On a semilinear elliptic equation in <span class="mathjax-formula">$\mathbb {H}^n$</span>

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On a semilinear elliptic equation in H

ⁿ

GIANNIMANCINI ANDKUNNATHSANDEEP

Dedicated to Prof. Yadava

Abstract. We prove existence/nonexistence and uniqueness of positive entire solutions for some semilinear elliptic equations on the Hyperbolic space.

Mathematics Subject Classification (2000):35J60 (primary); 35B05, 35A15 (secondary).

1.

Introduction

In this paper we discuss existence/nonexistence and uniqueness of positive entire (i.e. finite energy) solutions of the following semilinear elliptic equation on

H

⁼

H

ⁿ^{, the}ⁿdimensional Hyperbolic space:

_Hu+λu+u^p=0. (Eq_λ)

Here _H denotes the Laplace-Beltrami operator on

H

^, λis a real parameter and p>1 ifn=2 while 1< p≤2^∗−1 ifn>2 , where 2^∗:= _n²ⁿ₋₂.

A basic information is that the bottom of the spectrum of−_Hon

H

^is

λ1(−_H):= inf

u∈H¹(H)\0

H|∇_Hu|²d V

H

H|u|²d V_H

= (n−1)²

4 . (1.1)

A straightforward consequence of (1.1) is the following non existence result:

Theorem 1.1. Let n ≥2andλ > ⁽ⁿ⁻₄¹⁾². Then(Eq_λ)has no positive solution. If λ= ⁽ⁿ⁻₄¹⁾² there are no positive solutions of(Eq_λ)in H¹(

H

).

Received January 15, 2008; accepted in revised form May 28, 2008.

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So, in the sequel, we will assumeλ≤ ⁽ⁿ⁻₄¹⁾². Another consequence of (1.1) is that ifλ < ⁽ⁿ⁻₄¹⁾² then

u_λ:=

H

|∇_Hu|²−λu²

d V ¹

2 , u∈C_c^∞(

H

⁾

is a norm, equivalent to the H¹(

H

⁾norm. This is no longer true ifλ= ⁽ⁿ⁻₄¹⁾². A suitable definition of entire (or finite energy) solution in this case can be given taking into account an inequality which can be derived from Hardy-Sobolev-Maz’ya inequalities (see Appendix B).

A sharp Poincar´e-Sobolev inequality and the space

H

For everyn≥3 and everyp∈

1,ⁿ_n⁺₋²₂

there isS_n_,_p>0 such that

S_n_,_p

H|u|^p⁺¹d V_H

p+12

≤

H

|∇_Hu|²−(n−1)² 4 u²

d V_H (1.2)

for everyu∈C₀^∞(

H

^).^Ifⁿ⁼^{2 any} ^p^>1 is allowed.

Inequality (1.2) implies thatu_(n−1)²

4

is a norm as well onC^∞₀ (

H

⁾and we will denote by

H

the closure ofC_c^∞(

H

⁾with respect to this norm.

Definition 1.2. We will say that a positive solutionuof(Eq_λ)is an entire solution if it belongs to the closure ofC^∞_c (

H

⁾with respect to the norm._λ.

Before stating our main results, let us recall what it is known in the Euclidean case (

H

replaced by

R

ⁿ^, ⁿ ^≥ 3): in the subcritical case, a positive entire solution exists iff λ < 0 ; for suchλs the solution is radially symmetric and unique (up to translations); in the critical case a positive entire solution exists iffλ= 0, it is unique (up to translations and dilations) and it is explicitely known.

At our best knowledge, not much is known about(Eq_λ). It naturally appears when dealing with the Euler-Lagrange equations associated to Hardy-Sobolev- Maz’ya inequalities (see Section 6). It is also related (see [9]) to the Yamabe equation on the Heisenberg group and hence to the Webster scalar curvature equation (see [14]-[19], and, for a more general setting, [7]).

