AbstractThe main purpose of this paper is to prove several sharp singular
Trudinger–Moser-type inequalities on domains in {\mathbb{R}^{N}} with infinite
volume on the Sobolev-type spaces {D^{N,q}(\mathbb{R}^{N})}, {q\geq 1}, the completion of {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm {\|\nabla u\|_{N}+\|u\|_{q}}. The case {q=N} (i.e., {D^{N,q}(\mathbb{R}^{N})=W^{1,N}(\mathbb{R}^{N})}) has been well studied to date. Our goal is to investigate
which type of Trudinger–Moser inequality holds under different norms when q
changes. We will study these inequalities under two types of constraint:
semi-norm type {\|\nabla u\|_{N}\leq 1} and full-norm type
{\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1}, {a>0}, {b>0}. We will show that the Trudinger–Moser-type inequalities
hold if and only if {b\leq N}. Moreover, the relationship between these
inequalities under these two types of constraints will also be investigated.
Furthermore, we will also provide versions of exponential type inequalities
with exact growth when {b>N}.
Approximate solutions of equations defined by complex spherical multiplier operators
