I have a question about a Sobolev inequality of Moser's type.
In the following, $B(r)$ denotes open ball of $\mathbb{R}^{d}$ centered at origin and with radius $r>0$. $H^{1}\bigl( B(r)\bigr)$ deontes the first-order Sobolev space on $B(r)$ with Neumann boundary condition. I want to prove the following inequality:
There exists a constant $c_d$ which depends on $d$ and the choice of $c_0$ such that \begin{align*} &\left(r^{-d}\int_{B(r)}|f|^{2\kappa}\,dx \right)^{1/2\kappa} \le c_{d} \left\{\left( r^{2-d}\int_{B(r)}|\nabla f|^{2}\,dx \right)^{1/2}+\left( r^{-d}\int_{N}|f|^{2}\,dx\right)^{1/2} \right\}, \end{align*} for every $1 \le \kappa \le d/(d-2)$. Here $N$ is any measurable set in $B(r)$ of measure $m(N) \ge c_{0}^{-1}r^{d}$. When $d=2$, $d/(d-2)$ is any number in $[1,\infty)$.
Question
I can prove this inequality when $\kappa \in [1,d/(d-2))$. But I don't know how to prove when $\kappa=d/(d-2)$. Do you know how to prove this inequality when $\kappa=d/(d-2)$?
If necessary, I will try to write proof when $\kappa \in [1,d/(d-2))$. But this proof is very long...