Improved Friedrichs inequality for a subhomogeneous embedding

For a smooth bounded domain $\mathrm{\Omega }$ and $p\ge q\ge 2$, we establish quantified versions of the classical Friedrichs inequality $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_p^p-{\lambda }_1\mathrm{\parallel }u{\mathrm{\parallel }}_q^p\ge 0$, $u\in W_0^{1,p}(\mathrm{\Omega })$, where ${\lambda }_1$ is a generalized least frequency. We apply one of the obtained quantifications to show that the resonant equation $-{\mathrm{\Delta }}_{p}u={\lambda }_1\mathrm{\parallel }u{\mathrm{\parallel }}_q^{p-q}\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+f$ coupled with zero Dirichlet boundary conditions possesses a weak solution provided $f$ is orthogonal to the minimizer of ${\lambda }_1$.