^{1,2}

^{1}

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^{1}

^{2}

^{3}

We establish the nonexistence of solution for the following nonlinear elliptic problem with weights:

Liouville theorems are very important in proving a priori bound of solutions for elliptic equations. As far as we know, the most powerful tool for proving the a priori bound is the blow-up method. During the blow-up process, we first suppose that the solutions are unbounded; then we can scale the sequence of solutions. Finally, we get a nontrivial solution for the limit equation. On the other hand, if we can prove that the limit equation does not possess a nontrivial solution, then we get a contradiction. So the solutions must be bounded. From the above statements, it is easy to see that the most important ingredient in proving the a priori bound is the nonexistence result for the limit equation. These kinds of nonexistence results are usually called Liouville type theorems.

For elliptic equations, the first Liouville theorem was proved by Gidas and Spruck in [

At the same time, elliptic equations with weights

On the other hand, we note that the above-mentioned results only claim that the above equations do not possess positive solution. A natural and more difficult question is whether the above equations possess sign-changing solution. However, this question is also completely open up to now. A partial answer was given in [

In this paper, inspired by the above works, we study another problem, that is, the following elliptic equation with weight:

Suppose that

The rest of this paper is devoted to the proof of the above theorem. We first deduce some inequality based on finite Morse index; then we derive some integral conditions on this solution. Finally, we use the Pohozaev inequality to prove the above theorem. In the following, we denote by

In this section, we always assume the conditions in Theorem

Let

The proof is the same as [

The next lemma is the key ingredient in the proof of Theorem

Let

We will use the information of finite Morse index of

Next, we show that

Finally, we show that

In order to complete the proof of Theorem

Suppose that

The proof of this lemma is standard; we give the details to keep this paper self-contained.

Multiplying (

Combining the above two equations together, then we get the above local Pohozaev identity for problem (

With the above preparations, we can prove Theorem

First, since

Next, multiplying (

The authors declare that there are no competing interests regarding the publication of this paper.