New results concerning Moser-type inequalities in Lorentz-Sobolev spaces

Abstract

Let $n\in \mathbb {N}$, $n\ge 2$, $q\in (1,\infty ]$ and let $\Omega \subset \mathbb {R}^n$ be an open bounded set. We obtain sharp constants concerning the Moser-type inequalities corresponding to the Lorentz-Sobolev space $W_0^1L^{n,q}(\Omega )$ equipped with the norm

$$\begin{aligned} \Vert \nabla u\Vert _{(n,q)}:= {\left\{ \begin{array}{ll}\Vert t^{\frac{1}{n}-\frac{1}{q}}|\nabla u|^{**}(t)\Vert _{L^q((0,\infty ))}&{}\text {for }q\in (1,\infty )\\ \sup _{t\in (0,\infty )}t^{\frac{1}{n}}|\nabla u|^{**}(t)&{}\text {for }q=\infty . \end{array}\right. } \end{aligned}$$

We also derive the key estimate for the Concentration-Compactness Principle in the case $q\in (1,\infty )$ with respect to the above norm.