A regularity criterion for positive weak solutions of \(- {\Delta}u=u^{\alpha}\)
Let \(\Omega\subset \mathbb{R}^ n\) be open, \(\alpha> n/(n-2)\) and \(u\in L^ \alpha (\Omega)\) a positive weak solution to \(-\Delta u= u^ \alpha\). The author shows: If the map \[ x\in \Omega \mapsto \int_ \Omega {{u(y)^{\alpha-1}} \over {| x-y|^{n-2}}} dy\in (- \infty,\infty] \] is continuous then the Hausdorff dimension of the singular set of \(u\) is less than or equal to \(n- 2\alpha/ (\alpha-1)\).
scientific article; zbMATH DE number 218288