Reduced limit for semilinear boundary value problems with measure data

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We study boundary value problems for semilinear elliptic equations of the form -Δu+g∘ u=μ in a smooth bounded domain Ω⊂ Rⁿ. Let {μₙ} and {τₙ} be sequences of measure in Ω and ∂ Ω respectively. Assume that there exists a solution uₙ of the equation with μ=μₙ subject to boundary data τₙ. Further assume that the sequences of measures converge in a weak sense to μ and τ respectively and {uₙ} converges to u in L¹(Ω). In general u is not a solution of the boundary value problem with data (μ,τ). However there exist measures (μ*,τ*) such that u satisfies the equation with μ replaced by μ* and with u=τ* on the boundary. The pair (μ*,τ*) is called the reduced limit of the sequence {(μₙ,τₙ)}. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence.