  In this paper, we study a free boundary problem modeling solid tumor growth
with vasculature which supplies nutrients to the tumor; this is characterized
in the Robin boundary condition. It was recently established [Discrete Cont.
Dyn. Syst. 39 (2019) 2473-2510] that for this model, there exists a threshold
value $\mu^\ast$ such that the unique radially symmetric stationary solution is
linearly stable under non-radial perturbations for $0<\mu<\mu^\ast$ and
linearly unstable for $\mu>\mu^\ast$. In this paper we further study the
nonlinear stability of the radially symmetric stationary solution, which
introduces a significant mathematical difficulty: the center of the limiting
sphere is not known in advance owing to the perturbation of mode 1 terms. We
prove a new fixed point theorem to solve this problem, and finally obtain that
the radially symmetric stationary solution is nonlinearly stable for
$0<\mu<\mu^\ast$ when neglecting translations.
