Reduced limit for semilinear boundary value problems with measure data

We study boundary value problems for semilinear elliptic equations of the form $-Δu+g\circ u=μ$ in a smooth bounded domain $Ω\subset R^N$. Let $\{μ_n\}$ and $\{τ_n\}$ be sequences of measure in $Ω$ and $\partial Ω$ respectively. Assume that there exists a solution $u_n$ of the equation with $μ=μ_n$ subject to boundary data $τ_n$. Further assume that the sequences of measures converge in a weak sense to $μ$ and $τ$ respectively and $\{u_n\}$ converges to $u$ in $L^1(Ω)$. 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 $\{(μ_n,τ_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence.

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