数理解析研究所講究録
第 1896 巻 2014 年 34-52

34

Remarks on Liouville theorem

for H\’enon type equation on the hyperbolic space
東北大学大学院理学研究科 長谷川翔一
Shoichi Hasegawa

Mathematical InstituteTohoku University

1

Introduction

The paper is devoted to a H\’enon type equation on the hyperbolic space. In particular,
we shall prove an existence of solutions to the elliptic equation. Furthermore we announce a Liouville theorem for the equation on the hyperbolic space, which is obtainedin [21]. In order to state a motivation of our research, first we mention known results
for semilinear elliptic equations.
To begin with, we introduce known results for the following elliptic equation in the
Euclidean space:(E)$-\Delta u=|x|^{\alpha}|u|^{p-1}u$in$\mathbb{R}^{N},$is called a weight. The equation (E)where $\alpha>-2,$ $N\geq 3$ and $p>1$ . Here,$\alpha=0$
in 1869 and is well known as Lanewas posed by J.H. Lane ([27]) for the casewidelyequationhas beenstudied in the mathematicalEmden-Fowler equation. The
literature ([6, 7, 15, 19, 20, 26, 30]). Moreover, the equation was appeared in theastrophysical study of the structure of a singular star ([8, 14, 17]). In 1973, (E) forthe case $\alpha>-2$ was posed by M. H\’enon to study rotating stellar structures ([24]) and(E) is called H\’enon equation. Although he defined the equation only in 3-dimensional
unit ball with Dirichlet boundary condition, the equation has been studied for moregeneral setting by mathematical interest ([11, 18, 28, 31, 32, 35, 36]).
Regarding the exponent in (E), there exist certain critical exponents which chartypical exponent is Sobolev’s criticalacterize the structure of solutions to (E).
exponent:$|x|^{\alpha}$

$p$

$A$

$p_{S}(N):= \frac{N+2}{N-2}.$

For example,$p_{S}$

characterizes the solution of (E) with respect to the positivity:

Theorem 1.1 (B. Gidas and J. Spruck [18, 19]). Let $1<p<p_{s}(N)$ and $p\neq(N+2+$$2\alpha)/(N-2)$ . If the solutionof (E) is nonnegative, then $u=0.$$u\in C^{2}(\mathbb{R}^{N})$

Remark that Theorem 1.1 implies that there is no positive solution of (E) when$\alpha>-2,1<p<p_{s}(N)$ and $p\neq(N+2+2\alpha)/(N-2)$ . Moreover, it is sufficient toconsider only the case $\alpha>-2$ and $p\geq p_{s}(N)$ , because the nonexistence of positive
solution of (E) for the case $\alpha<-2$ was showed by B. Gidas and J. Spruck ([19]).

35
The other critical exponent, which characterizes the solution with respect to thestability, has been attracting a great interest in recent years. Indeed, the following
results were proved by Farina in 2007 for $\alpha=0$ ([15]) and by Dancer, Du and Guo in2011 for $\alpha>-2$ ([11]).
Theorem 1.2 ([11, 15]). Let$u\in C^{2}(\mathbb{R}^{N})$be a stable solution of (E). If $p>1$ satisfies$\{\begin{array}{ll}1<p<+\infty if N\leq 10+4\alpha,1<p<p(\alpha, N) if N>10+4\alpha,\end{array}$then$u\equiv 0$in$\mathbb{R}^{N}$. Here, $p(\alpha, N)$ is given by the following:$p( \alpha, N):=\frac{(N-2)^{2}-2(\alpha+2)(\alpha+N)+2\sqrt{2(\alpha+2)^{3}(\alpha+2N-2)}}{(N-2)(N-4\alpha-10)}.$On the other hand, $ifp\geq p(\alpha, N)$ , then the equation (E) has stable,
solutions.

$p_{0\mathcal{S}}$itive, and radial

The assertion in Theorem 1.2 is called a Liouville type theorem. Remark that they
proved Theorem 1.2 without any other assumption except stability, such as positivity,
radial symmetry and so on. Moreover, Theorem 1.2 implies that $p(\alpha, N)$ is critical.
Here, we define the stability of solutions to (E) as follows:
Definition 1.1.

$A$

solution$u\in C^{2}(\mathbb{R}^{N})$of (E) is stable if the inequality$\int_{\mathbb{R}^{N}}\{|\nabla\psi|^{2}-p|x|^{\alpha}|u|^{p-1}\psi^{2}\}dx\geq0$

holds for any$\psi\in C_{c}^{1}(\mathbb{R}^{N})$.

We mention some remark on Definition 1.1. One can observe that the equation (E)is formally derived as Euler-Lagrange equation for the functional$E(u):= \int_{\mathbb{R}^{N}}\{\frac{1}{2}|\nabla u|^{2}-|x|^{\alpha}\frac{|u|^{p+1}}{p+1}\}dx.$

Recall that the stability is defined forsolutions of (E) in Definition 1.1. Obviously
there exist
solutions with infinite energy. However Definition 1.1 is available forsuch solutions. Indeed, for each $R>0$ and any
solution of (E), the functional$C^{2}$$C^{2}$$C^{2}$$E_{R}(u):= \int_{B_{R}}\{\frac{1}{2}|\nabla u|^{2}-|x|^{\alpha}\frac{|u|^{p+1}}{p+1}\}dx$

is finite, where $B_{R}=\{x\in \mathbb{R}^{N} : |x|<R\}$ . Then, the second variational formula for, which is expressed as$E_{R}$$QR$ $[u]( \psi):=\int_{B_{R}}\{|\nabla\psi|^{2}-p|x|^{\alpha}|u|^{p-1}\psi^{2}\}dx,$$\forall\psi\in C_{c}^{1}(B_{R})$,36
is well-defined for any
solution of (E). Since $R>0$ is arbitrary, Definition 1.1
is equivalent to the following: “A solutionof (E) is stable if”
By making use of the concept of Definition
is non-negative for any
1.1, Liouville type theorems have been proved for many kinds of elliptic equations([9, 10, 11, 12,13, 16,25,36]).
On the other hand, recently semilinear parabolic and elliptic equations in the hyperbolic space have been studied ([1, 2, 3, 4, 5, 22, 29, 33, 34]). For example, theequation (E) for the case of $\alpha=0$ can be written as$C^{2}$$u$$Q_{R}[u](\psi)$$u\in C^{2}(\mathbb{R}^{N})$.