A relation between differential operators with mixed homogeneity and hyperbolic geometry was earlier observed by Beckner [2] in dealing with sharp Grushin estimates. In connection with Beckner work, and trying to extend previous results by Garofalo-Vassilev on Yamabe-type equations on groups of Heisenberg type [15], Monti and Morbidelli [23] enlightened the role of hyperbolic symmetry built in Grushin-type equations.

Motivated by these papers, we started investigating, among other things, uniqueness for(Eq_λ)and the main result in this paper, already announced in [9], is the following

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Theorem 1.3. Letλ≤ ⁽ⁿ⁻₄¹⁾² if n ≥3andλ≤ ²₍⁽_p^p₊⁺₃¹₎2⁾ if n =2. Then(Eq_λ)has at most one entire positive solution, up to hyperbolic isometries.

We also establish sharp existence results:

Theorem 1.4. Let p>1if n =2and1< p<2^∗−1if n ≥3. Then(Eq_λ)has a positive entire solution for anyλ≤ ⁽ⁿ⁻₄¹⁾².

In the critical case the situation is more complicated. We have the following Theorem 1.5. Let n ≥4, p= 2^∗−1and ⁿ⁽ⁿ₄⁻²⁾ < λ≤ ⁽ⁿ⁻₄¹⁾². Then(Eq_λ)has a positive entire solution.

The restriction onλin Theorem 1.5 is in fact sharp:

Theorem 1.6. Let n ≥3, p=2^∗−1andλ≤ ⁿ⁽ⁿ₄⁻²⁾. Then(Eq_λ)does not have any entire positive solution.

We also observed a surprising low dimension phenomena

Theorem 1.7. Let n=3and p=5. Then(Eq_λ)has no entire positive solution.

The paper is organized as follows.

In Section 2 we improve a symmetry result given in [1] as a first step towards a sharp uniqueness analysis.

In Section 3 we establish decay estimates for positive entire solutions, a crucial step to prove in Section 4 our uniqueness result.

Section 5 and 6 are devoted to existence/nonexistence and applications.

In Appendix A we recall notations and basic facts on the half space model

R

ⁿ₊ and the ball model

B

ⁿ^for

H

ⁿ and in Appendix B we derive the Poincar´e-Sobolev inequality in

H

ⁿ^.

Added in proof. After this paper was completed, we got to know from R. Musina of a paper by Benguria, Frank and Loss [3] concerning critical Hardy-Sobolev- Maz’ya inequality in the three dimensional upper half space, where the equivalent formulation (1.2) is also given. Among other things, they prove that the best constant in (1.2) is given by the Sobolev constant and it is not achieved. Theorem 1.7 above improves this result.

In addition, Yan Yan Li informed us that, in a recent work with D. Cao [11], they obtained results on locally finite energy solutions for a related PDE, which lead to conjecture uniqueness of such solutions. Indeed our result applies and the conjecture turns out to be true.

ACKNOWLEDGEMENTS. This work was mainly done while the first author was visiting TIFR, Bangalore, India. All the people there deserve gratitude for their warm hospitality. The first author was supported by MIUR, national project ‘Metodi variazionali ed equazioni differenziali non lineari’.

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2.

Hyperbolic symmetry

The main purpose in this section is to prove symmetry properties of solutions of (Eq_λ). The result we need, a refinement of a result in [1], is the following

Theorem 2.1. Letλ≤

n−1 2

2

,n≥2.Let u be a positive entire solution of(Eq_λ). Then u has Hyperbolic symmetry, i.e. there is x₀ ∈

H

such that u is constant on hyperbolic spheres centered at x₀.

The proof relies on moving planes techniques in connection with sharp Sobolev inequalities, and follows closely [1].

We start pointing out the main difference with respect to [1]. The proof therein relies on the classical sharp Sobolev inequality in H¹(

H

ⁿ^),ⁿ^≥3 (see [17]):

n(n−2)ω²ⁿ 4

H|u|ⁿ⁻²²ⁿ d V_H

n−2n

≤

H|∇_Hu|²d V_H−n(n−2) 4

Hu²d V_H. (2.1) This allows to get symmetry for H¹(

H

⁾positive solutions of (Eq_λ)in caseλ ≤

n(n−2) 4 .

