The paper deals with the (singular) weighted semilinear eigenvalue problem \[ - \text{div}(|x|^\beta \nabla u)= \lambda|x|^\sigma u+ |x|^\alpha u^{p- 1},\;u> 0\text{ in } \Omega,\quad u|_{\partial\Omega}= 0.\tag{1} \] Here \(\Omega\subset \mathbb{R}^n\) is a bounded smooth domain, \(0\in \Omega\), \(\beta\leq 0\), \(\alpha> - n\), \(p\geq 2\), \(\alpha\leq {p\over 2} \beta\), \(\sigma> \beta- 2\), and \(\lambda\in \mathbb{R}\) is the eigenvalue parameter. It is assumed that the nonlinear term has critical growth with respect to a related Sobolev-Hardy inequality, i.e. \(p= 2(n+ \alpha)/(n+ \beta- 2)\) for \(n> 2-\beta\).
The case \(\alpha= \beta= \sigma= 0\) was extensively studied in the famous paper of H. Brezis and L. Nirenberg [Commun. Pure Appl. Math. 36, 437-477 (1983; Zbl 0541.35029)]. According to a notion introduced by P. Pucci and J. Serrin [J. Math. Pures Appl. 69, No. 1, 55-83 (1990; Zbl 0717.35032)] the authors call a dimension \(n\) critical with respect to the Dirichlet problem (1) iff there is a positive bounded \(\overline\lambda> 0\) such that a necessary condition for a positive solution to (1) to exist is \(\lambda> \overline\lambda\). In the present paper, the results of H. Brezis and L. Nirenberg [loc. cit.] are extended to the weighted problem (1) as follows: It is shown (among other things) that in (strictly) star shaped domains \(\Omega\) for any \(n> 2- \beta\) a solution to (1) may exist only for \(\lambda> 0\) and, moreover, that the critical dimensions \(n\) are precisely those with \(n< 4- 2\beta+ \sigma\).
The case \(\beta= 0\) has already been studied by L. I. Nicolaescu [Differ. Integral Equations 4, No. 3, 653-671 (1991; Zbl 0736.35049)].