New weighted sharp Trudinger–Moser inequalities defined on the whole euclidean space $$ {\mathbb {R}}^N $$ and applications

Singular weighted sharp Trudinger-Moser inequalities defined on $ \mathbb{R}^N $ and applications to elliptic nonlinear equations

“…These same aspects of novelty were also highlighted for the nonsingular Trudinger-Moser inequalities established in a recently published work of ours (see [12]). In fact, for the same weight (10) and under assumptions ( 12), ( 13), ( 14) and ( 15), we proved a similar result to Theorem 1.11 for the case σ = 0.…”

“…So, the value k = 1 is excluded and this has a very important role for the correctness of the inequalities established in [12,Theorem 1.13]. In the present work, this value k = 1 is included which gives an aspect of harmony with hypothesis (16).…”

“…Thus, we can see Theorem 1.11 as a generalization (which is not trivial) of [12,Theorem 1.2]. However, Theorem 1.12 is not an immediate generalization of [12,Theorem 1.13]. The difference can be detected in three essential points.…”

“…In fact, for the same weight (10) and under assumptions ( 12), ( 13), ( 14) and ( 15), we proved a similar result to Theorem 1.11 for the case σ = 0. Thus, we can see Theorem 1.11 as a generalization (which is not trivial) of [12,Theorem 1.2]. However, Theorem 1.12 is not an immediate generalization of [12,Theorem 1.13].…”

“…where the supremum is taken over all balls B in R N . For the proof, we can refer to [11,Appendix] for the case N = 2 and to [12] for the general case. The importance of this property of the weight w β lies in the fact that it implies that C ∞ 0 (R N ) is dense in the space E β (see, for instance, [20,27,35] and references therein).…”