To prove symmetry for anyλ < ⁽ⁿ⁻₄¹⁾² (andn≥2) it will be enough to replace Sobolev inequality with the sharp Poincar´e-Sobolev inequality, which by density, holds true in H¹(

H

⁾, as well as in

H

^.

To prove our theorem up to the limiting caseλ=

n−1 2

2

, we need first some facts about

H

^{. Let}^D_cyl¹ (

R

^k^×

R

ⁿ⁻¹⁾be the closure inD¹(

R

^k⁺ⁿ⁻¹⁾^of^C0^∞ y-radial functionsu=u(|y|,z),y∈

R

^k^,^z^∈

R

ⁿ⁻¹. We have the following

Lemma 2.2 (An isometric model for

H

^). ^{Given u}∈C₀^∞(

R

ⁿ₊), let (T u)(y,z):= 1

√2π |y|⁻ⁿ⁻¹² u(|y|,z), (y,z)∈

R

²^×

R

ⁿ⁻¹^.

Then T uD¹(Rⁿ⁺¹) = u_H and hence T extends to an isometric isomorphism between

H

^{and D}¹cyl(

R

²^×

R

ⁿ⁻¹⁾^.

Moreover(

H

,u_H)is a Hilbert space with the inner product,_Hgiven by u₁,u₂_H= 1

2π

Rⁿ⁺¹∇T u₁∇T u₂d x. Proof. Letv:=T u. Then

2π|∇v|²=r⁻⁽ⁿ⁻¹⁾|∇zu|²+r⁻⁽ⁿ⁻¹⁾u_r²+

n−1 2

2

r⁻⁽ⁿ⁺¹⁾u²−n−1

2 r⁻ⁿ(u²)r.

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Integrating in polar coordinates and then by parts

Rⁿ⁺¹|∇v|²d yd z=

R⁺×Rⁿ⁻¹

|∇u|² rⁿ⁻² +

n−1 2

2 u²

rⁿ − n−1

2 r⁻ⁿ⁺¹(u²)r

dr d z

=

R⁺×Rⁿ⁻¹

|∇u|² rⁿ⁻² −

n−1 2

2u² rⁿ

dr d z= u²_H.

SinceT(C₀^∞(

H

⁾⁾contains all the cylindrically symmetricC^∞functions with compact support in (

R

² ^{\ {}⁰^})^×

R

ⁿ⁻¹, and they are dense in D_cyl¹ (

R

²^×

R

ⁿ⁻¹⁾^{, by} density T extends to an isometry from

H

^onto ^Dcyl¹ (

R

²^×

R

ⁿ⁻¹⁾. The same ar- guments show that T preserves the scalar product. Finally, since convergence in

H

^{and in} ^Dcyl¹ (

R

²^×

R

ⁿ⁻¹⁾imply, up to subsequences, pointwise convergence, we can in particular conclude that anyu ∈

H

can be written as rⁿ⁻¹² v(r,z) for some v∈D¹_cyl(

R

²^×

R

ⁿ⁻¹⁾^.

Lemma 2.3. Let u∈

H

be compactly supported in

H

^{. Then u}^∈^H¹⁽

H

⁾^.

Proof. Letu_n ∈C_c^∞(

H

⁾^{such that}^||^uⁿ⁻^u^||H→ 0.We may assume that support ofu_n’s are contained in a fixed compact subsetK of

H

. From the Poincar´e-Sobolev inequality (1.2), we getu_nis Cauchy inL^p(

H

⁾and hence converges touinL^p(

H

⁾ for any p > 2.This implies u_n → u in L²(

H

⁾thanks to the assumption on the support ofu_n’s. Convergence inL²and convergence in||.||_Htogether implies∇u_n is Cauchy in L²(

H

⁾and hence the convergence ofu_n touin H¹(

H

⁾^.

