Singular Supercritical Trudinger-Moser Inequalities and the Existence of Extremals

Abstract

In this paper, we present the singular supercritical Trudinger-Moser inequalities on the unit ball B in ℝⁿ, where n ≥ 2. More precisely, we show that for any given α > 0 and 0 < t < n, then the following two inequalities hold for ∀ u ∈ W^1,n_0,r (B),

$$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {\left( {{\alpha _{n,t}} + {{\left| x \right|}^\alpha }} \right){{\left| u \right|}^{{n \over {n - 1}}}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty $$

and

$$\mathop {\sup }\limits_{\int_B {{{\left| {\nabla u} \right|}^n}dx \le 1} } \,\int_B {{{\exp \left( {{\alpha _{n,t}} + {{\left| u \right|}^{^{{n \over {n - 1}}} + {{\left| x \right|}^\alpha }}}} \right)} \over {{{\left| x \right|}^t}}}} dx < \infty .$$

We also consider the problem of the sharpness of the constant α_n,t. Furthermore, by employing the method of estimating the lower bound and using the concentration-compactness principle, we establish the existence of extremals. These results extend the known results when t = 0 to the singular version for 0 < t < n.