  In this paper we study a free boundary problem modeling the growth of solid
tumor spheroid. It consists of two elliptic equations describing nutrient
diffusion and pressure distribution within tumor, respectively. The new feature
is that nutrient concentration on the boundary is less than external supply due
to a Gibbs-Thomson relation and the problem has two radial stationary
solutions, which differs from widely studied tumor spheroid model with surface
tension effect. We first establish local well-posedness by using a functional
approach based on Fourier multiplier method and analytic semigroup theory. Then
we investigate stability of each radial stationary solution. By employing a
generalized principle of linearized stability, we prove that the radial
stationary solution with a smaller radius is always unstable, and there exists
a positive threshold value $\gamma_*$ of cell-to-cell adhesiveness $\gamma$,
such that the radial stationary solution with a larger radius is asymptotically
stable for $\gamma>\gamma_*$, and unstable for $0<\gamma<\gamma_*$.
