LIOUVILLE TYPE RESULTS FOR SYSTEMS OF ELLIPTIC INEQUALITIES WITH GRADIENT TERMS

DANIELE CASTORINA (UNIVERSITÀ DI PADOVA, ITALIA)

We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of -Δ_pu = g(u) in a smooth bounded domain Ω ⊂ ℝⁿ. In particular, we obtain new L^r and W^1,r bounds for the extremal solution u^* when the domain is strictly convex. More precisely, we prove that u^* ∈ L^∞(Ω) if n ≤ p + 2 and u^* ∈ L^np∕(n-p-2)(Ω) ∩ W₀^1,p(Ω) if n > p + 2. This is a joint work with Manel Sanchon (UAB Barcelona).