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