$\psi\in C_{c}^{1}(B_{R}).$( $LH$ )$-\triangle_{\mathbb{H}}u=|u|^{p-1}u$in$\mathbb{B}^{N},$denotes a unit ball $\{x\in \mathbb{R}^{N} : |x|<1\}$ endowed
where $p>1$ and $N\geq 3$ . Here,with the following Riemannian metric:$\mathbb{B}^{N}$$g_{ij}=( \frac{2}{1-|x|^{2}})^{2}\delta_{ij},$

whereis Kronecker’s delta. The geodesic distance from the origin to
given by$\delta_{ij}$$d_{\mathbb{H}}(0, x):= \int_{0}^{|x|}\frac{2}{1-s^{2}}ds=\log(\frac{1+|x|}{1-|x|})$Furthermore,$\triangle_{\mathbb{H}}$is the Laplace-Beltrami operator on$\mathbb{B}^{N}$$x\in \mathbb{B}^{N}$is.

and is written by$\triangle_{\mathbb{H}}u=(\frac{1-|x|^{2}}{2})^{2}\triangle u+(N-2)(\frac{1-|x|^{2}}{2})x\cdot\nabla u.$

Although it is obvious that the metric affects the geodesic distance and differential
operators, it might affect the structure of solutions. Indeed, [29] shows that there
solution for $1<p<p_{s}(N)$ by using theexists at most one positive radial
variational method. Furthermore, Bonforte, Gazzola, Grillo, and V\’azquez proved theexistence of solutions with infinite energy for $1<p<p_{s}(N)$ :$H^{1}(\mathbb{B}^{N})$Theorem 1.3 ([5, 29]). Let $1<p<p_{S}(N)$ . Then, there exists a positive radial solutionof ( $LH$ ).

$u\in C^{2}(\mathbb{B}^{N})$

Although Theorem 1.1 showed the nonexistence of positive solution of (E) for $1<$
$p<p_{s}(N)$ , Theorem 1.3 shows the existence of positive solution of ( $LH$ ) for $1<p<$$p_{s}(N)$ .
holds
The difference is strongly related that Poincar\’e’s inequality in$((N-1)/2)^{2}$, i.e., positive. Making use of theissince the first eigenvalue of
positivity, Berchio, Ferrero, and Grillo showed the following result:$L^{2}(\mathbb{B}^{N})$$-\triangle_{\mathbb{H}}$

Theorem 1.4 ([3]). Let $p>1$ . Then, for each $\beta>0$ , there exists a unique radialsolutionof ( $LH$ ) satisfying the following conditions:$u_{\beta}$$u_{\beta}(0)=\beta, u_{\beta}’(0)=0.$

Moreover, there exists some positive constant$\beta_{0}$such that$u_{\beta}$

is stable for any$\beta\leq\beta_{0}.$37Here, denotes the geodesic distancefrom the origin to. Regarding, they proved thatis bounded when $1<p<p(O, N)$ . In [3], the stability ofsolutions of ( $LH$ ) is defined by the same manner as in Definition 1.1:
$r$$x\in \mathbb{B}^{N}$$d_{\mathbb{H}}(O, x)$$\beta_{0}$$\beta_{0}$Definition 1.2. The solution$u\in C^{2}(\mathbb{B}^{N})$of ( $LH$ ) is stable if the inequality$\int_{B^{N}}\{|\nabla_{\mathbb{H}}\psi|_{\mathbb{H}}^{2}-p|u|^{p-1}\psi^{2}\}dV_{\mathbb{H}}\geq 0$

holds for any$\psi\in C_{c}^{1}(\mathbb{B}^{N})$

.

Here,
and
are the gradient operator and the volume element on the hyperbolic space, respectively. Also,
denotes the inner product ofwith itself,
where this inner product is induced from the metric onfollows:$\nabla_{\mathbb{H}}$$dV_{\mathbb{H}}$$|\nabla_{\mathbb{H}}\psi|_{\mathbb{H}}^{2}$$\nabla_{\mathbb{H}}\psi$$\mathbb{B}^{N}a_{l}’s$

(1.1)

Here

.

$| \nabla_{\mathbb{H}}\psi(x)|_{\mathbb{H}}^{2}=\langle\nabla_{\mathbb{H}}\psi(x), \nabla_{\mathbb{H}}\psi(x)\rangle_{\mathbb{H}} :=(\frac{2}{1-|x|^{2}})^{2}(\nabla_{\mathbb{H}}\psi(x), \nabla_{\mathbb{H}}\psi(x))$

denotes the usual inner product in. Theorem 1.4 implies that there is no$LH$$p(\alpha,N)$
critical exponent for ( ) such asin Theorem 1.2. This fact also arises fromthe structure of spectrum. Indeed, letting the value of origin less than thefirst eigenvalue sufficiently, they first proved that the inequality in Definition 1.2 holds.
Furthermore they also constructed non-trivial stable solution. Comparing Theorem 1.4
with Theorem 1.2, we are interested in the following question:$\mathbb{R}^{N}$$(\cdot, \cdot)$$of-\triangle_{\mathbb{H}}$Problem 1.1. Does Liouville theorem hold for the equation ( $LH$ ) with some weight?
To consider this problem, first we introduce an typical weight for ( $LH$ ). From theanalogue of the weight in (E), we can choose the power of geodesic distance as weight:(1.2)$-\triangle_{\mathbb{H}}u=(d_{\mathbb{H}}(0, x))^{\alpha}|u|^{p-1}u$in$\mathbb{B}^{N}.$Actually, He and Wang proved the existence of solutions and its asymptotic behaviorfor (1.2) ([22, 23]). However, any Liouville type theorem with respect to the stability
has not been proved yet. Indeed, we couldn’t prove the Liouville type theorem for (1.2)
although we make use of the same method as the proof of Theorem 1.2.
In order to give an affirmative answer to Problem 1.1, we consider the following
equation:(H)$- \triangle_{\mathbb{H}}u=(\frac{2|x|}{1-|x|^{2}})^{\alpha}|u|^{p-1}u$in$\mathbb{B}^{N},$where $\alpha>0,$ $p>1$ and $N\geq 3$ . Remark that we can write the weight as follows:$w(x):= \frac{2|x|}{1-|x|^{2}}=\sinh r,$

where. The reason why we choose this weight is that $\sinh r$ , which hasstrong singularity in the infinity, arises in the volume element in the hyperbolic space.
$r=d_{\mathbb{H}}(O, x)$38
By making use of the fact, we can obtain an affirmative answer to Problem 1.1. Indeed,
we shall announce a Liouville theorem which is stated in concise form as follows: “For”sufficiently small $p>1$ , if is stable solution of (H), then $u=0.$ For the precise
thesis, see Section 3. As a first step of our study for (H), we start with an existence ofsolution of (H) with small $p>1$ :$u$Theorem 1.5. The equation (H) admits a radial positive solution in$H^{1}(\mathbb{B}^{N})\cap C^{2}(\mathbb{B}^{N})$if$p \in(\frac{N-1+2\alpha}{N-1}, N+N2-+22\alpha)$

We shall construct this nontrivial solution by using variational methods. Moreover, Sobolev’s embedding implies that the solution obtained in Theorem 1.5 has finite

energy.
This paper is organized as follows. In Section 2, we shall prove Theorem 1.5. The
proof is a modification of the proof of Theorem 6 in [31]. Finally, in Section 3, we state

the Liouville theorem and asymptotic behavior of radial solutions of (H) for $p>1$ big
enough. We shall show you an outline of proof of the Liouville theorem. For the precise
proof, see [21].

