As an exercise from my book I wanted to prove:
For $n>2$ and for any $u \in W_0^{1,2}\left(\mathbb{R}^n\right) \cap L^1\left(\mathbb{R}^n\right)$, $$ \int_{\mathbb{R}^n}|\nabla u|^2 d x \geq c\left(\int_{\mathbb{R}^n} u^2 d x\right)^{\frac{n+2}{n}}\left(\int_{\mathbb{R}^n}|u| d x\right)^{-\frac{4}{n}}, $$ where $c=c(n)>0$.
I solved a similar problem and used an appropriate Hölder inequality and the Sobolev inequality in the form $$ \left(\int_{\mathbb{R}^n}|u|^{\frac{2 n}{n-2}} d x\right)^{\frac{n-2}{n}} \leq C \int_{\mathbb{R}^n}|\nabla u|^2 d x $$ and I think it works here too, I just cannot find the fitting exponents. Any hints on how to tackle this one?
By Hölder's inequality $$ \int |u|^2 = \int |u|^{\frac{4}{n+2} + \frac{2n}{n-2}\frac{n-2}{n+2}} \\ \leq \left(\int |u|\right)^\frac{4}{n+2} \left(\int |u|^\frac{2n}{n-2}\right)^\frac{n-2}{n+2} $$ By Sobolev's inequality, which reads (notice a mistake in your question about the exponent) $$ \left(\int |u|^\frac{2n}{n-2}\right)^\frac{n-2}{n} \leq C_n \int |\nabla u|^2 $$ this implies $$ \int |u|^2 \leq C_n^\frac{n}{n+2}\left(\int |u|\right)^\frac{4}{n+2} \left(\int |\nabla u|^2\right)^\frac{n}{n+2} $$ or equivalently $$ \left(\int |u|^2\right)^\frac{n+2}{n} \leq C_n\left(\int |u|\right)^\frac{4}{n} \int |\nabla u|^2. $$