Solutions of semilinear elliptic problems based only on weak supersolutions

We consider the problems (E)E(f,u)  :=  Δu+f(r,u)=0   in   ℝn, https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367812027/499b2676-17a6-4805-9fd8-78c8baf0b813/content/eq950.tif"/> (E_Ω)E(f,u)  :=  Δu+f(r,u)=0   in   Ω,https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780367812027/499b2676-17a6-4805-9fd8-78c8baf0b813/content/eq951.tif"/>