Existence of solution

2. MoreIn this section, we shall prove an existence of solution to (H) in the class
6
proof
Theorem
of
the
of
by
a
modification
provedover, the following Theorem 1.5 isin [31]. We prove Theorem 1.5 by making use of Mountain Pass Theorem:$H^{1}(\mathbb{B}^{N})$

Proposition 2.1 (Mountain Pass Lemma). Let be a Banach space and let $J\in$satisfy the Palais-Smale condition. Suppose that (A) $J(O)=0$ and $J(e)=0$
on
for some $e\neq 0$ in , and (B) there exists $\rho\in(0, |e|)$ and $\alpha>0$ such that
$S_{\rho}=\{u\in E:|u|=\rho\}$ . Then
has a positive critical value$E$$C^{1}(E, \mathbb{R})$$J\geq\alpha$

$E$

$J$

$c= inf\max J(h(t))\geq\alpha>0$$h\in\Gamma t\in[0,1]$

where $\Gamma=\{h\in C([O, 1], E) : h(O)=0, h(1)=e\}.$satisfies the Palais-Smale condition if any sequencebounded and $J’(u_{n})arrow 0$ has a convergent subsequence.
$J$

Let $E$ be the completion of radially symmetricnorm, where$C_{0}^{\infty}$$\Vert u\Vert_{E}^{2}=\int_{B^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}.$$\{u_{n}\}\subseteq E$with $\{J(u_{n})\}$

functions with respect to the39
Since the bottom of the spectrum$of-\triangle_{\mathbb{H}}$is given by$\lambda_{1}(-\triangle_{\mathbb{H}}) := inf\underline{\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}}=\frac{(N-1)^{2}}{4},$$u \in H^{1}(\mathbb{B}^{N})\backslash \{0\} \int_{\mathbb{B}^{N}}|u|^{2}dV_{\mathbb{H}}$it is easy to verify that
that$\Vert\cdot\Vert_{E}$is equivarent to the norm of$H^{1}(\mathbb{B}^{N})$. Indeed we observe$\Vert u\Vert_{E}^{2}\leq\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}+\int_{\mathbb{B}^{N}}|u|^{2}dV_{\mathbb{H}}$$\leq(1+\frac{4}{(N-1)^{2}})\Vert u\Vert_{E}^{2}.$In the following, we shall prepare the proposition which we need in order to show theexistence of solution of (H) in:$H^{1}(\mathbb{B}^{N})$

Lemma 2.1. Let $u\in E$ . Then it holds that
(2.1)$|u(x)| \leq\frac{1|_{E}}{\sqrt{w_{N}(N-2)}(\sinh(2arnh|x|))^{\frac{N-2}{2}}},$(2.2)$|u(x)| \leq\frac{1E}{\sqrt{w_{N}(N-1)}(\sinh(2arnh|x|))^{\frac{N-1}{2}}},$

is the surface area of the unit ball in
Proof. Since $u\in E$ , it holds thatwhere$w_{N}$$\mathbb{R}^{N}.$$u(1)-u(|x|)= \int_{|x|}^{1}u’(t)dt.$

By H\’older’s inequality, we have
(2.3)$|u(x)| \leq\int_{|x|}^{1}|u’(t)|dt$$\leq(\int_{|x|}^{1}|u’(t)|^{2}t^{N-1}(\frac{2}{1-t^{2}})^{N-2}dt)^{\frac{1}{2}}(\int_{|x|}^{1}t^{-(N-1)}(\frac{2}{1-t^{2}})^{-(N-2)}dt)^{\frac{1}{2}}$$:=I_{1}+I_{2}.$First we estimate$I_{1}$

as follows:$I_{1}= \frac{1}{w_{N}}\int_{\partial B(0,1)}(\int_{|x|}^{1}(\frac{1-t^{2}}{2})^{2}|u’|^{2}(\frac{2}{1-t^{2}})^{N}t^{N-1}dt)dS$$= \frac{1}{w_{N}}\int_{|x|\leq|y|\leq 1}(\frac{1-|y|^{2}}{2})^{2}|\nabla u|^{2}(\frac{2}{1-|y|^{2}})^{N}dy$$= \frac{1}{w_{N}}\int_{|x|\leq|y|\leq 1}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}(y)$$\leq\frac{1}{w_{N}}\Vert u\Vert_{E}^{2}.$40Regarding$I_{2}$, we find$I_{2}= \int_{2arc\tanh|x|}^{\infty}(\tanh\frac{s}{2})^{-(N-1)}(2\cosh^{2}\frac{s}{2})^{-(N-2)}(2\cosh^{2}\frac{s}{2})^{-1}ds$$= \int_{2arctanh|x|}^{\infty}(\sinh s)^{-(N-1)}ds$$\leq\int_{2arc\tanh|x|}^{\infty}(\sinh s)^{-(N-1)}\cosh sds$$=- \frac{1}{N-2}[(\sinh s)^{-(N-2)}]_{2arctanh|x|}^{\infty}=\frac{1}{N-2}(\sinh(2arc\tanh|x|))^{-(N-2)}$

Then (2.1) is followed from this estimate and (2.3). Moreover, we can also estimate$I_{2}$

as follows:$I_{2}= \int_{2arctanh|x|}^{\infty}(\frac{1}{\sinh s})^{N-1}ds$$\leq\int_{2arctanh|x|}^{\infty}(\frac{1}{\sinh s})^{N-1}\frac{1}{\tanh s}ds$$= \int_{2arc\tanh|x|}^{\infty}(\frac{1}{\sinh s})^{N}$

cosh $sds$$=- \frac{1}{N-1}[(\sinh s)^{-(N-1)}]_{2arctanh|x|}^{\infty}=\frac{1}{N-1}(\sinh(2arc\tanh|x|))^{-(N-1)}$

Combining this estimate with (2.3), we find (2.2).
Lemma 2.2. Let $0<m<(N-1)/2$ . Then for any$\tau\in(\frac{2(N-1)}{N-1-2m},\hat{m})$there exists a constant $C=C(N, \tau, m)$ such that
(2.4)$\Vert w^{m}u\Vert_{L^{\tau}(B^{N})}\leq C\Vert u\Vert_{E}$where$\hat{m}=\{\begin{array}{ll}\frac{2N}{N-2-2m} when m<\frac{N-2}{2}.\infty when \frac{N-2}{2}\leq m<\frac{N-1}{2}.\end{array}$