Lemma 2.4 (Poincar´e-Sobolev inequality in

H

^). ^{Let p} > 1if n = 2 and1 <

p≤ ⁿ_n⁺₋²₂if n ≥3. There exists S_n_,_p >0such that S_n_,_p

H|u|^p⁺¹d V_H

p+12

≤ u²_H ∀u∈

H

. (2.2) Equivalently, for everyv∈ D_cyl¹ (

R

²^×

R

ⁿ⁻¹⁾

(2π)^p−2^p S_n_,_p

Rⁿ⁺¹

|v|^p⁺¹

|y|^{n+3−p(n−1)}² _p+1²

d x ≤ v²_D1(Rⁿ⁺¹). (2.3) Remark 2.5. Notice thatn+3−p(n−1) <0 if ⁿ_n⁺₋³₁ < p≤ ⁿ_n⁺₋²₂.

Proof. (2.2) follows from (1.2) and (2.3) follows from (2.2), Lemma 2.2 and

R²×Rⁿ⁻¹

|y|⁻ⁿ⁻¹² u(|y|,z)|p+1

|y|^{n+3−p(n−1)}² d yd z=2π

R⁺×Rⁿ⁻¹

|u|^p⁺¹ rⁿ dr d z

=2π

H|u|^p⁺¹d V_H.

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Furthermore, a direct computation gives

Lemma 2.6. Let u ∈

H

be a positive solution of(Eq_λ),λ=

n−1 2

2

, n≥2. Let v:=√

2π T u. Thenvis a D¹(

R

ⁿ⁺¹⁾solution of

−v(y,z)= v^p

|y|^{n+3−p(n−1)}² in

R

²^×

R

ⁿ⁻¹^. ^(2.4) Proof of Theorem2.1. The proof is in the same lines as in [1]; so, we will only give an outline. The main difference is in the use of sharp Poincar´e-Sobolev inequalities (2.2) and (1.2) instead of sharp Sobolev inequality (2.1).

LetA_tbe a one parameter group of isometries of

H

^{which is}^C¹⁽

R

^×

H

^,

H

⁾^and I be a reflection (i.e., I is an isometry andI²=Identity) satisfying the invariance conditionA_tI A_t = I, ∀t ∈

R

^.^Define^I^t ⁼ ^A^t^{I A}−tand LetU_tbe the hypersurface of

H

which is fixed by I_t. We also assume that∪t1<t<t2U_t is open for allt₁,t₂∈

R

and∪t∈RU_t =

H

.Fort ∈

R

^define

Q_t = ∪_−∞<s<tU_s, and Q^t = ∪t<s<∞U_s. Then I_t(Q_t)⊂Q^t andI_t(Q^t)⊂ Q_t for allt ∈

R

. Now define fort ∈

R

u_t(x)=u(I_t(x)), x∈

H

.

The theorem will be proved once we show the existence of a t₀ ∈

R

^{such that} u_t₀ =uinQ_t₀(see [1] for details.) As in [1] we will prove this in three steps. We will only give details for step 1; the other two steps follow as in [1]. Let

= {t ∈

R

^{: ∀}^s ^>^t^,^u^≥^u^sⁱⁿ ^Q^s^}.

Step 1. is nonempty.

Proof of Step1. It is enough to show that fort large enoughu_t ≥uin Q^t. Since I_t is an isometry,u_t ∈

H

^{and solves}⁽^Eqλ). Write, as in Lemma 2.2,v=√

2πT u, vt = √

2πT u_t. From Lemma 2.2, they are in D¹^,²(

R

ⁿ⁾and, furthermore, they solve (2.4). Now define

t = {(y,z)∈

R

²×

R

ⁿ⁻¹ : (|y|,z)∈ Q^t}.

Then Step 1 will be proved once we show thatvt ≥ vint. Now, t is open in

R

ⁿ⁺¹^and^v⁼^v^t ^on^∂^t. We also have

−(v−vt)= |y|^−τ(v^p−vt^p)

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where τ = ²⁽ⁿ⁺¹⁾⁻₂^p⁽ⁿ⁻¹⁾ as in (2.4). Now multiplying the above equation by (v−vt)⁺integrating by parts and applying (2.2) to(v−vt)⁺we get,

t

|∇(v−vt)⁺|²d x =

t

v^p−vt^p

|y|^τ(v−vt)((v−vt)⁺)²d x

≤C

t

v^p⁺¹

|y|^τ

^p−1_p+1

t

((v−vt)⁺)^p⁺¹

|y|^τ

_p+1²

≤C

t

v^p⁺¹

|y|^τ

^p−1_p+1

t

|∇(v−vt)⁺|²d x.

