Equivalence Between Uniform $$L^{p^*}$$ L p ∗ A Priori Bounds and Uniform $$L^{\infty }$$ L ∞ A Priori Bounds for Subcritical p -Laplacian Equations

We establish sufficient conditions for a uniform $$L^{p^\star }(\Omega )$$ bound to imply a uniform $$L^\infty (\Omega )$$ bound for positive weak solutions of subcritical 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 $$L^\infty (\Omega )$$ a priori bounded, and also obtain an alternative proof of the existence of a priori bounds for subcritical power like nonlinearities.