Minimization problem related to a Lyapunov inequality

In this study, we consider the following minimization problem on a bounded smooth domain Ω in RNRN:S′:=inf⁡{‖∇u‖22‖u‖2⁎2|u∈H1(Ω)∖{0},∫Ω|u|2⁎−2u=0}. This minimization problem plays a crucial role in the study of LpLp-Lyapunov type inequalities (1≤p≤∞1≤p≤∞) for linear partial differential equations with Neumann boundary conditions. In this study, we prove the existence of minimizers of S′S′. As a consequence, we prove the LpLp-Lyapunov type inequality in the critical case, which was left open in [4].