Bifurcation of Steady-state Solutions for a Tumor Model with a Necrotic Core

We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R there exists a radially symmetric stationary solution with free boundary r = R and interior boundary r = ρ.The system depends on a positive parameter μ, and for a sequence of values μ2 < μ3 < · · · there also exist branches of symmetric breaking stationary solutions, which bifurcate from these values. We use a homotopy method on the polynomial system associated to the discretization of the free boundary problem to compute the nonradial symmetric solutions and discuss their linear stability.