Ground States and Dirichlet Problems for — Δu = f(u) in R²

Abstract

In an earlier paper [1] we considered the existence of solutions of the problem

$$egin{gathered} - elta u = feft( u ight),u > 0,peratorname{in} {peratorname{R} ^N} fill ueft( x ight) o 0peratorname{as} |x| o nfty fill nd{gathered} $$

(I)

in which f(u) is to be positive for large u, but not for all u > O. Such solutions are sometimes called “ground states”, a term borrowed from the physical context (nonlinear field equations) in which Problem I arises. Following in part the approach of [5], we used a “shooting method” (in place of variational arguments) to prove under suitable conditions the existence of such a ground state. The principal difficulty lay in showing that if u(0) were chosen sufficiently large, then the associated radially symmetric solution had a zero, i.e. that the Dirichlet problem on some finite ball had a solution.