Proof. Let$0<m< \frac{N-1}{2}.$$\square$41
To prove (2.4), we divide the integral int two parts:$0$$\int_{B}w^{m\tau}|u|^{\tau}dV_{\mathbb{H}}=\int_{0\leq|x|\leq\frac{1}{2}}(\frac{2|x|}{1-|x|^{2}})^{m\tau}|u|^{\tau}(\frac{2}{1-|x|^{2}})^{N}dx$$+ \int_{2}\leq|x|\leq 1(\frac{2|x|}{1-|x|^{2}})^{m\tau}|u|^{\mathcal{T}}(\frac{2}{1-|x|^{2}})^{N}dx$$=:X+Y.$First we estimate the term X. By (2.1), we have
$X \leq C\Vert u\Vert_{E}^{\tau}\int_{0\leq|x|\leq\frac{1}{2}}(\frac{2|x|}{1-|x|^{2}})^{m\tau}(\sinh(2arc\tanh|x|))^{-\frac{N-2}{2}\tau}(\frac{2}{1-|x|^{2}})^{N}dx$$=C \Vert u\Vert_{E}^{\tau}\int_{0}^{2arctanh\frac{1}{2}}(\sinh s)^{m\tau+N-1-\frac{N-2}{2}\tau_{d_{S}}}$$\leq C\Vert u\Vert_{E}^{\tau}\int_{0}^{2arctanh\frac{1}{2}}(\sinh s)^{m\tau+N-1-\frac{N-2}{2}\tau}\cosh sds$$=C \Vert u\Vert_{E}^{\mathcal{T}}\int_{0}^{\sinh(2arc\tanh\frac{1}{2})_{t^{m\mathcal{T}+N-1-\frac{N-2}{2}\tau}}}dt.$

Since the relation$m \tau+N-1-\frac{N-2}{2}\tau>-1$

holds if and only if$\tau<\hat{m}$. Thus we see that if$\tau<\hat{m}$then it holds that

$X\leq C\Vert u\Vert_{E}^{\mathcal{T}},$

where $C$ depends only on $N,$$\tau$

and $m$ . On the other hand (2.2) gives us that$Y \leq C\Vert u\Vert_{E}^{\tau}\int_{2}\leq|x|\leq 1(\frac{2|x|}{1-|x|^{2}})^{m\tau}(\sinh(2arc\tanh|x|))^{-\frac{N-1}{2}\mathcal{T}}(\frac{2}{1-|x|^{2}})^{N}dx$$=C \Vert u\Vert_{E}^{\tau}\int_{2arctanh\frac{1}{2}}^{\infty}(\sinh s)^{m\tau+N-1-\frac{N-1}{2}\tau_{ds}}$$\leq C\Vert u\Vert_{E}^{\tau}\int_{2arc\tanh\frac{1}{2}}^{\infty}(\sinh s)^{m\tau+N-1-\frac{N-1}{2}\tau}\frac{1}{\tanh s}ds$$\leq C\Vert u\Vert_{E}^{\tau}\int_{2arc\tanh\frac{1}{2}}^{\infty}(\sinh s)^{m\tau+N-2-\frac{N-1}{2}\tau}$cosh $sds$$=C \Vert u\Vert_{E}^{\tau}\int_{\sinh(2arc\tanh\frac{1}{2})^{t^{m\tau+N-2-\frac{N-1}{2}\tau}}}^{\infty}dt.$

It is easy to verify that$m \tau+N-2-\frac{N-1}{2}\tau<-1$

42
is equivalent to(2.5)$\tau>\frac{2(N-1)}{N-1-2m}.$

Hence we see that
$Y\leq C\Vert u\Vert_{E}^{\tau}$

if$\tau$

satisfies (2.5). Therefore we obtain the conclusion.

$\square$

Making use of Lemma 2.2, we shall prove a compactness.

Lemma 2.3. Let $0<m<(N-1)/2$ . Let satisfy the condition given in Lemma 2.2.
is compact.
Then the map $u\mapsto w^{m}u$ from to$\tau$$E$$L^{\tau}(\mathbb{B}^{N})$Proof. Let $0<m<(N-1)/2$ and arbitrarily fix$\tau$

satisfying the condition given in

Lemma 2.2. Then Lemma 2.2 asserts that$\Vert w^{m}u\Vert_{L^{\tau}(B^{N})}\leq C\Vert u\Vert_{E}.$is continuous. Now we shallThis shows that the map $u\mapsto w^{m}u$ from $E$ tocompact.
prove that the map isis compact for any $q\in$We first note that the embedding$E$ is equivalent tothat
Recalling
Theorem3.1).
$(2,2N/(N-2))$ (see [29],is also compact, we see that the embedding
with respect to the norm
for any $q\in(2,2N/(N-2))$ .
Let us fix $q \in(2, \min\{\tau, 2N/(N-2)\})$ arbitrarily. By H\"older’s inequality, we have$L^{\tau}(\mathbb{B}^{N})$$H_{rad}^{1}(\mathbb{B}^{N})arrow L^{q}(\mathbb{B}^{N})$$H_{rad}^{1}(\mathbb{B}^{N})$$Earrow L^{q}(\mathbb{B}^{N})$$\Vert\cdot\Vert_{E}$(2.6)$|w^{m}u|_{L^{\tau}(B^{N})}=( \int_{B^{N}}|w^{m}u|^{\tau})^{\frac{1}{\tau}}$$=( \int_{J\beta^{N}}w^{m\tau}|u|^{\tau-qa}|u|^{qa})^{\frac{1}{\tau}}$$\leq(\int_{N}|u|^{q})^{\frac{a}{\tau}}(\int_{J\beta^{N}}(w^{m\tau}|u|^{\tau-qa})^{\frac{1}{1-a}})^{\frac{1-a}{\tau}}$$=|u|_{L^{q}(B^{N})}^{\Delta}a\tau|w^{\frac{m\tau}{\tau-qa}}|u||_{\frac{-qa\tau-qa\tau}{1-a}}^{\frac{\tau}{L}}(B^{N})$where $a\in(0,1)$ . In the following, setting$m^{*}:= \frac{m\tau}{\tau-qa}, \tau^{*}:=\frac{\tau-qa}{1-a},$and making use of Lemma 2.2, we shall verify that
(2.7)$\Vert w^{m^{*}}u\Vert_{L^{\tau^{*}}(B^{N})}\leq C\Vert u\Vert_{E}$’43holds. If $m\geq(N-2)/2$ , then the relation $m<m^{*}$ implies $m^{*}\geq(N-2)/2$ . Since$\frac{2(N-1)}{N-1-2m^{*}}<\tau^{*}\Leftrightarrow\frac{2(N-1)+(q-2)(N-1)a}{N-1-2m}<\tau$for sufficiently small $a>0$ , Lemma 2.2 asserts that (2.7) holds for each $\tau>2(N-$$1)/(N-1-2m)$ . Regarding the case of $0<m<(N-2)/2$ , it is sufficient to consider
the case of $0<m^{*}<(N-2)/2$ since the case of $m^{*}\geq(N-2)/2$ is contained in theabove case. Recalling$\tau^{*}<\frac{2N}{N-2-2m^{*}}\Leftrightarrow\tau<\frac{qa(N-2)+2N(1-a)}{N-2-2m}$

and $2N-(N-2)q>0$ , we observe from Lemma 2.2 that for $a\in(0,1)$ small enough(2.7) holds for each satisfying$\tau$$\frac{2(N-1)}{N-1-2m}<\tau<\frac{2N}{N-2-2m}.$

