Trudinger-Moser Inequalities with the Exact Growth Condition in ℝ^N and Applications

In a recent paper [19], the authors obtained a sharp version of the Trudinger-Moser inequality in the whole space ℝ², giving necessary and sufficient conditions for the boundedness and the compactness of general nonlinear functionals in W ^{1, 2}(ℝ²). We complete this study showing that an analogue of the result in [19] holds in arbitrary dimensions N ≥2. We also provide an application to the study of the existence of ground state solutions for quasilinear elliptic equations in ℝ^N.