On decaying entire solutions of second order sublinear elliptic equations
The elliptic partial differential equation \(\Delta u+a(x)u^{\sigma}=0\) in \({\mathbb{R}}^ n\) is considered; here, \(x=(x_ 1,...,x_ n)\), \(n\geq 3\), \(0<\sigma <1\), and a is a positive locally Hölder continuous function. The aim of the present paper is to prove the existence of positive functions u satisfying the subsequent conditions (i)-(iii): (i) u is a solution of the above equation at every point of \({\mathbb{R}}^ n\), (ii) \(u\in C^{2+\sigma}_{loc}({\mathbb{R}}^ n)\), (iii) u decays to zero at infinity. Mainly using results by \textit{W.-M. Ni} [Indiana Univ. Math. J. 31, 493-529 (1982; Zbl 0496.35036)] and \textit{N. Kawano} [Hiroshima Math J. 14, 125-158 (1984; Zbl 0555.35043)], the author proceeds by applying the supersolution-subsolution method, by constructing solutions which are squeezed between supersolutions and subsolutions tending to zero as \(| x| \to \infty\). Furthermore, analogous investigations are undertaken for a system of two related equations.
scientific article; zbMATH DE number 3903155