Sharp Constant and Extremal Function for the Improved Moser-trudinger Inequality Involving L Norm in Two Dimension

Let Ω ⊂ R 2 be a smooth bounded domain, and H 1 0 (Ω) be the standard Sobolev space. Define for any p > 1, λp(Ω) = inf u∈H 1 0 (Ω),u ≡0 ∇u 2 2 /u 2 p , where · p denotes L p norm. We derive in this paper a sharp form of the following improved Moser-Trudinger inequality involving the L p-norm using the method of blow-up analysis: sup u∈H 1 0 (Ω),∇u 2 =1 Ω e 4π(1+αu 2 p)u 2 dx < +∞ for 0 ≤ α < λp(Ω), and the supremum is infinity for all α ≥ λp(Ω). We also prove the existence of the extremal functions for this inequality when α is sufficiently small.