Improved Friedrichs inequality for a subhomogeneous embedding

Abstract

For a smooth bounded domain Ω and p⩾q⩾2, we establish quantified versions of the classical Friedrichs inequality ‖∇u‖pp−λ1‖u‖qp⩾0, u∈W01,p(Ω), where λ1 is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation −Δpu=λ1‖u‖qp−q|u|q−2u+f coupled with zero Dirichlet boundary conditions possesses a weak solution provided f is orthogonal to the minimizer of λ1.