Liouville-type theorems for the fourth order nonlinear elliptic equation

In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equationΔ2u=|x|a|u|p−1uin Ω, where a⩾0a⩾0, p>1p>1 and Ω⊂RnΩ⊂Rn is an unbounded domain of RnRn, n⩾5n⩾5. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence, which is used to obtain sharp results.