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summary:Let $\Omega \subset \mathbb R^n$ be a domain and let $\alpha <n-1$. We prove the Concentration-Compactness Principle for the embedding of the space $W_0^1L^n\log ^{\alpha }L(\Omega )$ into an Orlicz space corresponding to a Young function which behaves like $\exp (t^{{n}/{(n-1-\alpha )}})$ for large $t$. We also give the result for the embedding into multiple exponential spaces. \endgraf Our main result is Theorem \ref {lions4} where we show that if one passes to unbounded domains, then, after the usual modification of the integrand in the Moser functional, the statement of the Concentration-Compactnes Principle is very similar to the statement in the case of a bounded domain. In particular, in the case of a nontrivial weak limit the borderline exponent is still given by the formula $$ P:=(1-\|\Phi (|\nabla u|)\|_{L^1(\mathbb R^n)})^{-{1}/{(n-1)}}. $

Trudinger type inequalities in $\mathbf {R}^N$ and their best exponents