Solutions with high dimensional singular set, to a conformally invariant elliptic equation in \(\mathbb{R}^ 4\) and in \(\mathbb{R}^ 6\)
Using similar techniques as in his previous paper [Calc. Var. Partial Differ. Equ. 1, No. 3, 243-265 (1993; reviewed above)], the author proves that for \(n=4\) or 6, for any \(d\in [(n- 2)/2, n]\) there exists a positive weak solution of \(-\Delta u= u^{(n+2)/ (n-2)}\) in \(\mathbb{R}^ n\), belonging to \(L^{(n+2)/ (n-2)} (\mathbb{R}^ n, d\mu)\), such that the singular set of this solution has Hausdorff dimension \(d\). Here \(d\mu\) denotes the measure \(d\mu= (1+| x|^ 2 )^{-n} dx\). This gives a counter-example to a conjecture of \textit{R. Schoen} and \textit{S. T. Yau} [Invent. Math. 92, 47-71 (1988; Zbl 0658.53038)] concerning the singular set of solutions to a related problem on the sphere.
positive solution with prescribed singular set
scientific article; zbMATH DE number 509913