Supercritical Elliptic Equation in Hyperbolic Space

HE Haiyang

College of Mathematics and Computer Science,Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China), Hunan Normal University,Changsha 410081,China.

Supercritical Elliptic Equation in Hyperbolic Space

HE Haiyang?

College of Mathematics and Computer Science,Key Laboratory of High Performance Computing and Stochastic Information Processing(Ministry of Education of China), Hunan Normal University,Changsha 410081,China.

.In this paper,we study the following semi-linear elliptic equation

in the whole Hyperbolic space Hn,wheren≥3,p＞2n/(n?2).We obtain some regularity results for the radial singular solutions of problem(0.1).We show that the singular solutionu?withbelongs to the closure (in the natural topology given byof the set of smooth classical solutions to the Eq.(0.1).In contrast,we also prove that any oscillating radialsolutionsfails to be in the space

AMS Subject Classifications:58J05,35J60

Chinese Library Classifications:O175,O176

Supercritical;singularity;hyperbolic space.

1 Introduction and main results

In this paper,we consider the following nonlinear elliptic equation

on the simplest example of manifold with negative curvature,the hyperbolic spacewhere?Hndenotes the Laplace-Betrami operator on Hn,n≥3 andpis greater than the critical exponent 2n/(n?2).

The corresponding equation in the Euclidean space is called the Emden-Fowler equation and the study goes back to[1–5]and others.Attention was focused on the existence and the description of radial solutions.Recently,there is a complete description of radial solutions in[6].

We are interested in solutions which depend only on the hyperbolic distance from a fixed center.In order to express(1.1)for such radial solutions,we recall that the hyperbolic space Hnis an embedded hyperboloid in Rn+1,endowed with the inherited metric. Problem(1.1)is often considered on the unit ball in Rnwith the Poincare metric.Indeed, it is obtained by a stereographic projection from Hnonto Rn.However,we introduce a parametert≥0 such that sinh2If Eq.(1.1)possesses a solution depending only onxn+1,i.e.u=u(t),thenusatisfies the following O.D.E.

As an analogue of Gidas,Niand Nirenberg[7,8],the radial symmetry ofpositive solution to the problem on the hyperbolic space was established by Kumaresan and Prajapat[9] and Mancini and Sandeep[10].Thus the radial solutions play an important role,and problem(1.2)is worth studying.

In this article,the goal is to present a complete description of radial solutions of problem(1.2).The structures of positive radial solution to this problem has been wellinvestigated,see[10,11],Bonforte,Gazzola,Grillo and Vazquez[12],Bandle and Kabeya [13],Wu.Chen and Chern[14].Moreover,they proved that all radial solutionsuto(1.2) withu(0)＞0,u′(0)=0 are every positive and decay polynomially at infinity with the following rate

This result is different from the resultin Euclidean space with the asymptotic decayu(t)=Fora bounded domain case,see[15]and Stapelkampe[16,17].However, concerning all of the radial solutions which are singular att=0,studies are not well done. Here we show the classification of the singular radial solutions.

Theorem 1.1.Suppose that u∈C2(0,∞)solves(1.2)with

Then either(i)

From(ii)of Theorem 1.1,we know that there exists a radial solutionwith a mild singularlyIndeed,we show in the next theorem that any radial singular solutionuwhich does not satisfywill change sign infinitely many times and blow up at the origin with a predetermined rate which,in particular,implies that

Theorem 1.2.Suppose thatsolves(1.2)with

Then u changes sign infinitely many times on any interval(0,T),T＞0,and there exists a number c0＞0such that

where

Sincen+d＞n＞2p/(p?2)forp＞2?,we see that the the symmetric solutionsudescribed in Theorem 1.1 are the only radially symmetric solutions of(1.2)which are locally inH1.

This paper is organized as follows.In Section 2,we give the proof of Theorem 1.1 and 1.2.

2 Proof of main results

2.1Proof of Theorem 1.1.

Proof.Let sinhwe definewhich impliesFrom the definitions ofsandv,we have thatcosht/sinht,

Hence Eq.(1.2)becomes

Multiplying(2.1)with 2v′,we find that

Now we define

Then,by(2.2),for any suchvthe functionis non-increasing.

In the case(i),we have that

In the latter case(ii)it follows thatin particular,

Similar as[5],we can obtain regularity foruatx=0.Thus,by[10,12],we have

2.2Proof of Theorem 1.2.

Proof.Consider the quantity

which satisfies

Remark that for the solutions of(1.2)which satisfy(1.3),the difference Φ?Φ0is small compared with Φ.Indeed,for any numberδ∈(0,1),we have

for sufficiently smallt＞0.Hence we also have

for 0＜t＜tδ.From Eq.(2.5),we get the differential inequality

which implies that

ont∈(0,tδ)with numberC1,2＞0 depending onδ.

Inserting the bound(2.6)in(2.4),we can obtain

This bound gives the improved differential inequality

Integrating this inequality for a fixed numberδ∈(0,d),we conclude that there exists

By Eqs.(2.4),(2.8),then

Thus,we have

Observe that the limitc0must be positive,otherwise Eq.(2.6)would be violated.

ii)Now,we will show that any solutionuof(1.2)satisfying(1.3)and(1.4),will change sign infinitely many times ast→0.Assume by contradiction that there are someuwhich do not change sign on an interval(0,t1)for somet1＞0.So,we can assume thatu(t)≥0 for 0＜t＜t1.

We first claim thatumust be decreasing in(0,r1).Indeed,the maximum principle implies that suchucan not achieve non-negative local minima,souis increasing on an interval(0,r2)for some 0＜r2＜r1.However,for 0≤u≤u(r2)and the asymptotic of Φ0,it show thatwhich contradicts the assumption thatu≥0.

whose union contains the whole interval[T,2T],providedT＞0 is sufficiently small.

Sinceu′≤0 andu＞0 near zero,for the 1?dimensional Lebesgue measureμ(U1(T)), we obtain the inequality

with a constantc1＞0 depending onc0but independent onT.In view of the estimate

On the other hand,integrating Eq.(1.2),we find that

Sinceu′(t)(sinht)n?1→0 ast→0,then the left hand side is equal to?u′(2T)(2T)n?1.In view of the estimate

for smallT＞0,which contradicts the fact that

Acknowledgement

The author would like to thank Professor Yuxin Ge for his encouragement when she visit University Paris-Est-Creteil Val de Marne.This work is supported by the scholarship from China Scholarship Council(CSC)under the Grant CSC(No:201308430355),the National Natural Sciences Foundations of China(No:11201140),and Hunan Provincial Natural Science Foundation of China(No:14JJ6005).She would also like to thank the referees very much for their valuable comments and suggestions.

References

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Received 4 December 2014;Accepted 31 January 2015

?Corresponding author.Email address:hehy917@hotmail.com(H.He)