Since

Rⁿ⁺¹|y|^−τv^p⁺¹ < ∞,

t|y|^−τv^p⁺¹ → 0 ast → ∞.Hence there exist a t₀such that

t|∇(v−vt)⁺|²d x =0 for allt >t₀.This proves Step 1.

To complete the proof, one can show, following [1], that Step 2. is bounded below.

Step 3. u˜_t₀= ˜uin Q_t₀wheret₀=inf.

3.

Decay estimates

The proof of our uniqueness result relies on decay properties of entire positive solutions we are going to prove here. Letube a positive symmetric solution of(Eq_λ). As a function on

B

^:=

B

ⁿ^,^u⁼^u^(|ξ|)^,^ξ^∈

R

ⁿ^,^|ξ|^<^{1 and}

1− |ξ|² 2

₂

u+(n−2)

1− |ξ|² 2

<∇u, ξ >+λu+u^p=0. (3.1)

Setting |ξ| := tanh^t₂, u(t) := u(tanh^t₂), q := (sinht)ⁿ⁻¹, it is easy to see that

B|u|^pd V_B =ωn

_∞

0 q|u|^p,

B|∇_Bu|²d V_B =ωn

_∞

0 q|u|²so that u∈H¹(

B

ⁿ)⇔ωn

_∞

0

|u|²+ |u|² q=

B

|u|²+ |∇_Bu|²

d V_B <+∞. (3.2) Poincar´e-Sobolev inequality reads as follows:∀p∈(1,2^∗−1], ∃c(n,p) >0:

c(n,p) _+∞

0

q|w|^p⁺¹ _p+1²

≤ _+∞

0

q|w|²−(n−1)² 4

_+∞

0

qw² (3.3)

∀w∈C₀^∞([0,+∞)); ifn=2 any p>1 is allowed. In addition, (3.1) rewrites u+n−1

tanht u+λu+u^p =0, u(0)=0 (3.4)

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as well as

(qu)+λq u+qu^p =0, u(0)=0 (3.5) and ifu ∈ H¹(

B

)solves (3.1), then_∞

0 q[|u|²−λu²] = _∞

0 qu^p⁺¹. The above relations are no longer true if u ∈/ H¹(

B

). However, we can derive an analogue of (3.2) and (3.3) for functions in

H

possessing hyperbolic symmetry. As above, and if there is no confusion, we will writeu(tanh₂^t)asu(t).

Lemma 3.1. Let u∈

H

be a symmetric function. Then

||u||²_H=ωn

_∞

0







u+n−1 2 tanh t

2 u

2

+ (n−1)u²

2 cosh t 2

2





q <∞ (3.6)

and hence the Poincar´e - Sobolev inequality for u rewrites as

c(n,p) _∞

0

q|u|^p⁺¹ _p+1²

≤ _∞

0







u+n−1 2 tanht

2u

2

+ (n−1)u²

2 cosht 2

2





q. (3.7)

If u satisfies(3.4)and(3.6)then for everywsatisfying(3.6), we have _∞

t₀







u+ n−1 2 tanht

2u

w+n−1 2 tanh t

2w + (n−1)uw

2 cosht 2

2





q

= _∞

t₀

qu^pw where either t₀=0 or w(t₀)=0.

(3.8)

Proof. Letu∈C₀^∞(

B

)andv:=

2 cosh²(t/2)ⁿ⁻¹₂

u. Integrating by parts, u²_H: =ωn

_∞

0

(u)²−

n−1 2

2

u²

(sinht)ⁿ⁻¹dt

=ωn

_∞

0

(v)²+n−1 4

v cosh(t/2)

2

(tanh(t/2))ⁿ⁻¹dt.

