This paper gives a survey over the existence of uniform L ∞ a priori bounds for positive solutions of subcritical elliptic equationswidening the known ranges of subcritical nonlinearities for which positive solutions are a-priori bounded.Our arguments rely on the moving planes method, a Pohozaev identity, W 1,q regularity for q > N , and Morrey's Theorem.In this part I, when p = 2, we show that there exists a-priori bounds for classical, positive solutions of (P) 2 with f (u) = u 2 * -1 [ln(e + u)] α , with 2 * = 2N/(N -2), and α > 2/(N -2).Appealing to the Kelvin transform, we cover non-convex domains.In a forthcoming paper containing part II, we extend our results for Hamiltonian elliptic systems (see [22]), and for the p-Laplacian (see [10]).We also study the asymptotic behavior of radially symmetric solutions u α = u α (r) of (P) 2 as α → 0 (see [24]).