Combining (2.6) with (2.7), we obtain(2.8)$\Vert w^{m}u\Vert_{L^{\mathcal{T}}(\mathbb{B}^{N})}\leq C\Vert u\Vert_{Lq(\mathbb{B}^{N})}^{\tau}\underline{a}q\Vert u\Vert_{E^{\mathcal{T}}}\underline{\tau}g^{-}\underline{a}$for sufficiently small $a\in(O, 1)$ . Thus the map $u\mapsto w^{m}u$ from $E$ tois continuous.
Finally we show that the map $u\mapsto w^{m}u$ is compact. Let
be bounded sequence
in $E$ . Since
is compact, there exists a subsequenceand afunction $u\in E$ such that$L^{\tau}(\mathbb{B}^{N})$$\{u_{n}\}$$Earrow L^{q}(\mathbb{B}^{N})$$\{u_{nj}\}\subset\{u_{n}\}$$u_{nj}arrow u$

in$L^{q}(\mathbb{B}^{N})$.

By (2.8), we see that$|w^{m}(u_{nj}-u)|_{L^{\tau}(\mathbb{B}^{N})}\leq|u_{nj}-u|_{L^{q}(\mathbb{B}^{N})}^{\tau}|\nabla_{\mathbb{H}}(u_{nj}-u)|_{L^{2}(B^{N})}^{\tau}\underline{a}q\underline{\tau}-p\underline{a}$$\leq C|u_{nj}-u|_{L^{q}(\mathbb{B}^{N})}^{\tau}(|u_{nj}|_{E^{\tau}}a\Delta\underline{\tau}-\Delta^{\underline{a}}+|u|_{E^{\tau}})\underline{\mathcal{T}}-L^{a}$$\leq C|u_{nj}-u|_{L^{q}(\mathbb{B}^{N})}^{\tau}a\Delta$

Therefore, we complete the proof.

$\square$

We are in a position to prove the following theorem by using above propositions:

Theorem 2.1. Let$p \in(\frac{N-1+2\alpha}{N-1}, \frac{N+2+2\alpha}{N-2})$

Then, the equation (H) has a positive radial solution.

$u\in H^{1}(\mathbb{B}^{N})\cap C^{2}(\mathbb{B}^{N})$.

44
Proof. Instead of the equation (H), we prove that$\{\begin{array}{ll}-\triangle_{\mathbb{H}}u=w^{\alpha}(u^{+})^{p} in \mathbb{B}^{N}\lim_{|x|arrow 1}u=0 \end{array}$has a nontrivial solution inLet$H^{1}(\mathbb{B}^{N})$by using Mountain Pass Theorem.

$J(u):= \frac{1}{2}\int_{B^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}-\int_{B^{N}}w^{\alpha}F(u)dV_{\mathbb{H}},$

where$F(u) := \frac{1}{p+1}(u^{+})^{p+1} u^{+}:=\max\{u, 0\}.$To begin with, we verify that the functional$J$

is well-defined. Since$\frac{2N-2}{N-1-2\frac{\alpha}{p+1}}<p+1\Leftrightarrow\frac{N-1+2\alpha}{N-1}<p,$and$p+1< \frac{2N}{N-2-2\frac{\alpha}{p+1}}\Leftrightarrow p<\frac{N+2+2\alpha}{N-2},$

Lemma 2.3 implies that
(2.9)$\int_{B^{N}}w^{\alpha}F(u)dV_{\mathbb{H}}\leq C\int_{\mathbb{B}^{N}}|w^{\frac{\alpha}{p+1}}u|^{p+1}dV_{\mathbb{H}}\leq C\Vert u\Vert_{E}^{p+1}$

Next we show that
(2.9) yields that$J$

satisfies the hypothesis of the Proposition 2.1. The relation$J(u)= \frac{1}{2}\int_{B^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}-\int_{B^{N}}w^{\alpha}F(u)dV_{\mathbb{H}}$$\geq\frac{1}{2}\Vert u\Vert_{E}^{2}-C\Vert u\Vert_{E}^{p+1}$

Thus, setting$f( \rho):=\frac{1}{2}\rho^{2}-C\rho^{p+1},$

we see that for $\rho>0$ sufficiently small$f(\rho)>f(0)=0.$

45
Therefore, (B) is fulfilled. We turn to the condition (A). It is clear that $J(O)=0.$
Since$J(tu)= \frac{t^{2}}{2}\int_{\mathbb{B}^{N}}|\nabla_{\mathbb{H}}u|_{\mathbb{H}}^{2}dV_{\mathbb{H}}-t^{p+1}\int_{\mathbb{B}^{N}}w^{\alpha}F(u)dV_{\mathbb{H}}arrow-\infty$as$tarrow\infty$

we observe that there exists $e\in E$ such that $J(e)=0$ . Thus, (A) is fulfilled.
Next we prove that satisfies the Palais-Smale condition. Define a map : $Earrow E$
by
$J$

$T$

$(Tu, v)_{E}= \int_{B^{N}}w^{\alpha}(u^{+})^{p}v, v\in E.$$T$may be decomposed as follows:$T:u\mapsto w^{\frac{\alpha}{p}}u\mapsto w^{\frac{\alpha}{p}}u^{+}\mapsto(w^{\frac{\alpha}{p}}u^{+})^{p}\mapsto w^{\alpha}(u^{+})^{p}\mapsto Tu$$Earrow L^{pq}\tau_{1}arrow L^{pq}\tau_{2} arrow L^{q}\tau_{3} arrow H^{-1}\tau_{4} arrow E\tau_{5},$where$q= \{\begin{array}{lll}\frac{2N}{N+2} if p\in 2 if p\in\end{array}\}\frac{N+2}{2\alpha^{N}},\frac{N+2.+2\alpha}{-22\alpha N-2).=I_{2}}.):=I_{1},$In the following we shall show that the map is compact. To begin with, we verify
thatis compact by using Lemma (2.3). To do so, setting, weandcheck thatandsatisfy the condition in Lemma (2.3). Remark that$p>(N-1+2\alpha)/(N-1)$ implies $\tilde{m}<(N-1)/2$ . To begin with, we check that$T$

$T_{1}$$\tilde{m}=\alpha/p$$\tilde{m}=\alpha/p$(2.10)

When$\tilde{\tau}=pq$$\tilde{\tau}=pq$$\frac{2N-2}{N-1-2\tilde{m}}<\tilde{\tau}.$$p\in I_{1}$, one can verify that (2.10) is equivalent to$\frac{2\alpha}{N-1}+\frac{N+2}{2}<p.$

On the other hand, (2.10) is equivalent to$\frac{N-1+2\alpha}{N-1}<p,$

if $p\in I_{2}$ . Hence (2.10) is satisfied. Since$\tilde{m}<(N-2)/2$ then
(2.11)$\tilde{\tau}<+\infty$$\tilde{\tau}<\frac{2N}{N-2-2\tilde{m}}.$, it is sufficient to show that if

