1 Introduction

In the studying of the existence results for non-variational elliptic equations,we usually use the topological method such as the Leray-Schauder degree theory to get the existence result.In order to apply such a theory,the most important step is to get a priori bound for the solutions.As far as we know,the blow up method is the most powerful tool in proving priori bounds for elliptic equations.The idea of the blow up method is very simple and its essence is the proof by contradiction.More precisely,suppose on the contrary that there exists a sequence of solutions{un}with Mn=un(xn)= ‖un‖L∞(Ω)→ ∞,then we make a scaling on this sequence of solutions and getClearly,after the scaling,{vn}is bounded in the L∞norm.Hence,by the regularity theory of elliptic equations,we can assume thatfor some 0 ＜ γ ＜ 1.Moreover,we have‖v‖L∞ =1 and it satisfies some limit equation in Ω∞,where either Ω∞ =RNor Ω∞ =depending on the speed of xngoes to the boundary of Ω.On the other hand,if we can prove the limit equations do not possess nontrivial solution,then we get a contradiction,hence the solutions of the original problem must be bounded.From the descriptions of the blow-up procedure,it is easy to see that in order to get a contradiction,it is essential to prove the Liouville type theorems for the limit equations.

For the above mentioned reasons,Liouville theorems for elliptic equations have drawn much attention of scientists during the past few decades and there are many results on this subject up to now.For example,in two seminal articles[13,14],Gidas and Spruck studied the nonexistence of positive solution for the following nonlinear elliptic problem

They proved that for the subcritical case,that is,1＜p＜,this problem possesses no positive solution.This is the so-called Liouville type theorem for positive solution of problem(1.1).Later,in order to get the priori bound for elliptic equations in bounded domains,they studied a similar equation in the half space,

Similar nonexistence result was established in[14]for positive solution of the subcritical problem in the half space.The proof of Gidas and Spruck is very complicated.Later,W.Chen and C.Li[4]simplified their proofs and got similar results by using the moving plane method.After their results,the moving plane method and its variant,the moving sphere method were widely used in proving the Liouville theorems for elliptic equations;we refer the readers to[2,3,5–8,11,19,20,23]and we can not list all of them.

We note that all the results mentioned above only claim that the subcritical problems do not possess positive solution.A natural and more complicated question is that whether these problems possess sign-changing solutions.However,this problem is completely open up to now.The main difficulty lies in that the moving plane method does not work for sign-changing solutions.Hence,we must turn to other methods.A great progress on this area is the work[1],in which the authors studied the nonexistence of solution with finite Morse index for problems(1.1)and(1.2).They proved problems(1.1)and(1.2)do not possess nontrivial bounded solution with finite Morse index provided 1＜p＜.This result extended the nonexistence results of positive solution to finite Morse index solution.After the work[1],there are plenty of works concerning the finite Morse index solutions for elliptic equations.For example,A.Harrabi,S.Rebhi,and S.Selmi extended their results to more general nonlinear problems in[17,18].Recently,A.Harrabi,M.Ahmedou,S.Rebhi,and A.Selmi studied the nonexistence result for Neumann boundary value problems in[15];X.Yu studied the mixed boundary problems in[24],the nonlinear boundary value problem in[25],and fractional Laplacian equation in[26].X.Zhao and X.Wang obtained the nonexistence result for Robin boundary value problems in[27].Other results can be found in[10,12,16]and the references therein.

The rest of this article is devoted to the proof of the above theorem.In the following,we denote C by a positive constant,which may vary from line to line.

Recently,we studied the nonexistence of positive solution for the following nonlocal equation

in[28].This kind of equation is usually called the Choquard type equation since in 1976,a similar equation as(1.3)was used by P.Choquard to describe an electron trapped in its own hole,in a certain approximation to Hartree-Fock theory of one component plasma[21].In some contexts,equation of type(1.3)is also called the nonlinear Schrödinger-Newton equation.In[28],we proved this equation does not possess positive solution for 0＜ p＜ by using the moving plane method.In this article,we continue to study the nonexistence of finite Morse index solution for problem(1.3).To state our result,we first define the Morse indices of solutions to problem(1.3).The Morse index of a solution u is defined by

Theorem 1.1 Suppose that N≥3,0＜α＜min{4,N},and u is a solution of problem(1.3)with finite Morse index,if 2＜ p＜ ,then u≡0;if p=t

and hence

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As this solution can be sign-changing,the moving plane method does not work.We use the method of energy estimation combining the Pohozaev identity to prove our result.

Now,we state our main result as the following

where

hendxdy＜ ∞ and

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2 Proof of Theorem 1.1

In this section,we establish the nonexistence of finite Morse index solution for problem(1.3).For convenience,we denote

For s ＞ 2r ＞ 0,we define a cut-offfunction φr,sas

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moreover,we requireThen,by finite Morse indices,we have the following result.

Lemma 2.1 Let u be a solution of(1.3)with finite Morse index,then there exists R0＞0 such that

for any R＞2R0and m＞0.

Proof The proof of this lemma is the same as[27],we omit the details.

Next,we show that finite Morse index implies u satisfying some integrable conditions.More precisely,we have the following key lemma.

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Lemma 2.2 Under the assumptions of Theorem 1.1,if u is a solution of problem(1.3)with finite Morse index,then we have

Proof First,we prove thatdxdy ＜ ∞.By Lemma 2.1,there exists R0＞0,such that

for any R＞2R0and m＞0 to be determined later.That is

从表2中可以看出，量表中3个维度的Cronbach'a系数都大于0.70，总量表的信度系数高于0.80，说明具有良好的信度。

A direct calculation shows that the right hand side of equation(2.1)equals to Z

which is a contradiction.Hence,we must have

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Since p≥2,in particular,we have

Suppose on the contrary thatdxdy= ∞,then for sufficiently large R,we have

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Insert this equation into equation(2.1),then we have

Now,we choose m to be an integer with m≥,then we have m≤p(m−1),which further implies that

that is

Now,let R → ∞ and use the assumption that p≤ ,then we deduce from the above inequality that

On the other hand,multiplying equation(1.3)by and integrating by parts,then we get

This completes the proof of this lemma.

Next,we prove thatIn fact,if we multiply equation(1.3)by u and integrate on RN,then we get

The next Lemma is the well-known Pohozaev identity for Choquard equation,which can be found in[22].In order to keep this article self-contained,we sketch the proof of this identity.

Lemma 2.3 Let u be a solution of(1.3)with

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then the following identity holds

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Proof We first choose a cut-offfunction ϕ ∈ C10(RN)such that ϕ =1 in B1(0)and ϕ =0 outside B2(0).For λ ∈ (0,∞)and x ∈ RN,we define

If we multiply equation(1.3)by vλand integrate by parts,then we obtain

A direct calculation show that

Asthe Lebesgue dominated convergence theorem implies that

Similarly,for the right hand side of equation(2.7),we have

Asdxdy＜∞,we infer from the dominated convergence theorem that

The conclusion of this Lemma follows from equations(2.7)(2.9)and equation(2.11).

With the above preparations,we can prove Theorem 1.1 now.

Proof of Theorem 1.1 The proof of Theorem 1.1 is a direct consequence of Lemma 2.2 and Lemma 2.3.In fact,aswe test equation(1.3)by u(x)and integrate by parts,then we get

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However,Lemma 2.3 implies that

If 2＜ p＜ ,the only possibility for the above two equations to hold is

which implies u≡0.While for p=,we have already proved that

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Thus,this completes the proof of Theorem 1.1.

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