  The paper studies compactness properties of the affine Sobolev inequality of
Gaoyong Zhang et al in the case $p=2$, and existence and regularity of related
minimizers, in particular, solutions to the nonlocal Dirichlet problems \[
-\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial x_j}=f
\mbox{ in }\Omega\subset\mathbb R^N, \] and \[
-\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial
x_j}=u^{q-1}\,,\quad u>0,\mbox{ in }\Omega\subset\mathbb R^N, \] where
$A_{ij}[u]=\int_\Omega\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial
x_j}\mathrm{d}x$ and $q\in(2,\frac{2N}{N-2})$.

Analysis of PDEs