46
For the case of$p\in I_{1},$$(2.12)$is equivalent to$p< \frac{N+2+2\alpha}{N-2},$

and while if$p\in I_{2}$, then (2.12) is equivalent to$p<^{\underline{N+2\alpha}}$$N-2$

Therefore we can apply Lemma 2.3 to the map . Then Lemma 2.3 asserts thatis compact. The map
is clearly continuous. Regarding , since the map is a
Nemitski operator, we see thatis continuous. Next we turn to . Let us define$T_{1}$$T_{1}$$T_{2}$$T_{3}$$T_{3}$$T_{4}:L^{q}arrow(L^{\hat{q}})^{*}$$T_{4}$

by$(T_{4}(w^{\alpha}(u^{+})^{p}))(v)= \int_{B^{N}}w^{\alpha}(u^{+})^{p}v, v\in L^{\hat{q}},$where$\hat{q}=\{\begin{array}{lll}\frac{2N}{N-2} if p\in 2 if p\in\end{array}\},$

H\"older’s inequality yields that:is continuous. Since$L^{q}arrow H^{-1}$, we see that:is also continuous. Therefore,$(L^{\hat{q}})^{*}arrow H^{-1}$$H^{1}arrow L^{\hat{q}}$$L^{q}arrow(L^{\hat{q}})^{*}$$T_{4}$$T_{4}$

is continuous. Finally we show that$(T_{5}(f), v)_{E}=f(v)$$T_{5}$

implies:
is continuous. Define $T_{5}:H^{-1}arrow H^{1}$ by

for$f\in H^{-1}$and$T_{4}$$L^{q}arrow H^{-1}$$v\in H^{1}.$Then we have$|(T_{5}(f), v)_{E}|\leq\Vert f\Vert_{H^{-1}}\Vert v\Vert_{H^{1}}\leq C\Vert f\Vert_{H^{-1}}\Vert v\Vert_{E},$so that,$|T_{5}(f)|_{H^{1}}\leq\hat{C}\Vert f\Vert_{H^{-1}}.$

Therefore$T_{5}$

is continuous. In particular, we observe that$(T_{5}(T_{4}(w^{\alpha}(u^{+})^{p})), v)_{E}=(T_{4}(w^{\alpha}(u^{+})^{p}))(v)=\int_{B^{N}}(w^{\alpha}(u^{+})^{p})v=(Tu, v)_{E}.$

Thus, $T=T_{5}oT_{4}oT_{3}oT_{2}\circ T_{1}$ is compact from $E$ to $E.$Let
be a sequence satisfying $|J(u_{n})|\leq d$ and $J’(u_{n})arrow 0$ . For
large enough, we have$\{u_{n}\}\subset E$$d+ \Vert u_{n}\Vert_{E}\geq J(u_{n})-\frac{1}{\tau+1}J’(u_{n})(u_{n})$$=( \frac{1}{2}-\frac{1}{\tau+1})\Vert u_{n}\Vert_{E}^{2}.$$n\in \mathbb{N}$47This implies thatis bounded. Then there exists a subsequence$u\inE$functionsuch that$\Vert u_{n}\Vert_{E}^{2}$(2.12)$u_{n_{j}}arrow u$

Furthermore, since

$T$

in

$\hat{u}\in E$

$E.$

in

$E$

up to a subsequence. Recalling that$(u_{n}-Tu_{n}, v)_{E}=J’(u_{n})(v)arrow 0$

as$narrow\infty$

for any $v\in E$ , it must hold $\hat{u}=u$ . In the following we write
short. By a simple calculation, we have
(2.13)and ais compact operator, it follows from (2.12) that
$Tu_{n_{j}}arrow\hat{u}$for a function$u_{n_{j}}\subset u_{n}$$u_{n}$

instead of$u_{n_{j}}$for$\Vert u_{n}-u\Vert_{E}=J’(u_{n})(u_{n}-u)-J’(u)(u_{n}-u)+(Tu_{n}-Tu, u_{n}-u)_{E}$$=:I_{1}+I_{2}+I_{3},$and then$I_{1}\leq\Vert J’(u_{n})\Vert_{E^{*}}\Vert u_{n}-u\Vert_{E}\leq\Vert J’(u_{n})\Vert_{E^{*}}(\Vert u_{n}\Vert_{E}+\Vert u\Vert_{E})arrow 0,$$I_{2}=J’(u)(u_{n}-u)arrow 0,$$I_{3}=(Tu_{n}-u, u_{n}-u)_{E}+(u-Tu, u_{n}-u)_{E}$$\leq\Vert Tu_{n}-u\Vert_{E}(\Vert u_{n}\Vert_{E}+1u\Vert_{E})+(u-Tu, u_{n}-u)_{E}arrow 0.$

Therefore (2.13) yields that$u_{n}arrow u$

in

$E.$

This implies thathas a convergent subsequence, i.e., satisfies the Palais-Smalecondition. Then, the Mountain Pass Lemma assures that has a nontrivial criticalvalue, hence, a nontrivial critical point $u\in E$ . In particular, function satisfies$J$$\{u_{n}\}$

$J$

$u$

(2.14)$J’(u)(v)= \int_{B^{N}}\langle\nabla_{\mathbb{H}}u,$ $\nabla_{\mathbb{H}}v\rangle_{\mathbb{H}}dV_{\mathbb{H}}-\int_{B^{N}}w^{\alpha}(u^{+})^{p}vdV_{\mathbb{H}}=0$

Taking$u^{-}$

for$v\in E.$

as in (2.14), we have$v$

$0= \int_{\mathbb{B}^{N}}\langle\nabla_{\mathbb{H}}u, \nabla_{\mathbb{H}}u^{-}\rangle_{\mathbb{H}}dV_{\mathbb{H}}-\int_{B^{N}}w^{\alpha}(u^{+})^{p}u^{-}dV_{\mathbb{H}}=\Vert u^{-}\Vert_{E},$so that $u^{-}=0$ a.e. in. Therefore, combining this fact with (2.14), we see that is
a nonnegative and nontrivial
solution of (H).
By an elliptic regularity theorem, $u\in C^{2}$ . Finally we shall prove that is a positive
solution. Suppose not, there exists
such that $u(x_{0})=0$ . For any $r>0$ , it$\mathbb{B}^{N}$$u$$H^{1}(\mathbb{B}^{N})$$u$$x_{0}\in \mathbb{B}^{N}$

holds that$-\triangle_{\mathbb{H}}u=w^{\alpha}(u^{+})^{p}\geq 0$

whereimplies that $u\equiv 0$ inThis leads a contradiction.

