Compactness properties and ground states for the affine Laplacian

TL;DR

This paper investigates the compactness and ground states for the affine Laplacian, focusing on existence, regularity, and solutions to related nonlocal Dirichlet problems in the context of affine Sobolev inequalities.

Contribution

It establishes new compactness properties and existence results for minimizers and solutions to nonlocal PDEs involving the affine Laplacian, extending previous inequalities.

Findings

Proved compactness properties of the affine Sobolev inequality for p=2.

Established existence and regularity of minimizers and solutions.

Analyzed solutions to nonlocal Dirichlet problems involving the affine Laplacian.

Abstract

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 $q\in(2,\frac{2N}{N-2})$.