The structure of nonlinear elliptic equations on unbounded domains in dimensions 1 and 2 — A probabilistic approach

Suppose that D is an unbounded domain in R2 with a compact boundary ∂D and k(x) is a strictly positive Hölder continuous function on D such that for some constant a > 0. In this paper, we study the nonlinear elliptic equation (1/2)Δu = k(x)uα(x) on D, where α ε (1,2] is a constant. First, we give explicit expressions in terms of super-Brownian motions for positive solutions of the above equation with the boundary conditions: u ∂D = 0 and lim x →∞(u(x)/log( x )) = c (0 c < ∞). Then we give a complete classification of all positive solutions of the above equation with the boundary condition u ∂D= 0 when k behaves like x −2(log( x ))−l near ∞ for some constant l > 1 + α. In the one-dimensional case, we also have similar results.