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

✍ Scribed by Yan-Xia Ren

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 825 KB
• Volume: 45
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(03)00038-5

✦ Synopsis

that D is an unbounded domain in R2 with a compact boundary aD and k(z) is a strictly positive Holder continuous function on D such that .I (log (11~11))" k(r) dr < 00, IIZll>Q for some constant a > 0. In this paper, we study the nonlinear elliptic equation (1/2)A~ = Ic(x)u"(z) on D, where o E (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: I&D = 0 and limllsll-too (u(r)/log(]]z]])) = c (0 < c 5 co). Then we give a complete classification of all positive solutions of the above equation with the boundary condition u]a~ = 0 when k behaves like ll~ll-2(10g(llrll))-'

near co for some constant 1 > 1 + a. In the one-dimensional case, we also have similar results.