On minimization problems which approximate Hardy \(L^{p}\) inequality. (Q1406773)

The author deals with a family of problems, that is \[ \inf_{0\neq v\in W^{1,p}_0(\Omega\setminus\{0\})} {\int_\Omega(|\nabla v|^p- \lambda\eta{| v|^p\over| x|^p})dx\over \int_\Omega {| v|^{p-\varepsilon}\over| x|^p}\,dx} \] for \(\varepsilon> 0\), where \(\eta\in C(\overline\Omega)\), \(\eta\geq 0\), \(\eta\neq 0\), \(\eta(0)= 0\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(\lambda\in \mathbb{R}\). The author studies the asymptotic behaviour as \(\varepsilon\to 0\), of the positive minimizers \(\{u_\varepsilon\}\) which are normalized by \(\| u_\varepsilon\|_{L^p(\Omega)}= 1\). He shows that \(u_\varepsilon\) converges to \(u_*\) in \(\bigcap_{1< q< p} W^{1,q}_0(\Omega\setminus\{0\})\), where \(u_*\) is the unique solution of (2) \[ \begin{cases} -\Delta_p u= ({u^{p-1}\over| x|^p})(\widetilde c_{p,N}+ \lambda\eta(x))\quad &\text{in }\Omega\setminus\{0\},\\ u> 0\quad\text{and}\quad u= 0\quad &\text{on }\partial\Omega.\end{cases}\tag{2} \]