In this paper, we establish Liouville type theorem for boundedness solutions with finite Morse index of the following mixed boundary value problems: − Δ u = | u | p − 1 u in R + N , ∂ u ∂ ν = | u | q − 1 u on Γ 1 , ∂ u ∂ ν = 0 on Γ 0 , and − Δ u = | u | p − 1 u in R + N , ∂ u ∂ ν = | u | q − 1 u on Γ 1 , u = 0 on Γ 0 , where R + N = { x ∈ R N : x N > 0 } $\mathbb{R}^{N}_{+} =\{x\in\mathbb{R}^{N}:x_{N}>0\}$ , Γ 1 = { x ∈ R N : x N = 0 , x 1 < 0 } $\Gamma_{1}=\{x\in \mathbb{R}^{N}:x_{N}=0,x_{1}<0\}$ and Γ 0 = { x ∈ R N : x N = 0 , x 1 > 0 } $\Gamma_{0}=\{x\in\mathbb{R}^{N}:x_{N}=0,x_{1}>0\}$ . The exponents p, q satisfy the conditions in Theorem 1.1.