Degenerate Elliptic PDEs on a Network with Kirchhoff Conditions

In this article, we are interested in semilinear, possibly degenerate elliptic equations posed on a general network, with nonlinear Kirchhoff-type conditions for its interior vertices and Dirichlet boundary conditions for the boundary ones. The novelty here is the generality of the equations posed on each edge that is incident to a particular vertex, ranging from first-order equations to uniformly elliptic ones. Our main result is a strong comparison principle, i.e., a comparison result between discontinuous viscosity sub and supersolutions of such problems, from which we conclude the existence and uniqueness of a continuous viscosity by Perron’s method. Further extensions are also discussed.