Concentration-compactness principle associated with Adams' inequality in Lorentz-Sobolev space

Abstract

Abstract The concentration-compactness principle of Lions type in Euclidean space relies on the Pólya-Szegö inequality, which is not available in non-Euclidean settings. The first proof of the concentration-compactness principle in non-Euclidean setting, such as the Heisenberg group, was given by Li et al. by using a symmetrization-free nonsmooth truncation argument. In this article, we study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space W 0 2 L 2 , q ( Ω ) {W}_{0}^{2}{L}^{2,q}(\Omega ) for all 1 < q < ∞ 1\lt q\lt \infty . Due to the absence of the Pólya-Szegö inequality with respect to the second-order derivatives, we will use a symmetrization-free argument to study the concentration-compactness principle of second-order Adams’ inequality in Lorentz-Sobolev space. Furthermore, we show the sharpness of result by constructing a test function sequence. Our result is even new in the first-order case when q > 2 q\gt 2 .