Weighted estimates involving a function and its derivatives (Q2563791)

Let \(1<p,q<\infty\), let \(\Omega\) be a domain in \(\mathbb{R}^N\), and let \(w,v_1,\dots,v_N\) be weights on \(\Omega\). The author proves that the Hardy inequality \[ \Biggl(\int_\Omega |u(x)|^q w(x)dx\Biggr)^{1/q}\leq \Biggl(\sum^N_{i=1} \int_\Omega \Biggl|{\partial u\over\partial x_i} (x)\Biggr|^p v_i(x)dx\Biggr)^{1/p} \] holds for all \(u\in C^\infty_0(\Omega)\) provided that there exists a solution \(y= y(x)\) on \(\Omega\) of the partial differential equation \[ \sum^N_{i=1} {\partial\over\partial x_i} \Biggl[v_i(x) \Biggl|{\partial u\over\partial x_i} (x)\Biggr|^{p-1}\Biggr]+ M(x)|y(x)|^{p-1}= 0,\tag{1} \] where \[ M(x)= w(x)|u(x)|^{q-p}q^{-1}+\Biggl(\sum^N_{i=1} \int_\Omega \Biggl|{\partial u\over\partial x_i} (x)\Biggr|^p v_i(x)dx\Biggr)^{q'/p'} (q'N|u|^p|\Omega|)^{-1},\tag{2} \] such that for a.e. \(x\in\Omega\), \(y(x)\neq 0\), \({\partial y_i\over\partial x_i}\neq 0\), \(i=1,\dots,N\). Note that this result does not extend that of \textit{B. Opic} and \textit{A. Kufner} [``Hardy-type inequalities'' (1990; Zbl 0698.26007)] (which concerns the case \(p=q\)) since the function \(u\) is involved in (1) (cf. (2)). There are some defects in the paper (for example, the assumptions \(u\in C^\infty_0(\mathbb{R}^+)\), \(u'\geq 0\) in Theorem 1 cannot be satisfied unless \(u\equiv 0\); note that the condition \(u'\geq 0\) can be omitted).