Bifurcation for a free-boundary tumor model with angiogenesis

We consider a free boundary tumor model with vasculature which supply nutrients to the tumor, so that ∂σ∂n+β(σ−σ¯)=0 holds on the boundary, where a positive constant β is the rate of nutrient supply to the tumor and σ¯ is the nutrient concentration outside the tumor. The tumor cells proliferate at a rate μ. We show that for each μ2,μ4,μ6,μ8,⋯, symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions.