Take $ \alpha\gt0$ and consider $\Omega=\{ x \in R^2: x_1^2+x_2^2 \lt 1, x_i\gt 0 \}$ (first quadrant of unit ball in plane). I am interested in optimal (so I am looking for the range of $p$) embeddings of the form $$ \left( \int_\Omega | u(x)|^p |x|^\alpha x_1^{m-1} x_2^{n-1} dx \right)^\frac{2}{p} \le C \int_\Omega \left( u_{x_1}^2 + u_{x_2}^2 \right) x_1^{m-1} x_2^{n-1} dx $$ for all smooth functions $u$ which are zero on the curved portion of the boundary and where $m,n$ are positive integers.
Any comments would be appreciated. I asked a very similar question here Sobolev imbedding result; proof
