Limits at infinity of superharmonic functions and solutions of semilinear elliptic equations of Matukuma type (Q1014002)

This paper concerns the behaviour at infinity of a particular type of positive superharmonic functions \(u\) defined outside a compact set in \(\mathbb{R}^{n}\). More precisely, if \(n\geq 3\), the distributional Laplacian of \(u\) is required to satisfy \(-\Delta u\leq cgu^{p}\lambda \) for some \(c>0\), where \(\lambda \) is the Lebesgue measure and \(g(x)=\left| x\right| ^{-2}\). It is shown that, if \(0\leq p<n/(n-2)\), then \(u\) must have a finite limit at infinity. This result is close to being sharp in the sense that it is shown to fail for \(p>n/(n-2)\). An analogue of the positive result is also established for the case where \(n=2\). The proof relies on the notion of minimal thinness and techniques of the author from an earlier paper [Math. Ann. 340, No. 3, 625--645 (2008; Zbl 1133.31003)]. The main result applies, in particular, to positive solutions of a semilinear elliptic equation first studied by Matukuma.