We study the global stability of quasi-steady solutions for a simple mathematical model describing the growth of a spherical vascularized tumour consisting only of living cells. By assuming the rates of proliferation and absorption to be increasing nonlinear functions of the nutrient concentration, we establish the existence of a non-trivial steady solution and conditions for the existence and uniqueness of a quasi-steady solution for each initial configuration. Also, we prove that all these quasi-steady solutions converge uniformly to a non-trivial steady solution. The quasi-steady approach is justified by the smallness of the parameter that measures the ratio between the timescales for the diffusion of nutrients and growth of the tumour.

We prove a result of existence and localization of positive solutions of the Dirich- let problem for -Δ

We prove a result of existence of positive solutions of the Dirichlet problem for −∆pu = w(x)f (u, ∇u) in a bounded domain Ω ⊂ R N , where ∆p is the p-Laplacian and w is a weight function. As in previous results by the authors, and in contrast with the hypotheses usually made, no asymptotic behavior is assumed on f , but simple geometric assumptions on a neighborhood of the first eigenvalue of the p-Laplacian operator. We start by solving the problem in a radial domain by applying the Schauder Fixed Point Theorem and this result is used to construct an ordered pair of sub-and super-solution, also valid for nonlinearities which are superlinear both at the origin and at +∞. We apply our method to the Dirichlet problem −∆pu = λu(x) q−1 (1+|∇u(x)| p ) in Ω and give examples of superlinear nonlinearities which are also handled by our method. * The authors were supported in part by FAPEMIG and CNPq-Brazil.

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