A sharp inequality for Sobolev functions (Q1598470)

Let \(N\geq 5\), \(b> 0\), \(\alpha\geq 0\), \(\Omega\) be a smooth bounded domain in \(\mathbb{R}^N\), \(2^*= {2N\over N-2}\) and \(2^\#= {2(N- 1)\over N-2}\). Here, \(b\) is fixed and \(\alpha\) as a parameter and the \(L^p\), \(H^1\) norms of \(u\) in \(\Omega\) are given by \[ |u|:= (\int|u|^p)^{1/p},\quad\|u\|_1:= (|\nabla u|^2_2+ b|u|^2_2)^{1/2}. \] The author proves that, there exists and \(\alpha_0> 0\) such that, for all \(u\in H^1(\Omega)\setminus\{0\}\) \[ {S\over 2^{2/N}}\leq {\|u\|^2_1\over|u|^2_{2^*}} \Biggl(1+ \alpha_0{|u|^{2^\#}_{2^\#}\over\|u\|\cdot|u|^{2^*/2}_{2^*}}\Biggr) \] which in turn implies Cherriers inequality.