In this paper, we are concerned with Liouville-type theorems for the nonlinear elliptic equationwhere a ≥ 0, p > 1 and Ω ⊂ R n is an unbounded domain of R n , n ≥ 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 sharper results.Keywords: Liouville-type theorem; stable or finite Morse index solutions; monotonicity formula; blowing down sequence where a ≥ 0, p > 1 and Ω ⊂ R n is an unbounded domain of R n , n ≥ 5. We are interested in the Liouville-type theorems-i.e., the nonexistence of the solution u which is stable or finite Morse index, and the underlying domain Ω is an arbitrarily unbounded domain of R n .The idea of using the Morse index of a solution of a semilinear elliptic equation was first explored by Bahri and Lions [1] to obtain further qualitative properties of the *
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