabla v \right). \]

Hence

\[ \mathcal{Q}_u(u'\eta)=\int_{\mathbb{R}^n}{\lambda_2^{k+1}\left\lbrace\left\vert\left\vert x\right\vert\nabla\eta+v\frac{x\eta}{\left\vert x\right\vert}\right\vert^2-\left(v^2+\frac{n}{k}v+\frac{n}{k}-1-\left(x,\nabla v\right)\right)\right\rbrace \eta^2}, \]

which concludes the proof.

Multiplying the equation (1.2) by $r^{n+\frac {n}{k}-1}\lambda _2$, we obtain

\[ \left(\left(r^{\frac{n}{k}}\lambda_2\right)^{k+1}\right)'=(k+1)(kc_{n,k})^{{-}1}wg(u)r^{n+\frac{n}{k}-1}\lambda_2. \]

Since $r\lambda _2=u'$, the above equation is equivalent to

(2.8)\begin{equation} \left(\left(r^{\frac{n}{k}-1}u'\right)^{k+1}\right)'=(k+1)(kc_{n,k})^{{-}1}r^{n-1}wg(u)r^{\frac{n}{k}-1}u'. \end{equation}

On the other hand, from (1.4) and (1.7), we have

(2.9)\begin{equation} v(r)\geq v_{\infty}:=\lim_{r\to +\infty}{v(r)}=\frac{k-1+\gamma}{k+1},\quad\forall r\geq 0. \end{equation}

Thus, if $n>k(k+1)/(2k+\gamma )$, it follows that

(2.10)\begin{equation} \frac{n}{k}v(r)+\frac{n}{k}-1\geq\frac{n}{k}\left(v_{\infty}+1\right)-1=\frac{n(2k+\gamma)}{k(k+1)}-1>0,\quad\forall r\geq 0. \end{equation}

Proof of proposition 1.3. We follow an argument similar to that of proposition 1 in [4]. Let $n>k(k+1)/(2k+\gamma )$ and let $u$ be a semi-stable nonconstant radial solution of (1.1). Arguing by contradiction, assume that $u'(r_0) = 0$ for some $r_0 > 0$. Now let $\chi _{B_{r_0}}$ be the indicator function of the open ball of radius $r_0$ centred at the origin, so $\chi _{B_{r_0}}\in H_c^1(\mathbb {R}^n)\cap L_{loc}^\infty (\mathbb {R}^n)$. From (2.1), we obtain

\begin{align*} \mathcal{Q}_u\left(u'\chi_{B_{r_0}}\right)& =kc_{n,k}\left(\int_{{B_{r_0}}}+\int_{\mathbb{R}^n{\setminus} B_{r_0}}\right)\\ & {\left\lbrace\lambda_2^{k+1}\left(\left\vert\left\vert x\right\vert\nabla\chi_{B_{r_0}}\!\!+v\frac{x\chi_{B_{r_0}}}{\left\vert x\right\vert}\right\vert^2\!\!-\left(v^2+\frac{n}{k}v\!+\!\frac{n}{k}\!-1\!-\!\left(x,\nabla v\right)\right)\chi_{B_{r_0}}^2\right) \right\rbrace }\\ & ={-}kc_{n,k}\int_{{B_{r_0}}}{\lambda_2^{k+1}\left\lbrace\frac{n}{k}v+\frac{n}{k}-1-\left(x,\nabla v\right)\right\rbrace }+0\leq 0, \end{align*}

where we have used the fact that $\lambda _2\geq 0$, (2.10) and the monotonicity of $v$ to get the above inequality. Thus, the semi-stability of $u$ implies that $u(r)=u_0$ for all $r\in [0,\,r_0]$.

