Concentration-Compactness Principle for Moser-type Inequalities in Lorentz-Sobolev Spaces

Abstract

Let n ∈ 𝜖, n ≥ 2, q ∈ (1, ∞) and let ${\Omega }\subset \mathbb {R}^{n}$ be an open bounded set. We study the Concentration-Compactness Principle for the embedding of the Lorentz-Sobolev space ${W_{0}^{1}}L^{n,q}({\Omega })$ into an Orlicz space corresponding to a Young function $t\mapsto \exp (t^{q^{\prime }})-1$. The results are stated with respect to the (quasi-)norm

$$||\nabla u||_{n,q}:= ||t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{*}(t)||_{L^{q}((0,|{\Omega}|))} \ . $$