Reduced limit approach to semilinear partial differential equations (PDEs) involving the fractional Laplacian with measure data

Abstract

We study the following partial differential equation (PDE) (−∆)s u + g(x, u) = µ in Ω, u = 0 in R N Ω, (0.1) where (−∆)s is the fractional Laplacian operator, Ω is a bounded domain in R N with ∂Ω being the boundary of Ω, g(., .) is a nonlinear function defined over Ω × R. Let (µn)n be a sequence of measure in Ω. Assume that there exists a solution un with data µn , i.e. un satisfies the equation (0.1) with µ = µn . We further assume that the sequence of measures weakly converges to µ, while (un)n converges to u in L 1 (Ω). In general, u is not a solution to the partial differential equation in (0.1) with datum (µ, 0). However, there exists a measure µ # such that u is a solution of the partial differential equation with this data. µ # is called the reduced limit of the sequence (µn)n . We investigate the relation between weak limit µ and the reduced limit µ # and the dependence of µ # to the sequence (µn)n . A closely related problem was studied by Bhakta and Marcus [3] and then by Giri and Choudhuri [15] but for the case of a Laplacian and a general second order linear elliptic differential operator, respectively instead of a fractional Laplacian.