Liouville theorem for Robin boundary value problems and finite Morse indices

In this paper, we study the nonexistence result of the following nonlinear elliptic equation with Robin boundary condition { − Δ u = | u | p − 1 u in R + N , ∂ u ∂ ν + c u = 0 on ∂ R + N . Under some assumptions on c and p, we will show that this problem does not possess nontrivial bounded solution with finite Morse index.