Let $v(r)=(r^{\frac {n}{k}-1}u')^{k+1}$. From (2.8), we have the following problem

\[ \left\lbrace\begin{array}{@{}rcll} u'=r^{1-\dfrac{n}{k}}v^{\dfrac{1}{k+1}} & r>r_0>0,\\ v'=(k+1)(kc_{n,k})^{{-}1}r^{n-1}wg(u)v^{\dfrac{1}{k+1}} & r>r_0>0,\\ u(r_0)=u_0, & \\ v(r_0)=0. \end{array}\right. \]

Finally, by Cauchy's theorem, we get that $u=u_0$ for any $r>0$, a contradiction.

We adapt some estimates given by Villegas in [9] for the nonweighted semilinear equation. Here is one of our main integral estimates.

Lemma 2.2 Let $w\in C^1(\mathbb {R}^n{\setminus} \{0\})$ be a radial function that satisfies (1.4) , ${n>k(k+1)/ (2k+\gamma)}$ and $u$ be a semi-stable nonconstant radial solution of (1.1). Then, there exists $K>0$ depending on $u$ and $w$ and such that

(2.11)\begin{equation} \int_r^R{\frac{ds}{s^{n-k}({u'(s)})^{k+1}}}\leq Kr^{{-}2\alpha(r)}\quad \forall R>r\geq 1. \end{equation}

Proof. From proposition 1.3, we have that $u'(r)\neq 0$ for all $r>0$. Furthermore, $u'>0$ on $(0,\,\infty )$ (see remark 1.1).

Next we show that $\alpha (r)$ defined in (1.5) is strictly positive for all $r\geq 1$. For this, let $\kappa \geq 0$ and define the function

\[ \Psi(r,\kappa):=v(r)+\sqrt{v(r)^2+\frac{n}{k}v(r)+\frac{n}{k}-1 +\kappa}. \]

From (2.10), we have

(2.12)\begin{equation} v(r)+\frac{n}{2k}>\frac{k}{n}\left(1-\frac{n}{k}\right)+\frac{n}{2k}=\frac{k^2+(n-k)^2}{2kn}>0. \end{equation}

Then

(2.13)\begin{equation} \frac{\partial\Psi}{\partial r}(r,\kappa)=\left(1+\frac{v(r)+\frac{n}{2k}}{\sqrt{v(r)^2+\frac{n}{k}v(r)+\frac{n}{k}-1 +\kappa}}\right)v'(r)\leq 0\mbox{ for }r>0. \end{equation}

Thus, from (1.4), (2.10), (2.12) and (2.13), we obtain

\[ \Psi(r,-rv'(r))>v(r)+\left\vert v(r)\right\vert\geq 0,\quad\forall r\geq 0, \]

and

\begin{align*} \Psi(r,-rv'(r))& \geq\Psi(r,0)=v(r)+\sqrt{\left(\frac{v(r)+1}{k}\right)\left(k\left(v(r)-1\right)+n\right)}\\ & \geq\lim_{r\to +\infty}{\Psi(r,0)}=v_{\infty}+\sqrt{\frac{n}{k}(v_{\infty}+1)+(v_{\infty}+1)(v_{\infty}-1)}\\ & >\frac{k-1+\gamma}{k+1}+\sqrt{1+\frac{(2k+\gamma)(\gamma-2)}{(k+1)^2}}\\ & =\frac{k-1+\gamma+\left\vert k-1+\gamma\right\vert}{k+1}\geq 0. \end{align*}

Therefore by the previous inequalities, we get

(2.14)\begin{equation} \begin{aligned} \alpha(r) & =\frac{\displaystyle\int_1^r{\left\lbrace \frac{v(s)+\sqrt{v(s)^2+\frac{n}{k}v(s)+\frac{n}{k}-1 -sv'(s)}}{s}\right\rbrace\,{\rm d}s}}{\log r}\\ & \geq v_{\infty}+\sqrt{\frac{n}{k}(v_{\infty}+1)+(v_{\infty}+1)(v_{\infty}-1)}>0\mbox{ for }r\geq 1. \end{aligned} \end{equation}

