Wang Youde Zhang Aiqi

(1.School of Mathematics and Information Sciences,Guangzhou University,Guangzhou 510006,China;2.Hua Loo-Keng Key Laboratory of Mathematics,Institute of Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China;3.School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China)

Abstract In this paper we consider the gradient estimates on the positive solutions to the elliptic equation Δv+vrvs=0, defined on a complete Riemannian manifold (M,g),where r and s are two real constants.When(M,g)satisfies Ric ≥-(n-1)κ(where n ≥2 is the dimension of M and κ is a nonnegative constant),we employ the Nash-Moser iteration technique to derive a Cheng-Yau type gradient estimate for the positive solutions to the above equation under some suitable geometric and analysis conditions.Moreover,it is shown that when the Ricci curvature of M is nonnegative,this elliptic equation does not admit any positive solutions except for v ≡1 if r ＜s and

Key words Elliptic equation Riemannian manifold Gradient estimate

1 Introduction

In the last half century the following semi-linear elliptic equation defined on Rn:

whereh:Rn×R→R is a continuous or smooth function,attracted many mathematicians to pay attention to the study on the existence,sigularity and various symmetries of its solutions.For instance,Caffarelli,Gidas and Spruck in[2]discussed the solutions to some special form of the above equation:

with an isolated singularity at the origin,and studied the nonnegative smooth solutions of the conformal invariant equation

wheren ≥3.Later,Chen and Li classified the solutions to the following equation

in the critical or subcritical case in[3],and Li[7]simplified and further exploited the “measure theoretic” variation,introduced in [2],of the method of moving planes.On the other hand,one also studied the following prescribed scalar curvature equation(PSE)on Rn:

wheren ≥3 andK: Rn→R is a smooth function.It was shown by Ni in[9]that ifKis bounded and|K|decays in a three-dimensional subspace faster thanC/|x|2at∞for some constantC ＞0,then the above equation has infinitely many bounded solutions in Rn.It was also shown in[9]that ifKis negative and decays slower than-C/|x|2at∞,then the PSE has no positive solutions on R.Later,Lin(cf.[8])improved this result to the case whenK ≤-C/|x|2at∞.This gives an essentially complete picture for the negativeKcase.WhenK ≥0,the situation is much more complicated.It was proved in[9]that ifK ≥C|x|2,then the PSE admits no positive solutions on R.For the case thatKis bounded,this problem was studied by Ding and Ni in[4]using the “finite domain approximation”.In particular,they proved the following theorem.

Theorem 1.1([4])For anyb ＞0,the equation

on Rn,wherer ≥(n+2)/(n -2) andais a positive constant,possesses a positive solutionvwith‖v‖L∞=b.

Very recently,Wang and Wei [12] adopted the Nash-Moser iteration to study the nonexistence of positive solutions to the above Lane-Emden equation with a positive constanta,i.e.,

defined on a noncompact complete Riemannian manifold(M,g)with dim(M)=n ≥3,and improved some results in [10].Later,inspired by the work of Wang and Zhang [11],He,Wang and Wei [6] also discussed the gradient estimates and Liouville type theorems for the positive solutions to the following generalized Lane-Emden equation

Especially,the results obtained in[12]are also improved.It is shown in[6]that,if the Ricci curvature of the underlying manifold is nonnegative and

then the above equation witha ＞0 andp=2 does not admit any positive solutions.

It is worthy to point out that the casea ＜0 is also discussed in[6].Inspired by[6],one would like to ask naturally what happens if the nonlinear term in the Lane-Emden equation is replaced byvr -vs.More precisely,one would like to know howrandsaffect each other.So,in this paper we focus on the following elliptic equation defined on a complete Riemannian manifold(M,g):

whererandsare real constants,Δ is the Laplace-Beltrami operator on(M,g)with respect to the metricg.In other words,hereh(x,u)is of the special formh(x,u)≡ur -us.In fact,in the caser=1 ands=3 this equation is just the well-known Allen-Cahn equation(see[5])

It is worthy to point out that the method adopted here can be used to deal with the general equation Δu+h(x,u)=0 under some technical conditions and we will discuss it in a forthcoming paper.

In the sequel,we always let(M,g)be a complete Riemannian manifold with Ricci curvatureRic ≥-(n-1)κ.For the sake of convenience,we need to make some conventions firstly.Throughout this paper,unless otherwise mentioned,we always assumeκ ≥0,n ≥2 is the dimension ofM,randsare two real constants.Moreover,we denote:

Now,we are ready to state our results.

Immediately,we have the following direct corollary.

Corollary 1.1Let(M,g)be a noncompact complete Riemannian manifold with nonnegative Ricci curvature and dim(M)≥2.Then equation(1.1)admits a unique positive solutionv≡1 ifr ＜sand

Moreover,according to the above corollary we have the following conclusion:

Corollary 1.2Let(M,g)be a noncompact complete Riemannian manifold with nonnegative Ricci curvature and dim(M)=2.Then,the Allen-Cahn equation on (M,g) does not admit any positive solutions except forv≡1.

It is worthy to point out that our method is useful for the equation(1.1)on n-dimensional complete Riemannian manifolds for anyn ≥2.We will firstly discuss the casen ≥3 concretely,then briefly discuss the casen=2.

The rest of this paper is organized as follows.In Section 2,we will give a detailed estimate of the Laplacian of

wherevis the positive solution of equation(1.1)withrandssatisfying the conditions in Theorem 1.2,and then recall Saloff-Coste’s Sobolev embedding theorem.In Section 3,we use the Moser iteration to prove Theorem 1.2 in the casen ≥3,then we briefly discuss the casen=2 using the same approach,and finally we give the proof of Corollary 1.1.