$B_{\mathbb{H}}(x_{0}, r)=\{x\in \mathbb{B}^{N} :$B_{\mathbb{H}}(x_{0}, r)$in$B_{\mathbb{H}}(\xi_{0}, r)$,. Then the strong maximum principle. Since $r>0$ is arbitrary, we see that $u\equiv 0$ in

d_{\mathbb{H}}(x, x_{0})<r\}$$\mathbb{B}^{N}.$$\square$48

3

Liouville Theorem

In this section, we prove a Liouville theorem corresponding to (H). First, in order tostate the result, we define the stability of solutions. The stability of solutions of (H) isdefined by the same manner as in Definition 1.1:

Definition 3.1. The solution$u\in C^{2}(\mathbb{B}^{N})$of (H) is stable if the inequality$Q[u]( \psi):=\int_{B^{N}}\{|\nabla_{\mathbb{H}}\psi|_{\mathbb{H}}^{2}-pw^{\alpha}|u|^{p-1}\psi^{2}\}dV_{\mathbb{H}}\geq 0$

holds for any$\psi\in C_{c}^{1}(\mathbb{B}^{N})$.

Then we state the Liouville theorem corresponding to the equation (H):

Theorem 3.1 ([21]). Let$u\in C^{2}(\mathbb{B}^{N})$be a stable solution of (H). If $p>1$ satisfies$\{\begin{array}{ll}1<p<+\infty if N\leq 1+4\alpha,1<p<p_{c}(\alpha, N) if N>1+4\alpha,\end{array}$then$u\equiv 0$in$\mathbb{B}^{N}$. Here,$p_{c}(\alpha, N)$is given by the following:$p_{c}( \alpha, N):=\frac{(N-1)^{2}-2\alpha(N-1)-2\alpha^{2}+2\alpha\sqrt{2\alpha(N-1)+\alpha^{2}}}{(N-1)(N-4\alpha-1)}.$

Theorem 3.1 gives us an affirmative answer to Problem 1.1. And if we find a nontrivial stable solution when $p\geq p_{c}$ , thenis critical. Although we have not proved
this fact yet, we obtained the following result which suggests thatis critical:$p_{c}$$p_{c}$

Theorem 3.2 ([21]). Let $p>(N+2+2\alpha)/(N-2)$ . Then, there exists a positive
radial solution $u=u(r)$ of (H) satisfying$\lim_{rarrow+\infty}u(r)(\sinh r)^{\frac{\alpha}{p-1}}=\{\frac{\alpha}{p-1}(N-1-\frac{\alpha}{p-1})\}^{\frac{1}{p-1}} :=L.$

Now, using Theorem 3.2, we can give some consideration to. Let $p\geq$
and $N>1+4\alpha$ . We assume that there exists a radial solution $u=u(r)$ of$p_{c}(\alpha, N)$$p_{c}(\alpha, N)$(H) satisfying(3.1)$u(r)(\sinh r)^{\frac{\alpha}{p-1}}\leq L (\forall r>0)$.

Then, by some calculations, we see that the solution satisfying (3.1) is stable. From
Theorem 3.2, one can notice that the condition 3.1 is valid. Therefore we can expectthat the exponent
is critical.
Next, we state the outline of proof of Theorem 3.1. First, we prepare the following
proposition:$u$$p_{c}(\alpha, N)$49
Proposition 3.1. Let

be a stable solution of (H). Then, for any$[1,2p+2\sqrt{p(p-1)}-1)$ and for any integer $m \geq\max\{_{p-1}E^{+}\Delta,2\}$ , there exists somepositive constant $C=C(p, m, \alpha, \gamma)$ such that for any
with$u\in C^{2}(\mathbb{B}^{N})$$\gamma\in$$\psi\in C_{c}^{2}(\mathbb{B}^{N})$$|\psi|\leq 1,$$\int_{\mathbb{B}^{N}}w^{\alpha}|u|^{p+\gamma}\psi^{2m}dV_{\mathbb{H}}\leq C\int_{\mathbb{B}^{N}}w^{p^{\frac{+1}{-1}\alpha}}-f|\nabla_{\mathbb{H}}\psi|_{\mathbb{H}}^{2_{p-1}^{L+:f}}dV_{\mathbb{H}}.$We can prove this assertion by a modification of the proof in Proposition 1.4 of[10] and Proposition 1.7 of [11]. In the following, we prove Theorem 3.1 by using
Proposition 3.1.

Proof. Here, the essential matter of Proposition 3.1 is that one can estimate the integralof by the integral being independent of . Therefore, we expect that the stable
solution can be characterized by the test function. Indeed, in order to prove Theorem
3.1, we set the following test function
for each $R>0$ :
$u$

$u$

$u$

$\psi_{R}$$\psi_{R}(x):=\varphi(\frac{\sinh(d_{\mathbb{H}}(0,x))}{R})$where$\varphi\in C_{C}^{2}(\mathbb{R})$satisfies$0\leq\varphi\leq 1$,

and$\varphi(t)=\{\begin{array}{ll}1 if |t|\leq 1,0 if |t|\geq 2.\end{array}$In the following, we write$q= \frac{p+\gamma}{p-1}, \overline{q}=\frac{\gamma+1}{p-1}$

for short and we set$A(R)=$ arc $\sinh R,$$B(R)=$ arc sinh $2R.$

Then, notice that$\psi_{R}(x)=\{\begin{array}{l}1 if d_{\mathbb{H}}(O, x)\leq A(R) ,0 if d_{\mathbb{H}}(O, x)\geq B(R) .\end{array}$Since the change of variable
it follows from Proposition 3.1 that$r=d_{\mathbb{H}}(0, x)$(3.2)

yields $w(x)=\sinh r$ and $dV_{\mathbb{H}}=(\sinh r)^{N-1}dr,$$\int_{d_{\mathbb{H}}(0,x)\leq A(R)}w^{\alpha}|u|^{p+\gamma}dV_{\mathbb{H}}\leqC\int_{A(R)\leq d_{\mathbb{H}}(0,x)\leq B(R)}w^{-\overline{q}\alpha}|\nabla_{\mathbb{H}}\psi_{R}|_{\mathbb{H}}^{2q}dV_{\mathbb{H}}$$\leq\frac{C}{R^{2q}}\int_{A(R)}^{B(R)}(\sinh r)^{N-1-\overline{q}\alpha+2q}dr$$\leq CR^{N-1-\overline{q}\alpha}.$50
On the other hand, $p<p_{c}(\alpha, N)$ if and only if there exists some $\gamma\in[1,2p+$$2\sqrt{p(p-1)}-1)$ such that

(3.3)

$N-1-\overline{q}\alpha<0.$

Hence, we can choose $\gamma\in[1,2p+2\sqrt{p(p-1)}-1)$ satisfying (3.3). And then, (3.2)
implies that$\int_{d_{H}(0,x)\leq A(R)}w^{\alpha}|u|^{p+\gamma}dV_{\mathbb{H}}arrow 0$

Since $A(R)arrow+\infty$ as $Rarrow+\infty$ , we see thatcompletes the proof of Theorem 1.2.

