Improved Friedrichs inequality for a subhomogeneous embedding

Abstract

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

25 pages

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