On the existence of entire solutions to a class of semilinear elliptic equations on noncompact Riemannian manifolds (Q2460521)

Let \(M\) be a smooth connected noncompact Riemannian manifold without boundary. The Liouville property holds for the equation \[ \Delta u-u=0 \tag{1} \] on the manifold \(M\) if any of its bounded solutions is trivial. Similarly, the Liouville property holds for the equation \[ \Delta u=u\varphi(| u| ), \tag{2} \] where \(\varphi(\xi)>0\) is a monotone nondecreasing continuously differentiable function on \([0,\infty),\) if any bounded solution of this equation is trivial. In the paper under review, the author studies the relationship between the Liouville property for (1) and (2) on a manifold \(M\) and, as a corollary, obtains an existence theorem for nontrivial entire bounded solutions to (2).