Interpolation inequality on $L^p$ space

Question

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. If I already know the interpolation inequality that said $$ \|u\|_{L^q(\Omega)} \leq \|u\|^{\lambda}_{L^p(\Omega)} \|u\|^{1-\lambda}_{L^r(\Omega)} $$ for $1 \leq p \leq q \leq r$ satisfying $\frac{1}{q} = \frac{\lambda}{p} + \frac{1-\lambda}{r}$ with $0 \leq \lambda \leq 1$. How can I derive the following inequality $$ \|u\|_{L^{\frac{2q}{q-1}}(B_1)} \leq \varepsilon \|u\|_{L^{\frac{2n}{n-2}}(B_1)} + C(n,q) \varepsilon^{- \frac{n}{2q-n}} \|u\|_{L^{2}(B_1)} $$ for $2 < \frac{2q}{q-1} < \frac{2n}{n-2}$, and $B_1$ being the unit ball in $\mathbb{R}^n$? I think this is standard, but I do not have a good idea to split the multiplication to fit the power.