Compactness properties and ground states for the affine Laplacian

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 $$-\sum _{i,j=1}^N(A^{-1}[u]{)}_{ij}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}=fin\mathrm{\Omega }\subset {\mathbb{R}}^N,$$ and $$-\sum _{i,j=1}^N(A^{-1}[u]{)}_{ij}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}_i\mathrm{\partial }{x}_j}=u^{q-1},u>0,in\mathrm{\Omega }\subset {\mathbb{R}}^N,$$ where $A_{ij}[u]=\int_\Omega\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}$\mathrm{d}$x$and$q\in(2,\frac{2N}{N-2})$.