Equivalence between uniform Lp∗ a priori bounds and uniform L∞ a priori bounds for subcritical p-laplacian equations

Abstract

We establish sufficient conditions for a uniform Lp⋆ (Ω) bound to imply a uniform L∞(Ω) bound for positive weak solutions of sub- critical p-Laplacian equations. We also provide an equivalent result for sequences of boundary-value problems. As consequences, we prove that any set of solutions with finite energy is L∞(Ω) a priori bounded, and also obtain an alternative proof of the existence of a priori bounds for subcritical power like nonlinearities.