# 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 d*x*. 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* ∈ *W*_{0}^{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}.

Publisher URL: https://link.springer.com/article/10.1007/s11854-018-0040-0

Open URL: http://arxiv.org/pdf/1507.06040

DOI: 10.1007/s11854-018-0040-0

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