An elliptic semilinear equation with source term and boundary measure data: The supercritical case

We give new criteria for the existence of weak solutions to an equation with a super linear source term−Δu=uqin Ω,u=σon ∂Ω where Ω is either a bounded smooth domain or R+N, q>1q>1 and σ∈M+(∂Ω)σ∈M+(∂Ω) is a nonnegative Radon measure on ∂Ω. One of the criteria we obtain is expressed in terms of some Bessel capacities on ∂Ω. We also give a sufficient condition for the existence of weak solutions to equation with source mixed terms.−Δu=|u|q1−1u|∇u|q2in Ω,u=σon ∂Ω where q1,q2≥0q1,q2≥0, q1+q2>1q1+q2>1, q2<2q2<2, σ∈M(∂Ω)σ∈M(∂Ω) is a Radon measure on ∂Ω.