We now fix $R>r\geq 1$ and consider the function

\[ \eta (t)=\begin{cases} 1 & \mbox{if } 0\leq t \leq 1, \\ t^{-\alpha(t)} & \mbox{if } 1< t\leq r,\\ \frac{r^{-\alpha(r)}\displaystyle\int_t^R{\frac{{\rm d}s}{s^{n-k}(u'(s))^{k+1}}}}{\displaystyle\int_r^R{\frac{ds}{s^{n-k}(u'(s))^{k+1}}}} & \mbox{if } r< t\leq R,\\ 0 & \mbox{if } R< t<\infty. \end{cases} \]

Since, $u$ is a semi-stable solution, from (2.1), we have

(2.15)\begin{equation} \begin{aligned} 0 & \leq (\omega_n kc_{n,k})^{{-}1}\mathcal{Q}_u(u'\eta)= I_1+I_2+I_3\\ & =\left(\int_0^1+\int_1^r+\int_r^R\right){\left\lbrace t^{n-1}\lambda_2^{k+1}\left( \left(t\eta'\right)^2+2v(t)\left(t\eta\eta'\right)\right.\right. }\\ & \quad-{\left.\left. \left(\frac{n}{k}v(t)+\frac{n}{k}-1-tv'(t)\right)\eta^2\right)\,{\rm d}t\right\rbrace }, \end{aligned} \end{equation}

where $\omega _n$ is the measure area of the $n-1$ dimensional unit sphere $S^{n-1}$.

Then by (2.10), we have

(2.16)\begin{equation} \begin{aligned} I_1 & ={-}\int_0^1{t^{n-1}\lambda_2^{k+1}\left(\left(\frac{n}{k}v(t)+\frac{n}{k}-1-tv'(t)\right)\right)\,{\rm d}t}\\ & \leq{-}\left(\frac{n}{k}\left(v_{\infty}+1\right)-1\right)\int_0^1{t^{n-1}\lambda_2^{k+1}\,{\rm d}t}, \end{aligned} \end{equation}

and by (1.5), we obtain

(2.17)\begin{equation} \begin{aligned} I_2 & =\int_1^r{t^{n-2\alpha(t)-1}\lambda_2^{k+1}\left( \left(t\left(\alpha(t)\log t\right)'-v(t)\right)^2\right.}\\ & -{\left. \left(v(t)^2+\frac{n}{k}v(t)+\frac{n}{k}-1-tv'(t)\right)\right)\,{\rm d}t }\\ & =0. \end{aligned} \end{equation}

Now to estimate $I_3$, we need to consider two cases according to the sign of $v$ in the interval $[r,\,R]$. For this, we rewrite $I_3$ as

(2.18)\begin{equation} I_3=I_{3}^+{+}I_{3}^{-}=\int_{\{t\in[r,R]:v(t)\geq0\}}+\int_{\{t\in[r,R]:v(t)<0\}}, \end{equation}

and we define $\Phi _R: [r,\,R]\rightarrow \mathbb {R}$ as

\[ \Phi_R(t)=\int_t^R{s^{-(n+1)}\lambda_2^{-(k+1)}\,ds}, \]

and recall that $\lambda _2=u'(s)/s$ for $s>0$.

Thus for $t\in [r,\,R]$:

• • If $v(t)\geq 0$, then by (2.10), it follows that

  (2.19)\begin{equation} \begin{aligned} I_3^+ & =\int_r^R{t^{n-1}\lambda_{2}^{k+1}\left((t\eta')^2+2v(t)(t\eta\eta')\right)\,{\rm d}t}\\ & \quad-\int_r^R{t^{n-1}\lambda_{2}^{k+1}\left(\left(\frac{n}{k}v(t)+\frac{n}{k}-1-tv'(t)\right)\eta^2\right)\,{\rm d}t}\\ & \leq\int_r^R{t^{n-1}\lambda_{2}^{k+1}\frac{r^{{-}2\alpha(r)}}{\Phi^2_{R}(r)}t^{{-}2n}\lambda_{2}^{{-}2(k+1)}\,{\rm d}t}\\ & =\frac{r^{{-}2\alpha(r)}}{\Phi^2_{R}(r)}\int_{r}^R t^{-(n+1)}\lambda_2^{-(k+1)}\,{\rm d}t=\frac{r^{{-}2\alpha(r)}}{\Phi_{R}(r)}. \end{aligned} \end{equation}