(3.9)

Now, givenu∈

H

, there is a sequence of radial functionsu_m ∈ C_c^∞(

B

)such that u_m →m uin

H

and a.e. Hence, ifvm :=

2 cosh²(t/2)ⁿ⁻¹₂

u_m, thenvm →m v:=

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2 cosh²(t/2)ⁿ⁻¹₂ uand

||u_m||²_H=ωn

_∞

0

(vm )²+ n−1 4

vm

cosh(t/2)

2

(tanh(t/2))ⁿ⁻¹dt. (3.10) Sinceu_n is a Cauchy sequence in

H

, we can then pass to the limit in (3.10) and see that (3.9) actually holds for everyu ∈

H

. Now (3.6) follows just substituting v=

2 cosh²(t/2)ⁿ⁻¹₂

uin (3.9).

The Poincar´e-Sobolev inequality (3.7) follows from (3.3) and (3.6).

As for (3.8), letwsatisfy (3.6). Then there is a sequencewn ∈C₀^∞([0,∞)) withw_n(0)=0 such thatwn →win the

H

norm given by (3.6). Sinceusatisfies (3.5), multiplying this relation bywnand integrating by parts we see that (3.8) holds forwn. Sincewsatisfies (3.6) we can pass to the limit proving (3.8).

Now, let us notice that ifusolves (3.4) andE_u(t):= ^u₂² + ^λ₂u²+^|^u_p^|₊^p+1₁ ,then d

dtE_u(t)= −n−1

tanhtu²≤0 ∀t >0.

In particular, u and u remain bounded and hence u is defined for everyt. En- ergy considerations also lead to monotonicity properties and exponential decay of positive solutions of (3.4) as can be seen below.

Lemma 3.2. Let n ≥2, p>1. Letv >0be a solution of(3.4). Ifλ <0we also assume thatvsatisfies(3.2). Thenv(t) <0for every t > 0andlim_t_→+∞v(t)= lim_t_→+∞v(t)=0.

Proof. Ifλ≥ 0, equation (3.5) implies (qv)(t) < 0 ∀t > 0 and thenv(t) < 0

∀t > 0 because v(0) = 0. In particular, it existsv(∞) := lim_t_→+∞v(t)and, since E_vis decreasing,v(∞):=lim_t_→+∞v(t)exists as well and is zero because vis positive. Finally, (3.4) givesv(∞)= −[λv(∞)+v^p(∞)]with, necessarily, v(∞)=0 and hencev(∞)=0.

Ifλ <0,vis not decreasing and does not vanish at infinity, in general. How- ever, assuming (3.2), we have that lim inf_t_+∞q(t)[v²(t)+v²(t)] = 0. In particular, 0 = lim_t_→+∞E_v(t) < E_v(t)∀t by monotonicity. Now, let by contradiction v(t₀)= 0 for somet₀ > 0. Then 0 < E_v(t₀) = ¹₂λv²(t₀)+ ^|v|_p₊^p+1₁ (t₀)and hence λv(t₀)+v^p(t₀) >0 and hence, by equation (3.4),v(t₀) < 0. Thust₀is the only zero ofvandv >0 in(0,t₀). SinceE_v(0) >0 andv(0)=0,λv(t)+v^p(t) >0 for t small, and hencev < 0 fort small and hencev is negative fort small, a contraddiction. This and (3.2) implyv(∞) = 0. As above,v(∞)exists and it is necessarily zero.

Remark 3.3. In caseλ <0 and assumingv→t→+∞0 instead of (3.2), the same argument as above implies againv(t) <0 fort >0 andv(∞)=0.

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Lemma 3.4. Let n ≥ 2, p > 1,λ < ⁽ⁿ⁻₄¹⁾². Let v > 0be a solution of (3.4) satisfying(3.2). Then

t→+∞lim logv²

t = lim

t→+∞

logv² t = lim

t→+∞

log[v²+v²]

t =−

n−1+

(n−1)²−4λ v

v →t→+∞−n−1+

(n−1)²−4λ

2 .

