Pointwise bounds and blow-up for Choquard–Pekar inequalities at an isolated singularity

We study the behavior near the origin in Rn,n≥3Rn,n≥3, of nonnegative functionsequation(0.1)u∈C2(Rn\{0})∩Lλ(Rn)u∈C2(Rn\{0})∩Lλ(Rn) satisfying the Choquard–Pekar type inequalitiesequation(0.2)0≤−Δu≤(|x|−α⁎uλ)uσ in B2(0)\{0} where α∈(0,n)α∈(0,n), λ>0λ>0, and σ≥0σ≥0 are constants and ⁎ is the convolution operation in RnRn. We provide optimal conditions on α, λ, and σ such that nonnegative solutions u of (0.1), (0.2) satisfy pointwise bounds near the origin.