• • If $v(t)<0$, then for all $\epsilon >0$

  \[{-}2v(t)\Phi_R(t)t^{{-}n}\lambda_2^{-(k+1)}\leq \epsilon^2v(t)^2\Phi_R(t)^2+\frac{t^{{-}2n}\lambda_2^{{-}2(k+1)}}{\epsilon^2}, \]
  and
  \begin{align*} I_3^{-}& \leq \left(1+\frac1{\epsilon^2}\right) \frac{r^{{-}2\alpha(r)}}{\Phi_R(r)}-\frac{r^{{-}2\alpha(r)}}{\Phi_R(r)^2}\\ & \quad\times\int_{\{t\in[r,R]:v(t)<0\}}{t^{n-1}\lambda_2^{k+1}\left( \!-\epsilon^2v(t)^2\!+\!\frac{n}{k}v(t)\!+\frac{n}{k}-1\!-\!tv'(t)\right)\Phi_R(t)^2\,{\rm d}t }. \end{align*}

  Moreover, by (1.4) and (1.7), we have that

  \[ \left\vert v_{\infty}\right\vert>\left\vert v(t)\right\vert>0\mbox{ for any }t\in\{t\in[r,R]:v(t)<0\}. \]

  By (2.10) we can pick $\epsilon \in (0,\,\sqrt {\frac {n}{k}v_{\infty }+\frac {n}{k}-1}/\left \vert v _{\infty }\right \vert )$. Then applying the Young's inequality with $\varepsilon$ to $-v(t)\Phi _R(t) t^{-n}\lambda _2^{-(k+1)}$ with exponents 2 and 2, and using again (2.10), we have

  \[ -\epsilon^2v(t)^2+\frac{n}{k}v(t)+\frac{n}{k}-1-tv'(t)\geq{-}\epsilon^2v_{\infty}^2+\frac{n}{k}v_{\infty}+\frac{n}{k}-1>0, \]
  and therefore
  (2.20)\begin{equation} I_3^{-}\leq\left(1+\frac{1}{\epsilon^2}\right) \frac{r^{{-}2\alpha(r)}}{\Phi_R(r)}. \end{equation}

Finally, from (2.15)–(2.17), (2.19) and (2.20), the lemma follows.

Applying lemma 2.2 enables us to prove the following pointwise estimate.

Proposition 2.3 Let $w\in C^1(\mathbb {R}^n{\setminus} \{0\})$ be a radial function that satisfies (1.4) , $n>k(k+1)/(2k+\gamma )$ and $u$ be a semi-stable nonconstant radial solution of (1.1). Then, there exists $K'>0$ depending on $u$ and $w$ and such that

(2.21)\begin{equation} \left\vert u(2r)-u(r)\right\vert\geq K'r^{\delta(r)},\quad\forall r\geq 1. \end{equation}

Proof. Fix $r\geq 1$. Applying Hölder's inequality, lemma 2.2 with $R=2r$ and recalling that $u'$ does not vanish in $(0,\,\infty )$, we deduce

\begin{align*} \left (\int_1^{2}{ t^{-\frac{n-k}{k+2}}\,{\rm d}t}\right )r^{-\frac{n-2k-2}{k+2}}& =\int_r^{2r}{ t^{-\frac{n-k}{k+2}}\,{\rm d}t}\\ & \leq \left (\int_r^{2r}{ \frac{{\rm d}t}{t^{n-k}({u'(t)})^{k+1}}}\right )^{\frac1{k+2}} \left (\int_r^{2r}{{u'(t)}\,{\rm d}t}\right )^{\frac{k+1}{k+2}}\\ & \leq \left(Kr^{{-}2\alpha(r)} \left\vert u(2r)-u(r)\right\vert^{k+1}\right)^\frac{1}{k+2}, \end{align*}

which gives (2.21).

