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In this paper, we present a rigorous mathematical analysis of a free boundaryproblem modeling the growth of a vascular solid tumor with a necrotic core. Ifthe vascular system supplies the nutrient concentration $\sigma$ to the tumorat a rate $\beta$, then $\frac{\partial\sigma}{\partial\bfn}+\beta(\sigma-\bar\sigma)=0$ holds on the tumor boundary, where $\bf n$ isthe unit outward normal to the boundary and $\bar\sigma$ is the nutrientconcentration outside the tumor. The living cells in the nonnecrotic regionproliferate at a rate $\mu$. We show that for any given $\rho>0$, there existsa unique $R\in(\rho,\infty)$ such that the corresponding radially symmetricsolution solves the steady-state necrotic tumor system with necrotic coreboundary $r=\rho$ and outer boundary $r=R$; moreover, there exist a positiveinteger $n^{**}$ and a sequence of $\mu_n$, symmetry-breaking stationarysolutions bifurcate from the radially symmetric stationary solution for each$\mu_n$ (even $n\ge n^{**})$.

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core