Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

Abstract

Abstract Let {\mathcal{L}} be the Laplace operator on {\mathbb{R}^{d}} , {d\geq 3} , or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation {\mathcal{L}u-\varphi(\,\cdot\,,u)=0} we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.