Proof. By Lemma 3.2,v→t→+∞0. Then it existst >0 such that cotht ≤1+, v^p(t)≤v(t), ∀t ≥t. Since, again by Lemma 3.2,vis negative, we have, fort ≥t,

v+(n−1)(1+)v+λv ≤v+(n−1)cothtv+λ v+v^p =0 (3.11) v+(n−1)v+(λ+)v ≥v+(n−1)cothtv+λ v+v^p =0. (3.12) Letµ^±():= ⁻⁽ⁿ⁻¹⁾⁽¹^+)±√

(n−1)²(1+)²−4λ

2 ,ν^±():= ⁻⁽ⁿ⁻¹^)±√

(n−1)²−4(λ+)

2 be

the characteristic roots of the differential polinomials in the left hand side of (3.11)- (3.12), respectively (notice thatµ^±()are real and distincts forλ≤ ⁽ⁿ⁻₄¹⁾²; to have ν^±() real and distinct we need to choose < ⁽ⁿ⁻₄¹⁾² −λand with this choice µ⁻() < ν⁻()). Let

w:=v−µ⁺()v, z:=v−ν⁺()v. (3.13) Then, from (3.11) and (3.12), we have, fort ≥t,

w−µ⁻()w=v+(n−1)(1+)v+λv≤0 (3.14) z−ν⁻()z=v+(n−1)v+(λ+)v≥0. (3.15) From (3.14)-(3.15) we derive

e^−µ⁻⁽⁾^tw

≤0≤

e^−ν⁻⁽⁾^tz

for everyt ≥t and hence, integrating such inequalities in[τ,t], t ≤τ ≤t, we get, respectively,

v(t)−µ⁺()v(t)=w(t)≤

e^−µ⁻^()τw(τ)

e^µ⁻⁽⁾^t :=c(τ)e^µ⁻⁽⁾^t (3.16) v(t)−ν⁺()v(t)=z(t)≥

e^−ν⁻^()τz(τ)

e^ν⁻⁽⁾^t :=d(τ)e^ν⁻⁽⁾^t (3.17) for everyt ≥τ ≥t. Again by integration, we finally get, for everyt ≥τ ≥t,

v(t)≤

e^−µ⁺^()τv(τ)

e^µ⁺⁽⁾^t+c(τ)e^µ⁻⁽⁾^t −e^(µ⁻^()−µ⁺^())τ^+µ⁺⁽⁾^t µ⁻()−µ⁺()

=

e^−µ⁺^()τv(τ)− c(τ)

µ⁻()−µ⁺()e^(µ⁻^()−µ⁺^())τ

e^µ⁺⁽⁾^t + c(τ)e^µ⁻⁽⁾^t

µ⁻()−µ⁺()

(3.18)

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v(t)≥

e^−ν⁺^()τv(τ)

e^ν⁺⁽⁾^t +d(τ)e^ν⁻⁽⁾^t −e^(ν⁻^()−ν⁺^())τ^+ν⁺⁽⁾^t ν⁻()−ν⁺()

=

e^−ν⁺^()τv(τ)− d(τ)

ν⁻()−ν⁺()e^(ν⁻^()−ν⁺^())τ

e^ν⁺⁽⁾^t + d(τ)e^ν⁻⁽⁾^t

ν⁻()−ν⁺().

(3.19)

Now,vpositive,µ⁻() < µ⁺()and (3.18) imply e^−µ⁺^()τv(τ)− c(τ)

µ⁻()−µ⁺()e^(µ⁻^()−µ⁺^())τ ≥0 ∀τ ≥t that is

v(τ)≥ c(τ)e^µ⁻^()τ

µ⁻()−µ⁺() = w(τ)

µ⁻()−µ⁺() = v(τ)−µ⁺()v(τ)

