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

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We give new criteria for the existence of weak solutions to an equation with a super linear source term align*-Δu = uq in Ω, u=σ on ∂Ωalign*where Ω is a either a bounded smooth domain or R_+ⁿ, q1 and 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. align* -Δu = |u|q_¹⁻¹u|∇ u|q_² in Ω, u=σ on ∂Ωalign* where q_1,q_2≥ 0, q_1+q_21, q_22, σ∈ M(∂Ω) is a Radon measure on ∂Ω.