Existence and nonexistence of positive singular solutions for semilinear elliptic problems with applications in astrophysics
The first main theorem establishes the existence of infinitely many positive solutions \(u\in C^2(\mathbb{R}^n\backslash O)\) of the semilinear elliptic problem \[ -\Delta u= f(|x|) u^p\text{ in }\mathbb{R}^n\backslash O,\;u(x)\sim|x|^{2-n}\text{ as }|x|\to 0\text{ and as }|x|\to \infty,\tag{1} \] where \(f\) is a nonnegative locally Hölder continuous function in \((0, \infty)\) such that \(\int^\infty_0 r^a f(r)dr< \infty\), \(a= n-1- p(n- 2)\), \(n\geq 3\), \(p> 1\). Such a solution \(u\) automatically has ``finite total mass'', i.e., \(\int_{\mathbb{R}^n} |\Delta u|dx< \infty\), but infinite energy. The necessity of the integrability condition on \(f\) follows from an oscillation criterion of \textit{E. S. Noussair} and the reviewer [Proc. R. Soc. Edinburgh, Sect. A 75, 67-81 (1976; Zbl 0372.35004)]. Additional theorems yield existence or nonexistence of positive singular solutions to Dirichlet problems for Hénon type equations in a deleted neighborhood of the origin. A typical such equation has the singularity \(f(r)= 0(r^{- b})\) as \(r\to O+\) for \(b< 2\). Previous results of similar type are described by the first author and \textit{W.-M. Ni} [Arch. Ration. Mech. Anal. 108, 175-194 (1989; Zbl 0705.35039)]. The problems studied here correspond to the Matukuma and Hénon models in astrophysics for a globular stellar cluster with a black hole at the origin.
scientific article; zbMATH DE number 763472