Proof of theorem 1.4. Let $\delta (r)$ and $\delta _\infty (\gamma )$ as in (1.6). By (2.9) and (2.10), we obtain

\[ v_{\infty}+1=\frac{2k+\gamma}{k+1}>0\mbox{ and }k(v_{\infty}-1)+n>\frac{(n-k)^2}{n}\geq 0. \]

Thus, from (1.4), (1.5), (1.7), the L'Höspital rule and the elementary equality $2\sqrt {ab}=a+b-(\sqrt {a}-\sqrt {b})^2$ for $a,\,b\geq 0$, we have

(2.22)\begin{equation} \begin{aligned} \delta_{\infty}(\gamma) & =\frac{-n+2\lim\limits_{r\to +\infty}{\alpha(r)}+2k+2}{k+1}\\ & =\frac{-n+2\left(v_{\infty}+\sqrt{\left(\frac{v_{\infty}+1}{k}\right)\left(k(v_{\infty}-1)+n\right)}\right)+2(k+1)}{k+1}\\ & =\frac{(k+1)^2\left(\frac{v_{\infty}+1}{k}\right)-\left(\sqrt{k(v_{\infty}-1)+n}-\sqrt{\frac{v_{\infty}+1}{k}}\right)^2}{k+1}\\ & =\frac{\left(2\left(k+\frac{2(k+1)(v_{\infty}+1)}{k}\right)-n\right)\left(k\sqrt{\frac{v_{\infty}+1}{k}}+\sqrt{k(v_{\infty}-1)+n}\right)}{(k+1)\left((k+2)\sqrt{\frac{v_{\infty}+1}{k}}+\sqrt{k(v_{\infty}-1)+n}\right)}\\ & =C\left(2\left(k+\frac{2\gamma}{k}+4\right)-n\right), \end{aligned} \end{equation}

where $C$ is a positive constant depending on $n,\,k$ and $\gamma$.

Additionally, from (2.14) and (1.6), we obtain

(2.23)\begin{equation} \delta(r)\geq \delta_{\infty}(\gamma),\,\forall r\geq1. \end{equation}

Now, according to the dimension $n$, we consider three cases:

• • Case $n>2(k+\frac {2\gamma }{k}+4)$. By (2.22), $\delta _{\infty }(\gamma )<0$, and by continuity there exists $r_0>1$ such that $\delta (r)<0$ for any $r\geq r_0$. At this point, we have two subcases:

  • • $\lim \limits _{r\to +\infty }{\left \vert u (r)\right \vert }\in (0,\,\infty ]$. From (2.23), we have $0>\delta (r)\geq \delta _{\infty }(\gamma )$ for any $r\geq r_0$. Then, $1>r^{\delta (r)}\geq r^{\delta _{\infty }(\gamma )}$, $\forall r\geq r_0$ and it follows that

    \[ \lim\limits_{r\to +\infty}r^{\delta(r)}\in[0,1]. \]
    If $\lim \limits _{r\to +\infty }r^{\delta (r)}\in (0,\,1]$, we have a contradiction. Hence, $\lim \limits _{r\to +\infty }r^{\delta (r)}=0$ and (1.8) follows immediately.

  • • $\lim \limits _{r\to +\infty }{\left \vert u (r)\right \vert }=0$. Let $R\geq 2r$ and $r\geq 1$. Thus, by the monotony of $u$ and proposition 2.3 there exists $K'>0$ such that:

    (2.24)\begin{equation} \begin{aligned} \left\vert u(R)-u(r)\right\vert & =\left\vert u(R)-u(2r)\right\vert+\left\vert u(2r)-u(r)\right\vert\\ & \geq\left\vert u(2r)-u(r)\right\vert\geq K'r^{\delta(r)}. \end{aligned} \end{equation}
    Letting $R\to +\infty$, (1.8) is proved for $r_0=1$.

