Non-homogeneous fourth order elliptic inequalities with a convolution term motivated by the suspension bridge problem

We are concerned with the study of the twin non-local inequalities featuring non-homogeneous differential operators -Δ² u + λΔ u ≥ (Kα, β * uᵖ)uq in RN (N≥ 1), and Δ² u - λΔ u ≥ (Kα, β * uᵖ)uq in RN (N≥ 1), with parameters λ, p, q >0, 0≤ α ≤ N and β>α-N. In the above inequalities the potential Kα,β is given by Kα, β(x) = |x|-αlogβ(1 + |x|) while Kα, β * uᵖ denotes the standard convolution operator in RN. We discuss the existence and non-existence of non-negative solutions in terms of N, p, q, λ, α and β.