Existence results for a nonlinear transmission problem

Let ΩoΩo and ΩiΩi be open bounded regular subsets of RnRn such that the closure of ΩiΩi is contained in ΩoΩo. Let fofo be a regular function on ∂Ωo∂Ωo and let F and G   be continuous functions from ∂Ωi×R∂Ωi×R to RR. By exploiting an argument based on potential theory and on the Leray–Schauder principle we show that under suitable and completely explicit conditions on F and G   there exists at least one pair of continuous functions (uo,ui)(uo,ui) such that{Δuo=0in Ωo∖clΩi,Δui=0in Ωi,uo(x)=fo(x)for all x∈∂Ωo,uo(x)=F(x,ui(x))for all x∈∂Ωi,νΩi⋅∇uo(x)−νΩi⋅∇ui(x)=G(x,ui(x))for all x∈∂Ωi, where the last equality is attained in certain weak sense. A simple example shows that such a pair of functions (uo,ui)(uo,ui) is in general neither unique nor locally unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we prove the existence of at least one classical solution which is in addition locally unique.