2 Prelimanary

Letvbe a positive smooth solution to the elliptic equation:

whererandsare two real constants.Setu=-lnv.We compute directly and obtain

For convenience,we denoteh=|∇u|2.By a direct calculation we can verify

Lemma 2.1Leth=|∇u|2andu=-lnvwherevis a positive solution to(1.1).Ifr ≤sand

ProofBy the Bochner formula we have

Ifr ≤s,from the above inequality we see that there holds

we substitute the above two inequalities into(2.5)and(2.6)respectively to obtain

Now the discussion will follow in two cases.

Sincek(α)is a monotone decreasing function with respect toαon[1,∞)and

we obtain that for anyα ≥1,

Obviously,we obtain the required(2.2).The proof of Lemma 2.1 is complete.

Next,we recall Saloff-Coste’s Sobolev embedding theorem(Theorem 3.1 in[1]),which plays a key role in the following arguments(Moser iteration).

Theorem 2.1(Saloff-Coste’s Sobolev embedding theorem) Let(M,g)be a complete Riemannian manifold withRic ≥-(n-1)κ.Then for anyn ＞2,there exists a constantcn,depending only onn,such that for allB ⊂Mwe have

3 Proof of main results

In this section we first provide the proof of Theorem 1.2.We need to discuss two cases: the casen ≥3 and the casen=2.After that,we will give the proof of Corollary 1.1.

3.1 The case n ≥3

Throughout this subsection,unless otherwise mentioned,n ≥3,randsare two real constants satisfyingr ≤sand

Lemma 3.1Letvbe a positive solution to (1.1),u=-lnvandf=|∇u|2.Then,there existssuch that for any 0≤η∈and anyι ≥ι0large enough it holds that

ProofLet

Direct computation shows that

By substituting it into(3.2),we derive that

and it follows that

Hence

By rearranging the above inequality,we have

By passingϵto 0 we obtain

which follows that

Furthermore,by the choice ofιwe know

On the other hand,by Young’s inequality we can derive that

and

Now,by takingιsuch that

we see easily that(3.3)can be rewritten as

Besides,we have

and it follows that

Noticing that

we obtain

According to Theorem 2.1,we deduce from the above inequality that

Then(3.5)can be rewritten as

Using the above inequality we will infer a local estimate ofhstated in the following lemma,which will play a key role in the proofs of the main theorems.

For simplicity,we denote the first term on theRHSof(3.8)byR1(R2,L1,L2are understood similarly).

It means that one can find a universal constantcindependent of any parameters,such that

By the Hölder inequality,we have

Furthermore,for anyt ＞0,we use Young’s inequality to obtain

Letting

we can see that

and

Immediately,it follows that

Hence,

Substituting(3.9)and(3.10)into(3.8),we obtain

which implies

Thus,we arrive at

where we applied the inequality that for any two positive numbersaandb,

Hence,(3.7)follows immediately.The proof is complete.

Now,we are in the position to give the proof of Theorem 1.2 in the casen ≥3 by applying the Nash-Moser iteration method.

ProofAssumevis a smooth positive solution of(2.1)withr ≤son a complete Riemannian manifold(M,g)with Ricci curvatureRic(M)≥-2(n-1)κ.When

by the above arguments onf=|∇u|2,whereu=-lnv,we obtain

which is equivalent to

wherecis a universal positive constant independent of any parameters.

By lettingι+1=ιkandη=ηkin(3.11),we derive that

Thus,

which implies that

By iteration we have

In view of

and

by lettingk→∞in(3.12)we obtain the following inequality:

By Lemma 3.1,we conclude from the above inequality that

By the definition ofι0it follows that

Thus,we complete the proof of Theorem 1.2 in the casen ≥3.

3.2 The case n=2

Now,we focus on the positive solutions of (1.1) defined on a 2-dimensional complete Riemannian manifold withRic ≥-κ.According to Lemma 2.1,we have the following lemma.

Besides,according to Theorem 2.1,forn=2,by lettingn′=2m,wherem∈N*andm ＞1,we can get the following direct corollary.

Corollary 3.1Let (M,g) be a 2-dimensional complete Riemannian manifold withRic ≥-κ.Then there exists a constantc2m,depending only onm,such that for allB ⊂Mwe have

By following almost the same argument as in the casen ≥3,we can easily get the following lemmas.

ProofSimilar to the argument to Lemma 3.1,according to Lemma 3.3,for anyι′ ＞max{1, 2(α-1)},we have

Inaddition,by Young’s inequality we can derive that

Substituting the above identity into(3.14)leads to

Since

then

Noticing that

we obtain

According to Corollary 3.1,we obtain

The proof of Lemma 3.4 is complete.

To prove Lemma 3.5,we just need to follow almost the same argument with respect to Lemma 3.2,and we omit the details here.

Now,according to Lemma 3.5,we can use the Moser iteration technique to deduce Theorem 1.2 for the casen=2.Thus,Theorem 1.2 is proved.

3.3 The Proof of Corollary 1.1

Now,we turn to prove Corollary 1.1.

ProofLet(M,g)be a noncompact complete Riemannian manifold with nonnegative Ricci curvature and dim(M)≥2.Assumevis a smooth and positive solution to(1.1).Ifr ＜sand

then by Theorem 1.2 we have that for anyBR ⊂M,

By lettingR→∞we obtain∇v=0.Therefore,vis a positive constant onM.Furthermore,sincer ＜s,we have that unlessv=1

which means that the only positive solution of(1.1)isv≡1.Hence(1.1)admits a unique positive solutionv≡1.The proof of Corollary 1.1 is complete.