$u$

as

$Rarrow+\infty.$

must be identically equal to . This$0$$\square$Here, in order to obtain the estimate just as (3.2) in this proof, we have to select thein terms of the volume element
weight and test function. Hence, since theweight of the equation (1.2) is the power of the geodesic distance, the above argument
does not work for the equation (1.2). This is the reason why we choose the weight of
(H).
$w$

$\psi_{R}$$dV_{\mathbb{H}}$

References[1] C. Bandle and Y. Kabeya, On the positive, “radial”
of a semilinear
elliptic equation in, Adv. Nonlinear Anal. 1 (2012), no. 1, 1-25.
$solution\mathcal{S}$$\mathbb{H}^{N}$[2] C. Bandle, M.A. Pozio, and A. Tesei, The Fujita exponent for the Cauchy problem
in the hyperbolic space, J. Differential Equations 251 (2011), 2143-2163.

[3] E. Berchio, A. Ferrero, and G. Grillo, Stability and qualitative properties of radialsolutions of the Lane-Emden-Fowler equation on Riemannian models, J. Math.
Pure. Appl., to appear.
[4] M. Bhakta and K. Sandeep, Poincar\’e-Sobolev equations in the hyperbolic space,
Calc. Var. Partial Differential Equations 44 (2012), no. 1-2, 247-269.

[5] M. Bonforte, F. Gazzola, G. Grillo, and J.L. V\’azquez, Classification of radialsolutions to the Emden-Fowler equation on the hyperbolic space, Calc. Var. Partial
Differential Equations 46 (2013), no. 1-2, 375-401.
[6] H. Br\’ezis, Elliptic equations with limiting Sobolev exponents-the impact of topology,
Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S17-S39.

[7] H. Br\’ezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4,

437-477.
[8] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, NewYork (1967).

51[9] C. Cowan and M. Fazly, On stable entire solutions of semi-linear elliptic equations
with weights, Proc. Amer. Math. Soc. 140 (2012), no. 6, 2003-2012.
[10] L. Damascelli, A. Farina, B. Sciunzi, and E. Valdinoci, Liouville results for mLaplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincar\’e Anal. Non
Lin\’eaire 26 (2009), no. 4, 1099-1119.

[11] E.N. Dancer, Y. Du, and Z. Guo, Finite Morse index solutions of an elliptic
equation with supercritical exponent, J. Differential Equations 250 (2011), 3281-

3310.

[12] Y. Du and Z. Guo, Positive solutions of an elliptic equation with negative exponent:
stability and critical power, J. Differential Equations 246 (2009), no. 6, 2387-2414.

[13] J. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear
elliptic equations with convex nonlinearities, Nonlinear Anal. 70 (2009), no. 8,2882-2888.

[14] V.R. Emden, Gaskugeln, Anwendungen der mechanischen Warmentheorie auf
Kosmologie und meteorologische Probleme, Teubner, Leipzig (1907), Chap. XII.

[15] A. Farina, On the classification of solutions of the Lane-Emden equation on unbouded domains of, J. Math. Pure. Appl. 87 (2007), 537-561.
$\mathbb{R}^{N}$[16] A. Farina, Y. Sire, and E. Valdinoci, Stable solutions of elliptic equations on Rie-

mannian manifolds, J. Geom. Anal. 23 (2013), no. 3, 1158-1172.

[17] R. H. Fowler, Further studies of Emden’s and similar differential equations, Q. J.
Math. (Oxford Series) 2 (1931), 259-288.
[18] B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic
equations, Comm. Partial Differential Equations 6 (1981), no. 8, 883-901.

[19] B. Gidas and J. Spruck, Global and local behavior of positive solution of nonlinear
elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598.
[20] C. Gui, W.-M. Ni, and X. Wang, On the stability and instability of positive steadystates of a semilinear heat equation in, Comm. Pure Appl. Math. 45 (1992),
no. 9, 1153-1181.
$\mathbb{R}^{n}$[21] S. Hasegawa, Liouville theorem for H\’enon type equation on the hyperbolic space,
preprint.
[22] H. He, The existence of solutions for H\’enon equation in hyperbolic space, Proc.
Japan Acad., 89 (2013), Ser. A no. 2, 24-28.
[23] H. He and W. Wang, Existence and asymptotic behavior of solutions for H\’enon
equations in hyperbolic spaces, Electron. J. Differential Equations 208 (2013), 1-

13.

52
[24] M. H\’enon, Numerical experiments on the stability of spherical stellar systems,
Astronom. Astrophys. Lib. 24 (1973), 229-238.
[25] W. Jeong and Y. Lee, Stable solutions and finite Morse index solutions of nonlinear
elliptic equations with Hardy potential, Nonlinear Anal. 87 (2013), 126-145.
[26] D.D. Joseph, and T.S. Lundgren, Quasilinear Dirichlet problems driven by positive
sources, Arch. Rational Mech Anal. 49 (1972/73), 241-269.

[27] J.H. Lane, In the theoretical temperature of the Sun under the hypothesis of agaseous mass maintaining ots volume by its internal heat and depending on thelaws of gases known to terrestrial experiment, Am. J. Sci. Ser. II 50 (1869), 57-74.

[28] Y. Li, Asymptotic behavior of positive solutions of equation $\triangle u+K(x)u^{p}=0$ in, J. Differential Equations 95 (1992), 304-330.
$\mathbb{R}^{n}$[29] G. Mancini and K. Sandeep, On a semilinear elliptic equation in
Norm. Super. Pisa Cl. Sci. 5 (2008), no. 7, 635-671.

$\mathbb{H}^{N}$, Ann. Sci.

[30] W.-M. Ni, On the elliptic equation $\triangle u+K(x)u^{\frac{n+2}{n-2}}=0$ , its generalizations, andapplications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493-529.

[31] W.-M. Ni, A nonlinear Dirichlet problem on the unit ball and its applications,Indiana Univ. Math. J. 31 (1982), no. 6, 801-807.
[32] N. Kawano, E. Yanagida, and S. Yotsutani, Structure theorems for positive radialsolutions to $\triangle u+K(|x|)u^{p}=0$ in , Funkcial. Ekvac. 36 (1993), no. 3, 557-579.
$\mathbb{R}^{n}$[33] F. Punzo, On well-posedness of semilinear parabolic and elliptic problems in thehyperbolic space, J. Differential Equations 251 (2011), 1972-1989.
. Existence and uniqueness[34] S. Stapelkamp, The Brezis-Nirenberg problem onof solutions. In: Elliptic and Parabolic Problems (Rolduc/Gaeta, 2001), World
Science Publications, River Edge (2002), 283-290.
$\mathbb{H}^{n}$[35] X. Wang, On the Cauchy problem for reaction-diffusion equations, $n_{ans}$ . Amer.
Math. Soc. 337 (1993), no. 2, 1705-1727.
[36] C. Wang and D. Ye, Some Liouville theorems for H\’enon type elliptic equations, J.

Functional Analysis 262 (2012), 1705-1727.