Liouville theorems for a fourth order Hénon equation in the half-space

We investigate here the nonlinear elliptic Hénon-type equation Δ²u= |x|ᵃ|u|ᵖ⁻¹u in Rⁿ₊, u ={∂ u}{∂ xₙ}= 0 in ∂ Rⁿ₊, where p>1 , a≥ 0 and n≥ 5 . Based on the approach of Hu [J. Differential Equations 256 (2014), 1817–1846], we prove Liouville-type theorems for stable solutions and solutions which are stable outside a compact set possibly unbounded and sign-changing. In contrast with the results of Hu (2014), we apply a new method to provide an implicit existence of the fourth-order Joseph–Lundgren exponent. To classify finite Morse index solutions in the supercritical case, we adopt a new method of monotonicity formula together with blowing down sequence. In addition, a difficulty stems from the fact that applying the doubling lemma leads to the singularity. For this reason, we use a more delicate approach to the interval (n+4+2a, pJL2(n,0)) . Our analysis uses a combination of some integral estimates, Pohozaev-type identity, and monotonicity formula of solutions.