Runge property and multiplicity of solutions for elliptic equations in ℝ^N with a monotone nonlinearity

In this paper, we describe a method for extending (in someapproximated sense) solutions of a nonlinear P.D.E. on a domain Ω, to solutionsin a domain Ω′ containing Ω. Such an extension property, the Runge property, iswell known for a large class of linear problems including elliptic equations.We prove here the Runge property for semilinear problems of the kind-Δu+g(u)=f, with f ∈ L¹loc(ℝ^N). (As a consequence,we get infinitely many solutions for these problems). The proof is based on a“homotopy method”, and requires a refinement of the linear results: We provethat the “Runge extension” v on Ω′ of a solution u in Ω for a linear ellipticequation Lu=f can be choosen in order to depend continuously on u and thecoefficients of L.