The Choquard equation in the half-space with a Robin boundary condition

We are concerned with the study of C2, γloc C loc 2 , γ -positive solutions to the Choquard equation -Δ u=λ (|x|-α*uᵖ)uq in RN₊=RN-1× (0, ∞ )\, , N≥ 2, - Δ u = λ ( | x | - α ∗ u p ) u q in R + N = R N - 1 × ( 0 , ∞ ) , N ≥ 2 , subject to the Robin boundary condition β u+∂ u/∂ ν =g β u + ∂ u ∂ ν = g on ∂ RN₊ ∂ R + N . Here p,q, λ >0 p , q , λ > 0 , α ∈ (0,N) α ∈ ( 0 , N ) , β ∈ R β ∈ R and 0≤ g∈ C1, γloc(RN-1) 0 ≤ g ∈ C loc 1 , γ