Concentration-Compactness principle of singular Trudinger - Moser inequalities in ℝⁿ and n-Laplace equations

In this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger-Moser inequality in ℝⁿ. Then we prove a compact embedding theorem, which states that W1,n(ℝn) is compactly embedded into Lp(ℝⁿ, x-β dx) for p ≥ n and 0 < β < n. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the following n-Laplace equation with critical nonlinearity (Formula Presented) where V(x) ≥ c0 for some positive constant c0 and f(x, t) behaves like exp(αt n n-1 ) as t → +∞ This work improves substantially related results found in the literature.