We study the following minimization problem: $$\inf_{u\in {\mathcal D}^{1,2}(\Omega)\setminus \{0\}} \frac{\int_{\Omega} (|\nabla u|^2 + a(x) |u|^2) \hbox{ } dx}{\|u\|_{L^{2^*}}^2}, \hbox{ } \Omega \subset {\mathbf R}^n$$ in any dimension $n\geq 4$ and under suitable assumptions on $a(x)$. \noindent Mainly we assume that $a(x)$ belongs to the Lorentz space $L^{\frac n2, d}(\Omega)$ and the set $${\mathcal N}\equiv \{x\in \Omega|a(x)<0\}$$ has positive Lebesgue measure. Notice that this last condition is satisfied when the set $\mathcal N$ has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set $\mathcal N$).
On a minimization problem involving the critical sobolev exponent
PRINARI, Francesca Agnese;
2007
Abstract
We study the following minimization problem: $$\inf_{u\in {\mathcal D}^{1,2}(\Omega)\setminus \{0\}} \frac{\int_{\Omega} (|\nabla u|^2 + a(x) |u|^2) \hbox{ } dx}{\|u\|_{L^{2^*}}^2}, \hbox{ } \Omega \subset {\mathbf R}^n$$ in any dimension $n\geq 4$ and under suitable assumptions on $a(x)$. \noindent Mainly we assume that $a(x)$ belongs to the Lorentz space $L^{\frac n2, d}(\Omega)$ and the set $${\mathcal N}\equiv \{x\in \Omega|a(x)<0\}$$ has positive Lebesgue measure. Notice that this last condition is satisfied when the set $\mathcal N$ has a nontrivial interior part (in fact this is the typical assumption imposed in the literature on the set $\mathcal N$).I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