• • Case $n<2(k+\frac {2\gamma }{k}+4)$. We have three subcases:

  • • $\lim \limits _{r\to +\infty }{\left \vert u (r)\right \vert r ^{-\delta (r)}}=0$. From (2.21), we have that $K'=0$, a contradiction.

  • • $\lim \limits _{r\to +\infty }{\left \vert u (r)\right \vert r ^{-\delta (r)}}=L\in (0,\,+\infty )$. Then, $\forall \epsilon >0$ there exists $r_0\geq 1$ such that $\left \vert \left \vert u (r)\right \vert r ^{-\delta (r)}-L\right \vert <\epsilon$ for any $r\geq r_0$ and (1.8) is proved for $r_0$.

  • • $\lim \limits _{r\to +\infty }{\left \vert u (r)\right \vert r ^{-\delta (r)}}=+\infty$. Then there exists $r_0\geq 1$ and $K>0$ such that $\left \vert u (r)\right \vert r ^{-\delta (r)}\geq K$ for any $r\geq r_0$ and (1.8) is proved for $r_0$.

• • Case $n=2(k+\frac {2\gamma }{k}+4)$. Let $r\geq 1$. Then there exists $m\in \mathbb {N}$ and $1\leq r_1<2$ such that $r=2^{m-1}r_1$. Thus, by the monotony of $u$ and proposition 2.3, it follows that

  (2.25)\begin{equation} \begin{aligned} \left\vert u(r)\right\vert & \geq\left\vert u(r)-u(r_1)\right\vert-\left\vert u(r_1)\right\vert\\ & =\sum_{j=1}^{m-1}\left\vert u(2^j r_1)-u(2^{j-1} r_1)\right\vert-\left\vert u(r_1)\right\vert\\ & \geq \sum_{j=1}^{m-1}{K'(2^{j-1}r_1)^{\delta(2^{j-1}r_1)}}-\left\vert u(r_1)\right\vert. \end{aligned} \end{equation}

  By (2.23), we have

  \[ \left(2^{j-1}r_1\right)^{\delta(2^{j-1}r_1)}\geq\left(2^{j-1}r_1\right)^{\delta_{\infty}(\gamma)}=1, \]
  for any $j\in \{1,\,\ldots,\,m-1\}$ and together with (2.25), we get
  (2.26)\begin{equation} \left\vert u(r)\right\vert\geq K'(m-1)-\left\vert u(r_1)\right\vert=\left(\frac{K'}{\log 2}\right)\left(\log r-\log r_1\right) -\left\vert u(r_1)\right\vert, \end{equation}
  and (1.8) follows easily.

Proof of theorem 1.6. From (1.8) of theorem 1.4, it follows that $n\neq 2(k+\frac {2\gamma }{k}+4)$.

Recall that

\[ \delta(r)=\frac{-n+2\alpha(r)+2k+2}{k+1}. \]

Let $R\geq 2r$ and $r\geq 1$. From (2.23) and (2.24), we have

\[ \left\vert u(R)-u(r)\right\vert\geq K'r^{\delta(r)}\geq K'r^{\delta_{\infty}(\gamma)}. \]

Thus, letting $r\to +\infty$, we conclude that $\delta _{\infty }(\gamma )$ must be negative, which is equivalent to $n>2(k+\tfrac {2\gamma }{k}+4)$. This prove $i)$.

Finally, letting $R\to +\infty$, we have

\[ \left\vert u(r)-u_{\infty}\right\vert\geq K'r^{\delta(r)},\,\forall r\geq 1, \]

which is $ii)$ with $M=K'$. The proof is complete.