Improved Friedrichs inequality for a subhomogeneous embedding

Abstract

For a smooth bounded domain $Ω$ and $p \geq q \geq 2$, we establish quantified versions of the classical Friedrichs inequality $\|\nabla u\|_p^p - λ_1 \|u\|_q^p \geq 0$, $u \in W_0^{1,p}(Ω)$, where $λ_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-Δ_p u = λ_1 \|u\|_q^{p-q} |u|^{q-2} u + f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of $λ_1$.