A Weighted Trudinger–Moser Inequality on ℝN and Its Application to Grushin Operator

Abstract

Let x = (x’,x”) with x’∈ ℝk and x” ∈ ℝN-k and Ω be a x’-symmetric and bounded domain in ℝN (N ≥ 2). We show that if 0 ≤a ≤ k−2, then there exists a positive constant C > 0 such that for all x’-symmetric function u ∈ C∞0 (Ω) with ∫Ω∇u(x)N−ax’−adx ≤ 1, the following uniform inequality holds $$\frac{1}{{{{\int_\Omega {|x|} }^{ - a}}dx}}\int_\Omega {{e^{\beta a|u|\tfrac{{N - a}}{{N - a - 1}}}}} |x'{|^{ - a}}dx \leqslant C,$$ where $${\beta _a} = N(N - a)(\frac{{2\pi \tfrac{N}{2}\Gamma (\tfrac{{k - a}}{2})}}{{\Gamma (\tfrac{k}{2})\Gamma (\tfrac{{N - a}}{2})}})\tfrac{1}{{N - a - 1}}$$. Furthermore, βa can not be replaced by any greater number. As the application, we obtain some weighted Trudinger–Moser inequalities for x-symmetric function on Grushin space.