Let $\Omega\subset \mathbb{R}^n, n\geq 3$ be a nice bounded domain and $2^*=2n/(n-2)$ the critical Sobolev exponent. One may expect $\forall \epsilon>0$, $\exists C_\epsilon$ s.t. $$ \|u\|_{2^*}\leq \epsilon \|\nabla u\|_2+C_\epsilon \|u\|_2, \quad \forall u\in H^1(\Omega), $$ where $2^*=2n/(n-2)$ is the critical Sobolev exponent. Due to the lack of compact embedding from $H^1$ into $L^{2^*}$. The above inequality is inedd incorrect, see " mathoverflow.net/questions/81034/…" Now, I hope to get a stengthened version of it: given $p\in (2,2(n+2)/n)$ (or $p\in(2,2^*)$ in the worst case), $\forall \epsilon>0$, $\exists C_\epsilon$ s.t. $$ \|u\|_{2^*}\leq \epsilon (\|\nabla u\|_2+\|u\|_p^{\frac{p}{2}})+C_\epsilon(1+ \|u\|_2), \quad \forall u\in H^1(\Omega). \tag{MCIS} $$ I tried the example listed in mathoverflow.net/questions/81034/…., which does not give a counterexample. Also, arguing by contradiction seems not to work. Any help is greatly acknowledged