Moser iteration applied to elliptic equations with critical growth on the boundary. (March 2019) - Marino, Greta; Winkert, Patrick
Abstract: This paper deals with boundedness results for weak solutions of the equation (P) − div A ( x, u, ∇ u ) = B ( x, u, ∇ u ) in Ω, A ( x, u, ∇ u ) ⋅ ν = C ( x, u ) on ∂ Ω, where the functions A : Ω × R × R N → R N, B : Ω × R × R N → R, and C : ∂ Ω × R → R are Carathéodory functions satisfying certain p -structure conditions that have critical growth even on the boundary. Based on a modified version of the Moser iteration we are able to prove that every weak solution of(P) is bounded up to the boundary. Under some additional assumptions on the functions A and C this leads directly to regularity for weak solutions of(P) .