A priori bounds and global existence of solutions of the steady-state Sel'kov model

We consider the system of reaction-diffusion equations known as the Sel'kov model. This model has been applied to various problems in chemistry and bi ology. We obtain a priori bounds on the size of the positive steady-state s olutions of the system defined on bounded domains in R-n, 1 less than or eq ual to n less than or equal to 3 (this is the physically relevant case). Pr eviously, such bounds had been obtained in the case n = 1 under more restri ctive hypotheses. We also obtain regularity results on the smoothness of su ch solutions and show that non-trivial solutions exist for a wide range of parameter values.