Positive solutions of nonlinear elliptic equations in the Euclidean plane

Abstract

In the present paper, we study the existence of solutions to the problem \[ { Δ u + f ( x , u ) = 0 a m p ; in  D u > 0 a m p ; in  D u = 0 a m p ; on  ∂ D \begin {cases} \Delta u+f(x,u)=0 & \text {in $D$}\\ u>0&\text {in $D$}\\ u=0 & \text {on $\partial D$} \end {cases} \] where D D is an unbounded domain in R 2 \mathbb {R}^2 with a compact nonempty boundary ∂ D \partial D consisting of finitely many Jordan curves. The goal is to prove an existence theorem for the above problem in a general setting by using Brownian path integration and potential theory.