On a singular minimizing problem.

Abstract

For each q ∈ (0, 1) let λq(Ω) := inf k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and Z Ω |v| q dx = 1, where p > 1 and Ω is a bounded and smooth domain of R N , N ≥ 2. We first show that 0 < μ(Ω) := lim q→0+λq(Ω)|Ω| p q < ∞, where |Ω| = R Ω dx. Then, we prove that μ(Ω) = min (k∇vk p Lp(Ω) : v ∈ W1,p 0 (Ω) and lim q→0+ 1 |Ω| Z Ω |v| q dx 1 q = 1) and that μ(Ω) is reached by a function u ∈ W1,p 0 (Ω), which is positive in Ω, belongs to C 0,α(Ω), for some α ∈ (0, 1), and satisfies − div(|∇u| p−2 ∇u) = μ(Ω)|Ω| −1 u −1 in Ω, and Z Ω log udx = 0. We also show that μ(Ω)−1 is the best constant C in the following log-Sobolev type inequality exp 1 |Ω| Z Ω log |v| p dx ≤ C k∇vk p Lp(Ω) , v ∈ W1,p 0 (Ω) and that this inequality becomes an equality if, and only if, v is a scalar multiple of u and C = μ(Ω)−1.