µ⁻()−µ⁺() ∀τ ≥t. Hence

v(τ)≥µ⁻()v(τ), ∀τ ≥t (3.20)

and integrating on[t,t]we find v(t)≥

v(t)e^−µ⁻⁽⁾^t

e^µ⁻⁽⁾^t ∀t ≥t. (3.21) We notice, for future reference, that in caseλ≥0 (3.20) and (3.21) hold true for any positive solution: assumption (3.2) is used, at this stage, to insure monotonicity and decay properties ofvand hence it is required just in caseλ <0 (see Lemma 3.2;

accordingly with Remark 3.3, we might have asked, alternatively, tovto vanish at infinity). We similarly infer from (3.19) that

dˆ(τ):=e^−ν⁺^()τv(τ)− d(τ)

ν⁻()−ν⁺()e^(ν⁻^()−ν⁺^())τ ≤0 ∀τ ≥t. Otherwise, sinceν⁻() < ν⁺(), (3.19) givesv(t)≥ ¹₂dˆ(τ)e^ν⁺⁽⁾^t. But, sinceλ <

(n−1)²

4 and hencen−1+2ν⁺() >0, this inequality implies that_∞

0 qv²= +∞, violating (3.2) (in caseλ <0, and henceν⁺() >0, it would even follow thatvis unbounded). Thus,dˆ(τ)≤0 ∀τ ≥t, that is

v(τ)≤ d(τ)e^ν⁻^()τ

ν⁻()−ν⁺() = z(τ)

ν⁻()−ν⁺() = v(τ)−ν⁺()v(τ)

ν⁻()−ν⁺() ∀τ ≥t. Hence

v(τ)≤ν⁻()v(τ), ∀τ ≥t (3.22)

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and integrating on[t,t]we find v(t)≤

v(t)e^−ν⁻⁽⁾^t

e^ν⁻⁽⁾^t ∀t ≥t. (3.23) We see from (3.21) and (3.23) that

2µ⁻()≤lim inf

t→+∞

logv²

t ≤lim sup

t→+∞

logv²

t ≤2ν⁻(), ∀ >0 and hence lim_t_→+∞^log_t^v² = −

n−1+

(n−1)²−4λ .

From (3.20)-(3.22) and (3.21)-(3.23) we see that similar bounds hold true for vas well and hence

t→+∞lim

log(v)²

t = lim

t→+∞

log(v²+(v)²)

t = −

n−1+

(n−1)²−4λ . Finally, taking lim sup and lim inf in (3.20)-(3.22) and then sending to zero, we see that ^v_v →t→+∞−ⁿ⁻¹⁺√

(n−1)²−4λ

2 .

Remark 3.5. In caseλ <0 assumption (3.2) can be replaced by lim_t_→+∞v= 0.

Thus, ifλ <0 a positive solution vanishes at infinity iff it is inH¹.

Let us now consider the limit case λ = ⁽ⁿ⁻₄¹⁾². Here we have more precise estimates, which will be actually needed later on.

Lemma 3.6. Let n≥2,λ= ⁽ⁿ⁻₄¹⁾², p>1. Letv >0be a solution of(3.4). Then

t→+∞lim logv²

t = lim

t→+∞

log(v)²

t = lim

t→+∞

log[v²+(v)²]

t = −(n−1). (3.24) Let, in addition,vsatisfy(3.6). Then there is A>0such that

eⁿ⁻¹² ^tv(t)→ A and eⁿ⁻¹² ^tv(t)→ −n−1

2 A ast → ∞. (3.25)

Proof. Since,µ^±(see the proof of Lemma 3.4) are real and distinct, (3.14)-(3.18) and hence the bound from belowv(t) ≥ce^µ⁻⁽⁾^t given in (3.21), still holds true, because, at this stage, it was just required thatvdecreases to zero, a property satis- fied here (Lemma 3.2). For the same reason, an upper bound forvand (3.20) will give a similar upper bound for−v.

A bound from above forvcan be obtained as follows. First observe thatv(t) cannot be definitely negative and then it is definitely positive because, otherwise, it has a sequencet_j →j +∞of zeros, that is, denotedw :=v, thenw < 0 and w(t_j)=0. Taking the derivative of (3.4), we get

w+(n−1)cothtw+

λ− n−1

sinh²t + pv^p⁻¹ w=0