On the existence of bounded positive solutions of Schrödinger equations in two-dimensional exterior domains

We prove under quite general assumptions the existence of a bounded positive solution of the semilinear Schrödinger equation Δu+f(x,u)=0Δu+f(x,u)=0 in a two-dimensional exterior domain. Our results are independent of the behavior of f(x,u)f(x,u) when uu is sufficiently small or sufficiently large and just require some knowledge about the nonlinearity f(x,u)f(x,u) for a≤u≤ba≤u≤b, for some a,b>0a,b>0. We obtain solutions with a prescribed positive lower bound.