Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ=σ(x) and the inner pressure p=p(x). Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20]. In this paper we study existence of nonradial solutions by using the bifurcation method. Let {γk}k=2∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k⁎⩾2 such that in the two dimension case for any k⩾k⁎, γk is a bifurcation point, and in the three dimension case for any even k⩾k⁎, γk is also a bifurcation point.