Sharp Singular Trudinger–Moser Inequalities Under Different Norms

Abstract

Abstract The main purpose of this paper is to prove several sharp singular Trudinger–Moser-type inequalities on domains in ℝ N {\mathbb{R}^{N}} with infinite volume on the Sobolev-type spaces D N , q ⁢ ( ℝ N ) {D^{N,q}(\mathbb{R}^{N})} , q ≥ 1 {q\geq 1} , the completion of C 0 ∞ ⁢ ( ℝ N ) {C_{0}^{\infty}(\mathbb{R}^{N})} under the norm ∥ ∇ ⁡ u ∥ N + ∥ u ∥ q {\|\nabla u\|_{N}+\|u\|_{q}} . The case q = N {q=N} (i.e., D N , q ⁢ ( ℝ N ) = W 1 , N ⁢ ( ℝ N ) {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 ∥ ∇ ⁡ u ∥ N ≤ 1 {\|\nabla u\|_{N}\leq 1} and full-norm type ∥ ∇ ⁡ u ∥ N a + ∥ u ∥ q b ≤ 1 {\|\nabla u\|_{N}^{a}+\|u\|_{q}^{b}\leq 1} , a > 0 {a>0} , b > 0 {b>0} . We will show that the Trudinger–Moser-type inequalities hold if and only if b ≤ N {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 {b>N} .