On a singular minimizing problem

Abstract

For each q ∈ (0, 1), let

$${\lambda _q}(\Omega ): = inf\{ ||{\nabla _v}||_{{L^P}(\Omega )}^P:v \in W_0^{1,P}(\Omega )and\int_\Omega {|v{|^q}dx = 1\} } $$

where p > 1 and Ω is a bounded and smooth domain of R^N, N ≥ 2. We first show that

$$0 < \mu (\Omega ): = \mathop {\lim }\limits_{q \to {0^ + }} {\lambda _q}(\Omega )|\Omega {|^{p/q}} < \infty $$

where |Ω| = ∫Ω dx. Then we prove that

$$\mu (\Omega ) = min\left\{ {||{\nabla _v}||_{{l^p}(\Omega )}^p:v \in W_0^{1,p}(\Omega )and\mathop {lim}\limits_{q \to {0^ + }} {{(\frac{1}{{|\Omega |}}\int_\Omega {|v{|^q}dx} )}^{1/q}} = 1} \right\}$$

and that μ(Ω) is attained by a function u ∈ W₀^1,p (Ω) which is positive in Ω, belongs to ${C^{0,a}}(\bar \Omega )$ for some α ∈ (0, 1), and satisfies

$$ - div\left( {|{\nabla _u}{|^{p - 2}}{\nabla _u}} \right) = \mu (\Omega )|\Omega {|^{ - 1}}{u^{ - 1}}in\Omega ,and\int_\Omega {\log udx = 0.} $$

We also show that μ(Ω)⁻¹ is the best constant C in the log-Sobolev type inequality

$$\exp \left( {\frac{1}{{|\Omega |}}\int_\Omega {\log |v{|^p}dx} } \right) \leqslant C||{\nabla _V}||_{{L^P}(\Omega )}^P,v \in W_0^1(\Omega )$$

and that this inequality becomes an equality if and only if v is a scalar multiple of u and C = μ(Ω)⁻¹.