On the existence of a priori bounds for positive solutions of elliptic problems, I

This paper gives a survey over the existence of uniform L∞ a priori bounds for positive solutions of subcritical elliptic equations (P)p -\Delta_p u =f(u), in \Omega, u = 0, on \partial\Omega widening 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, W1,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) = u2∗−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 uα = uα(r) of (P)2 as α → 0 (see [24])